Elliptic hypergeometric integrals, superconformal indices, and anomalies

Elliptic hypergeometric integrals (introduced by the  speaker in 2000) represent the
top known level of special functions  of hypergeometric type. They appear to describe
superconformal   indices of four-dimensional supersymmetric gauge field theories.
 Superconformal indices of theories related by the Seiberg  duality coincide as a
consequence of nontrivial identities  for corresponding integrals. In the talk we show
that 't Hooft  anomaly matching conditions for dual theories are related to
 SL(3,Z) modular transformation properties of the kernels of  dual indices (this is a
joint work with G.S. Vartanov).