roceedings of the Summer School
Geometric Methods for Quantum Field Theory
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Editors Hernan Ocampo, Sylvie Paycha & Andres Reyes World Scientific
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Proceedings of the Summer School
Geometric Methods for Quantum Field Theory
Proceedings of the Summer School
Geometric Methods for— Quantum Field Theory Villa de Leyva, Colombia
12-30 July 1999
Editors
Hernan Ocampo Universidad del Valle, Call, Colombia
Sylvie Paycha Universite Blaise Pascal, Clermont-Ferrand, France
Andres Reyes Universidad de los Andes, Bogota, Colombia
World Scientific Singapore • New Jersey • London • Hong Kong
Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
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Proceedings of the Summer School on GEOMETRIC METHODS FOR QUANTUM FIELD THEORY Copyright © 2001 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
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ISBN 981-02-4351-0
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INTRODUCTION This volume offers an introduction to some basic mathematical and physical tools as well as methods from mathematics and physics required to follow the recent developments in several active topics at the interface between geometry and quantum field theory including duality, gauge field theory, geometric quantization, Seiberg-Witten theory, spectral properties and families of Dirac operators, and the geometry of loop groups. It is based on lectures delivered during a Summer School "Geometric methods for Quantum Field Theory" held at Villa de Leyva, Colombia in July 1999, complemented by some short communications by participants of the school. Since these lecture notes are aimed at beginning graduate students in physics or mathematics, the chapters are organized in an order of gradually increasing level of difficulty. Each chapter is self-contained and can be read independently; references to other chapters are made only to help the reader relate the different chapters. The volume starts with an introductory course by Tilmann Wurzbacher on differentiable manifolds and symplectic geometry, followed by a presentation by Oussama Hijazi of the Dirac operator and its spectral properties, the Dirac operator playing a fundamental part in the sequel. The third lecture by Edwin Langmann offers an introduction to second quantization of fermions, pointing out along the way how the one dimensional Dirac operator arises in the quantization of loops. The Dirac operator on the circle is the main object of the fourth lecture by Krzysztof Wojciechowski who investigates its zetadeterminant. Zeta-determinants are defined using regularization techniques which are described in their geometric context by Sylvie Paycha in the fifth lecture. Using some of the geometric tools introduced in the first two lectures, in the sixth one, Tsou Sheung Tsun gives a concise introduction to gauge theory and a glimpse into duality via the electro-magnetic model. The last lecture by Hernan Ocampo, starting from electro-magnetic duality, leads the reader into the realm of Seiberg-Witten theory. The short communications at the end of the volume, the topics of which are closely related to those of the preceding lectures, were selected by way of a strict refereeing procedure. We thank the referees most warmly for the wonderful job they did dedicating a lot of their time to help improve the short communications published here. We are indebted to various organizations for their financial support. Let us thank first of all the French organization "C.I.M.P.A.". We are specially grateful to its former director Claude Lobry who showed his constant support v
during the preparation of the school and who was kind enough to accept our invitation to attend part of it. We also thank ECOS-Nord, this school being part of a long term scientific program between the Universite Blaise Pascal in Clermont-Ferrand and the Universidad de Los Andes in Bogota in the areas of mathematics and physics. We are also grateful to the Universidad de Los Andes, which was our main source of financial support in Colombia. In addition we are indebted to the I.C.T.P., "Banco de la Republica", "Academia Colombiana de Ciencias", I.C.E.T.E.X, and I.C.F.E.S. who also contributed to the financial support needed for this school. Special thanks to Sergio Adarve (Universidad de Los Andes) coorganizer of the school who dedicated much time and energy to make this school possible in a country like Colombia where many difficulties are bound to arise along the way due to social, political and economic problems. Let us also thank Daniel Bennequin most warmly for his scientific advice when preparing the program of the school. We would furthermore like to express our gratitude to Rolando Roldan (Universidad de los Andes), coorganizer, and Jose Rafael Toro (Universidad de los Andes), whose contribution and support was essential for the success of the school. We also thank Monica Vargas most warmly for doing a wonderful job for the practical organization of the school. Special thanks to Andrea Bernal, Andres Garcia, Marta Kovacsics, Diego Laverde and Alexandra Parra. Without all the people named here, all of whom helped with the organization in some way or another, before, during and after the school, this scientific event would not have left such vivid memories in the lecturers' and the participants' minds. Last but not least, thanks to all the participants who gave us all, lecturers, contributors of short communications and editors, the impulse to prepare this volume through the enthusiasm they showed us during the school. We hope that these lectures will give -as much as the school itself seems to have given- young students the desire to pursue what might be their first acquaintance with some of the problems on the edge of mathematics and physics presented in this volume. On the other hand, we hope that the more advanced reader will find some pleasure reading about different outlooks on related topics and seeing how well-known geometric tools prove to be very useful in some areas of quantum field theory.
