# Lagrangian subgroup of a nonabelian Lie group

My post here concerns the concept of Lagrangian subgroup for a non-abelian Lie group, such as a semi-simple non-abelian Lie group for gauge theory.

See a previous post for other background information: Lagrangian subgroups/submanifolds, 2d topological boundary and 3d "non-abelian" Chern–Simons theory

First, the symplectic form $$\omega$$ is given by (with the restricted $$a_{\parallel,I}$$ on $${\partial \mathcal M}$$ ), $$\omega=\frac{K_{IJ}}{4\pi} \int_{ \mathcal M} (\delta a_{\parallel,I}) \wedge d (\delta a_{\parallel,J}).$$ for a varyation of the differential of this 1-form on an abelian Chern-Simons action $$\delta S_{bluk}$$.

For the well-known abelian gauge group such as the bulk gauge group $$\text{U(1)}^N \cong \mathbb{T}_\Lambda$$ as the torus, is the quotient space of $$N$$-dimensional vector space $$\bf{V}$$ by a subgroup $$\Lambda \cong \mathbb Z^N$$. Locally the gauge field $$a$$ is a 1-form, which has values in the Lie algebra of $$\mathbb{T}_\Lambda$$, we can denote this Lie algebra $$\mathbf{t}_\Lambda$$ as the vector space $$\mathbf{t}_\Lambda =\Lambda \otimes \mathbb{R}$$.

For topological gapped boundary, $$a_{\parallel,I} \quad$$ lies in a Lagrangian subspace of $$\mathbf{t}_\Lambda$$ implies that the boundary gauge group ($$\equiv \mathbb{T}_{\Lambda_{0}}$$) is a Lagrangian subgroup.

We can rephrase it in terms of the exact sequence for the vector space of Abelian group $$\Lambda \cong \mathbb Z^N$$ and its subgroup $$\Lambda_0$$:

These form an exact sequence: $$0 \to \Lambda_0 \overset{\mathbf{h}}{\to} \Lambda \to \Lambda/\Lambda_0 \to 0.$$ Here $$0$$ means the trivial Abelian group with only the identity, or the zero-dimensional vector space. Here $$0$$ means the trivial zero-dimensional vector space and $$\mathbf{h}$$ is an injective map from $$\Lambda_0$$ to $$\Lambda$$.

We can also rephrase it in terms of the exact sequence for the vector space of Lie algebra by

$$0 \to \mathbf{t}_{(\Lambda/\Lambda_0)}^* \to \mathbf{t}_{\Lambda}^* \to \mathbf{t}_{\Lambda_0}^* \to 0.$$

My Question: For topological gapped boundary, do we have a direct generalization such that the $$a_{\parallel,I}$$ lies in a Lagrangian subspace of $$\mathbf{t}_\Lambda$$ implies that the boundary gauge group is still a Lagrangian subgroup? How do we define the Lagrangian subgroup of a nonabelian Lie group?

• References along this direction are very welcome.