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.