The Editors Hernan Ocampo, Sylvie Paycha, Andres Reyes.
vi
CONTENTS Introduction
v
Lectures • Lecture 1: Introduction to differentiable manifolds and symplectic geometry Tilmann Wurzbacher
1
• Lecture 2: Spectral properties of the Dirac operator and geometrical structures Oussama Hijazi
116
• Lecture 3: Quantum theory of fermion systems: Topics between physics and mathematics Edwin Langmann
170
• Lecture 4: Heat equation and spectral geometry. Introduction for beginners Krzysztof Wojciechowski
238
• Lecture 5: Renormalized traces as a geometric tool Sylvie Paycha
293
• Lecture 6: Concepts in gauge theory leading to electricmagnetic duality Tsou Sheung Tsun
361
• Lecture 7: An introduction to Seiberg-Witten theory
421
Hernan Ocampo Short Communications • Remarks on duality, analytic torsion and gaussian integration in antisymmetric field theories Alexander Cardona • Multiplicative anomaly for the ^-regularized determinant Catherine Ducourtioux
451 467
• On cohomogeneity one Riemannian manifolds S. M. B. Kashani
483
• A differentiable calculus on the space of loops and connections Martin Reiris
489
• Quantum Hall conductivity and topological invariants Andres Reyes
498
• Determinant of the Dirac operator over the interval [0,0\ Fabian Torres-Ardila
509
Proceedings of the Summer School on Geometric Methods for Quantum Field Theory edited by H. O. Campo, A. Reyes & S. Paycha © World Scientific Publishing Co.
I N T R O D U C T I O N TO D I F F E R E N T I A B L E M A N I F O L D S A N D SYMPLECTIC GEOMETRY TILMANN WURZBACHER Institut de Recherche Mathematique Avancee Universite Louis Pasteur et C.N.R.S. 7, rue Rene Descartes, F-67084 Strasbourg Cedex, France E-mail:
[email protected] Assuming only undergraduate level knowledge of linear algebra, analysis including ordinary differential equations and rudimentary topology, we develop the basics of the theory of symplectic manifolds and Hamiltonian dynamical systems, that is pivotal in geometric considerations in theoretical physics. The material on multilinear and symplectic algebra as well as on differentiable manifolds (vector fields, differential forms and de Rham cohomology) necessary to bridge the gap from the prerequisites to symplectic geometry is thoroughly covered in the first chapters of the text.
Contents Introduction
3
0 Motivation
5
0.1 An example from mechanics: from Newton to Lagrange to Hamilton 0.2 An infinite dimensional example: the wave equation
5 6
0.3 Solving the Hamilton equation for one degree of freedom
8
1 Multilinear and symplectic algebra
13
1.1 Multilinear forms 1.2 Volume and orientation 1.3 Symplectic vector spaces 1.4 Linear symplectic geometry 1.5 Complex structures on real symplectic vector spaces 2 Elementary differential topology
13 19 23 29 31 37
2.1 2.2 2.3 2.4 2.5
Differentiable manifolds Lie groups and smooth actions Vector bundles The tangent bundle Vector fields on manifolds 1
37 43 48 51 55
2.6 Differential forms and the Lie derivative 2.7 The exterior derivative of differential forms and de Rham cohomology
62
2.8 Integration of differential forms on manifolds
81
3 Symplectic geometry 3.1 3.2 3.3 3.4
68 90
Symplectic manifolds Maps and submanifolds of symplectic manifolds Kahlerian and almost Kahlerian manifolds Hamiltonian dynamical systems on symplectic manifolds
References
90 93 101 105 113
2
Introduction These lecture notes are based on courses I gave in Villa de Leyva in Colombia and in Hamburg in Germany, mainly for students of mathematics and/or theoretical physics, during the second half of the year 1999. Since most of the more advanced material of the course in Villa de Leyva is available as part of the text [Wu], I concentrated here on the foundations of the theory of differentiable manifolds (and of symplectic geometry), being at the basis of most considerations in the field of geometry related to theoretical physics. After a short motivation of the Hamiltonian approach to mechanics, the main body of the text proceeds as follows: In Chapter 1, we complement standard knowledge in linear algebra by a thorough development of multilinear algebra, indispensable for the calculus of differential forms on manifolds, as well as of "symplectic algebra", i.e., the basic results on symplectic vector spaces as, e.g., normal forms and the existence of compatible complex structures. The second chapter develops the theory of finite dimensional manifolds from scratch. We give complete proofs of all crucial points of the text, with the only exceptions of the construction of partitions of unity, the proof of Stokes' theorem and of "Moser's formula". We include de Rham cohomology in our presentation since it plays a prominent role in theoretical physics (and of course in geometry), though admittedly physicists tend to describe it in a different language. We strongly believe that learning the general mathematical formulation at an early stage is well worth the effort since it unifies several important notions. Chapter 3 is an