arXiv research

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 quantum D-module of a symplectic manifold is isomorphic to the equivariant Floer cohom…

We use virtual neighborhood technique to establish GW-invariants, Quantum cohomology, equivariant GW-invariants, equivariant quantum cohomology and Floer cohomology for general symplectic manifold. We also establish GW-invariants for a family of symplectic manifolds. As a consequence, we prove Arnold conjecture for non…

Using mirror symmetry as described by Hori and Vafa, we compute the quantum equivariant cohomology ring of toric manifolds. This ring arises naturally in topological gauged sigma-models and is related to the Hamiltonian Gromov-Witten invariants of the target manifold.

Derives localization formulas in Batalin-Vilkovisky formalism.

We define quantum exterior product wedge_h and quantum exterior differential d_h on Poisson manifolds (of which symplectic manifolds are an important class of examples). Quantum de Rham cohomology, which is a deformation quantization of de Rham cohomology, is defined as the cohomology of d_h. We also define quantum Dol…

We define quantum exterior product wedge_h and quantum exterior differential d_h on Poisson manifolds, of which symplectic manifolds are an important class of examples. Quantum de Rham cohomology is defined as the cohomology of d_h. We also define quantum Dolbeault cohomology. Quantum hard Lefschetz theorem is proved. …

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 c_1(M) of the tangent bundle is nef. In this paper, even when c_1(M) is not nef, w…

We develop a Chern character map for twisted equivariant non-abelian cohomology.

Refines quantum annular homology using stable homotopy methods.

Study of 2d gauged linear sigma models to derive difference equations and spectral data.

We investigate an equivariant generalization of Morse theory for a general class of integrable models. In particular, we derive equivariant versions of the classical Poincaré-Hopf and Gauss-Bonnet-Chern theorems and present the corresponding path integral generalizations. Our approach is based on equivariant cohomology…

We present an equivariant extension of the Thom form with respect to a vector field action, in the framework of the Mathai-Quillen formalism. The associated Topological Quantum Field Theories correspond to twisted $N=2$ supersymmetric theories with a central charge. We analyze in detail two different cases: topological…

We construct a global geometric model for complex analytic equivariant elliptic cohomology for all compact Lie groups. Cocycles are specified by functions on the space of fields of the two-dimensional sigma model with background gauge fields and $\mathcal{N} = (0, 1)$ supersymmetry. We also consider a theory of free fe…

In this short note we prove an equivariant version of the formality of multidiffirential operators for a proper Lie group action. More precisely, we show that the equivariant Hochschild-Kostant-Rosenberg quasi-isomorphism between the cohomology of the equivariant multidifferential operators and the complex of equivaria…

We construct smooth concordance invariants of knots which take the form of piecewise linear maps from [0,1] to R, one for each n greater than or equal to 2. These invariants arise from sl(n) knot cohomology. We verify some properties which are analogous to those of the invariant Upsilon (which arises from knot Floer ho…

We exploit the Fedosov-Weinstein-Xu (FWX) resolution proposed in q-alg/9709043 to establish an isomorphism between the ring of Hochschild cohomology of the quantum algebra of functions on a symplectic manifold M and the ring H(M, C((h))) of De Rham cohomology of M with the coefficient field C((h)) without making use of…

New proof for 4D symplectic manifolds: equivariant cohomology determines diffeotype.

On the basis of Brylinski's work, we introduce a notion of equivariant smooth Deligne cohomology group, which is a generalization of both the ordinary smooth Deligne cohomology and the ordinary equivariant cohomology. Using the cohomology group, we classify equivariant circle bundles with connection, and equivariant ge…

We study complex Chern-Simons theory on a Seifert manifold $M_3$ by embedding it into string theory. We show that complex Chern-Simons theory on $M_3$ is equivalent to a topologically twisted supersymmetric theory and its partition function can be naturally regularized by turning on a mass parameter. We find that the d…

For a compact Lie group acting on a smooth manifold, we define the differential cohomology of a certain quotient stack involving principal bundles with connection. This produces differential equivariant cohomology groups that map to the Cartan-Weil equivariant forms and to Borel's equivariant integral cohomology. We sh…

Paper develops equivariant basic cohomology for Lie groupoids.

Study Berry connections for 2d GLSMs, linking to cohomology theories.

Constructs equivariant cohomology models for differentiable stacks.

This is the third of a series of papers on a new equivariant cohomology that takes values in a vertex algebra, and contains and generalizes the classical equivariant cohomology of a manifold with a Lie group action a la H. Cartan. In this paper, we compute this cohomology for spheres and show that for any simple connec…

We describe the Cartan and Weil models of twisted equivariant cohomology together with the Cartan homomorphism among the two, and we extend the Chern-Weil homomorphism to the twisted equivariant cohomology. We clarify that in order to have a cohomology theory, the coefficients of the twisted equivariant cohomology must…

We construct for an equivariant cohomology theory for proper equivariant CW-complexes an equivariant Chern character, provided that certain conditions about the coefficients are satisfied. These conditions are fulfilled if the coefficients of the equivariant cohomology theory possess a Mackey structure. Such a structur…

Study stabilizes Morse-Bott cohomology for equivariant manifolds.

Develops a framework for designing quantum neural networks that respect symmetries.

Unified approach to invariants in equivariant geometry.

Develops equivariant Seiberg-Witten-Floer cohomology for 3-spheres.

This paper introduces persistent equivariant cohomology and applies it to circle actions.

The equivariant cohomology of a space with a group action is not only a ring but also an algebra over the cohomology ring of the classifying space of the acting group. We prove that toric manifolds (i.e. compact smooth toric varieties) are isomorphic as varieties if and only if their equivariant cohomology algebras are…

We construct a new equivariant cohomology theory for a certain class of differential vertex algebras, which we call the chiral equivariant cohomology. A principal example of a differential vertex algebra in this class is the chiral de Rham complex of Malikov-Schechtman-Vaintrob of a manifold with a group action. The ma…

Quantum cohomology connects quantum physics with classical math.

The Witten class is derived from equivariant cohomology of a conformal loop space.

2d GLSM connects Berry connections to Coulomb branch via difference equations.

Equivariant cohomology simplifies symplectic manifold integrals with group actions.

This is a survey on the equivariant cohomology of Lie group actions on manifolds, from the point of view of de Rham theory. Emphasis is put on the notion of equivariant formality, as well as on applications to ordinary cohomology and to fixed points.

We introduce and study a new spectral sequence associated with a Poisson group action on a Poisson manifold and an equivariant momentum mapping. This spectral sequence is a Poisson analog of the Leray spectral sequence of a fibration. The spectral sequence converges to the Poisson cohomology of the manifold and has the…

New neural networks for non-commutative data.

The equivariant cohomology ring of a GKM manifold is isomorphic to the cohomology ring of its GKM graph. In this paper we explore the implications of this fact for equivariant fiber bundles for which the total space and the base space are both GKM and derive a graph theoretical version of the Leray-Hirsch theorem. Then…

Enhances quantum computing for symmetrical systems, proving a new class of problems.

Explains model structures for higher orbifolds and applies them to quantum cohomology.

New equivariant version of Khovanov homology for annuli.

Unified theory of orbifolds and cohomology.

Relates quantum cohomology to tt*-Toda equations for minuscule flag manifolds.

Explains quantum cohomology of Grassmannians using tt* equations.

Spin networks boost quantum algorithms solving SU(2) symmetric problems.