Vector bundles on Riemann surfaces and Conformal Field Theory

Arnaud Beauville

Description

Niveau: Supérieur, Doctorat, Bac+8
Vector bundles on Riemann surfaces and Conformal Field Theory Arnaud BEAUVILLE (*) Introduction The main character of these lectures is a finite-dimensional vector space, the space of generalized (or non-Abelian) theta functions, which has recently appeared in (at least) three different domains: Conformal Field Theory (CFT), Topological Quantum Field Theory (TQFT), and Algebraic Geometry. The fact that the same space appears in such different frameworks has some fascinating consequences, which have not yet been fully explored. For instance the dimension of this space can be computed by CFT-type methods, while algebraic geometers would have never dreamed of being able to perform such a computation. In the Kaciveli conference I had focussed (apart from the Algebraic Geometry) on the TQFT point of view. Here I have chosen instead to explain the CFT aspect. The main reason is that there is an excellent account of the TQFT part in the little book [A], which anyone wishing to learn about the subject should read. On the other hand the CFT is the most relevant part for algebraic geometers, and it is not easily accessible in the literature. This is an introductory survey, intended for mathematicians with little back- ground in Algebraic Geometry or Quantum Field Theory. In the first part I define a rational CFT as a way of associating to each marked Riemann surface a finite- dimensional vector space, so that certain axioms are satisfied.

• tqft


• dimensional vector


• conformal field


• compact riemann


• riemann surface


• rational conformal

Sujets

• Europe
• Beauville
• Riemann surface

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 19
• Langue: English

Extrait

Vector bundles on Riemann surfaces and Conformal Field Theory

Introduction

Arnaud B UVILEALE (*)

Themaincharacteroftheselecturesisa¯nite-dimensionalvectorspace,the space of generalized (or non-Abelian) theta functions, which has recently appeared in (at least) three di®eren t domains: Conformal Field Theory (CFT), Topological Quantum Field Theory (TQFT), and Algebraic Geometry. The fact that the same space appears in such di®eren t frameworks has some fascinating consequences, which have not yet been fully explored. For instance the dimension of this space can be computed by CFT-type methods, while algebraic geometers would have never dreamed of being able to perform such a computation.

In the Kaciveli conference I had focussed (apart from the Algebraic Geometry) on the TQFT point of view. Here I have chosen instead to explain the CFT aspect. The main reason is that there is an excellent account of the TQFT part in the little book [A], which anyone wishing to learn about the subject should read. On the other hand the CFT is the most relevant part for algebraic geometers, and it is not easily accessible in the literature.

This is an introductory survey, intended for mathematicians with little back- ground in Algebraic Geometry or Quantum Field Theory. In the ¯rst part I de¯ ne arationalCFTasawayofassociatingtoeachmarkedRiemannsurfacea¯nite- dimensional vector space, so that certain axioms are satis¯ ed. I explain how the dimensions of these spaces can be encoded in a ¯nite-dimensionalZ-algebra, the fusion ringof the theory. Then I consider a particular RCFT, the WZW model, associated to a simple Lie algebra and a positive integer, and I show how the dimen- sions can be computed in that case.

In the second part I try to explain what is the space of non-abelian theta functions, and why it coincides with the spaces which appear in the WZW model. This allows to give an explicit formula for the dimension of this space. Then I discuss how such a formula can be used in Algebraic Geometry. I would like to thank the organizers of the Conference for providing such a warm and stimulatingatmosphereduringtheConference–despiteallthematerialdi±cultiestheyhad to face.

(*) Partially supported by the HCM project “Algebraic Geometry in Europe” (AGE).

1

Part I: Conformal Field Theory

1. The de¯nition of a RCFT

Therearevariousde¯nitionsintheliteratureofwhatis(orshouldbe)a Rational Conformal Field Theory (see e.g. [B-K-Z], [F-S], [M-S 1], [S]); unfortunately they do not seem to coincide. In the following I will follow the approach of [F-S], i.e. I will deal only withcatocpmalgebraic curves. I suppose given an auxiliary ¯ nite set ¤ , endowed with an involution¸7→¸ ¤ (in practice ¤ will be a set of representations of the symmetry algebra of the theory). ~ By amarked Riemann surface(C¸p~ mean a compact Riemann surface (not nec-) I essarily connected) C with a ¯ nite number of distinguished points~p= (p 1  . . .  p n ) , eachp i having attached a “label”¸ i ∈¤ . Then a RCFT is a functor which asso- ~ ciates to any marked Riemann surface (C¸~p complex ¯nite-dimensional vector) a ~ V C (~ ¸) , satisfying the following axioms: spacep A 0.V P 1 (∅) =C(the symbol∅means no marked points). A 1.There is a canonical isomorphism

~ ~ V C (p~¸)¡ » →V C (~p ¸

)

~ with¸ ¤ = (¸ ¤ 1  . . .  ¸ ¤ n ) . ~ ~ A 2.Let (C~¸p (C) be the disjoint union of two marked Riemann surfaces 0 ~p 0  ¸ 0 ) 0 ~ 00 and (C 00 ~p 0  ¸) . Then

V C ( ~ ¸p~) = V C 0 (p~ 0  ¸~ 0 )

V C 00 (~p 00  ¸~ 00 ).

A 3.Let (C t ) t∈D be a holomorphic family of compact Riemann surfaces, parametri- zed by the unit disk D½C, with marked pointsp 1 (t) . . .  p n (t) depending holo- morphically ont(¯g. 1 Then below). for anyt∈D there is a canonical isomor- phism »

~ ~ V C t (p~(t) ¸)¡ →V C 0 (~p(0) ¸). A 4. CSame picture, but assume now that the “special ¯bre” 0 acquires a nodes e (¯g. 2aand 2b); we assume that the pointsp i  away from(0) stays. Let C 0 be thenormalizationof C 0 , i.e. the Riemann surface obtained by separating the two branches atsto get two distinct pointss 0 ands 00  is an isomorphism. There

V C t (p~(t) ¸) » X V e C 0 (~p(0) s 0  s 00 ; ¸~  º   ~ ¡ → º∈¤

2

).