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

Cartilage on the Move: Cartilage Lineage Tracing During Tadpole Metamorphosis

The reorganization of cranial cartilages during tadpole metamorphosis is a set of complex processes. The fates of larval cartilage-forming cells (chondrocytes) and sources of adult chondrocytes are largely unknown. Individual larval cranial cartilages may either degenerate or remodel, while many adult cartilages appear to form de novo during metamorphosis. Determining the extent to which adult chondrocytes/cartilages are derived from larval chondrocytes during metamorphosis requires new techniques in chondrocyte lineage tracing. We have developed two transgenic systems to label cartilage cells throughout the body with fluorescent proteins. One system strongly labels early tadpole cartilages only. The other system inducibly labels forming cartilages at any developmental stage. We examined cartilages of the skull (viscero- and neurocranium), and identified larval cartilages that either resorb or remodel into adult cartilages. Our data show that the adult otic capsules, tecti anterius and posterius, hyale, and portions of Meckel\u27s cartilage are derived from larval chondrocytes. Our data also suggest that most adult cartilages form de novo, though we cannot rule out the potential for extreme larval chondrocyte proliferation or de- and re-differentiation, which could dilute our fluorescent protein signal. The transgenic lineage tracing strategies developed here are the first examples of inducible, skeleton-specific, lineage tracing in Xenopus

A shared role for sonic hedgehog signalling in patterning chondrichthyan gill arch appendages and tetrapod limbs.

Chondrichthyans (sharks, skates, rays and holocephalans) possess paired appendages that project laterally from their gill arches, known as branchial rays. This led Carl Gegenbaur to propose that paired fins (and hence tetrapod limbs) originally evolved via transformation of gill arches. Tetrapod limbs are patterned by asonic hedgehog(Shh)-expressing signalling centre known as the zone of polarising activity, which establishes the anteroposterior axis of the limb bud and maintains proliferative expansion of limb endoskeletal progenitors. Here, we use loss-of-function, label-retention and fate-mapping approaches in the little skate to demonstrate that Shh secretion from a signalling centre in the developing gill arches establishes gill arch anteroposterior polarity and maintains the proliferative expansion of branchial ray endoskeletal progenitor cells. These findings highlight striking parallels in the axial patterning mechanisms employed by chondrichthyan branchial rays and paired fins/limbs, and provide mechanistic insight into the anatomical foundation of Gegenbaur's gill arch hypothesis.This research was supported by a Royal Society University Research Fellowship [UF130182 to JAG], by Plum foundation John E. Dowling and Laura and Arthur Colwin Endowed Summer Research Fellowships at the Marine Biological Laboratory to JAG, by a grant from the University of Cambridge Isaac Newton Trust to [14.23z to JAG], and by a grant from the Natural Sciences and Engineering Research Council of Canada [A5056 to BKH].This is the final version of the article. It first appeared from The Company of Biologists via http://dx.doi.org/10.1242/dev.13388

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.

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

Coherent states on spheres

We describe a family of coherent states and an associated resolution of the identity for a quantum particle whose classical configuration space is the d-dimensional sphere S^d. The coherent states are labeled by points in the associated phase space T*(S^d). These coherent states are NOT of Perelomov type but rather are constructed as the eigenvectors of suitably defined annihilation operators. We describe as well the Segal-Bargmann representation for the system, the associated unitary Segal-Bargmann transform, and a natural inversion formula. Although many of these results are in principle special cases of the results of B. Hall and M. Stenzel, we give here a substantially different description based on ideas of T. Thiemann and of K. Kowalski and J. Rembielinski. All of these results can be generalized to a system whose configuration space is an arbitrary compact symmetric space. We focus on the sphere case in order to be able to carry out the calculations in a self-contained and explicit way.Comment: Revised version. Submitted to J. Mathematical Physic

The Network Structure of Posttraumatic Stress Disorder Among Filipina Migrant Domestic Workers: Comorbidity With Depression

Background Labour migrants are exposed to potentially traumatic events throughout the migration cycle, making them susceptible to developing mental disorders. Posttraumatic stress disorder (PTSD) is often comorbid with depression. Comorbidity worsens the course of illness, prognosis, treatment response, and increases suicidal risk. Using network analysis, this study examined the structure of PTSD and depression in a sample of migrant domestic workers, an especially vulnerable community of labour migrants. This study sought to derive the central or most important symptoms, strongest edges or relationships among symptoms, and bridge symptoms between PTSD and depression. Methods Data were obtained from 1,375 Filipina domestic workers in Macao SAR, China. Data from a subsample of 1,258 trauma-exposed participants were analysed using R software. Results Most of the strongest edges were within the same disorder and, for PTSD, within the same symptom cluster. Highest node centrality were PCL-5’s ‘avoid thoughts’, ‘lose interest’, ‘negative emotions’, and ‘not concentrate’, and PHQ-9’s ‘sleep difficulties’. The bridge symptoms were PHQ-9’s ‘sleep difficulties,’ ‘psychomotor agitation/retardation,’ and ‘fatigue,’ PCL-5’s ‘not concentrate’, and PHQ-9’s ‘worthlessness’ and ‘anhedonia’. Limitations Results may not generalize to Filipino migrant workers in other occupations and to male migrant workers. Potentially relevant symptoms like somatic symptoms and fear of somatic and mental symptoms were not included. Conclusions Central and bridge symptoms are the most important nodes in the network. Developing interventions targeting these symptoms, particularly depression symptoms, is a promising alternative to PTSD treatment given substantial barriers to specialist care for this population