Elliptic Genera and Vertex Operator Super-Algebras

Name: Elliptic Genera and Vertex Operator Super-Algebras
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ISBN: 9783540660064

Lecture Notes in Mathematics

By: H. Tamanoi

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Paperback Published: July 1999
ISBN: 9783540660064
Number Of Pages: 396

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This monograph deals with two aspects of the theory of elliptic genus: its topological aspect involving elliptic functions, & its representation theoretic aspect involving vertex operator super-algebras. For the second aspect, elliptic genera are shown to have the structure of modules over certain vertex operator super-algebras. The vertex operators corresponding to parallel tensor fields on closed Riemannian Spin Kahler manifolds such as Riemannian tensors & Kahler forms are shown to give rise to Virasoro algebras & affine Lie algebras. This monograph is chiefly intended for topologists & it includes accounts on topics outside of topology such as vertex operator algebras.

Introduction and summary of results	p. 1
Towards the signature of loop spaces: Elliptic genus	p. 1
Elliptic genera	p. 4
Vertex operator super-algebras	p. 7
G-invariant vertex operator super-subalgebras and their vertex operators	p. 9
Classifying spaces of geometric structures on manifolds	p. 13
Elliptic genera as modules over vertex operator super- algebras	p. 16
Infinite dimensional symmetries of elliptic genera for Kähler manifolds	p. 17
Elliptic genera	p. 22
Elliptic genera	p. 22
Theta functions and elliptic functions	p. 29
Proof of Theorem 4 of §1.1	p. 32
The modular properties of elliptic genera	p. 35
On the divisibility of coefficients of the elliptic genus ϕellR	p. 42
Spinors on loop spaces: the Witten genus	p. 46
Vertex operator super-algebras	p. 49
Vertex operator super-algebras: A brief introduction	p. 49
The infinite dimensional Clifford modules as vertex oper- ator super-algebras and their modules	p. 60
Clifford algebras, Virasoro algebras and affine Lie algebras as subalgebras of the vertex operator super-algebra V	p. 65
The adjoint operators of vertex operators	p. 72
G-invariant vertex operator super-subalgebras	p. 93
VG as vertex operator super-algebra	p. 93
Lie algebras as Lie subalgebras of Clifford algebras	p. 94
Commutants of Lie subalgebras in o(2N) and Casimir elements	p. 104
Generating G-invariant vectors in (V)_*	p. 111
Vertex operators associated to G-invariant vectors of low weight	p. 122
Unitary Virasoro algebras	p. 129
Vertex operators generated by complex volume forms	p. 139
Symplectic Virasoro algebras	p. 147
Kähler forms in quaternionic vector spaces	p. 174
Unitary Virasoro algebras in quaternionic vector spaces	p. 185
Vertex operators for products of Kähler forms and their commutators	p. 188
Determination of G-invariant vectors of low weight in the vertex operator super-algebra (V )_*	p. 211
Geometric structures in vector spaces and reductions of structure groups on manifolds	p. 222
Some geometric structures in vector spaces	p. 222
Moduli of geometric structures as G-orbits in representa- tion spaces: Some examples of interest	p. 236
A general theory of geometric structures in vector spaces and their compatibilities	p. 250
On connections and parallelism	p. 268
Reductions of structure groups on manifolds and parallel sections	p. 285
Infinite dimensional symmetries in elliptic genera for Kahler manifolds	p. 303
Kählerian manifolds	p. 303
Generalizations of Riemannian metrics, Riemannian vol- ume forms, Kähler forms, complex volume forms, and quaternion-Kähler forms	p. 313
Twisted G-Dirac operators	p. 326
Elliptic genera as modules over vertex operator super- algebras of parallel sections	p. 337
Infinite dimensional symmetries in elliptic genera for Kahler manifolds	p. 357
References	p. 379
Index of Notation	p. 383
Subject Index	p. 387
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9783540660064
ISBN-10: 3540660062
Series: Lecture Notes in Mathematics
Audience: General
Format: Paperback
Language: English
Number Of Pages: 396
Published: July 1999
Publisher: SPRINGER VERLAG GMBH
Country of Publication: DE
Dimensions (cm): 23.39 x 15.6 x 2.11
Weight (kg): 0.56