Twisted representation rings and Dirac induction

AbstractExtending ideas of twisted equivariant K-theory, we construct twisted versions of the representation rings for Lie superalgebras and Lie supergroups, built from projective Z2-graded representations with a given cocycle. We then investigate the pullback and pushforward maps on these representation rings (and their completions) associated to homomorphisms of Lie superalgebras and Lie supergroups. As an application, we consider the Lie supergroup Π(T*G), obtained by taking the cotangent bundle of a compact Lie group and reversing the parity of its fibers. An inclusion H↪G induces a homomorphism from the twisted representation ring of Π(T*H) to the twisted representation ring of Π(T*G), which pulls back via an algebraic version of the Thom isomorphism to give an additive homomorphism from KH(pt) to KG(pt) (possibly with twistings). We then show that this homomorphism is in fact Dirac induction, which takes an H-module U to the G-equivariant index of the Dirac operator /∂⊗U on the homogeneous space G/H with values in the homogeneous bundle induced by U

On General Off-Shell Representations of Worldline (1D) Supersymmetry

Every finite-dimensional unitary representation of the N-extended worldline supersymmetry without central charges may be obtained by a sequence of differential transformations from a direct sum of minimal Adinkras, simple supermultiplets that are identifiable with representations of the Clifford algebra. The data specifying this procedure is a sequence of subspaces of the direct sum of Adinkras, which then opens an avenue for classification of the continuum of so constructed off-shell supermultiplets.Comment: 21 pages, 5 illustrations; references update