Exponential formula for reconstructing the wave function by topological recursion

Let $\boldsymbol{x}$ be the spectral-curve projection, let $z^i(x)$ denote the sheets above $x$, and let $\rho$ be the determinantal solution of the differential system. Write $\rho$ in a local basis with blocks of sizes $d_k$, and let $\boldsymbol{\beta}=\boldsymbol{\beta}_k$ be the corresponding points with local coordinates $\beta_k$. Let $u_{g,n}$ be the topological-recursion correlation differentials, let $u_{0,1}$ be the leading differential, and define the prime form on the Riemann sphere by

$$E(z,z')=\frac{z-z'}{\sqrt{dz\,dz'}}.$$

For each $k$, define

$$V_k(z)=\operatorname{Res}_{z'\to\beta_k}\nu_{0,1}(z')\ln\left(1-\frac{\boldsymbol{x}(z)^{1/d_k}}{\boldsymbol{x}(z')^{1/d_k}}\right),\qquad t_k=\operatorname{Res}_{\beta_k}\nu_{0,1},$$

and

$$\Phi_k(z)=\int_{\beta_k}^{z}\left(\nu_{0,1}-dV_k+\frac{t_k}{d_k}\frac{d\boldsymbol{x}}{\boldsymbol{x}}\right)+V_k(z)-\frac{t_k}{d_k}\ln\boldsymbol{x}(z).$$

Exponential formula. The solution should have the WKB expansion

$$\left(\frac{\rho(x')^{-1}\rho(x)}{x-x'}\sqrt{dx\,dx'}\right)_{j,i}=\frac{\exp\left(\hbar^{-1}\int_{z^j(x')}^{z^i(x)}\nu_{0,1}\right)}{E(z^i(x),z^j(x'))}\exp\left(\sum_{2g-2+n>0}\frac{\hbar^{2g-2+n}}{n!}\int_{z^j(x')}^{z^i(x)}\cdots\int_{z^j(x')}^{z^i(x)}\nu_{g,n}\right).$$

Consequently,

$$\rho(x)_{k,1;i}=\frac{\exp\left(\hbar^{-1}\Phi_k(z^i(x))\right)}{z^i(x)-\beta_k}\exp\left(\sum_{2g-2+n>0}\frac{\hbar^{2g-2+n}}{n!}\int_{\beta_k}^{z^i(x)}\cdots\int_{\beta_k}^{z^i(x)}\nu_{g,n}\right),$$

and, for $1\leq j\leq d_k$,

$$\left(v^{-1}\rho(x)\right)_{k,j;i}=\left.\frac{d^{j-1}}{dz'^{j-1}}\left(\frac{\exp\left(\hbar^{-1}\Phi_k(z^i(x))\right)}{z^i(x)-z'}\exp\left(\sum_{2g-2+n>0}\frac{\hbar^{2g-2+n}}{n!}\int_{z'}^{z^i(x)}\cdots\int_{z'}^{z^i(x)}\nu_{g,n}\right)\right)\right|_{z'=\beta_k}.$$

This conjecture proposes a reconstruction of the solution of the differential system from the topological-recursion correlation functions, extending the established direction from the solution to topological recursion. The source gives no resolution status for the reconstruction formulas, so their general validity remains open.