Disputation Steffen Schmidt On Unitarity for sl(m|n)-Supermodules: Dirac Cohomology, Superdimension, Indices

This dissertation investigates unitarizable supermodules over certain Lie superalgebras of type sl(m|n) and their simple classical counterparts of type A(m|n). The focus lies on questions of structure, classification, and applications in mathematics and theoretical physics. The first part develops a classification of unitarizable supermodules based on the Dirac inequality and Dirac cohomology, which emerges as a central tool for characterizing unitarity. A formal superdimension is then introduced for infinite-dimensional unitarizable supermodules, which vanishes in general but is nonzero precisely in the maximally atypical case—thus generalizing a conjecture of Kac and Wakimoto. In a physical context, the so-called superconformal index is analyzed, which is assigned to unitarizable supermodules over Lie superalgebras such as su(2,2|n) and related to a Q-type Witten index. Finally, the theory is extended to cubic Dirac operators, and a super-analogue of the Casselman–Osborne theorem is formulated, including explicit computations of Dirac cohomology for various classes of supermodules.