introduction to symplectic geometry and Hamiltonian dynamical systems. The concise formulation and easy proofs of the foundational results of analytical mechanics as, e.g., the theorems of Darboux and Noether, the existence of symplectic structures on the total space of the cotangent bundle of a manifold and the properties of the Poisson structure on a symplectic manifolds here show the usefulness of the preceding chapter. We also give some basic material of contemporary symplectic differential geometry as the notion of a Kahlerian manifold and rudiments of the theory of non-linear symplectic maps. Though there are a few references scattered throughout the text, we conclude each section with some "Bibliographical remarks", where hints on related literature are given, as an incitation for further reading and self-study. Let me take the opportunity to thank all participants of the courses in Villa de Leyva and Hamburg for their interest and "feedback". Last but not 3
least I would like to thank Sergio Adarve, Dorothea Glasenapp, Sylvie Paycha, Peter Slodowy, Andres Reyes, Rolando Roldan and Monica Vargas without whose efforts these courses and lecture notes would not have been possible and with whom working together was always a pleasure for me. Tilmann Wurzbacher Strasbourg, June 26, 2000.
4
0 Motivation 0.1 An example from mechanics: Hamilton
from Newton
to Lagrange
to
Let us consider a particle of mass m (m > 0, small compared to the mass of the earth), "close" to the earth, subject to the gravitational field of the earth. We can assume the surface of the earth to be the plane qz = 0 and describe the trajectory of the particle by q(t) = {qi(t),q2(t),q3(t)) with qz > 0. The exterior force acting on the particle is given by F = —mg ez with g denoting the (strictly positive) "gravitational constant" and ez standing for the third unit vector in E 3 . N e w t o n i a n description: The Newtonian equation of motion is F = mq with initial conditions q(0) = q° with q% > 0 and q(0) = v° . Here we have —mgez = mq so that the trajectory is given by t2 q(t) = q° + tv° - g-e3 . Observation. function
The above force field F is "conservative", i.e. there is a
U : {q E R3 | qz > 0} -»• E such that F = - W . This function, here we can take e.g. U(q) = mgqz, is called the "potential energy", whereas the function T = ^(q)2 = y ( ( g i ) 2 + fe)2 + (Q3)2) is considered as the "kinetic energy". In the case of a conservative force field we can go to the Lagrangian description: Let L = L(q,q,t) := T—U be the "Lagrange function" (that might depend explicitly on
for * = 1,2,3
with initial condition as above q(0) = q° with q% > 0 and q(0) = v° . 5
In our example these equations are obviously the same as in the Newtonian approach. Observation. We have - at least in our example dL Q~=rnqk=Pk, the "(linear) momentum" of the particle. We can thus write the "total energy" H = H{q,p) = T + U as a function of q and p! (This transition from L to H is called "Legendre transformation" and is not always possible! Since later on we will not be concerned with Lagrangian mechanics we do not go into this question more deeply.) For such a "Hamilton function" or "Hamiltonian" H we have the following Hamiltonian equations of motion dH . , dH . r , -r— = qk and ^— = -pk for fc = 1,2,3 opk oqk with initial condition q(0) = q° and p(0) = p° . In the example we have H = ^ p 2 + mgqs (setting of course p2 = (pi) 2 + (P2)2 + (P3)2) and p(0) = rnv°. The equation of motion then reads as follow dH 1 OH qk = ^— = —Pit and pk--—= -mg5k,3 . aph m dqk Differentiating the first equation with respect to time t and inserting the result into the second we find the Newtonian equation of motion for q = q{t) and we get p = p(t) then in the case of the example trivially from q(t) and the first equation. Remark. The choice between the Lagrangian and the Hamiltonian approach (in a situation where both can be applied) depends on further details: the advantages of the Hamiltonian approach lie in the equal treatment of the variables q and p, the first order of the equations, and a simple transition to quantum mechanics. On the other hand, in relativistic mechanics or in the transition from classical field theory to quantum field theory the Lagrangian might often be more useful, at least for theoretical physicists. 0.2 An infinite
dimensional
example:
the wave
equation
Without going into the (functional-analytic) problems of domains crucial in infinite dimensional situations we will give here a simple class of a Hamiltonian equations, which are "equivalent" to certain non-linear wave equations. 6
Let for n > 1
<S(E")={/ : E n ->E| lim„s||_f0OP(a;)g (JA
f(x) = 0 VP,Q €
M[Tt,...,Tn)\
(with Q{-§-) — gf - ) viewed as a scalar partial differential operator with constant coefficients). Given a function U in, e.g., <S(E) we define the vector space E = <S(E3) e <S(E3) and a function H : E -»• E by
By analogy we can consider the following Hamiltonian equation dH • . — = and
dH — = -*.
