The Duistermaat-Heckman formula with application to circle actions and Poincar\'{e} $q$-polynomials in twisted equivariant K-theory

TL;DR

This paper explores the Duistermaat-Heckman formula's proof sketch, its specialization to symplectic orbit spaces, and its applications to elliptic genera, Poincaré q-polynomials, and twisted equivariant K-theory, linking geometry with spectral functions.

Contribution

It provides a detailed analysis of the Duistermaat-Heckman formula's proof and applies it to circle actions and Poincaré q-polynomials in twisted equivariant K-theory, including spectral function representations.

Findings

Derivation of the Duistermaat-Heckman formula proof sketch

Application to elliptic genera and characteristic q-series

Representation of Poincaré q-polynomials via spectral functions

Abstract

In this paper we deduce the sketch of proof of the Duistermaat-Heckman formula and investigate how the known Duistermaat-Heckman result could be specialized to the symplectic structure on the orbit space. The theorems of localization in equivariant cohomology not only provide us with beautiful mathematical formulas and stimulate achievements in algorithmic computations, but also promote progress in theoretical and mathematical physics. We present the elliptic genera and the characteristic $q$-series for the circle actions and twisted equivariant K-theory, with the case of the symmetric group of $n$ symbols separately analyzed. We show that the Poincar\'{e} $q$-polynomials admit presentation in terms of the Patterson-Selberg (or the Ruelle-type) spectral functions.