The statistical mechanics of near-BPS black holes

To obtain the bulk action of $\mathcal{N}=4$ super-JT gravity we start by considering a BF theory with a $\mathfrak{p}\mathfrak{s}\mathfrak{u}(1,1\vert 2)$ superalgebra,

$$\begin{align} [{L}_{m},{L}_{n}]& =(m-n){L}_{m-n},\qquad \qquad [{T}_{i},{T}_{j}]=i{{\epsilon}}_{ijk}{T}_{k}, \\ [{L}_{m},{{G}_{p}}^{\alpha }]& =\left(\frac{m}{2}-\alpha \right){{G}_{p}}^{\alpha +m},\qquad [{L}_{m},{{\bar{G}}^{p}}_{\alpha }]=\left(\frac{m}{2}-\alpha \right){{\bar{G}}^{p}}_{\alpha +m}, \\ [{T}_{i},{{G}_{p}}^{\alpha }]& =-\frac{1}{2}{{\left({\sigma }^{i}\right)}_{p}}^{q}{{G}_{q}}^{\alpha }\enspace ,\qquad \qquad [{T}_{i},{{\bar{G}}^{p}}_{\alpha }]=\frac{1}{2}{{{\left({\sigma }^{i}\right)}^{\ast }}^{p}}_{q}{{G}^{q}}_{\alpha }, \\ \left\{{{G}_{p}}^{\alpha },{{\bar{G}}^{q}}_{\beta }\right\}& =2{{\delta }^{q}}_{p}{L}_{\alpha +\beta }-2(\alpha -\beta ){{({\sigma }^{i})}^{q}}_{p}{T}_{i}, \end{align} \tag{ 2.1 }$$

with all other commutators/anti-commutators between the generators vanishing. Here L₀ and L_±1 are bosonic generators forming an $\mathfrak{s}\mathfrak{l}(2,\mathbb{R})$ sub-algebra, T_i , i = 1, ..., 3, are bosonic generators forming an $\mathfrak{s}\mathfrak{u}(2)$ sub-algebra, and ${{G}_{p}}^{\alpha }$ and ${{\bar{G}}^{q}}_{\beta }$ are the fermionic generators with p, q = 1, 2 and $\alpha ,\beta =-\frac{1}{2},\enspace \frac{1}{2}$.

The action of $\mathcal{N}=4$ super-JT gravity is given by,

$$\begin{equation}{I}_{BF}=-i\int \text{Str}\enspace \enspace \phi F,\quad F=dA-A\wedge A,\end{equation} \tag{ 2.2 }$$

where A is a $\mathfrak{p}\mathfrak{s}\mathfrak{u}(1,1\vert 2)$ gauge field with a field strength F and ϕ is a zero-form field transforming in the adjoint of $\mathfrak{p}\mathfrak{s}\mathfrak{u}(1,1\vert 2)$. Above, Str ϕF = ⟨ϕ, F⟩ is a quadratic bilinear form invariant under adjoint transformations. Such a form can be obtained from the quadratic Casimir of $\mathfrak{p}\mathfrak{s}\mathfrak{u}(1,1\vert 2)$. ¹³ We define the gauge field in terms of the supermultiplet of the frame e^a and spin connection ω, also consisting of the SU(2) gauge field Bⁱ and the gravitinos ${{\psi }_{p}}^{\alpha }$ as ¹⁴ :

$$\begin{align} A(x)& =\sqrt{\frac{{\Lambda}}{2}}\left[{e}^{1}(x){L}_{0}+\frac{{e}^{2}(x)}{2}\left({L}_{1}-{L}_{-1}\right)\right]-\frac{\omega (x)}{2}\left({L}_{1}+{L}_{-1}\right)+{B}^{i}(x){T}_{i} \\ & \quad +{\left(\frac{{\Lambda}}{2}\right)}^{\frac{1}{4}}\left({{\bar{\psi }}^{p}}_{\alpha }(x){{G}_{p}}^{\alpha }+{{\psi }_{p}}^{\alpha }(x){{\bar{G}}^{p}}_{\alpha }\right). \end{align} \tag{ 2.3 }$$

The zero-form field ϕ(x) is fixed in terms of the supermultiplet of the $\mathrm{S}\mathrm{L}(2,\mathbb{R})$ Lagrange multipliers ϕ^1,2 and ϕ⁰,

$$\begin{align} \phi (x)& =2{\phi }^{1}(x){L}_{0}+{\phi }^{2}(x)\left({L}_{1}-{L}_{-1}\right)-{\phi }^{0}(x)\left({L}_{1}+{L}_{-1}\right)+{b}^{i}(x){T}_{i} \\ & \quad +{\left(\frac{{\Lambda}}{2}\right)}^{-\frac{1}{4}}\left({{\bar{\lambda }}^{p}}_{\alpha }(x){{G}_{p}}^{\alpha }+{{\lambda }_{p}}^{\alpha }(x){{\bar{G}}^{p}}_{\alpha }\right). \end{align} \tag{ 2.4 }$$

Here, λ and $\bar{\lambda }$, and ψ and $\bar{\psi }$ should be understood as independent components of A and are not related to the Hermitian conjugates of each other. In such a case, the action can be written as,

$$\begin{align} {I}_{\text{JT}}^{\mathcal{N}=4}& =i\int \left(\underset{\sim \;\text{Dilaton}}{\underbrace{{\phi }^{0}}}\left[\underset{\frac{{d}^{2}x}{2}\sqrt{g}(R+{\Lambda})}{\underbrace{\;\;d\omega +\frac{{\Lambda}}{4}{{\epsilon}}_{ab}{e}^{a}\wedge {e}^{b}}}-\underset{\begin{subarray}{c}\text{Gravitino}\enspace \enspace {{\psi }_{p}}^{\alpha }\enspace \enspace \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{b}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\\ \text{multiplying}\;\text{the}\;\text{dilaton}\end{subarray}}{\underbrace{\sqrt{2{\Lambda}}{{\bar{\psi }}^{p}}_{\alpha }\wedge {{\psi }_{p}}^{\alpha }}}\right]\right. \\ & \quad \left.-\sqrt{\frac{{\Lambda}}{2}}\underset{\begin{subarray}{c}\text{Super}\mbox{-}\text{torsion}\\ \text{Lagr.multiplier}\end{subarray}}{\underbrace{{\phi }^{a}}}\underset{\text{super}\mbox{-}\text{torsion}\;\text{component}\;{\tau }^{a}}{\underbrace{\left[d{e}^{a}-{{\epsilon}}_{ab}\enspace \omega \wedge {e}^{b}+2{{\left({\bar{\gamma }}_{a}\right)}^{\alpha }}_{\beta }{{\bar{\psi }}^{p}}_{\alpha }\wedge {{\psi }_{p}}^{\beta }\right]}}\right. \\ & \quad \left.-{\mathrm{t}\mathrm{r}}_{SU(2)}\underset{SU(2)\;\text{BF}\;\text{theory}}{\underbrace{\left[b\left(dB+B\wedge B\right)\right]}}\enspace \enspace +\enspace \enspace \sqrt{\frac{{\Lambda}}{2}}\underset{SU(2)\;\text{BF}\;\text{super}\mbox{-}\text{partner}}{\underbrace{{{b}^{q}}_{p}{\bar{\psi }}^{p}\wedge {\bar{\gamma }}_{3}{\psi }_{q}}}\right.\quad \left.+\enspace \underset{\begin{subarray}{c}\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{v}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{o}\enspace \enspace {{\mathit{\psi }}^{p}}_{\alpha }\enspace \enspace \mathrm{a}\mathrm{n}\mathrm{d}\\ \mathrm{d}\mathrm{i}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{o}\enspace \enspace {{\mathit{\lambda }}_{p}}^{\alpha }\enspace \enspace \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{b}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\end{subarray}}{\underbrace{2\lambda \mathcal{D}\bar{\psi }+2{\mathcal{D}}^{\ast }\psi \bar{\lambda }}}\right), \end{align} \tag{ 2.5 }$$

where we have outlined the important terms in the action which will ease the comparison with the near-horizon action which we shall obtain in section 4.1. ¹⁵

The equations of motion for ϕ^1,2 act as Lagrange multipliers and imposes that the super-torsion vanishes. After integrating out ϕ^1,2, one can replace e¹, e² and ω in terms of the metric g_μν to obtain the supergravity action (2.5) in the second order formalism. The JT gravity dilaton is given by −iϕ⁰ ≡ Φ. When comparing the gravitational theory obtained from the dimensional reduction of the near-horizon region of near-BPS black holes to the $\mathcal{N}=4$ super-JT action we will use this latter form. For simplicity, in the remainder of this section we will fix the cosmological constant, Λ = 2. Later, when discussing the effective action for the near-horizon region of the near-BP black holes we will revert to a general cosmological constant, determined by the radius of the black hole.

We can complete the dictionary between the BF theory (2.5) and the second-order super-JT gravity by noting that infinitesimal PSU(1, 1|2) gauge transformations in the former, map to infinitesimal diffeomorphisms in the latter. In particular, the infinitesimal supersymmetric transformations on all the fields in (2.5) can be obtained by considering the infinitesimal gauge transformation whose gauge parameter takes the form ${\Lambda}={{\epsilon}}^{p\alpha }{G}_{p\alpha }+{\bar{{\epsilon}}}^{p\alpha }{\bar{G}}_{p\alpha }$.

With this mapping between the BF theory (2.5) and $\mathcal{N}=4$ super-JT gravity in mind, we can now analyze the supersymmetric boundary conditions applied to the theory (2.5) which will be necessary in section 4 to understand the gluing of the near-horizon region of near-BPS black holes to asymptotic flatspace. As we will see, these boundary conditions will reduce the gravitational path integral to that of the $\mathcal{N}=4$ super-Schwarzian.

We begin, just like in the non-supersymmetric case, by imposing that the boundary metric is fixed and proper boundary length is large, L = β/ɛ, with ɛ small. Working in Fefferman–Graham gauge, where the metric can be written as

$$\begin{equation}\mathrm{d}{s}^{2}=\mathrm{d}{r}^{2}+\left(\frac{1}{4}{e}^{2r}-\tilde{\mathcal{L}}(\tau )+\dots \enspace \right)\mathrm{d}{\tau }^{2},\end{equation} \tag{ 2.6 }$$

we consider the boundary to be at fixed, but large, r. To satisfy the boundary conditions we identify τ ∼ τ + β and cut-off the geometry at e^−r = ɛ/2. We fix the leading component of the boundary metric and allow the sub-leading component, $\tilde{\mathcal{L}}(\tau )$, to vary. The ... represent terms that are further sub-leading in r which we do not need to fix.

Similarly, we require that the dilaton takes an asymptotically large and constant value at the boundary, ${\Phi}{\vert }_{\partial \mathcal{M}}\equiv -i{\phi }_{0}{\vert }_{\partial \mathcal{M}}={{\Phi}}_{r}/\varepsilon$. Next, we discuss the boundary conditions for the super-partners of the frame and spin-connection. If working in the Fefferman–Graham gauge (2.6), then, in order to preserve supersymmetry, we impose that the leading component of the gravitino is fixed and vanishes ψ^pα = O(e^−r/2) and ${\bar{\psi }}^{q\alpha }=O({e}^{-r/2})$. Similarly, to again preserve supersymmetry, we impose that the leading contribution of the dilatino vanishes at asymptotically large values λ = O(e^−r/2). Finally, we describe the boundary conditions for the SU(2) gauge field and its Lagrange multiplier. From the perspective of the higher dimensional black holes which we will study in section 4, the SU(2) gauge field appears from the isometry of S² along which we perform the dimensional reduction. We would therefore, like to fix Dirichlet boundary conditions at the boundary of the asymptotically flat region. As we will describe in detail in section 4, imposing these boundary conditions in the asymptotically flat region translates to fixing a linear combination of the zero-form field b and the SU(2) gauge field B at the boundary of the near-horizon region.

We would now like to translate the boundary conditions discussed above in the second-order formalism to boundary conditions in the BF theory (2.5). We will follow the steps outlined in [65] (explained also recently in [64]) in non-supersymmetric JT gravity and will obtain results similar to [66], which studied boundary conditions in JT supergravity with an $OSp(2,\mathcal{N})$ isometry group. Fixing the Fefferman–Graham gauge for the metric (2.6) yields the value of the frame e^1,2 and spin connection ω [64]

$$\begin{equation}{e}^{1}=\mathrm{d}r,\qquad {e}^{2}=\left(\frac{1}{2}{e}^{r}-\tilde{\mathcal{L}}(\tau ){e}^{-r}\right)\mathrm{d}\tau ,\qquad \omega =-\left(\frac{1}{2}{e}^{r}+\tilde{\mathcal{L}}(\tau ){e}^{-r}\right)\mathrm{d}\tau .\end{equation} \tag{ 2.7 }$$

Fixing the decaying piece of the gravitino we can gauge fix all other components of the PSU(1, 1|2) gauge field along the boundary to be

$$\begin{align} {A}_{\tau }(\tau )& ={L}_{1}\frac{{e}^{r}}{2}+{L}_{-1}\tilde{\mathcal{L}}(\tau ){e}^{-r}+{B}_{\tau }^{i}(\tau ){T}_{i} \\ & \quad +{e}^{-\frac{r}{2}}\frac{{\bar{\psi }}^{p}(\tau )}{2}{G}_{p,\enspace -\frac{1}{2}}+{e}^{-\frac{r}{2}}\frac{{\psi }^{p}(\tau )}{2}{\bar{G}}_{p,\enspace -\frac{1}{2}}+O({e}^{-2r}), \end{align} \tag{ 2.8 }$$

where we have used the shorthand notation ${\psi }^{p}(\tau )\equiv {\psi }^{p,\enspace \frac{1}{2}}$ and ${\bar{\psi }}^{p}(\tau )\equiv {\bar{\psi }}^{p,\enspace \frac{1}{2}}$ on the boundary $\partial \mathcal{M}$. We can now impose the equations of motion of A_τ in the proximity of the boundary D_τ ϕ = 0, or rather due to only fixing the leading orders of A_τ and ϕ, D_τ ϕ = O(e^−r ), where D denotes the PSU(1, 1|2) covariant derivative. This implies that the zero-form field ϕ should be constrained to take the form

$$\begin{align} \phi (\tau )& =-i{{\Phi}}_{r}{L}_{1}{e}^{r}+2i{{\Phi}}_{r}^{\prime }{L}_{0}+(\bar{\psi }\lambda +\psi \bar{\lambda }-2i\tilde{\mathcal{L}}{{\Phi}}_{r}-2i{{\Phi}}_{r}^{{\prime\prime}}){L}_{-1}{e}^{-r}+{b}^{i}(x){T}_{i} \\ & \quad +{\bar{\lambda }}^{p}{G}_{p,\frac{1}{2}}{e}^{r/2}-\left(2{({\bar{\lambda }}^{p})}^{\prime }-i{{\Phi}}_{r}{\bar{\lambda }}^{p}-{B}_{\tau }^{i}{{({\sigma }^{i\ast })}^{p}}_{q}\enspace {\bar{\lambda }}^{q}\right){G}_{p,-\frac{1}{2}}{e}^{-r/2} \\ & \quad +{\lambda }^{p}{\bar{G}}_{p,\frac{1}{2}}{e}^{r/2}-\left(2{({\lambda }^{p})}^{\prime }-i{{\Phi}}_{r}{\lambda }^{p}-{B}_{\tau }^{i}{{({\sigma }^{i}\enspace )}^{p}}_{q}\enspace {\lambda }^{q}\right){\bar{G}}_{p,-\frac{1}{2}}{e}^{-r/2}+O({e}^{-2r}), \end{align} \tag{ 2.9 }$$

where, again, we have used the short-hand notation ${\lambda }^{p}(\tau )={\lambda }^{p,\frac{1}{2}}$ and ${\bar{\lambda }}^{p}(\tau )={\bar{\lambda }}^{p,\frac{1}{2}}$ on the boundary $\partial \mathcal{M}$ and where ${{\Phi}}_{r}^{\prime }\equiv {\partial }_{\tau }{{\Phi}}_{r}(\tau )$ denotes the derivative with respect to the boundary time τ. ¹⁶

If we want to impose the boundary conditions for the dilaton (Φ_r = const), the dilatinos (${\lambda }^{p,\enspace \frac{1}{2}}=0$ and ${\bar{\lambda }}^{p,\enspace \frac{1}{2}}=0$) and the zero-form field bⁱ (x) we can then relate the gauge field in (2.8) to the zero-form field ϕ(τ) as −2iΦ_r A_τ (τ) = ϕ(τ), where Φ_r is the renormalized value of the dilaton. Thus, in the first-order formalism and in the gauge in which A_τ takes the form (2.8), the boundary condition that we want to impose is $\delta (2i{{\Phi}}_{r}{A}_{\tau }(\tau )+\phi (\tau )){\vert }_{\partial \mathcal{M}}=0$. The boundary term necessary for such a boundary condition to have a well defined variational principle is [66]:

$$\begin{equation}{I}_{BF,\enspace \text{bdy.}}=\frac{i}{2}{\int }_{\partial \mathcal{M}}\text{Str}\enspace \phi A={{\Phi}}_{r}{\int }_{\partial \mathcal{M}}\mathrm{d}\tau \enspace \text{Str}\enspace {A}_{\tau }^{2}.\end{equation} \tag{ 2.10 }$$

Integrating out the field Φ in the bulk we find that the bulk term yields no contribution. Replacing A_τ in (2.10) we find that the action can then be rewritten as

$$\begin{equation}{I}_{BF,\enspace \text{bdy.}}=-{{\Phi}}_{r}{\int }_{\partial \mathcal{M}}\mathrm{d}\tau \left[\tilde{\mathcal{L}}(\tau )+\frac{1}{2}\left({({B}_{\tau }^{1})}^{2}+{({B}_{\tau }^{2})}^{2}+{({B}_{\tau }^{3})}^{2}\right)\right]\end{equation} \tag{ 2.11 }$$

Thus, it is convenient to define

$$\begin{align} & \mathcal{L}(\tau )=\tilde{\mathcal{L}}(\tau )+\frac{1}{2}\left({({B}_{\tau }^{1})}^{2}+{({B}_{\tau }^{2})}^{2}+{({B}_{\tau }^{3})}^{2}\right)\enspace ,\enspace \enspace \;\text{from}\;\text{which}\;\enspace \enspace \\ & {I}_{BF,\enspace \text{bdy.}}=-{{\Phi}}_{r}{\int }_{\partial \mathcal{M}}\mathrm{d}\tau \enspace \mathcal{L}(\tau ). \end{align} \tag{ 2.12 }$$

We will not determine $\mathcal{L}(\tau )$ explicitly. Rather we will see how $\mathcal{L}(\tau )$ (and all other variables in (2.8)) transform under the gauge transformations that preserve the asymptotic form of (2.8). In general in a BF theory with gauge group G, boundary gauge transformations are parametrized by functions g in loop(G)/G. However, since we are preserving the asymptotic form (2.8) which comes from working in the Fefferman–Graham gauge (2.7) the space of gauge transformations is instead parametrized by Diff(S^1|4)/PSU(1, 1|2). Therefore, the way in which $\mathcal{L}(\tau )$ transforms under this special class of gauge transformations will yield how the boundary Lagrangian transforms under Diff(S^1|4)/PSU(1, 1|2). From here, we will show in section 2.3.3 that the boundary Lagrangian can be identified with the $\mathcal{N}=4$ super-Schwarzian derivative. Thus, to do this identification, we note that $\mathcal{L}(\tau )$, B_τ (τ), and ψ^p (τ) transform as

$$\begin{align} {\delta }_{{\Lambda}}\mathcal{L}& =\xi {\mathcal{L}}^{\prime }+2\mathcal{L}{\xi }^{\prime }+\xi {\prime\prime\prime}-{B}_{\tau }^{i}{({t}^{i})}^{\prime }+\frac{1}{2}\left(3\bar{\psi }{{\epsilon}}^{\prime }+{\bar{\psi }}^{\prime }{\epsilon}-3{\bar{{\epsilon}}}^{\prime }\psi -\bar{{\epsilon}}{\psi }^{\prime }\right), \\ {\delta }_{{\Lambda}}{\psi }^{p}& =\xi {({\psi }^{p})}^{\prime }+\frac{3}{2}{\psi }^{p}{\xi }^{\prime }-{{\epsilon}}^{p}\mathcal{L}-2{({{\epsilon}}^{p})}^{{\prime\prime}}-\frac{1}{2}{t}^{i}{{({\sigma }^{i})}^{p}}_{q}\enspace {\psi }^{q}+{({B}_{\tau }^{i})}^{\prime }{({\sigma }^{i})}_{q}^{p}{{\epsilon}}^{q}+2{B}_{\tau }^{i}{{({\sigma }^{i})}^{p}}_{q}\enspace {({{\epsilon}}^{q})}^{\prime }, \\ {\delta }_{{\Lambda}}{B}_{\tau }^{i}& ={\left(\xi {B}_{\tau }^{i}\right)}^{\prime }-{({t}^{i})}^{\prime }+\frac{1}{2}\bar{\psi }{\sigma }^{i}{\epsilon}+\frac{1}{2}\bar{{\epsilon}}{\sigma }^{i}\psi +i{{\epsilon}}^{ijk}\enspace {t}^{j}\enspace {B}_{\tau }^{k}, \end{align} \tag{ 2.13 }$$

under the gauge transformation which preserves the form of A_τ in (2.8):

$$\begin{align} {\Lambda}(\tau )& =\frac{\xi }{2}{L}_{1}-{\xi }^{\prime }{L}_{0}+\left(\frac{1}{2}\bar{\psi }{\epsilon}+\frac{1}{2}\psi \bar{{\epsilon}}-\frac{1}{2}\xi {({B}^{a})}^{2}+\mathcal{L}\xi +{\xi }^{{\prime\prime}}\right){L}_{-1}-({t}^{i}-\xi {B}_{\tau }^{i}){T}_{i} \\ & \quad +\frac{1}{2}{\bar{{\epsilon}}}^{p}{G}_{p,\frac{1}{2}}-\left({({\bar{{\epsilon}}}^{p})}^{\prime }-\frac{1}{2}\xi {\bar{\psi }}^{p}-\frac{1}{2}{B}_{\tau }^{i}{{({\sigma }^{i\ast })}^{p}}_{\hspace{-4pt}q}\enspace {\bar{{\epsilon}}}^{q}\right){G}_{p,-\frac{1}{2}}+\frac{1}{2}{{\epsilon}}^{p}{\bar{G}}_{p,\frac{1}{2}} \\ & \quad -\left({{\epsilon}}^{p}-\frac{1}{2}\xi {\psi }^{p}-\frac{1}{2}{B}_{\tau }^{i}{({\sigma }^{i})}_{q}^{p}\enspace {{\epsilon}}^{q}\right){\bar{G}}_{p,-\frac{1}{2}}. \end{align} \tag{ 2.14 }$$

It is not a coincidence that Λ(τ) takes the same form (up to the redefinition (2.12)) as ϕ(τ) in (2.9). This follows from requiring that the leading result in both D_τ Λ (since we require that the gauge transformation Λ not change the asymptotic form of A_τ (2.8)) and D_τ ϕ (since we impose the equation of motion for A by also using (2.8)) both vanish. With these transformations in mind, we now proceed by introducing the necessary technology to define the $\mathcal{N}=4$ Schwarzian derivative. Following that, we will finally show that this Schwarzian derivative can be identified with the boundary Lagrangian $\mathcal{L}(\tau )$ from (2.12).

