Partition Functions and Automorphic Forms

We give a brief account of the key properties of elliptic hypergeometric integrals—a relatively recently discovered top class of transcendental special functions of hypergeometric type. In particular, we describe an elliptic generalization of Euler’s and Selberg’s beta integrals, elliptic analogue of the Euler–Gauss hypergeometric function and some multivariable elliptic hypergeometric functions on root systems. The elliptic Fourier transformation and corresponding integral Bailey lemma technique is outlined together with a connection to the star-triangle relation and Coxeter relations for a permutation group. We review also the interpretation of elliptic hypergeometric integrals as superconformal indices of four dimensional supersymmetric quantum field theories and corresponding applications to Seiberg type dualities.

These lecture notes discuss some of the basics of elliptic hypergeometric functions. These are fairly recent generalizations of ordinary hypergeometric functions. In this chapter we first discuss both ordinary hypergeometric functions and elliptic functions, as you need to know both to define elliptic hypergeometric series. We subsequently discuss some of the important properties these series satisfy, in particular we consider the biorthogonal functions found by Spiridonov and Zhedanov, both with respect to discrete and continuous measure. In doing so we naturally encounter the most important evaluation and transformation formulas for elliptic hypergeometric series, and for the associated elliptic beta integral.

Rarefied elliptic hypergeometric functions

We present the fermionic representation for the q-deformed hypergeometric functions related to Schur polynomials. We show that these multivariate hypergeometric functions are tau-functions of the KP hierarchy, and at the same time they are the ratios of Toda lattice tau-functions, considered by Takasaki, evaluated at certain values of higher Toda lattice times. The variables of the hypergeometric functions are related to the higher times of those hierarchies via a Miwa change of variables. The discrete Toda lattice variable shifts parameters of hypergeometric functions. Hypergeometric functions of type pΦs can also be viewed as a group 2-cocycle for the ΨDO on the circle (the group times are higher times of TL hierarchy and the arguments of a hypergeometric function). We obtain the determinant representation and the integral representation of a special type of KP tau-functions, these results generalize some of the results of Milne concerning multivariate hypergeometric functions. We write down a system of partial differential equations for these tau-functions (string equations).

A multivariable hypergeometric function considered recently by Niukkanen and Srivastava (1984, 1985) provides an interesting unification of the generalised hypergeometric function pFq of one variable, Appell and Kampe de Feriet functions of two variables, and Lauricella functions of n variables, and also of many other hypergeometric series which arise naturally in various physical and quantum chemical applications. As pointed out by Srivastava, the multivariable hypergeometric function is an obvious special case of the generalised Lauricella function of n variables, which was first introduced and studied by Srivastava and Daoust (1969). By employing such connections of this multivariable hypergeometric function with the more general multiple hypergeometric functions several interesting and useful properties of this function have been studied, many of which have not been given by Niukkanen. The author derives a number of new reduction formulae for the multivariable hypergeometric function from substantially more general identities involving multiple series with essentially arbitrary terms. Some interesting summation formulae for the multivariable hypergeometric function with x1= . . . =xn=1 and x1= . . . =xn=-1 are also presented.

The multivariable hypergeometric function $$F_{q_0 :q_1 ;...;q_n }^{P_0 :P_1 ;...;P_n } \left( {\begin{array}{*{20}c} {x_1 } \\ \vdots \\ {x_n } \\ \end{array} } \right),$$ considered recently by A. W. Niukkanen and H.M. Srivastava, is known to provide an interesting unification of the generalized hypergeometric functionp F q of one variable, Appell and Kampe de Feriet functions of two variables, and Lauricella functions ofn variables, as also of many other hypergeometric series which arise naturally in various physical, astrophysical, and quantum chemical applications. Indeed, as already pointed out by Srivastava, this multivariable hypergeometric function is an obvious special case of the generalized Lauricella function ofn variables, which was first introduced and studied by Srivastava and M. C. Daoust. By employing such fruitful connections of this multivariable hypergeometric function with much more general multiple hypergeometric functions studied in the literature rather systematically and widely, Srivastava presented several interesting and useful properties of this function, most of which did not appear in the work of Niukkanen. The object of this sequel to Srivastava's work is to derive a further reduction formula for the multivariable hypergeometric function from substantially more general identities involving multiple series with essentially arbitrary terms. Some interesting connections of the results considered here with those given in the literature, and some indication of their applicability, are also provided.

