Vertex algebras capture physicists' notion of OPEs in chiral CFTs, in complex dimension one. For various motivations, one would like to have analogs of vertex algebras in higher dimensions. Chiral algebras, in the sense of Beilinson-Drinfeld and Francis-Gaitsgory, provide a promising framework here, because they re-express the vertex algebra axioms (which are rather sui generis, and therefore hard to generalize) as something more recognizable (a chiral algebra is a Lie algebra, of a sort).
I will review this, and then go on to introduce a certain concrete model of the unit chiral algebra in higher complex dimensions. We shall see that in going to higher dimensions, one naturally moves from Lie algebras to their homotopy analogs, L-infinity algebras, and from chiral algebras to homotopy chiral algebras in a sense recently introduced by Malikov-Schechtman. The main tool in the talk will be a new model -- the polysimplicial model -- of derived sections of the sheaf of functions on higher configuration spaces. The hope is that this model will prove well-adapted to doing concrete calculations.
This is joint work in preparation with Zhengping Gui and Laura Felder.