# Value of the Chern-Simons functional for flat connections on $S^3/\Gamma$

Let $\Gamma$ be a finite subgroup of SU(2) and consider the quotient of $S^3$ by $\Gamma$ via its left action. Pick a simply connected compact Lie group $G$ and take a flat connection on this quotient. Or equivalently, choose a homomorphism $\Gamma \to G$. The Chern-Simons action functional should assign a value in $\mathbb{R}/\mathbb{Z}$. How do I compute it?

The paper http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002334585 by Kronheimer and Nakajima has the formula for the eta invariant which is closely related, but I think it gives the CS functional only as a value in $\mathbb{R}/(\mathbb{Z}/h^\vee(G))$.

## 2 Answers

For $G=SU(N)$, there is a paper SU(n)–Chern–Simons invariants of Seifert fibered 3–manifolds (Int. J. Math., 09, 295-330 (1998))

• Thank you for the info. For SU(n) I realized that you can compute the CS functional using the Kronheimer-Nakajima formula by considering the eta invariant of the Dirac operator associated to the the defining representation of the principal bundle. The cases that can't be reduced to Kronheimer-Nakajima are type B,D,E, F and G. – Yuji Tachikawa Dec 30 '16 at 0:49
• I went through the paper cited more carefully. I believe the first method described there can be straightforwardly generalized to all simply-connected groups (although Lemma 3.3 can't be used to simplify the final expression.) So I accept this as an answer. – Yuji Tachikawa Jan 2 '17 at 7:09

For $G = SU(2)$ and the representation given by including $\Gamma$, the value of the Chern-Simons invariant is computed by Millson (Examples of nonvanishing Chern-Simons invariants, J. Differential Geom. Volume 10, Number 4 (1975), 589-600) in topological terms. He explicitly works out the case of lens spaces (cyclic $\Gamma$). It seems that you could get the same method to work for general representations, but it might be fairly involved, because it would depend on a fairly detailed knowledge of the representation theory of your group $\Gamma$.

For $\Gamma$ cyclic, this is probably fairly straightforward, using the fact that any representation splits as a sum of $1$-dimensional representations. There are some additional computations (for $G=SU(2)$ but non-abelian $\Gamma$) in a paper of Tsuboi, On Chern-Simons invariants of spherical space forms, (Japanese journal of mathematics. New series Vol. 10 (1984) No. 1, 9-28).

• Thanks, for cyclic $\Gamma$ I realized that you can easily write down an explicit connection one-form and compute the CS. – Yuji Tachikawa Dec 29 '16 at 10:38