Gravitational actions in two dimensions and the Mabuchi functional

Abstract

The Mabuchi energy is an interesting geometric functional on the space of Kähler metrics that plays a crucial rôle in the study of the geometry of Kähler manifolds. We show that this functional, as well as other related geometric actions, contribute to the effective gravitational action when a massive scalar field is coupled to gravity in two dimensions in a small mass expansion. This yields new theories of two-dimensional quantum gravity generalizing the standard Liouville models.

Introduction

When a matter quantum field theory is coupled to gravity, the metric dependence of the partition function induces an effective gravitational action. On a two-dimensional Euclidean space–time, when the quantum field theory is conformally invariant, this effective action is always proportional to the Liouville action [1]. In the so-called conformal gauge, for which the metric g is proportional to a fixed background metric $g_0$,

$$g=e^{2\sigma }{g}_0,$$

the Liouville action takes the simple form

$$S_{\mathrm{L}}(g_0,g)=\int {\mathrm{d}}^{2}x\sqrt{g_0}(g_0^{ab}{\partial }_a\sigma {\partial }_b\sigma +R_0\sigma ),$$

where $R_0$ is the Ricci scalar for the metric $g_0$. The resulting Liouville models for two-dimensional quantum gravity have been extensively studied in the literature.

The gravitational coupling of non-conformally invariant matter has been much less studied (see e.g. [2]). In this case, one expects a complicated and model-dependent gravitational action. It is unclear a priori if new interesting and simple actions, yielding models with nice geometrical and physical features, can emerge.

Recently, the authors have explained that two-dimensional quantum gravity could be naturally formulated in the context of Kähler geometry [3], [4]. In this context, mathematicians have much used and studied natural functionals defined on the space of Kähler potentials, the most salient being the Mabuchi action [5], also called the Mabuchi energy in the mathematical literature. Our aim in the present work is to investigate whether such actions could contribute to the gravitational effective action in some simple physical models. We shall see that they do, yielding new models of two-dimensional quantum gravity with potentially very rich mathematical and physical features.

Our space–time will always be a compact Riemann surface ${\Sigma }_h$ of genus h. An arbitrary Riemannian metric on ${\Sigma }_h$ can be parametrized by a finite set of complex moduli τ and the conformal factor $e^{2\sigma }$, with a background metric $g_0(\tau )$ in (1.1) of the form $g_0=2g_{0z\overline{z}}{|\mathrm{d}z|}^2$. In the Kähler framework, we write the conformal factor $e^{2\sigma }$ in terms of the area A (which is the unique Kähler modulus in two dimensions) and the Kähler potential ϕ,

$$e^{2\sigma }=\frac{A}{A_0}-\frac{1}{2}A{\mathrm{\Delta }}_0\varphi ,$$

where $A_0$ and ${\mathrm{\Delta }}_0$ are the area and the positive Laplacian for the metric $g_0$ respectively. The $\partial \overline{\partial }$ lemma shows that the differential equation (1.3) can always be solved for A and ϕ in terms of σ, and the solution is unique up to constant shifts in ϕ.

There are many reasons to use the Kähler potential ϕ in the context of two-dimensional quantum gravity. In particular it is possible to rigorously define in terms of ϕ a regularized version of the gravitational path integrals [3], [4]. For our present purposes, the variable ϕ is useful because interesting functionals of the metric are most naturally written down in terms of ϕ. A famous example is the Mabuchi action [5],

$$S_{\mathrm{M}}(g_0,g)=\underset{{\Sigma }_h}{\int }{\mathrm{d}}^{2}x\sqrt{g_0}[-2\pi (1-h)\varphi {\mathrm{\Delta }}_0\varphi +(\frac{8\pi (1-h)}{A_0}-R_0)\varphi +\frac{4}{A}\sigma e^{2\sigma }].$$

This action and its higher dimensional generalizations play a central rôle in Kähler geometry. It is well defined on the space or metrics, because it is unchanged if ϕ is shifted by a constant. It is bounded from below and convex, which makes it a good candidate for an action to be used in a path integral. It satisfies the so-called cocycle conditions,

$$S_{\mathrm{M}}(g_1,g_2)=-S_{\mathrm{M}}(g_2,g_1),$$

$$S_{\mathrm{M}}(g_1,g_3)=S_{\mathrm{M}}(g_1,g_2)+S_{\mathrm{M}}(g_2,g_3),$$

which can be checked straightforwardly from the definition (1.4). The same cocycle identities are also satisfied by the Liouville action. As we shall review below, these identities actually are fundamental consistency conditions that any effective gravitational action must satisfy. Finally, the critical points of the Mabuchi action are the metrics of constant scalar curvature, another property shared in two dimensions with the Liouville action. This property is also valid for the higher dimensional generalizations of (1.4), which goes a long way in explaining the central rôle played by the Mabuchi action in the study of such metrics on general Kähler manifolds. For more details and references on the profound geometrical properties of the space of metrics on a Kähler manifold, we refer the reader to [4], [6], [7] and the comprehensive recent review [8].

