Real categorical representation theory in topology and physics

Abstract: I will give an overview of the theory of Real categorical representations of a finite group, as developed in [1, 2, 3]. In this theory, a $\mathbb Z_2$-graded finite group acts on a category by autoequivalences or anti-autoequivalences, according to the $\mathbb Z_2$-grading. There is a natural geometric character theory of such representations which is most naturally formulated in terms of unoriented mapping spaces. In this way, one obtains an unoriented generalization of the 2-character theory of Ganter–Kapranov and a candidate for a Hopkins–Kuhn–Ravenel-type character theory for a conjectural Real equivariant elliptic cohomology theory. Time permitting, I will explain how Real categorical representation theory is related to unoriented Dijkgraaf–Witten theory, a three dimensional topological quantum field theory [4].

References:

[1] D. Rumynin and M. Young. Burnside rings for Real 2-representations: The linear theory. Commun. Contemp. Math., to appear. arXiv:1906.11006.

[2] B. Noohi and M. Young. Twisted loop transgression and higher Jandl gerbes over finite groupoids. arXiv:1910.01422, 2019.

[3] M. Young. Real representation theory of finite categorical groups. Higher Struct., to appear. arXiv:1804.09053.

[4] M. Young. Orientation twisted homotopy field theories and twisted unoriented Dijkgraaf–Witten theory. Commun. Math. Phys., 374(3):1645–1691, 2020.

Mathematics

Audience: researchers in the topic

Opening Workshop (IRP Higher Homotopy Structures 2021, CRM-Bellaterra)