Selberg Integral Research Articles

Proof of An AGT conjecture at β = 1

AGT conjecture reveals a connection between 4D N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 gauge theory and 2D conformal field theory. Though some special instances have been proven, others remain elusive and the attempts on its full proof never stop. When the Ω background parameters satisfy −ϵ1/ϵ2 ≡ β = 1, the story can be simplified a bit. A proof of the correspondence in the case of A1 gauge group was given in 2010 by Mironov et al., while the An extension is verified by Matsuo and Zhang in 2011, with an assumption on the Selberg integral of n + 1 Schur polynomials. Then in 2020, Albion et al. obtained the rigorous result of this formula. In this paper, we show that the conjecture on the Selberg integral of Schur polynomials is formally equivalent to their result, after applying a more complicated complex contour, thus leading to the proof of the An case at β = 1. To perform a double check, we also directly start from this formula, and manage to show the identification between the two sides of AGT correspondence.

Proof of 5D An AGT conjecture at β = 1

In this paper, we give a proof of 5D An AGT conjecture at β = 1, where the gauge theory side is one dimension higher than the original 4D case, and corresponds to the q-deformation of the 2D conformal field theory side. We define a q-deformed An Selberg integral, which generalizes the An Selberg integral and the q-deformed A1 Selberg integral in the literature. A q-deformed An Selberg average formula with n + 1 Schur polynomials is proposed and proved to complete the proof.

Complex and rational hypergeometric functions on root systems

We consider some new limits for the elliptic hypergeometric integrals on root systems. After the degeneration of elliptic beta integrals of type I and type II for root systems An and Cn to the hyperbolic hypergeometric integrals, we apply the limit ω1→−ω2 for their quasiperiods (corresponding to b→i in the two-dimensional conformal field theory) and obtain complex beta integrals in the Mellin–Barnes representation admitting exact evaluation. Considering type I elliptic hypergeometric integrals of a higher order obeying nontrivial symmetry transformations, we derive their descendants to the level of complex hypergeometric functions and prove the Derkachov–Manashov conjectures for functions emerging in the theory of non-compact spin chains. We describe also symmetry transformations for a type II complex hypergeometric function on the Cn-root system related to the recently derived generalized complex Selberg integral. For some hyperbolic beta integrals we consider a special limit ω1→ω2 (or b→1) and obtain new hypergeometric identities for sums of integrals of rational functions.

ARCHIMEDEAN NEWFORM THEORY FOR

Abstract We introduce a new invariant, the conductor exponent, of a generic irreducible Casselman–Wallach representation of $\operatorname {\mathrm {GL}}_n(F)$ , where F is an archimedean local field, that quantifies the extent to which this representation may be ramified. We also determine a distinguished vector, the newform, occurring with multiplicity one in this representation, with the complexity of this vector measured in a natural way by the conductor exponent. Finally, we show that the newform is a test vector for $\operatorname {\mathrm {GL}}_n \times \operatorname {\mathrm {GL}}_n$ and $\operatorname {\mathrm {GL}}_n \times \operatorname {\mathrm {GL}}_{n - 1}$ Rankin–Selberg integrals when the second representation is unramified. This theory parallels an analogous nonarchimedean theory due to Jacquet, Piatetski-Shapiro, and Shalika; combined, this completes a global theory of newforms for automorphic representations of $\operatorname {\mathrm {GL}}_n$ over number fields. By-products of the proofs include new proofs of Stade’s formulæ and a new resolution of the test vector problem for archimedean Godement–Jacquet zeta integrals.

The Singularities of Selberg- and Dotsenko–Fateev-Like Integrals

We discuss the meromorphic continuation of certain hypergeometric integrals modeled on the Selberg integral, including the 3-point and 4-point functions of BPZ’s minimal models of 2D CFT as described by Felder & Silvotti and Dotsenko & Fateev (the “Coulomb gas formalism”). This is accomplished via a geometric analysis of the singularities of the integrands. In the case that the integrand is symmetric (as in the Selberg integral itself) or, more generally, what we call “DF-symmetric,” we show that a number of apparent singularities are removable, as required for the construction of the minimal models via these methods.

