On unitary 2-representations of finite groups and topological quantum field theory

3.3 MB·English

by  Bruce Bartlett

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n On unitary 2-representations of (cid:28)nite groups and
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J topological quantum (cid:28)eld theory

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Bruce Bartlett
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m A thesis submitted in ful(cid:28)lment of the requirements
[ for the degree of Doctor of Philosophy

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v University of She(cid:30)eld
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Department of Pure Mathematics
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9 Oct 2008
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Abstract
This thesis contains various results on unitary 2-representations of (cid:28)nite
groups and their 2-characters, as well as on pivotal structures for fusion
categories. The motivation is extended topological quantum (cid:28)eld theory
(TQFT), where the 2-category of unitary 2-representations of a (cid:28)nite group
is thought of as the ‘2-category assigned to the point’ in the untwisted (cid:28)nite
group model.
The (cid:28)rst result is that the braided monoidal category of transformations
oftheidentityonthe2-categoryofunitary2-representationsofa(cid:28)nitegroup
computes as the category of conjugation equivariant vector bundles over the
groupequippedwiththefusiontensorproduct. Thisresultisconsistentwith
the extended TQFT hypotheses of Baez and Dolan, since it establishes that
the category assigned to the circle can be obtained as the ‘higher trace of
the identity’ of the 2-category assigned to the point.
The second result is about 2-characters of 2-representations, a concept
which has been introduced independently by Ganter and Kapranov. It
is shown that the 2-character of a unitary 2-representation can be made
functorial with respect to morphisms of 2-representations, and that in fact
the 2-character is a unitarily fully faithful functor from the complexi(cid:28)ed
Grothendieck category of unitary 2-representations to the category of uni-
tary equivariant vector bundles over the group.
The (cid:28)nal result is about pivotal structures on fusion categories, with
a view towards a conjecture made by Etingof, Nikshych and Ostrik. It
is shown that a pivotal structure on a fusion category cannot exist unless
certain involutions on the hom-sets are plus or minus the identity map, in
whichcaseapivotalstructureisthesamethingasatwistedmonoidalnatural
transformation of the identity functor on the category. Moreover the pivotal
structure can be made spherical if and only if these signs can be removed.
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Dedication
To mom and dad, granny and gramps, nanny and pa!
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Acknowledgements
Firstly my heartfelt thanks goes to my supervisor Simon Willerton whom I
had the great fortune of learning so many ideas from these last three years;
he has been my mentor and my friend. I wish to thank all my fellow Houses
of Maths inmates for the good times that we had, and all the postgrads,
present and past, the young ’uns and the old guns.
As far as work is concerned, I wish to especially thank the ‘locals’ David
Gepner,JamesCranch,PaulBuckingham,AlmarKaid,DaveStern,Eugenia
Cheng, Tom Bridgeland, Vic Snaith and Richard Hepworth for stimulating
conversations and for helping me at various points during this thesis.
I would like to acknowledge Michael Shulman , Nick Gurski and Tom
Leinster who have helped me with various categorical notions. I am also
grateful to Scott Carter for patiently explaining to me the movie moves
needed in Appendix C.
I am sincerely grateful to John Baez whose column ‘This Week’s Finds’
has had such a profound e(cid:27)ect on me, and to Urs Schreiber, whom I count
myself very lucky to know.
I would also like to thank Jamie Vicary, Mathieu Anel, Frank Neumann,
Hellen Colman and Steve Lack.
Finally, I would like to acknowledge the support of the Excellence Ex-
change Scheme, as well as the Glasgow Mathematical Journal Trust.
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Contents
Abstract 3
Dedication 5
Acknowledgements 7
