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We are finalizing an article:
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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 – 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 $\mathbb{A}$-type singularity evaluate to the twisted holomorphic de Rham cohomology which Feigin, Schechtman & Varchenko showed realizes $\widehat{\mathfrak{sl}_2}$-conformal blocks, here in degree 1 – 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 “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.
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Comments are welcome. If you do have a look, please grab our latest pdf version from behind the above link.
Quick typos:
1-twisted de Rham cohomology already of N-punctured planes (why ’already’?)
Fugure (twice)
Thanks. Am on my phone now will fix the Fugures a little later.
The “already” is meant to be short for “already for an example as simple as”(!).
Maybe there is room to clarify.
Interesting to see F-theory’s appearance in the article.
it supports the Hypothesis H that the latter is (at least partly) the missing rigorous definition of F/M-theory
Won’t there need to be a proper dual to Hypothesis H, along the lines of
F-theory apparently wants to lift the $S^4$-coefficient of M-theory to its branched cover by $CP^1 \times CP^1$?
Yes, this article’s ambition is to present a TED-K-theoretic computation of a configuration space, and to make plausible it’s F&M-theoretic meaning under Hypothesis H and the pertinent F/M-duality folkore, but is not to make rigorous that F/M-duality. That’s for another time. Though it will help to have seen, hereby, where the pieces of the puzzle want to go.
For instance, knowing from the math+HypothesisH that there must be the $\mathfrak{su}(2)$-WZW model at shifted level $k = \kappa - 2$ (instead of the unshifted level, considered in most contemporary literature) that controls the gauge theory “in” the $\mathbb{A}_{\kappa-1}$-singularity made us (re-)discover the crucial articles [Le00][LLS02] (above Figure F on p. 23), whose insight may not have found due attention before (cf Rem. 4.7).
Thanks!
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