On the Dotsenko–Fateev complex twin of the Selberg integral and its extensions

The Selberg integral has a twin (‘the Dotsenko–Fateev integral’) of the following form. We replace real variables xk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x_k$$\\end{document} in the integrand ∏|xk|σ-1|1-xk|τ-1∏|xk-xl|2θ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\prod |x_k|^{\\sigma -1}\\,|1-x_k|^{\ au -1} \\prod |x_k-x_l|^{2\ heta }$$\\end{document} of the Selberg integral by complex variables zk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$z_k$$\\end{document}, integration over a cube we replace by an integration over the whole complex space Cn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {C}}^n$$\\end{document}. According to Dotsenko, Fateev, and Aomoto, such integral is a product of Gamma functions. We define and evaluate a family of beta integrals over spaces Cm×Cm+1×⋯×Cn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {C}}^m\ imes {\\mathbb {C}}^{m+1}\ imes \\dots \ imes {\\mathbb {C}}^n$$\\end{document}, which for m=n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m=n$$\\end{document} gives the complex twin of the Selberg integral (with three additional integer parameters).

Jacobi Beta Ensemble and $b$-Hurwitz Numbers

We express correlators of the Jacobi $\beta$ ensemble in terms of (a special case of) $b$-Hurwitz numbers, a deformation of Hurwitz numbers recently introduced by Chapuy and Dołęga. The proof relies on Kadell's generalization of the Selberg integral. The Laguerre limit is also considered. All the relevant $b$-Hurwitz numbers are interpreted (following Bonzom, Chapuy, and Dołęga) in terms of colored monotone Hurwitz maps.

Iterated integrals, multiple zeta values and Selberg integrals

Classical multiple zeta values can be viewed as iterated integrals of the differentials [Formula: see text] from [Formula: see text] to [Formula: see text]. In this paper, we reprove Brown’s theorem: For [Formula: see text], the iterated integral of the form [Formula: see text] is a [Formula: see text]-linear combination of multiple zeta values of weight [Formula: see text] if convergent. What is more, we show that if [Formula: see text] are in a [Formula: see text]-algebra generated by multiple polylogarithms and their dual, and if [Formula: see text], are in a [Formula: see text]-algebra generated by logarithm, then the iterated integral [Formula: see text] is a [Formula: see text]-linear combination of multiple zeta values. As an application of our main results, we show that the coefficients of the Taylor expansions of the Selberg integrals [Formula: see text] (with respect to [Formula: see text]) at the integral points in some product of right half complex plane are [Formula: see text]-linear combinations of multiple zeta values for any [Formula: see text] This statement generalizes Terasoma’s original result.

The AFLT q-Morris constant term identity

It is well-known that the Selberg integral is equivalent to the Morris constant term identity. More generally, Selberg type integrals can be turned into constant term identities for Laurent polynomials. In this paper, by extending the Gessel–Xin method of the Laurent series proof of constant term identities, we obtain an AFLT type q-Morris constant term identity. That is a q-Morris type constant term identity for a product of two Macdonald polynomials.

Eigenfunctions of the van Diejen model generated by gauge and integral transformations

We present how explicit eigenfunctions of the principal Hamiltonian for the BCm relativistic Calogero-Moser-Sutherland model, due to van Diejen, can be constructed using gauge and integral transformations. In particular, we find that certain BC-type elliptic hypergeometric integrals, including elliptic Selberg integrals, of both Type I and Type II arise as eigenfunctions of the van Diejen model under some parameter restrictions. Among these are also joint eigenfunctions of so-called modular pairs of van Diejen operators. Furthermore, these transformations are related to reflections of the E8 Weyl group acting on the space of model parameters.

Rankin–Selberg Integrals and L-Functions for Covering Groups of General Linear Groups

Abstract Let ${\operatorname {GL}}_c^{(m)}$ be the covering group of ${\operatorname {GL}}_c$, obtained by restriction from the $m$-fold central extension of Matsumoto of the symplectic group. We introduce a new family of Rankin–Selberg integrals for representations of ${\operatorname {GL}}_c^{(m)}\times {\operatorname {GL}}_k^{(m)}$. The construction is based on certain assumptions, which we prove here for $k=1$. Using the integrals, we define local $\gamma $-, $L$-, and $\epsilon $-factors. Globally, our construction is strong in the sense that the integrals are truly Eulerian. This enables us to define the completed $L$-function for cuspidal representations and prove its standard functional equation.

Calculus of archimedean Rankin–Selberg integrals with recurrence relations

Let n n and n ′ n’ be positive integers such that n − n ′ ∈ { 0 , 1 } n-n’\in \{0,1\} . Let F F be either R \mathbb {R} or C \mathbb {C} . Let K n K_n and K n ′ K_{n’} be maximal compact subgroups of G L ( n , F ) \mathrm {GL}(n,F) and G L ( n ′ , F ) \mathrm {GL}(n’,F) , respectively. We give the explicit descriptions of archimedean Rankin–Selberg integrals at the minimal K n K_n - and K n ′ K_{n’} -types for pairs of principal series representations of G L ( n , F ) \mathrm {GL}(n,F) and G L ( n ′ , F ) \mathrm {GL}(n’,F) , using their recurrence relations. Our results for F = C F=\mathbb {C} can be applied to the arithmetic study of critical values of automorphic L L -functions.

