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
***
**superalgebra** and (synthetic
) **supergeometry**
## Background
* algebra
* geometry
* graded object
## Introductions
* geometry of physics -- superalgebra
* geometry of physics -- supergeometry
## Superalgebra ##
* super vector space, SVect
* super algebra
* supercommutative superalgebra
* Grassmann algebra
* Clifford algebra
* superdeterminant
* super Lie algebra
* super Poincare Lie algebra
* super L-infinity algebra
## Supergeometry ##
* superpoint
* super Cartesian space
* supermanifold, SDiff
* NQ-supermanifold
* super vector bundle
* complex supermanifold
* Euclidean supermanifold
* super spacetime
* super Minkowski spacetime
* integration over supermanifolds
* Berezin integral
* super Lie group
* super translation group
* super Euclidean group
* super ∞-groupoid
* super formal smooth ∞-groupoid
* super line 2-bundle
## Supersymmetry
supersymmetry
* division algebra and supersymmetry
* super Poincare Lie algebra
* supermultiplet
* BPS state
* M-theory super Lie algebra, type II super Lie algebra
* supergravity Lie 3-algebra, supergravity Lie 6-algebra
## Supersymmetric field theory
* superfield
* supersymmetric quantum mechanics
* superparticle
* adinkra
* super Yang-Mills theory
* supergravity
* gauged supergravity
* superstring theory
* type II string theory
* heterotic string theory
## Applications
* geometric model for elliptic cohomology

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:

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 $(2|1)$-dimensional EFT is a modular form. Hence $(2|1)$-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
(195.37.209.182)