New modular invariants in $$ \mathcal{N} $$ = 4 Super-Yang-Mills theory
Shai M. Chester, Michael B. Green, Silviu S. Pufu, Yifan Wang, Congkao Wen
Abstract
A bstract We study modular invariants arising in the four-point functions of the stress tensor multiplet operators of the $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 4 SU( N ) super-Yang-Mills theory, in the limit where N is taken to be large while the complexified Yang-Mills coupling τ is held fixed. The specific four-point functions we consider are integrated correlators obtained by taking various combinations of four derivatives of the squashed sphere partition function of the $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 2 ∗ theory with respect to the squashing parameter b and mass parameter m , evaluated at the values b = 1 and m = 0 that correspond to the $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 4 theory on a round sphere. At each order in the 1 /N expansion, these fourth derivatives are modular invariant functions of ( τ, $$ \overline{\tau} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>τ</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> ). We present evidence that at half-integer orders in 1 /N , these modular invariants are linear combinations of non-holomorphic Eisenstein series, while at integer orders in 1 /N , they are certain “generalized Eisenstein series” which satisfy inhomogeneous Laplace eigenvalue equations on the hyperbolic plane. These results reproduce known features of the low-energy expansion of the four-graviton amplitude in type IIB superstring theory in ten-dimensional flat space and have interesting implications for the structure of the analogous expansion in AdS 5 × S 5 .