Higher current algebras and chiral algebras

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 natural 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 dimensions. In higher dimensions one is forced to work up to coherent homotopy in some fashion; in this model it turns out to be in the mildest fashion one could hope for: namely, one moves from Lie algebras to their homotopy analogs, L-infinity algebras, and from chiral algebras to homotopy chiral algebras in a sense introduced by Malikov-Schechtman.

The main tool in the talk will be a strict cdga model — the polysimplicial model — of derived global sections of the structure sheaf on configuration space. The hope is that this model will prove well-adapted to doing concrete calculations, and in that direction, I will gesture towards a homotopy version of the usual Arnold/Orlik-Solomon relations for broken circuits.

This is joint work with Zhengping Gui and Laura Felder and is based largely on the preprint 2506.09728

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