Finally, we can extend this analysis to the case when the SU(2) chemical potential, α, is turned on. In order to do this we can generalize the previous boundary condition from ${A}_{\tau }-\frac{i\phi }{2{{\Phi}}_{r}}=0$ to ${A}_{\tau }-\frac{i\phi }{2{{\Phi}}_{r}}=\frac{2\pi i}{\beta }\alpha {T}^{3}$, where α ∼ α + 1. This does not modify the boundary conditions of gravity and the fermions but now the SU(2) gauge field boundary condition is ${B}_{\tau }-{{\Phi}}_{r}^{-1}b=\frac{2\pi i}{\beta }\alpha {T}^{3}$, supporting the identification of α with a chemical potential. In section 4.4 we explain how from the 4D near-extremal black hole perspective this boundary condition is equivalent to fixing the holonomy of the gauge field in the asymptotically flat region, which is related to fixing the boundary 4D angular velocity.

So far we have seen that $\mathcal{N}=4$ super-JT gravity can be reduced to a boundary theory. In this section, we will see this theory can be recasted as a $\mathcal{N}=4$ super-reparametrization mode with a super-Schwarzian action. We will first review in section 2.3.1 the definition of super-diffeomorphisms. In section 2.3.2 we will define the super-Schwarzian derivative that will be the action and in section 2.3.3 we will put everything together to write down the final boundary action.

2.3.1. Super-diffeomorphisms

To match with the $\mathcal{N}=4$ super-JT theory defined previously we will be interested in SU(2) extended $\mathcal{N}=4$ supersymmetry ¹⁷ . We will begin by giving a super-space description of $\mathcal{N}=4$ super-diffeomorphisms.

Consider an $\mathcal{N}=4$ super-line parametrized by a bosonic variable τ and fermionic variables θ^p and ${\bar{\theta }}_{q}$, where p, q = 1, 2. The coordinates θ^p and ${\bar{\theta }}_{q}$ transform respectively in the fundamental and antifundamental representations of a local SU(2) symmetry. We will omit the indices and simply write θ and $\bar{\theta }$ when it is clear by context. We define the four super-derivatives as

$$\begin{equation}{D}_{p}=\frac{\partial }{\partial {\theta }^{p}}+\frac{1}{2}{\bar{\theta }}_{p}{\partial }_{\tau },\qquad {\bar{D}}^{q}=\frac{\partial }{\partial {\bar{\theta }}_{q}}+\frac{1}{2}{\theta }^{q}{\partial }_{\tau }.\end{equation} \tag{ 2.15 }$$

The SU(2) indices will be raised and lowered by the antisymmetric tensors ɛ_pq and ɛ^pq with ɛ₁₂ = ɛ²¹ = 1. We will denote by σⁱ with i = 1, 2, 3 the Pauli matrices and indices will be contracted from 'left-bottom' to 'right-up', e.g. $\bar{\theta }{\sigma }^{i}\theta ={\bar{\theta }}_{p}{({\sigma }^{i})}_{q}^{p}{\theta }^{q}$. We will sometimes group the coordinates of the $\mathcal{N}=4$ super-line as $Z=(\tau ,{\theta }^{p},{\bar{\theta }}_{p})$.

We will study general reparametrizations of the super-line, which will become the degrees of freedom in the path integral that defines the Schwarzian theory. These have the following form

$$\begin{equation}\tau \to {\tau }^{\prime }(\tau ,\theta ,\bar{\theta }),\qquad {\theta }^{p}\to {{\theta }^{\prime }}^{p}(\tau ,\theta ,\bar{\theta }),\qquad {\bar{\theta }}_{q}\to {\bar{\theta }}_{q}^{\prime }(\tau ,\theta ,\bar{\theta }),\end{equation} \tag{ 2.16 }$$

and satisfy a set of constrains given by

$$\begin{equation}{D}_{p}{\bar{\theta }}_{q}^{\prime }=0,\qquad {\bar{D}}^{p}{{\theta }^{\prime }}^{b}=0,\end{equation} \tag{ 2.17 }$$

$$\begin{equation}{D}_{p}{\tau }^{\prime }-\frac{1}{2}({D}_{p}{{\theta }^{\prime }}^{q}){\bar{\theta }}_{q}^{\prime }=0,\qquad {\bar{D}}^{p}{\tau }^{\prime }-\frac{1}{2}({\bar{D}}^{p}{\bar{\theta }}_{q}^{\prime }){{\theta }^{\prime }}^{q}=0\end{equation} \tag{ 2.18 }$$

analyzed in [62, 63]. They guarantee that the superderivatives transform homogeneously and preserve the global SU(2) symmetry. We will refer to the space of solutions of these constrains as $\mathcal{N}=4$ super-diffeomorphisms and denote it as Diff(S^1|4), indicating one bosonic and four fermionic directions.

Next we will look at some examples. The simplest super-reparametrizations are purely bosonic. They are given in terms of an arbitrary function f(τ) and an arbitrary SU(2) matrix g(τ). The solutions of the constraints have the form

$$\begin{equation}\tau \to f(\tau )+\frac{1}{8}{f}^{{\prime\prime}}(\tau ){(\bar{\theta }\theta )}^{2},\end{equation} \tag{ 2.19 }$$

$$\begin{equation}{\theta }^{p}\to g{\left(\tau -\frac{1}{2}\bar{\theta }\theta \right)}_{q}^{p}{\theta }^{q}\sqrt{{f}^{\prime }\left(\tau -\frac{1}{2}\bar{\theta }\theta \right)},\end{equation} \tag{ 2.20 }$$

$$\begin{equation}{\bar{\theta }}_{p}\to {\bar{\theta }}_{q}\bar{g}{\left(\tau +\frac{1}{2}\bar{\theta }\theta \right)}_{p}^{q}\sqrt{{f}^{\prime }\left(\tau +\frac{1}{2}\bar{\theta }\theta \right)}.\end{equation} \tag{ 2.21 }$$

Another simple example are the 'chiral' and 'anti-chiral' fermionic transformations. The chiral ones are parametrized in terms of two fermionic functions η^p (τ) in the fundamental of SU(2) and the reparametrization is given by

$$\begin{equation}\tau \to \tau +\frac{1}{2}\bar{\theta }\eta \left(\tau +\frac{1}{2}\bar{\theta }\theta \right)+\frac{1}{4}{\partial }_{\tau }{(\bar{\theta }\eta (\tau ))}^{2},\end{equation} \tag{ 2.22 }$$

$$\begin{equation}{\theta }^{p}\to {\theta }^{p}+{\eta }^{p}\left(\tau -\frac{1}{2}\bar{\theta }\theta \right),\end{equation} \tag{ 2.23 }$$

$$\begin{equation}{\bar{\theta }}_{p}\to {\bar{\theta }}_{p}\left(1+\bar{\theta }{\eta }^{\prime }\left(\tau -\frac{1}{2}\bar{\theta }\theta \right)\right).\end{equation} \tag{ 2.24 }$$

The anti-chiral are parametrized by ${\bar{\eta }}_{p}(\tau )$ in the antifundamental, and the reparametrization is given by

$$\begin{equation}\tau \to \tau +\frac{1}{2}\theta \bar{\eta }\left(\tau -\frac{1}{2}\bar{\theta }\theta \right)+\frac{1}{4}{\partial }_{\tau }{(\theta \bar{\eta }(\tau ))}^{2},\end{equation} \tag{ 2.25 }$$

$$\begin{equation}{\theta }^{p}\to {\theta }^{p}\left(1+\theta {\bar{\eta }}^{\prime }\left(\tau +\frac{1}{2}\bar{\theta }\theta \right)\right),\end{equation} \tag{ 2.26 }$$

$$\begin{equation}{\bar{\theta }}_{p}\to {\bar{\theta }}_{p}+{\bar{\eta }}_{p}\left(\tau +\frac{1}{2}\bar{\theta }\theta \right).\end{equation} \tag{ 2.27 }$$

The most general element of Diff(S^1|4) is obtained by a general fermionic transformation followed by a bosonic one ¹⁸ . We parametrize Diff(S^1|4) in terms of the degrees of freedom $f(\tau ),g(\tau )\in SU(2),{\eta }^{p}(\tau ),{\bar{\eta }}_{p}(\tau ),$ where p = 1, 2. It is hard to determine the finite form that has all parameters turned on since such answer contains a large number of terms. Therefore, we will not write it down explicitly although its straightforward to get from the results presented so far. Instead, it is useful to analyze the most general infinitesimal transformation. Up to $\mathcal{O}(\eta )$ is given by

$$\begin{equation}\tau \to f\left(\tau -\frac{1}{2}\bar{\eta }\left(\tau -\frac{1}{2}\bar{\theta }\theta \right)\theta +\frac{1}{2}\bar{\theta }\eta \left(\tau +\frac{1}{2}\bar{\theta }\theta \right)\right)+\frac{1}{8}{f}^{{\prime\prime}}(\tau ){\left((\bar{\theta }+\bar{\eta })(\theta +\eta )\right)}^{2}\end{equation} \tag{ 2.28 }$$

$$\begin{equation}{\theta }^{p}\to {{F}^{p}}_{q}\left(\tau -\frac{1}{2}\bar{\theta }\theta \right)\left({\theta }^{q}+{\eta }^{q}\left(\tau -\frac{1}{2}\bar{\theta }\theta \right)\right)+{\partial }_{\tau }\left({{F}^{p}}_{q}\left(\tau \right){\theta }^{q}\theta \bar{\eta }\left(\tau \right)\right)\end{equation} \tag{ 2.29 }$$

$$\begin{equation}{\bar{\theta }}_{p}\to \left({\bar{\theta }}_{q}+{\bar{\eta }}_{q}\left(\tau +\frac{1}{2}\bar{\theta }\theta \right)\right){{F}^{q}}_{p}\left(\tau +\frac{1}{2}\bar{\theta }\theta \right)+{\partial }_{\tau }\left(\eta \left(\tau \right)\bar{\theta }{\bar{\theta }}_{q}{{F}^{q}}_{p}\left(\tau \right)\right),\end{equation} \tag{ 2.30 }$$

where we use ${{F}^{p}}_{q}(\tau )=\sqrt{{f}^{\prime }(\tau )}{{g}^{p}}_{q}(\tau )$.

We introduce now an important sub-group of the super-reparametrizations, the global superconformal group PSU(1, 1|2). It is generated by six bosonic variables (three from SL(2) and three from SU(2)) and eight fermionic. The general case can be found in [63]. Instead we will write down the bosonic generators

$$\begin{equation}\tau \to \frac{a\tau +b}{c\tau +d}-\frac{c}{4{(c\tau +d)}^{3}}{(\bar{\theta }\theta )}^{2},\end{equation} \tag{ 2.31 }$$

$$\begin{equation}{\theta }^{p}\to {[{\text{e}}^{\text{i}\overrightarrow{t}\cdot \overrightarrow{\sigma }}]}_{q}^{p}{\theta }^{q}\frac{1}{c\left(\tau -\frac{1}{2}\bar{\theta }\theta \right)+d},\end{equation} \tag{ 2.32 }$$

$$\begin{equation}{\bar{\theta }}_{p}\to {\bar{\theta }}_{q}{[{\text{e}}^{-\text{i}\overrightarrow{t}\cdot \overrightarrow{\sigma }}]}_{p}^{q}\frac{1}{c\left(\tau +\frac{1}{2}\bar{\theta }\theta \right)+d},\end{equation} \tag{ 2.33 }$$

with $a,b,c,d\in \mathbb{R}$ such that ad − bc = 1 and c > 0. This is precisely of the form (2.19) where f(τ) is a global conformal transformation $PSL(2,\mathbb{R})$ while (t¹, t², t³) parametrize a global SU(2) transformation. The fermionic generators can be parametrized by two constant spinor doublets η and $\tilde{\eta }$ and act as

$$\begin{equation}\tau \to \tau +\frac{1}{2}(\bar{\theta }-\bar{\tilde{\eta }})\eta -\frac{1}{2}\bar{\eta }(\theta -\tilde{\eta }),\end{equation} \tag{ 2.34 }$$

$$\begin{equation}{\theta }^{p}\to {\theta }^{p}+{\eta }^{p}-{\tilde{\eta }}^{p},\end{equation} \tag{ 2.35 }$$

$$\begin{equation}{\bar{\theta }}_{p}\to {\bar{\theta }}_{p}+{\bar{\eta }}_{p}-{\bar{\tilde{\eta }}}_{p}.\end{equation} \tag{ 2.36 }$$

As anticipated there are eight fermionic generators. As explained above, the most general case is a composition of bosonic and fermionic.

2.3.2. Super-Schwarzian action

The Schwarzian derivative associated to reparametrizations of the $\mathcal{N}=4$ super-circle was defined by Matsuda and Uematsu in [63] (see also [62]). The Schwarzian derivative is given in terms of superspace variables as

$$\begin{equation}{S}^{i}(Z;{Z}^{\prime })=-2D{\sigma }^{i}\bar{D}\enspace \mathrm{log}\left(\frac{1}{2}({D}_{p}{{\theta }^{\prime }}^{q})({\bar{D}}^{p}{\bar{\theta }}_{q}^{\prime })\right).\end{equation} \tag{ 2.37 }$$

It satisfies the following chain rule

$$\begin{equation}{S}^{i}(Z;{Z}^{\prime })=\frac{1}{2}{({\sigma }^{i})}_{q}^{p}{({\sigma }^{j})}_{{p}^{\prime }}^{{q}^{\prime }}({D}_{p}{{\theta }^{{\prime\prime}}}^{{p}^{\prime }})({\bar{D}}^{q}{\bar{\theta }}_{{q}^{\prime }}^{{\prime\prime}}){S}^{j}({Z}^{{\prime\prime}};{Z}^{\prime })+{S}^{i}(Z;{Z}^{{\prime\prime}}).\end{equation} \tag{ 2.38 }$$

Another defining property is the fact that Sⁱ (Z; Z') = 0 whenever the super-reparametrization is an element of the global PSU(1, 1|2) as in (2.31). This will prove consequential in section 2.4 when studying the global symmetries of the $\mathcal{N}=4$ super-Schwarzian action which we shall define shortly.

The derivative Sⁱ is a superfield with several components. We can extract the bosonic piece we want to associate to the Schwarzian action. In the notation of [63] it is

$$\begin{equation}{S}^{i}(Z;{Z}^{\prime })\supset -\bar{\theta }{\sigma }^{i}\theta \hspace{2.84544pt}{S}_{b}(\tau ,\theta ,\bar{\theta };{\tau }^{\prime },{\theta }^{\prime },{\bar{\theta }}^{\prime }).\end{equation} \tag{ 2.39 }$$

One motivation of this choice is to look at purely bosonic transformations defined in equation (2.19). For simplicity let us briefly consider ${Z}^{\prime }=({\tau }^{\prime },{\theta }^{\prime },{\bar{\theta }}^{\prime })$ where τ', θ' and ${\bar{\theta }}^{\prime }$ take the special form (2.19), in terms of an arbitrary f(τ) and set g(τ) = 1. Then the definition above gives the super-Schwarzian

$$\begin{equation}{S}^{i}(Z,{Z}^{\prime })=-\mathrm{S}\mathrm{c}\mathrm{h}(f,\tau )\bar{\theta }{\sigma }^{i}\theta .\end{equation} \tag{ 2.40 }$$

Another motivation is that when we interpret Sⁱ as a superconformal generator, that component generates bosonic translations along the circle and we want to identify this as the action of the Schwarzian theory.

The super-Schwarzian satisfies the constrains ${D}_{p}{D}_{q}{S}^{i}={\bar{D}}^{p}{\bar{D}}^{q}{S}^{i}=0$. Therefore, as a superfield it can be expanded in the following components

$$\begin{align} {S}^{i}& =2{S}_{T}^{i}+\bar{\theta }{\sigma }^{i}{S}_{f}+{\bar{S}}_{f}{\sigma }^{i}\theta -\bar{\theta }{\sigma }^{i}\theta {S}_{b}+i{{\epsilon}}^{ijk}\bar{\theta }{\sigma }^{j}\theta {\partial }_{\tau }{S}_{T}^{k} \\ & \quad -\frac{1}{2}(\bar{\theta }\theta )\bar{\theta }{\sigma }^{i}{\partial }_{\tau }{S}_{f}+\frac{1}{2}(\bar{\theta }\theta ){\partial }_{\tau }{\bar{S}}_{f}{\sigma }^{i}\theta +\frac{1}{4}{(\bar{\theta }\theta )}^{2}{\partial }_{\tau }^{2}{S}_{T}^{i}. \end{align} \tag{ 2.41 }$$

Then the super-Schwarzian action is

$$\begin{equation}{I}_{\mathcal{N}=4}=-{{\Phi}}_{r}\int \mathrm{d}\tau \enspace {S}_{b}[f(\tau ),g(\tau ),\eta (\tau )],\end{equation} \tag{ 2.42 }$$

where Φ_r can be viewed as a coupling constant whose role we shall soon discuss and the factor of 12 above is chosen such that it simplifies the factor in (2.40).

We can rewrite the action in super-field notation by defining $S=\bar{\theta }{\sigma }^{i}\theta {S}^{i}$, where we sum over i = 1, 2, 3. Then, using the expansion of Sⁱ gives ¹⁹ ,

$$\begin{equation}S=2\bar{\theta }{\sigma }^{i}{S}_{T}^{i}\theta -3(\bar{\theta }\theta )\bar{\theta }{S}_{f}+3(\bar{\theta }\theta ){\bar{S}}_{f}\theta +3{S}_{b}{(\bar{\theta }\theta )}^{2}.\end{equation} \tag{ 2.43 }$$

Then the action (2.42) can be rewritten as ${I}_{\mathcal{N}=4}\sim {{\Phi}}_{r}\int \mathrm{d}\tau \enspace {\mathrm{d}}^{4}\theta S$. Note that in terms of the S[f(τ), g(τ), η(τ)] there is no obvious chain rule analogous to (2.38). For this reason it will oftentimes be easier to work with Sⁱ instead of the super-field S. To make things concrete, it is informative to write the Schwarzian action when focusing on purely bosonic components, when setting η(τ) = 0 in (2.42):

$$\begin{equation}{I}_{\mathcal{N}=4,\text{bosonic}}[f(\tau ),g(\tau ),\eta (\tau )=0]=-{\int }_{0}^{\beta }\mathrm{d}\tau \enspace {{\Phi}}_{r}\left[\text{Sch}(f,\tau )+Tr{({g}^{-1}{\partial }_{\tau }g)}^{2}\right].\end{equation} \tag{ 2.44 }$$

Since η(τ) = 0 is a solution to the equations of motion we will soon use the above action to extract the classical saddle point when quantizing the super-Schwarzian action at the level of the path integral.

2.3.3. Transformation law and a match with the JT boundary term

In this section, we shall derive explicitly the infinitesimal transformation rules of the $\mathcal{N}=4$ super-Schwarzian (2.41). We will expand f(τ) ≈ τ + ξ(τ), g ≈ 1 + itⁱ (τ)σⁱ , η ≈ (τ) and $\bar{\eta }\approx \bar{{\epsilon}}(\tau )$ and work to linear order in ξ, tⁱ , and $\bar{{\epsilon}}$. A convenient way to encode the infinitesimal reparametrizations of the super line (2.16) that automatically satisfies the constraints (2.17) is to use a super-field as in [62, 63]:

$$\begin{align} E(Z)& =\xi \left(\tau +\frac{1}{2}\bar{\theta }\theta \right)+\xi \left(\tau -\frac{1}{2}\bar{\theta }\theta \right)+\bar{\theta }{\epsilon}\left(\tau -\frac{1}{2}\bar{\theta }\theta \right) \\ & \quad -\bar{{\epsilon}}\left(\tau +\frac{1}{2}\bar{\theta }\theta \right)\theta +\frac{1}{2}\bar{\theta }{\sigma }^{i}\theta {t}^{i}(\tau ). \end{align} \tag{ 2.45 }$$

Under such a reparametrization (2.45), the super-Schwarzian (2.41) transforms in the same way as the super-energy momentum tensor, which is given in [63]:

$$\begin{align} {\delta }_{E}{S}^{i}& ={\partial }_{\tau }\left(E(Z){S}^{i}\right)+DE(Z)\bar{D}{S}^{i}+\bar{D}E(Z)D{S}^{i} \\ & \quad -i{{\epsilon}}^{ijk}\left(D{\sigma }^{j}\bar{D}E\right){S}^{k}-2D{\sigma }^{i}\bar{D}{\partial }_{\tau }E(z). \end{align} \tag{ 2.46 }$$

Now substitute (2.41) and (2.45) into (2.46), and collect in components:

$$\begin{equation}{\delta }_{E}{S}^{i}=2{\delta }_{E}{S}_{T}^{i}+\bar{\theta }{\sigma }^{i}{\delta }_{E}{S}_{f}+{\delta }_{E}{\bar{S}}_{f}{\sigma }^{i}\theta -\bar{\theta }{\sigma }^{i}\theta {\delta }_{E}{S}_{b}+i{{\epsilon}}^{ijk}\bar{\theta }{\sigma }^{j}\theta {\partial }_{\tau }{\delta }_{E}{S}_{T}^{k}+\cdots \enspace ,\end{equation} \tag{ 2.47 }$$

we obtain the infinitesimal transformation of ${S}_{T}^{i},{S}_{f},{\bar{S}}_{f},{S}_{b}$. Note that the terms in ... are purely determined by lower components, and thus it is enough to focus on the terms up to $\mathcal{O}(\bar{\theta }\theta )$. As a result, the transformations of ${S}_{T}^{i},{S}_{f},{\bar{S}}_{f},{S}_{b}$ are given by:

$$\begin{align} {\delta }_{E}{S}_{b}& =\xi {S}_{b}^{\prime }+2{S}_{b}{\xi }^{\prime }+{\xi }^{\prime \prime \prime }-{S}_{T}^{i}{({t}^{i})}^{\prime }+\frac{1}{2}\left(3{\bar{S}}_{f}{{\epsilon}}^{\prime }+{\bar{S}}_{f}^{\prime }{\epsilon}-3{\bar{{\epsilon}}}^{\prime }{S}_{f}-\bar{{\epsilon}}{S}_{f}^{\prime }\right), \\ {\delta }_{E}{S}_{f}^{p}& =\xi {({S}_{f}^{p})}^{\prime }+\frac{3}{2}{S}_{f}^{p}{\xi }^{\prime }-{{\epsilon}}^{p}{S}_{b}-2{({{\epsilon}}^{p})}^{{\prime\prime}}-\frac{1}{2}{t}^{i}{({\sigma }^{i})}_{q}^{p}{S}_{f}^{q}+{({S}_{T}^{i})}^{\prime }{({\sigma }^{i})}_{q}^{p}{{\epsilon}}^{q}+2{S}_{T}^{i}{({\sigma }^{i})}_{q}^{p}{({{\epsilon}}^{q})}^{\prime }, \\ {\delta }_{E}{S}_{T}^{i}& ={\left(\xi {S}_{T}^{i}\right)}^{\prime }-{({t}^{i})}^{\prime }+\frac{1}{2}{\bar{S}}_{f}{\sigma }^{i}{\epsilon}+\frac{1}{2}\bar{{\epsilon}}{\sigma }^{i}{S}_{f}+i{{\epsilon}}^{ijk}{t}^{j}{S}_{T}^{k}. \end{align} \tag{ 2.48 }$$

We note that they exactly agree with the infinitesimal transformation deduced from the boundary action of the BF theory (2.13) with the field identification:

$$\begin{align} \mathcal{L}& {\leftrightarrow}{S}_{b}, \\ {\psi }^{p}& {\leftrightarrow}{S}_{f}^{p}, \\ {B}_{\tau }^{i}& {\leftrightarrow}{S}_{T}^{i}. \end{align} \tag{ 2.49 }$$

Since the infinitesimal transformations and PSU(1, 1|2) invariance suffices to determine the global form of the action as the $\mathcal{N}=4$ super-Schwarzian, it then follows that the boundary action (2.12) from BF theory agrees with our definition of the Schwarzian action (2.42). Therefore, the path integral in the $\mathcal{N}=4$ super-JT gravity with the boundary conditions discussed in section 2.2 can be reduced to that for the $\mathcal{N}=4$ super-Schwarzian action defined by (2.42).

