# How to calculate partition function of a QFT by summing over irreducible representations of the symmetry group?

By definition computing the partition function of a QFT amounts to doing a Feynman Path Integral exactly. At a schematic level I can see why this can become a question of summing/integrating over characters of the irreducible representations of the symmetry group.

But I don't understand this precisely enough to do a calculation of this kind.

In QFT books I have never come across such a calculation. There one is always bothered about doing a perturbative evaluation of that using Feynman diagrams. One runs into this kind of a calculation in only papers like this.

I would like to know if there is an expository/introductory reference about this translation and computational technology. Something which say explains this representation theory approach of exact computation of partition function starting from simple field theories which will eventually help me understand the exotic ones seen in the papers like the above.

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The following article by A.B. Balantekin describes in section III the structure of thermal grand canonical partition functions (i.e., with chemical potentials) for systems with Lie group symmetries (which is the partition function of the type described in the reference given in the question).

The sum over the group charcters originates from the presence of the chemical potentials, which are the (analytical continuations of the) coordinates of the maximal torus of the group.

The Balantekin article describes the "sum over states" formulas of the partition functions in which the summation over the group representations is explicit, in contrast to the Feynman's "sum over paths" (mentioned in the question) in which the summation over the representations is implicit.

I'll describe here a simple example of an explicit calculation of the grand canonical partition function over the two-sphere.

Let $-H$ be the scalar Laplacian on the two sphere. The state Hilbert space is spanned the spherical representations (with multiplicity 1), corresponding to integer spins only. The symmetry group $SU(2)$ is generated by the usual set of generators $[J_i, J_j] = \epsilon_{ijk} J_k$

The grand canonical partition function is given by:

$Z = \textrm{Tr}(exp(-\beta H + \mu J_3)) = \sum_{j=0}^{\infty}\chi_j(i \mu)exp(-\beta j(j+1))$

$= \sum_{j=0}^{\infty}\frac{sinh((2j+1) \mu)}{sinh(\mu)}exp(-\beta j(j+1))$

$= \frac{e^{\frac{\beta}{4}}}{2sinh(\mu) }\theta_1(\mu, \beta)$

The actual computation (of the N=4 SYM) referred to in the question needs much more work and it relies on several assumptions and approximations. The introduction section of the following article by Yamada and Yaffe and references therein may be helpful.

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Thanks for your reply and the great references. Can you kindly define what is a "spherical representation"? I have seen this terminology earlier too but couldn't track that in the representation theory books. And if you could enlighten what formula you have used to do the summation in the last line? What is this $\theta _1$ function? –  Anirbit Jun 30 '10 at 14:41
The spherical representations are the representations containing a zero weight. Scalar fields on coset spaces decompose into the spherical representations, since the trivial bundle is induced from the trivial representation of the isotropy group. The formula used is equation (7) from mathworld.wolfram.com/JacobiThetaFunctions.html –  David Bar Moshe Jun 30 '10 at 15:03