On Unitarity for sl(m/n)-Supermodules: Dirac Cohomology, Indices, Superdimension

Abstract This thesis investigates unitarizable supermodules over special linear Lie superalgebras sl(m|n) and their basic classical counterparts A(m|n), denoted g, with a focus on their structure, classification, and applications in both mathematics and theoretical physics. It is structured in four main parts, each exploring a distinct but interrelated aspect of the theory. The first part develops a general framework for understanding unitarity and provides a concise classification of unitarizable simple g-supermodules, derived using the Dirac inequality and decomposition under the even Lie subalgebra. The Dirac operator and its associated Dirac cohomology serve as central tools in this study, capturing essential aspects of unitarity. We demonstrate that Dirac cohomology can uniquely determine unitarizable supermodules, and compute it explicitly of unitarizable simple supermodules. This leads to a refined characterization of unitarity, forming the basis for our novel classifica- tion of unitarizable simple supermodules. Furthermore, we establish a connection between Dirac cohomology and Kostant’s cohomology of Lie superalgebras, derive a decomposition of formal characters, and introduce a Dirac index. In the second part, we construct a formal superdimension for infinite-dimensional unitarizable supermodules, inspired by the theory of relative discrete series representations. We show that this superdimension vanishes for most simple supermodules but is non-trivial precisely when the infinitesimal character has maximal degree of atypicality. In particular, our result aligns with the Kac–Wakimoto conjecture for finite-dimensional supermodules. The third part investigates applications to theoretical physics, focusing in particular on the so-called “superconformal index” – a character-valued invariant assigned by physicists to unitarizable supermodules of Lie superalgebras, such as su(2, 2|n), which appear in the context of certain quantum field theories. The index is computed as a supertrace over a Hilbert space and remains constant across families of representations that arise from varying physical parameters. This invariance is due to the fact that only “short” simple supermodules contribute to the index, making it stable under recombination phenomena occurring at the boundary of the unitarity region. We develop these notions for unitarizable supermodules over g. 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 g-supermodule to the character-valued Q-Witten index, expressed as a supertrace over the full supermodule. The formal superdimension of part 2 can also be formulated in this framework. The final part is an addendum that extends the Dirac operator and cohomology to their cubic counterparts. We develop a theory of cubic Dirac operators associated to parabolic subalgebras and prove a super-analog of the Casselman–Osborne theorem. We show that Dirac cohomology is trivial unless for highest weight supermodules, and demonstrate, under suitable conditions, an embedding of Dirac cohomology into Kostant’s (co)homology. This embedding becomes an isomorphism in the unitarizable case. We also provide complete computations of Dirac cohomology for finite-dimensional simple supermodules with typical highest weight and for supermodules in the parabolic BGG category.