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Type: Thesis
Title: Regularized equivariant Euler classes and gamma functions.
Author: Lu, Rongmin
Issue Date: 2008
School/Discipline: School of Mathematical Sciences : Pure Mathematics
Abstract: We consider the regularization of some equivariant Euler classes of certain infinite-dimensional vector bundles over a finite-dimensional manifold M using the framework of zeta-regularized products [35, 53, 59]. An example of such a regularization is the Atiyah–Witten regularization of the T-equivariant Euler class of the normal bundle v(TM) of M in the free loop space LM [2]. In this thesis, we propose a new regularization procedure — W-regularization — which can be shown to reduce to the Atiyah–Witten regularization when applied to the case of v(TM). This new regularization yields a new multiplicative genus (in the sense of Hirzebruch [26]) — the ^Γ-genus — when applied to the more general case of a complex spin vector bundle of complex rank ≥ 2 over M, as opposed to the case of the complexification of TM for the Atiyah–Witten regularization. Some of its properties are investigated and some tantalizing connections to other areas of mathematics are also discussed. We also consider the application of W-regularization to the regularization of T²- equivariant Euler classes associated to the case of the double free loop space LLM. We find that the theory of zeta-regularized products, as set out by Jorgenson–Lang [35], Quine et al [53] and Voros [59], amongst others, provides a good framework for comparing the regularizations that have been considered so far. In particular, it reveals relations between some of the genera that appeared in elliptic cohomology, allowing us to clarify and prove an assertion of Liu [44] on the ˆΘ-genus, as well as to recover the Witten genus. The ^Γ₂-genus, a new genus generated by a function based on Barnes’ double gamma function [5, 6], is also derived in a similar way to the ^Γ-genus.
Advisor: Varghese, Mathai
Buchdahl, Nicholas
Dissertation Note: Thesis (Ph.D.) - University of Adelaide, School of Mathematical Sciences, 2008
Subject: Loop spaces. Characteristic classes. Gamma functions.
Keywords: loop spaces; characteristic classes; equivariant; cohomology; zeta-function; regularization; double gamma function; elliptic genera
Appears in Collections:Research Theses

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