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.