We consider a special degeneration limit $ω_1\to - ω_2$ (or $b\to {\rm i}$ in the context of $2d$ Liouville quantum field theory) for the most general univariate hyperbolic beta integral. This limit is also applied to the most general hyperbolic analogue of the Euler-Gauss hypergeometric function and its $W(E_7)$ group of symmetry transformations. Resulting functions are identified as hypergeometric functions over the field of complex numbers related to the ${\rm SL}(2,\mathbb{C})$ group. A new similar nontrivial hypergeometric degeneration of the Faddeev modular quantum dilogarithm (or hyperbolic gamma function) is discovered in the limit $ω_1\to ω_2$ (or $b\to 1$).
We interpolate matrix beta-integrals of Siegel, Hua Loo Keng and Gindikin types with respect to dimension of the field. The domain of integration (Rayleigh triangles) imitates collections of all the eigenvalues of all the principal minors of a self-adjoint matrix. We also interpolate the Hua--Pickrell measures on inverse limits of symmetric spaces. Our family of integrals also contains the Selberg integral.
This article is devoted to the asymptotic expansion of the generalized Harish Chandra-Itzykson-Zuber matrix integral for non-unitary symmetries characterized by a parameter beta(as usual beta =1,2 and 4 correspond to the orthogonal, unitary and symplectic group integrals). A WKB-expansion for f is derived from the heat kernel differential equation, for general values of k and beta. From an expansion in terms of zonal polynomials, one obtain an expansion in powers of the tau's for beta=1, and generalizations are considered for general beta. A duality relation, and a transformation of products o…
We construct an infinite-dimensional solution of the Yang-Baxter equation (YBE) of rank 1 which is represented as an integral operator with an elliptic hypergeometric kernel acting in the space of functions of two complex variables. This R-operator intertwines the product of two standard L-operators associated with the Sklyanin algebra, an elliptic deformation of sl(2)-algebra. It is built from three basic operators $\mathrm{S}_1, \mathrm{S}_2$, and $\mathrm{S}_3$ generating the permutation group of four parameters $\mathfrak{S}_4$. Validity of the key Coxeter relations (including the star-tri…
A decomposition of identity is given as a complex integral over the coherent states associated with a class of shape-invariant self-similar potentials. There is a remarkable connection between these coherent states and Ramanujan's integral extension of the beta function.
The univariate elliptic beta integral was discovered by the author in 2000. Recently Bazhanov and Sergeev have interpreted it as a star-triangle relation (STR). This important observation is discussed in more detail in connection to author's previous work on the elliptic modular double and supersymmetric dualities. We describe also a new Faddeev-Volkov type solution of STR, connections with the star-star relation, and higher-dimensional analogues of such relations. In this picture, Seiberg dualities are described by symmetries of the elliptic hypergeometric integrals (interpreted as superconfo…
Gaussian and Chiral Beta-Ensembles, which generalise well known orthogonal (Beta=1), unitary (Beta=2), and symplectic (Beta=4) ensembles of random Hermitian matrices, are considered. Averages are shown to satisfy duality relations like {Beta,N,n} <--> {4/Beta,n,N} for all Beta>0, where N and n respectively denote the number of eigenvalues and products of characteristic polynomials. At the edge of the spectrum, matrix integrals of the Airy (Kontsevich) type are obtained. Consequences on the integral representation of the multiple orthogonal polynomials and the partition function of the formal o…
The HarishChandra-Itzykson-Zuber integral over the unitary group U(k) (beta=2) is present in numerous problems involving Hermitian random matrices. It is well known that the result is semi-classically exact. This simple result does not extend to other symmetry groups, such as the symplectic or orthogonal groups. In this article the analysis of this integral is extended first to the symplectic group Sp(k) (beta=4). There the semi-classical approximation has to be corrected by a WKB expansion. It turns out that this expansion stops after a finite number of terms ; in other words the WKB approxim…
We give a brief review of the main results of the theory of elliptic hypergeometric functions -- a new class of special functions of mathematical physics. We prove the most general univariate exact integration formula generalizing Euler's beta integral, which is called the elliptic beta integral. An elliptic analogue of the Gauss hypergeometric function is constructed together with the elliptic hypergeometric equation for it. Biorthogonality relations for this function and its particular subcases are described. We list known elliptic beta integrals on root systems and consider symmetry transfo…
Particles and $p-$adic integrals of Spin$\left(\frac{1}{2}\right)$: spin Lie group, $\mathcal{R}(ρ,q)-$gamma and $\mathcal{R}(ρ,q)-$ beta functions, ghost and applications
We rewrite the recently constructed q-hypergeometric integral Bailey pair in a general form. Then with the help of the Bailey pair and $q$-beta hypergeometric sum-integral, we construct the star-triangle relation.
The (quantum) pentagon relation underlies the existing constructions of three dimensional quantum topology in the combinatorial framework of triangulations. Following the recent works \cite{KashaevLuoVartanov2012,AndersenKashaev2013}, we discuss a special type of integral pentagon relations and their relationships with the Faddeev type operator pentagon relations.
