Schreiber Anyonic defect branes in TED K-theory

An article that we have written:

---

• Hisham Sati$\,$ and $\,$ Urs Schreiber:

  
  

  Anyonic Defect Branes and Conformal Blocks in

  Twisted Equivariant Differential (TED) K-Theory

  
  

  Reviews in Mathematical Physics

  35 06 (2023) 2350009

  • doi:10.1142/S0129055X23500095

  
  

  download:

  • arXiv:2203.11838

  • pdf (more references added on $D_4 \!\perp\! NS_5 \rightsquigarrow M_5 \!\perp\! M_5$ as codim=2 defects; some typose fixed)

---

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 and 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.

Related articles:

• Hisham Sati and Urs Schreiber:

  Anyonic topological order in TED K-theory

• Hisham Sati and Urs Schreiber:

  Topological Quantum Programming in TED-K

• Hisham Sati and Urs Schreiber:

  Differential Cohomotopy implies intersecting brane observables

  arXiv:1912.10425

• Hisham Sati, Urs Schreiber:

  Equivariant principal $\infty$-bundles

  arXiv:2112.13654

• David Corfield, Hisham Sati and Urs Schreiber:

  Fundamental weight systems are quantum states

  arXiv:2105.02871

• Hisham Sati and Urs Schreiber:

  Topological Quantum Computation in TED-K

• David Jaz Myers, Hisham Sati and Urs Schreiber:

  Topological Quantum Gates in Homotopy Type Theory

• Hisham Sati and Urs Schreiber:

  Abelian Anyons on Flux-Quantized M5-Branes

For more see at Hypothesis H.

Expository talk:

• via Seminario de Categorías UNAM, 13 April 2022

• via RIND MathPhys & StringTheory seminar, 9 May 2022

• download:

  talk slides pdf

Related talks:

• Knots for quantum computation from defect branes

• TED K-theory of Cohomotopy moduli spaces and Anyonic Topological Order

• Anyonic Defect Branes and Conformal Blocks in TED K

• Proper Orbifold Cohomotopy for M-Theory

• Some Quantum States of M-Branes under Hypothesis H

---