In this talk, I will first describe a reformulation of a construction by Gorbounov, Malikov and Schechtman in terms of global geometric data. This will then be applied to define vertex algebraic versions of Dolbeault complexes and obtain a geometric description of the Witten and elliptic genera of complex manifolds.
Links:
[1] https://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/node/3444
[3] https://www.mpim-bonn.mpg.de/node/249