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).
functorial quantum field theory
Reshetikhin?Turaev model? / Chern-Simons theory
FQFT and cohomology
noncommutative topology, noncommutative geometry
noncommutative stable homotopy theory
genus, orientation in generalized cohomology
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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 $H$.
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
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.
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:
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
A. N. Schellekens, Nicholas P. Warner, Anomalies and modular invariance in string theory, Physics Letters B 177 (3-4), 317-323, 1986 (doi:10.1016/0370-2693(86)90760-4)
A. N. Schellekens, Nicholas P. Warner, Anomalies, characters and strings, Nuclear Physics B Volume 287, 1987, Pages 317-361 (doi:10.1016/0550-3213(87)90108-8)
Wolfgang Lerche, Bengt Nilsson, A. N. Schellekens, Nicholas P. Warner, Anomaly cancelling terms from the elliptic genus, Nuclear Physics B Volume 299, Issue 1, 28 March 1988, Pages 91-116 (doi:10.1016/0550-3213(88)90468-3)
and then strictly originates with:
Edward Witten, Elliptic genera and quantum field theory, Comm. Math. Phys. Volume 109, Number 4 (1987), 525-536. (euclid:cmp/1104117076)
Edward Witten, On the Landau-Ginzburg Description of $N=2$ Minimal Models, Int. J. Mod. Phys.A9:4783-4800,1994 (arXiv:hep-th/9304026)
Toshiya Kawai, Yasuhiko Yamada, Sung-Kil Yang, Elliptic Genera and $N=2$ Superconformal Field Theory, Nucl. Phys. B414:191-212, 1994 (arXiv:hep-th/9306096, doi:10.1016/0550-3213(94)90428-6)
Sujay K. Ashok, Jan Troost, A Twisted Non-compact Elliptic Genus, JHEP 1103:067, 2011 (arXiv:1101.1059)
Matthew Ando, Eric Sharpe, Elliptic genera of Landau-Ginzburg models over nontrivial spaces, Adv. Theor. Math. Phys. 16 (2012) 1087-1144 (arXiv:0905.1285)
Review in:
Miranda Cheng, (Mock) Modular Forms in String Theory and Moonshine, lecture notes 2016 (pdf)
Katrin Wendland, Section 2.4 in: Snapshots of Conformal Field Theory, in: Mathematical Aspects of Quantum Field Theories Mathematical Physics Studies. Springer 2015 (arXiv:1404.3108, doi:10.1007/978-3-319-09949-1_4)
Formulation via super vertex operator algebras:
Hirotaka Tamanoi, Elliptic Genera and Vertex Operator Super-Algebras, Springer 1999 (doi:10.1007/BFb0092541)
Chongying Dong, Kefeng Liu, Xiaonan Ma, Elliptic genus and vertex operator algebras, Algebr. Geom. Topol. 1 (2001) 743-762 (arXiv:math/0201135, doi:10.2140/agt.2001.1.743)
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:
Tentative interpretation as indices of Dirac-Ramond operators as would-be Dirac operators on smooth loop space:
Edward Witten, The Index Of The Dirac Operator In Loop Space, in: Elliptic Curves and Modular Forms in Algebraic Topology, Lecture Notes in Mathematics 1326, Springer (1988) 161-181 [doi:10.1007/BFb0078045, spire]
originating from:
Edward Witten, p. 92-94 in: Global anomalies in string theory, in: W. Bardeen and A. White (eds.) Symposium on Anomalies, Geometry, Topology, World Scientific (1985) 61-99 [pdf, spire:214913]
Orlando Alvarez, T. P. Killingback, Michelangelo Mangano, Paul Windey, The Dirac-Ramond operator in string theory and loop space index theorems, Nuclear Phys. B Proc. Suppl., 1A:189–215, 1987, in: Nonperturbative methods in field theory, 1987 (doi"10.1016/0920-5632(87)90110-1)
Orlando Alvarez, T. P. Killingback, Michelangelo Mangano, Paul Windey, String theory and loop space index theorems, Comm. Math. Phys., 111(1):1–10, 1987 (euclid:cmp/1104159462)
