The KP decomposition conjecture for tropical amplitudes

Let $\mathcal{M}_{g,n}$ be the moduli space of genus-$g$ curves with $n$ marked points, and let $\mathcal{D}_G$ be KP cells associated with tropical graphs $\Gamma=(G,\ell,w)$ of combinatorial type $G$. Let $\mathcal{D}_0$ be an outer domain, let $\mathcal{F}_{g,n}$ be the string-amplitude integrand, and let

$$\mathcal{M}_{g,n}=\left(\bigsqcup_{i=1}^N\mathcal{D}_G\right)\sqcup\mathcal{D}_0.$$

KP decomposition conjecture. There exists such a KP decomposition for which, as $\alpha'\to0$, the following hold: (i) integrating over the outer domain gives only a subleading contribution,

$$\int_{\mathcal{D}_0}\mathrm d\mu_{\mathrm{bos}}\,\mathcal{F}_{g,n}=O(\alpha');$$

and (ii) for every KP cell $\mathcal{D}_G$, there is a function $F_{g,n}$ on the tropical moduli space $\mathcal{M}^{\mathrm{trop}}(\Gamma)$ such that

$$\int_{\mathcal{D}_G}\mathrm d\mu_{\mathrm{bos}}\,\mathcal{F}_{g,n}=\int_{\mathcal{M}^{\mathrm{trop}}(\Gamma)}\mathrm d\mu_{\mathrm{trop}}\,F_{g,n}+O(\alpha'),$$

where

$$\mathrm d\mu_{\mathrm{trop}}:=\frac{\prod_{i\in E(G)}\mathrm d\ell(i)}{(\det K)^5},$$

and $K$ is the period matrix of $\Gamma$.

This conjecture formalizes the localization of string amplitudes onto KP cells in the tropical limit, separating subleading massive-state contributions from the leading tropical amplitude. The source gives no evidence of a resolution, so the assertion remains open.