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The Heat Kernel Lefschetz Fixed Point Formula for the Spin-c Dirac Operator

Part of the Modern Birkhäuser Classics Book Series

DOI: 10.1007/978-0-8176-8247-7

ISBN: 978-0-8176-8246-0

Publisher: Birkhäuser Boston - Springer Journals

Jan 1, 2011

1 - 10 of 16 Chapters

LitStream

Chapter 1Introduction

Duistermaat, J. J.

Pages: 1 –5

[Let M be an almost complex manifold of real dimension 2n, provided with a Hermitian structure. Furthermore, let L be a complex vector bundle over M, provided with a Hermitian connection. We also assume that K*, the dual bundle of the so-called canonical line bundle K of M, is provided with a...
LitStream

Chapter 2The Dolbeault-Dirac Operator

Duistermaat, J. J.

Pages: 7 –17

[In this chapter we set the stage, by introducing complex and almost structures, the Dolbeault complex and Hermitian structures. The holomorphic Lefschetz number, defined as the alternating sum of the trace of the automorphism acting on the cohomology of the sheaf of holomorphic sections, will be...
LitStream

Chapter 3Clifford Modules

Duistermaat, J. J.

Pages: 19 –33

[In general, if the manifold is not Kähler, then the Dolbeault-Dirac operator D = 2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek}...
LitStream

Chapter 4The Spin Group and the Spin-c Group

Duistermaat, J. J.

Pages: 35 –40

[We begin the definition of the group Spinc(V), introduced by Atiyah-BottShapiro [4], with the definition of the slightly smaller spin group Spin(V).]
LitStream

Chapter 5The Spin-c Dirac Operator

Duistermaat, J. J.

Pages: 41 –51

[In this chapter we start by viewing E as a principal bundle for the group Spinc (2n), which contains the unitary group. The fact that Spinc (2n) also contains the spin group Spin(2n), which is a twofold cover of SO(2n), allows us to introduce a connection in this principal bundle which has the...
LitStream

Chapter 6Its Square

Duistermaat, J. J.

Pages: 53 –68

[The main goal of this chapter is Theorem 6.1, which says that the square of the spin-c Dirac operator D is equal to the Laplace operator plus a zero order term, given by curvature expressions. The contribution from the curvature of L will be responsible for the Chern characters ch (Lj) in...
LitStream

Chapter 7The Heat Kernel Method

Duistermaat, J. J.

Pages: 69 –75

[The goal of the heat kernel method is to express (2.40) as an integral over the fixed point set Mγ in M of the transformation γ. Here Mγ = M if γ is the identity. The method is based on the following observations about arbitrary elliptic differential operators D, acting on sections of a smooth...
LitStream

Chapter 8The Heat Kernel Expansion

Duistermaat, J. J.

Pages: 77 –97

[Our operator Q = D2, the square of the spin-c Dirac operator, has scalar principal symbol. So for the discussion of the asymptotic expansion of its heat kernel, we may restrict ourselves to the case that Q is a second order differential operator, acting on sections of a complex vector bundle F...
LitStream

Chapter 9The Heat Kernel on a Principal Bundle

Duistermaat, J. J.

Pages: 99 –115

[We now use that the operator Q = D2 acts on sections of the vector bundle E ⊗ L, where E is the bundle (5.7), associated to the principal SpinC (2n)bundle Spinc F M. The covariant differentiation of sections of this bundle is defined by (3.45).]
LitStream

Chapter 10The Automorphism

Duistermaat, J. J.

Pages: 117 –129

[The goal here is Theorem 10.1 below, which is the modification of Theorem 9.1 in the presence of an automorphism γ. The only contributions to the asymptotic expansion come from neighborhoods of fixed points in M ofγ, which in the principal G-bundle P correspond to fixed points modulo the action...

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