I have been reading Kontsevich and Soibelman's "Airy structures and symplectic geometry of topological recursion" (https://arxiv.org/pdf/1701.09137.pdf) and having trouble understanding their Section 7.2 on Quantum Hamiltonian reduction. In particular, I'd like to understand how to compute $\psi_{\hat{\mathcal{B}}}$.
The following is what I think the paper says: Let $(W, \omega)$ be a symplectic vector space, $G\subset W$ a coisotropic subspace ($G^\perp \subset G$) and $L \subset W$ a Lagrangian submanifold. Given a point $x \in L \cap G$ such that $T_xL + T_xG = T_xW$ then $\mathcal{H} := G/G^\perp$ is a symplectic vector space and $\mathcal{B}_x := L_x \cap G \hookrightarrow \mathcal{H}$ is embedded as a Lagrangian submanifold (I interpreted the germ $L_x$ as 'small neighbourhood of $L$ around $x$', which is probably wrong?). Then $G$ is naturally embedded into $\mathcal{H}\times \bar{W}$, where $(\bar{W},\omega) = (W,-\omega)$ as a Lagrangian subspace. Let the coordinates of $W$ be $(q,p)$ and the coordinates of $\mathcal{H}$ be $(q',p')$. From general theory (Section 2.4?) we have the wave function $\psi_{G}(q,q') = \exp(Q_2(q,q')/\hbar)$ quantizing $G$ where $Q_2$ is a quadratic polynomial. Then the Hamiltonian reduction of $L\subset M$ to $\hat{\mathcal{B}}_x \subset \mathcal{H}$ at the level of wave functions becomes \begin{equation} \psi_{\hat{\mathcal{B}}}(q') := \int \psi_{G}(q,q')\psi_L(q)dq \end{equation} where $\psi_L(q)$ is the wave function quantizing the quadratic Lagrangian $L$ as studied in Section 2.4, 2.5.
My Attempts
The only natural way I can think of to embed $G \hookrightarrow \mathcal{H}\times \bar{W}$ is by writing $G = T_x\mathcal{B}\oplus V$, where $V$ is a Lagrangian complement to $T_xL$, then embed $G$ via $T_x\mathcal{B}\hookrightarrow \mathcal{H}, V \hookrightarrow \bar{W}$.
However, this would mean I can write $Q_2(q,q') = Q_{T_x\mathcal{B}}(q') + Q_W(q)$. Evaluating the integral we would get $\psi_{\hat{\mathcal{B}}} = \text{constant}\times \exp(Q_{T_x\mathcal{B}}(q'))$. So $\psi_{\hat{\mathcal{B}}}$ only going to quantize $T_x\mathcal{B}$ in $\mathcal{H}$ for a choice of $x$ instead of the entire Lagrangian submanifold $\mathcal{B}\subset \mathcal{H}$ as I would expect the result of this section to be about.Perhape the embedding $G \hookrightarrow \mathcal{H}\times \bar{W}$ meant to be such that the image in $\mathcal{H}$ is actually $\mathcal{B}_x$ (and the image in $\bar{W}$ is $V$). If that is the case then $G$ is embedded as Lagrangian submanifold not subspace (as stated in the paper). But then I'm still going to have $Q_2(q,q') = Q_{\mathcal{B}}(q') + Q_W(q)$ where $Q_{\mathcal{B}}(q')$ is no longer just quadratic in $q'$ and probably can be found using Section 2.4. But then I'm still going to have $\psi_{\hat{\mathcal{B}}} = \text{constant}\times \exp(Q_{\mathcal{B}}(q'))$ which make me wonder why don't I just directly quantizing $\mathcal{B}_x \subset \mathcal{H}$ since the start instead of looking at $G\hookrightarrow \mathcal{H}\times \bar{W}$ and do a quantum Hamiltonian reduction. Quantizing $\mathcal{B}_x \subset \mathcal{H}$ directly seems difficult and I thought Hamiltonian reduction will help me with it.
Obviously, I have missed a lot of important things. If someone could help me understanding this section better or guid me to good references for quantum Hamiltonian reduction I would be really appreciated. Thank you.
