We present a cocycle model for elliptic cohomology with complex coefficients in which methods
from
–dimensional
quantum field theory can be used to rigorously construct cocycles. For example,
quantizing a theory of vector-bundle-valued fermions yields a cocycle representative
of the elliptic Thom class. This constructs the complexified string orientation of
elliptic cohomology, which determines a pushforward for families of rational string
manifolds. A second pushforward is constructed from quantizing a supersymmetric
–model.
These two pushforwards agree, giving a precise physical interpretation for the elliptic
index theorem with complex coefficients. This both refines and supplies further
evidence for the long-conjectured relationship between elliptic cohomology and
–dimensional
quantum field theory. Analogous methods in supersymmetric mechanics recover path
integral constructions of the Mathai–Quillen Thom form in complexified
–theory and a cocycle
representative of the
–class
for a family of oriented manifolds.
Keywords
elliptic cohomology, topological modular forms,
supersymmetric field theories, Witten genus, Mathai-Quillen
forms
