Fortuity and relevant deformation

TL;DR

This work analyzes the $Q$-cohomology of a mass-deformed $ N=4$ SYM theory (the Leigh–Strassler IR fixed point) using a cohomological field redefinition that integrates out massive modes, and maps IR cohomologies to UV data. By exploiting a field redefinition that yields a commutator-based $Q$-action, the authors classify cohomologies into Coulomb and non-Coulomb types and construct both monotone (KK) gravitons and infinitely many fortuitous non-graviton cohomologies, with detailed SU(2) results. They compute the BMN sector indices in UV and IR, derive a Coulomb graviton index, and analyze hairy (non-Coulomb) operators that contribute at higher orders, uncovering a stringy-exclusion-like behavior for certain graviton towers. The UV-to-IR analysis reveals that fortuitous cohomologies can arise due to the relevant deformation while monotonicity is not guaranteed, and the SU(2) case shows nontrivial mappings between UV gravitons and IR cohomologies, suggesting deep connections between moduli space structure and cohomology stability under RG flow.

Abstract

We investigate the supercharge cohomology of an $\mathcal{N}=1$ relevant deformation of $\mathcal{N}=4$ super Yang-Mills. By introducing a field redefinition, we integrate out massive fields in a cohomological sense. Then, we construct the monotone cohomologies corresponding to the Kaluza-Klein particles of the dual supergravity solution. Some of the monotone cohomologies obey stringy exclusion principle analogous to that of $AdS_3$. Relatedly, they vanish on the diagonal field configurations, unlike $\mathcal{N}=4$ monotone cohomologies. We also construct infinitely many fortuitous cohomologies for gauge group SU(2). We find that unlike $\mathcal{N}=4$ fortuitous cohomologies, they can either be non-vanishing or vanishing on the diagonal fields. By undoing the field redefinition and taking a suitable UV limit, we show that non-vanishing ones reduce to monotone cohomologies of $\mathcal{N}=4$ SYM, while vanishing ones reduce to fortuitous cohomologies of $\mathcal{N}=4$ SYM. This implies that the fortuity can arise due to the relevant deformation, while monotonicity is not.