For the type A surface singularity orbifold X=[C2/μn]\mathcal{X}=[\mathbb{C}^2/\mu_n] and its minimal (crepant) resolution YY, Coates-Corti-Iritani-Tseng proved the crepant resolution conjecture: the quantum cohomology of X\mathcal{X} becomes isomorphic to that of YY after analytic continuation in the B-model moduli space of the mirror Landau-Ginzburg model. The resulting affine-linear identification of flat coordinates depends on the homotopy class of the continuation path; the ambiguity is governed by the orbifold fundamental group G≅A~n−1⋊μnG\cong\widetilde{A}_{n-1}\rtimes\mu_n (affine braid group extended by nn-th roots of unity), which also appears as the subgroup of autoequivalences of the derived category DZb(Y)D^b_Z(Y) of sheaves supported on the exceptional locus, generated by spherical twists and line-bundle twists. The open problem is to make precise, and prove, the mirror-symmetric correspondence between the monodromy action of GG on the analytic-continuation identifications and this group of derived autoequivalences, yielding a monodromy-equivariant form of the crepant resolution conjecture for type A surface singularities.

Background

A type An−1A_{n-1} surface singularity is the quotient C2/μn\mathbb{C}^2/\mu_n, where μn⊂SL2(C)\mu_n\subset\mathrm{SL}_2(\mathbb{C}) is the group of nn-th roots of unity acting by ζ⋅(z1,z2)=(ζz1,ζ−1z2)\zeta\cdot(z_1,z_2)=(\zeta z_1,\zeta^{-1}z_2); it is also called a Kleinian or Du Val singularity. The quotient orbifold X=[C2/μn]\mathcal{X}=[\mathbb{C}^2/\mu_n] has a crepant resolution π:Y→C2/μn\pi:Y\to\mathbb{C}^2/\mu_n, namely the minimal resolution, on which the exceptional locus ZZ is a chain of n−1n-1 rational curves; DZb(Y)D^b_Z(Y) denotes the bounded derived category of coherent sheaves on YY supported on ZZ. Quantum cohomology deforms the usual cup product on cohomology using genus-zero Gromov-Witten invariants (counts of holomorphic curves); for an orbifold one uses Chen-Ruan orbifold Gromov-Witten theory, where twisted sectors indexed by conjugacy classes of the group contribute additional classes. The crepant resolution conjecture of Ruan and Bryan-Graber predicts that the quantum cohomology of an orbifold X\mathcal{X} with crepant resolution YY becomes isomorphic to the quantum cohomology of YY after analytic continuation in the quantum parameters, followed by specialization of some parameters to roots of unity. Coates-Corti-Iritani-Tseng (arXiv:0704.2034) proved this for type A surface singularities using mirror symmetry for toric orbifolds: both quantum cohomologies are computed from a mirror Landau-Ginzburg model, that is, a holomorphic potential function WXW_{\mathcal{X}} on an algebraic torus, and the isomorphism is obtained by analytic continuation in the B-model moduli space MB\mathcal{M}_B of complex structures of the mirror, along a path joining the large-radius limit points of YY and X\mathcal{X}. Flat coordinates are distinguished coordinates on MB\mathcal{M}_B in which the Frobenius-manifold metric is constant; transition functions between overlapping flat-coordinate systems are affine-linear. Changing the continuation path changes the resulting affine-linear coordinate change, hence the linear identification of cohomology bases; the paper shows that this ambiguity is governed by the orbifold fundamental group G=π1orb(MB∖{discriminant locus of WX})G=\pi_1^{\mathrm{orb}}(\mathcal{M}_B\setminus\{\text{discriminant locus of }W_{\mathcal{X}}\}), which acts simply-transitively on homotopy classes of paths and transitively, but not effectively, on the set of coordinate changes, and that G≅A~n−1⋊μnG\cong\widetilde{A}_{n-1}\rtimes\mu_n, where A~n−1\widetilde{A}_{n-1} is the affine braid group. Independently, an object of a derived category is called spherical (in the sense of Seidel-Thomas) if it has only two self-Ext groups, in degrees 00 and dim⁡\dim, as for the structure sheaf of a (−2)(-2)-curve, i.e. a rational curve of self-intersection −2-2; the associated spherical twist functor is an autoequivalence, and Seidel-Thomas showed these twists realize braid group actions modeled on symplectic Dehn twists. Ishii-Uehara computed the autoequivalences of derived categories on minimal resolutions of AnA_n-singularities, and together with Bridgeland's and Ishii-Ueda-Uehara's work on stability conditions for Kleinian singularities this exhibits A~n−1⋊μn\widetilde{A}_{n-1}\rtimes\mu_n as the subgroup of Aut⁡(DZb(Y))\operatorname{Aut}(D^b_Z(Y)) generated by spherical twists about the exceptional sheaves and tensoring by line bundles. Prior research links the B-model of WXW_{\mathcal{X}} to the derived side: homological mirror symmetry for the orbifold X\mathcal{X}, composed with the McKay correspondence, yields the derived equivalence DZb(Y)≃Db(coh⁡0μn C2)D^b_Z(Y)\simeq D^b(\operatorname{coh}^{\mu_n}_0\,\mathbb{C}^2) between coherent sheaves on YY supported on the exceptional locus ZZ and the μn\mu_n-equivariant coherent sheaves on C2\mathbb{C}^2 supported at the origin. That the same group governs both the analytic-continuation ambiguity of the quantum-cohomology identification and these derived autoequivalences is an intriguing coincidence noticed by the authors, and understanding its mirror-symmetric meaning is the problem posed here.

