The chamber formula conjecture for the classical Gromov–Witten potentials

Let $u,p,q$ be the chamber variables, let $\zeta$ be the variable associated with the Cartan directions of the Lie algebra $\mathfrak{g}$, and let $m$ and $s$ be as in the source. Write $(\,\ )$ for the invariant bilinear form, let $c_0(0,\gamma)$ be the coefficients of the associated Jacobi form, and define

$$\log\zeta=2\pi\sqrt{-1}\,z,$$

and

$$\mathcal{I}_2=\sum_{\gamma>0}c_0(0,\gamma)(\gamma,\gamma).$$

The chamber formula conjecture. In the fundamental chamber,

$$\begin{aligned} F_0^{(0)}={}&\log u\left\{\log p\log q-\frac{m}{2}(\log\zeta,\log\zeta)\right\}\\ &+\left(\frac{m}{2}-\frac{\mathcal{I}_2}{24s}\right)\log p\,(\log\zeta,\log\zeta)\\ &+\frac{1}{3}(\log q)^3-\frac{\mathcal{I}_2}{24s}\log q\,(\log\zeta,\log\zeta)\\ &+\frac{1}{12}\sum_{\gamma>0}c_0(0,\gamma)(\gamma,\log\zeta)^3, \end{aligned}$$

and

$$F_1^{(0)}=-\frac{1}{24}(24\log u+24\log p+44\log q)+\frac{1}{24}\sum_{\gamma>0}c_0(0,\gamma)(\gamma,\log\zeta).$$

These formulas conjecturally express the classical genus-zero and genus-one potentials in terms of the Jacobi-form data. The source attributes the formulas to earlier work and explains that they are compatible with the expected exchange symmetry of the potentials, with corroboration in concrete examples for $g=0,1$; a general proof is not supplied.