  Trace map on deformation quantized algebra leads to the algebraic index
theorem. In this paper, we investigate a two-dimensional chiral analogue of the
algebraic index theorem via the theory of chiral algebras developed by
Beilinson and Drinfeld. We construct a trace map on the elliptic chiral
homology of the free beta gamma-bc system using the BV quantization framework.
As an example, we compute the trace evaluated on the unit constant chiral chain
and obtain the formal Witten genus in the Lie algebra cohomology. We also
construct a family of elliptic trace maps on coset models.
