partition function



Quantum field theory

Index theory



Special and general types

Special notions


Extra structure





The partition function is a certain assignment that may be extracted from a system in statistical mechanics, or in quantum field theory. If the quantum field theory ZZ is presented as an FQFT, that is, as a functor on a category of dd-dimensional cobordisms, then the partition function is the assignment to dd-dimensional tori TT of the values Z(T)Z(T) assigned to these by the QFT.

By the axioms of functoriality and symmetric monoidalness of a QFT, this means that the partition function is the trace over the value of the QFT in the cylinder obtained by cutting the torus open.

This is where the partition function originally derives its name from: typically for QFTs on Riemannian cobordisms the value of the QFT on a cylinder of length tt is a linear operator of the form exp(tH)\exp(- t H) for some operator HH.

Origin of the term

When one thinks of the QFT — under Wick rotation — as describing a physical system in statistical mechanics, then the vector space that HH acts on is the vector space of all states of the system and HH is the operator whose eigenstates are the states of definite energy. The expression

tr(exp(tH)) tr(exp(-t H))

then is interpreted as

sum over all states Ψ\Psi of the system and weigh each one by its energy E ΨE_\Psi.

This involves, conversely, counting for each fixed energy E ΨE_\Psi the number of states of that energy. This will typically be a sum over certain partitions of various particles of an ensemble into various “bins” of partial energies. Therefore the term partition function.

In fact, the common letter ZZ uses to denote QFTs (or at least TQFTs) also derives from this: in German the partition function is called Zustandssumme — from German Zustand for “state” .

The Mellin transform of the partition function is known in quantum field theory as the Schwinger parameter-formulation which takes the worldline theory to its zeta regulated Feynman propagator.


Partition function for the superparticle: K-theory index.

Partition function for the type II superstring: elliptic genus.

Partition function for the heterotic string: Witten genus.

For some discussion of partition functions of 1-dimensional QFTs see (1,1)-dimensional Euclidean field theories and K-theory.

For some discussion of partition functions of 2-dimensional QFTs see (2,1)-dimensional Euclidean field theories and tmf

partition functions in quantum field theory as indices/genera/orientations in generalized cohomology theory:

ddpartition function in dd-dimensional QFTsuperchargeindex in cohomology theorygenuslogarithmic coefficients of Hirzebruch series
0push-forward in ordinary cohomology: integration of differential formsorientation
1spinning particleDirac operatorKO-theory indexA-hat genusBernoulli numbersAtiyah-Bott-Shapiro orientation MSpinKOM Spin \to KO
endpoint of 2d Poisson-Chern-Simons theory stringSpin^c Dirac operator twisted by prequantum line bundlespace of quantum states of boundary phase space/Poisson manifoldTodd genusBernoulli numbersAtiyah-Bott-Shapiro orientation MSpin cKUM Spin^c \to KU
endpoint of type II superstringSpin^c Dirac operator twisted by Chan-Paton gauge fieldD-brane chargeTodd genusBernoulli numbersAtiyah-Bott-Shapiro orientation MSpin cKUM Spin^c \to KU
2type II superstringDirac-Ramond operatorsuperstring partition function in NS-R sectorOchanine elliptic genusSO orientation of elliptic cohomology
heterotic superstringDirac-Ramond operatorsuperstring partition functionWitten genusEisenstein seriesstring orientation of tmf
self-dual stringM5-brane charge
3w4-orientation of EO(2)-theory


  • Addison Ault, “The partition function: If that’s what it is Why don’t they say so!” (pdf)

Last revised on January 1, 2020 at 17:29:26. See the history of this page for a list of all contributions to it.