Elliptic Gamma Function Research Articles

Modified Elliptic Gamma Functions and 6d Superconformal Indices

We construct a modified double elliptic gamma function which is well defined when one of the base parameters lies on the unit circle. A model consisting of 6d hypermultiplets coupled to a gauge field theory living on a 4d defect is proposed whose superconformal index uses the double elliptic gamma function and obeys W(E 7)-group symmetry.

Comment on star–star relations in statistical mechanics and elliptic gamma-function identities

We prove a recently conjectured star–star relation, which plays the role of an integrability condition for a class of 2D Ising-type models with multicomponent continuous spin variables. Namely, we reduce this relation to an identity for elliptic gamma functions, previously obtained by Rains.

S-duality and 2d topological QFT

We study the superconformal index for the class of $$ \mathcal{N} = 2 $$ 4d superconformal field theories recently introduced by Gaiotto [1]. These theories are defined by compactifying the (2, 0) 6d theory on a Riemann surface with punctures. We interpret the index of the 4d theory associated to an n-punctured Riemann surface as the n-point correlation function of a 2d topological QFT living on the surface. Invariance of the index under generalized S-duality transformations (the mapping class group of the Riemann surface) translates into associativity of the operator algebra of the 2d TQFT. In the A 1 case, for which the 4d SCFTs have a Lagrangian realization, the structure constants and metric of the 2d TQFT can be calculated explicitly in terms of elliptic gamma functions. Associativity then holds thanks to a remarkable symmetry of an elliptic hypergeometric beta integral, proved very recently by van de Bult [2].

Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type I. The Eigenfunction Identities

In this series of papers we study Hilbert-Schmidt integral operators acting on the Hilbert spaces associated with elliptic Calogero-Moser type Hamiltonians. As shown in this first part, the integral kernels are joint eigenfunctions of differences of the latter Hamiltonians. On the relativistic (difference operator) level the kernel is built from the elliptic gamma function, whereas the building block in the nonrelativistic (differential operator) limit is basically the Weierstrass sigma-function. For the AN−1 case we consider all of the commuting Hamiltonians at once, the eigenfunction properties reducing to a sequence of elliptic identities. For the BCN case we only treat the defining Hamiltonians. The functional identities encoding the eigenfunction properties have a remarkable corollary in the relativistic BC1 case: They imply that the sum over eight-fold products of the four Jacobi theta functions is invariant under the Weyl group of E8.

A gerbe for the elliptic gamma function

The identities for elliptic gamma functions discovered by Felder and Varchenko [8] are generalized to an infinite set of identities for elliptic gamma functions associated to pairs of planes in 3-dimensional space. The language of stacks and gerbes gives a natural framework for a systematic description of these identities and their domain of validity. A triptic curve is the quotient of the complex plane by a subgroup of rank three. (It is a stack.) Our identities can be summarized by saying that elliptic gamma functions form a meromorphic section of a hermitian holomorphic abelian gerbe over the universal oriented triptic curve

Zero-Eigenvalue Eigenfunctions for Differences of Elliptic Relativistic Calogero-Moser Hamiltonians

Letting A l (x) denote the commuting analytic difference operators of elliptic relativistic Calogero-Moser type, we present and study zero-eigenvalue eigenfunctions for the operators A l (x) - A l (-y) (with I = 1,2,..., N and x, y ∈ C N ). The eigenfunctions are products of elliptic gamma functions. They are invariant under permutations of x 1 , x N and y 1 ,..., y N and under interchange of the step-size parameters.

Even powers of divisors and elliptic zeta values

We introduce and study elliptic zeta values , a two-parameter deformation of the values of Riemann's zeta function at positive integers. They are essentially Taylor coeffcients of the logarithm of the elliptic gamma function, and inherit the functional equations of this function. Elliptic zeta values at even integers are related to Eisenstein series and thus to sums of odd powers of divisors. The elliptic zeta values at odd integers can be expressed in terms of generating series of sums of even powers of divisors.

The modular properties and the integral representations of the multiple elliptic gamma functions

We show the modular properties of the multiple “elliptic” gamma functions, which are an extension of those of the theta function and the elliptic gamma function. The modular property of the theta function is known as Jacobi's transformation, and that of the elliptic gamma function was provided by Felder and Varchenko. In this paper, we deal with the multiple sine functions, since the modular properties of the multiple elliptic gamma functions result from the equivalence between two ways to represent the multiple sine functions as infinite products. We also derive integral representations of the multiple sine functions and the multiple elliptic gamma functions. We introduce correspondences between the multiple elliptic gamma functions and the multiple sine functions.

Absolute tensor products

We calculate the absolute tensor product ζ(s,Fp1)⊗···⊗ζ(s,Fpr) by observing a relation between the multiple sine function of Shintani's type and Appell's O-function, or equivalently, the multiple elliptic gamma function. This allows us to give an Euler product expression for an absolute tensor product.

