IAS/PARK CITY MATHEMATICS SERIES

Volume 1

Geometry an d Quantum

Field Theory Daniel S. Freed

Karen K. Uhlenbeck Editors

American Mathematica l Societ y Institute fo r Advanced Stud y

https://doi.org/10.1090//pcms/001

I A S / P a r k Cit y Ma thema t i c s Ins t i tu t e run s m a t h e m a t i c s educat io n p rogram s t h a t brin g together hig h schoo l ma themat i c s teachers , researchers , g radua t e s tudents , an d under -graduates t o par t ic ipa t e i n dist inc t bu t overlappin g program s o f researc h an d educat ion . This volum e contain s t h e lectur e note s fro m th e G r a d u a t e Summe r Schoo l p rogra m o n the Geometr y an d Topolog y o f Manifold s an d Q u a n t u m Fiel d Theory , hel d J u n e 2 2 -July 20 , 1991 , i n P a r k City , Utah .

Suppor ted b y th e Nat iona l Scienc e Foundat ion .

1991 Mathematics Subject Classification. P r i m a r y 58-XX , 81-XX , 70 -XX , 53-XX ; Secondary 16-XX , 18-XX , 22 -XX , 3 4 - X X , 35-XX , 49 -XX .

L i b r a r y o f C o n g r e s s C a t a l o g i n g - i n - P u b l i c a t i o n D a t a

Geometry an d quantu m field theor y : Jun e 22-Jul y 20 , 1991 , Park City , Uta h / Danie l S . Freed , Karen K . Uhlenbeck , editors .

p. cm . — (IAS/Par k Cit y mathematic s series , ISS N 1079-5634 ; v . 1 ) Includes bibliographica l references . ISBN 0-8218-0400- 6 (acid-free ) 1. Li e groups—Congresses . 2 . Symplecti c groups—Congresses . 3 . Quantu m field theory —

Congresses. 4 . Mathematica l physics—Congresses . I . Freed , Danie l S . II . Uhlenbeck , Kare n K. III . Series . QC20.7.L54G46 199 5 530.1'43'0151604-dc20 94-4673 1

CIP

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10 9 8 7 6 5 4 3 0 0 9 9 9 8 9 7

Preface

The IAS/Park Cit y Mathematics Institute (PCMI ) was founded i n 199 1 as par t of the "Regiona l Geometr y Institute" initiativ e o f the National Science Foundation . In mi d 199 3 i t foun d a n institutiona l hom e a t th e Institut e fo r Advance d Stud y (IAS) i n Princeton . Th e PCM I wil l continu e t o hol d summe r program s i n bot h Park Cit y an d i n Princeton .

The IAS/Par k Cit y Mathematic s Institut e encourage s researc h an d educatio n in mathematic s an d foster s interactio n betwee n th e two . Th e month-lon g sum -mer institut e offer s program s fo r researcher s an d postdoctora l scholars , graduat e students, undergraduates , an d hig h schoo l teachers . On e o f ou r mai n goal s i s t o make al l o f the participant s awar e o f th e tota l spectru m o f activities tha t occu r i n mathematics educatio n an d research : w e wis h t o involv e professiona l mathemati -cians in education and t o bring modern concept s in mathematics to the attention of educators. T o tha t en d th e summe r institut e feature s genera l session s designe d t o encourage interaction amon g th e variou s groups . In-yea r activitie s a t site s aroun d the countr y for m a n integra l par t o f the Progra m fo r Hig h Schoo l Teachers .

Each summe r a differen t topi c i s chose n a s th e focu s o f the Researc h Progra m and Graduat e Summe r School . (Activitie s i n th e Undergraduat e Progra m dea l with thi s topi c a s well. ) Lectur e note s fro m th e Graduat e Summe r Schoo l wil l be publishe d eac h yea r i n thi s series . Thi s firs t volum e contain s note s fro m th e 1991 Summer Schoo l o n th e Geometry and Topology of Manifolds and Quantum Field Theory. Volume s from the 199 2 Summer School Nonlinear Partial Differential Equations in Differential Geometry , th e 199 3 Summer Schoo l Higher Dimensional Algebraic Geometry, an d the 199 4 Summer School Gauge Theory and the Topology of Four-Manifolds ar e i n preparation . Th e 199 5 Researc h Progra m an d Graduat e Summer Schoo l topi c i s Nonlinear Wave Phenomena.