We focus in the following on the simple model of a massive scalar field X with action

$$S_{\mathrm{mat}}(X,g;q,m)=\frac{1}{8\pi }\int {\mathrm{d}}^{2}x\sqrt{g}(g^{ab}{\partial }_{a}X{\partial }_{b}X+qRX+m^2{X}^2).$$

Our main result is to show that the Mabuchi action and other simple functionals of the Kähler potential ϕ, like the so-called Aubin–Yau action, contribute to the gravitational effective action for this model in a small mass expansion. We also study the gravitational dressing of operators of the form

$${\mathcal{O}}_k=e^{kX}$$

and show that it involves the Aubin–Yau action on top of the familiar dressing factors found in the conformal field theory limit.

A striking feature is that it is always possible to cancel out the familiar Liouville term in the effective action, for example by coupling with a suitable spectator conformal field theory. The resulting pure Mabuchi theories are entirely new and intriguing two-dimensional quantum gravity models.

We start in Section 2 with a brief review on two-dimensional quantum gravity, gravitational effective actions and gravitational dressing. In Section 3, we discuss the basic properties of various actions, in particular Mabuchiʼs and Aubin–Yauʼs. We derive fundamental identities relating these actions to the variations of various functionals associated with the Laplace operator. In Section 4, we apply the results of Section 3 to compute the gravitational effective action and the gravitational dressing of the operators $e^{kX}$ for the model (1.7). We also compute the trace of the stress–energy tensor. Finally, in Section 4, we conclude and discuss possible extensions of our work. We have tried to make the presentation as elementary and self-contained as possible. In particular we have included Appendix A On the derivatives of the Greenʼs function, Appendix B Greenʼs function and containing simple derivations of results used in the main text.

To a metric $g=2g_{z\overline{z}}{|\mathrm{d}z|}^2$ on ${\Sigma }_h$ we associate its Kähler form,

$$\omega =i{g}_{z\overline{z}}\mathrm{d}z\wedge \mathrm{d}\overline{z}.$$

In two dimensions, the Kähler form coincides with the volume form and thus in particular the area is given by

$$A(g)=\underset{{\Sigma }_h}{\int }\omega .$$

The positive Laplacian for the metric g, whose determinant we also denote by g, is defined as usual by

$$\mathrm{\Delta }f=-\frac{1}{\sqrt{g}}{\partial }_a\left(\sqrt{g}{g}^{ab}{\partial }_{b}f\right).$$

In terms of the standard Dolbeault operators ∂ and $\overline{\partial }$ we have

$$\partial \overline{\partial }f=\frac{i}{2}\mathrm{\Delta }f\omega ,$$

from which it can be seen that the relation (1.3) is equivalent to

$$\omega =\frac{A}{A_0}{\omega }_0+iA\partial \overline{\partial }\varphi .$$

The space of Kähler potentials ϕ is simply

$${\mathcal{K}}_{{\omega }_0}=\{\varphi :\Sigma \to \mathbb{R}|A_0{\mathrm{\Delta }}_0\varphi <2\},$$

the condition on ϕ ensuring the strict positivity of the metrics, $g_{z\overline{z}}>0$. The scalar curvature, or Ricci scalar R, is such that

$$R\omega =-i\partial \overline{\partial }\ln g.$$

The integral of the scalar curvature is a topological invariant,

$$\underset{{\Sigma }_h}{\int }R\omega =8\pi (1-h).$$

This is a useful formula, from which for instance the invariance of (1.4) under constant shifts of ϕ is derived. Finally, the Laplacians Δ and ${\mathrm{\Delta }}_0$ and scalar curvatures R and $R_0$ of two metrics g and $g_0$ related by a formula of the form (1.1) are themselves linked by the simple relations

$$e^{2\sigma }\mathrm{\Delta }={\mathrm{\Delta }}_0,$$

$$e^{2\sigma }R=R_0+2{\mathrm{\Delta }}_0\sigma .$$