Congruence Property In Conformal Field Theory

The congruence subgroup property is established for the modular representations associated to any modular tensor category. This result is used to prove that the kernel of the representation of the modular group on the conformal blocks of any rational, C_2-cofinite vertex operator algebra is a congruence subgroup. In particular, the q-character of each irreducible module is a modular function on the same congruence subgroup. The Galois symmetry of the modular representations is obtained and the order of the anomaly for those modular categories satisfying some integrality conditions is determined.Comment: References are updated. Some typos and grammatical errors are correcte

From 2-sequents and Linear Nested Sequents to Natural Deduction for Normal Modal Logics

We extend to natural deduction the approach of Linear Nested Sequents and 2-sequents. Formulas are decorated with a spatial coordinate, which allows a formulation of formal systems in the original spirit of natural deduction---only one introduction and one elimination rule per connective, no additional (structural) rule, no explicit reference to the accessibility relation of the intended Kripke models. We give systems for the normal modal logics from K to S4. For the intuitionistic versions of the systems, we define proof reduction, and prove proof normalisation, thus obtaining a syntactical proof of consistency. For logics K and K4 we use existence predicates (following Scott) for formulating sound deduction rules

Geometry of Prym semicanonical pencils and an application to cubic threefolds

In the moduli space $\mathcal{R}_{\mathrm{g}}$ of double étale covers of curves of a fixed genus $g$, the locus formed by covers of curves with a semicanonical pencil consists of two irreducible divisors $\mathcal{T}_g{ }^e$ and $\mathcal{T}_g^o$. We study the Prym map on these divisors, which shows significant differences between them and has a rich geometry in the cases of low genus. In particular, the analysis of $\mathcal{T}_5^o$ has enumerative consequences for lines on cubic threefolds

Painlev\'e III′' and the Hankel Determinant Generated by a Singularly Perturbed Gaussian Weight

In this paper, we study the Hankel determinant generated by a singularly perturbed Gaussian weight $$w(x,t)=\mathrm{e}^{-x^{2}-\frac{t}{x^{2}}},\;\;x\in(-\infty, \infty),\;\;t>0.$$ By using the ladder operator approach associated with the orthogonal polynomials, we show that the logarithmic derivative of the Hankel determinant satisfies both a non-linear second order difference equation and a non-linear second order differential equation. The Hankel determinant also admits an integral representation involving a Painlev\'e III$'$. Furthermore, we consider the asymptotics of the Hankel determinant under a double scaling, i.e. $n\rightarrow\infty$ and $t\rightarrow 0$ such that $s=(2n+1)t$ is fixed. The asymptotic expansions of the scaled Hankel determinant for large $s$ and small $s$ are established, from which Dyson's constant appears.Comment: 22 page

Random Matrix Models, Double-Time Painlev\'e Equations, and Wireless Relaying

This paper gives an in-depth study of a multiple-antenna wireless communication scenario in which a weak signal received at an intermediate relay station is amplified and then forwarded to the final destination. The key quantity determining system performance is the statistical properties of the signal-to-noise ratio (SNR) \gamma\ at the destination. Under certain assumptions on the encoding structure, recent work has characterized the SNR distribution through its moment generating function, in terms of a certain Hankel determinant generated via a deformed Laguerre weight. Here, we employ two different methods to describe the Hankel determinant. First, we make use of ladder operators satisfied by orthogonal polynomials to give an exact characterization in terms of a "double-time" Painlev\'e differential equation, which reduces to Painlev\'e V under certain limits. Second, we employ Dyson's Coulomb Fluid method to derive a closed form approximation for the Hankel determinant. The two characterizations are used to derive closed-form expressions for the cumulants of \gamma, and to compute performance quantities of engineering interest.Comment: 72 pages, 6 figures; Minor typos corrected; Two additional lemmas added in Appendix

Fuzzy Scalar Field Theory as a Multitrace Matrix Model

We develop an analytical approach to scalar field theory on the fuzzy sphere based on considering a perturbative expansion of the kinetic term. This expansion allows us to integrate out the angular degrees of freedom in the hermitian matrices encoding the scalar field. The remaining model depends only on the eigenvalues of the matrices and corresponds to a multitrace hermitian matrix model. Such a model can be solved by standard techniques as e.g. the saddle-point approximation. We evaluate the perturbative expansion up to second order and present the one-cut solution of the saddle-point approximation in the large N limit. We apply our approach to a model which has been proposed as an appropriate regularization of scalar field theory on the plane within the framework of fuzzy geometry.Comment: 1+25 pages, replaced with published version, minor improvement