We interpret the partial derivatives as "L 2 -gradients" in the following sense
( ( I f ) (^0'7ro)'f)L2(R3) =
(Z?2
"Wo) W = | | 0 ^ o . - o +«f)
and analogously for . A direct calculation yields: -5—) (^o,7To),7f )
=
( t x ) (^o.^o),^ \0(j> J
(KO,K)L2(R3)
and
= ( - A 0 o + C/'(0o)^) r2m3 ^ ,
/Z,2(R3)
\
/L2(R3)
where U' denotes the derivative of U. In the case at hand the Hamiltonian equation thus reads as follows: ; 9H 0 = — = 7r
and
. oH n =-—=
Afo-U
_ r,. , {cj)0).
(Let us leave aside here the discussion of the initial condition as well as the existence and uniqueness questions for the solutions of this equation.) Thus a curve t t-4 (<j>, <j>) £ E satisfies the Hamilton equation if and only if the function (t,x) H->- i-e- as l o n S a8 K') > 0-
sin(wt) mu>
Observation. The above solution is a priori only locally denned, i.e. for t close to t0. In the case at hand we can immediately extend it to a solution for all real t. Case 3. p° = 0 and ^(q°) £ 0 We leave the analysis of this case as an exercise. Remark. The aim to find explicit formulas for the solutions of Hamiltonian systems led to the discovery of many important special functions in the 19th century. Notably the theory of analytic functions in one complex variable and of "Riemann surfaces" was highly stimulated by this search. Bibliographical remarks. Our - highly subjective - choice of physics texts on classical mechanics include [Arl], [Gol] and [Sch]. For the mathematical approach to infinite dimensional Hamiltonian mechanics see, e.g., [AMR] and [CM]. A good german reference is [Lau] which we followed in Section 0.2. 12
1
Multilinear and symplectic algebra
In Chapter 1 all vector spaces will be finite dimensional over a field K which is R or C if not explicitely stated otherwise. 1.1
Multilinear
forms
Definition. A "bilinear form" on a K-vector space V is a map B : V x V -» IK such that (i) B(v +
v',w)=B(v,w)+B(y',w)
B(X • v,w) = A • B(v,w)
and
(ii) B(v,w + w') = B{v,w) + B(v,w') B(v, \x • w) = fi • B(v, w) for all v, v',w, w' in V and for all A, \i in K. Remark. Using the canonical basis {ei,...,e„} of Kn a bilinear form B on K™ can be represented in a unique way by a square matrix Q = QB in Mat(n x n , K ) as follows: n
n
n
B(x,y) = BlJTxje^^ykek) j=l
= ] P XjB(ej,ek)yk = fe=l
j,k-l n
= ] P XjQjkVk
-tx-QB-y-
j,k=l
Lemma. The map B(V) := {B : V x V —>• K| B is bilinear} ->- Mat(n x n,K),B i->- QB is a K-vector space isomorphism. Proof. Exercise.
