Super-Chevalley Restriction and Relative Lie Algebra Cohomology over the 2|3 Algebra

Abstract

Let $A:=\mathbb{C}[z_{+},z_{-}]\otimes \Lambda ({\theta }_1,{\theta }_2,{\theta }_3)$, with $z_{\pm }$ even and ${\theta }_1,{\theta }_2,{\theta }_3$ odd. For a reductive Lie algebra $\mathfrak{g}$, let $\mathfrak{g}[A]:=\mathfrak{g}\otimes A$ be the corresponding current Lie superalgebra. Motivated by the Chang--Yin description of weak-coupling $1/16$-BPS cohomology in $\mathcal{N}=4$ super-Yang--Mills, we study the relative Lie algebra cohomology $H^{\bullet }(\mathfrak{g}[A],\mathfrak{g};\mathbb{C})$. We isolate three finite-rank phenomena. First, the natural $3|2$ super-commuting restriction map, viewed as a super analogue of Chevalley restriction and its commuting-scheme variants, already fails to be an isomorphism for $\mathfrak{g}={\mathfrak{so}}_7$; the obstruction is a non-Cartan class. Second, the same algebra produces explicit fortuitous classes for ${\mathfrak{sl}}_2$ and ${\mathfrak{so}}_7$, giving concrete counterexamples to naive stable-image expectations suggested by the type-A Loday--Quillen--Tsygan theorem and its current-algebra refinements. Third, the classical relative cohomologies for the Langlands-dual pair $({\mathfrak{so}}_7,{\mathfrak{sp}}_6)$ are not isomorphic. We then record the conjectural quantum deformation of the differential expected to restore duality, together with first-order evidence pairing the fortuitous and non-Cartan ${\mathfrak{so}}_7$ classes.

18 pages, no figures