Before analyzing the quantization of the super-Schwarzian theory, it is useful to study the space-time and global symmetries present in this action.

A useful way to discuss symmetries of a quantum field theory is to cast it in terms of internal symmetries and spacetime symmetries. The continuous internal symmetries of (2.42) form PSU(1, 1|2), ²⁰ generated by (2.31)–(2.36), directly acting on $\left(f(\tau ),g(\tau ),{\bar{\eta }}_{p}(\tau ),{\eta }^{p}(\tau )\right)$. They are zero modes of the $\mathcal{N}=4$ super-Schwarzian derivative, and the action (2.42) is invariant due to the chain rule (2.38). Specifically,

$$\begin{equation}{S}^{i}(Z,h{\circ}{Z}^{\prime }(Z))={S}^{i}(Z,{Z}^{\prime }(Z)),\end{equation} \tag{ 2.50 }$$

where h is a composition of the bosonic and fermionic transformations in (2.31)–(2.36) which we apply to the super-reparametrization Z'(Z). These transformations are the supersymmetric generalization of the $\mathrm{S}\mathrm{L}(2,\enspace \mathbb{R}):f(\tau )\to \frac{af(\tau )+b}{cf(\tau )+d}$, and we will have to quotient out such transformations as we proceed to compute the partition function [13–15], in order to obtain a well-defined partition function. Furthermore, aside from the PSU(1, 1|2) that has an SU(2) subgroup which acts on the left of the field g(τ) there is an additional SU(2) symmetry which acts on the right of the field g(τ). This additional symmetry does not act on f(τ) or on the fermionic fields $\bar{\eta }$ and η. Thus, to summarize the continuous internal symmetry is, up to discrete factors, PSU(1, 1|2) × SU(2). ²¹

The $\mathcal{N}=4$ super-Schwarzian theory also admits spacetime symmetries. To see this consider the transformation (2.48); for the transformation ξ(τ) = ξ corresponding to τ → τ + ξ (time translations), tⁱ (τ) = tⁱ corresponding to constant infinitesimal rotation of θ^p (R-symmetry rotations), and (τ) = corresponding to θ → θ + (super-translations), the action S_b is invariant up to a total derivative. Together, all such infinitesimal transformations generate the $\mathcal{N}=4$ super-Poincaré spacetime symmetry. They satisfy the algebra

$$\begin{equation}\left\{{\bar{Q}}_{p},{Q}^{q}\right\}=2{\delta }_{p}^{q}H,\quad \text{with}\;\enspace {\bar{Q}}_{p}=i\frac{\partial }{\partial {\theta }^{p}}+{\bar{\theta }}_{p}{\partial }_{\tau },\enspace {Q}^{p}=-i\frac{\partial }{\partial {\bar{\theta }}_{p}}-{\theta }^{p}{\partial }_{\tau },\enspace H={\partial }_{\tau }\end{equation} \tag{ 2.51 }$$

while all other commutators vanish. It is straightforward to identify these $\mathcal{N}=4$ supercharges from the Noether procedure using the transformation (2.48): S_b is the charge generating time translations (and is thus the Hamiltonian of the theory), S_f is the generator of super-translations, and ${S}_{T}^{i}$ is the generator of R-symmetry rotations. Note that the Hamiltonian here is not one of the generators of PSU(1, 1|2), since it is in fact the super-Schwarzian itself, and thus proportional to the quadratic Casimir of the PSU(1, 1|2). This is quite analogous to the non-supersymmetric case [8, 69, 70].

We would like to stress that even though PSU(1, 1|2) plays a big role in constructing the Schwarzian action, the theory is not invariant under spacetime superconformal PSU(1, 1|2) symmetries. The spectrum is not organized according to PSU(1, 1|2) representations. Only the $\mathcal{N}=4$ super-Poincaré sub-group is a spacetime symmetry.

It is also useful to briefly discuss some of the discrete internal and spacetime symmetries of the theory. The $\mathcal{N}=4$ super-Schwarzian has a time reversal symmetry $\mathcal{T}$ which acts on the fermionic fields as $\mathcal{T}\eta (\tau )=i\eta (-\tau )$ and $\mathcal{T}\bar{\eta }(\tau )=i\bar{\eta }(-\tau )$ and on the bosonic fields as $\mathcal{T}f(\tau )=f(-\tau )$ and $\mathcal{T}g(\tau )=g(-\tau )$; in such a case, ${\mathcal{T}}^{2}=1$ and the symmetry is ${\mathbb{Z}}_{2}^{\mathcal{T}}$. There is also a ${\mathbb{Z}}_{2}^{F}$ fermionic symmetry (−1)^F which solely acts on the fermionic fields ²² . Thus, the symmetry is ${\mathbb{Z}}_{2}^{F}\times {\mathbb{Z}}_{2}^{\mathcal{T}}$. We can now address discrete factors for the internal symmetry group of the theory. In (2.31)–(2.36) we note that the transformation given by the center of $\mathrm{S}\mathrm{L}(2,\mathbb{R})$ with a = d = −1 and b = c = 0 (acting as η → −η and $\bar{\eta }\to -\bar{\eta }$) is redundant with the composition of the two center transformations for the two SU(2), the first of which acts as g → −g, η → −η, $\bar{\eta }\to -\bar{\eta }$, and the second of which acts solely on g → −g. Furthermore, this transformation is again redundant with the (−1)^F symmetry mentioned previously. Thus, the bosonic subgroup of the symmetry group for the theory is given by

$$\begin{equation}\frac{\mathrm{S}\mathrm{L}(2,\mathbb{R})\times SU(2)\times SU(2)\times {\mathbb{Z}}_{2}^{F}}{{\mathbb{Z}}_{2}}\times {\mathbb{Z}}_{2}^{\mathcal{T}}.\end{equation} \tag{ 2.52 }$$

Consequently, we note that even-dimensional representations of SU(2) (half-integer spins) need to be fermionic and odd-dimensional representations of SU(2) (integer spins) need to be bosonic. Since we will be able to decompose our partition function as a sum over SU(2) characters, this fact will play an important role in easily obtaining the supersymmetric index from the theory by using the result for the partition function.

The $\mathcal{N}=4$ super-line can be parametrized in superspace by $(\tau ,{\theta }^{p},{\bar{\theta }}_{p})$, p = 1, 2, where θ and $\bar{\theta }$ are Grassman variables transforming as fundamental and anti-fundamental of an SU(2) symmetry. $\mathcal{N}=4$ super-reparametrizations are parametrized by a bosonic field f(τ) ∈ Diff(S¹), a local transformation g(τ) ∈ SU(2) (or more precisely the loop group) and fermionic fields η^p (τ) and ${\bar{\eta }}_{p}(\tau )$. In terms of a super-reparametrization these fields can be roughly written as

$$\begin{equation}\tau \to f(\tau )+\cdots \enspace ,\end{equation} \tag{ 3.1 }$$

$$\begin{equation}{\theta }^{p}\to {g}_{\;\;q}^{p}(\tau ){\theta }^{q}\sqrt{{f}^{\prime }(\tau )}+{\eta }^{p}(\tau )+\cdots \enspace ,\end{equation} \tag{ 3.2 }$$

$$\begin{equation}{\bar{\theta }}_{p}\to {\bar{\theta }}_{q}\hspace{2.84544pt}{\bar{g}}_{\;\;p}^{q}(\tau )\sqrt{{f}^{\prime }(\tau )}+{\bar{\eta }}_{p}(\tau )+\cdots \enspace .\end{equation} \tag{ 3.3 }$$

The dots correspond to terms that are fixed by the super-reparametrization constrains and can be found in the previous section. We also defined a Schwarzian action ${I}_{\mathcal{N}=4}=-{{\Phi}}_{r}\int \mathrm{d}\tau {S}_{b}[f,g,\eta ,\bar{\eta }]$ invariant under PSU(1, 1|2) transformations acting on the fields $(f,g,\eta ,\bar{\eta })$. The bosonic component of this action is

$$\begin{equation}{I}_{\mathcal{N}=4}=-{{\Phi}}_{r}{\int }_{0}^{\beta }\mathrm{d}\tau \left[\text{Sch}(f,\tau )+\mathrm{Tr}{({g}^{-1}{\partial }_{\tau }g)}^{2}+(\mathrm{f}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s})\right]\end{equation} \tag{ 3.4 }$$

which gives the usual Schwarzian action and a particle moving on SU(2). The extra terms involve the fermions η and $\bar{\eta }$.

In this section, we will compute the Euclidean path integral giving the partition function ²³ ,

$$\begin{equation}Z(\beta ,\alpha )=\int \frac{\mathcal{D}f\mathcal{D}g\mathcal{D}\eta \mathcal{D}\bar{\eta }}{PSU(1,1\vert 2)}\enspace \mathrm{exp}\left({{\Phi}}_{r}\int \mathrm{d}\tau \enspace {S}_{b}[f,g,\eta ,\bar{\eta }]\right),\end{equation} \tag{ 3.5 }$$

where Φ_r is a dimensionful coupling constant of the theory. The inverse temperature β and chemical potentials α appears in the path integral through the boundary conditions of the fields:

$$\begin{equation}f(\tau +\beta )=f(\tau ),\qquad g(\tau +\beta )={\text{e}}^{2\pi \text{i}\alpha {\sigma }^{3}}g(\tau ),\qquad \eta (\tau +\beta )=-{\text{e}}^{2\pi \text{i}\alpha {\sigma }^{3}}\eta (\tau ),\end{equation} \tag{ 3.6 }$$

and similarly for $\bar{\eta }$. In the rest of this section we will evaluate this path integral as a function of β, α and the coupling Φ_r , and rewrite it as a trace over a Hilbert space with a possibly continuous spectrum.

3.2.1. Method 1: fermionic localization

In this section, we will solve the theory following [72]. The integration space is a coadjoint orbit and the super-Schwarzian action generates a U(1) symmetry. Even though we will not work out the measure and symplectic form in detail, we will assume it is chosen such that we can apply the Duistermaat–Heckman theorem. Therefore, we will compute the classical saddles, the one-loop determinants, and put everything together into the final answer (3.18).

The $\mathcal{N}=4$ saddle-point: as previously mentioned, the bosonic part of the $\mathcal{N}=4$ Schwarzian action is given by

$$\begin{equation}{I}_{\mathcal{N}=4,\text{bosonic}}=-{\int }_{0}^{\beta }\mathrm{d}\tau \enspace {{\Phi}}_{r}\left[\text{Sch}(f,\tau )+\mathrm{Tr}{({g}^{-1}{\partial }_{\tau }g)}^{2}\right].\end{equation} \tag{ 3.7 }$$

The equations of motion for f(τ) and g(τ) imply that:

$$\begin{equation}{\partial }_{\tau }\text{Sch}(f,\tau )=0,\qquad {\partial }_{\tau }\mathrm{Tr}{({g}^{-1}{\partial }_{\tau }g)}^{2}=0.\end{equation} \tag{ 3.8 }$$

The solution for the Schwarzian is well-known and is given by f(τ) = tan(πτ/β). The solution for the SU(2) adjoint field takes the form g = exp(it_i ⁱ τ), where ⁱ is a constant that needs is set by the boundary conditions for the field g(τ). To make the computation easier we note that all solutions can be transformed to the diagonal form (g = exp(iσ₃ ³ τ)) using an SU(2) transformation. If require that the field g be periodic, than we have that ³ = 2πn/β, with $n\in \mathbb{Z}$. More generally, the SU(2) symmetry could have a fugacity which would imply that the field g is no longer periodic; rather, it has g(β) = zg(0) where z ∈ SU(2) is the fugacity. Once again, since the partition function only depends on the conjugacy class of z, we can perform an SU(2) transformation to diagonalize z = exp(2πiασ₃). The solution for h is then given by $g=\mathrm{exp}\left(2\pi i{\sigma }_{3}(n+\alpha )\frac{\tau }{\beta }\right)$.

In such a case, the on-shell value of the action ${I}_{\mathcal{N}=4,\text{bosonic}}$ is given by:

$$\begin{equation}{I}_{\mathcal{N}=4,\text{bosonic}}^{\enspace \text{on}\mbox{-}\text{shell}}=-\frac{2{\pi }^{2}{{\Phi}}_{r}}{\beta }\left(1-4{(n+\alpha )}^{2}\right),\end{equation} \tag{ 3.9 }$$

Now that we have the on-shell action we can proceed by computing the one-loop determinant which is sufficient for fully computing the partition function.

The one-loop determinant: to compute the one-loop determinant, we have to account for all quadratic fluctuations in the theory. The quadratic fluctuations of the Schwarzian field have been analyzed in great detail [14, 72], and its contributions to the one-loop determinant is given, up to an overall proportionality constant, by

$$\begin{equation}{\mathrm{det}}_{\text{Schw.,one}\mbox{-}\text{loop}}={\left(\frac{{{\Phi}}_{r}}{\beta }\right)}^{\frac{3}{2}}.\end{equation} \tag{ 3.10 }$$

The quadratic fluctuation around the saddle-point of the SU(2)-group element can be param-etrized as $g(\tau )=\mathrm{exp}\left({\sigma }_{3}\left[2\pi i(n+\alpha )\frac{\tau }{\beta }+{{\epsilon}}^{3}(\tau )\right]\right)\mathrm{exp}\left({{\epsilon}}^{2}(\tau ){\sigma }_{2}\right)\mathrm{exp}\left({{\epsilon}}^{1}(\tau ){\sigma }_{1}\right)$, and yields a contribution to the action [73]

$$\begin{align} {I}_{SU(2),\text{quad}}& =\frac{8{\pi }^{2}{{\Phi}}_{r}}{\beta }{(n+\alpha )}^{2} \\ & \quad -2{{\Phi}}_{r}{\int }_{0}^{\beta }\mathrm{d}\tau \left({({{\epsilon}}^{1}{(\tau )}^{\prime })}^{2}+{({{\epsilon}}^{2}{(\tau )}^{\prime })}^{2}-{({{\epsilon}}^{3}{(\tau )}^{\prime })}^{2}+\frac{8\pi (n+\alpha )}{\beta }{{\epsilon}}^{2}(\tau ){{\epsilon}}^{1}{(\tau )}^{\prime }\right), \end{align} \tag{ 3.11 }$$

and the one-loop determinant obtained from integrating out these modes for each value of n is given by [73]:

$$\begin{equation}{\mathrm{det}}_{SU(2),\text{one}\mbox{-}\text{loop}}=\frac{{{\Phi}}_{r}^{3/2}(n+\alpha )}{{\beta }^{3/2}\enspace \text{sin}(2\pi \alpha )}.\end{equation} \tag{ 3.12 }$$

Finally, we discuss the quadratic contribution of the fermionic fields. By using the saddle-point solution for f(τ) and g(τ) and by quadratically expanding the super-Schwarzian action:

$$\begin{align} {I}_{\text{ferm.,quad.}}& ={{\Phi}}_{r}{\int }_{0}^{\beta }\mathrm{d}\tau \enspace \left({\eta }^{p}\left[\frac{2{\pi }^{2}}{{\beta }^{2}}\left(1+2{(n+\alpha )}^{2}\right){\partial }_{\tau }-{\partial }_{\tau }^{3}\right]{\bar{\eta }}_{p}\right. \\ & \quad \left.+{\partial }_{\tau }{\eta }^{p}\left[\frac{12{\pi }^{2}{(n+\alpha )}^{2}}{{\beta }^{2}}+\frac{8i\pi (n+\alpha ){\partial }_{\tau }}{\beta }-3{\partial }_{\tau }^{2}\right]{\bar{\eta }}_{p}\right). \end{align} \tag{ 3.13 }$$

Expanding the fermionic fields in Fourier modes,

$$\begin{equation}{\eta }^{1}(\tau )={\text{e}}^{\text{i}\frac{2\pi (n+\alpha )\tau }{\beta }}\sum\limits _{{m}_{1}\in \dots ,-\frac{1}{2},\enspace \frac{1}{2},\dots }\sqrt{\frac{\beta }{2\pi }}\enspace {\eta }_{{m}_{1}}^{1}{\text{e}}^{-\text{i}\frac{2\pi {m}_{1}\tau }{\beta }}\end{equation} \tag{ 3.14 }$$

$$\begin{equation}{\eta }^{2}(\tau )={\text{e}}^{-\text{i}\frac{2\pi (n+\alpha )\tau }{\beta }}\sum\limits _{{m}_{2}\in \dots ,-\frac{1}{2},\enspace \frac{1}{2},\dots }\sqrt{\frac{\beta }{2\pi }}\enspace {\eta }_{{m}_{2}}^{2}{\text{e}}^{\text{i}\frac{2\pi {m}_{2}\tau }{\beta }},\end{equation} \tag{ 3.15 }$$

where we impose anti-periodic boundary conditions for the fermionic fields when α = 0 and we impose boundary conditions consistent with the introduction of the fugacity z when α ≠ 0. We can then rewrite the action as,

$$\begin{equation}{I}_{\text{ferm.,quad.}}=\frac{2{\pi }^{2}i{{\Phi}}_{r}}{\beta }\left[\sum\limits _{p=1,2}\enspace \sum\limits _{{m}_{p}\in \dots ,-\frac{1}{2},\enspace \frac{1}{2},\dots }({m}_{p}-n-\alpha )(4{m}_{p}^{2}-1){\eta }_{{m}_{p}}^{p}{\bar{\eta }}_{-{m}_{p}}^{p}\right].\end{equation} \tag{ 3.16 }$$

We are interested in computing the dependence of the one-loop determinant on n, α, β and Φ_r . The β and Φ_r dependence is captured by the existence of the four-fermionic zero modes with m_p = ±1/2. As in [72], to compute the rest of the one-loop determinant, we will regularize this result by dividing the result by the one-loop determinant with n = 0 and α = 0. We thus find that the regularized one-loop determinant is given, again up to a proportionality constant, by

$$\begin{equation}\underset{\text{ferm.,one}\mbox{-}\text{loop}}{\mathrm{det}}=\frac{{\beta }^{4}}{{{\Phi}}_{r}^{4}}\prod\limits _{p=1,2}\enspace \enspace \prod\limits _{{m}_{p}\in \dots ,-\frac{5}{2},-\frac{3}{2},\frac{3}{2},\frac{5}{2},\dots }\frac{m-n-\alpha }{m}=\frac{{\beta }^{4}}{{{\Phi}}_{r}^{4}}\frac{\text{cos}{(\pi \alpha )}^{2}}{{(1-4{(n+\alpha )}^{2})}^{2}}.\end{equation} \tag{ 3.17 }$$

Final answer: thus, accounting for the saddle-point value of the action (3.9) together with the one-loop determinants (3.10), (3.12), and (3.17), we find that the partition function of the $\mathcal{N}=4$ Schwarzian theory is given, up to an overall proportionality constant, by:

$$\begin{align} {Z}_{\mathcal{N}=4\enspace \enspace \enspace \text{Schw.}}& =\sum\limits _{n\in \mathbb{Z}}\underset{\text{Schw.,}\;\text{one}\mbox{-}\text{loop}}{\mathrm{det}}\enspace \enspace \underset{SU(2),\;\text{one}\mbox{-}\text{loop}}{\mathrm{det}}\enspace \underset{\text{ferm.,}\;\text{one}\mbox{-}\text{loop}}{\mathrm{det}}{e}^{-{I}_{\mathcal{N}=4,\text{bosonic}}^{\enspace \text{on}\mbox{-}\text{shell}}} \\ & =\sum\limits _{n\in \mathbb{Z}}\frac{\beta \text{cot}(\pi \alpha )(\alpha +n)}{{{\Phi}}_{r}{(1-4{(n+\alpha )}^{2})}^{2}}{e}^{\frac{2{\pi }^{2}{{\Phi}}_{r}}{\beta }\left(1-4{(n+\alpha )}^{2}\right)}. \end{align} \tag{ 3.18 }$$

We will thus continue by matching this result using the completely distinct method of canonical quantization, after which we will come back to a detailed analysis of the spectrum associated to (3.18) in section 3.4.

3.2.2. Method 2: canonical quantization

In this section, we will compute the partition function of the $\mathcal{N}=4$ super-Schwarzian theory using the canonical quantization approach of [69] (for a very recent discussion explaining the connection to the localization approach see also [74]).

