modular forms from partition functions

**functorial quantum field theory** ## Contents * cobordism category * cobordism * extended cobordism * (∞,n)-category of cobordisms * Riemannian bordism category * cobordism hypothesis * generalized tangle hypothesis * classification of TQFTs * FQFT * extended TQFT * CFT * vertex operator algebra * TQFT * Reshetikhin–Turaev model / Chern-Simons theory * HQFT * TCFT * A-model, B-model, Gromov-Witten theory * homological mirror symmetry * FQFT and cohomology * (1,1)-dimensional Euclidean field theories and K-theory * (2,1)-dimensional Euclidean field theory * geometric models for tmf * holographic principle of higher category theory * holographic principle * AdS/CFT correspondence * quantization via the A-model

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*** **superalgebra** and **supergeometry** ## Formal context ## * superpoint * super ∞-groupoid, smooth super ∞-groupoid, synthetic differential super ∞-groupoid * synthetic differential supergeometry ## Superalgebra ## * super vector space, SVect * super algebra * Grassmann algebra * Clifford algebra * superdeterminant * super Lie algebra * super Poincare Lie algebra ## Supergeometry ## * supermanifold, SDiff * super Lie group * super translation group * super Euclidean group * NQ-supermanifold * super vector bundle * complex supermanifold * Euclidean supermanifold * integration over supermanifolds * Berezin integral ## Structures * supersymmetry * division algebra and supersymmetry ## Applications ## * supergravity * supersymmetric quantum mechanics * geometric model for elliptic cohomology

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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.




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|1)(2|1)-dimensional EFT is a modular form. Hence (2|1)(2|1)-dimensional EFTs do yield the correct cohomology ring of tmf over the point.

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Created on September 24, 2009 15:21:39 by Urs Schreiber (