Seiberg-Witten Gauge Theory

235 Pages·1999·17.309 MB·English

by  Matilde Marcolli (auth.)

TEXTS AND READINGS 17
IN MATHEMATICS
Seiberg-Witten Gauge Theory
Texts and Readings in Mathematics
Advisory Editor
C. S. Seshadri, Chennai Mathematical Institute, Chennai.
Managing Editor
Rajendra Bhatia, Indian Statistical Institute, New Delhi.
Editors
R. B. Bapat, Indian Statistical Institute, New Delhi.
V. S. Borkar, Tata Inst. of Fundamental Research, Mumbai.
Probal Chaudhuri, Indian Statistical Institute, Kolkata.
V. S. Sunder, Inst. of Mathematical Sciences, Chennai.
M. Vanninathan, TIFR Centre, Bangalore.
Seiberg-Witten
Gauge Theory
Matilde Marcolli
Massachusetts Institute of Technology
Massachusetts
~HINDUSTAN
U ULJ UB OOK AGENCY
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ISBN 978-81-85931-22-7 ISBN 978-93-86279-00-2 (eBook)
DOI 10.1007/978-93-86279-00-2
Copyright © 1999, Hindustan Book Agency (India)
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ISBN 978-93-80250-23-6
Contents
1 Introduction 1
2 Seiberg-Witten on four-manifolds 9
2.1 Preliminary Notions . . . . . . . . . . . . . . 10
2.1.1 Clifford Algebras and Dirac Operators 10
2.1.2 Spin and Spine Structures . 13
2.1.3 Spinor Bundles . . . . . . . . . . . 14
2.1.4 Topology of the gauge group ... 15
2.1.5 Symplectic and Kiihler Manifolds. 15
2.1.6 The index theorem. . . 19
2.1.7 Equivariant cohomology 21
2.1.8 Sobolev norms . . . 24
2.1.9 Fredholm properties . . 26
2.1.10 Exercises ........ 27
2.2 The Functional and the Equations 29
2.2.1 The Equations . . . . . . . 29
2.2.2 The Gauge Group . . . . . 29
2.2.3 The Seiberg-Witten Functional and the Variational
Problem. . . . . . . . . . . . . . . . 30
2.2.4 Analytic properties of the functional 34
2.2.5 Exercises ..... 39
2.3 Invariants of 4-manifolds . 41
2.3.1 The Moduli Space 41
2.3.2 The Invariants . . 49
2.3.3 Finiteness..... 51
2.3.4 A Cobordism Argument 52
3 Seiberg-Witten on three-manifolds 57
3.1 Three-manifolds ...... 58
3.2 A three-manifold invariant . . . . . 60
vi CONTENTS
3.2.1 Dimensional reduction ........ . 60
3.2.2 The moduli space and the invariant . 62
3.2.3 Cobordism and wall crossing formulae 66
3.2.4 Casson invariant and Alexander polynomial 69
3.3 Seiberg-Witten Floer homology ...... . 77
3.3.1 The Chern-Simons-Dirac functional 78
3.3.2 Hessian and relative index . 80
3.3.3 Flow lines: asymptotics . 83
3.3.4 Flow lines: moduli spaces 86
3.3.5 Homology ........ . 90
3.3.6 The cobordism argument 94
3.3.7 Equivariant Floer homology and wall crossing . 98
3.3.8 Exact triangles . . . . . . . . . . . . . . . 103
3.3.9 The relation to instanton Floer homology 106
3.3.10 Summary 108
3.3.11 Exercises ... 108
4 Topology and Geometry 115
4.1 Computing Seiberg-Witten invariants 116
4.1.1 Connected Sum theorem. 116
4.1.2 The blowup formula . 116
4.1.3 Kiihler Surfaces . . . . . . 118
4.1.4 Symplectic Manifolds 122
4.1.5 Pseudo-holomorphic curves 127
4.1.6 Beyond the symplectic world 130
4.1. 7 Algebraic Surfaces . . . 132
4.2 Topology of embedded surfaces 135
4.2.1 The Thom Conjecture . 136
4.2.2 Contact structures ... 140
4.2.3 Three-manifolds: Thurston norm 143
4.3 Further applications 144
4.3.1 Exercises ..... . 145
5 Seiberg-Witten and Physics 153
5.1 Mathai-Quillen formalism . 154
5.1.1 The finite dimensional case 154
5.1.2 The infinite dimensional case 159
5.1.3 Euler numbers in Seiberg-Wit ten theory. 163
=
5.1.4 N 2 symmetry and the Euler characteristic 170
5.1.5 Quantum Field Theory and Floer homology . 175
5.1.6 Exercises ................... . 177
CONTENTS vii
5.2 Seiberg-Witten and Donaldson theory 179
5.2.1 The Physics way: S-duality 181
5.2.2 The Mathematics way ... 188
6 Appendix: a bibliographical guide 201
Chapter 1
Introduction
Where the fire is churned, where the wind wafts, wher'e the Soma juice
flows over -there the mind is born.
Svetasvatara Upanii?ad 2.61
In the fall of 1994 E. Witten announced a "new gauge theory of 4-
manifolds", capable of giving results analogous to the earlier theory of
