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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMar 15th 2022
   • (edited Mar 15th 2022)
   • PermaLink
   Author: Urs
   Format: MarkdownItexWe are finalizing an article: * H. Sati & U.S.: **[[schreiber:Anyonic defect branes in TED-K-theory|Anyonic defect branes in TED-K-theory]]** $\,$ > **Abstract**: We demonstrate that [[twisted K-theory|twisted]] [[equivariant K-theory|equivariant]] [[differential K-theory|differential]] [[topological K-theory|K-theory]] of transverse complex curves accommodates exotic charges of the form expected of [[codimension]]=2 [[defect branes]], such as of [[D7-branes]] in [[F-theory|IIB/F-theory]] on [[ADE-singularity|$\mathbb{A}$-type]] [[orbifold]] [[singularities]], but also of their [[duality in string theory|dual]] [[3-brane in 6d|3-brane]] [[defect QFT|defects]] of class-S theories on [[M5-branes]]. These branes have been argued, within [[F-theory]] and the [[AGT correspondence]], to carry special [[modular group|$\mathrm{SL}(2)$]]-[[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 brane|"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 [[Dirac charge quantization|charge quantization law]] given [[D-brane charge quantization in K-theory|by K-theory]]. > Here we observe that it is the subtle (and previously somewhat neglected) [[twisted equivariant K-theory|twisting of equivariant K-theory]] by [[flat connection|flat]] complex line bundles appearing inside [[orbifold singularity|orbi-singularities]] ("inner local systems") that makes the secondary [[Chern character]] on a punctured plane inside an [[ADE singularity|$\mathbb{A}$-type singularity]] evaluate to the [[twisted de Rham cohomology|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 "[[schreiber:Hypothesis H|Hypothesis H]]" about brane [[Dirac charge quantization|charge quantization]] in M-theory. Since [[conformal blocks]] -- and hence these twisted equivariant secondary Chern characters -- solve the [[Knizhnik-Zamolodchikov equation]] and thus constitute [[braid representation|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 theory|string]]/[[M-theory]]. $\,$ Comments are welcome. If you do have a look, please grab our latest pdf version from behind [[schreiber:Anyonic defect branes in TED-K-theory|the above link]].

   We are finalizing an article:

   • H. Sati & U.S.: Anyonic defect branes in TED-K-theory

   $\,$

   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.

   $\,$

   Comments are welcome. If you do have a look, please grab our latest pdf version from behind the above link.

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeMar 15th 2022
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexQuick typos: > 1-twisted de Rham cohomology already of N-punctured planes (why 'already'?) > Fugure (twice)

   Quick typos:

   1-twisted de Rham cohomology already of N-punctured planes (why ’already’?)

   Fugure (twice)

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMar 15th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks. 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.

   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.

4. • CommentRowNumber4.
   • CommentAuthorDavid_Corfield
   • CommentTimeMar 16th 2022
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexInteresting 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](https://twitter.com/SchreiberUrs/status/1468861885739151364) of > F-theory apparently wants to lift the $S^4$-coefficient of M-theory to its branched cover by $CP^1 \times CP^1$?

   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 $CP^1 \times CP^1$?

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeMar 16th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexYes, 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](http://128.2.67.219/schreiber/files/DefectBranes_220316b.pdf#page=23)), whose insight may not have found due attention before (cf Rem. 4.7).

   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).

6. • CommentRowNumber6.
   • CommentAuthorDavid_Corfield
   • CommentTimeMar 17th 2022
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexThanks!

   Thanks!