We plan t o publish materia l from othe r part s o f the IAS/Park Cit y Mathemat -ics Institute i n the future . Unfortunately , th e initia l volume s in thi s serie s may no t reflect th e interactio n betwee n researc h an d educatio n whic h i s a primar y focu s o f the PCMI. At the summer institute late afternoons ar e devoted t o seminars of com-mon interest t o al l participants . Man y dea l with curren t issue s in education ; other s treat mathematica l topic s a t a level whic h encourage s broad participation . Severa l popular evenin g program s ar e als o wel l attended . Thes e includ e lectures , pane l discussions, compute r demonstrations , an d videos . Th e PCM I ha s als o spawne d interactions betwee n universitie s an d hig h school s a t a loca l level . W e hop e t o share thes e activitie s wit h a wide r audienc e i n futur e volumes .

Dan Freed , Serie s Edito r October, 199 4

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Contents

Introduction 1

Robert L . Bryant , A n Introductio n t o Li e Group s an d Symplecti c Geometry 5

Introduction 7 Background Materia l an d Basi c Terminolog y 8

Lecture 1 . Introduction : Symmetr y an d Differentia l Equation s 1 3 Symmetry an d conservatio n law s 1 3 Classical integratio n technique s 1 4

Lecture 2 . Li e Group s an d Li e Algebra s 1 9 Lie groups 1 9 Lie algebras 3 0 Left-invariant form s an d th e structur e equation s 3 5 The 1- , 2-, and 3-dimensiona l Li e algebras 3 6 Exercises 4 1

Lecture 3 . Grou p Action s o n Manifold s 4 7 Lie group action s 4 7 Group action s an d vecto r field s 5 3 Equations o f Lie typ e 5 5 Appendix: Lie' s transformatio n groups , I 6 1 Appendix: connection s and curvatur e 6 3 Exercises 6 6

Lecture 4 . Symmetrie s an d Conservatio n Law s 7 1 Variational problem s 7 1 Symmetries 7 5 Hamiltonian for m 8 3 The cotangen t bundl e 8 5 Poincare recurrenc e 8 6 Exercises 8 9

Lecture 5 . Symplecti c Manifolds , I 9 3 Symplectic Algebr a 9 3 Symplectic Manifold s 9 7 Exercises 10 8

Lecture 6 . Symplecti c Manifolds , I I 11 3

vi CONTENT S

Obstructions 11 3 Deformations o f symplectic structure s 11 6 Submanifolds o f symplectic manifold s 11 8 Appendix: Lie' s transformatio n groups , I I 12 2 Lie pseudo-group s 12 4 Exercises 12 5

Lecture 7 . Classica l Reductio n 12 9 Symplectic grou p action s 12 9 The momen t ma p 13 2 Reduction 13 4 Exercises 13 8

Lecture 8 . Recen t Application s o f Reduction 14 3 Riemannian holonom y 14 3 Kahler manifold s an d algebrai c geometr y 14 5 Kahler reductio n 14 9 Hyper kahler manifold s 15 3 Exercises 16 0

Lecture 9 . Th e Gromo v Schoo l o f Symplecti c Geometr y 16 3 Soft technique s i n symplecti c manifold s 16 3 Hard technique s i n symplecti c manifold s 16 7 Epilogue 17 4 Exercises 17 5

Bibliography 17 9

Jeffrey M . Rabin , Introductio n t o Quantu m Fiel d Theor y fo r Math -ematicians 18 3

Introduction 18 5 Acknowledgements 18 6

Lecture 1 . Classical Mechanic s 18 7

The Geometr y o f Hamiltonia n Mechanic s 19 2

Lecture 2 . Classical Fiel d Theor y 19 5

Lecture 3 . The Lorent z Grou p an d Spinor s 20 3

Lecture 4.'Quantu m Mechanic s 20 9 Symmetries i n Quantu m Mechanic s 21 3

Heisenberg Pictur e 21 3

Lecture 5 . The Simpl e Harmoni c Oscillato r 21 5

Lecture 6 . The Pat h Integra l Formulatio n o f Quantum Mechanic s 21 9

Lecture 7 . The Harmoni c Oscillato r vi a Pat h Integral s 22 7

Lecture 8 . Quantu m Fiel d Theory : Fre e Field s 23 3 Axioms 23 9 The Dira c Fiel d 24 1

CONTENTS vi i

Lecture 9 . Interacting Fields , Feynma n Diagrams , and Renormalizatio n 24 5

Exercises 25 7 Self-adjointness an d tim e evolutio n i n quantu m mechanic s 25 7 Gaussian measur e problem s 25 9 Miscellaneous physica l problem s 26 4

