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Geometry of canonical bases and mirror symmetry

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Abstract

A decorated surface \(S\) is an oriented surface with boundary and a finite, possibly empty, set of special points on the boundary, considered modulo isotopy. Let \(\mathrm{G}\) be a split reductive group over \({\mathbb Q}\). A pair \((\mathrm{G}, S)\) gives rise to a moduli space \({\mathcal A}_{\mathrm{G}, S}\), closely related to the moduli space of \(\mathrm{G}\)-local systems on \(S\). It is equipped with a positive structure (Fock and Goncharov, Publ Math IHES 103:1–212, 2006). So a set \({\mathcal A}_{\mathrm{G}, S}({\mathbb Z}^t)\) of its integral tropical points is defined. We introduce a rational positive function \({\mathcal W}\) on the space \({\mathcal A}_{\mathrm{G}, S}\), called the potential. Its tropicalisation is a function \({\mathcal W}^t: {\mathcal A}_{\mathrm{G}, S}({\mathbb Z}^t) \rightarrow {\mathbb Z}\). The condition \({\mathcal W}^t\ge 0\) defines a subset of positive integral tropical points \({\mathcal A}^+_{\mathrm{G}, S}({\mathbb Z}^t)\). For \(\mathrm{G=SL}_2\), we recover the set of positive integral \({\mathcal A}\)-laminations on \(S\) from Fock and Goncharov (Publ Math IHES 103:1–212, 2006). We prove that when \(S\) is a disc with \(n\) special points on the boundary, the set \({\mathcal A}^+_{\mathrm{G}, S}({\mathbb Z}^t)\) parametrises top dimensional components of the fibers of the convolution maps. Therefore, via the geometric Satake correspondence (Lusztig, Astérisque 101–102:208–229, 1983; Ginzburg,1995; Mirkovic and Vilonen, Ann Math (2) 166(1):95–143, 2007; Beilinson and Drinfeld, Chiral algebras. American Mathematical Society Colloquium Publications, vol. 51, 2004) they provide a canonical basis in the tensor product invariants of irreducible modules of the Langlands dual group \(\mathrm{G}^L\):

$$\begin{aligned} (V_{\lambda _1}\otimes \ldots \otimes V_{\lambda _n})^{\mathrm{G}^L}. \end{aligned}$$
(1)

