12:00 pm Monday, December 4, 2017
Junior Geometry and String Theory: Verlinde ring for non-connected Lie groups and gauging finite group symmetries by Valentin Zakharevich (UT Austin) in RLM 12.166
Given a short exact sequence of finite groups 1→ H→ G→ K→ 1 and a K -invariant class α H ∈ H n ( BH,U(1) ) , there is an action of K on the n -dimensional Dijkgraaf-Witten theory associated with the pair ( H, α H ) . Extending the symmetry action in a higher-categorical sense is equivalent to extending the class α H to a class α G ∈ H n ( BG,U(1) ) . Gauging this action one obtains the Dijkgraaf-Witten theory corresponding to the pair ( G, α G ) . We ask: “Does the analogous statement holds if the group H is allowed to be a compact Lie group and the Dijkgraaf-Witten theory is replaced by the Chern-Simons theory?” The difficulty lies in understanding the Chern-Simons theory for the non-connected Lie group G . By a theorem of Freed, Hopkins, and Teleman, the corresponding Verlinde ring is isomorphic to the twisted equivariant K-theory K G τ (G) where G acts on itself by conjugation and the fusion is given by the Pontryagin product, i.e. induced from the multiplication map of G. Comparing it to the Verlinde ring of the gauged Chern-Simons theory of H by K , as developed by Barkeshli, Bonderson, Cheng, and Wang, suggests that the answer to the question above is affirmative. Submitted by
|
|