Ambar N. Sengupta

Gauge theory on compact surfaces

Introduction Terminology and basic facts The structure of bundles and connections over compact surfaces Quantum gauge theory on the disk A conditional probability measure The Yang-Mills measure Invariants of systems of curves Loop expectation values I Some tools for the Abelian case Loop expectation values II Appendix Figures 1, 2, 3 References.

The Yang-Mills measure for S2

Abstract The Yang-Mills measure for gauge fields over the two-sphere is constructed using a conditioned white-noise process. Stochastic differential equations are used to study parallel translation under this measure. Expectation values of a broad class of Wilson loop configurations are computed explicitly and it is shown that these values are invariant under area-preserving diffeomorphisms of the sphere.

Parallel Transport over Path Spaces

We develop a differential geometric framework for parallel transport over path spaces and a corresponding discrete theory, an integrated version of the continuum theory, using a category-theoretic framework.

The semiclassical limit of the two‐dimensional quantum Yang–Mills model

The semiclassical limit of the quantum Yang–Mills partition function on a compact oriented surface is related to the symplectic volume of the moduli space of flat connections, by using an explicit expression for the symplectic form. This gives an independent proof of some recent results of Witten and Forman.

An explicit description of the symplectic structure of moduli spaces of flat connections

The moduli space of flat connections on a principal G‐bundle over a compact oriented surface of genus g≥1 is considered herein. Using the holonomies around noncontractible loops, the moduli space is described as a quotient of a submanifold of G2g. An explicit expression is obtained for the symplectic form on the smooth part of moduli space, and several properties of this form are established.

The Moduli Space of Yang–Mills Connections Over a Compact Surface

Yang–Mills connections over closed oriented surfaces of genus ≥1, for compact connected gauge groups, are constructed explicitly. The resulting formulas for Yang–Mills connections are used to carry out a Marsden–Weinstein type procedure. An explicit formula is obtained for the resulting 2-form on the moduli space. It is shown that this 2-form provides a symplectic structure on appropriate subsets of the moduli space.

Quantum free Yang–Mills on the plane

Abstract We construct a free-probability quantum Yang–Mills theory on the two dimensional plane, determine the Wilson loop expectation values, and show that this theory is the N = ∞ limit of U ( N ) quantum Yang–Mills theory on the plane. Our model provides an example of a stochastic geometry, motivated by quantum field theory, based on free probability theory.

The volume measure for flat connections as limit of the Yang–Mills measure

Abstract We prove that integration over the moduli space of flat connections can be obtained as a limit of integration with respect to the Yang–Mills measure defined in terms of the heat-kernel for the gauge group. In doing this we also give a rigorous proof of Witten’s formula for the symplectic volume of the moduli space of flat connections. Our proof uses an elementary identity connecting determinants of matrices along with a careful accounting of certain dense subsets of full measure in the moduli space.

A symplectic structure for connections on surfaces with boundary

For compact surfaces with one boundary component, and semisimple gauge groups, we construct a closed gauge invariant 2-form on the space of flat connections whose boundary holonomy lies in a fixed conjugacy class. This form descends to the moduli space under the action of the full gauge group, and provides an explicit description of a symplectic structure for this moduli space.

A support theorem for a Gaussian Radon transform in infinite dimensions

We prove that in infinite dimensions, if a bounded continuous function has zero Gaussian integral over all hyperplanes outside a closed bounded convex set then the function is zero outside this set. This is an infinite-dimensional form of the well-known Helgason support theorem for Radon transforms in finite dimensions.