When \(\mathrm{G=GL}_m\), \(n=3\), there is a special coordinate system on \({\mathcal A}_{\mathrm{G}, S}\) (Fock and Goncharov, Publ Math IHES 103:1–212, 2006). We show that it identifies the set \({\mathcal A}^+_{\mathrm{GL_m}, S}({\mathbb Z}^t)\) with Knutson–Tao’s hives (Knutson and Tao, The honeycomb model of GL(n) tensor products I: proof of the saturation conjecture, 1998). Our result generalises a theorem of Kamnitzer (Hives and the fibres of the convolution morphism, 2007), who used hives to parametrise top components of convolution varieties for \(\mathrm{G=GL}_m\), \(n=3\). For \(\mathrm{G=GL}_m\), \(n>3\), we prove Kamnitzer’s conjecture (Kamnitzer, Hives and the fibres of the convolution morphism, 2012). Our parametrisation is naturally cyclic invariant. We show that for any \(\mathrm{G}\) and \(n=3\) it agrees with Berenstein–Zelevinsky’s parametrisation (Berenstein and Zelevinsky, Invent Math 143(1):77–128, 2001), whose cyclic invariance is obscure. We define more general positive spaces with potentials \(({\mathcal A}, {\mathcal W})\), parametrising mixed configurations of flags. Using them, we define a generalization of Mirković–Vilonen cycles (Mirkovic and Vilonen, Ann Math (2) 166(1):95–143, 2007), and a canonical basis in \(V_{\lambda _1}\otimes \ldots \otimes V_{\lambda _n}\), generalizing the Mirković–Vilonen basis in \(V_{\lambda }\). Our construction comes naturally with a parametrisation of the generalised MV cycles. For the classical MV cycles it is equivalent to the one discovered by Kamnitzer (Mirkovich–Vilonen cycles and polytopes, 2005). We prove that the set \({\mathcal A}^+_{\mathrm{G}, S}({\mathbb Z}^t)\) parametrises top dimensional components of a new moduli space, surface affine Grasmannian, generalising the fibers of the convolution maps. These components are usually infinite dimensional. We define their dimension being an element of a \({\mathbb Z}\)-torsor, rather then an integer. We define a new moduli space \(\mathrm{Loc}_{G^L, S}\), which reduces to the moduli spaces of \(G^L\)-local systems on \(S\) if \(S\) has no special points. The set \({\mathcal A}^+_{\mathrm{G}, S}({\mathbb Z}^t)\) parametrises a basis in the linear space of regular functions on \(\mathrm{Loc}_{G^L, S}\). We suggest that the potential \({\mathcal W}\) itself, not only its tropicalization, is important—it should be viewed as the potential for a Landau–Ginzburg model on \({\mathcal A}_{\mathrm{G}, S}\). We conjecture that the pair \(({\mathcal A}_{\mathrm{G}, S}, {\mathcal W})\) is the mirror dual to \(\mathrm{Loc}_{G^L, S}\). In a special case, we recover Givental’s description of the quantum cohomology connection for flag varieties and its generalisation (Gerasimov et al., New integral representations of Whittaker functions for classical Lie groups, 2012; Rietsch, A mirror symmetric solution to the quantum Toda lattice, 2012). We formulate equivariant homological mirror symmetry conjectures parallel to our parametrisations of canonical bases.

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Notes

  1. Inspite of the name, it is not an affine variety.

  2. We would like to stress that the positive structures and potentials on configuration spaces which we employ for parametrization of canonical bases do not depend on any extra choices, like pinning etc., in the group. See Sect. 6.3.

  3. Indeed, \(\omega _i({\alpha }(\mathrm{A}_1, \mathrm{A}_2))=0\) if and only if the corresponding pair of flags belongs to the codimension one \(\mathrm{G}\)-orbit corresponding to a simple reflection \(s_i\).

  4. Indeed, it follows from Lemma 11.3 and an explicit description of cluster structure on \(\mathrm{Conf}_n({\mathcal A})\) that the form \(\Omega _{\mathcal A}\) can not have non-zero residues anywhere else the divisors \(D^E_i\). One can show that the residues at these divisors are non-zero.

  5. The set \({\mathcal Y}({\mathbb Z}^t)\), the tropicalization \({\mathcal W}^t\), and thus the subset \({\mathcal Y}^+_{\mathcal W}({\mathbb Z}^t)\) can also be determined by the volume form \(\Omega _{\mathcal Y}\), without using the positive structure on \({\mathcal Y}\).

  6. Laminations on decorated surfaces were investigated in [17, Section 12], and [19]. However the two types of laminations considered there, the \({\mathcal A}\)- and \({\mathcal X}\)-laminations, are different then the ones in Definition 1.3. Indeed, they parametrise canonical bases in \({\mathcal O}({\mathcal X}_{PGL_2, S})\) and, respectively, \({\mathcal O}({\mathcal A}_{SL_2, S})\), while the latter parametrise a canonical basis in \({\mathcal O}(\mathrm{Loc}_{SL_2, S})\). Notice that a lamination in Definition 1.3 can not end on a boundary circle.

  7. In our main examples the symplectic structure is exact, \(\omega = d\alpha \). So averaging the form \(\alpha \) by the action of the compact group \({\mathbb T}_K\) we can assume that it is \({\mathbb T}_K\)-invariant. Therefore the action is Hamiltonian: the Hamiltonian at \(x\) for a one parametric subgroup \(g^t\) is given by the formula \(\alpha (\frac{d}{dt}g^t(x))\).

