Scaling Lee-Yang Model on a Sphere. I. Partition Function

Some general properties of perturbed (rational) CFT in the background metric of symmetric 2D sphere of radius $R$ are discussed, including conformal perturbation theory for the partition function and the large $R$ asymptotic. The truncated conformal space scheme is adopted to treat numerically perturbed rational CFT's in the spherical background. Numerical results obtained for the scaling Lee-Yang model lead to the conclusion that the partition function is an entire function of the coupling constant. Exploiting this analytic structure we are able to describe rather precisely the ``experimental'' truncated space data, including even the large $R$ behavior, starting only with the CFT information and few first terms of conformal perturbation theory.Comment: Extended version of a talk presented at the NATO Advanced Research Workshop on Statistical Field Theories, Como 18--23 June 200

Correlation functions in conformal Toda field theory I

Two-dimensional sl(n) quantum Toda field theory on a sphere is considered. This theory provides an important example of conformal field theory with higher spin symmetry. We derive the three-point correlation functions of the exponential fields if one of the three fields has a special form. In this case it is possible to write down and solve explicitly the differential equation for the four-point correlation function if the fourth field is completely degenerate. We give also expressions for the three-point correlation functions in the cases, when they can be expressed in terms of known functions. The semiclassical and minisuperspace approaches in the conformal Toda field theory are studied and the results coming from these approaches are compared with the proposed analytical expression for the three-point correlation function. We show, that in the framework of semiclassical and minisuperspace approaches general three-point correlation function can be reduced to the finite-dimensional integral.Comment: 54 pages, JHEP styl

Interlaced particle systems and tilings of the Aztec diamond

Motivated by the problem of domino tilings of the Aztec diamond, a weighted particle system is defined on $N$ lines, with line $j$ containing $j$ particles. The particles are restricted to lattice points from 0 to $N$, and particles on successive lines are subject to an interlacing constraint. It is shown that marginal distributions for this particle system can be computed exactly. This in turn is used to give unified derivations of a number of fundamental properties of the tiling problem, for example the evaluation of the number of distinct configurations and the relation to the GUE minor process. An interlaced particle system associated with the domino tiling of a certain half Aztec diamond is similarly defined and analyzed.Comment: 17 pages, 4 figure

Heat-kernels and functional determinants on the generalized cone

We consider zeta functions and heat-kernel expansions on the bounded, generalized cone in arbitrary dimensions using an improved calculational technique. The specific case of a global monopole is analysed in detail and some restrictions thereby placed on the $A_{5/2}$ coefficient. The computation of functional determinants is also addressed. General formulas are given and known results are incidentally, and rapidly, reproduced.Comment: 26p,LaTeX.(Cosmetic changes and eqns (9.8),(11.2) corrected.

Unit circle elliptic beta integrals

We present some elliptic beta integrals with a base parameter on the unit circle, together with their basic degenerations.Comment: 15 pages; minor corrections, references updated, to appear in Ramanujan

Liouville Perturbation Theory

A comparison is made between proposals for the exact three point function in Liouville quantum field theory and the nonperturbative weak coupling expansion developed long ago by Braaten, Curtright, Ghandour, and Thorn. Exact agreement to the order calculated (i.e. up to and including corrections of order O(g^{10})) is found.Comment: 6 pages, LaTe

Spectral analysis and zeta determinant on the deformed spheres

We consider a class of singular Riemannian manifolds, the deformed spheres $S^N_k$, defined as the classical spheres with a one parameter family $g[k]$ of singular Riemannian structures, that reduces for $k=1$ to the classical metric. After giving explicit formulas for the eigenvalues and eigenfunctions of the metric Laplacian $\Delta_{S^N_k}$, we study the associated zeta functions $\zeta(s,\Delta_{S^N_k})$. We introduce a general method to deal with some classes of simple and double abstract zeta functions, generalizing the ones appearing in $\zeta(s,\Delta_{S^N_k})$. An application of this method allows to obtain the main zeta invariants for these zeta functions in all dimensions, and in particular $\zeta(0,\Delta_{S^N_k})$ and $\zeta'(0,\Delta_{S^N_k})$. We give explicit formulas for the zeta regularized determinant in the low dimensional cases, $N=2,3$, thus generalizing a result of Dowker \cite{Dow1}, and we compute the first coefficients in the expansion of these determinants in powers of the deformation parameter $k$.Comment: 1 figur

Liouville Correlation Functions from Four-dimensional Gauge Theories

We conjecture an expression for the Liouville theory conformal blocks and correlation functions on a Riemann surface of genus g and n punctures as the Nekrasov partition function of a certain class of N=2 SCFTs recently defined by one of the authors. We conduct extensive tests of the conjecture at genus 0,1.Comment: 32 pages, 8 figures; v2: minor corrections, published versio

Level-Spacing Distributions and the Bessel Kernel

The level spacing distributions which arise when one rescales the Laguerre or Jacobi ensembles of hermitian matrices is studied. These distributions are expressible in terms of a Fredholm determinant of an integral operator whose kernel is expressible in terms of Bessel functions of order $\alpha$. We derive a system of partial differential equations associated with the logarithmic derivative of this Fredholm determinant when the underlying domain is a union of intervals. In the case of a single interval this Fredholm determinant is a Painleve tau function.Comment: 18 pages, resubmitted to make postscript compatible, no changes to manuscript conten

The quantum dilogarithm and representations quantum cluster varieties

We construct, using the quantum dilogarithm, a series of *-representations of quantized cluster varieties. This includes a construction of infinite dimensional unitary projective representations of their discrete symmetry groups - the cluster modular groups. The examples of the latter include the classical mapping class groups of punctured surfaces. One of applications is quantization of higher Teichmuller spaces. The constructed unitary representations can be viewed as analogs of the Weil representation. In both cases representations are given by integral operators. Their kernels in our case are the quantum dilogarithms. We introduce the symplectic/quantum double of cluster varieties and related them to the representations.Comment: Dedicated to David Kazhdan for his 60th birthday. The final version. To appear in Inventiones Math. The last Section of the previous versions was removed, and will become a separate pape