We will illustrate briefly the idea first. The localization formula we used above can be applied to integrals that generally have the following form

$$\begin{equation}Z=\int \mathrm{d}q\enspace \mathrm{d}p{e}^{-H(p,q)},\end{equation} \tag{ 3.19 }$$

where the integral is over a symplectic space (a classical phase space) with coordinates (q, p) and H(q, p) generates via the Poisson brackets a U(1) symmetry. In the case of the bosonic Schwarzian theory the integration manifold is $\mathrm{D}\mathrm{i}\mathrm{ff}({S}^{1})/\mathrm{S}\mathrm{L}(2,\mathbb{R})$ which a coadjoint orbit of the Virasoro group (and, therefore, symplectic), and H(p, q) is the Schwarzian action. Instead of using localization, we can obtain this integral using the following identity

$$\begin{equation}\underset{\hslash \to 0}{\mathrm{lim}}\enspace \mathrm{Tr}\left({e}^{-H(p,q)}\right)=\int \mathrm{d}q\enspace \mathrm{d}p{e}^{-H(p,q)},\end{equation} \tag{ 3.20 }$$

where the left-hand side is ℏ → 0 limit of the trace evaluated over the Hilbert space obtained by quantizing the phase space. In the case of the Schwarzian, the quantization of $\mathrm{D}\mathrm{i}\mathrm{ff}({S}^{1})/\mathrm{S}\mathrm{L}(2,\mathbb{R})$ is the identity representation of the Virasoro algebra with central charge c ∼ 1/ℏ. The left-hand side of (3.20) can by very easily computed at finite c as a Virasoro vacuum character by counting descendants, and a very simple calculation gives the Schwarzian path integral [69]. In the bosonic case, the main advantage of this method is the possibility to compute correlation functions which are not available using localization. In this case, we will use it as a double check on our previous result.

For the case of the $\mathcal{N}=4$ super-Schwarzian the integration space is Diff(S^1|4)/PSU(1, 1|2) which is a coadjoint orbit of super-Virasoro and, therefore, also symplectic. We will assume that the quantization of this phase space, the Hilbert space in (3.20), is equivalent to the identity representation of the small $\mathcal{N}=4$ Virasoro algebra with central charge c ∼ 1/ℏ. The $\mathcal{N}=4$ super-Schwarzian partition function is then the semiclassical limit of the vacuum character.

Lets begin then by recalling the super-Virasoro algebra involved in this problem. The bosonic generators are L_n and ${T}_{n}^{i}$ where n is an integer and i = 1, 2, 3 label the generators of a Kac–Moody SU(2) at level k. Their algebra is

$$\begin{equation}[{L}_{m},{L}_{n}]=(m-n){L}_{m+n}+\frac{k}{2}m({m}^{2}-1){\delta }_{n+m,0}\end{equation} \tag{ 3.21 }$$

$$\begin{equation}[{T}_{m}^{i},{T}_{n}^{j}]=i{{\epsilon}}^{ijk}{T}_{m+n}^{k}+\frac{k}{2}m{\delta }_{m+n,0}{\delta }_{i,j}\end{equation} \tag{ 3.22 }$$

$$\begin{equation}[{L}_{m},{T}_{n}^{i}]=-n{T}_{m+n}^{i}.\end{equation} \tag{ 3.23 }$$

The central charge of the bosonic Virasoro sector is c = 6k, which is fixed by a Jacobi identity. The fermionic generators are ${G}_{r}^{p}$ and ${\bar{G}}_{s}^{p}$, p = 1, 2. They transform in the fundamental and antifundamental of the SU(2). The Fourier mode parameter r, s are integer in the Ramond sector or half-integer in the Neveu–Schwarz sector. The rest of the algebra, involving the fermionic generators, can be found for example in [75], and is given by

$$\begin{align} & \left\{{G}_{r}^{p},{\bar{G}}_{s}^{q}\right\}=2{\delta }^{pq}{L}_{r+s}-2(r-s){\sigma }_{pq}^{i}{T}_{r+s}^{i}+\frac{k}{2}(4{r}^{2}-1){\delta }_{r+s,0}, \\ & [{T}_{m}^{i},{G}_{r}^{p}]=-\frac{1}{2}{\sigma }_{pq}^{i}{G}_{m+r}^{q},\enspace [{T}_{m}^{i},{\bar{G}}_{r}^{p}]=\frac{1}{2}{{\sigma }_{pq}^{i}}^{\star }{\bar{G}}_{m+r}^{q},\enspace \left\{{G}_{r}^{p},{G}_{s}^{q}\right\}=\left\{{\bar{G}}_{r}^{p},{\bar{G}}_{s}^{q}\right\}=0 \\ & [{L}_{m},{G}_{r}^{p}]=\left(\frac{m}{2}-r\right){G}_{m+r}^{p},\quad [{L}_{m},{\bar{G}}_{r}^{p}]=\left(\frac{m}{2}-r\right){\bar{G}}_{m+r}^{p}, \end{align} \tag{ 3.24 }$$

where ${\sigma }_{pq}^{i}$ are the Pauli matrices. In that reference, Eguchi and Taormina also construct the unitary representations of the algebra.

For the application we have in mind in this paper, the Schwarzian path integral, we will only need the massless representations in the NS sector, due to the fact that we want the Schwarzian fermions to be antiperiodic as explained in [69]. ²⁴ General representations are labeled by h, the eigenvalue of L₀, and ℓ, the spin of the SU(2) representation, and the massless sector has (h = ℓ, ℓ) with half-integer $\ell =0,\frac{1}{2},\dots ,\frac{k}{2}$. For the Schwarzian path integral we will need the ℓ = 0 representation. The characters are defined by

$$\begin{equation}{\chi }_{\ell }(k;q,z)\equiv {\mathrm{Tr}}_{\mathrm{N}\mathrm{S}}\left[{(-1)}^{F}{q}^{{L}_{0}-\frac{c}{24}}{z}^{{T}_{0}^{3}}\right],\end{equation} \tag{ 3.25 }$$

over a representation ℓ of the algebra. We need to insert (−1)^F such that the fermions along the quantization direction are periodic and survive the semiclassical limit (see discussion in [69]).

These characters were computed by Eguchi and Taormina [76] by simply counting states. They are given by the following expression ²⁵

$$\begin{equation}{\chi }_{\ell }(k;q,z={\text{e}}^{2\pi \text{i}y})={\text{e}}^{2\pi \text{i}k\frac{{y}^{2}}{\tau }}{q}^{-\frac{k}{4}}{q}^{\ell +\frac{1}{4}}\frac{i{\theta }_{3}(q,-z){\theta }_{3}(q,-{z}^{-1})}{\eta {(q)}^{3}{\theta }_{1}(q,{z}^{2})}\left[\mu (z,q)-\mu ({z}^{-1},q)\right],\end{equation} \tag{ 3.26 }$$

where θ₃(q, z) is the Jacobi theta function and we defined the function

$$\begin{equation}\mu (z,q)\equiv \sum\limits _{n\in \mathbb{Z}}\frac{{q}^{(k+1){n}^{2}+(2\ell +1)n}{z}^{2(k+1)n+2\ell +1}}{\left(1-z{q}^{n+\frac{1}{2}}\right)\left(1-z{q}^{n+\frac{1}{2}}\right)}.\end{equation} \tag{ 3.27 }$$

Now we have all the ingredients to extract the Schwarzian partition function from the ℏ ∼ 1/k → 0 (c → ∞) limit applied to the above expression for the identity ℓ = 0 representation. As explained in [69] we need to consider the following scaling

$$\begin{equation}z={\text{e}}^{2\pi \text{i}\alpha \tau }\hspace{-2pt},\qquad q={\text{e}}^{2\pi \text{i}\tau }\quad \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\enspace \tau =\frac{i}{k}\frac{4\pi {{\Phi}}_{r}}{\beta }.\end{equation} \tag{ 3.28 }$$

We then take τ ⋅ k fixed in the limit and this constant is related to the ratio Φ_r /β in the Schwarzian theory. This choice of z and q is written directly in terms of α and β which will become the chemical potential and inverse temperature in the Schwarzian limit.

When taking the Schwarzian limit we will only keep track of the dependence on α and β since any prefactor can be absorbed in a redefinition of the zero-point entropy and energy. We will not go over all the details but some useful intermediate steps are

$$\begin{equation}\mu (z,q)-\mu ({z}^{-1},q)\sim \frac{8{e}^{\frac{2{\pi }^{2}{{\Phi}}_{r}}{\beta }4{\alpha }^{2}}}{{\pi }^{2}\vert \tau {\vert }^{2}}\sum\limits _{n\in \mathbb{Z}}\frac{(\alpha +n)}{{(1-4{(\alpha +n)}^{2})}^{2}}{e}^{-\frac{2{\pi }^{2}{{\Phi}}_{r}}{\beta }4{(n+\alpha )}^{2}}.\end{equation} \tag{ 3.29 }$$

Using equation (3.15) of [78] gives the following limit

$$\begin{equation}i{\left(\frac{{\theta }_{4}(q,z)}{\eta {(q)}^{3}}\right)}^{2}\frac{\eta {(q)}^{3}}{{\theta }_{4}\left(q,{z}^{2}{q}^{\frac{1}{2}}\right)}\sim \frac{\tau }{\mathrm{tan}\enspace \pi \alpha },\end{equation} \tag{ 3.30 }$$

which is related in a simple way to the Jacobi theta functions appearing in the character. Including the rest of the terms the semiclassical k → ∞ limit of the vacuum character is

$$\begin{equation}{\chi }_{\ell =0}(k\to \infty ;q,z)\sim \sum\limits _{n\in \mathbb{Z}}\frac{\beta \enspace \mathrm{cot}(\pi \alpha )(\alpha +n)}{{{\Phi}}_{r}{(1-4{(n+\alpha )}^{2})}^{2}}{e}^{\frac{2{\pi }^{2}{{\Phi}}_{r}}{\beta }(1-4{(n+\alpha )}^{2})}.\end{equation} \tag{ 3.31 }$$

This precisely reproduces the partition function computed by localization, given in equation (3.18) (An analogous match was checked in [69] for the case of $\mathcal{N}=1$ and $\mathcal{N}=2$ super-Schwarzian).

We can mention some interesting features of this expression. First of all the factor of cot πα is important for the formula to make sense. When α → 0 or 1 it is crucial to include this factor for the final answer to be finite. The same happens when α → 1/2 since otherwise the sum would be divergent.

From the 2D CFT perspective the identity representation is invariant under the generators of the global PSU(1, 1|2) algebra. In terms of the Virasoro algebra those generators are

$$\begin{equation}\text{Bosonic:}\enspace {L}_{-1},{L}_{0},{L}_{1},\enspace {T}_{0}^{1},{T}_{0}^{2},{T}_{0}^{3},\qquad \text{Fermionic:}\enspace {G}_{\pm \frac{1}{2}}^{a},\enspace {\bar{G}}_{\pm \frac{1}{2}}^{a},\end{equation} \tag{ 3.32 }$$

which satisfy the same superalgebra as (2.1). It is important to take the fermionic generators in the NS sector. These produce the pre-factor of β¹ in the character. In the localization calculation this factor basically counts the number of bosonic and fermionic zero modes Z ∼ β^{(#fermion)/2−(#bosons)/2}. In the case of the small $\mathcal{N}=4$ algebra there are eight fermionic zero modes and six bosonic zero modes, giving a factor of β.

Before extracting the spectrum from the exact partition function we first explain what properties we expect it to have. The super-Schwarzian theory we are studying captures the explicit breaking of the superconformal symmetry group PSU(1, 1|2). Still, as we have seen in section 2.4, translations, super-translations and rigid SU(2) rotations are symmetries. We can write the fermionic generators as Q_p and ${\bar{Q}}^{p}$ with p = 1, 2. Then a part of the algebra that we will use here is

$$\begin{equation}\left\{{Q}_{p},{\bar{Q}}^{q}\right\}=2{\delta }_{a}^{b}H,\qquad \left\{{Q}_{p},{Q}_{q}\right\}=\left\{{\bar{Q}}_{p},{\bar{Q}}_{q}\right\}=0.\end{equation} \tag{ 3.33 }$$

These generators can be written in terms of the Schwarzian fields but we will not need it for the manipulations here. Imagine we first diagonalize H and look at some states with energy E. Then as long as E ≠ 0 the operators $Q\sim \hat{a}$ act as a SU(2) doublet of lowering fermionic operators and $\bar{Q}\sim {\hat{a}}^{{\dagger}}$ as a SU(2) doublet of rising fermionic operators. To construct a representation we can begin with a state |J⟩ which transforms as a spin J representation of SU(2), constructed such that Q_p |J⟩ = 0. The supermultiplet will have states acting with a single charge ${\bar{Q}}^{q}\vert J\rangle$, which can be expanded into (1/2) ⊗ J = (J − 1/2) ⊕ (J + 1/2); and acting with two charges ${\bar{Q}}_{1}{\bar{Q}}_{2}\vert J\rangle$ of spin J. Therefore, the supermultiplet with E ≠ 0, starting with J ≠ 0 is made of (J − 1/2) ⊕ 2(J) ⊕ (J + 1/2). When we construct a supermultiplet starting with a singlet |0⟩, the ${\bar{Q}}^{q}\vert 0\rangle$ transforms as a doublet and ${\bar{Q}}_{1}{\bar{Q}}_{2}\vert 0\rangle$ as another singlet, giving 2(0) ⊕ (1/2). Labeling the supermultiplet by the state with highest SU(2) spin, the E ≠ 0 part of the spectrum should organize as

$$\begin{equation}\mathbf{J}=(J)\oplus 2(J-1/2)\oplus (J-1),\quad J\geqslant 1\end{equation} \tag{ 3.34 }$$

$$\begin{equation}1/2=(1/2)\oplus 2(0).\end{equation} \tag{ 3.35 }$$

Finally we might also have states with E = 0. Starting with a spin-J representation |J⟩, having H|J⟩ = 0 implies that all supercharges annihilate the state and, therefore, that's the whole supermultiplet.

Taking these considerations into account, we can expect the partition function of the $\mathcal{N}=4$ super-Schwarzian theory to be expanded in the following way

$$\begin{align} Z(\beta ,\alpha )& =\sum\limits _{J}{\chi }_{J}(\alpha ){\rho }_{\text{ext}}(J)+\int \mathrm{d}E\enspace {e}^{-\beta E}\left({\chi }_{1/2}(\alpha )+2{\chi }_{0}(\alpha )\right){\rho }_{\text{cont}}(1/2,E) \\ & \quad +\sum\limits _{J\geqslant 1}\int \mathrm{d}E\enspace {e}^{-\beta E}\left({\chi }_{J}(\alpha )+2{\chi }_{J-\frac{1}{2}}(\alpha )+{\chi }_{J-1}(\alpha )\right){\rho }_{\text{cont}}(J,E), \end{align} \tag{ 3.36 }$$

where the sums are over half-integer J and ${\chi }_{J}(\alpha )\equiv {\sum }_{m=-J}^{J}\;{\text{e}}^{4\pi \mathrm{i}\alpha m}=\frac{\mathrm{sin}(2J+1)2\pi \alpha }{\mathrm{sin}\enspace 2\pi \alpha }$ is the character of a spin-J representation of SU(2). In the first line, the first term corresponds to states with E = 0 while the second term to the E ≠ 0 multiplet 1/2. The second line corresponds to the sum over all other E ≠ 0 supermultiplets. Therefore, ρ(J, E) is the density of supermultiplets with energy E ≠ 0 and highest spin J, while ρ_ext(J) is the density of E = 0 states of spin J.

We will see in the next section that the spectrum of the $\mathcal{N}=4$ super-Schwarzian derived from the exact partition function we computed above has precisely this form (although with only singlet J = 0 zero energy states).

The final answer for the exact $\mathcal{N}=4$ super-Schwarzian theory partition function is given by the following function of inverse temperature β and SU(2) chemical potential α as

$$\begin{equation}Z(\beta ,\alpha )={e}^{{S}_{0}}\sum\limits _{n\in \mathbb{Z}}\frac{\beta }{{{\Phi}}_{r}}\frac{2\enspace \mathrm{cot}(\pi \alpha )(\alpha +n)}{{\pi }^{3}{(1-4{(n+\alpha )}^{2})}^{2}}{e}^{\frac{2{\pi }^{2}{{\Phi}}_{r}}{\beta }(1-4{(n+\alpha )}^{2})}.\end{equation} \tag{ 3.37 }$$

We have fixed the overall normalization in a way that will be convenient later. We will write this answer as a trace over a Hilbert space (albeit with continuous spectrum) realizing it has precisely the form (3.36).

To understand the physics of this partition function we want to extract the density of states as a function of energy at fixed SU(2) charge, which we will refer to as angular momentum (anticipating the application to near extremal black holes in 4D). To do that we begin by performing an inverse Laplace transform and define the fixed-chemical-potential density of states

$$\begin{equation}Z(\beta ,\alpha )=\int \mathrm{d}E{e}^{-\beta E}D(\alpha ,E).\end{equation} \tag{ 3.38 }$$

Applying this to our result (3.37) gives

$$\begin{equation}D(\alpha ,E)={D}_{E=0}(\alpha )\delta (E)+{D}_{\text{cont}}(\alpha ,E),\end{equation} \tag{ 3.39 }$$

where we separate the BPS and continuous part of the spectrum ²⁶ ,

$$\begin{equation}{D}_{E=0}(\alpha )={e}^{{S}_{0}}\sum\limits _{n\in \mathbb{Z}}\frac{4(\alpha +n)}{\pi \enspace \mathrm{tan}\enspace \pi \alpha (1-4{(\alpha +n)}^{2})}={e}^{{S}_{0}}\end{equation} \tag{ 3.40 }$$

$$\begin{equation}{D}_{\text{cont}}(\alpha ,E)={e}^{{S}_{0}}\sum\limits _{n\in \mathbb{Z}}\frac{4(\alpha +n)}{\pi \enspace \mathrm{tan}\enspace \pi \alpha }\frac{{I}_{2}\left(2\pi \sqrt{2{{\Phi}}_{r}E(1-4{(\alpha +n)}^{2})}\right)}{E(1-4{(\alpha +n)}^{2})}.\end{equation} \tag{ 3.41 }$$

We see the first line corresponding only to states with zero energy is independent of α. This means it only gets contributions from zero charge (angular momentum) states. We chose the normalization of the partition function such that this gives $\mathrm{exp}\left({S}_{0}\right)$ and can be interpreted as the degeneracy of ground states.

To find the density of states we use the following identity to rewrite (3.41) as

$$\begin{equation}{D}_{\text{cont}}(\alpha ,E)={e}^{{S}_{0}}\sum\limits _{m=1}^{\infty }\frac{m\enspace \mathrm{sin}\enspace 2\pi m\alpha }{\mathrm{tan}\enspace \pi \alpha }\frac{\mathrm{sinh}\left(2\pi \sqrt{2{{\Phi}}_{r}E-\frac{1}{4}{m}^{2}}\right)}{2{\pi }^{2}{{\Phi}}_{r}{E}^{2}}{\Theta}\left(E-\frac{{m}^{2}}{8{{\Phi}}_{r}}\right).\end{equation} \tag{ 3.42 }$$

We defined the Heaviside function Θ(x) such that Θ(x > 0) = 1 and Θ(x < 0) = 0. The dependence with the chemical potential can be expanded in SU(2) characters in the following simple way

$$\begin{equation}\frac{2\enspace \mathrm{sin}\enspace 2\pi m\alpha }{\mathrm{tan}\enspace \pi \alpha }={\chi }_{J}(\alpha )+2{\chi }_{J-\frac{1}{2}}(\alpha )+{\chi }_{J-1}(\alpha ),\quad J\equiv m/2,\enspace \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\enspace m > 1\end{equation} \tag{ 3.43 }$$

$$\begin{equation}\frac{2\enspace \mathrm{sin}\enspace 2\pi \alpha }{\mathrm{tan}\enspace \pi \alpha }={\chi }_{1/2}(\alpha )+2{\chi }_{0}(\alpha ),\quad J\equiv m/2,\enspace \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\enspace m=1,\end{equation} \tag{ 3.44 }$$

where in the right-hand side we defined the angular momentum J in terms of the integer m. In principle we can use this formula to extract the density of states for each SU(2) representation. Instead we will notice this is precisely the combination in equation (3.36) where J now labels the supermultiplet J. The second line with m = 1 involves the special case 1/2. Comparing (3.36) with (3.42) we can extract the density of supermultiplets ρ_cont(J, E) for E ≠ 0 and using (3.40) we can write the density of E = 0 states ρ_ext(J). The final answer is given by

$$\begin{equation}{\rho }_{\text{ext}}(J)={e}^{{S}_{0}}{\delta }_{J,0}.\end{equation} \tag{ 3.45 }$$

$$\begin{equation}{\rho }_{\text{cont}}(J,E)=\frac{{e}^{{S}_{0}}J}{4{\pi }^{2}{{\Phi}}_{r}{E}^{2}}\enspace \mathrm{sinh}\left(2\pi \sqrt{2{{\Phi}}_{r}(E-{E}_{0}(J))}\right){\Theta}\left(E-{E}_{0}(J)\right),\quad \text{for}\enspace \enspace J\geqslant \frac{1}{2},\end{equation} \tag{ 3.46 }$$

where the gap for each supermultiplet labeled by J is given by E₀(J) ≡ J²/(2Φ_r ).

Using this result we can get a simple picture of the shape of the spectrum. First we have a number ${e}^{{S}_{0}}$ of states at exactly E = 0 which are all in the supermultiplet 0, an SU(2) singlet. These are the extremal BPS states of the black hole as we will see in the next section. For small energies there are no states until we reach the gap in the spectrum given by the threshold energy for the supermultiplet $\frac{1}{2}=(\frac{1}{2})\oplus 2(0)$, given by

$$\begin{equation}{E}_{\text{gap}}=\frac{1}{8{{\Phi}}_{r}},\end{equation} \tag{ 3.47 }$$

and for E > E_gap we have a continuum of states. Something similar is true for higher multiplets J > 1/2, but now the continuum starts at a supermultiplet-dependent gap

$$\begin{equation}{E}_{0}(J)=\frac{1}{2{{\Phi}}_{r}}{J}^{2}.\end{equation} \tag{ 3.48 }$$

It is perhaps not surprising that states with spin J start at E₀(J). The surprising feature is that there are no states with J = 0 at energies 0 < E < E_gap.

In figure 3 we depict the shape of the spectrum of the $\mathcal{N}=4$ super-Schwarzian theory. In the left panel we show the degeneracy of supermultiplets, with the supermultiplet-dependent gap mentioned above. In the right we show the degeneracy of states (not supermultiplets) with J = 0 as an illustration. We see the delta function from 0 but also contributions from the continuum coming from (0) ⊂ 1/2 and (0) ⊂ 1.