The asymptotic casuality condition of Screaton (1969) is generalized to two dimensions and applied to the vacuum expectation value of the retarded commutator, R( phi (x), phi Dagger ( y)), in a two-dimensional field theory. It is proved that the asymptotic causality condition (0 mod R( phi (x), phi Dagger (y))0) approximately 0 for large (x-y) not in V+, implies the microscopic causality condition, (0 mod R( phi (x), phi Dagger (y)) mod 0)=0 unless (x-y) in V+, where V+ is the forward light cone. In a two dimensional quantum field theory, therefore, the microscopic causality condition can be replaced by the weaker asymptotic causality condition.

We discuss the duality in three dimensional quantum field theory at infrared limit. The starting point is to use a conjecture of a duality between the free fermion and the interacting scalar field theories at the Wilson–Fisher fixed point. The conjecture is useful for deriving various dualities in three dimensions to obtain a duality web. The study is also interesting for understanding the dualities, or equivalence of different theories from the perspective of the renormalization group flow. We first discuss the “derivation” without losing the holonomy. Furthermore, we also derive these dualities from the mean-field study, and consider the extension of the conjecture or dualities at finite temperature.

The Indian govt. celebrated 125𝑡ℎ anniversary of the great Mathematician of Indian soil Srinivasa Ramanujan on 22 December in the year 2012. Without any formal education and extreme poverty conditions, he emerged as one of great mathematician of India. His mathematical ideas transformed and reshaped 20th century mathematics and their ideas are inspiration for 21𝑠𝑡 century mathematicians. Srinivasa Ramanujan made substantial contributions to the analytical theory of numbers and worked on ‘elliptic functions’, ‘continued fractions’, and ‘infinite series’. He was a great Mathematician, who became world famous at the tender age of twenty-six. He was born into a family that had a humble background and that had no distinguished professional achievement, yet his mathematical ideas transformed and reshaped century mathematics and continues to inspire modern day mathematicians. Considered to be a mathematical genius, Srinivasa Ramanujan, was regarded at par with the likes of Leonhard Euler and Carl Jacobi. In spite of having almost no formal training in mathematics, Ramanujan’s knowledge of mathematics was astonishing. Even though he had no knowledge of the modern developments in the subject, he effortlessly worked out the Riemann series, the elliptic integrals, hypergeometric series, and the functional equations of the zeta function. The purpose of this paper is to introduce some of the contributions of Srinivasa Ramanujan in the field of mathematics.

This article is devoted to the investigation of the interplay of supersymmetric Yang–Mills theories (SYM) and supergravity (SUGRA). The topic is studied from two points of view: Firstly from the point of view of AdS/CFT correspondence, which realises the coupling of four dimensional superconformal 𝒩 = 4 SYM theory and ten dimensional type IIB SUGRA in a holographic way. In order to arrive at theories that resemble quantum chromodynamics (QCD) more closely, fundamental fields are introduced using probe D7‐branes and non‐trivial background configuration are considered. In particular supergravity solutions that are only asymptotically anti‐de Sitter and break supersymmetry are used. This allows the description of spontaneous chiral symmetry breaking. The meson spectrum is calculated and the existence of an associated Goldstone mode is demonstrated. Moreover it is shown that highly radially excited mesons are not degenerate. Additionally instanton configurations on the D7‐branes are investigated, which lead to a holographic description of the dual field theory's Higgs branch. Finally a holographic description of heavy‐light mesons is developed, which are mesons consisting of quarks with a large mass difference, such that a treatment of B mesons can be achieved The second approach is the technique of so‐called space‐time dependent couplings (also known as “local couplings”), where coupling constants are promoted to external sources. This allows to explore the conformal anomaly of quantum field theories coupled to a classical gravity background. The technique is extended to the superfield description of 𝒩 = 1 supergravity, a complete basis for the anomaly is given and the consistency conditions that arise from a cohomological treatment are calculated. Possible implications for an extension of Zamolodchikov's c‐theorem to four dimensional supersymmetric quantum field theories are discussed.

