Suppose $Y$ is a 3-manifold and the boundary $\Sigma:=\partial Y$ is non-empty. Let $G$ be a Lie group with trivial center. Let $\overline {\mathcal A}_{flat}(\Sigma)$ and $\overline {\mathcal A}_{flat}(Y)$ denote the gauge equivalence classes of flat connections on the trivial bundles $\Sigma \times G$ and $Y \times G$ respectively. The space $\overline {\mathcal A}_{flat}(\Sigma)$ is a finite dimensional symplectic manifold with singularitites. I am trying to understand why the image of the restriction map $\overline {\mathcal A}_{flat}(Y) \to \overline {\mathcal A}_{flat}(\Sigma)$ is a Lagrangian submanifold. I know why the image is an isotropic subspace, but I am having a problem with the dimensions. In particular, the space $$\overline {\mathcal A}_{flat}(\Sigma)\simeq \operatorname{Hom}(\pi_1(\Sigma),G)/G$$ has dimension $(2\operatorname{genus}(\Sigma)-2)\dim(G)$, but I can not see why the space $\operatorname{Hom}(\pi_1(Y),G)/G$ has half that dimension.

For completeness, I include a proof of why the image of $\overline {\mathcal A}_{flat}(Y)$ in $\overline {\mathcal A}_{flat}(\Sigma)$ under the restriction map is isotropic. Let $\mathcal A(Y)$ be the space of connections on $Y \times G$. There is a one-form $\mathcal F$ on $\mathcal A(Y)$ defined as $${\mathcal F}_A(\alpha):=\int_Y \langle \alpha \wedge F_A\rangle,$$ where $A \in \mathcal A(Y)$ is a connection, $F_A \in \Omega^2(Y,\mathfrak g)$ is its curvature and $\alpha \in T_A\mathcal A(Y)$. We can see by some calculations that $$d\mathcal F_A(\alpha,\beta)=\int_\Sigma \langle \alpha \wedge \beta \rangle,$$ where $\alpha, \beta \in T_A\mathcal A(Y)$, which coincides with the symplectic form on $\mathcal A(\Sigma)$ the space of connections on $\Sigma \times G$. When restricted to $\mathcal A_{flat}(Y)$, $\mathcal F$ vansihes, and so does $d\mathcal F$. Therefore the restriction map takes $\mathcal A_{flat}(Y)$ to an isotropic submanifold of $\mathcal A_{flat}(\Sigma)$. By gauge invariance, the restriction map takes $\overline {\mathcal A}_{flat}(Y)$ to an isotropic submanifold of $\overline {\mathcal A}_{flat}(\Sigma)$.

  • 1
    $\begingroup$ See arxiv.org/pdf/0902.2589.pdf. The image of the restriction map need not be smooth even in the smooth part of the surface group character variety. If you treat the image as a variety and not a scheme then it is unknown if the image of half-dimensional. However, if you work on the scheme-theoretic level then indeed the image is Lagrangian. $\endgroup$
    – Misha
    Commented May 12, 2016 at 6:27
  • $\begingroup$ Thanks @misha. The paper you refer to takes care of the case when $G$ is complex reductive. Can we say something when $G$ is compact, especially when it is $SU(2)$? In that case, the result seems to be taken for granted by many authors. $\endgroup$
    – Anon
    Commented May 12, 2016 at 8:04
  • $\begingroup$ If you work with schemes, this is the same computation. $\endgroup$
    – Misha
    Commented May 12, 2016 at 20:53

2 Answers 2


Another good reference is Chris Herald's paper. Legendrian cobordism and Chern-Simons theory on 3-manifolds with boundary. Comm. Anal. Geom. 2 (1994), no. 3, 337–413.

It is an easy exercise with Poincare duality to see that the image of the map in cohomology (so at the level of "Zaraski" tangent spaces) $$ H^1(Y;ad_\rho) \to H^1(\Sigma;ad_\rho) $$ is half dimensional. Indeed this map appears in the long exact sequence of the pair $(Y,\Sigma)$ $$ H^1(Y;ad_\rho) \to H^1(\Sigma;ad_\rho)\to H^2(Y, \Sigma;ad_\rho) $$ and the two arrows are dual by PD so have the same rank. The exactness tells you the sum of these ranks is $dim(H^1(\Sigma;ad_\rho))$.

Here $\rho$ is a representation of $\pi_1(Y)\to G$ corresponding to the given flat connection and $ad_\rho$ is the local system with fiber $\mathfrak{g}$ corresponding to the adjoint action of $\rho$ on $\mathfrak{g}$.

  • 2
    $\begingroup$ It might be helpful to comment here that since $G$ is assumed to be semisimple, the adjoint representation is self-dual, and hence Poincaré duality applies. $\endgroup$
    – Ian Agol
    Commented May 16, 2016 at 15:37

For the case of compact $G$, see Proposition 3.27 in: Freed, Daniel S. Classical Chern-Simons theory. I. Adv. Math. 113 (1995), no. 2, 237–303.

For the case of complex reductive $G$, see Theorem 61 in: Sikora, Adam S. Character varieties. Trans. Amer. Math. Soc. 364 (2012), no. 10, 5173–5208.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.