Anyonic Defect Branes and Conformal Blocks in Twisted Equivariant Differential (TED) K-theory

TL;DR

This paper links twisted equivariant differential K-theory to exotic defect brane charges in string theory, revealing how conformal blocks and braid group representations emerge, with implications for topological quantum computation.

Contribution

It uncovers the role of twisted equivariant K-theory and inner local systems in realizing conformal blocks and anyon statistics for defect branes in string/M-theory.

Findings

Identification of exotic brane charges via twisted K-theory.

Connection between secondary Chern characters and conformal blocks.

Realization of braid group representations for defect branes.

Abstract

We demonstrate that twisted equivariant differential K-theory of transverse complex curves accommodates exotic charges of the form expected of codimension=2 defect branes, such as of D7-branes in IIB/F-theory on A-type orbifold singularities, but also of their dual 3-brane defects of class-S theories on M5-branes. These branes have been argued, within F-theory and the AGT correspondence, to carry special SL(2)-monodromy charges not seen for other branes, but none of these had previously been identified in the expected brane charge quantization law given by K-theory. Here we observe that it is the subtle (and previously somewhat neglected) twisting of equivariant K-theory by flat complex line bundles appearing inside orbi-singularities ("inner local systems") that makes the secondary Chern character on a punctured plane inside an A-type singularity evaluate to the twisted holomorphic de Rham cohomology which Feigin, Schechtman & Varchenko showed realizes sl(2,C)-conformal blocks, here in degree 1 -- in fact it gives the direct sum of these over all admissible fractional levels. The remaining higher-degree conformal blocks appear similarly if we assume our previously discussed "Hypothesis H" about brane charge quantization in M-theory. Since conformal blocks -- and hence these twisted equivariant secondary Chern characters -- solve the Knizhnik-Zamolodchikov equation and thus constitute representations of the braid group of motions of defect branes inside their transverse space, this provides a concrete first-principles realization of anyon statistics of -- and hence of topological quantum computation on -- defect branes in string/M-theory.