nLab
modular forms from partition functions

This is a sub-entry of geometric models for elliptic cohomology and A Survey of Elliptic Cohomology

See there for background and context.

This entry here is about the fact and its derivation that the partition function of a (2,1)-dimensional Euclidean field theories is a modular form.

Previous:

• Axiomatic field theories and their motivation from topology

• (1,1)-dimensional Euclidean field theories and K-theory

• (2,1)-dimensional Euclidean field theories and tmf

• bordism categories following Stolz-Teichner

• (2,1)-dimensional Euclidean field theories

Idea

As described at (2,1)-dimensional Euclidean field theories and tmf, the idea is that (2,1)-dimensional Euclidean field theories are a geometric model for tmf cohomology theory.

While there is no complete proof of this so far, here we discuss the construction and proof – due to Stephan Stolz and Peter Teichner – for the situation over the point: the partition function of a $(2\mid 1)$-dimensional EFT is a modular form. Hence $(2\mid 1)$-dimensional EFTs do yield the correct cohomology ring of tmf over the point.

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