3,015 research outputs found
Aspects of elliptic hypergeometric functions
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
-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 brief survey of the developments in this field.Comment: 15 pp., 1 fig., accepted in Proc. of the Conference "The Legacy of
Srinivasa Ramanujan" (Delhi, India, December 2012
Derivatives of Horn-type hypergeometric functions with respect to their parameters
We consider the derivatives of Horn hypergeometric functions of any number
variables with respect to their parameters. The derivative of the function in
variables is expressed as a Horn hypergeometric series of infinite
summations depending on the same variables and with the same region of
convergence as for original Horn function. The derivatives of Appell functions,
generalized hypergeometric functions, confluent and non-confluent Lauricella
series and generalized Lauricella series are explicitly presented. Applications
to the calculation of Feynman diagrams are discussed, especially the series
expansion in within dimensional regularization. Connections with
other classes of special functions are discussed as well.Comment: 27 page
Rarefied elliptic hypergeometric functions
Two exact evaluation formulae for multiple rarefied elliptic beta integrals
related to the simplest lens space are proved. They generalize evaluations of
the type I and II elliptic beta integrals attached to the root system . In
a special case, the simplest limit is shown to lead to a new
class of -hypergeometric identities. Symmetries of a rarefied elliptic
analogue of the Euler-Gauss hypergeometric function are described and the
respective generalization of the hypergeometric equation is constructed. Some
extensions of the latter function to and root systems and
corresponding symmetry transformations are considered. An application of the
rarefied type II elliptic hypergeometric function to some eigenvalue
problems is briefly discussed.Comment: 41 pp., corrected numeration of formula
Elliptic hypergeometric terms
General structure of the multivariate plain and q-hypergeometric terms and
univariate elliptic hypergeometric terms is described. Some explicit examples
of the totally elliptic hypergeometric terms leading to multidimensional
integrals on root systems, either computable or obeying non-trivial symmetry
transformations, are presented.Comment: 20 pp., version to appear in a workshop proceeding
Multiple (inverse) binomial sums of arbitrary weight and depth and the all-order epsilon-expansion of generalized hypergeometric functions with one half-integer value of parameter
We continue the study of the construction of analytical coefficients of the
epsilon-expansion of hypergeometric functions and their connection with Feynman
diagrams. In this paper, we show the following results:
Theorem A: The multiple (inverse) binomial sums of arbitrary weight and depth
(see Eq. (1.1)) are expressible in terms of Remiddi-Vermaseren functions.
Theorem B: The epsilon expansion of a hypergeometric function with one
half-integer value of parameter (see Eq. (1.2)) is expressible in terms of the
harmonic polylogarithms of Remiddi and Vermaseren with coefficients that are
ratios of polynomials. Some extra materials are available via the www at this
http://theor.jinr.ru/~kalmykov/hypergeom/hyper.htmlComment: 24 pages, latex with amsmath and JHEP3.cls; v2: some typos corrected
and a few references added; v3: few references added
The 1/2 BPS Wilson loop in ABJ(M) at two loops: The details
We compute the expectation value of the 1/2 BPS circular Wilson loop operator
in ABJ(M) theory at two loops in perturbation theory. Our result turns out to
be in exact agreement with the weak coupling limit of the prediction coming
from localization, including finite N contributions associated to non-planar
diagrams. It also confirms the identification of the correct framing factor
that connects framing-zero and framing-one expressions, previously proposed.
The evaluation of the 1/2 BPS operator is made technically difficult in
comparison with other observables of ABJ(M) theory by the appearance of
integrals involving the coupling between fermions and gauge fields, which are
absent for instance in the 1/6 BPS case. We describe in detail how to
analytically solve these integrals in dimensional regularization with
dimensional reduction (DRED). By suitably performing the physical limit to
three dimensions we clarify the role played by short distance divergences on
the final result and the mechanism of their cancellation.Comment: 54 pages, 2 figure
S-duality and 2d Topological QFT
We study the superconformal index for the class of N=2 4d superconformal
field theories recently introduced by Gaiotto. 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.Comment: 25 pages, 11 figure
- …