Modular invariant holographic correlators for $$ \mathcal{N} $$ = 4 SYM with general gauge group

Abstract

We study the stress tensor four-point function for $$ \mathcal{N} $$ N = 4 SYM with gauge group G = SU(N), SO(2N + 1), SO(2N) or USp(2N) at large N . When G = SU(N), the theory is dual to type IIB string theory on AdS5× S5 with complexified string coupling τs, while for the other cases it is dual to the orbifold theory on AdS5× S5/ℤ2. In all cases we use the analytic bootstrap and constraints from localization to compute 1-loop and higher derivative tree level corrections to the leading supergravity approximation of the correlator. We give perturbative evidence that the localization constraint in the large N and finite complexified coupling τ limit can be written for each G in terms of Eisenstein series that are modular invariant in terms of τs ∝ τ, which allows us to fix protected terms in the correlator in that limit. In all cases, we find that the flat space limit of the correlator precisely matches the type IIB S-matrix. We also find a closed form expression for the SU(N) 1-loop Mellin amplitude with supergravity vertices. Finally, we compare our analytic predictions at large N and finite τ to bounds from the numerical bootstrap in the large N regime, and find that they are not saturated for any G and any τ , which suggests that no physical theory saturates these bootstrap bounds.

Exact results for duality-covariant integrated correlators in $\mathcal{N}=4$ SYM with general classical gauge groups

“…In this paper we will consider the extension of the SU (N ) results of [1,2] to the other classical Lie groups, SO(2N ), SO(2N + 1), and U Sp(2N ). Some aspects of the perturbative expansions of the integrated correlators for these groups, and their large-N expansions in the 't Hooft limit were considered in [10] starting from the localised partition function of N = 2 * SYM described in [4]. Our analysis will include the nonperturbative instanton contributions, leading to expressions for the integrated correlators for N = 4 SYM with any classical gauge group that take the form of two-dimensional lattice sums 3…”

“…In section 2 we will present some properties of the integrated correlator, C GN , defined in (1.1) in terms of derivatives of the partition function of N = 2 * SYM on S 4 in the m → 0 limit. These results are based on methods outlined in appendix B, which includes a brief summary of the perturbative structure of integrated correlators given in [10], and an overview of instanton calculations based on the Nekrasov partition function [18] generalisied to arbitrary classical gauge groups [19,20]. The perturbation expansions for C GN (τ 2 ) with finite N are presented in section 2.1.…”

“…The perturbative sectors of the integrated correlators, C pert GN derived from the localised partition function, were discussed in [10], where they were expressed in terms of generalised Laguerre polynomials as reviewed in appendix B.2. One of the primary interests in [10] was to use this perturbative data to determine terms in the large-N expansion order by order in 1/N or, more precisely, order by order in the inverse central charges, 1/c GN .…”

“…The perturbative sectors of the integrated correlators, C pert GN derived from the localised partition function, were discussed in [10], where they were expressed in terms of generalised Laguerre polynomials as reviewed in appendix B.2. One of the primary interests in [10] was to use this perturbative data to determine terms in the large-N expansion order by order in 1/N or, more precisely, order by order in the inverse central charges, 1/c GN . However, here we will study the perturbation expressions at finite N in more detail, which will motivate the form of a set of Laplace-difference equations that generalises the analysis in [1,2] of the SU (N ) correlators, as well as the modular covariant expressions (1.3) that are well-defined for all values of N and τ .…”

“…Our starting point is the explicit result for the perturbative sector C pert GN obtained in [10], and for convenience summarised in appendix B. The expansions of the expressions in (B.15)-(B.18) in powers of g 2 Y M can be organised in a striking manner by defining the expansion parameters, a GN , for each gauge group in the following manner 5…”