The $\hatΓ$-genus and a regularization of an $S^1$-equivariant Euler class

An Analytical Approach to the equivariant index and Witten genus on spin manifolds

This work is divide in two cases. In the first case, we consider a spin manifold $M$ as the set of fixed points of an $S^{1}$-action on a spin manifold $X$, and in the second case we consider the spin manifold $M$ as the set of fixed points of an $S^{1}$-action on the loop space of $M$. For each case, we build on $M$ a vector bundle, a connection and a set of bundle endomorphisms. These objects are used to build global operators on $M$ which define an analytical index in each case. In the first case, the analytical index is equal to the topological equivariant Atiyah Singer index, and in the s…

On Equivariant Gromov--Witten Invariants of Resolved Conifold with Diagonal and Anti-Diagonal Actions

We propose two conjectural relationships between the equivariant Gromov-Witten invariants of the resolved conifold under diagonal and anti-diagonal actions and the Gromov-Witten invariants of $\mathbb{P}^1$, and verify their validity in genus zero approximation. We also provide evidences to support the validity of these relationships in genus one and genus two.

所有亏格的开镜像对称性对于射影线

All genus open mirror symmetry for the projective line

本文证明了射影线的镜像曲线递归关系编码了所有亏格的开Gromov-Witten不变量，揭示了其等效开镜像对称性。

The (anti-)holomorphic sector in $\mathbb{C}/Λ$-equivariant cohomology, and the Witten class

Atiyah's classical work on circular symmetry and stationary phase shows how the $\hat{A}$-genus is obtained by formally applying the equivariant cohomology localization formula to the loop space of a simply connected spin manifold. The same technique, applied to a suitable ''antiholomorphic sector'' in the $\mathbb{C}/Λ$-equivariant cohomology of the conformal double loop space $\mathrm{Maps}(\mathbb{C}/Λ,X)$ of a rationally string manifold $X$ produces the Witten genus of $X$. This can be seen as an equivariant localization counterpart to Berwick-Evans supersymmetric localization derivation o…

Supersymmetric localization, modularity and the Witten genus

Equivariant localization techniques give a rigorous interpretation of the Witten genus as an integral over the double loop space. This provides a geometric explanation for its modularity properties. It also reveals an interplay between the geometry of double loop spaces and complex analytic elliptic cohomology. In particular, we identify a candidate target for the elliptic Bismut-Chern character.

Givental's Lagrangian Cone and S^1-Equivariant Gromov-Witten Theory

In the approach to Gromov-Witten theory developed by Givental, genus-zero Gromov-Witten invariants of a manifold X are encoded by a Lagrangian cone in a certain infinite-dimensional symplectic vector space. We give a construction of this cone, in the spirit of S^1-equivariant Floer theory, in terms of S^1-equivariant Gromov-Witten theory of the product X \times P^1. This gives a conceptual understanding of the "dilaton shift": a change-of-variables which plays an essential role in Givental's theory.