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For each supermultiplet, the density of states has a spectral edge ${\rho }_{\text{cont}}(J,E)\sim {(E-{E}_{0}(J))}^{1/2}$ characteristic of a Hermitian random matrix model, although we will not follow this direction in this paper.

We can make some comments regarding the calculation of the $\mathcal{N}=4$ super-Schwarzian Witten index. First it is straightforward to show that the fermion number can be computed in the following way

$$\begin{equation}{(-1)}^{F}={\text{e}}^{2\pi \text{i}{T}^{3}}\end{equation} \tag{ 3.49 }$$

acting on a SU(2) representation of spin J. ²⁷ Using the density of states derived above it is easy to extract the index. First all states with E = 0 have J = 0 and, therefore, (−1)^F = 1. States with J > 1/2 contribute in the following way

$$\begin{equation}{\text{e}}^{2\pi \text{i}J}\left[{\chi }_{J}(\alpha )-2{\chi }_{J-1/2}(\alpha )+{\chi }_{J-1}(\alpha )\right].\end{equation} \tag{ 3.50 }$$

It is easy to see that when α = 0 this combination exactly vanishes since there is the same number of bosonic and fermionic states in the supermultiplet. The same is true for 1/2 which gives χ_1/2(α) − 2χ₀(α) and also vanishes for α = 0. Therefore, the Witten index of the $\mathcal{N}=4$ super-Schwarzian theory is given by ${e}^{{S}_{0}}$ and counts the number of ground states.

As a final comment, there are two different definitions of the $\mathcal{N}=2$ super-Schwarzian theory that differ on the presence of a 't Hooft anomaly, as we review in appendix A. Since the gauge group of the $\mathcal{N}=4$ super-Schwarzian is SU(2) we think there cannot be such anomaly [17] and the theory is unique, but this deserves further investigation.

To finish this section we would like to compare this solution to a non-supersymmetric version of the theory, such as the one in [7]. Imagine we have a bosonic Schwarzian theory coupled to an SU(2) mode. The action is

$$\begin{equation}I=-{{\Phi}}_{r}\int \mathrm{d}\tau \enspace \text{Sch}(f,\tau )+K\int \mathrm{d}\tau \enspace \mathrm{Tr}{\left({g}^{-1}{\partial }_{\tau }g\right)}^{2},\end{equation} \tag{ 3.51 }$$

where K and Φ_r are independent parameters. This theory can be solved exactly [79]. The density of states as a function of energy and angular momentum J is given by

$$\begin{align} & {\rho }_{\mathrm{b}\mathrm{o}\mathrm{s}.}(J,E)={e}^{{S}_{0}}\enspace \mathrm{sinh}\left(2\pi \sqrt{2{{\Phi}}_{r}\left(E-{E}_{\mathrm{b}\mathrm{o}\mathrm{s}.}(J)\right)}\right){\Theta}(E-{E}_{\mathrm{b}\mathrm{o}\mathrm{s}.}(J)), \\ & \enspace \enspace \enspace {E}_{\mathrm{b}\mathrm{o}\mathrm{s}.}(J)\equiv \frac{J(J+1)}{2K}. \end{align} \tag{ 3.52 }$$

The bosonic sector of the supersymmetric theory is special in two ways. First of all it necessarily has K = Φ_r . Therefore, at least semiclassically one can compute the gap scale by measuring the following quantity

$$\begin{equation}{\left(\frac{\partial J}{\partial {\Omega}}\right)}_{T=0,{\Omega}=0}=K={{\Phi}}_{r},\end{equation} \tag{ 3.53 }$$

where Ω = iα/β is the potential conjugated to J (from 4D perspective, angular velocity). The second, and more important, feature is the fact that for the bosonic theory (3.51), even if K = Φ_r , there are states with J = 0 for any energy E > 0, namely ρ(J = 0, E > 0) ≠ 0 and ρ(J = 0, E = 0) = 0. The $\mathcal{N}=4$ supersymmetric Schwarzian theory is completely different. We find a delta function at E = 0 describing ${e}^{{S}_{0}}$ states. Moreover, even for J = 0, there are no states in the range $0< E< \frac{1}{8{{\Phi}}_{r}}$. This is the surprising feature we are emphasizing in this paper.

We begin with some notions and notations for gravity in 4D. Some of these conventions will be different than those used in section 2 since we will be working in four-dimensional notation. In the end we will explain how to translate the results to be compatible with the two-dimensional BF theory.

We use a mostly plus Lorentzian metric, and will eventually Wick rotate to Euclidean signature in 2D. We denote 4D curved space indices M, N = 0, ..., 3 and the tangent space indices A, B = 0, ..., 3. The 4D metric ${G}_{MN}={\eta }_{AB}{E}_{M}^{A}{E}_{N}^{B}$ is expressed in terms of the vierbein ${E}_{M}^{A}$. In 2D, the curved and tangent space indices are respectively μ, ν = 0, 1 and a, b = 0, 1, and we use the vielbein ${e}_{\mu }^{a}$ and dualized spin connection ω_μ . We will occasionally use indices for the internal S² space, which will take values m, n = 2, 3 for the coordinate indices and a', b' = 2, 3 for the frame indices. When we are working in the near-horizon region, with the AdS₂ × S² product manifold, the vielbein will decompose as:

$$\begin{equation}{E}_{M}^{A}={e}_{\mu }^{a}\enspace ,\enspace {e}_{m}^{{a}^{\prime }}.\end{equation} \tag{ 4.1 }$$

We will generally suppress spinor indices when possible, but our spinor conventions are given in appendix B.

Pure, ungauged $\mathcal{N}=2$ supergravity is a minimal theory containing only the graviton, G_MN ; an SU(2)_R doublet of gravitinos, ${{\Psi}}_{M}^{I}$, I = 1, 2; and a U(1) gauge field A_M under which a black hole solution is electrically or magnetically charged. Without the addition of extra vector or hypermultiplets, this gravity theory does not have a precise embedding in string theory ²⁸ . Nevertheless, we argue that universal features of supersymmetric black holes are already present in this model, through the relationship with 2D super-JT gravity.

We write the Lagrangian of pure 4D $\mathcal{N}=2$ supergravity, following the conventions of [80]:

$$\begin{align} {E}^{-1}\mathcal{L}& ={\kappa }^{-2}\left(\frac{1}{2}R-{\bar{{\Psi}}}_{IM}{{\Gamma}}^{MNP}{D}_{N}{{\Psi}}_{P}^{I}-\frac{1}{4}{F}_{MN}{F}^{MN}\right. \\ & \quad \left.+\frac{{\varepsilon }^{IJ}}{2\sqrt{2}}{\bar{{\Psi}}}_{I}^{M}({F}_{MN}+i\enspace \star {F}_{MN}{{\Gamma}}_{5}){{\Psi}}_{J}^{N}+4\;\;\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{o}\right), \end{align} \tag{ 4.2 }$$

with the standard definitions ${D}_{M}={\partial }_{M}+\frac{1}{4}{\omega }_{M}^{AB}{{\Gamma}}_{AB}$ and F = dA. We also write the 4D Newton constant as 8πG_N = κ². In the above action and throughout this subsection, we follow the convention that a spinor ^I with an upper SU(2)_R index has positive chirality, while a lower index has negative chirality:

$$\begin{equation}{{\Gamma}}_{5}{{\epsilon}}^{I}=+{{\epsilon}}^{I},\qquad {{\Gamma}}_{5}{{\epsilon}}_{I}=-{{\epsilon}}_{I}.\end{equation} \tag{ 4.3 }$$

In analyzing the supersymmetry and dimensional reduction later, we will often instead use Dirac spinors where the chirality will be implicit.

While we have explicitly displayed the Pauli terms in equation (4.2) proportional to the antisymmetric symbol ɛ^IJ , we have omitted the more complicated 4-gravitino term. The form of this term is known in general ²⁹ , but it is vanishing in the bosonic backgrounds corresponding to extremal black hole solutions. In the dimensional reduction to 2D, we will use other methods to fix the higher order fermion terms. With the understanding that some expressions are augmented by these terms higher order in fermions, equation (4.2) is invariant under the following local supersymmetry transformations:

$$\begin{equation}{\delta }_{{\epsilon}}{E}_{M}^{A}=\frac{1}{2}{\bar{{\epsilon}}}^{I}{{\Gamma}}^{A}{{\Psi}}_{MI}+\mathrm{h}.\mathrm{c}.,\end{equation} \tag{ 4.4 }$$

$$\begin{equation}{\delta }_{{\epsilon}}{A}_{M}=\frac{1}{\sqrt{2}}{\varepsilon }^{IJ}{\bar{{\epsilon}}}_{I}{{\Psi}}_{MJ}+\mathrm{h}.\mathrm{c}.,\end{equation} \tag{ 4.5 }$$

$$\begin{equation}{\delta }_{{\epsilon}}{{\Psi}}_{M}^{I}=\left({\partial }_{M}+\frac{1}{4}{\omega }_{M}^{AB}{{\Gamma}}_{AB}\right){{\epsilon}}^{I}-\frac{1}{4\sqrt{2}}{{\Gamma}}^{AB}{F}_{AB}{{\Gamma}}_{M}{\varepsilon }^{IJ}{{\epsilon}}_{J}.\end{equation} \tag{ 4.6 }$$

The Lagrangian and supersymmetry transformations above make manifest the SU(2)_R symmetry of $\mathcal{N}=2$ supergravity. In the application of this theory to nearly AdS₂ × S² black holes described in section 2, we considered a theory with PSU(1, 1|2) symmetry and bosonic subgroup $\mathrm{S}\mathrm{L}(2,\mathbb{R})\times SU(2)$. This latter SU(2) is realized geometrically by the spatial rotations of the black hole, and is independent of the SU(2)_R symmetry present in 4D. This feature is characteristic of the small $\mathcal{N}=4$ superconformal algebra, which has the R-symmetry SU(2) × SU(2)_outer. In the context of this paper, it is possible to show that the outer R-symmetry corresponds to the 4D SU(2) global symmetry ³⁰ . Because the outer symmetry plays no role in our analysis, it will be convenient to use a formulation of $\mathcal{N}=2$ supergravity with Dirac rather than Majorana gravitinos. This will hide the outer SU(2) symmetry, but simplify many formulas in what follows. This version of the theory is presented in [39, 83], and the passage to this formalism involves introduction of the Dirac spinors ${{\Psi}}_{M}={{\Psi}}_{M}^{1}+i{{\Psi}}_{M}^{2}$.

The extremal black hole of $\mathcal{N}=2$ supergravity is a standard solution of Einstein–Maxwell theory which preserves 4 real supercharges of the 8 total. This is in contrast to the near horizon limit of this solution, Ad S₂ × S², which preserves all supersymmetries. The supersymmetry enhancement [47, 49] corresponds to the addition of conformal supercharges, and the AdS background should have a full superconformal set of isometries. In this section, we briefly review these two solutions, which are by now somewhat standard.

The bosonic part of our action using our normalization conventions from equation (4.2) is the typical Einstein–Maxwell action:

$$\begin{equation}{S}_{4\text{D}}^{(\text{bosonic})}=\frac{1}{{\kappa }^{2}}\int {\mathrm{d}}^{4}x\sqrt{-G}\left(\frac{1}{2}R-\frac{1}{4}{F}_{MN}{F}^{MN}\right).\end{equation} \tag{ 4.7 }$$

Solving the equations of motion, one can obtain the metric and gauge field of the celebrated extremal Reissner–Nordström black hole solution, here given by:

$$\begin{align} \mathrm{d}{s}^{2}& =-{\left(\frac{{r}_{0}}{r}-1\right)}^{2}\mathrm{d}{t}^{2}+{\left(\frac{{r}_{0}}{r}-1\right)}^{-2}\mathrm{d}{r}^{2}+{r}^{2}\enspace \mathrm{d}{{\Omega}}_{2}^{2}, \\ {F}_{tr}& =-\frac{\kappa Q}{\sqrt{4\pi }{r}^{2}}. \end{align} \tag{ 4.8 }$$

We also have the relationships between the extremal radius, mass, and charge, r₀ = G_N M and $Q=\sqrt{{r}_{0}M}$. For simplicity, we have chosen an electrically charged black hole and set the magnetic charge P = 0, but the results of this section can be easily generalized to this case. In terms of the charge which we are keeping fixed, we have chosen units such that the extremal mass and extremal Bekenstein–Hawking entropy are ³¹ :

$$\begin{equation}{M}_{0}=Q/\sqrt{{G}_{N}},\qquad {S}_{0}=\frac{\pi {r}_{0}^{2}}{{G}_{N}}=\pi {Q}^{2}.\end{equation} \tag{ 4.9 }$$

In addition to the familiar bosonic symmetries of the solution (4.8) corresponding to translations and rotation, there are additional supersymmetries represented by covariantly constant spinors on the manifold. Since all fermions vanish on this background, we only need to analyze the gravitino supersymmetry transformation, equation (4.6), adapted to Dirac spinors. Preserved Killing spinors are spacetime dependent which are annihilated by the gravitino transformation rule ³² :

$$\begin{equation}{\delta }_{{\epsilon}}{{\Psi}}_{M}={\hat{D}}_{M}{\epsilon}=\left({\partial }_{M}+\frac{1}{4}{\omega }_{M}^{AB}{{\Gamma}}_{AB}\right){\epsilon}+\frac{i}{4\sqrt{2}}{{\Gamma}}^{AB}{F}_{AB}{{\Gamma}}_{M}{\epsilon}=0.\end{equation} \tag{ 4.10 }$$

We will typically use Dirac spinors and do not make use of the chiral components.

The extremal black hole of equation (4.8) can be thought of as a supergravity soliton which interpolates between two maximally supersymmetric vacuua; Minkowski space in the asymptotic far region, and Ad S₂ × S² in the near horizon region. Because we are interested in an effective AdS₂ theory approximated by supersymmetric JT gravity, we now turn our attention to the Ad S₂ × S² metric with the purely electric Bertotti–Robinson ansatz. Starting with equation (4.8), we perform standard manipulations to take the near horizon limit. We shift the location of the event horizon then rescale the radial coordinate with to get the Poincare patch metric:

$$\begin{equation}\mathrm{d}{s}^{2}=\frac{{r}_{0}^{2}}{{z}^{2}}(-\mathrm{d}{t}^{2}+\mathrm{d}{z}^{2})+{r}_{0}^{2}\enspace \mathrm{d}{{\Omega}}_{2}^{2},\qquad F=\frac{1}{2}\frac{\kappa Q}{\sqrt{4\pi }}\frac{1}{{z}^{2}}\mathrm{d}t\wedge \mathrm{d}z.\end{equation} \tag{ 4.11 }$$

To study this background, including the Killing spinors, we make the further change of coordinates $z={r}_{0}{e}^{-r/{r}_{0}}$:

$$\begin{equation}\mathrm{d}{s}^{2}=-{e}^{2r/{r}_{0}}\mathrm{d}{t}^{2}+\mathrm{d}{r}^{2}+{r}_{0}^{2}\enspace \mathrm{d}{{\Omega}}_{2}^{2},\qquad F=-\frac{1}{2}\frac{\kappa Q}{\sqrt{4\pi }{r}_{0}^{2}}{e}^{r/{r}_{0}}\mathrm{d}t\wedge \mathrm{d}r.\end{equation} \tag{ 4.12 }$$

At the risk of notational clutter, we denote frame indices with a hat, and in frame coordinates, the bosonic fields are:

$$\begin{align} {E}^{\hat{t}}& ={e}^{r/{r}_{0}}\mathrm{d}t,\enspace {E}^{\hat{r}}=\mathrm{d}r,\enspace {E}^{\hat{m}}=({r}_{0}\enspace \mathrm{d}\theta ,{r}_{0}\enspace \mathrm{sin}\enspace \theta \enspace \mathrm{d}\phi ),\enspace {F}_{\hat{t}\enspace \hat{r}}=-\frac{\kappa Q}{\sqrt{4\pi }{r}_{0}^{2}}=-\frac{\sqrt{2}}{{r}_{0}}. \\ {\omega }^{\hat{t}\enspace \hat{r}}& =\frac{1}{{r}_{0}}{E}^{\hat{t}}=\frac{1}{{r}_{0}}{e}^{r/{r}_{0}}\mathrm{d}t,\quad {\omega }^{\hat{\phi }\enspace \hat{\theta }}={\omega }_{{S}^{2}}^{\phi \theta }=\mathrm{cos}\enspace \theta \enspace \mathrm{d}\phi . \end{align} \tag{ 4.13 }$$

In the expression for the flux, we used the definitions for κ, Q, and r₀ in the extremal limit. The constant flux background is analogous to the Freund–Reubin ansatz [87] in that the spin connection also has a 1/r₀ behavior, and it is the cancellation between these two terms which will permit a maximally supersymmetric vacuum.

Before plugging the explicit background into equation (4.10), it is useful to first examine the integrability condition for the supercovariant derivative with a general background acting on a Killing spinor. A necessary condition for the existence of Killing spinors is

$$\begin{equation}[{\hat{D}}_{M},{\hat{D}}_{N}]{\epsilon}=0.\end{equation} \tag{ 4.14 }$$

Computing this in the case of pure ungauged $\mathcal{N}=2$ supergravity [83] gives

$$\begin{equation}\left(\left({R}_{MN}^{\enspace \enspace AB}{{\Gamma}}_{AB}-\frac{1}{8}\enspace F\hspace{-6pt}/\enspace {{\Gamma}}_{\left[\right.M}\enspace F\hspace{-6pt}/\enspace {{\Gamma}}_{N\left.\right]}\right)+\frac{i}{\sqrt{2}}{{\Gamma}}_{AB}{{\Gamma}}_{\left[\right.N}({\nabla }_{M\left.\right]}{F}^{AB})\right){\epsilon}=0.\end{equation} \tag{ 4.15 }$$

It is possible to further simplify the gamma matrix contractions for a general flux background, but the case of Ad S₂ × S² has further simplifications as the third term vanishes identically. Allowing a, b = 0, 1; a', b' = 2, 3 as frame indices and μ, ν; m, n as coordinate indices for AdS and the sphere respectively, we obtain the familiar conditions:

$$\begin{align} & \left({R}_{\mu \nu }^{\enspace \enspace ab}{\gamma }_{ab}+\frac{1}{{r}_{0}^{2}}({e}_{\mu }^{a}{e}_{\nu }^{b}-{e}_{\nu }^{a}{e}_{\mu }^{b}){\gamma }_{ab}\right){\epsilon}=0, \\ & \left({R}_{mn}^{\enspace \enspace {a}^{\prime }{b}^{\prime }}{\gamma }_{{a}^{\prime }{b}^{\prime }}-\frac{1}{{r}_{0}^{2}}({e}_{m}^{{a}^{\prime }}{e}_{n}^{{b}^{\prime }}-{e}_{n}^{{a}^{\prime }}{e}_{m}^{{b}^{\prime }}){\gamma }_{{a}^{\prime }{b}^{\prime }}\right){\epsilon}=0. \end{align} \tag{ 4.16 }$$

These are identically satisfied by the Bertotti–Robinson metric as the AdS and sphere are maximally symmetric. We do not obtain any new algebraic conditions on the Killing spinors. These conditions, as well as the form of the spinors themselves, must be determined from the first order equation, equation (4.10). Inserting ${F}_{ab}=-\frac{\sqrt{2}}{{r}_{0}}{{\epsilon}}_{ab}$ into this Killing spinor equation, we find that it splits into a pair of equations for the AdS and sphere parts:

$$\begin{equation}\left({\partial }_{\mu }+\frac{1}{4}{\omega }_{\mu }^{ab}{\gamma }_{ab}-\frac{i}{2{r}_{0}}{\gamma }_{3}{\gamma }_{\mu }\right){\epsilon}=0,\end{equation} \tag{ 4.17 }$$

$$\begin{equation}\left({\partial }_{m}+\frac{1}{4}{\omega }_{m}^{{a}^{\prime }{b}^{\prime }}{\gamma }_{{a}^{\prime }{b}^{\prime }}-\frac{i}{2{r}_{0}}{\gamma }_{m}\right){\epsilon}=0,\end{equation} \tag{ 4.18 }$$

where γ₃ = −γ₀ γ₁ = σ₃ ⊗ 1 is the pseudo-chirality operator on the AdS space. These can be obtained by applying the conventions of appendix B. We observe that these equations are of the same form studied in [85], and we will give an explicit construction of the Killing spinors.

To find expressions for the Killing spinors, we make a slight abuse of notation in writing _4D = ⊗ η. We will see that the complex 2 component and η are Killing spinors on the AdS₂ and S² factors. Inserting the expressions of equation (4.13), we find the four Killing spinor equations,

$$\begin{align} & \left({\partial }_{t}-\frac{1}{2{r}_{0}}{e}^{r/{r}_{0}}{\gamma }_{3}(1+i{\gamma }_{0})\right){\epsilon}=0, \\ & \left({\partial }_{r}+\frac{i}{2{r}_{0}}{\gamma }_{0}\right){\epsilon}=0, \\ & \left({\partial }_{\theta }-\frac{i}{2}{\sigma }_{1}\right)\eta =0, \\ & \left({\partial }_{\phi }-\frac{1}{2}\enspace \mathrm{cos}\enspace \theta {\sigma }_{13}-\frac{i}{2}\enspace \mathrm{sin}\enspace \theta {\sigma }_{3}\right)\eta =0. \end{align} \tag{ 4.19 }$$

To solve these, we first introduce constant anticommuting Dirac spinors ${{\epsilon}}_{\pm }^{0}$ and η^α . Here the α, β = 1, 2 spinor indices are those for the internal SU(2) on the sphere; this contrasts with the notation of section 2, where these indices were denoted by p, q.

The ${{\epsilon}}_{\pm }^{0}$ are eigenstates under the supersymmetric projector:

$$\begin{equation}{{\epsilon}}_{\pm }^{0}={\mathcal{P}}_{\pm }{{\epsilon}}_{\pm }^{0}=\frac{1}{2}(1\pm i{\gamma }_{0}){{\epsilon}}_{\pm }^{0}.\end{equation} \tag{ 4.20 }$$

A convenient basis for the η^α is the one of definite chirality on the S². From appendix B, we see that we may choose η^α = (η¹, η²) with σ₂ η¹ = η¹ and σ₂ η² = −η². For clarity, this explicit basis choice amounts to the familiar eigenstates: ${{\epsilon}}_{+}^{0}=(1,i)$, ${{\epsilon}}_{-}^{0}=(1,-i)$, η¹ = (1, i), η² = (1, −i).

