Universiteit van Amsterdam Extended Topological Gauge Theories in Codimension Zero and Higher

178 Pages·2010·1.09 MB·English

Universiteit van Amsterdam
Instituut voor Theoretische Fysica
Master’s Thesis
Extended Topological Gauge Theories
in Codimension Zero and Higher
Author: Supervisor:
Kevin Wray Prof. Dr. Robbert Dijkgraaf
May 26, 2010
(cid:176)c Kevin Wray
All rights reserved.
Author Thesis advisor
Kevin Wray Prof. dr. Robbert Dijkgraaf
Extended Topological Gauge Theories in Codimension Zero and Higher
Abstract
Topological field theories (TFT) have been extensively studied by physicists and mathe-
maticians ever since the seminal paper by Edward Witten. Not only can these theories
lead to new insights in the world of low-dimensional topology, but they have also been used
to better comprehend the mysterious structure of the WZW models and the (fractional)
quantum Hall effects, while at the same time alleviating the arduous lives lead by string
theorists. The mathematical structure of a topological field theory was completely laid
out by Atiyah with his “axioms of a TFT,” and later modernized into the “symmetric,
monoidal functor from the category of cobordisms to the category of vector spaces” which
we are most familiar. Although a beautiful theory, there are drawbacks to the definition of
a TFT set forth by Atiyah. Namely, with the Atiyah-type TFT, one can only talk about
the TFT living on manifolds of at most codimension one. Hence, there is no notion of the
action of an n-dimensional TFT on a (n−2)-submanifold. Most recently, there has been
an entire legion of mathematical physicists publishing copious amounts of research towards
developing the theory of extended TFT’s. These TFT’s can live on manifolds of any arbi-
trary codimension. Extended TFT’s have also found their way into the study of quantum
gravity; notable via the paper of Morton and Baez. The purpose of this paper is to lead
the reader from the usual notions of TFT’s all the way to extended TFT’s, while covering
the relevant mathematical and physical structures arising in between. We begin with a re-
view (or introduction, depending on the readers current knowledge) of all the mathematical
concepts required to study such theories; namely, category theory, (co)homological algebra,
principal bundles, connections and characteristic classes. Following this review, we then
introduce the classical 3-dimensional Chern-Simons theory with compact gauge group. We
then restrict to the case of a finite gauge group (also known as Dijkgraaf-Witten theories),
which are far simpler to rigorously quantize. Finally, we introduce the extended Dijkgraaf-
Wittentheoryandshowhow,underquantization,itleadstohighercategories. Weconclude
by explicitly carrying out several calculations of the quantum invariants associated, by the
quantum theory, to specific manifolds of varying codimension.
iii
Acknowledgments
First, I must thank my parents and the HSP Huygens scholarship from Nuffic
for, financially, making it possible to spend two years studying here at the Uni-
versiteit van Amsterdam. Also, I would like to thank ma belle Reine for her
encouragement to continue when times were difficult. I would like to thank my
advisor,ProfessorRobbertDijkgraaf,forhisabilitytoanswereveryquestion,no
matter how ill thought, with great clarity and for his comments on the direction
I have chosen for my thesis. Next, I would like to thank Dr. Kenneth Flurchick
for his willingness to talk about physics and mathematics and for helping spark
the love and appreciation that I have for these two sciences. Earl Hampton for
hisnever-endingquestforknowledgeandhiswonderfuldiscussionsofnewideas.
Lastly, I would like to thank Professors Jan de Boer and Nicolai Reshetikhin for
their insightful discussions, along with Professors Dan Freed and John Baez for
their invaluable email correspondences, and Urs Schreiber for his explanation of
higher algebraic structures.
