相关论文: Fourier analysis of equivariant quantum cohomology

We prove a decomposition theorem of the quantum cohomology D-module of the blowup of a smooth projective variety X along a smooth subvariety Z. The main tools we use are shift operators and Fourier analysis for equivariant quantum…

The objective of this paper is to clarify the relationships between the quantum D-module and equivariant Floer theory. Equivariant Floer theory was introduced by Givental in his paper ``Homological Geometry''. He conjectured that the…

Let $\mathcal X=[(\mathbb C^r\setminus Z)/G]$ be a toric Fano orbifold. We compute the Fourier transform of the $G$-equivariant quantum cohomology central charge of any $G$-equivariant line bundle on $\mathbb C^r$ with respect to certain…

We construct shift operators on equivariant symplectic cohomology which generalise the shift operators on equivariant quantum cohomology in algebraic geometry. That is, given a Hamiltonian action of the torus $T$, we assign to a cocharacter…

In the previous paper, the author defined equivariant Floer cohomology for a complete intersection in a toric variety and showed that it is isomorphic to the small quantum D-module after a mirror transformation when the first Chern class…

In this paper we identify the problem of equivariant vortex counting in a $(2,2)$ supersymmetric two dimensional quiver gauged linear sigma model with that of computing the equivariant Gromov-Witten invariants of the GIT quotient target…

In an earlier paper we conjectured a relation between the quantum $\mathcal D$-modules of a smooth variety $X$ and the projectivisation of a direct sum of line bundles over it. In this paper we prove the conjecture when $X$ is a complete…

Let X be a "nice" space with an action of a torus T. We consider the Atiyah-Bredon sequence of equivariant cohomology modules arising from the filtration of X by orbit dimension. We show that a front piece of this sequence is exact if and…

We propose a new point of view on quantum cohomology, strongly motivated by the work of Givental and Dubrovin, but closer to differential geometry than the existing approaches. The central object is the D-module which "quantizes" a…

In quantum mechanics, the momentum space and position space wave functions are related by the Fourier transform. We investigate how the Fourier transform arises in the context of geometric quantization. We consider a Hilbert space bundle H…

We use Floer theory to describe invariants of symplectic $\mathbb{C}^*$-manifolds admitting several commuting $\mathbb{C}^*$-actions. The $\mathbb{C}^*$-actions induce filtrations by ideals on quantum cohomology, as well as filtrations on…

We consider generalizations of equivariant volumes of abelian GIT quotients obtained as partition functions of 1d, 2d, and 3d supersymmetric GLSM on $S^1$, $D^2$ and $D^2 \times S^1$, respectively. We define these objects and study their…

We prove a decomposition theorem for the quantum cohomology of variations of GIT quotients. More precisely, for any reductive group $G$ and a simple $G$-VGIT wall-crossing $X_- \dashrightarrow X_+$ with a wall $S$, we show that the quantum…

A strong version of the quantization conjecture of Guillemin and Sternberg is proved. For a reductive group action on a smooth, compact, polarized variety (X,L), the cohomologies of L over the GIT quotient X // G equal the invariant part of…

We use differential cohomology to systematically construct a large class of topological actions in physics, including Chern-Simons terms, Wess-Zumino-Novikov-Witten terms, and theta terms (continuous or discrete). We introduce a notion of…

For $X$ a smooth projective variety, the quantum cohomology ring $QH^*(X)$ is a deformation of the usual cohomology ring $H^*(X)$, where the product structure is modified to incorporate quantum corrections. These correction terms are…

We first retell in the K-theoretic context the heuristics of $S^1$-equivariant Floer theory on loop spaces which gives rise to $D_q$-module structures, and in the case of toric manifolds, vector bundles, or super-bundles to their explicit…

In this paper, we study the Floer theory of equivariant Lagrangian correspondences and apply it to derive precise relations between the disc potential of an invariant Lagrangian submanifold and that of its quotient, thereby addressing a…

Let $G$ be a compact, connected Lie group and $T \subset G$ a maximal torus. Let $(M,\omega)$ be a monotone closed symplectic manifold equipped with a Hamiltonian action of $G$. We construct a module action of the affine nil-Hecke algebra…

Givental's $K$-theoretical $J$-function can be used to reconstruct genus zero $K$-theoretical Gromov--Witten invariants. We view this function as a fundamental solution of a $q$-difference system. In the case of projective spaces, we show…