Let S be a closed, orientable 2d manifold and G a finite group. Since a principal Gbundle over S is specified by maps $\phi : \pi_1(S) \rightarrow G$ modulo the adjoint action by G, the way to count how many such bundles exist is \begin{align} \dfrac{  Hom(\pi_1(S),G) }{G} = \sum_{\text{irreps V of }G} \bigg( \dfrac{d(V)}{G} \bigg)^{22g} \; , \end{align} where $d(V)$ is the dimension of the irrep V, and g is the genus of S. My question is how the formula above generalizes when we want to count principal Gbundles over S, when S has n boundaries with prescribed holonomies for the bundle specified by conjugacy classes $\{ k_i : i=1,...,n \}$ of G. Morally we should have something like \begin{align} \dfrac{  Hom(\pi_1(S_{g,k_i}),G) }{G} = \sum_{\text{irreps V of }G} \bigg( \dfrac{d(V)}{G} \bigg)^{22gn} \prod_{i=1}^n \chi_V(k_i) \; , \end{align} where $\chi_V(k_i)$ is the character of the representation V evaluated at any representative of the conjugacy class $k_i$. So, is the above formula completely correct? Are there factors missing? I would really appreciate any references that prove what the generalization is.
1 Answer
I get a slightly different formula:
$$\sum_{V} \frac{d(V)^{2g2n}}{G^{2g2}} \prod_{i=1}^n k_i\chi_V(k_i)$$
Here $k_i$ denotes the size of the conjugacy class.
I prefer to express this in a slightly different way. For a conjugacy class $k$ and irreducible representation $V$ of $G$, let $f^V_k$ denote the scalar by which the indicator function $k \in Z(\mathbb C[G])$ acts on $V$. In these terms the formula is:
$$\sum_V \left(\frac{d(V)}{G}\right)^{2g2} \prod_{i=1}^n f^V_{k_i}$$
To obtain the first formula from the second, note that the numbers $f^V_k$ are related to characters as follows:
$$f^V_k = \frac{k}{d(V)} \chi_V(k)$$
For me, these formulas come from general properties of commutative Frobenius algebras, or equivalently 2d topological field theory (TFT). Namely, given a commutative Frobenius algebra $A$, a genus $g$, and a collection of elements $k_1, \ldots , k_n$ one obtains a number
$$Z_A(g; k_1, \ldots , k_n)$$
recording the value of the TFT on an oriented surface of genus $g$ with $n$ punctures labelled by the elements $k_1, \ldots , k_n$. You can compute this number by first multiplying together the elements $k_1 \ldots k_n$ in $A$, then applying a sequence of $g$ comultipication followed by multiplication operations, finally followed by the Frobenius trace (see cartoon below).
In the case when $A$ is semisimple, one can be more explicit and write everything in terms of a basis of orthogonal idempotents.
In our case we take $A=Z(\mathbb C[G])$, the center of the group algebra, equipped with the trace $t$ (which takes the value $1/G$ at the identity element of $G$ and zero on all other elements). The numbers $f^V_k$ is just the change of basis matrix between the conjugacy classes $k$ and the orthogonal idempotents $e_V$ labelled by irreps.
I can't think of a reference for this right now. In this paper I explain some of this stuff in a related context, see e.g. Prop 2.13.

$\begingroup$ Thank you for your answer. Actually my question exactly came from studying 2d TQFT. However, I couldn't find Proportion 2.13 in your nice paper. Do you maybe mean Theorem 1.4, which has a similar expression for the symmetric group? $\endgroup$– JordanSep 15, 2020 at 19:00

1$\begingroup$ I am not sure if I should ask this as a separate question, but I was also wondering the following. The partition function of (untwisted) Dijkgraaf Witten theory for a finite group G just counts the number of principal bundles up to gauge transformations. However, when we have DW theory with defects, how do we count principal bundles with prescribed boundaries and defects?In arxiv.org/abs/1507.00941, they proposed a topological invariant for the case of defects but they are missing a representation theory formula that generalizes the ones above. Is it somewhere in the literature? $\endgroup$– JordanSep 15, 2020 at 19:36

$\begingroup$ @Jordan There should be a Proposition 2.13 on page 18 of the paper. It basically gives the formula for the linear map associated to a punctured surface in terms of the orthogonal idempotents. $\endgroup$ Sep 17, 2020 at 10:26

$\begingroup$ I personally haven't thought these kind of formulas for defects other than the ones where you label a boundary circle with a conjugacy class. $\endgroup$ Sep 17, 2020 at 10:26