With these projections, we find the AdS₂ × S² solutions to the Killing spinor equation (4.10) are:

$$\begin{equation}{{\epsilon}}_{-}^{\alpha }\equiv {\text{e}}^{\frac{r}{2{r}_{0}}}{\text{e}}^{\frac{\text{i}\theta }{2}{\sigma }_{1}}{\text{e}}^{\frac{\phi }{2}{\sigma }_{13}}({{\epsilon}}_{-}^{0}\otimes {\eta }^{\alpha }).\end{equation} \tag{ 4.21 }$$

$$\begin{equation}{{\epsilon}}_{+}^{\alpha }\equiv \left(1+\frac{t\enspace {\text{e}}^{r/{r}_{0}}}{{r}_{0}}{\gamma }_{3}\right){\text{e}}^{-\frac{r}{2{r}_{0}}}{\text{e}}^{\frac{\text{i}\theta }{2}{\sigma }_{1}}{\text{e}}^{\frac{\phi }{2}{\sigma }_{13}}({{\epsilon}}_{+}^{0}\otimes {\eta }^{\alpha }).\end{equation} \tag{ 4.22 }$$

Similar expressions hold for the Dirac conjugate ${\bar{{\epsilon}}}_{\pm }^{\alpha }$, which we will need to form Killing vectors ³³ .

The super-isometry algebra of Killing vectors and spinors is typically determined by forming bilinears of spinors and checking that these correspond to the bosonic symmetries of the metric. Knowledge of the Killing spinors may be used to construct spacetime supercharges as integrals of the supercurrent [86]; for our purposes it is sufficient to check that all bosonic isometries are generated by Killing spinor bilinears. This allows us to loosely interpret ${{\epsilon}}_{-}^{\alpha }\sim {G}_{-1/2}^{\alpha }$ and ${{\epsilon}}_{+}^{\alpha }\sim {G}_{1/2}^{\alpha }$, and we will compute the Killing spinor analog of $\left\{{\bar{G}}_{\pm }^{\alpha },{G}_{{\pm }^{\prime }}^{\beta }\right\}$ which were the anticommutators appearing in (3.24). Therefore, we consider objects of the form,

$$\begin{equation}{\bar{{\epsilon}}}_{\pm }^{\alpha }{{\Gamma}}^{M}{{\epsilon}}_{{\pm }^{\prime }}^{\beta }{E}_{M},\end{equation} \tag{ 4.23 }$$

which transforms as a vector field. In fact, if we form bilinears from Killing spinors, this result will automatically be a Killing vector. A typical bilinear of the form equation (4.23) may vanish depending on the commutation of ${\mathcal{P}}_{\pm }$ with the various gamma matrices.

To simplify the formulas, we will choose a unit normalization for the η^α and ${{\epsilon}}_{\pm }^{0}$ which solve equation (4.20), but in general one would find constant spinor contractions appearing on the right-hand side of the formulas to follow. As shown in appendix C, the Killing vectors determined by (4.23), (4.21), (4.22) are:

$$\begin{equation}H={\partial }_{t},\qquad D=t{\partial }_{t}+z{\partial }_{z},\qquad K=({z}^{2}+{t}^{2}){\partial }_{t}+2tz{\partial }_{z},\end{equation} \tag{ 4.24 }$$

$$\begin{equation}{T}_{1}=\mathrm{sin}\enspace \phi {\partial }_{\theta }+\mathrm{cot}\enspace \theta \enspace \mathrm{cos}\enspace \phi {\partial }_{\phi },\quad {T}_{2}=\mathrm{cos}\enspace \phi {\partial }_{\theta }-\mathrm{cot}\enspace \theta \enspace \mathrm{sin}\enspace \phi {\partial }_{\phi },\quad {T}_{3}={\partial }_{\phi }\end{equation} \tag{ 4.25 }$$

which form the SL(2) × SU(2) algebra, i.e. the bosonic part of the supergroup PSU(1, 1|2). This motivates the conclusion that the near horizon theory should have this symmetry.

Our goal is to now demonstrate that the effective 2D gravity theory describing the near extremal perturbations of equation (4.11) ultimately reduces to the supersymmetric completion of JT gravity described in section 2. We will start with a convenient Kaluza–Klein ansatz for 4D black holes with a dilaton χ capturing the fluctuating horizon area [84, 89]. Without the inclusion of extra massless fields, this ansatz does not generally satisfy all the higher dimensional Einstein equations [90]. However, as in the non-supersymmetric case [7], for sufficiently small masses and spins above extremality, the reduced effective 2D theory does capture the higher dimensional near extremal black holes.

In our discussion of the Kaluza–Klein ansatz, distorted 4D quantities will be hatted. We begin with the distorted metric, written in terms of coordinates x^M = (x^μ , y^m ):

$$\begin{equation}\mathrm{d}\enspace {\hat{s}}_{4\text{D}}^{2}=\frac{{r}_{0}}{{\chi }^{1/2}}{g}_{\mu \nu }\enspace \mathrm{d}{x}^{\mu }\enspace \mathrm{d}{x}^{\nu }+\chi \enspace {h}_{mn}(\mathrm{d}{y}^{m}+{T}_{i}^{m}{B}_{\mu }^{i}\enspace \mathrm{d}{x}^{\mu })(\mathrm{d}{y}^{n}+{T}_{j}^{n}{B}_{\nu }^{j}\enspace \mathrm{d}{x}^{\nu }),\end{equation} \tag{ 4.26 }$$

with an arbitrary 2D metric g_μν and the unit S² metric:

$$\begin{equation}{h}_{mn}\enspace \mathrm{d}{y}^{m\enspace }\mathrm{d}{y}^{n}=\mathrm{d}{\theta }^{2}+{\mathrm{sin}}^{2}\enspace \theta \enspace \mathrm{d}{\phi }^{2}.\end{equation} \tag{ 4.27 }$$

We parametrize the angular fluctuations with an SU(2) gauge field ${B}_{\mu }^{i}$, which transforms as a 3 of SO(3) and enters through the S² Killing vectors (4.25). ³⁴

The vierbein associated with this decomposition is

$$\begin{equation}{E}_{M}^{A}\enspace \mathrm{d}{X}^{M}\to {\hat{e}}^{\enspace a}=\frac{{r}_{0}^{\frac{1}{2}}}{{\chi }^{\frac{1}{4}}}{e}_{\mu }^{a}\enspace \mathrm{d}{x}^{\mu },\qquad {\hat{e}}^{\enspace {a}^{\prime }}={\chi }^{\frac{1}{2}}{e}_{m}^{{a}^{\prime }}(\mathrm{d}{y}^{m}+{T}_{i}^{m}{B}_{\mu }^{i}\enspace \mathrm{d}{x}^{\mu }),\end{equation} \tag{ 4.28 }$$

with the metrics for the arbitrary 2D theory and the explicit metric of S²:

$$\begin{equation}{g}_{\mu \nu }={\eta }_{ab}{e}_{\mu }^{a}{e}_{\nu }^{b}\enspace ,\qquad {h}_{mn}={\delta }_{{a}^{\prime }{b}^{\prime }}{e}_{m}^{{a}^{\prime }}{e}_{n}^{{b}^{\prime }}.\end{equation} \tag{ 4.29 }$$

The dimensional reduction of the Einstein–Hilbert term as well as the gravitino kinetic term both require the spin connection compatible with the vierbeins above. Even though the torsion tensor is not in general zero in supergravity due to fermions, for now we are free to use the torsion free first Cartan structure equations ³⁵ :

$$\begin{equation}d{\hat{E}}^{A}+{\hat{\omega }}^{AB}\wedge {\hat{E}}_{B}=0.\end{equation} \tag{ 4.30 }$$

We will ultimately pass to the first order formulation of gravity in the effective AdS₂ theory. This first order theory does not automatically enforce the torsion free constraint, so we will only use the 4D distorted spin connection as an intermediate step. The expressions for the distorted spin connection, Ricci tensors and Ricci scalar are given in appendix B.

We may now write the reduced gravitational action explicitly after dropping a total derivative on χ: ³⁶

$$\begin{equation}{S}_{2\text{D},\text{metric}}=\frac{2\pi }{{\kappa }^{2}}\int {\mathrm{d}}^{2}x\sqrt{-g}\left(\chi R+\frac{2{r}_{0}}{\sqrt{\chi }}-\frac{1}{6}\frac{{\chi }^{\frac{5}{2}}}{{r}_{0}}{H}_{ab}^{i}{H}_{i}^{ab}\right).\end{equation} \tag{ 4.31 }$$

Our ansatz for F in the distorted case is a term which generalizes the constant flux background by the inclusion of the dilaton; in local coordinates it is

$$\begin{equation}{\hat{F}}_{ab}=-\sqrt{2}\frac{{r}_{0}}{\chi }{{\epsilon}}_{ab}.\end{equation} \tag{ 4.32 }$$

As we will explain shortly, fixing the field strength in the near-horizon region is well motivated and we do not need to be concerned with quantum fluctuations of the U(1) gauge field around this saddle. Thus, such a mode will not contribute to the Euclidean path integral when studying black holes in the canonical ensemble.

Once we expand around the near extremal background, this will have the effect of introducing a cosmological constant term. Additionally, we will perform the Wick rotation to Euclidean time, which will give the gravitational path integral an interpretation as a statistical partition function. In total, the reduction of the 4D bosonic Euclidean action becomes:

$$\begin{equation}{S}_{2\text{D}}^{(\text{bosonic})}=-\frac{2\pi }{{\kappa }^{2}}\int {\mathrm{d}}^{2}x\sqrt{g}\left(\chi R+\frac{2{r}_{0}}{{\chi }^{1/2}}-\frac{2{r}_{0}^{3}}{{\chi }^{3/2}}-\frac{1}{3}\frac{{\chi }^{5/2}}{{r}_{0}}{\mathrm{t}\mathrm{r}}_{SU(2)}{H}_{ab}{H}^{ab}\right).\end{equation} \tag{ 4.33 }$$

This is the bosonic part of a more complicated 2D dilaton supergravity, see for example [91]. This model is difficult to quantize directly, so we must make use of the near extremal condition in order to make contact with an extension of the JT model. To do this, we expand the dilaton to first order around the extremal squared radius, with a multiple of G_N to obtain a convenient normalization:

$$\begin{equation}\frac{\chi (x)}{{G}_{N}}=\frac{{r}_{0}^{2}}{{G}_{N}}+2{\Phi}(x)+\mathcal{O}\left(\frac{{G}_{N}{{\Phi}}^{2}}{{r}_{0}^{2}}\right).\end{equation} \tag{ 4.34 }$$

To this order, the action becomes

$$\begin{align} {S}_{2\text{D}}^{(\text{bosonic})}& =-\int {\mathrm{d}}^{2}x\sqrt{g}\left[\frac{{r}_{0}^{2}}{4{G}_{N}}R-\frac{1}{12}\frac{{r}_{0}^{4}}{{G}_{N}}\enspace {\mathrm{t}\mathrm{r}}_{SU(2)}{H}_{ab}{H}^{ab}+\frac{{\Phi}}{2}\left(R+\frac{2}{{r}_{0}^{2}}-\frac{5}{6}{r}_{0}^{2}\enspace {\mathrm{t}\mathrm{r}}_{SU(2)}{H}_{ab}{H}^{ab}\right)\right]\hspace{-3pt}. \end{align} \tag{ 4.35 }$$

The first term does not involve the dilaton, and is proportional to the Euler characteristic once we restore the Gibbons–Hawking–York boundary term containing the extrinsic curvature. This term thus sets the extremal entropy, and this will remain true independently of how we supersymmetrize the model. Thus, we find:

$$\begin{equation}{S}_{0}=\frac{\pi {r}_{0}^{2}}{{G}_{N}}=\pi {Q}^{2},\end{equation} \tag{ 4.36 }$$

which is precisely the extremal entropy of equation (4.9).

To make contact with the bosonic part of the BF description presented in section 2.1, we must introduce Lagrange multiplier fields for this gauge field. This is a scalar field b_i transforming in the adjoint of SU(2); we will additionally look for a form of the BF coupling ib_i Hⁱ . In a large r₀ expansion, this part of the action becomes:

$$\begin{equation}{S}_{2\text{D}}^{(\text{bosonic})}\supset -i\int \enspace {\mathrm{t}\mathrm{r}}_{SU(2)}bH+\int {\mathrm{d}}^{2}x\sqrt{g}\enspace \frac{3{G}_{N}}{2{r}_{0}^{4}}\enspace \left(1+\mathcal{O}\left(\frac{{G}_{N}{\Phi}}{{r}_{0}^{2}}\right)\right){b}_{i}{b}^{i}.\end{equation} \tag{ 4.37 }$$

We see the quadratic terms in b_i are suppressed by large powers of the extremal radius, so in the large mass near extremal limit $({r}_{0}\gg 1/{G}_{N}^{1/2})$, the effective description is the unit normalized BF theory for SU(2). ³⁷ Consequently, the dimensionally reduced bosonic part of the action is

$$\begin{equation}{S}_{2\text{D}}^{(\text{bosonic})}={S}_{0}-\frac{1}{2}\int {\mathrm{d}}^{2}x\sqrt{g}\enspace {\Phi}\left(R+\frac{2}{{r}_{0}^{2}}\right)-i\int {\mathrm{t}\mathrm{r}}_{SU(2)}bH\enspace +\mathcal{O}\left(\frac{{G}_{N}}{{r}_{0}^{2}}\right).\end{equation} \tag{ 4.38 }$$

This has the form of the standard JT bulk term plus a decoupled BF term. This matches the fact that the extremal AdS₂ × S² metric possesses $\mathrm{S}\mathrm{L}(2,\mathbb{R})\enspace \times \enspace SU(2)$ as bosonic symmetries. Of course, to the action (4.38), one in principle has to add the GHY term and the boundary term for the SU(2) gauge field. Nevertheless, since the correct boundary term was derived in directly in the first order formalism in section 2.1, we will solely discuss the matching of bulk terms in this section.

Our next goal is to restore the appropriate fermionic terms to the 2D action. Kaluza–Klein supergravity reductions on spheres are important for the AdS/CFT correspondence, but performing this procedure for the full, consistent, nonlinear supergravity is somewhat more involved than what we will attempt here ³⁸ . Rather than performing a full Kaluza–Klein reduction, we will instead derive only the terms that are linear (or less than linear) in the dilaton Φ. Higher powers of Φ will come suppressed by factors of ${G}_{N}/{r}_{0}^{2}$ (i.e. similar to the higher powers in Φ in (4.38)) and can thus be neglected for sufficiently large black holes.

The basic fermion which must be included is the 2D gravitino. To ultimately match PSU(1, 1|2) BF theory, which naturally uses complex fermions, we will use the Dirac formulation of $\mathcal{N}=2$ supergravity as discussed above. After dimensional reduction on S², the Dirac gravitinos ${\psi }_{\mu }^{\alpha }$, ${\bar{\psi }}_{\mu \alpha }$ will transform in the 2 and $\bar{2}$ representations of the SU(2) symmetry, which in this section, is parametrized by the spinor index α = 1, 2. The form of the gravitinos follows from the standard Kaluza Klein ansatz ${{\Psi}}_{\mu }\sim {\psi }_{\mu }^{\alpha }{\eta }_{\alpha }$, where η_α is a Dirac spinor which solves the angular equation (4.18), i.e. just the S² Killing spinor equation.

The angular solution is given by taking only the angular part of (4.21) and (4.22) in terms of constant η^α spinors. A similar relation holds for the dilatino λ^α , ${\bar{\lambda }}_{\alpha }$ which come from the S² vector components of the gravitino ³⁹ .

The standard gravitino kinetic term ${\bar{\psi }}_{\mu \alpha }{\gamma }^{\mu \nu \rho }{D}_{\nu }{\psi }_{\rho }^{\alpha }$ vanishes in two dimensions. This is related to the Einstein–Hilbert term R being a topological invariant, and the supersymmetric variation of this is automatically zero. The basic term arising from dimensional reduction after choosing the correct normalization for fermions and dropping higher order terms in r₀ and the dilaton fields is:

$$\begin{equation}{S}_{2\text{D}}\supset 2i\int {\mathrm{d}}^{2}x\enspace {\varepsilon }^{\mu \nu }{\lambda }^{\alpha }{\mathcal{D}}_{\mu }{\bar{\psi }}_{\nu \alpha },+\mathrm{h}.\mathrm{c}.\end{equation} \tag{ 4.39 }$$

for ${\mathcal{D}}_{\mu }$ a suitable gauge supercovariant derivative whose form we will address directly in the first-order formalism ⁴⁰ . The next fermionic term, arising from the dimensional reduction, couples the dilaton Φ to the 2D gravitinos ψ and $\bar{\psi }$. Such a term comes from the 4D terms $-{\bar{{\Psi}}}_{IM}{{\Gamma}}^{MNP}{D}_{N}{{\Psi}}_{P}^{I}+\frac{{\varepsilon }^{IJ}}{2\sqrt{2}}{\bar{{\Psi}}}_{I}^{M}({F}_{MN}+i\star {F}_{MN}{{\Gamma}}_{5}){{\Psi}}_{J}^{N}$ in the full supergravity Lagrangian (4.2). After passing to the Dirac spinors ${{\Psi}}_{M}={{\Psi}}_{M}^{1}+i{{\Psi}}_{M}^{2}$, the resulting term in the 2D action is

$$\begin{equation}{S}_{2\text{D}}\supset \int \frac{2}{{r}_{0}}{\Phi}\enspace ({\bar{\psi }}_{\alpha }\wedge {\psi }^{\alpha }),\end{equation} \tag{ 4.40 }$$

with all higher powers in Φ suppressed by factors of ${G}_{N}^{1/2}/{r}_{0}$. The final term that involves quadratic fermionic terms couples ψ and $\bar{\psi }$ to the SU(2) field strength H arising from the KK reduction. Such a term arises from the dimensional reduction of $-{\bar{{\Psi}}}_{IM}{{\Gamma}}^{MNP}{D}_{N}{{\Psi}}_{P}^{I}$ by using the dependence of the 4D spin connection on the field strength H (see appendix B), again dropping higher order terms. After integrating-in the zero form field b, the 2D gravitinos then couple to b instead of to the field strength H, resulting in the term

$$\begin{equation}{S}_{2\text{D}}\supset i\int \frac{{b}^{i}}{{r}_{0}}{\bar{\psi }}_{\alpha }({\gamma }_{3}^{\prime }){{({\sigma }^{i})}^{\alpha }}_{\beta }\wedge {\psi }^{\beta }.\end{equation} \tag{ 4.41 }$$

In principle integrating-out b generates a four gravitino term, which is suppressed in r₀ and can be absorbed in a shift of the four gravitino term which we have only schematically written in the action (4.2). Because such higher power fermionic terms are suppressed by factors of ${G}_{N}^{1/2}/{r}_{0}$ we will not be concerned with them when computing the linearized gravitational action.

We can now put together the bosonic and fermionic terms discussed above. In passing to the first order formulation, ${\omega }_{\mu }^{ab}$ will be promoted to a dynamical variable. Additionally, we will introduce standard one-form notation for all fields, ${e}^{a}={e}_{\mu }^{a}\enspace \mathrm{d}{x}^{\mu }$, etc. We can now also now move to working in Euclidean signature, for which we will use the appropriate gamma matrices γ'^a , as outlined in appendix B. In terms of these, the first order bulk action (which will soon need to be modified) is:

$$\begin{align} {S}_{2\text{D}}& \supset -\int {\Phi}\left(\mathrm{d}\omega +\frac{1}{2{r}_{0}^{2}}{{\epsilon}}_{ab}{e}^{a}\wedge {e}^{b}-\frac{2}{{r}_{0}}{\bar{\psi }}_{\alpha }\wedge {\psi }^{\alpha }\right) \\ & \quad +\frac{i}{2}{b}_{i}\left({H}^{i}-{\frac{2}{r}}_{0}{\bar{\psi }}_{\alpha }({\gamma }_{3}^{\prime }){({\sigma }^{i})}_{\beta }^{\alpha }\wedge {\psi }^{\beta }\right)-2i\lambda \mathcal{D}\bar{\psi }+\mathrm{h}.\mathrm{c}., \end{align} \tag{ 4.42 }$$

up to orders of $\mathcal{O}({G}_{N}/{r}_{0}^{2})$ which we have neglected. Here, the supercovariant gauge exterior derivative is

$$\begin{equation}\mathcal{D}{\psi }^{\alpha }=\mathrm{d}{\psi }^{\alpha }+\frac{1}{2}\omega {\gamma }_{3}^{\prime }\wedge {\psi }^{\alpha }-\frac{1}{2{r}_{0}}{\gamma }_{3}^{\prime }{\gamma }_{a}^{\prime }{e}^{a}\wedge {\psi }^{\alpha }+i{B}^{i}{{({\sigma }_{i})}^{\alpha }}_{\beta }\wedge {\psi }^{\beta }.\end{equation} \tag{ 4.43 }$$

The problem with the action (4.42) is that there is nothing to ensure the vanishing of the torsion tensor which has bosonic part τ^a = de^a + ɛ^ab ω ∧ e_b . In deriving equation (B.15), this was implicitly assumed. Additionally, in supergravity, it is only the supertorsion which vanishes. This will be remedied by introducing extra Lagrange multipliers to fix the supertorsion constraint explicitly. Therefore, we can add to the action an additional term ∫ϕ_a τ^a , where ϕ_a are new scalars.

The various one-forms and scalars can now be collected into super-multiplets as:

$$\begin{equation}(\omega ,{e}^{a},{B}^{i},{\psi }^{\alpha },{\bar{\psi }}_{\alpha }),\qquad ({\Phi},{\phi }^{a},{b}^{i},{\lambda }^{\alpha },{\bar{\lambda }}_{\alpha }),\end{equation} \tag{ 4.44 }$$

which we recognize as the matter content of equations (2.3) and (2.4), respectively.

Thus, after adding the super-torsion terms, the total linearized (in the dilaton) action arising from the dimensional reduction is ⁴¹ :

$$\begin{align}{S}_{2\mathrm{D}}=-\int & {\Phi}\left(d\omega +\frac{1}{2{r}_{0}^{2}}{\varepsilon }_{ab}{e}^{a}\wedge {e}^{b}-\frac{2}{{r}_{0}}{\bar{\psi }}_{\alpha }\wedge {\psi }^{\alpha }\right)\\\ & +i\frac{{\phi }_{a}}{{r}_{0}}\left(d{e}^{a}+{\varepsilon }^{ab}\omega \wedge {e}_{b}-2\enspace {\bar{\psi }}_{\alpha }{\gamma }^{\prime a}\wedge {\psi }^{\alpha }\right)\\ & +\frac{i}{2}{b}_{i}\left({H}^{i}-\frac{2}{{r}_{0}}{\bar{\psi }}_{\alpha }({\gamma }_{3}^{\prime }){{({\sigma }^{i})}^{\alpha }}_{\beta }\wedge {\psi }^{\beta }\right)-\left[2i{\lambda }_{\alpha }\mathcal{D}{\bar{\psi }}^{\alpha }+\mathrm{h}.\mathrm{c}.\right],\end{align} \tag{ 4.45 }$$

where the index α is an SU(2) index and the fermionic indices are contracted with the Euclidean gamma matrices ${\gamma }_{a}^{\prime }$ defined in appendix B.

