Yang-Mills theory and the Segal-Bargmann transform

We use a variant of the classical Segal-Bargmann transform to understand the canonical quantization of Yang-Mills theory on a space-time cylinder. This transform gives a rigorous way to make sense of the Hamiltonian on the gauge-invariant subspace. Our results are a rigorous version of the widely accepted notion that on the gauge-invariant subspace the Hamiltonian should reduce to the Laplacian on the compact structure group. We show that the infinite-dimensional classical Segal-Bargmann transform for the space of connections, when restricted to the gauge-invariant subspace, becomes the generalized Segal-Bargmann transform for the the structure group

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.Comment: Final version, minor typographical corrections. To appear in Comm. Math. Phy

The Brown measure of the free multiplicative Brownian motion

The free multiplicative Brownian motion $b_{t}$ is the large-$N$ limit of the Brownian motion on $\mathsf{GL}(N;\mathbb{C}),$ in the sense of $\ast$-distributions. The natural candidate for the large-$N$ limit of the empirical distribution of eigenvalues is thus the Brown measure of $b_{t}$. In previous work, the second and third authors showed that this Brown measure is supported in the closure of a region $\Sigma_{t}$ that appeared work of Biane. In the present paper, we compute the Brown measure completely. It has a continuous density $W_{t}$ on $\bar{\Sigma}_{t},$ which is strictly positive and real analytic on $\Sigma_{t}$. This density has a simple form in polar coordinates: $$W_{t}(r,\theta)=\frac{1}{r^{2}}w_{t}(\theta),$$ where $w_{t}$ is an analytic function determined by the geometry of the region $\Sigma_{t}$. We show also that the spectral measure of free unitary Brownian motion $u_{t}$ is a "shadow" of the Brown measure of $b_{t}$, precisely mirroring the relationship between Wigner's semicircle law and Ginibre's circular law. We develop several new methods, based on stochastic differential equations and PDE, to prove these results.Comment: Added references to subsequent works building on these results. Made a notational change, replacing the regularization parameter x with epsilo

Star Cluster Candidates in M81

We present a catalog of extended objects in the vicinity of M81 based a set of 24 Hubble Space Telescope Advanced Camera for Surveys (ACS) Wide Field Camera (WFC) F814W (I-band) images. We have found 233 good globular cluster candidates; 92 candidate HII regions, OB associations, or diffuse open clusters; 489 probable background galaxies; and 1719 unclassified objects. We have color data from ground-based g- and r-band MMT Megacam images for 79 galaxies, 125 globular cluster candidates, 7 HII regions, and 184 unclassified objects. The color-color diagram of globular cluster candidates shows that most fall into the range 0.25 < g-r < 1.25 and 0.5 < r-I < 1.25, similar to the color range of Milky Way globular clusters. Unclassified objects are often blue, suggesting that many of them are likely to be HII regions and open clusters, although a few galaxies and globular clusters may be among them.Comment: 35 pages, 11 figures, submitted to A

Invariant Natural Killer T-Cell Control of Type 1 Diabetes: A Dendritic Cell Genetic Decision of a Silver Bullet or Russian Roulette

OBJECTIVE: In part, activation of invariant natural killer T (iNKT)-cells with the superagonist alpha-galactosylceramide (alpha-GalCer) inhibits the development of T-cell-mediated autoimmune type 1 diabetes in NOD mice by inducing the downstream differentiation of antigen-presenting dendritic cells (DCs) to an immunotolerogenic state. However, in other systems iNKT-cell activation has an adjuvant-like effect that enhances rather than suppresses various immunological responses. Thus, we tested whether in some circumstances genetic variation would enable activated iNKT-cells to support rather than inhibit type 1 diabetes development. RESEARCH DESIGN AND METHODS: We tested whether iNKT-conditioned DCs in NOD mice and a major histocompatibility complex-matched C57BL/6 (B6) background congenic stock differed in capacity to inhibit type 1 diabetes induced by the adoptive transfer of pathogenic AI4 CD8 T-cells. RESULTS: Unlike those of NOD origin, iNKT-conditioned DCs in the B6 background stock matured to a state that actually supported rather than inhibited AI4 T-cell-induced type 1 diabetes. The induction of a differing activity pattern of T-cell costimulatory molecules varying in capacity to override programmed death-ligand-1 inhibitory effects contributes to the respective ability of iNKT-conditioned DCs in NOD and B6 background mice to inhibit or support type 1 diabetes development. Genetic differences inherent to both iNKT-cells and DCs contribute to their varying interactions in NOD and B6.H2(g7) mice. CONCLUSIONS: This great variability in the interactions between iNKT-cells and DCs in two inbred mouse strains should raise a cautionary note about considering manipulation of this axis as a potential type 1 diabetes prevention therapy in genetically heterogeneous humans

Coherent states for compact Lie groups and their large-N limits

The first two parts of this article surveys results related to the heat-kernel coherent states for a compact Lie group K. I begin by reviewing the definition of the coherent states, their resolution of the identity, and the associated Segal-Bargmann transform. I then describe related results including connections to geometric quantization and (1+1)-dimensional Yang--Mills theory, the associated coherent states on spheres, and applications to quantum gravity. The third part of this article summarizes recent work of mine with Driver and Kemp on the large-N limit of the Segal--Bargmann transform for the unitary group U(N). A key result is the identification of the leading-order large-N behavior of the Laplacian on "trace polynomials."Comment: Submitted to the proceeding of the CIRM conference, "Coherent states and their applications: A contemporary panorama.