# Are central charges equal or similar to irreducible spinor representations?

• 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? Apr 23 at 22:27
• 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... Apr 24 at 11:36
• 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? Apr 25 at 17:41