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Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Acknowledgments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
1 Introduction/Overview 1
I MATHEMATICAL BACKGROUND 3
2 Category Theory (Abstract Nonsense) 4
2.1 Categories and Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Natural Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Additional Structure: Symmetric Monoidal Categories and Functors . . . . 6
2.4 Higher Category Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 (Co)Homological Algebra: Homology and Cohomology 11
3.1 Chain Complexes and Homology . . . . . . . . . . . . . . . . . . . . . . . . 11
3.1.1 Exact Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.1.2 Singular Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.1.3 Fundamental Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 Cochain Complexes and Their Cohomology . . . . . . . . . . . . . . . . . . 20
3.2.1 Baby Steps: The de Rham Cohomology . . . . . . . . . . . . . . . . 20
3.2.2 Exact Sequences and Cohomology . . . . . . . . . . . . . . . . . . . 22
3.2.3 Singular Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3 Torsion and the Universal Coefficient Theorem . . . . . . . . . . . . . . . . 26
3.4 Derived Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4 Fibre Bundles 31
4.1 Motivation (“Glued Fibre Bundles”) . . . . . . . . . . . . . . . . . . . . . . 31
4.2 (Standard) Fibre Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.3 Associated Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.4 Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.5 Bundle Maps and Gauge Transformations . . . . . . . . . . . . . . . . . . . 40
5 Connections on Fibre Bundles 43
5.1 Ehresmann Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.2 Connection Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
v
Contents vi
5.2.1 Intermezzo: Vector-Valued Forms . . . . . . . . . . . . . . . . . . . . 47
5.3 Curvature Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.3.1 Universal Connections and Curvatures . . . . . . . . . . . . . . . . . 56
5.4 Gauge Potentials and Field Strengths . . . . . . . . . . . . . . . . . . . . . 57
6 Characteristic Classes 60
6.1 Motivation (Classification of U(1)-Bundles) . . . . . . . . . . . . . . . . . . 60
6.2 Invariant Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6.3 The Chern-Weil Homomorphism . . . . . . . . . . . . . . . . . . . . . . . . 66
6.4 Characteristic Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.5 Universal Characteristic Classes . . . . . . . . . . . . . . . . . . . . . . . . . 71
II TOPOLOGICAL FIELD THEORIES 73
7 Classical Chern-Simons Theory 74
7.1 Topological Field Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
7.2 Chern-Simons Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
7.3 Chern-Simons Action (Trivial Bundle) . . . . . . . . . . . . . . . . . . . . . 76
7.4 Classical Field Theory Construction (a.k.a. Pre-Quantization) . . . . . . . . 82
7.4.1 Invariant Section Construction . . . . . . . . . . . . . . . . . . . . . 83
7.5 Chern-Simons Action (General Theory) . . . . . . . . . . . . . . . . . . . . 88
7.5.1 Warm up (somewhat less-general case) . . . . . . . . . . . . . . . . . 88
7.5.2 Group Cohomology and the Chern-Simons Form . . . . . . . . . . . 89
7.5.3 General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7.6 Quantization: A Primer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
8 Topological Quantum Field Theories (TQFTs) 103
8.1 Atiyah’s Axiomatic Definition of a TQFT . . . . . . . . . . . . . . . . . . . 103
8.2 Invariant Section Construction of a TQFT . . . . . . . . . . . . . . . . . . . 106
8.3 Dijkgraaf-Witten Theory (Chern-Simons Theory with Finite Group) . . . . 108
8.3.1 Classical Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
8.3.2 Quantum Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
8.3.3 Untwisted Theories ([α] = 0) . . . . . . . . . . . . . . . . . . . . . . 120
8.3.4 Twisted Theories ([α] (cid:54)= 0) . . . . . . . . . . . . . . . . . . . . . . . 123
III EXTENDED TOPOLOGICAL FIELD THEORIES 127
9 Extended Dijkgraaf-Witten Theories 128
9.1 Extended (or Multi-Tiered) TQFTs . . . . . . . . . . . . . . . . . . . . . . 129
9.2 Classical Extended Dijkgraaf-Witten Theories . . . . . . . . . . . . . . . . . 130
9.2.1 G-Torsors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