Gregory Landweber, Dirac operators on loop space, PhD thesis (Harvard 1999) (pdf)
Orlando Alvarez, Paul Windey, Analytic index for a family of Dirac-Ramond operators, Proc. Natl. Acad. Sci. USA, 107(11):4845–4850, 2010 (arXiv:0904.4748)
Tentative formulation via conformal nets:
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):
Davide Gaiotto, Theo Johnson-Freyd, Edward Witten, p. 17 of: A Note On Some Minimally Supersymmetric Models In Two Dimensions, (arXiv:1902.10249) in S. Novikov et al. Integrability, Quantization, and Geometry: II. Quantum Theories and Algebraic Geometry, Proc. Symposia Pure Math., 103(2), 2021 (ISBN: 978-1-4704-5592-7)
Davide Gaiotto, Theo Johnson-Freyd, Mock modularity and a secondary elliptic genus (arXiv:1904.05788)
Theo Johnson-Freyd, Topological Mathieu Moonshine (arXiv:2006.02922)
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$]$
See also
Ying-Hsuan Lin, Du Pei, Holomorphic CFTs and topological modular forms [arXiv:2112.10724]
Jan Albert, Justin Kaidi, Ying-Hsuan Lin, Topological modularity of Supermoonshine $[$arXiv:2210.14923$]$
Yuji Tachikawa, Mayuko Yamashita, Kazuya Yonekura, Remarks on mod-2 elliptic genus $[$arXiv:2302.07548$]$
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:
Jacques Distler, Eric Sharpe, section 8.5 of Heterotic compactifications with principal bundles for general groups and general levels, Adv. Theor. Math. Phys. 14:335-398, 2010 (arXiv:hep-th/0701244)
Matthew Ando, Equivariant elliptic cohomology and the Fibered WZW models of Distler and Sharpe, talk 2007 (lecture notes pdf)
Speculations on physics aspects of lifting the Witten genus to topological modular forms:
Yuji Tachikawa, Topological modular forms and the absence of a heterotic global anomaly (arXiv:2103.12211)
Yuji Tachikawa, Mayuko Yamashita, Topological modular forms and the absence of all heterotic global anomalies (arXiv:2108.13542)
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:
Davide Gaiotto, Andrew Strominger, Xi Yin, The M5-Brane Elliptic Genus: Modularity and BPS States, JHEP 0708:070, 2007 (hep-th/0607010)
Davide Gaiotto, Xi Yin, Examples of M5-Brane Elliptic Genera, JHEP 0711:004, 2007 (arXiv:hep-th/0702012)
Further discussion in:
Murad Alim, Babak Haghighat, Michael Hecht, Albrecht Klemm, Marco Rauch, Thomas Wotschke, Wall-crossing holomorphic anomaly and mock modularity of multiple M5-branes, Comm. Math. Phys. 339 (2015) 773–814 $[$arXiv:1012.1608, doi:10.1007/s00220-015-2436-3$]$
Sergei Gukov, Du Pei, Pavel Putrov, Cumrun Vafa, 4-manifolds and topological modular forms, J. High Energ. Phys. 2021 84 (2021) $[$arXiv:1811.07884, doi:10.1007/JHEP05(2021)084, spire:1704312$]$
On the elliptic genus of M-strings inside M5-branes:
Stefan Hohenegger, Amer Iqbal, M-strings, Elliptic Genera and $\mathcal{N}=4$ String Amplitudes, Fortschritte der PhysikVolume 62, Issue 3 (arXiv:1310.1325)
Stefan Hohenegger, Amer Iqbal, Soo-Jong Rey, M String, Monopole String and Modular Forms, Phys. Rev. D 92, 066005 (2015) (arXiv:1503.06983)
M. Nouman Muteeb, Domain walls and M2-branes partition functions: M-theory and ABJM Theory (arXiv:2010.04233)
Kimyeong Lee, Kaiwen Sun, Xin Wang, Twisted Elliptic Genera [arXiv:2212.07341]
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)
On the elliptic genus of E-strings as M2-branes ending on M5-branes:
Last revised on November 6, 2021 at 22:16:43. See the history of this page for a list of all contributions to it.