Violation of S-duality in classical $Q$-cohomology
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 $A=\mathbb{C}[z^+,z^-]\otimes\Lambda(\theta_1,\theta_2,\theta_3)$.