1 Introduction 13
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2 Verifying Z(S1) (cid:39) Dim Z(pt) in the (cid:28)nite group model . . . . 21
1.3 The 2-character functor is unitarily fully faithful . . . . . . . 22
1.4 Characterizing pivotal structures on fusion categories . . . . . 25
1.5 Comparison with previous work . . . . . . . . . . . . . . . . . 27
1.6 Overview of thesis . . . . . . . . . . . . . . . . . . . . . . . . 28
2 Geometry of ordinary representations 31
2.1 Holomorphic hermitian line bundles and kernels . . . . . . . . 32
2.2 The Bergman kernel . . . . . . . . . . . . . . . . . . . . . . . 34
2.2.1 The Bergman kernel via modes of propagation . . . . 34
2.2.2 The Bergman kernel as the large time limit of the heat
kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2.3 The Bergman kernel as a path integral . . . . . . . . . 37
2.2.4 The Bergman kernel and coherent states . . . . . . . . 38
2.3 The category of line bundles and kernels . . . . . . . . . . . . 39
2.4 The geometric line bundle over projective space . . . . . . . . 41
2.5 An equivalence of categories . . . . . . . . . . . . . . . . . . . 47
2.6 Unitary representations and equivariant line bundles . . . . . 51
2.7 Geometric characters of equivariant line bundles . . . . . . . . 54
3 2-Hilbert spaces 59
3.1 H∗-categories . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.2 2-Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.3 The 2-category of 2-Hilbert spaces . . . . . . . . . . . . . . . 66
3.3.1 Characterizing linear ∗-functors . . . . . . . . . . . . . 66
3.3.2 The Hilbert space of natural transformations . . . . . 68
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10 CONTENTS
3.3.3 Unitary equivalence for 2-Hilbert spaces . . . . . . . . 70
3.3.4 Frobenius algebras from 2-Hilbert spaces . . . . . . . . 71
3.4 The geometry of 2-Hilbert spaces . . . . . . . . . . . . . . . . 72
4 String diagrams 77
5 Even-handed structures 81
5.1 De(cid:28)nition of an even-handed structure . . . . . . . . . . . . . 82
5.2 Geometric interpretation . . . . . . . . . . . . . . . . . . . . . 88
5.3 Relationship with other approaches . . . . . . . . . . . . . . . 90
5.3.1 Pivotal structures on monoidal categories . . . . . . . 90
5.3.2 The coherence theorem of Barrett and Westbury . . . 95
5.3.3 Monoidal 2-categories with duals . . . . . . . . . . . . 95
5.3.4 Comparison with even-handed structures. . . . . . . . 96
5.4 Even-handedness in terms of adjunction isomorphisms . . . . 97
5.5 Even-handed structures from traces . . . . . . . . . . . . . . . 100
5.5.1 Traces on linear categories . . . . . . . . . . . . . . . . 100
5.5.2 Even-handed structures from traces . . . . . . . . . . . 102
5.5.3 Even-handedness for semisimple categories . . . . . . . 103
5.6 2-Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.7 Derived categories of Calabi-Yau manifolds . . . . . . . . . . 108
6 Even-handed structures on fusion categories 111
6.1 Warm-up exercise . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.2 Fusion categories . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.3 The pivotal symbols of a fusion category . . . . . . . . . . . . 119
6.4 Facts about the pivotal tensor . . . . . . . . . . . . . . . . . . 123
6.5 Existence of pivotal and spherical structures . . . . . . . . . . 128
7 2-representations and their 2-characters 137
7.1 The 2-category of unitary 2-representations . . . . . . . . . . 139
7.1.1 Unitary 2-representations . . . . . . . . . . . . . . . . 139
7.1.2 Morphisms . . . . . . . . . . . . . . . . . . . . . . . . 141
7.1.3 2-morphisms . . . . . . . . . . . . . . . . . . . . . . . 142
7.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
7.2.1 Automorphisms of groups . . . . . . . . . . . . . . . . 144
7.2.2 The metaplectic representation . . . . . . . . . . . . . 144
7.2.3 2-representations from exact sequences . . . . . . . . . 145
7.2.4 Other examples of 2-representations . . . . . . . . . . 146
7.2.5 Morphismsof2-representationsfrommorphismsofex-
act sequences . . . . . . . . . . . . . . . . . . . . . . . 146
7.3 More graphical elements . . . . . . . . . . . . . . . . . . . . . 146
7.4 Even-handedness and unitary 2-representations . . . . . . . . 149
7.5 2-characters of 2-representations. . . . . . . . . . . . . . . . . 152