S. G. Rajeev

The Holomorphic Geometry of Closed Bosonic String Theory and Diff S1 / S1

Abstract We present a proposal for a classical non-perturbative bosonic closed string field theory based on Kahler geometry. Motivated by the observation that the loop space of Minkowski space-time is a Kahler manifold, we conjecture that infinite-dimensional complex (Kahler) geometry is the right setting for closed string field theory and that the correct dynamical variable (closed string field) is the Kahler potential. To incorporate reparametrization invariance, one must consider the space of complex structures Diff S1/S1. Geometrical considerations then lead us to a (non-linear) equation of motion for the Kahler potential which is that the curvature of a certain vector bundle over Diff S1/S1 vanish. This is basically the requirement of conformal invariance. Loops on flat Minkowski space are shown to be a solution only if the space-time dimension is 26. We also discuss geometric quantization since our approach can be viewed as an application of geometric quantization to string theory. Previously announced mathematical results that Diff S1/S1 is a homogeneous Kahler manifold are established in more detail and its curvature is computed explicitly. We also give an axiomatic formulation of the minimal geometric setting we require — this is an attempt to avoid basing the theory on loops of a given riemannian manifold. Einsteins field equations are derived in an adiabatic approximation. The relation of our work to some other approaches to string theory is briefly discussed.

Renormalization in quantum mechanics

We implement the concept of Wilson renormalization in the context of simple quantum-mechanical systems. The attractive inverse square potential leads to a

Current algebras in

\ensuremath{\beta}

d+1

function with a nontrivial ultraviolet stable fixed point and the Hulthen potential exhibits both ultraviolet and infrared stable fixed points. We also discuss the implementation of the Wilson scheme in the broader context of one-dimensional potential problems. The possibility of an analogue of Zamolodchikovs

-dimensions and determinant bundles over infinite-dimensional Grassmannians

C

YANG-MILLS THEORY ON A CYLINDER

function in these systems is also discussed.

Yangian Symmetries of Matrix Models and Spin Chains: The Dilatation Operator of N = 4 SYM

We extend the methods of Pressley and Segal for constructing cocycle representations of the restricted general linear group in infinite-dimensions to the case of a larger linear group modeled by Schatten classes of rank 1≦p<∞. An essential ingredient is the generalization of the determinant line bundle over an infinite-dimensional Grassmannian to the case of an arbitrary Schatten rank,p≧1. The results are used to obtain highest weight representations of current algebras (with the operator Schwinger terms) ind+1-dimensions when the space dimensiond is any odd number.

Quantum hadrodynamics in two-dimensions

Abstract Yang-Mills theory on a two dimensional cylinder is studied in the hamiltonian formalism, without using gauge conditions. Since the only gauge invariant variable is the Wilson loop (holonomy) this system is equivalent to a finite dimensional system. The eigenstates and eigenvalues of the hamiltonian are found exactly.

Quantization of contact manifolds and thermodynamics

We present an analysis of the Yangian symmetries of various bosonic sectors of the dilatation operator of

Non-abelian bosonization without Wess-Zumino terms. I. New current algebra

\cal N