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

TL;DR

This paper investigates the relative Lie algebra cohomology of a superalgebra related to $ ext{Super-Chevalley}$ restriction, revealing finite-rank phenomena, counterexamples to classical theorems, and conjectural quantum deformations.

Contribution

It identifies specific obstructions and counterexamples in the superalgebra cohomology, challenging classical stability expectations and proposing a quantum deformation to restore duality.

Findings

Failure of the super Chevalley restriction map for $rak{so}_7$ due to a non-Cartan class.

Explicit fortuitous classes for $rak{sl}_2$ and $rak{so}_7$ challenge naive stability expectations.

Classical relative cohomologies for dual pairs $(rak{so}_7,rak{sp}_6)$ are not isomorphic, with a conjectural quantum deformation proposed.

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.