Donaldson, but where the computations involved are "at least a thousand
times easier" (Taubes). The dawn of the new theory began with the
introduction of the monopole equations, whose roots lie in the depths of
the still rather mysterious notion of S-duality in N = 2 supersymmetric
Yang-Mills theory (IX: [59], [601f.
The equations are in terms of a section of a spinor bundle and a U (1)
connection on a line bundle L. The first equation just says that the
spinor section 1/J has to be in the kernel of the Dirac operator twisted
with a connection A. The second equation describes a relation between
the self-dual part of the curvature of the connection A and the section
1/J in terms of the Clifford action.
The mathematical setting for Witten's gauge theory is considerably
simpler than Donaldson's analogue: first of all it deals with connections
on a U(l)-principal bundle (hermitian line bundle) rather than on an
SU(2)-bundle, and the abelian structure group allows simpler calcula-
lThe quotes from the Upani~ads are from the Oxford University Press edition, in
the English translation of P. Olivelle.
2References listed in the Introduction refer to the bibliographical appendix.
2 CHAPTER 1. INTRODUCTION
tions; moreover the equation, which plays a role somehow analogous to
the previous anti-self-dual equation for SU(2)-instantons, involves Dirac
operators and Spine-structures, which are well known and long devel
oped mathematical tools.
The main differences between the two theories arise when it comes to
the properties of the moduli space of solutions of the monopole equation
up to gauge transformations. As was immediately observed, the moduli
space of Seiberg-Witten equations turns out to be always compact, es
sentially due to the Weitzenbock formula for the Dirac operator: a fact
that avoids the complicated analytic techniques that were needed for the
compactification of the moduli space of SU(2)-instantons.
In Witten's seminal paper (1:[19]) the monopole equation is intro
duced, and the main properties of the moduli space of solutions are
deduced. The dimension of the moduli space is computed by an index
theory technique, following an analogous proof for Donaldson's theory;
and the Seiberg-Witten invariant (which depends on the Chern class of
the line bundle L) is defined as the number of points, counted with ori
entation, in a zero-dimensional moduli space. The circumstances under
which the Seiberg-\Vitten invariants provide a topological invariant of
the four-manifold are illustrated in a similar way to the analogous result
regarding the Donaldson polynomials. Again as a consequence of the
Weitzenbock formula, the moduli space turns out to be empty for all
but finitely many choices of the line bundle L. Another advantage of
this theory is that the singularities of the moduli space only appear at
the trivial section 1jJ == 0, since elsewhere the action of the gauge group
is free. Thus, by perturbing the equation, it is possible to get a smooth
moduli space.
Shortly after the new equations were introduced and their properties
analysed, an impressIve series of new results were proven in the span of
a few months. As pointed out many times, the experts in the field had
all the right questions to ask, questions that were too difficult to attack
with the tools provided by Donaldson theory, and suddenly the new tool
of Seiberg-Witten theory became available. A very rapid advancement
in the field was thus possible.
The invariants were first computed for Kiihler manifolds, and then,
in extending the technique to the world of symplectic manifolds, Taubes
uncovered a deep relation between Seiberg-Wit ten invariant and pseudo
holoIllorphic curves. The non-vanishing results for Seiberg-Witten in
variants of symplectic manifolds led to conjecture that these constitute
the most "basic" (indecomposable) kind of four-manifolds. The con
jecture was later disproved, again by means of Seiberg-Witten theory,