An article that we are finalizing:
Hisham Sati$\,$ and $\,$ Urs Schreiber:
Anyonic Defect Branes and Conformal Blocks in
Twisted Equivariant Differential (TED) K-Theory
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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 $\mathbb{A}$-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 $\mathrm{SL}(2)$-monodromy charges not seen for other branes, at least partially reflected in conformal blocks of the $\widehat{\mathfrak{sl}_2}$-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 β 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 (βinner local systemsβ) that makes the secondary Chern character on a punctured plane inside an $\mathbb{A}$-type singularity evaluate to the twisted holomorphic de Rham cohomology which Feigin, Schechtman and Varchenko showed realizes $\widehat{\mathfrak{sl}_2}$-conformal blocks, here in degree 1 β in fact it gives the direct sum of these over all admissible fractional levels $k = - 2 + \kappa/r$. The remaining higher-degree $\widehat{\mathfrak{sl}_2}$-conformal blocks appear similarly if we assume our previously discussed βHypothesis Hβ about brane charge quantization in M-theory. Since conformal blocks β and hence these twisted equivariant secondary Chern characters β 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 β and hence of topological quantum computation on β defect branes in string/M-theory.
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For more see at Hypothesis H.
Expository talk:
via Seminario de CategorΓas UNAM, 13 April 2022
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Last revised on August 24, 2022 at 04:10:44. See the history of this page for a list of all contributions to it.