In any $\mathcal{N}=2$ supersymmetric conformal field theory in four dimensions, one can find on any $\mathbb{R}^2$ plane a chiral algebra (similar to what exists naturally in two-dimensional CFTs). This chiral algebra has the structure of a $\mathcal{W}$-algebra. This construction also applies to six-dimensional $\mathcal{N}=(2,0)$ theories. Here are some manuscript notes, based on the following papers : Four dimensions; Six dimensions; $\mathcal{W}$-algebras.

Notes

Clay Cordova mentions during his talk at String Maths 2017 nice examples of the correspondence between indices and chiral algebras:

  • The free vector multiplet is related to the $bc$ ghosts theory, and $I(q) = (q)_{\infty}^2$.
  • The $SU(2)$ theory with $N_f=4$ is related to the Kac-Moody $\widehat{SO(8)}_{-2}$ algebra, and the index is expressed in terms of modular forms.