Rankin–Selberg integrals for $$\mathrm{SO}_{2n+1}{\times }\mathrm{GL}_r$$ attached to newforms and oldforms

The conjectural newform theory for generic representations of $p$-adic ${\rm SO}_{2n+1}$ was formulated by P.-Y. Tsai in her thesis in which Tsai also verified the conjecture when the representations are supercuspidal. The main purpose of this work is to compute the Rankin-Selberg integrals for ${\rm SO}_{2n+1}\times{\rm GL}_r$ with $1\le r\le n$ attached to newforms and also oldforms under the validity of the conjecture.

Bivariate Selberg-beta type 1 distribution

By using the well known Selberg integral we define a bivariate beta distribution. Unlike other bivariate beta distributions which have support on the simplex {(x,y):x>0,y>0,x+y<1} this distribution has support on {(x,y):0<x<1,0<y<1}. Several properties such as marginal and conditional distributions, joint moments, coefficient of correlation, entropies, Fisher information matrix and estimation of parameters have been studied.

The $${{\mathbb {F}}}_p$$-Selberg Integral

We prove an $${{\mathbb {F}}}_p$$ -Selberg integral formula, in which the $${{\mathbb {F}}}_p$$ -Selberg integral is an element of the finite field $${{\mathbb {F}}}_p$$ with odd prime number p of elements. The formula is motivated by the analogy between multidimensional hypergeometric solutions of the KZ equations and polynomial solutions of the same equations reduced modulo p.

Amplitude recursions with an extra marked point

The recursive calculation of Selberg integrals by Aomoto and Terasoma using the Knizhnik-Zamolodchikov equation and the Drinfeld associator makes use of an auxiliary point and facilitates the recursive evaluation of string amplitudes at genus zero: open-string N-point amplitudes can be obtained from those at N-1 points. We establish a similar formalism at genus one, which allows the recursive calculation of genus-one Selberg integrals using an extra marked point in a differential equation of Knizhnik-Zamolodchikov-Bernard type. Hereby genus-one Selberg integrals are related to genus-zero Selberg integrals. Accordingly, N-point open-string amplitudes at genus one can be obtained from (N+2)-point open-string amplitudes at tree level. The construction is related to and in accordance with various recent results in intersection theory and string theory.

Local newforms for the general linear groups over a non-archimedean local field

AbstractIn [14], Jacquet–Piatetskii-Shapiro–Shalika defined a family of compact open subgroups ofp-adic general linear groups indexed by nonnegative integers and established the theory of local newforms for irreducible generic representations. In this paper, we extend their results to all irreducible representations. To do this, we define a new family of compact open subgroups indexed by certain tuples of nonnegative integers. For the proof, we introduce the Rankin–Selberg integrals for Speh representations.

AFLT-type Selberg integrals

In their 2011 paper on the AGT conjecture, Alba, Fateev, Litvinov and Tarnopolsky (AFLT) obtained a closed-form evaluation for a Selberg integral over the product of two Jack polynomials, thereby unifying the well-known Kadell and Hua--Kadell integrals. In this paper we use a variety of symmetric functions and symmetric function techniques to prove generalisations of the AFLT integral. These include (i) an $\mathrm{A}_n$ analogue of the AFLT integral, containing two Jack polynomials in the integrand; (ii) a generalisation of (i) for $\gamma=1$ (the Schur or GUE case), containing a product of $n+1$ Schur functions; (iii) an elliptic generalisation of the AFLT integral in which the role of the Jack polynomials is played by a pair of elliptic interpolation functions; (iv) an AFLT integral for Macdonald polynomials.

Rankin–Selberg integrals for principal series representations of GL(n)

Abstract We prove that the local Rankin–Selberg integrals for principal series representations of the general linear groups agree with certain simple integrals over the Rankin–Selberg subgroups, up to certain constants given by the local gamma factors.

The $${\mathbb {F}}_p$$-Selberg integral of type $$A_n$$

We prove an $\mathbb F_p$-Selberg integral formula of type $A_n$, in which the $\mathbb F_p$-Selberg integral is an element of the finite field $\mathbb F_p$ with odd prime number $p$ of elements. The formula is motivated by analogy between multidimensional hypergeometric solutions of the KZ equations and polynomial solutions of the same equations reduced modulo $p$. For the type $A_1$ the formula was proved in a previous paper by the authors.