Probing non-graviton spectra in $$ \mathcal{N}=4 $$ SYM via BMN truncation and S-duality

A bstract The one-loop cohomology of $$ \mathcal{N}=4 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> </mml:math> super Yang-Mills is conjectured to be isomorphic to the exact cohomology. As a result, its truncations are expected to be sub-rings of the exact cohomology. We study the superconformal index restricted over one such truncation known as the BMN truncation. We present a systematic algorithm to compute the BMN index using the method of residues. We compute the BMN index for SU( N ) $$ \mathcal{N}=4 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> </mml:math> SYM for N = 2, … , 6 in closed form. It is expressed as a rational function of the fugacity. A term of the type (1 − x ) in the denominator indicates the presence of a bosonic generator counted with fugacity x . We find a rich and universal set of such terms in the denominator showing an interesting bosonic Fock space in the spectrum of protected operators. This Fock space can not be explained as coming from the non-interacting supersymmetric graviton gas far away from the black hole as in the grey-galaxy solutions because the charges of the bosonic generators are not compatible with those of the supersymmetric gravitons. This suggests a novel micro-structure within the supersymmetric black hole itself. We also examine the indices of S-dual pairs SO(2 N + 1) and Sp( N ) SYM. Although their full 1/16-BPS indices coincide, we find discrepancies in their BMN-sector indices. As the BMN indices restricted to the graviton sector are expected to be the same, this mismatch allows us to identify non-graviton cohomologies. We explicitly find one of them in the SO(7) theory that is responsible for the mismatch of the BMN index. We also show that indices restricted to other one-loop cohomology truncations, in general, do not match under S-duality. If the conjecture of exactness of one-loop cohomology is correct, this suggests the presence of new cohomology sub-rings that match under S-duality with the letter-based truncations of the one-loop cohomology. This offers a way to check the one-loop exactness conjecture.