Comparing (4.45) to the $\mathcal{N}=4$ JT gravity action (2.5) with ${\Lambda}=2/{r}_{0}^{2}$, one finds an exact match after some simple field redefinitions having to do with the reality of dilaton and the choice of basis for the gamma matrices ⁴² . Before proceeding to obtain the full partition function of such black holes by using the results from section 3, we first address two subtleties about the dimensional reduction: how to obtain the boundary conditions presented in section 2.1, and why other KK modes (besides the massless KK modes we have included thus far) do not contribute to the partition function in a significant way.

Before presenting the final results regarding the near-extremal spectrum of black holes in these supergravity theories, we will mention some subtleties and clarifications about the dimensional reduction.

Boundary conditions

This paper aims to perform the path integral in the 4D geometry by separately analyzing the near-horizon contribution and the far away (asymptotically flat) region. The far away contribution is trivial, but importantly propagating the boundary conditions from infinity to the edge of the throat, we can derive the boundary conditions we should impose on the fields living in AdS₂ [22, 27]. We will argue that the boundary conditions derived in this way are the same as the one used in section 2 to analyze $\mathcal{N}=4$ super-JT.

This exercise has been done for the metric appearing in JT gravity and for the dilaton. We will follow the presentation of [7]. The throat and far-away regions are separated by an arbitrary curve with fixed dilaton value χ_b , fixed intrinsic boundary metric h_ττ = 1/ɛ² and proper length $\ell =\int \sqrt{h}=\beta {L}_{2}/\varepsilon$, where L₂ is the AdS₂ radius. Curves of constant dilaton are fixed at some radial distance r₀ + δr_bdy, with ${r}_{\text{bdy}}={L}_{2}^{2}/\varepsilon$. Looking at the extremal solution in the far-away region we can derive the boundary condition for the dilaton Φ_b = Φ_r /ɛ with

$$\begin{equation*}{{\Phi}}_{r}=\frac{{r}_{0}{L}_{2}^{2}}{{G}_{N}}.\end{equation*}$$

In our case the AdS₂ radius is L₂ = r₀. The definition of Φ_r through the dilaton boundary condition agrees with the one used in section 2, and this calculation relates it to 4D quantities.

Next, we can analyze the SU(2) gauge field boundary condition. We will derive this indirectly in the following way, ignoring for now its coupling to other fields. In [7] we integrated out the 2D gauge field everywhere. We can see from the exact solution which boundary condition should be chosen in the throat in order to reproduce the angular-momentum dependence of the partition function. This gives the mixed boundary condition between the zero-form bⁱ and the field strength Bⁱ : ${\left.\delta (2i{{\Phi}}_{r}{B}_{\tau }-b)\right\vert }_{\text{NHR}\;\text{bdy}}=0$. To prove this, we can consider the classical solution for the SU(2) gauge field, in the region far-away from the horizon. We fix the holonomy of the gauge field $h=\mathcal{P}\enspace \mathrm{exp}\left({\oint }_{r\to \infty }B\right)=\mathrm{exp}(i2\pi \alpha {\sigma }^{3})$ at the asymptotic boundary, with α ∼ α + 1. Following appendix A in [7], we will work in a gauge where B_r = 0, to obtain the general solution for the zero-form field b and the gauge field B:

$$\begin{equation}B=i\frac{2\pi \alpha {T}^{3}}{\beta }\left(1+\frac{\mathcal{C}}{{r}^{3}}\right)\mathrm{d}\tau ,\qquad {H}_{r\tau }=-i\frac{6\pi \alpha {T}^{3}}{\beta }\frac{\mathcal{C}}{{r}^{4}},\qquad b=\frac{4\pi \mathcal{C}\enspace \alpha \enspace {T}^{3}}{\beta },\end{equation} \tag{ 4.46 }$$

where $\mathcal{C}$ is an undetermined constant. Solving for $\mathcal{C}$, at the boundary that separates the near-horizon region from the asymptotic region we find that B_τ and b are related as

$$\begin{equation}{B}_{\tau }=\frac{2\pi i\alpha {T}^{3}}{\beta }+\frac{ib\enspace {\ell }_{\text{Pl}}^{2}\enspace }{{r}^{3}}=\frac{2\pi i\alpha {T}^{3}}{\beta }+\frac{ib}{2{{\Phi}}_{r}}.\end{equation} \tag{ 4.47 }$$

Thus, the boundary condition which we want to choose for the SU(2) field is

$$\begin{equation}\delta \left(2{{\Phi}}_{r}i{B}_{\tau }-b\right)=0,\end{equation} \tag{ 4.48 }$$

which is consistent with the boundary condition in $\mathcal{N}=4$ super-JT gravity discussed in section 2.2, given that τ ∼ τ + β in both the diffeomorphism gauges of (2.6) and for the Euclidean black hole metric. From a 4D perspective, the boundary holonomy of the SU(2) gauge field is proportional to the boundary angular velocity Ω ∼ iα/β.

Next, we mention why we could fix the flux of the U(1) gauge field in the near-horizon region when working in the canonical ensemble. In such a case, the field strength (and not the gauge field itself) is fixed as r → ∞. Once again, we can study how this happens by solving the equations of motion in the far-away region. The general solution takes the same form as (4.47) for non-abelian SU(2) gauge fields; however, since we now fix the field strength at r → ∞ the constant $\mathcal{C}$ in (4.47) is also fixed. Therefore, the field strength is completely fixed at r = r|_{NHR bdy}, instead of fixing the linear combination (4.48) as in the case of the SU(2) gauge field. Since no massless field in the gravity, sector is charged under the U(1) gauge field, one can integrate-out the 2D gauge field exactly and work in a sector of fixed charge. Since there is no temperature-dependent one-loop determinant when integrating out this field, the computation simply amounts to replacing the field strength in the near-horizon region with its classical values. Thus, this motivates our dimensional reduction in section (4.3), where the U(1) gauge field was absent from all supermultiplets.

Finally, we will choose boundary conditions in the asymptotically flat space region for the fermions to vanish. Then it is reasonable to impose the same boundary condition in the boundary of the throat. This is also consistent with the choice made in section 2.2.

Massive KK modes

Performing the reduction in the previous sections, we have identified the massless Kaluza–Klein modes in the spectrum when reducing the supergravity sector of the 4D theory to 2D. We have ignored so far the contribution from towers of massive KK modes or other matter fields present in 4D. The reason is that they do not affect the temperature dependence of the partition function to leading order [7].

The first step to see this is to realize that in the throat, all interactions are suppressed by factors of S₀. This means that in AdS₂ besides having the JT mode, we have a tower of free fields to integrate out. Let us begin with the case of massless fields in 2D, which arise from an s-wave reduction of a massless field in 4D. We can couple it to JT and integrate it out. The answer has the following form

$$\begin{equation}\mathrm{log}\enspace {Z}_{m=0}=\delta {E}_{0}\beta +\delta {S}_{0}+\mathcal{O}(\varepsilon ,{\beta }^{-2}),\end{equation} \tag{ 4.49 }$$

where δE₀ ∼ 1/ɛ and δS₀ ∼ log L₂. The shift in energy is divergent and can be removed by a counterterm. The parameter δS₀ simply shifts the overall prefactor of the partition function ${e}^{{S}_{0}}\to {e}^{{S}_{0}+\delta {S}_{0}}$. Since L₂ ∼ r₀ this correction is logarithmic in the extremal area. The terms of order ɛ include couplings between matter and gravity, through the Schwarzian mode [96]. They are multiplied by a factor of the cut-off ɛ and therefore suppressed. Any other correction is further suppressed at low temperatures. The same conclusions are true for massive fields, and their contributions have the same structure as (4.49), producing only a shift of S₀. Finally, we can ask what happens when we integrate out a tower of massive KK modes coming from dimensionally reducing an individual field in 4D. This has been studied extensively by Sen and collaborators [33–36]. The answer always has the form (4.49) and, in general, δS₀ ∼ log S₀ with a prefactor depending on the 4D matter spectrum.

The only exception to this rule are dimensional reductions of extra gauge fields in 4D. They reduce to gauge fields in 2D that can affect the temperature dependence in general. Nevertheless, one can focus on boundary condition of fixed charge for all these fields. This choice makes their contribution trivial, and temperature independent ⁴³ .

We can now put all results together. We argued the temperature dependence of the partition function of the near-extremal black hole is captured by $\mathcal{N}=4$ super-JT gravity, and in the previous sections, we precisely solved this theory. By matching boundary conditions, we have found the parameters of the JT theory in terms of their 4D origin

$$\begin{equation}{S}_{0}=\pi {Q}^{2},\qquad {{\Phi}}_{r}=\sqrt{{G}_{N}}\hspace{2.84544pt}{Q}^{3}.\end{equation} \tag{ 4.50 }$$

We are looking at the spectrum at fixed U(1) charge and, therefore, these are fixed parameters. To extract the near-extremal black hole spectrum, we can use this identification in equations (3.45) and (3.46). In these expressions, the parameter J is interpreted as the black hole angular momentum in 4D.

Since this spectrum was already analyzed in the introduction, so we will not repeat it here. The main features we want to point out are the following. If we look at states with zero angular momentum, we find a large extremal degeneracy and the presence of a mass gap in the black hole spectrum, given by

$$\begin{equation}\text{Degeneracy}\;\text{of}\;\text{extremal}\;\text{BH}={e}^{\pi {Q}^{2}},\qquad {E}_{\text{gap}}=\frac{1}{8\sqrt{{G}_{N}}\hspace{2.84544pt}{Q}^{3}}.\end{equation} \tag{ 4.51 }$$

Moreover, our analysis in section 3.4 shows that all extremal states are bosonic, and therefore a calculation of the index would match with the black hole degeneracy.

We can also look at other angular momentum sectors. The correction to the extremal energy we find when we turn on rotation is given by

$$\begin{equation*}{E}_{\text{ext}}(J)=\frac{{J}^{2}}{2{{\Phi}}_{r}}=\frac{{J}^{2}}{2\sqrt{{G}_{N}}\hspace{2.84544pt}{Q}^{3}},\end{equation*}$$

which is measured with respect to the J = 0 extremal mass ${M}_{0}=Q/\sqrt{{G}_{N}}$. This expression matches the expectation from 4D gravity: if we start with the Kerr–Newman metric, take the extremal limit with Q ≠ 0 J ≠ 0, we obtain precisely this correction when expanded to leading order in small J. Interestingly, we find the delta function giving the large extremal degeneracy at J = 0 disappears at J ≠ 0. This makes sense since extremal black holes with J ≠ 0 do not preserve any supersymmetry, and they resemble the results in non-supersymmetry gravity in [7].

Another interesting aspect of the theory is that the supersymmetry in the throat relates the heat capacity Φ_r of the black hole to a very different quantity ${\left(\frac{\partial J}{\partial {\Omega}}\right)}_{T=0,{\Omega}=0}={{\Phi}}_{r}$, where Ω is the black hole angular velocity. In non-supersymmetric theories, these are independent quantities. Moreover, by measuring the dependence of the angular momentum with the angular velocity at zero temperature, we can also determine the gap in the spectrum (since it is controlled by the heat capacity Φ_r ). We comment on this in section 3.5 as well.

Finally, if we decide to fix the metric in the asymptotically flat region and fix the angular velocity to vanish, it is equivalent to fixing the SU(2) chemical potential to zero. With this choice we can use the exact results (3.45) and (3.46) for the black hole spectrum to analyze how quantum corrections affect S(T) and E(T) when α = 0. The results are shown in figure 4. At large temperatures, the results above match the semiclassical result. At low temperatures, both the energy and the entropy behave exponentially as $E(T)\sim {e}^{-1/(8T{{\Phi}}_{r})}$ and $S(T)\sim {S}_{0}+{e}^{-1/(8T{{\Phi}}_{r})}$. This behavior is dictated by the fact that there is an actual gap in the spectrum.

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We will summarize features of supergravity in AdS₃ and compute the partition function with a torus boundary. We will focus on the case with $\mathcal{N}=(4,4)$ supersymmetry. These cases appear in string theory constructions that we will describe in the next section. We will also mention some features of theories with $\mathcal{N}=(4,0)$ in the end.

When these theories are obtained from a dimensional reduction of six-dimensional supergravity on AdS₃ × S³ we have a specific spectrum of matter and gauge fields coupled to gravity. In the near extremal limit, we already saw that as long as we pick boundary conditions with fixed charges for gauge fields, both them and the matter fields do not affect the partition function's temperature dependence [7]. Therefore, in this section, we will focus on the pure supergravity sector on AdS₃. We will also give an argument for this fact near the end combining AdS₃/CFT₂ with the modular bootstrap (which is a supersymmetric version of [37]).

Theories of supergravity in AdS₃ appearing in dimensional reductions of string theory can be written as a difference between two Chern–Simons theories [97]. We will follow the presentation in [98]. Generally for a set of left and right moving symmetry groups G_L × G_R we can define a gravity theory through the following action

$$\begin{equation}{I}_{{G}_{\mathrm{L}}\times {G}_{\mathrm{R}}}=\frac{k}{4\pi }\int \left(\mathcal{A}\wedge \mathrm{d}\mathcal{A}+\frac{2}{3}\mathcal{A}\wedge \mathcal{A}\wedge \mathcal{A}\right)-\frac{k}{4\pi }\int \left(\bar{\mathcal{A}}\wedge \mathrm{d}\bar{\mathcal{A}}+\frac{2}{3}\bar{\mathcal{A}}\wedge \bar{\mathcal{A}}\wedge \bar{\mathcal{A}}\right),,\end{equation} \tag{ 5.1 }$$

where $\mathcal{A}$ is a connection for G_L and $\bar{\mathcal{A}}$ a connection for G_R. We will begin studying the $\mathcal{N}=(4,4)$ case, which corresponds to G_L × G_R = PSU(1,1|2)_L × PSU(1,1|2)_R. Expanding the gauge connection essentially involves two copies of the expansion performed in section 2.1 for BF theory.

The field content is a three-dimensional metric, a set of gravitini, and a SU(2)_L × SU(2)_R Chern–Simons gauge field with level k. The action in terms of these fields can be found adapting the analysis of [99], for example. The metric part of the action in second-order formalism can be written as

$$\begin{equation}{I}_{{G}_{\mathrm{L}}\times {G}_{\mathrm{R}}}\supset \frac{1}{16\pi {G}_{3}}\int {\mathrm{d}}^{3}x\sqrt{g}\left(R+\frac{2}{{\ell }^{2}}\right),\end{equation} \tag{ 5.2 }$$

where ℓ is the AdS₃ radius and G₃ the three-dimensional Newton constant. The Chern–Simons level is given by k = ℓ/(4G₃). The fact that this is also a level of SU(2) implies k is quantized.

To study this theory in asymptotically AdS₃ we consider a supersymmetric generalization of the Brown–Henneaux boundary conditions [100]. The asymptotic symmetries in the (4, 4) case are given by two copies, left and right moving, of the small $\mathcal{N}=4$ Virasoro algebra. The generators of the bosonic Virasoro algebra are L_n , with integer n. The generators of the fermionic symmetries are ${G}_{r}^{a}$ where a = 1, 2 transforms in a double of SU(2) and r can be an integer (R-sector) or half-integer (NS-sector). Finally there is an SU(2) current algebra at level k generated by ${T}_{n}^{i}$, with integer n and i = 1, 2, 3 in the adjoint of SU(2). The commutation relations were already written down above in equations (3.21)–(3.24) and we will not repeat them here. The central charge of the algebra is c, and it is related through supersymmetry to the level k of the SU(2) algebra by c = 6k.

We will be interested in computing the partition function on three dimensional surfaces with torus boundaries

$$\begin{equation}\mathrm{d}{s}^{2}=\mathrm{d}{\tau }^{2}+\mathrm{d}{\varphi }^{2},\qquad (\tau ,\varphi )\sim (\tau ,\varphi +2\pi )\sim (\tau +\beta ,\varphi +\theta ).\end{equation} \tag{ 5.3 }$$

We take the moduli of the torus to be q = e^2πiτ and $\bar{q}={\text{e}}^{-2\pi \text{i}\bar{\tau }}$, where

$$\begin{equation}\tau =\frac{\theta +i\beta }{2\pi }=\frac{i{\beta }_{\mathrm{L}}}{2\pi }\quad \mathrm{a}\mathrm{n}\mathrm{d}\quad \bar{\tau }=\frac{\theta -i\beta }{2\pi }=-\frac{i{\beta }_{\mathrm{R}}}{2\pi },\end{equation} \tag{ 5.4 }$$

and it will be useful to think in terms of left and right moving temperatures. We also consider fixing the boundary chemical potential for the SU(2) Chern–Simons field. We denote its fugacity by z associated to the left-moving charge ${T}_{0}^{3}$ and $\bar{z}$ for the right-moving charge ${\bar{T}}_{0}^{3}$. Finally, we can pick the left- and right-moving fermions to be in the NS or R sector. In terms of AdS/CFT, the partition function with these boundary conditions can be interpreted as a trace over a CFT₂ Hilbert space as

$$\begin{equation}{Z}_{\text{NS/R}\text{--}\text{NS/R}}={\mathrm{Tr}}_{\text{NS/R}\text{--}\text{NS/R}}\left[{q}^{{L}_{0}-\frac{c}{24}}{\bar{q}}^{{\bar{L}}_{0}-\frac{c}{24}}{z}^{{T}_{0}^{3}}{\bar{z}}^{{\bar{T}}_{0}^{3}}\right],\end{equation} \tag{ 5.5 }$$

and we have four possible partition functions.

Pure AdS₃ : to begin, we start with considering the pure AdS₃ solution of this theory, and the fermions in the NS sector. In the bosonic case, the path integral of pure gravity including quantum fluctuations around this geometry can be exactly computed and it is one-loop exact [101]. An analogous argument applies for the supersymmetric generalizations considered here. The answer for the path integral around AdS₃ is given by the product of left- and right-moving vacuum super-Virasoro characters

$$\begin{equation}{Z}_{\text{NS}\text{--}\text{NS}}={\chi }_{\text{id},\text{NS}}^{\mathcal{N}=4}(q,z){\chi }_{\text{id},\text{NS}}^{\mathcal{N}=4}(\bar{q},\bar{z}).\end{equation} \tag{ 5.6 }$$

These characters were computed by Eguchi and Taormina [76] and are given by the following expression in the NS sector

$$\begin{equation}{\chi }_{\text{id},\text{NS}}^{\mathcal{N}=4}(q,z)={q}^{-\frac{k-1}{4}}\frac{i{\theta }_{3}(q,-z){\theta }_{3}(q,-{z}^{-1})}{\eta {(q)}^{3}{\theta }_{1}(q,{z}^{2})}\left[\mu (z,q)-\mu ({z}^{-1},q)\right],\end{equation} \tag{ 5.7 }$$

where θ₃(q, z) is the Jacobi theta function and we defined the function

$$\begin{equation}\mu (z,q)\equiv \sum\limits _{n\in \mathbb{Z}}\frac{{q}^{(k+1){n}^{2}+n}{z}^{2(k+1)n+1}}{\left(1-z{q}^{n+\frac{1}{2}}\right)\left(1-z{q}^{n+\frac{1}{2}}\right)}.\end{equation} \tag{ 5.8 }$$

The characters in the Ramond sector can be obtained by a simple replacement z → zq^1/2. A similar expression applies for the right-mover sector.

BTZ: now we can consider quantum fluctuations around the BTZ solution. Classically, the metric is parametrized by an energy E and angular momentum P (for simplicity we turn off the SU(2) fields now whose generators are denoted above by Tⁱ ). The metric is given by

$$\begin{equation}\mathrm{d}{s}^{2}=f\mathrm{d}{\tau }^{2}+\frac{{\ell }^{2}\enspace \mathrm{d}{r}^{2}}{f}+{r}^{2}{\left(\mathrm{d}\varphi -i\frac{{r}_{+}{r}_{-}}{{r}^{2}}\mathrm{d}\tau \right)}^{2},\quad f=\frac{({r}^{2}-{r}_{+}^{2})({r}^{2}-{r}_{-}^{2})}{{r}^{2}},\end{equation} \tag{ 5.9 }$$

where some useful thermodynamics quantities given in terms of the inner and outer horizon radii r_p m are given by

$$\begin{equation}E=\frac{{r}_{+}^{2}+{r}_{-}^{2}}{8{G}_{3}\ell },\qquad P=\frac{{r}_{+}{r}_{-}}{4{G}_{3}\ell },\qquad T=\frac{{r}_{+}^{2}-{r}_{-}^{2}}{2\pi \ell {r}_{+}},\qquad S=\frac{2\pi {r}_{+}}{4{G}_{3}}.\end{equation} \tag{ 5.10 }$$

The chemical potential for the momentum along the circle is $\theta =i\frac{1}{T}\frac{{r}_{-}}{{r}_{+}}$. For simplicity we will work with a boundary torus with all chemical potentials fixed (both for rotation and SU(2) charges). The partition function including quantum effects around the BTZ solution with these boundary conditions can be obtained from the pure AdS result (5.6) by a modular transformation. We will be interested in the R–R sector since the fermion becomes periodic in spatial S¹ and supersymmetry is preserved in the throat. The final answer is

$$\begin{equation}{Z}_{\text{R}\text{--}\text{R}}={\text{e}}^{2\pi \text{i}k\frac{{z}^{2}}{\tau }-2\pi \text{i}k\frac{{\bar{z}}^{2}}{\bar{\tau }}}{\chi }_{\text{id},\text{NS}}^{\mathcal{N}=4}({q}^{\prime },{z}^{\prime }){\chi }_{\text{id},\text{NS}}^{\mathcal{N}=4}({\bar{q}}^{\prime },{\bar{z}}^{\prime }),\end{equation} \tag{ 5.11 }$$

where we take into account that the modular transformation exchanges sectors and

$$\begin{equation}{q}^{\prime }={\text{e}}^{-2\pi \text{i}/\tau },\qquad {z}^{\prime }={\text{e}}^{2\pi \text{i}\alpha /\tau },\qquad {\bar{q}}^{\prime }={\text{e}}^{2\pi \text{i}/\bar{\tau }},\qquad {\bar{z}}^{\prime }={\text{e}}^{2\pi \text{i}\bar{\alpha }/\bar{\tau }},\end{equation} \tag{ 5.12 }$$

where α is the chemical potential for the SU(2). In the next section we will use these expressions in the near extremal near-BPS limit. The origin of the first two factors is explained in section 5 of [77]. We can rewrite this answer as follows in a more transparent way

$$\begin{equation}{Z}_{\text{R}\text{--}\text{R}}=\sum\limits _{n,\bar{n}\in \mathbb{Z}}{Z}_{\text{one}\mbox{-}\text{loop}}^{n,\bar{n}}{\text{e}}^{\frac{\text{i}\pi k}{2\tau }(1-4{(\alpha +n)}^{2})-\frac{\text{i}\pi k}{2\bar{\tau }}(1-4{(\bar{\alpha }+\bar{n})}^{2})},\end{equation} \tag{ 5.13 }$$

where the one-loop determinant is

$$\begin{align} {Z}_{\text{one}\mbox{-}\text{loop}}^{n,\bar{n}}& =\left\vert \frac{(1-{q}^{\prime })}{\eta ({q}^{\prime })}\frac{{z}^{\prime 2n}\left({q}^{\prime {\left(\frac{1}{2}+n\right)}^{2}}{z}^{\prime }-{q}^{\prime {\left(\frac{1}{2}-n\right)}^{2}}{{z}^{\prime }}^{-1}\right)}{{\theta }_{1}(q,-{z}^{\prime 2})}\right. \\ & \quad \times {\left.\frac{{\theta }_{3}^{2}({q}^{\prime },-{z}^{\prime })}{\eta {({q}^{\prime })}^{2}{\left(1-{z}^{\prime }{q}^{\prime n+\frac{1}{2}}\right)}^{2}{\left(1-{z}^{\prime -1}{q}^{\prime -n+\frac{1}{2}}\right)}^{2}}\right\vert }^{2}. \end{align} \tag{ 5.14 }$$

This way of rewritten the product of vacuum characters shows that the answer is given by a sum over saddles labeled by integers n and $\bar{n}$, with each contribution having a classical action written in (5.13) and a one-loop determinant (5.14) given by the product of the graviton, SU(2) Chern–Simons and gravitino contributions.

