Differential operators on the loop group via chiral algebras (Q2778024)

Starting from an affine group \(G\) with Lie algebra \(\mathfrak g\) and a given invariant symmetric bilinear form \(Q\) on \(\mathfrak g\), one can introduce the affine Lie algebra \(\widetilde g_Q\) and associate to the data a module \(V_{G,Q}\) over \(\widetilde g_Q\), which carries a natural structure of a vertex operator algebra. In this article the language of chiral algebras is used, and \(V_{G,Q}\) corresponds to a chiral algebra \(D_{G,Q}\). The \(\widetilde g_Q\)-module structure on \(D_{G,Q}\) comes from an embedding of chiral algebras \(l:A_{\mathfrak g,Q}\to D_{G,Q}\). Here \(A_{\mathfrak g,Q}\) is the (Kac-Moody) chiral algebra attached to the pair \((\mathfrak g,Q)\). NEWLINENEWLINENEWLINEThe authors point out that the main result of this paper is the observation that there is another embedding of chiral algebras \(r:A_{\mathfrak g,Q'}\to D_{G,Q}\), with \(Q'=-Q-Q_0\), where \(Q_0\) is the Killing form. Both embeddings commute with one another in the appropriate sense. NEWLINENEWLINENEWLINEThe authors explain the algebro-geometric meaning of the construction. It is related to defining the notion of \({\mathcal D}\)-modules on the \(\infty\)-dimensional variety \(Z((t))\), the \(ind\)-scheme that classifies maps from the formal punctured disc to the smooth affine variety \(Z\). For \(Z=G\) and chosen form \(Q\), there exists a specific chiral algebra of differential operators \(D_{G,O}\). The embeddings \(l\) and \(r\) are related with the embeddings of \(\mathfrak g((t))\) into the ``ring of differential operators'' by means of left and right-invariant vector fields. For the embedding \(r\) a suitable modification is necessary which yields a nontrivial correction term appearing as an anomaly. This correction term is related to the Killing form. NEWLINENEWLINENEWLINERelations between the semi-infinite cohomology of \(\mathfrak g((t))\) and the vertex algebra \(V_{G,Q}\) are discussed. Also a version of \(D_{G,O}\) enlarged by fermions (i.e. a super version) is studied. NEWLINENEWLINENEWLINEIn an appendix it is shown that when \(Q=0\) the category of chiral \(D_{G,O}\) might be a reasonable candidate for the category of \({\mathcal D}\)-modules on \(G((t))\).