nLab partition function

Contents

This entry is about partition functions in the sense of statistical mechanics and quantum field theory. For the function in number theory/combinatorics that assigns to a natural number the number of its partitions see at partition function (number theory).

Contents

Idea

A 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 $Z$ is presented as an FQFT, that is as a functor on a category of $d$-dimensional cobordisms, then the partition function is the assignment to $d$-dimensional tori $T$ of the values $Z(T)$ 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 $t$ is a linear operator of the form $\exp(- t H)$ for some operator $H$.

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 $H$ acts on is the vector space of all states of the system and $H$ is the operator whose eigenstates are the states of definite energy. The expression

$tr(exp(-t H))$

then is interpreted as

sum over all states $\Psi$ of the system and weigh each one by its energy $E_\Psi$.

This involves, conversely, counting for each fixed energy $E_\Psi$ 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 $Z$ 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.

Examples

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

$d$partition function in $d$-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 $M Spin \to KO$
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 $M Spin^c \to KU$
endpoint of type II superstringSpin^c Dirac operator twisted by Chan-Paton gauge fieldD-brane chargeTodd genusBernoulli numbersAtiyah-Bott-Shapiro orientation $M Spin^c \to KU$
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

References

General

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

Elliptic genera as super $p$-brane partition functions

The interpretation of elliptic genera (especially the Witten genus) as the partition function of a 2d superconformal field theory (or Landau-Ginzburg model) – and especially of the heterotic string (“H-string”) or type II superstring worldsheet theory has precursors in

and then strictly originates with:

Review in:

With emphasis on orbifold CFTs:

Formulations

Via super vertex operator algebra

Formulation via super vertex operator algebras:

and for the topologically twisted 2d (2,0)-superconformal QFT (the heterotic string with enhanced supersymmetry) via sheaves of vertex operator algebras in

based on chiral differential operators:

In relation to error-correcting codes:

• Kohki Kawabata, Shinichiro Yahagi, Elliptic genera from classical error-correcting codes $[$arXiv:2308.12592$]$
Via Dirac-Ramond operators on free loop space

Tentative interpretation as indices of Dirac-Ramond operators as would-be Dirac operators on smooth loop space:

Via conformal nets

Tentative formulation via conformal nets:

Conjectural interpretation in tmf-cohomology

The resulting suggestion that, roughly, deformation-classes (concordance classes) of 2d SCFTs with target space $X$ are the generalized cohomology of $X$ with coefficients in the spectrum of topological modular forms (tmf):

and the more explicit suggestion that, under this identification, the Chern-Dold character from tmf to modular forms, sends a 2d SCFT to its partition function/elliptic genus/supersymmetric index:

This perspective is also picked up in Gukov, Pei, Putrov & Vafa 18.

Discussion of the 2d SCFTs (namely supersymmetric SU(2)-WZW-models) conjecturally corresponding, under this conjectural identification, to the elements of $\mathbb{Z}/24$ $\simeq$ $tmf^{-3}(\ast) = \pi_3(tmf)$ $\simeq$ $\pi_3(\mathbb{S})$ (the third stable homotopy group of spheres):

Discussion properly via (2,1)-dimensional Euclidean field theory:

• Daniel Berwick-Evans, How do field theories detect the torsion in topological modular forms? $[$arXiv:2303.09138$]$

• Daniel Berwick-Evans, How do field theories detect the torsion in topological modular forms?, talk at QFT and Cobordism, CQTS (Mar 2023) $[$web, video:YT$]$

Occurrences in string theory

H-string elliptic genus

Further on the elliptic genus of the heterotic string being the Witten genus:

The interpretation of equivariant elliptic genera as partition functions of parametrized WZW models in heterotic string theory:

Proposals on physics aspects of lifting the Witten genus to topological modular forms:

M5-brane elliptic genus

On the M5-brane elliptic genus:

A 2d SCFT argued to describe the KK-compactification of the M5-brane on a 4-manifold (specifically: a complex surface) originates with

Discussion of the resulting elliptic genus (2d SCFT partition function) originates with:

Further discussion in:

M-string elliptic genus

On the elliptic genus of M-strings inside M5-branes:

E-string elliptic genus

On the elliptic genus of E-strings as wrapped M5-branes:

• J. A. Minahan, D. Nemeschansky, Cumrun Vafa, N. P. Warner, E-Strings and $N=4$ Topological Yang-Mills Theories, Nucl. Phys. B527 (1998) 581-623 (arXiv:hep-th/9802168)

• Wenhe Cai, Min-xin Huang, Kaiwen Sun, On the Elliptic Genus of Three E-strings and Heterotic Strings, J. High Energ. Phys. 2015, 79 (2015). (arXiv:1411.2801, doi:10.1007/JHEP01(2015)079)

Last revised on November 6, 2021 at 22:16:43. See the history of this page for a list of all contributions to it.