Twisted brane charges for non-simply connected groups

Abstract

Terry GannonDepartment of Mathematical Sciences, University of Alberta Edmonton, Alberta, Canada, T6G 2G1 E-mail: tgannon@math.ualberta.caAbstract: The charges of the twisted branes for strings on the group manifold SU(n)/Z d are determined. To this end we derive explicit (and remarkably simple) formulae for the relevant NIM-rep coefficients. The charge groups of the twisted and untwisted branes are compared and found to agree for the cases we consider.

Group dualities, T‐dualities, and twisted K‐theory

“…The final result that, when h = 3, the twisted K-groups in both even and odd degree are (Z/3) 3 was obtained previously in [32, (2.20)] from the physics perspective of D-brane charges in WZW theories for PSU (3). More complicated calculations for PSU (9), where the result is not so simple, appear in [33].…”

“…Remark The final result that, when , the twisted ‐groups in both even and odd degree are was obtained previously in [, (2.20)] from the physics perspective of D‐brane charges in WZW theories for . More complicated calculations for , where the result is not so simple, appear in . Theorem actually proves more: if with odd and relatively prime to 3, then , and if with even and relatively prime to 3, then .…”

Modular Invariants and Twisted EquivariantK-theory II: Dynkin diagram symmetries

“…However, in this description both integrality and nonnegativity are highly unobvious. Using the fixed-point factorisation of [48], [43] found a relatively simple expression (given below) for some of these coefficients, and from this could prove integrality, but nonnegativity remained out of reach. In Theorem 3 below, we use this to find a simple global description for the Ver k (G n )-module structure, making its relation to K-homology more evident, and allowing us to finally prove nonnegativity and establish the nimrep property.…”

“…To prove this, first derive (5.9), which follows straightforwardly from (5.7). The expressions for the coefficients of N (n,k,d) is much more difficult, but the Appendix of [43] uses the fixed-point factorisation of [48] to compute Since these two Ver k (G n )-modules have identical formulas for multiplication by the fundamental weights Λ m , which generate Ver k (G n ), they are isomorphic as Ver k (G n )modules, and in fact N λ = N (n,k,d) λ for all λ ∈ P k + (G n ). Therefore the coefficients of N (n,k,d) are nonnegative integers, and so N = N (n,k,d) is indeed a nimrep compatible with Z J n .…”

“…QED Theorem 3 provides a massive simplification of the original expressions for N (n,k,d) in [4,42]. Although integrality of the matrix entries of N (n,k,d) was established in [43], nonnegativity was out-of-reach. We were led to Theorem 3 by trying to match the conjectured N (n,k,d) to the K-homology τ K G×Z d * (G).…”