In this paper, we establish an equivariant version of Dai-Zhang's Toeplitz index theorem for compact odd-dimensional spin manifolds with even-dimensional boundary.

Abstract Consider a proper, isometric action by unimodular locally compact group $G$ on Riemannian manifold $M$ with boundary, such that $M/G$ is compact. For an equivariant, elliptic operator $D$ $M$, and element $g \in G$, we define numerical index ${\operatorname {index}}_g(D)$, in terms of parametrix for trace associated to $g$. We prove equivariant Atiyah–Patodi–Singer theorem this index. ...

Let D be a (generalized) Dirac operator on a non-compact complete Riemannian manifold M acted on by a compact Lie group G. Let v : M → g = LieG be an equivariant map, such that the corresponding vector field on M does not vanish outside of a compact subset. These data define an element of K-theory of the transversal cotangent bundle to M . Hence, by embedding of M into a compact manifold, one c...

The main purpose of this article is to extend some of the ideas from Schubert calculus to the more general setting of Hamiltonian torus actions on compact symplectic manifolds with isolated fixed points. Given a generic component Ψ of the moment map, which is a Morse function, we define a canonical class αp in the equivariant cohomology of the manifold M for each fixed point p ∈ M. When they ex...

We study complex Chern-Simons theory on a Seifert manifold M3 by embedding it into string theory. We show that complex Chern-Simons theory on M3 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 dimensional reduction of this theory to 2d gives the low energy dynamics of vortices...