Violation of S-duality in classical $Q$-cohomology

TL;DR

This work tests whether the tree-level $Q$-cohomology of ${\cal N}=4$ SYM is preserved by S-duality between Langlands-dual gauge groups, challenging the conjecture of one-loop exactness for the ${1\over16}$-BPS spectrum. It frames the cohomology in terms of the relative Lie algebra cohomology $H^\bullet(\mathfrak g[A],\mathfrak g)$ with $A=\mathbb{C}[z^+,z^-]\otimes\Lambda(\theta_1,\theta_2,\theta_3)$ and analyzes Coulomb-branch abelianization and monotone cohomology. The main result shows that for the ${\rm SO}(7)$/${\rm USp}(6)$ pair up to level $L=18$, there exist two extra cohomology classes in ${\rm SO}(7)$ not matched by ${\rm USp}(6)$ (one fermionic fortuitous and one bosonic monotone) which cancel in the index but prove a failure of isomorphism between the classical $Q$-cohomologies, thereby questioning either exact S-duality or tree-level exactness. These findings further indicate that monotone cohomology can exceed Coulomb-branch cohomology and suggest possible lifting by higher-loop or non-perturbative effects, with implications for the ${1\over 8}$-BPS chiral ring.

Abstract

We study the cohomology of a chiral supercharge $Q$ in the $\mathcal{N}=4$ super-Yang-Mills (SYM) theory at tree level. The cohomology classes correspond one-to-one to the $\frac1{16}$ Bogomol'nyi-Prasad-Sommerfield (BPS) states at one-loop. We argue that monotone classes on the Coulomb branch respect the S-duality between the theories with $\mathrm{SO}(2N+1)$ and $\mathrm{USp}(2N)$ gauge groups, but find an explicit example of a pair of cohomology classes that "violate" the S-duality in the sense that the tree-level $Q$-cohomologies are not isomorphic between the neighborhoods near the two free points. Within this pair, one is a fortuitous class and the other is a monotone chiral ring element. Assuming the non-perturbative validity of S-duality, our results disprove a long-standing conjecture on the one-loop exactness of the $\frac1{16}$-BPS spectrum (including the $\frac1{8}$-BPS chiral ring spectrum) in the $\mathcal{N}=4$ SYM. Mathematically, this shows that, the relative Lie algebra cohomology $H^\bullet(\mathfrak{g}[A],\mathfrak{g})$ is generally not graded-isomorphic to $H^\bullet({}^L\mathfrak{g}[A],{}^L\mathfrak{g})$, where $\mathfrak{g}$ and ${}^L\mathfrak{g}$ are a pair of Langlands dual Lie algebras and $A=\mathbb{C}[z^+,z^-]\otimesΛ(θ_1,θ_2,θ_3)$.

Figures (1)

• Figure 1: The space of supersymmetric deformations of the $\mathrm{SO}(2N+1)$/$\mathrm{USp}(2N)$$\mathcal{N}=4$ super-Yang-Mills, and the subspace of conformal deformations.