# Geometry of ag.algebraic – Quantum Hamiltonian reduction and Quantum Airy structure

I have been reading "Aerated Structures and Symplectic Geometry of the Topological Recursion of Kontsevich and Soibelman" (https://arxiv.org/pdf/1701.09137.pdf) and I am having trouble understanding Section 7.2 about the reduction of the Quantum Hamiltonian. In particular, I would like to understand how to calculate $$psi _ { hat { mathcal {B}}}$$.

The following is what I think the document says: $$(W, omega)$$ be a symplectic vector space, $$G subset W$$ a co-isotropic subspace ($$G ^ perp subset G$$) Y $$L subset W$$ a Lagrangian submanifold. Given a point $$x in L cap G$$ such that $$T_xL + T_xG = T_xW$$ so $$mathcal {H}: = G / G ^ perp$$ It 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 $$L_x$$ like & # 39; small neighborhood of $$L$$ around $$x$$& # 39 ;, which is probably wrong?). So $$G$$ is naturally embedded in $$mathcal {H} times bar {W}$$, 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 $$(q & # 39 ;, p & # 39;)$$. From the general theory (Section 2.4?) We have the wave function $$psi_ {G} (q, q & # 39;) = exp (Q_2 (q, q & # 39;) / hbar)$$ quantization $$G$$ where $$Q_2$$ It 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 it becomes
$$begin {equation} psi _ { hat { mathcal {B}}} (q & # 39;): = int psi_ {G} (q, q & # 39;) psi_L (q) end {equation}$$
where $$psi_L (q)$$ It is the wave function that quantifies 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. $$G hookrightarrow mathcal {H} times bar {W}$$ it is written $$G = T_x mathcal {B} oplus V$$, where $$V$$ is a Lagrangian complement for $$T_xL$$, then embed $$G$$ via $$T_x mathcal {B} hookrightarrow mathcal {H}, V hookrightarrow bar {W}$$.
However, this would mean that I can write $$Q_2 (q, q & # 39;) = Q_ {T_x mathcal {B}} (q & # 39;) + Q_W (q)$$. Evaluating the integral that we would obtain. $$psi _ { hat { mathcal {B}}} = text {constant} times exp (Q_ {T_x mathcal {B}} (q & # 39;))$$. So $$psi _ { hat { mathcal {B}}}$$ I'm just going to quantize $$T_x mathcal {B}$$ in $$mathcal {H}$$ for a choice of $$x$$ instead of all the Lagrangian submanifold $$mathcal {B} subset mathcal {H}$$ How would you expect the result of this section?

2. Perhape inlay $$G hookrightarrow mathcal {H} times bar {W}$$ intended to be such that the image in $$mathcal {H}$$ it's really $$mathcal {B} _x$$ (and the image in $$bar {W}$$ is $$V$$). If that is the case then $$G$$ is embedded as Lagrangian submanifold do not subspace (as indicated in the document). But then I'm still going to have $$Q_2 (q, q & # 39;) = Q _ { mathcal {B}} (q & # 39;) + Q_W (q)$$ where $$Q _ { mathcal {B}} (q & # 39;)$$ It is no longer just quadratic in $$q & # 39;$$ and can probably be found using Section 2.4. But then I'm still going to have $$psi _ { hat { mathcal {B}}} = text {constant} times exp (Q _ { mathcal {B}} (q & # 39;))$$ what makes me wonder why I do not simply quantify directly $$mathcal {B} _x subset mathcal {H}$$ From the beginning instead of looking $$G hookrightarrow mathcal {H} times bar {W}$$ and make a quantum Hamiltonian reduction. Quantization $$mathcal {B} _x subset mathcal {H}$$ It seems directly difficult and I thought that the Hamiltonian reduction would help me with that.

Obviously, I have missed many important things. If someone helped me to better understand this section or guide me towards good references for quantum Hamiltonian reduction, I would really appreciate it. Thank you.