  8. Although the claim (39) is not addressed in the paper, it can be deduced from (38).

  9. By abuse of notation, such a cell will always be denoted by \({\mathcal C}_l^\circ \). The tropical point \(l\) tells which space it lives.

  10. Here is a non-archimedean analog: A generic pair of flags \(\{\mathrm{B}_1, \mathrm{B}_2\}\) over \({\mathcal K}\) gives rise to an \(H({\mathcal K})/H({\mathcal O})\)-torsor in the affine Grassmannian—the projection of \(\mathrm{B}_1({\mathcal K})\cap \mathrm{B}_2({\mathcal K})\) to \(\mathrm{G}({\mathcal K})/\mathrm{G}({\mathcal O})\).

  11. In the archimedean case, a maximal compact subgroup \(\mathrm{K}\) is defined by using the Cartan involution. A generic pair \(\{\mathrm{A}, \mathrm{B}\}\) determines a pinning, and hence a Cartan involution.

  12. Every transcendental point \(C\in \mathrm{T}^{\circ }_l\) automatically satisfies such conditions.

  13. There is a positive Cartan group action on \(\mathrm{Conf}_3({\mathcal A})({\mathbb Z}^t)\) defined via

    $$\begin{aligned} \mathrm{H}\times \mathrm{Conf}_3({\mathcal A})\longrightarrow \mathrm{Conf}_3({\mathcal A}), \quad \quad h\times (\mathrm{A}_1, \mathrm{A}_2, \mathrm{A}_3)\longmapsto (\mathrm{A}_1, \mathrm{A}_2, \mathrm{A}_3\cdot h). \end{aligned}$$

    Its tropicalization determines a free \(\mathrm{H}({\mathbb Z}^t)\)-action on \(\mathrm{Conf}_3({\mathcal A})({\mathbb Z}^t)\). By definition, one can thus identify the \(\mathrm{H}({\mathbb Z}^t)\)-orbits of \(\mathrm{Conf}_3({\mathcal A})({\mathbb Z}^t)\) with points of \(\mathrm{Conf}({\mathcal A},{\mathcal A}, {\mathcal B})({\mathbb Z}^t)\). Note that each element in \(\mathbf{B}_{\lambda _1}^{\mu -\lambda _2}\) has a unique representative in \(\widetilde{\mathbf{B}}_{\lambda _1,\lambda _2}^{\mu }\). Hence the map \(\pi _3^t\) is a bijection.

  14. If the vertices of \(t\) coincide, one can first pull back to a sufficient big cover \(\widetilde{S}\) of \(S\), and then consider the restriction to a triangle \(\widetilde{t}\subset \widetilde{S}\) which projects onto \(t\). Clearly the result is independent of the pair \(\widetilde{t}\subset \widetilde{S}\) chosen.

  15. Although the decorated surface \(*S\) is isomorphic to \(S\), the isomorphism is not quite canonical.

  16. Taking the quotient by a unipotent group does not affect the category of equivariant sheaves. This is why we require the prounipotence condition here.

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Acknowledgments

This work was supported by the NSF grants DMS-1059129 and DMS-1301776. A.G. is grateful to IHES and Max Planck Institute fur Mathematic (Bonn) for the support. We are grateful to Mohammed Abouzaid, Joseph Bernstein, Alexander Braverman, Vladimir Fock, Alexander Givental, David Kazhdan, Joel Kamnitzer, Sean Keel, Ivan Mirkovic, and Sergey Oblezin for many useful discussions. We are especially grateful to Maxim Kontsevich for fruitful conversations on mirror symmetry during the Summer of 2013 in IHES. We are very grateful to the referee for many fruitful comments, remarks and suggestions.

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    Alexander Goncharov & Linhui Shen

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Goncharov, A., Shen, L. Geometry of canonical bases and mirror symmetry. Invent. math. 202, 487–633 (2015). https://doi.org/10.1007/s00222-014-0568-2

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