Generalized zeta function representation of groups and 2-dimensional topological Yang-Mills theory: the example of \(\operatorname{GL}(2,\mathbb{F}_q)\) and \(\operatorname{PGL}(2,\mathbb{F}_q)\)) (Q2798665)

This well-written paper deals with a certain class of functions generalizing the classic Riemann-zeta one to a representation-theoretic setting inspired by formulas from quantum field theory. The original motivation is the so-called topological limit of the partition function of the Yang-Mills theory on a Riemann surface, which turns out to be \(\sum_\pi \frac{1}{\dim^\chi(\pi)} \). Here the sum goes over the finite dimensional irreducible representations \(\pi\) of the group G (which goes into the definition of the Yang-Mills theory) and \(\chi\) is the genus of the surface. The above formula can be immediately generalized to the case when \(\chi\) is an arbitrary complex parameter. The author reviews and studies this generalized function in the case when the group under consideration is not necessarily a compact simply connected Lie group. Additionally, the above zeta function can be generalized a bit further to a form naturally corresponding to the case when the Riemann surface has boundaries (in this case the ``integral'' in the partition function is performed over the set of connections with prescribed holonomies around the boundary circles), and this case is dealt with in this work as well.