ISBN: 3-540-66006-2
TITLE: Elliptic Genera and Vertex Operator Super-Algebras
AUTHOR: Tamanoi, Hirotaka
TOC:

Introduction and summary of results 1
0.1 Towards the signature of loop spaces: Elliptic genus 1
0.2 Elliptic genera 4
0.3 Vertex operator super-algebras 7
0.4 G-invariant vertex operator super-subalgebras and their vertex operators 9
0.5 Classifying spaces of geometric structures on manifolds 13
0.6 Elliptic genera as modules over vertex operator super- algebras 16
0.7 Infinite dimensional symmetries of elliptic genera for Khler manifolds 17
Chapter I. Elliptic genera 22
1.1 Elliptic genera 22
1.2 Theta functions and elliptic functions 29
1.3 Proof of Theorem 4 of 1.1 32
1.4 The modular properties of elliptic genera 35
1.5 On the divisibility of coefficients of the elliptic genus varphi^R_ell 42
1.6 Spinors on loop spaces: the Witten genus 46
Chapter II. Vertex operator super-algebras 49
2.1 Vertex operator super-algebras: A brief introduction 49
2.2 The infinite dimensional Clifford modules as vertex operator super-algebras and their modules 60
2.3 Clifford algebras, Virasoro algebras and affine Lie algebras as subalgebras of the vertex operator super-algebra V 65
2.4 The adjoint operators of vertex operators 72
Chapter III. G-invariant vertex operator super-subalgebras 93
3.1 V^G as vertex operator super-algebra 93
3.2 Lie algebras as Lie subalgebras of Clifford algebras 94
3.3 Commutants of Lie subalgebras in o(2N) and Casimir elements 104
3.4 Generating G-invariant vectors in (V)_* 111
3.5 Vertex operators associated to G-invariant vectors of low weight 122
3.6 Unitary Virasoro algebras 129
3.7 Vertex operators generated by complex volume forms 139
3.8 Symplectic Virasoro algebras 147
3.9 Khler forms in quaternionic vector spaces 174
3.10 Unitary Virasoro algebras in quaternionic vector spaces 185
3.11 Vertex operators for products of Khler forms and their commutators 188
3.12 Determination of G-invariant vectors of low weight in the vertex operator super-algebra (V)_* 211
Chapter IV. Geometric structures in vector spaces and reductions of structure groups on manifolds 222
4.1 Some geometric structures in vector spaces 222
4.2 Moduli of geometric structures as G-orbits in representation spaces: Some examples of interest 236
4.3 A general theory of geometric structures in vector spaces and their compatibilities 250
4.4 On connections and parallelism 268
4.5 Reductions of structure groups on manifolds and parallel sections 285
Chapter V. Infinite dimensional symmetries in elliptic genera for Khler manifolds 303
5.1 Khlerian manifolds 303
5.2 Generalizations of Riemannian metrics, Riemannian volume forms, Khler forms, complex volume forms, and quaternion-Khler forms 313
5.3 Twisted G-Dirac operators 326
5.4 Elliptic genera as modules over vertex operator super-algebras of parallel sections 337
5.5 Infinite dimensional symmetries in elliptic genera for Khler manifolds 357
References 379
Index of Notation 383
Subject Index 387
END
