MathSciDoc: An Archive for Mathematician ∫

We prove a Givental-style mirror theorem for toric Deligne--Mumford stacks X. This determines the genus-zero Gromov--Witten invariants of X in terms of an explicit hypergeometric function, called the I-function, that takes values in the Chen--Ruan orbifold cohomology of X.

For a gerbe Y over a smooth proper Deligne-Mumford stack B banded by a finite group G, we prove a structure result on the Gromov-Witten theory of Y, expressing Gromov-Witten invariants of Y in terms of Gromov-Witten invariants of B twisted by various flat U(1)-gerbes on B. This is interpreted as a Leray-Hirsch type of result for Gromov-Witten theory of gerbes.

We show that the Virasoro conjecture in Gromov–Witten theory holds for the the total space of a toric bundle E → B if and only if it holds for the base B. The main steps are: (i) we establish a localization formula that expresses Gromov–Witten invariants of E, equivariant with respect to the fiberwise torus action, in terms of genus-zero invariants of the toric fiber and all-genus invariants of B; and (ii) we pass to the non- equivariant limit in this formula, using Brown’s mirror theorem for toric bundles.

We derive a formula for the virtual class of the moduli space of rubber maps to [P^1/G] pushed forward to the moduli space of stable maps to BG. As an application, we show that the Gromov-Witten theory of [P^1/G] relative to 0 and ∞ are determined by known calculations.

We use Seidel representation for symplectic orbifolds to compute the quantum cohomology ring of a compact symplectic toric orbifold (X , ω).

We present a generalization of the Bogomolov-Miyaoka-Yau inequality to Deligne-Mumford surfaces of general type.

We study the higher genus equivariant Gromov-Witten theory of the Hilbert scheme of n points of the plane. Since the equivariant quantum cohomology is semisimple, the higher genus theory is determined by an R-matrix via the Givental-Teleman classification of Cohomological Field Theories (CohFTs). We uniquely specify the required R-matrix by explicit data in degree 0. As a consequence, we lift the basic triangle of equivalences relating the equivariant quantum cohomology of the Hilbert scheme and the Gromov-Witten/Donaldson-Thomas correspondence for 3-fold theories of local curves to a triangle of equivalences in all higher genera. The proof uses the previously determined analytic continuation of the fundamental solution of the QDE of the Hilbert scheme. The GW/DT edge of the triangle in higher genus concerns new CohFTs defined by varying the 3-fold local curve in the moduli space of stable curves. The equivariant orbifold Gromov-Witten theory of the symmetric product of the plane is also shown to be equivalent to the theories of the triangle in all genera. The result establishes a complete case of the crepant resolution conjecture.

Using the mirror theorem [CCIT15], we give a Landau-Ginzburg mirror description for the big equivariant quantum cohomology of toric Deligne-Mumford stacks. More precisely, we prove that the big equivariant quantum D-module of a toric Deligne-Mumford stack is isomorphic to the Saito structure associated to the mirror Landau-Ginzburg potential. We give a GKZ-style presentation of the quantum D-module, and a combinatorial description of quantum cohomology as a quantum Stanley-Reisner ring. We establish the convergence of the mirror isomorphism and of quantum cohomology in the big and equivariant setting.

For each positive rational number ϵ, we define K-theoretic ϵ-stable quasimaps to certain GIT quotients $W\sslash G$. For ϵ>1, this recovers the K-theoretic Gromov-Witten theory of $W\sslash G$ introduced in more general context by Givental and Y.-P. Lee. For arbitrary ϵ1 and ϵ2 in different stability chambers, these K-theoretic quasimap invariants are expected to be related by wall-crossing formulas. We prove wall-crossing formulas for genus zero K-theoretic quasimap theory when the target $W\sslash G$ admits a torus action with isolated fixed points and isolated one-dimensional orbits.

We study Gromov-Witten theory of hypertoric Deligne-Mumford stacks from two points of view. From the viewpoint of representation theory, we calculate the operator of small quantum product by a divisor, following \cite{BMO}, \cite{MO}, \cite{MS}. From the viewpoint of Lawrence toric geometry, we compare Gromov-Witten invariants of a hypertoric Deligne-Mumford stack with those of its associated Lawrence toric stack.