Maximal U(1)Y-violating n-point correlators in $$ \mathcal{N} $$ = 4 super-Yang-Mills theory
Michael B. Green, Congkao Wen
Abstract
A bstract This paper concerns a special class of n -point correlation functions of operators in the stress tensor supermultiplet of $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 4 supersymmetric SU( N ) Yang-Mills theory. These are “maximal U(1) Y -violating” correlators that violate the bonus U(1) Y charge by a maximum of 2( n − 4) units. We will demonstrate that such correlators satisfy SL(2 , ℤ)-covariant recursion relations that relate n -point correlators to ( n − 1)-point correlators in a manner analogous to the soft dilaton relations that relate the corresponding amplitudes in flat-space type IIB superstring theory. These recursion relations are used to determine terms in the large- N expansion of n -point maximal U(1) Y -violating correlators in the chiral sector, including correlators with four superconformal stress tensor primaries and ( n − 4) chiral Lagrangian operators, starting from known properties of the n = 4 case. We concentrate on the first three orders in 1/ N beyond the supergravity limit. The Mellin representations of the correlators are polynomials in Mellin variables, which correspond to higher derivative contact terms in the low-energy expansion of type IIB superstring theory in AdS 5 × S 5 at the same orders as R 4 , d 4 R 4 and d 6 R 4 . The coupling constant dependence of these terms is found to be described by non-holomorphic modular forms with holomorphic and anti-holomorphic weights ( n − 4 , 4 − n ) that are SL(2 , ℤ)-covariant derivatives of Eisenstein series and certain generalisations. This determines a number of non-leading contributions to U(1) Y -violating n -particle interactions ( n > 4) in the low-energy expansion of type IIB superstring amplitudes in AdS 5 × S 5 .