First of, if any of the following below does not make sense, please feel free to leave a comment =)

Central charges in Chern Simons in in the Virasoro conformal blocks play an important role for generating Conformal and Lorentz anomalies as non-local modes. In Chern Simons, a flat Holonomy corresponds to a vanishing Monodromy. I was wondering if there was any way to connect central charges with the representations of the spin structure. If so, does that mean the irreducible represents of the spin structure with a non-flat Holonomy can be regarded as the non-local modes generating the various anomalies above.

  • $\begingroup$ A spinor is a representation of the extension of the orthogonal group. But this is a group extension, not an algebra extension. Central charges classify algebraic-theoretic extensions, not group-theoretic ones. So no, spinors and central charges are different objects. But this is math, really; should this be in math.SE instead? $\endgroup$ Apr 23 at 20:43
  • $\begingroup$ But central charges are given by the central extension for some sub group $N \subseteq G$ where N lies in the center of G where G is a group extension. So then, central extensions are related to group extensions and so are central charges and spinors right? $\endgroup$ Apr 23 at 22:27
  • $\begingroup$ Central charges lead to central extensions, but not all central extensions come from central charges. Spinors are due to an extension that does not come from a central charge (simple algebras have trivial homology in the appropriate degree, cf. Whitehead's lemmas). Virasoro and Kac-Moody do come from a central charge (Poincare and the loop algebra have $H^2=\mathbb Z$ so they do admit a central charge). This really feels like a math question, not a physics one... $\endgroup$ Apr 24 at 11:36
  • $\begingroup$ So the Poincare algebra admits both spinors and central charges. Please hear me out: For a zero cosmological constant, in Chern simons we can build a connection over some group G (in this case the group is $SO(1,3)$. We can calculate the expected value of the Wilson loop over this connection for a holographic decomposition with some boundary connections (Maybe non-chiral?) Furthermore, the connection admits a spin structure and thus for a zero cosmological constant. But can the representations of $SO(1,3)$ also be the elements in the expected value for the wilson loop? $\endgroup$ Apr 25 at 17:41
  • $\begingroup$ Also, I don't know how to transfer this post, or do it just re-do it in math? $\endgroup$ Apr 25 at 17:48

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