Twisted loop transgression and higher Jandl gerbes over finite groupoids

Abstract

Given a double cover π : G → Ĝ of finite groupoids, we explicitly construct twisted loop transgression maps, τ π and τ ref π , thereby associating to a Jandl n-gerbe λ on Ĝ a Jandl (n− 1)-gerbe τ π ( λ) on the quotient loop groupoid of G and an ordinary (n − 1)-gerbe τ ref π ( λ) on the unoriented quotient loop groupoid of G. For n = 1, 2, we interpret the character theory (resp. centre) of the category of Real λ-twisted n-vector bundles over Ĝ in terms of flat sections of the (n − 1)vector bundle associated to τ ref π ( λ) (resp. the Real (n−1)-vector bundle associated to τ π ( λ)). We relate our results to Real versions of twisted Drinfeld doubles and pointed fusion categories and to discrete torsion in orientifold string and M-theory.

Real Representations of $C_2$-Graded Groups: The Antilinear Theory

“…For example in C n ≤ D 2n , all Real conjugacy classes have size one, so a is an element of this form, but bab −1 = a −1 = a, so ba = ab. However, we can explicitly give a basis of the centre in terms of the conjugacy classes of G in the proof of the following proposition, also proven in [16,Cor. 13.6].…”

“…In Section 5 we undertake a further study of A-characters. The first main result, Theorem 5.1, is the only overlap of this section with [16], but our proof is again different. The second main result of this section is Theorem 5.4, which brings together our earlier results and resembles similar summaries of the classical theory -see the appendix by the first author [13].…”

“…Earlier studies dealt with the case where G was a semidirect product [3,11,5,18,9]. The recent paper by Noohi and Young [16] considers the general case and has some of our results with different proofs (see further references to this work throughout our paper). Some of our results are known in the split case [3,11,5,18,9] but our comprehensive exposition will be useful even for those interested only in the split case.…”

“…Next we prove Theorem 2.5, our first main result. The A-character theory was also developed by Noohi and Young [16] but our proofs are essentially different. To make paper self-contained we include them.…”

Real Representations of $C_2$-Graded Groups: The Linear and Hermitian Theories

“…Over the time they were actively studied by many scientists with a range of backgrounds (cf. [2,3,4,9,10,11,13,16]). The present paper is a sequel to our study of antilinear representations [12].…”