from the bordism ring.
The cobordism ring here may be replaced by rings of cobordisms with extra structure.
At least in some important cases, genera seem to be naturally understood as encoding sigma-model quantum field theories. For some structure, the Thom spectrum is the classifying space of manifolds with G-structure, and hence may be thought of as classifying target spaces for sigma-models. The codomain spectrum itself may then be thought of as a classifying space for a certain class of QFTs, and hence the genus can be thought of as assigning to any target space the corresponding sigma-model.
This is for instance the case at least over the point for the A-hat genus , which may be thought of as sending manifolds with spin structure to the corresponding (1,1)-supersymmetric EFT (“spinning particle”); and for the Witten genus , which can be thought of as sending a manifold with string structure to the corresponding (2,1)-supersymmetric EFT (“heterotic string”).
The signature genus;
|partition function in -dimensional QFT||index/genus in cohomology theory|
|0||push-forward in ordinary cohomology: integration of differential forms|
|1||spinning particle||K-theory index|
|endpoint of 2d Poisson-Chern-Simons theory string||space of quantum states of boundary phase space/Poisson manifold|
|endpoint of type II superstring||D-brane charge|
|2||type II superstring||elliptic genus|
|heterotic string||Witten genus|