<> "The repository administrator has not yet configured an RDF license."^^ . <> . . "On Unitarity for sl(m/n)-Supermodules: Dirac Cohomology, Indices, Superdimension"^^ . "Abstract\r\nThis thesis investigates unitarizable supermodules over special linear Lie superalgebras\r\nsl(m|n) and their basic classical counterparts A(m|n), denoted g, with a focus on their\r\nstructure, classification, and applications in both mathematics and theoretical physics. It\r\nis structured in four main parts, each exploring a distinct but interrelated aspect of the\r\ntheory.\r\n\r\nThe first part develops a general framework for understanding unitarity and provides a concise classification of unitarizable simple g-supermodules, derived using the Dirac inequality\r\nand 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\r\nunitarity. We demonstrate that Dirac cohomology can uniquely determine unitarizable\r\nsupermodules, and compute it explicitly of unitarizable simple supermodules. This\r\nleads to a refined characterization of unitarity, forming the basis for our novel classifica-\r\ntion of unitarizable simple supermodules. Furthermore, we establish a connection between\r\nDirac cohomology and Kostant’s cohomology of Lie superalgebras, derive a decomposition\r\nof formal characters, and introduce a Dirac index.\r\n\r\nIn the second part, we construct a formal superdimension for infinite-dimensional unitarizable supermodules, inspired by the theory of relative discrete series representations. We\r\nshow that this superdimension vanishes for most simple supermodules but is non-trivial\r\nprecisely when the infinitesimal character has maximal degree of atypicality. In particular,\r\nour result aligns with the Kac–Wakimoto conjecture for finite-dimensional supermodules.\r\nThe third part investigates applications to theoretical physics, focusing in particular on\r\nthe so-called “superconformal index” – a character-valued invariant assigned by physicists\r\nto unitarizable supermodules of Lie superalgebras, such as su(2, 2|n), which appear in the\r\ncontext of certain quantum field theories. The index is computed as a supertrace over\r\na Hilbert space and remains constant across families of representations that arise from\r\nvarying physical parameters. This invariance is due to the fact that only “short” simple\r\nsupermodules contribute to the index, making it stable under recombination phenomena\r\noccurring at the boundary of the unitarity region. We develop these notions for unitarizable\r\nsupermodules over g. Along the way, we provide a precise dictionary between various\r\nnotions from theoretical physics and mathematical terminology. Our final result is a kind of\r\n“index theorem” that relates the counting of atypical constituents in a general unitarizable\r\ng-supermodule to the character-valued Q-Witten index, expressed as a supertrace over the\r\nfull supermodule. The formal superdimension of part 2 can also be formulated in this\r\nframework.\r\n\r\nThe final part is an addendum that extends the Dirac operator and cohomology to their\r\ncubic counterparts. We develop a theory of cubic Dirac operators associated to parabolic\r\nsubalgebras and prove a super-analog of the Casselman–Osborne theorem. We show that\r\nDirac cohomology is trivial unless for highest weight supermodules, and demonstrate, under suitable conditions, an embedding of Dirac cohomology into Kostant’s (co)homology. This\r\nembedding becomes an isomorphism in the unitarizable case. We also provide complete\r\ncomputations of Dirac cohomology for finite-dimensional simple supermodules with typical\r\nhighest weight and for supermodules in the parabolic BGG category."^^ . "2025" . . . . . . . "Steffen"^^ . "Schmidt"^^ . "Steffen Schmidt"^^ . . . . . . "On Unitarity for sl(m/n)-Supermodules: Dirac Cohomology, Indices, Superdimension (PDF)"^^ . . . "Promotion_FINAL-1.pdf"^^ . . . "On Unitarity for sl(m/n)-Supermodules: Dirac Cohomology, Indices, Superdimension (Other)"^^ . . . . . . "indexcodes.txt"^^ . . . "On Unitarity for sl(m/n)-Supermodules: Dirac Cohomology, Indices, Superdimension (Other)"^^ . . . . . . "lightbox.jpg"^^ . . . "On Unitarity for sl(m/n)-Supermodules: Dirac Cohomology, Indices, Superdimension (Other)"^^ . . . . . . "preview.jpg"^^ . . . "On Unitarity for sl(m/n)-Supermodules: Dirac Cohomology, Indices, Superdimension (Other)"^^ . . . . . . "medium.jpg"^^ . . . "On Unitarity for sl(m/n)-Supermodules: Dirac Cohomology, Indices, Superdimension (Other)"^^ . . . . . . "small.jpg"^^ . . "HTML Summary of #36958 \n\nOn Unitarity for sl(m/n)-Supermodules: Dirac Cohomology, Indices, Superdimension\n\n" . "text/html" . . . "510 Mathematik"@de . "510 Mathematics"@en . .