•
Remark. Let T : U —> V be a K-linear map and B a bilinear form on V, then we define the "pullback of B unter T" by (T*B)(uuu2)
:=
B(T(Ul),T(u2))
for all ui,U2 in U. We observe that T*B is a bilinear form von U. Special C a s e . Let U — V = Kn and T = TA, the linear map a; i-> yl-a; associated to a (n x n)-matrix A. Then Q T * B = *-4 • QB • A. Proof. Exercise. D Definition. 13
(1) A bilinear form B on V is called "symmetric" if B(v,v')
= B{v',v)
for all v,v' in V.
(2) A bilinear form B on V is called "skew-symmetric" (or "anti-symmetric" or "alternating") if B(v,v')
= -B(v',v)
for all v,v' in V.
Remark. If B is skew-symmetric, then B(v,v) = 0 for all v in V. Lemma. (i) Each bilinear form B on V is uniquely decomposed into the sum of a symmetric and an alternating bilinear form. (ii) If B is a bilinear form on K71 and QB the associated matrix, then one has (B is symmetric if and only if
'(0
If { e i , . . . , e n } is an ordered basis of V and { e j , . . . , e*} the dual basis of V* then {e*h ® ... ® e*J h,...,ik
£{l,...,n}}
is a basis of 0 ' V* and thus its dimension equals nk. The multiplication on T(V*) is given as follows: let t G kV*,s € ®'V* and v±,..., ufc+; <E V. Then the element £ ® s of ® + V* is defined by (t ® s)(vi,..
.,vk,vk+i,.
• -,vk+i)
:= i ( u i , . . . ,vk)s{vk+i,..
.,vk+t).
Lemma. The K-vector space T(V*) together with the multiplication given by (3> is a non-commutative, associative, unital K-algebra. 14
Proof. Exercise.
•
The "symmetric group" Sk of all permutations of the set { 1 , . . . , k) acts on (g)ft V* as follows: a{t)(yi,...,vk)
:=t{v defined by (—l) r on a product a = &x o • • • o ar of transpositions crj for j = 1 , . . . , r. Idea of the proof. The group Sk is generated by transpositions and K\{0} is abelian. • Definition. (1) The space of "symmetric /c-forms" is given by SkV* = {t € ®kV*\o-(t) =t
Vo-eS f c }.
(2) The space of "skew-symmetric (or alternating) fc-forms" is given by AkV* = {te ®kV*\a(t) = sign(a) -t
V T{V*) with analogous properties. Combining Symm with the tensor product we obtain a multiplication on S(V*) = Q)SkV*
: tVs:=Symra(t®s)
for t,s in 5 ( V ) .
fe>0
It follows that <S(V*) is a commutative, associative, unital K-algebra. Let n + k— l\ , j if n is the
(
dimension of V, and that S(V*) is isomorphic to the space of polynomials on V as a K-algebra. (See [Gre] for proofs of this and more details on symmetric tensors.) (2) Analogously, we have a natural "alternator" or "anti-symmetrizer" map Alt : kV* -»• ®kV\
Alt(t) = -^ Y,
si
S n (*M*)
such that Alt o Alt = Alt and Alt(fcV*) = AkV*. Again Alt extends to a map
Alt : T(V*) -> A(V) = 0 A V . Definition. Let a be in kV* and /3 in <E>'V*, then the "wedge product of a with /?" is defined as follows: aA/3:=
^T7T Alt(a ®' 3) -
Remark. The factor in the above definition of the wedge product is chosen such that it relates in the easiest possible way to volumes: let {£1,62} be a basis of a vector space V and {e^Cj} t n e dual basis, then with the above definition e*1Ae*2(e1,e2)
= l.
Proposition. Let a be in ®kV*, fi in ®lV and 7 in ®mV*. Then (i) a A/3 = Alt(a) A (3 = a A Alt(/3) = Alt(a) A Alt(P). (ii) "A" is K-bilinear. (Hi) a A 13 = {-l)k4P
Aa.
(iv) a A (P A 7) = (a A (3) A 7. 16
Proof. Ad (i). Alt(Alt(a) /?) = A l t ( - ^ Yl sign((r)(7(a) (8)/?) o-eSi
= JkTlY
^
sign(/*)(- ^sign(j+i,..., w/+fc, u i , . . . , vt)
= a®/?(w W f c ( 1 ) ,...,t; W f c { , + f e ) ), where the permutation /j,i>k in S i+fc is uniquely denned by the last equality sign and s i g n e t ) = (-l)'' f c . Thus 0 a = tntk(a ® 0) and one finds
Alt(0 ® a) = = sign(/xi|fc)
= sign(/t
