People usually say that the quantum cohomology of $ \mathbb{P}^1$ Frobenius manifold $ QH^*(\mathbb{P}^1)$ , corresponds to dispersionless extended Toda hierarchy (e.g. page 6 of https://arxiv.org/pdf/math/0308152.pdf). I’m trying to understand this correspondence in more detail but I’m some having troubles. More specifically how do I recover the $ \tau$ -function? Should I be able to get the Frobenius potential $ F$ of $ QH^*(\mathbb{P}^1)$ from the $ \tau$ -function by \begin{equation} F = \frac{1}{2}(t^1)^2t^2 + e^{t^2} = \lim_{\epsilon \rightarrow 0}\epsilon^2\log \tau |_{t^{\alpha,p > 0} = 0, t^\alpha := t^{\alpha,0}} ? \end{equation} (Where $ \epsilon \rightarrow 0$ is the dispersionless limit. I’m sort of getting this from Theorem 6.1 of https://arxiv.org/pdf/hep-th/9407018.pdf, but the result is for KdV). If I didn’t set $ t^{\alpha,p > 0} = 0$ , should I get the Gromov-Witten potential of $ \mathbb{P}^1$ with descendences from $ \lim_{\epsilon \rightarrow 0}\epsilon^2\log \tau$ ?

Here is what I have gathered so far (mainly from https://arxiv.org/pdf/nlin/0306060.pdf), please correct me if I misunderstood something. The extended Toda hierarchy is given by the bihamiltonian structure on the loop space $ \mathcal{L}(\mathbb{R}^2)$ : \begin{align*} \{u(x),u(y)\}_1 = \{v(x), v(y)\}_1 &= 0, \ \{u(x), v(y)\}_1 &= \frac{1}{\epsilon}(e^{\epsilon \partial_x} – 1)\delta(x – y),\ \{u(x), u(y)\}_2 &= \frac{1}{\epsilon}(e^{\epsilon \partial_x}e^{v(x)} – e^{v(x)}e^{-\epsilon\partial_x})\delta(x-y)\ \{u(x),v(y)\}_2 &= \frac{1}{\epsilon}u(x)(e^{\epsilon\partial_x} – 1)\delta(x-y)\ \{v(x),v(y)\}_2 &= \frac{1}{\epsilon}(e^{\epsilon\partial_x} – e^{-\epsilon\partial_x})\delta(x-y). \end{align*} From this, I’m guessing some sort of Miura transformation can be used to put $ \{.,.\}_2 – \lambda\{.,.\}_1$ into the normal form and from that we get a set of Hamiltonian densitites $ \{h_{\alpha,p} = h_{\alpha,p}(x,u,v,u_x,v_x,…)\}$ . The Hamiltonians are given by $ H_{\alpha, p} := \int_0^{2\pi}h_{\alpha,p}dx$ . The $ t^{\alpha,p}$ -evolution is \begin{equation} \frac{\partial w^\alpha}{\partial t^{\beta,p}} = \{w^\alpha,H_{\beta,p}\}_1, \qquad w = u,v;\ \alpha,\beta = 1,2;\ p \geq 0. \end{equation} All of these came from Theorem 3.1 of https://arxiv.org/pdf/nlin/0306060.pdf. From here we can use the standard result of biharmiltonian theory that $ \partial h_{\alpha,p-1}/\partial t^{\beta,q} = \partial h_{\beta,q-1}/\partial t^{\alpha,p} =: \frac{1}{\epsilon}(e^{\epsilon\partial_x} – 1)\Omega_{\alpha,p;\beta,q}$ , therefore there is a function $ \log \tau$ such that \begin{equation} \Omega_{\alpha,p;\beta,q} = \epsilon^2\frac{\partial^2\log \tau}{\partial t^{\alpha,p}\partial t^{\beta,q}}. \end{equation}

Ideally, I would like to calculate $ \log\tau$ from here and directly and answer my question above. However, the explicit form of $ h_{\alpha,p}$ as given in Theorem 3.1 is complicated and I’m not sure how to proceed. Also, I have never seen anyone written down the equation $ F = \lim_{\epsilon \rightarrow 0}\epsilon^2\log \tau |_{t^{\alpha,p > 0} = 0, t^\alpha := t^{\alpha,0}}$ explicitely anywhere, so maybe it could be wrong. Could anyone give me some advices or correct my understand or provide me with a suitable reference?