Quantum field theory, Grassmannians, and algebraic curves
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''.
scientific article; zbMATH DE number 4036097
scientific article