Quantum field theory, Grassmannians, and algebraic curves
type: Item
instance of: scholarly article
title: Quantum field theory, Grassmannians, and algebraic curves
zbMATH Open document ID: 0636.22012
DOI: 10.1007/BF01223238
published in: Communications in Mathematical Physics
publication date: 1987-01-01T00:00:00Z
review text: This paper discusses the connections between quantum field theory, holomorphic vector bundles, and automorphic forms. While considerable mathematical background is provided, the author's intent is to give physical motivation for various recent constructions. The first part of the paper deals with the Dirac operator on a Riemann surface. Boundary conditions under which this operator is elliptic and skew adjoint lead to consideration of an infinite dimensional Grassmannian Gr. The determinant of the Dirac operator is a section of a line bundle over Gr. This identifies quantum field theories of free fermions with holomorphic sections of the determinant bundle. Multiplicative Ward identities are developed and their relation to the bozonation of fermions is discussed. The next section deals with operator valued differential forms and their residues. The Ward identities are reformulated in terms of a residue calculus. There follows an extensive generalization of the operator algebra of quantum field theory. The third major topic is quantum field theory over arbitrary ground fields. The space of all observables on an algebraic curve X is defined as an adelic product of spaces of local observables at each point of X, and a global Clifford algebra is constructed which acts on this product. The current algebra of X is investigated, and representations of the global current algebra are related to automorphic representations. In fact, the adelic space constructed above from local observables is an automorphic representation of a certain adele group GL(N,A), and the Dirac determinant is an automorphic form for this representation. The paper concludes with some remarks on string field theory and its ``adelization''.
Mathematics Subject Classification ID: 22E55
Mathematics Subject Classification ID: 81T08
Mathematics Subject Classification ID: 81T05
Mathematics Subject Classification ID: 58A25
Mathematics Subject Classification ID: 58J60
Mathematics Subject Classification ID: 81R10
Mathematics Subject Classification ID: 58D30
Mathematics Subject Classification ID: 55R65
zbMATH DE Number: 4036097
zbMATH Keywords: quantum field theory
zbMATH Keywords: holomorphic vector bundles
zbMATH Keywords: automorphic forms
zbMATH Keywords: Dirac operator
zbMATH Keywords: Riemann surface
zbMATH Keywords: infinite dimensional Grassmannian
zbMATH Keywords: line bundle
zbMATH Keywords: operator valued differential forms
zbMATH Keywords: Ward identities
zbMATH Keywords: global Clifford algebra
zbMATH Keywords: global current algebra
zbMATH Keywords: automorphic representations
zbMATH Keywords: adelic space
zbMATH Keywords: adele group
zbMATH Keywords: string field theory
reviewed by: Sidney C. Scull
MaRDI profile type: Publication
cites work: Loop groups and equations of KdV type
cites work: Characterization of Jacobian varieties in terms of soliton equations
cites work: Automorphic Forms on Adele Groups. (AM-83)
cites work: An elementary introduction to the Langlands program
cites work: Q5180619
cites work: Determinants of Cauchy-Riemann operators on a Riemann surface
cites work: Dirac operators coupled to vector potentials
cites work: Spectral asymmetry and Riemannian Geometry. I
cites work: Q3754832
cites work: Bosonization on higher genus Riemann surfaces
cites work: Infinite conformal symmetry in two-dimensional quantum field theory
cites work: Spinor representations of affine Lie algebras
cites work: Two constructions of affine Lie algebra representations and boson-fermion correspondence in quantum field theory
cites work: The arithmetic theory of loop algebras
author: Edward Witten
Recommended article: Q4524265
Recommended article: Quantum field theories on an algebraic curve
Recommended article: Quantum fields, periods and algebraic geometry
Recommended article: Algebraic Quantum Field Theory in Curved Spacetimes
Recommended article: Q4357377
Recommended article: Higher Structures in Algebraic Quantum Field Theory
Recommended article: Q3799185
Recommended article: Q4534589
Recommended article: Algebraic Quantum Field Theory
description: scientific article; zbMATH DE number 4036097