I. M. Gel’fand and his collaborators created the theory of general hypergeometric functions (multivariate hypergeometric functions on lattices and hypergeometric functions on Grassmannians). These functions are multivariate generalizations of hypergeometric functions of one variable. They are solutions of certain systems of differential equations (general hypergeometric systems) and are related to the Radon transform. At the present time, a deep connection of these hypergeometric functions with representations of groups is absent. But it is clear that this connection exists. Remark that the theory of q-analogues of Grassmannians and of related generalizations of Gel’fand hypergeometric functions is under elaboration (see, for example, [432]).

12 - Elliptic Integrals and Functions

This paper continues the study of recursion formulas of multivariable hypergeometric functions. In [Recursion formulas for multivariable hypergeometric functions, Asian-Eur. J. Math. 8(4) (2015) Article ID: 1550082, 50 pp.], the authors have given the recursion formulas for three-variable Lauricella functions, three Srivastava’s triple hypergeometric functions and four [Formula: see text]-variable Lauricella functions. We present here the recursion formulas for the general triple hypergeometric function.

We study the supersymmetric partition function of 4d supersymmetric gauge theories with a U(1) R-symmetry on Euclidean S<sup>3</sup> × S<sub>β</sub><sup>1</sup>, with S<sup>3</sup> the unit-radius squashed three-sphere, and β the circumference of the circle. For superconformal theories, this partition function coincides (up to a Casimir energy factor) with the 4d superconformal index. The partition function can be computed exactly using the supersymmetric localization of the gauge theory path-integral. It takes the form of an elliptic hypergeometric integral, which may be viewed as a matrix-integral over the moduli space of the holonomies of the gauge fields around S<sub>β</sub><sup>1</sup>. At high temperatures (β → 0, corresponding to the hyperbolic limit of the elliptic hypergeometric integral) we obtain from the matrix-integral a quantum effective potential for the holonomies. The effective potential is proportional to the temperature. Therefore the high-temperature limit further localizes the matrix-integral to the locus of the minima of the potential. If the effective potential is positive semi-definite, the leading high-temperature asymptotics of the partition function is given by the formula of Di Pietro and Komargodski, and the subleading asymptotics is connected to the Coulomb branch dynamics on R<sup>3</sup> × S<sup>1</sup>. In theories where the effective potential is not positive semi-definite, the Di Pietro-Komargodski formula needs to be modified. In particular, this modification occurs in the SU(2) theory of Intriligator-Seiberg-Shenker, and the SO(N) theory of Brodie-Cho-Intriligator, both believed to exhibit “misleading” anomaly matchings, and both believed to yield interacting superconformal field theories with c < a. Lastly, two new simple tests for dualities between 4d supersymmetric gauge theories emerge as byproducts of our analysis.

In recent years, there have been two independent but related developments in the study of irrelevant deformations in two dimensional quantum field theories (QFTs). The first development is the deformation of a two dimensional QFT by the determinant of the energy momentum stress tensor, commonly referred to as $T{\bar T}$ deformation. The second development is in two dimensional holographic field theories which are dual to string theory in asymptotically Anti-de Sitter (AdS) spacetimes. In this latter development, the deformation is commonly referred to as single-trace $T{\bar T}$ deformation. The single-trace $T{\bar T}$ deformation corresponds in the bulk to a string background that interpolates between AdS spacetime in the infrared (IR) and a linear dilaton spacetime (vacuum of little string theory (LST)) in the ultraviolet (UV). It serves as a useful tool and guide to better understand and explore holography in asymptotically AdS and non-AdS spacetimes in a controlled setting. In particular, it is useful to gain insights into holography in flat spacetimes. The dissertation is devoted to the study of single-trace $T{\bar T}$ deformation and its single-trace generalizations in theories with $U(1)$ currents, namely $J\bar T$ and $T\bar J$ deformations, in the context of gauge/gravity duality. In the dissertation I present new results in the study of holography in asymptotically non-AdS spacetimes. I discuss two point correlation functions in single-trace $T{\bar T}$ deformation, and entanglement entropy and entropic $c$-function in single-trace $T{\bar T}$, $J\bar T$ and $T\bar J$ deformations.