# Quantum Hamiltonian reduction and Quantum Airy structure

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 $$$$\psi_{\hat{\mathcal{B}}}(q') := \int \psi_{G}(q,q')\psi_L(q)dq$$$$ where $$\psi_L(q)$$ is the wave function quantizing the quadratic Lagrangian $$L$$ as studied in Section 2.4, 2.5.

My Attempts

1. 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.

2. 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.