Bruce K. Driver

Yang-Mills Theory and the Segal-Bargmann Transform

Abstract:We use a variant of the Segal–Bargmann transform to study canonically quantized Yang–Mills theory on a space-time cylinder with a compact structure group K. The non-existent Lebesgue measure on the space of connections is “approximated” by a Gaussian measure with large variance. The Segal–Bargmann transform is then a unitary map from the L2 space over the space of connections to a holomorphicL2 space over the space of complexified connections with a certain Gaussian measure. This transform is given roughly by followed by analytic continuation. Here is the Laplacian on the space of connections and is the Hamiltonian for the quantized theory.On the gauge-trivial subspace, consisting of functions of the holonomy around the spatial circle, the Segal–Bargmann transform becomes followed by analytic continuation, where ΔK is the Laplacian for the structure group K. This result gives a rigorous meaning to the idea that reduces to ΔK on functions of the holonomy. By letting the variance of the Gaussian measure tend to infinity we recover the standard realization of the quantized Yang–Mills theory on a space-time cylinder, namely, −½ΔK is the Hamiltonian and L2(K) is the Hilbert space. As a byproduct of these considerations, we find a new one-parameter family of unitary transforms from L2(K) to certain holomorphic L2-spaces over the complexification of K. This family of transformations interpolates between the two previously known unitary transformations.Our work is motivated by results of Landsman and Wren and uses probabilistic techniques similar to those of Gross and Malliavin.

YM2: continuum expectations, lattice convergence, and lassos

The two dimensional Yang-Mills theory (YM2) is analyzed in both the continuum and the lattice. In the complete axial gauge the continuum theory may be defined in terms of a Lie algebra valued white noise, and parallel translation may be defined by stochastic differential equations. This machinery is used to compute the expectations of gauge invariant functions of the parallel translation operators along a collection of curvesC. The expectation values are expressed as finite dimensional integrals with densities that are products of the heat kernel on the structure group. The time parameters of the heat kernels are determined by the areas enclosed by the collectionC, and the arguments are determined by the crossing topologies of the curves inC. The expectations for the Wilson lattice models have a similar structure, and from this it follows that in the limit of small lattice spacing the lattice expectations converge to the continuum expectations. It is also shown that the lasso variables advocated by L. Gross [36] exist and are sufficient to generate all the measurable functions on the YM2-measure space.

Integration by parts for heat kernel measures revisited

Abstract Stochastic calculus proofs of the integration by parts formula for cylinder functions of parallel translation on the Wiener space of a compact Riemannian manifold (M) are given. These formulas are used to prove a new probabilistic formula for the logarithmic derivative of the heart kernel on M. This new formula is well suited for generalizations to infinite dimensional manifolds.

Equivalence of heat kernel measure and pinned Wiener measure on loop groups

Abstract We show that heat kernel measure and pinned Wiener measure on loop groups over simply connected compact Lie groups are equivalent.

Curved Wiener Space Analysis

The purpose of these notes is to first provide some basic background to Riemannian geometry and stochastic calculus on manifolds and then to cover some of the more recent developments pertaining to analysis on “curved Wiener spaces.” Essentially no differential geometry is assumed, however, it is assumed that the reader is comfortable with stochastic calculus and differential equations on Euclidean spaces. Here is a brief description of what will be covered in the text below.

Three Proofs of the Makeenko–Migdal Equation for Yang–Mills Theory on the Plane

We give three short proofs of the Makeenko–Migdal equation for the Yang–Mills measure on the plane, two using the edge variables and one using the loop or lasso variables. Our proofs are significantly simpler than the earlier pioneering rigorous proofs given by Lévy and by Dahlqvist. In particular, our proofs are “local” in nature, in that they involve only derivatives with respect to variables adjacent to the crossing in question. In an accompanying paper with Gabriel, we show that two of our proofs can be adapted to the case of Yang–Mills theory on any compact surface.

The Makeenko–Migdal Equation for Yang–Mills Theory on Compact Surfaces

We prove the Makeenko–Migdal equation for two-dimensional Euclidean Yang–Mills theory on an arbitrary compact surface, possibly with boundary. In particular, we show that two of the proofs given by the first, third, and fourth authors for the plane case extend essentially without change to compact surfaces.

Constrained Rough Paths

We introduce a notion of rough paths on embedded submanifolds and demonstrate that this class of rough paths is natural. On the way we develop a notion of rough integration and an ecient and intrinsic theory of rough dierential equations (RDEs) on manifolds. The theory of RDEs is then used to construct parallel translation along manifold valued rough paths. Finally, this framework is used to show there is a one to one correspondence between rough paths on a d { dimensional manifold and rough paths on d { dimensional Euclidean space. This last result is a rough path analogue of Cartan’s development map and its stochastic version which was developed by Eells and Elworthy and Malliavin.

Surjectivity of the Taylor map for complex nilpotent Lie groups

A Hermitian form q on the dual space, g, of the Lie algebra, g, of a simply connected complex Lie group, G, determines a sub-Laplacian, �, on G. Assuming H¨ ormander’s condition for hypoellipticity, there is a smooth heat kernel measure, ρt , on G associated to e t�/ 4 . In a companion paper [6], we proved the existence of a unitary “Taylor” map from the space of holomorphic functions in L 2 (G ,ρ t ) onto J 0 t (a subspace of) the dual of the universal enveloping algebra of g. Here we give a very different proof of the surjectivity of the Taylor map under the assumption that G is nilpotent. This proof provides further insight into the structure of the Taylor map. In particular we show that the finite rank tensors are dense in J 0 t when the Lie algebra is graded and the Laplacian is adapted to the gradation. We also show how the Fourier–Wigner transform produces a natural family of holomorphic functions in L 2 (G ,ρ t ), for appropriate t, when G is the complex Heisenberg group.

Hypoelliptic heat kernels on infinite-dimensional Heisenberg groups

We study the law of a hypoelliptic Brownian motion on an infinite-dimensional Heisenberg group based on an abstract Wiener space. We show that the endpoint distribution, which can be seen as a heat kernel measure, is absolutely continuous with respect to a certain product of Gaussian and Lebesgue measures, that the heat kernel is quasi-invariant under translation by the Cameron-Martin subgroup, and that the Radon-Nikodym derivative is Malliavin smooth.