Wodzicki residue and anomalies of current algebras

The commutator anomalies (Schwinger terms) of current algebras in $3+1$ dimensions are computed in terms of the Wodzicki residue of pseudodifferential operators; the result can be written as a (twisted) Radul 2-cocycle for the Lie algebra of PSDO's. The construction of the (second quantized) current algebra is closely related to a geometric renormalization of the interaction Hamiltonian $H_I=j_{\mu} A^{\mu}$ in gauge theory.Comment: 15 pages, updated version of a talk at the Baltic School in Field Theory, September 199

From Gauge Anomalies to Gerbes and Gerbal Representations: Group Cocycles in Quantum Theory

In this paper I shall discuss the role of group cohomology in quantum mechanics and quantum field theory. First, I recall how cocycles of degree 1 and 2 appear naturally in the context of gauge anomalies. Then we investigate how group cohomology of degree 3 comes from a prolongation problem for group extensions and we discuss its role in quantum field theory. Finally, we discuss a generalization to representation theory where a representation is replaced by a 1-cocycle or its prolongation by a circle, and point out how this type of situations come up in the quantization of Yang-Mills theory

Elementary Derivation of the Chiral Anomaly

An elementary derivation of the chiral gauge anomaly in all even dimensions is given in terms of noncommutative traces of pseudo-differential operators.Comment: Minor errors and misprints corrected, a reference added. AmsTex file, 12 output pages. If you do not have preloaded AmsTex you have to \input amstex.te

Fractional Loop Group and Twisted K-Theory

We study the structure of abelian extensions of the group $L_qG$ of $q$-differentiable loops (in the Sobolev sense), generalizing from the case of central extension of the smooth loop group. This is motivated by the aim of understanding the problems with current algebras in higher dimensions. Highest weight modules are constructed for the Lie algebra. The construction is extended to the current algebra of supersymmetric Wess-Zumino-Witten model. An application to the twisted K-theory on $G$ is discussed.Comment: Final version in Commun. Math. Phy

Regularization and Quantization of Higher Dimensional Current Algebras

We present some recently discovered infinite dimensional Lie algebras that can be understood as extensions of the algebra Map(M,g) of maps from a compact p-dimensional manifold to some finite dimensional Lie algebra g. In the first part of the paper, we describe the physical motivations for the study of these algebras. In the second part, we discuss their realization in terms of pseudo-differential operators and comment on their possible representation theory.Comment: (Talk given at the G\"ursey Memorial Conference I on Strings and Symmetries, Istanbul, Turkey, June 6-10 1994.) 11 pages, plain TEX. ITP 94-1

Families index theorem in supersymmetric WZW model and twisted K-theory: The SU(2) case

The construction of twisted K-theory classes on a compact Lie group is reviewed using the supersymmetric Wess-Zumino-Witten model on a cylinder. The Quillen superconnection is introduced for a family of supercharges parametrized by a compact Lie group and the Chern character is explicitly computed in the case of SU(2). For large euclidean time, the character form is localized on a D-brane.Comment: Version 2: Essentially simplified proof of the main result using a map from twisted K-theory to gerbes modulo the twisting gerbe; references added + minor correction

Current Algebra in Three Dimensions

We study a three dimensional analogue of the Wess--Zumino--Witten model, which describes the Goldstone bosons of three dimensional Quantum Chromodynamics. The topologically non--trivial term of the action can also be viewed as a nonlinear realization of Chern--Simons form. We obtain the current algebra of this model by canonical methods. This is a three dimensional generalization of the Kac--Moody algebra.Comment: 11 pages, UR-1266, ER40685-72