Bibliography 26 9

Orlando Alvarez , Lecture s o n Quantu m Mechanic s an d th e Inde x Theorem 27 1

Introduction 27 3 References 27 4

Lecture 1 . Canonical Quantizatio n o f Bosonic System s 27 5 Lagrangian an d Hamiltonia n mechanic s 27 5 Naive quantizatio n 27 7 Noether's theore m 27 7 Review o f the simpl e harmoni c oscillato r 27 8

Lecture 2 . Canonical Quantizatio n o f Fermionic System s 28 1 The basic s 28 1 The fermioni c oscillato r 28 3 Quantization o f fermionic system s 28 4 Clifford algebr a fermion s 28 7

Lecture 3 . Supersymmetry 29 3 Getting spi n 1/ 2 particle s t o mov e 29 3 The simples t supersymmetr y 29 5 The Dira c equatio n o n a manifol d 29 8

Lecture 4 . The Inde x o f Operator s 30 1 The basic s 30 1 Supersymmetry an d th e inde x 30 1

Lecture 5 . Path Integral s 30 3 Path integra l fo r th e simpl e harmoni c oscillato r 30 3 Theory o f fermionic integratio n 30 9

Lecture 6 . The Atiyah-Singe r formul a 31 1 Atiyah-Singer formul a fo r th e Dira c operato r 31 1

Lecture 7 . Global Topologica l Issue s 31 9 Characteristic classe s 31 9 Global propertie s o f th e pat h integra l 32 1

Frank Quinn , Lecture s o n Axiomati c Topologica l Quantu m Fiel d Theory 32 3

Outline 32 6

Lecture 1 . A rough ide a 329

viii CONTENT S

Lecture 2 . Topologica l opportunitie s fo r TQF T 33 1

Lecture 3 . Th e Eule r theor y 33 5 Euler characteristic s 33 5

Lecture 4 . Finit e tota l homotop y TQF T 33 9 The ske w theor y 34 2 Witten's integra l fo r finite gaug e grou p 34 2 Calculations fo r finite group s 34 3

Lecture 5 . Twiste d finite homotop y TQF T 34 5 The answe r 34 7 Calculations fo r finite group s 34 8 The constructio n 35 5 Induced homomorphism s 35 8 Dependence o n th e cocycl e 36 0 The Chern-Simon s TQF T 36 1

Lecture 6 . Axiom s 36 5 Definition o f domain categor y 36 5 Examples 36 6 Bordism categorie s 36 8 Definition o f TQFT 36 9 Examples 37 0

Lecture 7 . Elementar y structur e 37 3 Pairings 37 3 Ambialgebras 37 8 Classification o n surface s an d graph s 380

Lecture 8 . Modula r example s 38 3 The ide a 38 3 The Eule r theor y 38 6 The finite tota l homotop y TQF T 38 7 The twiste d finite homotop y TQF T 39 0 The endomorphis m TQF T 39 3

Lecture 9 . Modula r axiom s and structur e 39 5 Modular domai n categorie s 39 5 Modular bordism s and TQF T 39 7 The corne r ambialgebra s 39 8 Classification o n surface s 40 1 Module structure s o n Z(Y) 40 3 Nondegenerate pairings 40 4 The ran k TQF T 40 5 Morita equivalenc e an d rationa l conforma l field theorie s 40 6 The Verlind e basi s an d lin k invariant s 40 7 The structur e o f Z(T 2) 40 8

Lecture 10 . Categorica l construction s 41 1 Tensor categorie s 41 2

CONTENTS IX

Semisimple tenso r categorie s 41 3 Examples o f finit e semisimpl e tenso r categorie s 41 4 Example: 51(2 ) mo d 5 41 5 State module s 41 6 Glueing tree s 41 7 The corne r ambialgebr a 41 9 State module s o f graphs 42 0 Slices throug h 2-complexe s 42 2 Beginning an d endin g a presentatio n 42 3 Beginning an d endin g a relatio n 42 3 The circulato r 42 6 The Interchang e 42 8 Doing th e calculatio n 43 0 Extensions 43 1

Appendix A . Algebr a 43 5 Tensor product s 43 5 Pairings an d trace s 43 7 Ambialgebras 43 9 Examples o f ambialgebra s 44 2 Complex ambialgebra s 44 7

Bibliography 451

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INDEX O F NOTATION S

Notations ar e liste d i n orde r o f appearance . Th e numbe r give n i s tha t o f th e largest numbere d ite m preceedin g th e appearance .