9.2.2 G-Gerbes (G2-Torsors). . . . . . . . . . . . . . . . . . . . . . . . . . 134
9.2.3 Action of a Group on Torsors and Gerbes . . . . . . . . . . . . . . . 135
Contents vii
9.2.4 Classical Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
9.3 Quantum Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
9.3.1 (Higher) Inner Product Spaces . . . . . . . . . . . . . . . . . . . . . 143
9.3.2 Quantization of the Extended Dijkgraaf-Witten Theory . . . . . . . 148
9.4 Tying Everything Together: Example with Γ =Z . . . . . . . . . . . . . . 154
2
10 Conclusion 167
Bibliography 169
Chapter 1
Introduction/Overview
A topological quantum field theory (TQFT) is a metric independent quantum
field theory that gives rise to topological invariants of the background manifold. That is,
a TQFT is a background-free quantum theory with no local degrees of freedom. TQFT’s
have been the main source for the interaction between physics and mathematics for the
past 25 years. For example mathematicians are interested in topological theories because
of the knot invariants they produce, while physicists are interested in topological theories
because they are, in a sense, the simplest examples of quantum field theories which are
exactly solvable and generally covariant. Although, in general, one can define TQFT’s in
any arbitrary dimension, most of the research currently being conducted is restricted to the
three dimensional Chern-Simons theory. Partly because Witten was able to show that the
expectation value of an observable (obtained as the product of the Wilson loops associated
with a link) gives the generalized Jones invariant of the link, and partly because of its
implications in 3-dimensional gravity.
Thepurposeofthisthesisistocompletelyworkoutthedetails, forthefinitestruc-
ture group case, required to construct an extended TQFT, all the way down to points. This
will require the introduction of many concepts from mathematics. In particular, we begin
with an overview of category theory, algebraic topology, and the theory of characteristic
classes on principal G-bundles. Following this discussion, we introduce the classical Chern-
π
Simons theory on trivial principal bundles over a 3-dimensional manifold, G (cid:44)→ P −→ M.
Here we take, for the Lagrangian, the pullback of the antiderivative of the Chern-Weil 4-
form α associated to a connection ω on P - this form α is also known as the Chern-Simons
3-form - via the section s : M → P. The classical action is then defined by integrating α
over the moduli space of connections
(cid:90)
k
S = s∗(α(ω)).
8π2
A/G
Furthermore, we show that this action is gauge-invariant when defined over closed 3-
manifolds (up to an integer), while on a compact manifold with boundary, a gauge transfor-
mation effects the action by the addition of a WZW term. Finally, we derive the expression
for the classical action in the case where the principal bundle is not trivial.
Following the discussion of the classical Chern-Simons theory, we then begin the
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Chapter 1: Introduction/Overview 2
study of the 3-dimensional Dijkgraaf-Witten theory (Chern-Simons theory with a finite
gauge group Γ). Here, rather than integrating the Chern-Simons 3-form, our classical
action is given an element in the degree 4 cohomology class [α] ∈ H4(BΓ;Z)
S = e2πi(cid:104)γM∗ ([α]),[M](cid:105).
Once the classical action has been defined, we define the quantum theory (i.e., partition
function) by summing this action over the moduli space of flat Γ-bundles over manifolds of
dimension three and two. We then show that this definition of the path integral obeys the
axioms,setforthbyAtiyah,definingaTQFT.Thatis,thepathintegraldefinesasymmetric
monoidalfunctorfromthecategoryofcobordismstothecategoryofvectorspaces-toclosed
2-dimensionalmanifoldsitassignsavectorspace, whiletocompact3-dimensionalmanifolds
it assigns an element in the vector space(s) associated to its boundary.
Finally, we show how to extend the classical action and the path integral to in-
corporate manifolds of codimensions higher than one. This will require us to define torsors
and gerbes, as well as higher Hilbert spaces and higher categories. We end this section, and
the thesis, by performing several explicit calculations, and we show that by assigning the
category of vector bundles over Γ to the point gives an extended TQFT down to points,
thus fulfilling our beginning objective.
Part I
MATHEMATICAL
BACKGROUND
3