Elliptic hypergeometric integrals describe superconformal indices of 4d supersymmetric field theories. We show that all t Hooft anomaly matching conditions for Seiberg dual theories can be derived from $SL(3,mathbb{Z})$-modular transformation properties of the kernels of dual indices.
In this paper we develop a supersymmetric version of unitarity cut method for form factors of operators from the chiral truncation of the the $mathcal{N}=4$ stress-tensor current supermultiplet $T^{AB}$. The relation between the superform factor with
supermomentum equals to zero and the logarithmic derivative of the superamplitude with respect to the coupling constant is discussed and verified at tree- and one-loop level for any MHV $n$-point ($n geq 4$) superform factor involving operators from chiral truncation of the stress-tensor energy supermultiplet. The explicit $mathcal{N}=4$ covariant expressions for n-point tree- and one-loop MHV form factors are obtained. As well, the ansatz for the two-loop three-point MHV superform factor is suggested in the planar limit, based on the reduction procedure for the scalar integrals suggested in our previous work. The different soft and collinear limits in the MHV sector at tree- and one-loop level are discussed.
An exact formula for partition functions in 3d field theories was recently suggested by Jafferis, and Hama, Hosomichi, and Lee. These functions are expressed in terms of specific $q$-hypergeometric integrals whose key building block is the double sin
e function (or the hyperbolic gamma function). Elliptic hypergeometric integrals, discovered by the second author, define 4d superconformal indices. Using their reduction to the hyperbolic level, we describe a general scheme of reducing 4d superconformal indices to 3d partition functions which imply an efficient way of getting 3d $mathcal{N}=2$ supersymmetric dualities for both SYM and CS theories from the parent 4d $mathcal{N}=1$ dualities for SYM theories. As an example, we consider explicitly the duality pattern for 3d $mathcal{N}=2$ SYM and CS theories with SP(2N) gauge group with the antisymmetric tensor matter.
Recently, Kim and Imamura and Yokoyama derived an exact formula for superconformal indices in three-dimensional field theories. Using their results, we prove analytically the equality of superconformal indices in some U(1)-gauge group theories relate
d by the mirror symmetry. The proofs are based on the well known identities of the theory of $q$-special functions. We also suggest the general index formula taking into account the $U(1)_J$ global symmetry present for abelian theories.
In this letter we would like to apply the superconformal index technique to give one more evidence for the theory proposed by Intriligator, Seiberg and Shenker (ISS) as being described by interacting conformal field theory in its IR fixed point.
Superconformal indices (SCIs) of 4d ${mathcal N}=4$ SYM theories with simple gauge groups are described in terms of elliptic hypergeometric integrals. For $F_4, E_6, E_7, E_8$ gauge groups this yields first examples of integrals of such type. S-duali
ty transformation for G_2 and F_4 SCIs is equivalent to a change of integration variables. Equality of SCIs for SP(2N) and SO(2N+1) group theories is proved in several important special cases. Reduction of SCIs to partition functions of 3d $mathcal{N}=2$ SYM theories with one matter field in the adjoint representation is investigated, corresponding 3d dual partners are found, and some new related hyperbolic beta integrals are conjectured.