Relativistic Lamé functions: Completeness vs. polynomial asymptotics

In earlier work we introduced and studied two commuting generalized Lamé operators, obtaining in particular joint eigenfunctions for a dense set in the natural parameter space. Here we consider these difference operators and their eigenfunctions in relation to the Hilbert space L 2((0, π/ r), w( x) dx), with r > 0 and the weight function w( x) a ratio of elliptic gamma functions. In particular, we show that the previously known pairwise orthogonal joint eigenfunctions need only be supplemented by finitely many new ones to obtain an orthogonal base. This completeness property is derived by exploiting recent results on the large-degree Hilbert space asymptotics of a class of orthonormal polynomials. The polynomials p n (cos( rx)), n ϵ N , that are relevant in the Lamé setting are orthonormal in L 2((0, π/ r), w P ( x) dx), with w p ( x) closely related to w(x).

Shintani–Barnes zeta and gamma functions

We show that Shintani's work on multiple zeta and gamma functions can be simplified and extended by exploiting difference equations. We re-prove many of Shintani's formulas and prove several new ones. Among the latter is a generalization to the Shintani–Barnes gamma functions of Raabe's 1843 formula ∫ 0 1 log Γ(x) dx= log 2π , and a further generalization to the Shintani zeta functions. These explicit formulas can be interpreted as “vanishing period integral” side conditions for the ladder of difference equations obeyed by the multiple gamma and zeta functions. We also relate Barnes’ triple gamma function to the elliptic gamma function appearing in connection with certain integrable systems.

Q-Deformed KZB Heat Equation: Completeness, Modular Properties and SL(3, [formula omitted

We study the properties of one-dimensional hypergeometric integral solutions of the q-difference (“quantum”) analogue of the Knizhnik–Zamolodchikov–Bernard equations on tori. We show that they also obey a difference KZB heat equation in the modular parameter, give formulae for modular transformations, and prove a completeness result, by showing that the associated Fourier transform is invertible. These results are based on SL(3, Z ) transformation properties parallel to those of elliptic gamma functions.

Multiple Gamma Function, Its $q$- and Elliptic Analogue

Vigneras's multiple gamma function is introduced as a function satisfying a generalization of the Bohr-Mollerup theorem. An infinite product representation and an asymptotic expansion of the function are given. Furthermore, its q- and elliptic analogue are introduced as relevant with the defining relations of q-gamma function and of elliptic gamma function.

An elliptic analogue of the multiple gamma function

A hierarchy of functions including the elliptic gamma function is introduced. It can be interpreted as an elliptic analogue of the multiple gamma function and its trigonometric limit coincides with a q-analogue of the multiple gamma function. Some properties of the functions are considered.

Elliptic Selberg integrals

We introduce new Selberg-type multidimensional integrals built of Ruijsenaars' elliptic gamma functions. We show that the vanishing of our integrals for a specific parameter hypersurface implies closed evaluation formulas valid for the full parameter space. The resulting integration formulas contain the Macdonald-Morris constant term identities for nonreduced root systems as special limiting cases.

The Elliptic Gamma Function and SL(3, [formula omitted])⋉[formula omitted

The elliptic gamma function is a generalization of the Euler gamma function and is associated to an elliptic curve. Its trigonometric and rational degenerations are the Jackson q-gamma function and the Euler gamma function, respectively. The elliptic gamma function appears in Baxter's formula for the free energy of the eight-vertex model and in the hypergeometric solutions of the elliptic qKZB equations. In this paper, the properties of this function are studied. In particular we show that elliptic gamma functions are generalizations of automorphic forms of G=SL(3, Z)⋉Z3 associated to a non-trivial class in H3(G, Z).

An Elliptic Macdonald-Morris Conjecture and Multiple Modular Hypergeometric Sums

We present an elliptic Macdonald-Morris constant term conjecture in the form of an evaluation formula for a Selberg-type multiple beta integral com- posed of elliptic gamma functions. By multivariate residue calculus, a summa- tion formula recently conjectured by Warnaar for a multiple modular (or elliptic) hypergeometric series is recovered. When the imaginary part of the modular pa- rameter tends to +∞, our elliptic Macdonald-Morris conjecture follows from a Selberg-type multivariate Nassrallah-Rahman integral due to Gustafson. As a consequence we arrive at a proof for the basic hypergeometric degeneration of Warnaar's sum, which amounts to a multidimensional generalization of Jackson's very-well-poised balanced terminating 8Φ7 summation formula. By exploiting its modular properties, the validity of Warnaar's sum at the elliptic level is moreover verified independently for low orders in log(q) (viz. up to order 10).