Weyl group multiple Dirichlet series constructed from quadratic characters

We construct multiple Dirichlet series in several complex variables whose coefficients involve quadratic residue symbols. The series are shown to have an analytic continuation and satisfy a certain group of functional equations. These are the first examples of an infinite collection of unstable Weyl group multiple Dirichlet series in greater than two variables.Comment: incorporated referee's comment

SL(2,Z) Multiplets in N=4 SYM Theory

We discuss the action of SL(2,Z) on local operators in D=4, N=4 SYM theory in the superconformal phase. The modular property of the operator's scaling dimension determines whether the operator transforms as a singlet, or covariantly, as part of a finite or infinite dimensional multiplet under the SL(2,Z) action. As an example, we argue that operators in the Konishi multiplet transform as part of a (p,q) PSL(2,Z) multiplet. We also comment on the non-perturbative local operators dual to the Konishi multiplet.Comment: 14 pages, harvmac; v2: published version with minor change

Generalized Involution Models for Wreath Products

We prove that if a finite group $H$ has a generalized involution model, as defined by Bump and Ginzburg, then the wreath product $H \wr S_n$ also has a generalized involution model. This extends the work of Baddeley concerning involution models for wreath products. As an application, we construct a Gelfand model for wreath products of the form $A \wr S_n$ with $A$ abelian, and give an alternate proof of a recent result due to Adin, Postnikov, and Roichman describing a particularly elegant Gelfand model for the wreath product \ZZ_r \wr S_n. We conclude by discussing some notable properties of this representation and its decomposition into irreducible constituents, proving a conjecture of Adin, Roichman, and Postnikov's.Comment: 29 page

Loop Quantum Gravity a la Aharonov-Bohm

The state space of Loop Quantum Gravity admits a decomposition into orthogonal subspaces associated to diffeomorphism equivalence classes of spin-network graphs. In this paper I investigate the possibility of obtaining this state space from the quantization of a topological field theory with many degrees of freedom. The starting point is a 3-manifold with a network of defect-lines. A locally-flat connection on this manifold can have non-trivial holonomy around non-contractible loops. This is in fact the mathematical origin of the Aharonov-Bohm effect. I quantize this theory using standard field theoretical methods. The functional integral defining the scalar product is shown to reduce to a finite dimensional integral over moduli space. A non-trivial measure given by the Faddeev-Popov determinant is derived. I argue that the scalar product obtained coincides with the one used in Loop Quantum Gravity. I provide an explicit derivation in the case of a single defect-line, corresponding to a single loop in Loop Quantum Gravity. Moreover, I discuss the relation with spin-networks as used in the context of spin foam models.Comment: 19 pages, 1 figure; v2: corrected typos, section 4 expanded

Higgs Bundles, Gauge Theories and Quantum Groups

The appearance of the Bethe Ansatz equation for the Nonlinear Schr\"{o}dinger equation in the equivariant integration over the moduli space of Higgs bundles is revisited. We argue that the wave functions of the corresponding two-dimensional topological U(N) gauge theory reproduce quantum wave functions of the Nonlinear Schr\"{o}dinger equation in the $N$-particle sector. This implies the full equivalence between the above gauge theory and the $N$-particle sub-sector of the quantum theory of Nonlinear Schr\"{o}dinger equation. This also implies the explicit correspondence between the gauge theory and the representation theory of degenerate double affine Hecke algebra. We propose similar construction based on the $G/G$ gauged WZW model leading to the representation theory of the double affine Hecke algebra. The relation with the Nahm transform and the geometric Langlands correspondence is briefly discussed.Comment: 48 pages, typos corrected, one reference adde

On absolute moments of characteristic polynomials of a certain class of complex random matrices

Integer moments of the spectral determinant $|\det(zI-W)|^2$ of complex random matrices $W$ are obtained in terms of the characteristic polynomial of the Hermitian matrix $WW^*$ for the class of matrices $W=AU$ where $A$ is a given matrix and $U$ is random unitary. This work is motivated by studies of complex eigenvalues of random matrices and potential applications of the obtained results in this context are discussed.Comment: 41 page, typos correcte

Automorphic Instanton Partition Functions on Calabi-Yau Threefolds

We survey recent results on quantum corrections to the hypermultiplet moduli space M in type IIA/B string theory on a compact Calabi-Yau threefold X, or, equivalently, the vector multiplet moduli space in type IIB/A on X x S^1. Our main focus lies on the problem of resumming the infinite series of D-brane and NS5-brane instantons, using the mathematical machinery of automorphic forms. We review the proposal that whenever the low-energy theory in D=3 exhibits an arithmetic "U-duality" symmetry G(Z) the total instanton partition function arises from a certain unitary automorphic representation of G, whose Fourier coefficients reproduce the BPS-degeneracies. For D=4, N=2 theories on R^3 x S^1 we argue that the relevant automorphic representation falls in the quaternionic discrete series of G, and that the partition function can be realized as a holomorphic section on the twistor space Z over M. We also offer some comments on the close relation with N=2 wall crossing formulae.Comment: 25 pages, contribution to the proceedings of the workshop "Algebra, Geometry and Mathematical Physics", Tjarno, Sweden, 25-30 October, 201

Effect of cytisine on some brain and hepatic biochemical parameters in spontaneously hypertensive rats

Tobacco smoking is a risk factor for variety of cardio-vascular diseases, such as hypertension, myocardial infarction, stroke and many others. It is of great importance for hypertensive patients to stop smoking. One of the medicines widely used for smoking cessation in Bulgaria is the original Bulgarian product Tabex®, which is developed on the basis of natural plant alkaloid cytisine. The aim of the following study was to ivestigate the effects of cytisine on some brain and hepatic biochemical parameters in spontaneously hypertensive rats (SHR), an widely used rodent model for human essential hypertension, and to compare the obtained results with their age-matched normotensive controls Wistar Kyoto (WKY). Multiple cytisine administration did not affect the activity of ethylmorphine-N-demethylase (EMND) and anylinehydroxylase (AH), as well as the quantity of cytochrome P 450, nor in WKY neither in SHR In the liver cytisine increased the MDA quantity both in SHR and in WKY, by 25% (p<0.05) and by 29% (p<0.05) respectively, while the GSH level was not significantly changed by the compound in both strains. In contrast, on the brain level, cytisine administration to SHR caused more prominent toxicity, resulted in GSH depletion and increased MDA quantity, while in WKY strain did not exert any toxicity. Cytisine did not significantly affect ALAT and ASAT activity in both strains. In conclusion, the results of our study suggest higher brain toxicity of cytisine in spontaneously hypertensive rats, that might be due to their pathophysiological characteristics

Toward verification of the Riemann hypothesis: Application of the Li criterion

We substantially apply the Li criterion for the Riemann hypothesis to hold. Based upon a series representation for the sequence \{\lambda_k\}, which are certain logarithmic derivatives of the Riemann xi function evaluated at unity, we determine new bounds for relevant Riemann zeta function sums and the sequence itself. We find that the Riemann hypothesis holds if certain conjectured properties of a sequence \eta_j are valid. The constants \eta_j enter the Laurent expansion of the logarithmic derivative of the zeta function about s=1 and appear to have remarkable characteristics. {\em On our conjecture}, not only does the Riemann hypothesis follow, but an inequality governing the values \lambda_n and inequalities for the sums of reciprocal powers of the nontrivial zeros of the zeta function.Comment: to appear in Math. Physics, Analysis and Geometry; 1 figur