A generalization of Selberg's beta integral involving Schur polynomials associated with partitions with entries not greater than 2 is explicitly computed. The complex version of this integral is given after proving a general statement concerning the complex extensions of Selberg-Schur integrals. All these results have interesting applications in both mathematics and physics, particularly in conformal field theory, since the conformal blocks for the $SL(2,\mathbb{R})$ Wess-Zumino-Novikov-Witten model can be obtained by analytical continuation of these integrals.
This is author's Habilitation Thesis (Dr. Sci. dissertation) submitted at the beginning of September 2004. It is written in Russian and is posted due to the continuing requests for the manuscript. The content: 1. Introduction, 2. Nonlinear chains with the discrete time and their self-similar solutions, 3. General theory of theta hypergeometric series, 4. Theta hypergeometric integrals, 5. Biorthogonal functions, 6. Elliptic hypergeometric functions with |q|=1, 7. Conclusion, 8. References. It contains an outline of a general heuristic scheme for building univariate special functions through se…
The invention of the "dual resonance model" N-point functions BN motivated the development of current string theory. The simplest of these models, the four-point function B4, is the classical Euler Beta function. Many standard methods of complex analysis in a single variable have been applied to elucidate the properties of the Euler Beta function, leading, for example, to analytic continuation formulas such as the contour-integral representation obtained by Pochhammer in 1890. Here we explore the geometry underlying the dual five-point function B5, the simplest generalization of the Euler Beta…
General elliptic hypergeometric functions are defined by elliptic hypergeometric integrals. They comprise the elliptic beta integral, elliptic analogues of the Euler-Gauss hypergeometric function and Selberg integral, as well as elliptic extensions of many other plain hypergeometric and $q$-hypergeometric constructions. In particular, the Bailey chain technique, used for proving Rogers-Ramanujan type identities, has been generalized to integrals. At the elliptic level it yields a solution of the Yang-Baxter equation as an integral operator with an elliptic hypergeometric kernel. We give a brie…
Recent studies in several interrelated areas -- from combinatorics and representation theory in mathematics to quantum field theory and topological string theory in physics -- have independently revealed that many classical objects in these fields admit a relatively novel one-parameter deformation. This deformation, known in different contexts under the names of Omega-background, refinement, or beta-deformation, has a number of interesting mathematical implications. In particular, in Chern-Simons theory beta-deformation transforms the classical HOMFLY invariants into Dunfield-Gukov-Rasmussen s…
Beijing Institute of Mathematical Sciences and Applications（北京国际数学研究中心）；Laboratory of Theoretical Physics, JINR（约飞联合核子研究所理论物理实验室）；Yerevan Physics Institute（埃里安物理研究所）；National Research University Higher School of Economics（国立研究大学高等经济大学）
Exact integral expressions of the skew orthogonal polynomials involved in Orthogonal (beta=1) and Symplectic (beta=4) random matrix ensembles are obtained: the (even rank) skew orthogonal polynomials are average characteristic polynomials of random matrices. From there, asymptotics of the skew orthogonal polynomials are derived.
We study the averaged product of characteristic polynomials of large random matrices in the Gaussian beta-ensemble perturbed by an external source of finite rank. We prove that at the edge of the spectrum, the limiting correlations involve two families of multivariate functions of Airy and Gaussian types. The precise form of the limiting correlations depends on the strength of the nonzero eigenvalues of the external source. A critical value for the latter is obtained and a phase transition phenomenon similar to that of arXiv:math/0403022 is established. The derivation of our results relies mai…
We consider multiple sums and multi-integrals as tau functions of the BKP hierarchy using neutral fermions as the simplest tool for deriving these. The sums are over projective Schur functions $Q_α$ for strict partitions $α$. We consider two types of such sums: weighted sums of $Q_α$ over strict partitions $α$ and sums over products $Q_αQ_γ$. In this way we obtain discrete analogues of the beta-ensembles ($β=1,2,4$). Continuous versions are represented as multiple integrals. Such sums and integrals are of interest in a number of problems in mathematics and physics.
We study the cumulants and their generating functions of the probability distributions of the conductance, shot noise and Wigner delay time in ballistic quantum dots. Our approach is based on the integrable theory of certain matrix integrals and applies to all the symmetry classes beta=1,2,4 of Random Matrix Theory. We compute the weak localization corrections to the mixed cumulants of the conductance and shot noise for beta=1,4, thus proving a number of conjectures of Khoruzhenko et al. (Phys. Rev. B, Vol. 80 (2009), 125301). We derive differential equations that characterize the cumulant gen…
We prove, for any $β>0$, a central limit theorem for the fluctuations of linear statistics in the Sine-$β$ process, which is the infinite volume limit of the random microscopic behavior in the bulk of one-dimensional log-gases at inverse temperature $β$. If $ϕ$ is a compactly supported test function of class $C^4$, and $\mathcal{C}$ is a random point configuration distributed according to Sine-$β$, the integral of $ϕ(\cdot / \ell)$ against the random fluctuation $d\mathcal{C} - dx$, converges in law, as $\ell$ goes to infinity, to a centered normal random variable whose standard deviation is p…