Preserving only (4, 0) SUSY: this theory is also given by a combination of Chern–Simons actions with group ${G}_{\mathrm{L}}\times {G}_{\mathrm{R}}=PSU{(1,1\vert 2)}_{\mathrm{L}}\times {\left(\mathrm{S}\mathrm{L}(2,\mathbb{R})\times SU(2)\right)}_{\mathrm{R}}$. In the (4, 0) case, we have as asymptotic a small $\mathcal{N}=4$ Virasoro algebra for left-movers and a bosonic Virasoro and an SU(2) Kac–Moody algebra for the right-mover. The expressions for the partition functions look similar to the cases above, but the right-moving character has to be replaced by the bosonic ones.

In this section, we will consider a near-extremal rotating black hole state, for the theory considered in the previous section, at fixed SU(2) charges.

We will take the limit directly for the exact pure gravity result. If matter is present, we need to be more careful, divide the geometry into the throat and far-away region, and dimensionally reduce in the throat to $\mathcal{N}=4$ super-JT. To simplify, we also assume we pick fixed charge boundary conditions for any other gauge field that is not in the gravity sector. Then for the same reasons as in 4D [7] the temperature dependence of the partition function is given by the $\mathcal{N}=4$ super-Schwarzian.

Using AdS₃/CFT₂, we can give another perspective of this universality using the modular bootstrap: in the near-extremal limit, the vacuum block in the modular-dual channel dominates up to corrections suppressed by the twist gap [37]. ⁴⁴ This is a special case of the extended Cardy regime studied for example in [103].

We will be interested in the following limit. First, we take large β_R → 0. The ensemble is then dominated by black holes with a very large spin P > 0. ⁴⁵ Second, we take large k ≫ 1 so that the gravity description in the bulk is accurate. Finally, we take β_L ∼ k ≫ 1, which implies that we are looking at very low temperatures or states with E ∼ P. Since the state without left-moving excitations preserves supersymmetry, this is also a near-BPS limit [104].

We will follow the calculation first in the fixed β_L, β_R ensemble. As explained in [37] when taking this near-extremal limit we can inverse Fourier transform to obtain fixed P ensemble by basically replacing ${\beta }_{\mathrm{R}}\to 2\pi \sqrt{c/(24P)}$ and β_L → 2β at the end of the calculation. We also keep the left-moving SU(2) chemical potential α fixed in this limit, and consider zero right-moving charge. Either taking the limit of the character or doing the reduction, the near-extremal near-BPS limit of the partition function is

$$\begin{equation}{Z}_{\text{R}\text{--}\text{R}}(\beta ,\alpha )\sim {e}^{2\pi \sqrt{kP}}\sum\limits _{n\in \mathbb{Z}}\frac{\beta }{k}\frac{\mathrm{cot}(\pi \alpha )(\alpha +n)}{{(1-4{(n+\alpha )}^{2})}^{2}}{e}^{\frac{{\pi }^{2}k}{2\beta }(1-4{(n+\alpha )}^{2})}.\end{equation} \tag{ 5.15 }$$

We will not repeat the calculation here since it is completely analogous to section 3.2.2. The first term ${e}^{2\pi \sqrt{kP}}$ comes from the right-moving identity character which is basically evaluated in the usual Cardy limit, since ${\bar{q}}^{\prime }\to 0$. The rest comes from the evaluation of the left-moving identity character, in the limit q' → 1.

The parameters describing the near-extremal near-BPS effective theory analyzed in section 3 are given in terms of the level k and the angular momentum P, by

$$\begin{equation}{S}_{0}=2\pi \sqrt{kP},\qquad {{\Phi}}_{r}=\frac{k}{4}.\end{equation} \tag{ 5.16 }$$

Taking the inverse Laplace transform of this, we obtain the same density of states as the $\mathcal{N}=4$ super-Schwarzian theory with these parameters. In particular, we find a large degeneracy of BPS states given by ${e}^{2\pi \sqrt{kP}}$, we find a gap to the first excited black hole state E_gap = 1/(2k), and this predicts the index matches with the black hole degeneracy.

This can be easily generalized to cases with non-zero SU(2)_R charge ${\bar{T}}_{0}^{3}={J}_{\mathrm{R}}$. In this cases states with ${T}_{0}^{3}=0$ are still BPS and the contribution from the right-moving sector replaces ${S}_{0}\to 2\pi \sqrt{kP-{J}_{\mathrm{R}}^{2}}$. With this modification, the spectrum as a function of temperature and SU(2)_L chemical potential is still given by (5.15), controlled by the $\mathcal{N}=4$ super-Schwarzian.

Alternative construction—preserving (4, 0) SUSY: in this case we obtain a similar conclusion for β_R → 0 and large β_L. The difference now is that we can take instead β_L → 0 and β_R large. This extremal limit breaks supersymmetry since the right-moving sector of the theory is purely bosonic. Therefore, we expect that in this case, the black hole spectrum has, to leading order, no extremal states and no gap, similar to [37] (or [7]).

In this section, we will very briefly mention some string theory constructions using D-branes that can be dimensionally reduced to the AdS₃ supergravity theories studied above.

For example, take type IIB string theory compactified on M⁴, being either T⁴ or K3. We consider the D1–D5 system, which at low energies can be described by either a (4, 4) 2D superconformal field theory or supergravity on AdS₃ × S³. Compactifying down to AdS₃ gives the (4, 4) supergravity theory we studied above. The bosonic part of the spectrum is a 3D metric on AdS₃, and a gauge symmetry coming from the S³ factor in the metric SO(4) ∼ SU(2)_L × SU(2)_R separated into left- and right-movers. For the reasons explained in the previous section, we expect the presence of other fields to leave the conclusions below unchanged. For this theory, we can derive the level of the SU(2) current algebra by matching the chiral anomaly

$$\begin{equation}k={Q}_{1}{Q}_{5}\quad \mathrm{f}\mathrm{o}\mathrm{r}\enspace {T}^{4},\qquad k={Q}_{1}{Q}_{5}+1\quad \mathrm{f}\mathrm{o}\mathrm{r}\enspace K3.\end{equation} \tag{ 5.17 }$$

For concreteness, we look at the T⁴ case below.

So far we have vacuum AdS₃. We can add some momentum P along the D1-string direction, which is identified in the BTZ gravitational description with the angular momentum P defined above [105]. The BPS extremal states correspond to no left-moving excitations of the string. Looking at low temperatures we have a near-extremal near-BPS black hole string with AdS₂ × S¹ × S³ horizon. The parameters of the effective low-energy AdS₂ theory from the microscopic model is

$$\begin{equation}{S}_{0}=2\pi \sqrt{{Q}_{1}{Q}_{5}P},\qquad {{\Phi}}_{r}=\frac{{Q}_{1}{Q}_{5}}{4}.\end{equation} \tag{ 5.18 }$$

Using our solution we see we have ${e}^{{S}_{0}}$ states at extremality, consistent with [6]. Our analysis also explains why the index matches with the black hole degeneracy. This can be easily generalized to near BPS states with non-zero SU(2)_R charges corresponding at zero temperature to BPS black holes with angular momentum in S³ [51].

From our gravitational analysis giving the low-temperature dependence of the partition function, we have also derived the gap to the first excited black hole, and it is E_gap = 1/(8Φ_r ). In terms of the microscopic model parameters, it is

$$\begin{equation}{E}_{\mathrm{g}\mathrm{a}\mathrm{p}}=\frac{1}{2{Q}_{1}{Q}_{5}}.\end{equation} \tag{ 5.19 }$$

This answer matches with the string theory approach from [4]. From this perspective, the extremal black hole states come from counting string configurations at the brane system with only left movers. The lowest energy excitation comes from the first excitation of long strings wound Q₁ Q₅ times along the branes.

Alternative construction—black holes in type I string theory: we can also analyze a similar model in type I string theory instead of type II. We consider a D1–D5 brane system but now the supergravity theory emerging in AdS₃ has (4, 0) supersymmetry [106, 107]. When we have an extremal black hole made out of right-movers, we expect the spectrum near-extremality to be analogous to the $\mathcal{N}=4$ super-Schwarzian. On the other hand, when the extremal black hole is made out of left-movers, supersymmetry is broken, and the near-extremal spectrum will look like the non-supersymmetric cases studied in [37] or [7].

Finally, there are other interesting compactifications which we do not analyze in this paper, but whose role we briefly mention in the discussion section.

We follow the definition of the $\mathcal{N}=2$ super-Schwarzian theory given in [59]. The theory can be described by superspace coordinates $(\tau ,\theta ,\bar{\theta })$. We will consider super-reparametrization $(\tau ,\theta ,\bar{\theta })\to ({\tau }^{\prime },{\theta }^{\prime },{\bar{\theta }}^{\prime })$, which satisfy the constrains $D{\bar{\theta }}^{\prime }=\bar{D}{\theta }^{\prime }=0$, $D{\tau }^{\prime }={\bar{\theta }}^{\prime }D{\theta }^{\prime }$ and $\bar{D}{\tau }^{\prime }={\theta }^{\prime }\bar{D}\enspace {\bar{\theta }}^{\prime }$. Here we used the super-derivative $D\equiv {\partial }_{\theta }+\bar{\theta }{\partial }_{\tau }$ and $\bar{D}\equiv {\partial }_{\bar{\theta }}+\theta {\partial }_{\tau }$. We will parametrize the solutions by

$$\begin{equation}{\tau }^{\prime }=f(\tau )+\cdots \enspace ,\end{equation} \tag{ A.1 }$$

$$\begin{equation}{\theta }^{\prime }={\text{e}}^{\text{i}\sigma (\tau )}\sqrt{{f}^{\prime }(\tau )}\theta +\eta (\tau )+\cdots \end{equation} \tag{ A.2 }$$

$$\begin{equation}{\bar{\theta }}^{\prime }={\text{e}}^{-\text{i}\sigma (\tau )}\sqrt{{f}^{\prime }(\tau )}\bar{\theta }+\bar{\eta }(\tau )+\cdots \enspace ,\end{equation} \tag{ A.3 }$$

where the dots denote higher order terms necessary to solve the super-reparametrization constrains. We introduce with this the time-dependent fields f(τ), e^iσ(τ) ∈ U(1) (or more precisely the loop group) and fermions η(τ) and $\bar{\eta }(\tau )$. These will become the degrees of freedom of the Schwarzian theory. The Schwarzian derivative is defined as

$$\begin{equation}S(f\hspace{-1.5pt},\sigma ,\eta ,\bar{\eta })=\frac{{\partial }_{\tau }\bar{D}\enspace {\bar{\theta }}^{\prime }}{\bar{D}\enspace {\bar{\theta }}^{\prime }}-\frac{{\partial }_{\tau }D{\theta }^{\prime }}{D{\theta }^{\prime }}-2\frac{{\partial }_{\tau }{\theta }^{\prime }{\partial }_{\tau }{\bar{\theta }}^{\prime }}{(\bar{D}{\bar{\theta }}^{\prime })(D{\theta }^{\prime })}=\cdots +\theta \bar{\theta }{S}_{b}(f\hspace{-2pt},\sigma ,\eta ,\bar{\eta }).\end{equation} \tag{ A.4 }$$

Finally, the $\mathcal{N}=2$ super-Schwarzian action is given by ${I}_{\mathcal{N}=2}=-{{\Phi}}_{r}\int \mathrm{d}\tau \enspace \mathrm{d}\theta \enspace \mathrm{d}\bar{\theta }S=-{{\Phi}}_{r}\int \mathrm{d}\tau \enspace {S}_{b}$. The explicit expression is complicated but the bosonic part is naturally given by

$$\begin{equation}{I}_{\mathcal{N}=2}={{\Phi}}_{r}\int \text{Sch}(f,\tau )+2{({\partial }_{\tau }\sigma )}^{2}+(\mathrm{f}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}),\end{equation} \tag{ A.5 }$$

and the terms we omit involve the fermions η and $\bar{\eta }$. Finally the action is invariant under a global SU(1, 1|1) acting on the fields $f,\sigma ,\eta ,\bar{\eta }$ which can be found explicitly in [59]. In this section, we will review the calculation of

$$\begin{equation}Z(\beta ,\alpha )=\int \frac{\mathcal{D}f\mathcal{D}\sigma \mathcal{D}\eta \mathcal{D}\bar{\eta }}{SU(1,1\vert 1)}{e}^{{{\Phi}}_{r}\int \mathrm{d}\tau {S}_{b}(f\hspace{-1.5pt},\sigma ,\eta ,\bar{\eta })}.\end{equation} \tag{ A.6 }$$

The partition function depends on the inverse temperature β and the U(1) chemical potential α. They appear in the path integral as boundary conditions for the fields f(τ + β) = f(τ), e^iσ(τ+β) = e^2πiα e^iσ(τ), η(τ + β) = −e^−2πiα η(τ) and similarly for $\bar{\eta }$. The partition also depends on the parameter $\hat{q}$, which is the periodicity of the U(1) field $\sigma \sim \sigma +2\pi \hat{q}$.

The partition function can be computed either from localization [72] or quantizing the (symplectic) integration space [69]. The answer depends on whether there is an anomaly or not.

$$\begin{equation}Z(\beta ,\alpha )={e}^{{S}_{0}}\sum\limits _{n\in \mathbb{Z}}{(-1)}^{n\nu }\frac{2\hat{q}\enspace \mathrm{cos}\left(\pi (\alpha +\hat{q}n)\right)}{\pi \left(1-4{(\alpha +\hat{q}n)}^{2}\right)}{e}^{\frac{2{\pi }^{2}{{\Phi}}_{r}}{\beta }(1-4{(\alpha +\hat{q}n)}^{2})}.\end{equation} \tag{ A.7 }$$

From a localization perspective this is a sum over saddles labeled by an integer n and the difference between both theories is whether they are weighted by 1 or (−1)ⁿ . Therefore, we introduced the parameter ν which take the values

$$\begin{equation}\nu =0:\enspace \mathrm{n}\mathrm{o}\enspace \mathrm{a}\mathrm{n}\mathrm{o}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{y},\end{equation} \tag{ A.8 }$$

$$\begin{equation}\nu =1:\enspace \mathrm{a}\mathrm{n}\mathrm{o}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{y}.\end{equation} \tag{ A.9 }$$

This will simplify some expressions below. For the theory ν = 1 we will see the spectrum of charges is half-integer and the partition function is not periodic under $\alpha \to \alpha +\hat{q}$.

Now we will extract the spectrum from the partition function above. This was done in section 6.2 of [69]. The goal is to rewrite the partition function as a sum over supermultiplets ⁴⁹ , which for $\mathcal{N}=2$ are (Q) ⊕ (Q − 1) for states with E ≠ 0 and just Q for states with E = 0. Then we aim at an expansion

$$\begin{align} Z(\beta ,\alpha )& =\sum\limits _{Q}{\text{e}}^{2\pi \text{i}\alpha Q}{\rho }_{\text{ext}}(Q) \\ & \quad +\sum\limits _{Q}\int \mathrm{d}E{\text{e}}^{-\beta E}\left({\text{e}}^{\text{i}2\pi \alpha Q}+{\text{e}}^{\text{i}2\pi \alpha (Q-1)}\right){\rho }_{\text{cont}}(Q,E). \end{align} \tag{ A.10 }$$

We will write down the range of Q below determined from the exact partition function. We will also find states at exact E = 0 which are in shorter representations.

We can start by doing an inverse Laplace transform of the partition function with respect to inverse temperature

$$\begin{equation}Z=\int \mathrm{d}E{\text{e}}^{-\beta E}\left({D}_{\text{ext}}(\alpha )\delta (E)+{D}_{\text{cont}}(\alpha ,E)\right).\end{equation} \tag{ A.11 }$$

The continuous can be given by two different expressions depending on the theory

$$\begin{align} {D}_{\text{cont}}(E,\alpha )& ={e}^{{S}_{0}}\sum\limits _{n\in \mathbb{Z}}{(-1)}^{n\nu }\frac{2\hat{q}\sqrt{2{{\Phi}}_{r}}\mathrm{cos}\left(\pi (\alpha +n\hat{q})\right)}{\sqrt{E(1-4{(\alpha +n\hat{q})}^{2})}}{I}_{1}\left(2\sqrt{2{\pi }^{2}{{\Phi}}_{r}E(1-4{(\alpha +n\hat{q})}^{2})}\right). \end{align} \tag{ A.12 }$$

To get an efficient description of the spectrum we can rewrite the density of states derived in [69] in the following way

$$\begin{align} {D}_{\text{cont}}(E,\alpha )& ={e}^{{S}_{0}}\sum\limits _{Q\in \frac{\mathbb{Z}}{\hat{q}}+\frac{\nu }{2\hat{q}}}\left({\text{e}}^{2\pi \text{i}\alpha Q}+{\text{e}}^{2\pi \text{i}\alpha (Q-1)}\right)\frac{\mathrm{sinh}\left(2\pi \sqrt{2{{\Phi}}_{r}(E-{E}_{0}(Q))}\right)}{2\pi E}{\Theta}(E-{E}_{0}(Q)). \end{align} \tag{ A.13 }$$

We define the function E₀(Q) giving the edge of the spectrum of each supermultiplet

$$\begin{equation}{E}_{0}(Q)=\frac{1}{8{{\Phi}}_{r}}{\left(Q-\frac{1}{2}\right)}^{2}.\end{equation} \tag{ A.14 }$$

From the expressions above a very simple picture emerges. The continuous component of the Q supermultiplet density of states as a function of Q and E is given by

$$\begin{equation}{\rho }_{\text{cont}}(Q,E)={e}^{{S}_{0}}\frac{\mathrm{sinh}\left(2\pi \sqrt{2{{\Phi}}_{r}(E-{E}_{0}(Q))}\right)}{2\pi E}{\Theta}(E-{E}_{0}(Q)).\end{equation} \tag{ A.15 }$$

First of all this is completely independent on $\hat{q}$ which only appears through the unit of charge. Second, this expression is valid for both type of theories with or without anomaly, the only difference being whether $Q\in \frac{1}{\hat{q}}\cdot \mathbb{Z}$ (no anomaly) or $Q\in \frac{1}{\hat{q}}\cdot \mathbb{Z}+\frac{1}{2\hat{q}}$ (anomaly).

The presence of a gap in the spectrum depends on both whether $\hat{q}$ is even or odd and whether we have an anomaly or not. If $\hat{q}$ is odd, then the non-anomalous theory has a gap controlled by the supermultiplets labeled by ${Q}_{0}=\frac{1}{2}\pm \frac{1}{2\hat{q}}$ with ${E}_{\text{gap}}\equiv {E}_{0}({Q}_{0})=1/(32{{\Phi}}_{r}{\hat{q}}^{2})$. This is the actual gap of the theory since any other supermultiplet Q has E₀(Q) > E_gap. If the theory with odd $\hat{q}$ is anomalous then the spectrum of Q is half-integers and there is no gap since for Q = 1/2, E₀(1/2) = 0. Similarly, for the case with $\hat{q}$ even, the non-anomalous theory has no gap, while this time the anomalous theory has a gap of the same scale as before.

We can check the same with the 'extremal' states with support at E = 0. From the partition function we get

$$\begin{equation}{D}_{\text{ext}}(\alpha )\delta (E)={e}^{{S}_{0}}\delta (E)\sum\limits _{n\in \mathbb{Z}}{(-1)}^{n\nu }\frac{2\hat{q}\enspace \mathrm{cos}\left(\pi (\alpha +n\hat{q})\right)}{\pi \left(1-4{(\alpha +n\hat{q})}^{2}\right)}.\end{equation} \tag{ A.16 }$$

This expression can be rewritten as a sum over charges in the following way

$$\begin{equation}{D}_{\text{ext}}(\alpha )=\sum\limits _{Q\in \frac{1}{\hat{q}}\mathbb{Z}+\frac{\nu }{2\hat{q}},\vert Q\vert < \frac{1}{2}}{e}^{2\pi \text{i}\alpha Q}\enspace {e}^{{S}_{0}}\enspace \mathrm{cos}\left(\pi Q\right)\end{equation} \tag{ A.17 }$$

which was found in [69] using the modular transform of the character. Finally, the density of states of extremal states per charge is given by

$$\begin{equation}{\rho }_{\text{ext}}(Q,E)={e}^{{S}_{0}}\enspace \mathrm{cos}\left(\pi Q\right){\Theta}\left(2\vert Q\vert -1\right).\end{equation} \tag{ A.18 }$$

This is always positive since the cosine is always evaluated between (−π/2, π/2). This formula, like before, is valid for either theory as long as Q is taken over the correct set.

As a summary, we show the density of supermultiplets for the anomalous and non-anomalous cases in figure 5. These results for the spectrum of the $\mathcal{N}=2$ super-Schwarzian theory have been also obtained analytically from double-scaled $\mathcal{N}=2$ SYK in [122].

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