FQFT and cohomology
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 is presented as an FQFT, that is, as a functor on a category of -dimensional cobordisms, then the partition function is the assignment to -dimensional tori of the values 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 is a linear operator of the form for some operator .
When one thinks of the QFT — under Wick rotation — as describing a physical system in statistical mechanics, then the vector space that acts on is the vector space of all states of the system and is the operator whose eigenstates are the states of definite energy. The expression
then is interpreted as
sum over all states of the system and weigh each one by its energy .
This involves, conversely, counting for each fixed energy 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 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” .
|partition function in -dimensional QFT||supercharge||index in cohomology theory||genus||logarithmic coefficients of Hirzebruch series|
|0||push-forward in ordinary cohomology: integration of differential forms|
|1||spinning particle||Dirac operator||KO-theory index||A-hat genus||Bernoulli numbers|
|endpoint of 2d Poisson-Chern-Simons theory string||Spin^c Dirac operator twisted by prequantum line bundle||space of quantum states of boundary phase space/Poisson manifold||Todd genus||Bernoulli numbers|
|endpoint of type II superstring||Spin^c Dirac operator twisted by Chan-Paton gauge field||D-brane charge||Todd genus||Bernoulli numbers|
|2||superstring||Dirac-Ramond operator||superstring partition function||elliptic genus/Witten genus||Eisenstein series|
|self-dual string||M5-brane charge|
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