An index for unitarizable $\mathfrak{sl}(m\vert n)$-supermodules

TL;DR

This work develops a rigorous, representation‑theoretic framework for the superconformal index of unitarizable ${rak{sl}}(m|n)$‑modules by translating physical notions into mathematics: the Witten index, Dirac operators, and the Duflo–Serganova functor. The authors prove an index theorem that equates counting atypical constituents in a general unitarizable module with a character‑valued ${Q}$‑Witten index and relate this to Harish‑Chandra formal dimensions; they also establish a correspondence with the KMMR index and formal superdimension. Central to the approach is a detailed analysis of the region of unitarity, fragmentation at its boundary, and the role of the Dirac inequality in guaranteeing unitarity via ${rak g}_{0}$‑constituents. The results bridge mathematical Lie superalgebra theory and the physics literature on short multiplets, providing tools to compute invariants that persist under continuous deformations with potential applications to superconformal field theories and their representation theory.

Abstract

The "superconformal index" is a character-valued invariant attached by theoretical physics to unitary representations of Lie superalgebras, such as $\mathfrak{su}(2,2\vert n)$, that govern certain quantum field theories. The index can be calculated as a supertrace over Hilbert space, and is constant in families induced by variation of physical parameters. This is because the index receives contributions only from "short" irreducible representations such that it is invariant under recombination at the boundary of the region of unitarity. The purpose of this paper is to develop these notions for unitarizable supermodules over the special linear Lie superalgebras $\mathfrak{sl}(m\vert n)$ with $m\ge 2$, $n\ge 1$. To keep it self-contained, we include a fair amount of background material on structure theory, unitarizable supermodules, the Duflo-Serganova functor, and elements of Harish-Chandra theory. Along the way, we provide a precise dictionary between various notions from theoretical physics and mathematical terminology. Our final result is a kind of "index theorem" that relates the counting of atypical constituents in a general unitarizable $\mathfrak{sl}(m\vert n)$-supermodule to the character-valued $Q$-Witten index, expressed as a supertrace over the full supermodule. The formal superdimension of holomorphic discrete series $\mathfrak{sl}(m\vert n)$-supermodules can also be formulated in this framework.

Figures (2)

• Figure 1: Root system of $\sl(2\vert 1)$ with respect to two canonical positive systems. The coordinates are the basis $(H,J)$ of the Cartan subalgebra used in the presentation in eq. \ref{['standardRelations']}. The green dots represent the positive roots, and the red dots, the negative roots. The blue circles are (some of) the weights of the oscillator supermodule ${\mathfrak{O}}^+$, the magenta circles, of ${\mathfrak{O}}^-$. The double circles are the highest weights. The black dot is the Weyl vector.
• Figure :

Theorems & Definitions (15)

• Definition 1.3.1
• Lemma 1.3.2
• Theorem 1.3.3: chen2021simple
• Definition 1.3.4
• Definition 1.4.1
• Definition 1.4.2
• Proposition 1.4.3
• Theorem 1.4.4: furutsu1991classificationneeb2011lie
• Proposition 1.4.5
• Proposition 2.1.1: huang2007dirac