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Monday, October 23, 2017

**Abstract:** The Witten genus is an invariant of a manifold $M$ that is morally gotten by doing index theory on $LM$. It would be lovely to turn ‘morally’ into ‘actually’, and hence the interest in geometric constructions of the Witten genus. One possible approach is via the ring of chiral differential operators (CDOs) on $M$, introduced for complex $M$ by Malikov, Schechtman, and Vaintrob. This is a sheaf of vertex algebras on $M$ whose character yields the Witten genus. Of course, any old bundle with the right numerical data can do this—the interesting part is to relate the construction to loop space. I’ll introduce the bare minimum about vertex algebras to discuss the obstruction theory to gluing together a global sheaf of CDOs, wherein loopy things like local lifts of Wess-Zumino forms and ‘curvature' 3-forms appear.