The tau-function as conformal block conjecture for W(sl_d)-symmetric Toda theories

Let $\mathcal M_{\mathcal W({\mathfrak{sl}_d})}^{\mathbb P^1}$ be the moduli space of marked points, weights, and times, and let $\mathfrak T_{\bold z,\bold \alpha,\bold t}$ be the tau-function defined by

$$\ln\mathfrak T_{\bold z,\bold \alpha,\bold t}=\sum_{g=0}^{\infty}\varepsilon^{2g-2}\mathcal F_g.$$

Here $\bold z=(z_1,\ldots,z_M)$ and $\boldsymbol\alpha=(\alpha_1,\ldots,\alpha_M)$ specify the insertions $V_{\alpha_j}(z_j)$ in the associated $\mathcal W(\mathfrak{sl}_d)$-symmetric conformal field theory. Tau-function as conformal block conjecture. The tau-function equals the conformal block

$$\mathfrak T_{\bold z,\bold \alpha,\bold t}=\left\langle\prod_{j=1}^{M}V_{\alpha_j}(z_j)\right\rangle.$$

The conjecture identifies the integrable-system tau-function constructed from the special-geometry data with the conformal-field-theory correlation function. The paper presents this as a natural conjecture; no resolution is supplied in the given text.