Orlando Alvarez

Topological quantization and cohomology

The relationships between topological charge quantization, Lagrangians and various cohomology theories are studied. A very general criterion for charge quantization is developed and applied to various physical models. The relationship between cohomology and homotopy is discussed.

Gravitational anomalies and the family's index theorem

We discuss the use of the familys index theorem in the study of gravitational anomalies. The geometrical framework required to apply the familys index theorem is presented and the relation to gravitational anomalies is discussed. We show how physics necessitates the introduction of the notion oflocal cohomology which is distinct from the ordinary topological cohomology. The recent results of Alvarez-Gaumé and Witten are derived by using the familys index theorem.

String theory and loop space index theorems

We study index theorems for the Dirac-Ramond operator on a compact Riemannian manifold. The existence of a group action on the loop space makes possible the definition of a character valued index which we calculate by using a two-dimensional sigma model withN=1/2 supersymmetry. We compute the Euler characteristic, the Hirzebruch signature and the Dirac-Ramond genus of loop space. We compare our results to the calculations made by using the Atiyah-Singer character-valued index theorem.

Fermion determinants, chiral symmetry, and the Wess-Zumino anomaly

Abstract A general method for constructing exactly solvable fermion determinants is discussed. A two-dimensional determinant is solved exactly. A new class of four-dimensional fermion models is presented. These theories are non-renormalizable yet the fermion determinant can be calculated and there is an analogue of the Adler-Bell-Jackiw anomaly.

Conformal Anomalies and the Index Theorem

Abstract In some two-dimensional field theories, the family index theorem can be used to calculate the conformal anomaly. This is surprising since there is no known underlying topological origin for the conformal anomaly. The index theorem approach provides a different way of understanding Friedans theorem relating the conformal anomaly to the Virasoro charge. Various aspects of string models are discussed within the context of the index theorem.

The supersymmetric σ-model and the geometry of the Weyl-Kač character formula

Abstract Field theoretic and geometric ideas are used to construct a chiral supersymmetric field theory for which the ground state is a specified irreducible representation of a centrally extended loop group. The character index of the associated supercharge (an appropriate Dirac operator on LG/T) is the Weyl-Kac character formula which we compute explicitly by the steepest-descent approximation.

Vertex operator construction of the SO(2n+1) Kac-Moody algebra and its spinor representation

Abstract An explicit representation of the B n (1) affine Lie algebra (Kac-Moody algebra) is constructed in terms of vertex operators associated with the Chevalley basis of the underlyingfinite-dimentsionnal Lie algebra. This construction, contrary to the simpler current algebra one, gives a concrete realization of the spinor representation of the algebra. The key feature is a partial bosonization of two-dimensional Weyl-Majorana free fermions. The vertex operators associated with the long and short roots of the B n algebra have fermion number zero and one, respectively.

Analytic index for a family of Dirac-Ramond operators.

We derive a cohomological formula for the analytic index of a family of Dirac–Ramond operators and exhibit its modular properties.

An Introduction to Polyakov’s String Model

This talk is an introduction to Polyakov’s string model. The objective is to demonstrate how the Liouville field theory enters into the string model. Thorn Curtright will discuss the quantum Liouville theory in the next talk.