P(phi)_2 quantum field theories and Segal's axioms

We show that P(phi)_2 quantum field theories satisfy axioms of the type advocated by Graeme Segal.Comment: final version; to appear in Comm. Math. Phy

Homogeneous Poisson Structures on Loop Spaces of Symmetric Spaces

This paper is a sequel to [Caine A., Pickrell D., Int. Math. Res. Not., to appear, arXiv:0710.4484], where we studied the Hamiltonian systems which arise from the Evens-Lu construction of homogeneous Poisson structures on both compact and noncompact type symmetric spaces. In this paper we consider loop space analogues. Many of the results extend in a relatively routine way to the loop space setting, but new issues emerge. The main point of this paper is to spell out the meaning of the results, especially in the SU(2) case. Applications include integral formulas and factorizations for Toeplitz determinants

Ewens measures on compact groups and hypergeometric kernels

On unitary compact groups the decomposition of a generic element into product of reflections induces a decomposition of the characteristic polynomial into a product of factors. When the group is equipped with the Haar probability measure, these factors become independent random variables with explicit distributions. Beyond the known results on the orthogonal and unitary groups (O(n) and U(n)), we treat the symplectic case. In U(n), this induces a family of probability changes analogous to the biassing in the Ewens sampling formula known for the symmetric group. Then we study the spectral properties of these measures, connected to the pure Fisher-Hartvig symbol on the unit circle. The associated orthogonal polynomials give rise, as $n$ tends to infinity to a limit kernel at the singularity.Comment: New version of the previous paper "Hua-Pickrell measures on general compact groups". The article has been completely re-written (the presentation has changed and some proofs have been simplified). New references added

Multiplicity, Invariants and Tensor Product Decomposition of Tame Representations of U(\infty)

The structure of r-fold tensor products of irreducible tame representations of the inductive limit U(\infty) of unitary groups U(n) are are described, versions of contragredient representations and invariants are realized on Bargmann-Segal-Fock spaces.Comment: 48 pages, LaTeX file, to appear in J. Math. Phy

Surface normal photonic crystal waveguide coupling for N^3 distributed optoelectronic crossbar

The realization of the N^3 distributed optoelectronic crossbar requires the incorporation of bidirectional transceiver modules. The current design philosophy of these modules in their single wavelength configuration consist of the integration of VCSEL and RCE detection devices monolithically integrated with a bidirectional common waveguide. Coupling into this common waveguide is currently under investigation utilizing two methods 1.) surface normal coupling using a buried grating coupler external but monolithic surface normal coupling utilizing photonic crystal. This paper will briefly discuss the first method and its drawbacks which motivate the second photonic crystal implementation method. Our initial design work has been accomplished at 980 nm. The measure reflectance spectrum of the VCSEL/PD epitaxy structure prior to the fabrication of the photonic crystal coupler and waveguide layer

Loops in SU(2), Riemann Surfaces, and Factorization, I

In previous work we showed that a loop g:S¹→SU(2) has a triangular factorization if and only if the loop g has a root subgroup factorization. In this paper we present generalizations in which the unit disk and its double, the sphere, are replaced by a based compact Riemann surface with boundary, and its double. One ingredient is the theory of generalized Fourier-Laurent expansions developed by Krichever and Novikov. We show that a SU(2) valued multiloop having an analogue of a root subgroup factorization satisfies the condition that the multiloop, viewed as a transition function, defines a semistable holomorphic SL(2,C) bundle. Additionally, for such a multiloop, there is a corresponding factorization for determinants associated to the spin Toeplitz operators defined by the multiloop

Fractional Loop Group and Twisted K-Theory

We study the structure of abelian extensions of the group $L_qG$ of $q$-differentiable loops (in the Sobolev sense), generalizing from the case of central extension of the smooth loop group. This is motivated by the aim of understanding the problems with current algebras in higher dimensions. Highest weight modules are constructed for the Lie algebra. The construction is extended to the current algebra of supersymmetric Wess-Zumino-Witten model. An application to the twisted K-theory on $G$ is discussed.Comment: Final version in Commun. Math. Phy

Schwinger Terms and Cohomology of Pseudodifferential Operators

We study the cohomology of the Schwinger term arising in second quantization of the class of observables belonging to the restricted general linear algebra. We prove that, for all pseudodifferential operators in 3+1 dimensions of this type, the Schwinger term is equivalent to the ``twisted'' Radul cocycle, a modified version of the Radul cocycle arising in non-commutative differential geometry. In the process we also show how the ordinary Radul cocycle for any pair of pseudodifferential operators in any dimension can be written as the phase space integral of the star commutator of their symbols projected to the appropriate asymptotic component.Comment: 19 pages, plain te

Epilepsy mortality in Wales during COVID-19

Purpose: The COVID-19 pandemic has increased mortality worldwide and those with chronic conditions may have been disproportionally affected. However, it is unknown whether the pandemic has changed mortality rates for people with epilepsy. We aimed to compare mortality rates in people with epilepsy in Wales during the pandemic with pre-pandemic rates. Methods: We performed a retrospective study using individual-level linked population-scale anonymised electronic health records. We identified deaths in people with epilepsy (DPWE), i.e. those with a diagnosis of epilepsy, and deaths associated with epilepsy (DAE), where epilepsy was recorded as a cause of death on death certificates. We compared death rates in 2020 with average rates in 2015–2019 using Poisson models to calculate death rate ratios. Results: There were 188 DAE and 628 DPWE in Wales in 2020 (death rates: 7.7/100,000/year and 25.7/100,000/year). The average rates for DAE and DPWE from 2015 to 2019 were 5.8/100,000/year and 23.8/100,000/year, respectively. Death rate ratios (2020 compared to 2015–2019) for DAE were 1.34 (95%CI 1.14–1.57, p<0.001) and for DPWE were 1.08 (0.99–1.17, p = 0.09). The death rate ratios for non-COVID deaths (deaths without COVID mentioned on death certificates) for DAE were 1.17 (0.99–1.39, p = 0.06) and for DPWE were 0.96 (0.87–1.05, p = 0.37). Conclusions: The significant increase in DAE in Wales during 2020 could be explained by the direct effect of COVID-19 infection. Non-COVID-19 deaths have not increased significantly but further work is needed to assess the longer-term impact