Anyonic defect branes and conformal blocks in twisted equivariant differential (TED) K-theory

• Published in: Reviews in mathematical physics Vol. 35; no. 6
• Main Authors: Sati, Hisham, Schreiber, Urs
• Format: Journal Article
• Language: English
• Published: Singapore World Scientific Publishing Company 01.07.2023; World Scientific Publishing Co. Pte., Ltd
• Subjects: Braid theory; Branes; Defects; Field theory; First principles; Gauge theory; Homology; M theory; Quantum computing; Singularities; String theory; String theory; algebraic topology; conformal field theory; branes; K-theory
• ISSN: 0129-055X; 1793-6659
• DOI: 10.1142/S0129055X23500095

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 -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, at least partially reflected in conformal blocks of the 2 -WZW model over their transverse punctured complex curve. Indeed, it has been argued that all “exotic” branes of string theory are defect branes carrying such U-duality monodromy charges — 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 -type singularity evaluate to the twisted holomorphic de Rham cohomology which Feigin, Schechtman and Varchenko showed realizes 2 k ̂ -conformal blocks, here in degree 1 — in fact it gives the direct sum of these over all admissible fractional levels k = − 2 + κ / r . The remaining higher-degree 2 k ̂ -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.