Problem Statement

For μn⊂SL2(C)\mu_n\subset\mathrm{SL}_2(\mathbb{C}) acting by ζ⋅(z1,z2)=(ζz1,ζ−1z2)\zeta\cdot(z_1,z_2)=(\zeta z_1,\zeta^{-1}z_2), let X=[C2/μn]\mathcal{X}=[\mathbb{C}^2/\mu_n] be the type An−1A_{n-1} surface singularity orbifold, let π:Y→C2/μn\pi:Y\to\mathbb{C}^2/\mu_n be its minimal (crepant) resolution with exceptional locus ZZ, and let G=π1orb(MB∖{discriminant locus of WX})≅A~n−1⋊μnG=\pi_1^{\mathrm{orb}}\big(\mathcal{M}_B\setminus\{\text{discriminant locus of }W_{\mathcal{X}}\}\big)\cong\widetilde{A}_{n-1}\rtimes\mu_n be the orbifold fundamental group of the B-model moduli space of the mirror Landau-Ginzburg model, which governs the dependence of the Coates-Corti-Iritani-Tseng identification of the quantum cohomologies of X\mathcal{X} and YY on the homotopy class of the analytic-continuation path. The problem, stated as deserving further study by the source, is to make the mirror-symmetric relationship between the two occurrences of GG precise, as one coherent objective, both parts of which are required: (1) describe explicitly the monodromy action of GG on the flat coordinates and on the set of affine-linear coordinate changes realizing the crepant resolution conjecture identification, determining the kernel through which this non-effective action factors; and (2) prove the conjectural correspondence between elements of GG and geometric/derived symmetries of YY: under a mirror equivalence relating the B-model of WXW_{\mathcal{X}} to the derived category DZb(Y)D^b_Z(Y), show that analytic continuation along a loop representing an element of GG corresponds to the associated autoequivalence of DZb(Y)D^b_Z(Y) generated by spherical twists and line-bundle twists, in a way compatible with the crepant resolution conjecture identification of quantum cohomologies, namely by agreeing on the induced monodromy actions on the quantum DD-module, i.e. the flat bundle of the quantum differential equation whose monodromy records analytic continuation of quantum cohomology. Together, the two parts constitute a monodromy-equivariant form of the crepant resolution conjecture for type A surface singularities.

References

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