x(X), Eule r characteristic , 3.1. Map, Spac e o f maps , 4.4 . #n, homotop y order , 4.5 . Hmy, Holonom y group , 5.4 . [h, g]c, commutato r i n G , 5.17 . lim, invers e limit , 5.28 . lim, direc t limit , 5.31 . TTQV, component s o f a category , 5.28 . "bar", involution , 6.1 . SDiffd+1, Domai n categor y o f smooth manifolds , 6.2 . FHty, 5 H t y d + 1 homotop y domai n categories , 6.3 . F C x d + 1 , finite C W domai n category , 6.3 . o, Produc t i n a tenso r category , 10.1. SL{2) mo d 5 , 10.7 . ®, tenso r product , Al . t r , TV , trace, A2.5 . r k t r , Trace-rank , A2.7 . End(JB), endomorphis m ring , A4.1.

455

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INDEX O F TERMINOLOG Y

Terms ar e liste d alphabetically . Th e numbe r give n i s tha t o f th e larges t num -bered ite m preceedin g th e appearance .

Additive (category ) 9.3 . Ambialgebra, A3.1 . Andrews-curtis conjecture , 2.0 . Antiinvolutions, A 1.3. Arrangement, 10.1 .

Bar constructio n (Be), 5.7 . Beginrelation matrix , 10.31 . Bordism

category, 6.S. modular, 8.3 .

Boundary, 1.0 , 6.1 . Braided monoida l category , 10.51 .

Cellular chains , 5.9 . Central homomorphis m (ambialgebra ) A4.6 . Characteristic numbers , 5.2 . Chern-Simons TQFT , 5.36 . Circulator matrix , 10.36 . Cocycle relation , 5.11 . Collars, 6.1. Colorings, o f links , 9.25 . Commutator, 5.17 . Comultiplication, A3 . Conjugate (module ) A 1.3. Copairing, A2 . Corners, 8.1 .

ambialgebras, 9.7 . Cylinders, i n a domai n category , 6.1 .

Direct limit , 5.31 . Domain category , 6.1.

45 7

458 F . QUINN , AXIOMATI C TOPOLOGICA L QUANTU M FIEL D THEOR Y

modular, 9.1 .

Endomorphism ring , A4.1. Endrelation matrix , 10.31 . Euler (TQFT , characteristic ) 3.1 .

Four-manifolds, 2.0 . Finite tota l homotopy , 4.2 . Framed link , 9.25 .

Gelfand-Kazhdan category , 10.51 . Glueing

in a domai n category , 6.1. endpoints i n a grapn , 10.13 .

Group ring , A4.5 .

Hermitian ambialgebra, A3.15 . form, A2 . structure, 6.11. TQFT, 6.11 .

Holonomy, 5.4 , 5.28 . Homotopy order , 4.5 . Hopf algebra , 10.51 , A3.3.

Incoming (boundary ) 1.0 , 6.8 . Interchange matrix , 10.43 . Inverse limit , 5.27 . Isomorphism (o f TQFT) , 3.6 .

Mapping cylinders , 6.10 . Modular

bordism, 8.2 , 9.5 . domain category , 9.1. TQFT, 9.5 .

Monoidal (category ) 10.2 . Morita equivalenc e

of TQFT, 9.23 . of categories , A5.6 .

Natural maxima l decompositions , (i n a category ) 9.24 . Nondegenerate (pairings) , A2.1.

Opposite ring , A1.2 . Oriented complexes , 5.1. Outgoing (boundary ) 1.0 , 6.8 .

Pairing, A2 .

Quantum group , 10.51 . Quasitensor category , 10.51 .

Rank

INDEX O F TERMINOLOG Y

TQFT, 9.20 . trace, A2.7 .

Rational conforma l field theory , 9.23 . Relation (i n a presentatio n o r 2-complex ) 10.28 .

Skew Eule r TQFT , 3.4 . Space, Introduction . Spacetime, Introduction . Special (ambialgebra ) A4.7 . Superspecial (ambialgebra) , A4.7 . Symmetric monoida l (category ) 10.2 .

Tensor category , 10.1 . Tensor products , Al . Thickenings, 7.17 . Topological quantu m fiel d theor y (TQFT ) Introduction , 1.0 , 6.11 . Trace

ambialgebra, A4.7 . group, functio n A2.4 . rank, A2.7 .

Trees, 10.9 . Twisted (finit e homotop y TQFT) , 5.0 .

Vacuum vectors , 7.8 . Verlinde basis , 9.25 .