Let \(\overline{\mathcal{M}}_{g,n}\) be the moduli space of stable algebraic curves of genus \(g\) with \(n\) distinct marked points, where \(g\) and \(n\) are non-negative integers satisfying the stability condition \(2g-2+n>0\). For \(1\le k\le n\) and \(0\le j\le g\), denote by \(\psi_k\) the first Chern class of the \(k\)-th tautological line bundle \(\mathbb{L}_k\) on \(\overline{\mathcal{M}}_{g,n}\), and by \(\lambda_j\) the \(j\)-th Chern class of the Hodge bundle \(\mathbb{E}_{g,n}\) on \(\overline{\mathcal{M}}_{g,n}\). The rational numbers defined by the formula \[ \int_{\overline{\mathcal{M}}_{g,n}} \psi_1^{i_1}\cdots\psi_n^{i_n}\lambda_{j_1}\cdots\lambda_{j_m} \] are called the Hodge integrals. These numbers take zero value unless the degree-dimension counting matches, i.e. \[ i_1+\dots +i_n+ j_1+\dots +j_m = 3g-3+n. \tag{1} \] Denote by \(\mathcal{C}_g(z):= \sum_{j=0}^g \lambda_j z^j\) the Chern polynomial of \(\mathbb{E}_{g,n}\). We will be particularly interested in the following class of Hodge integrals defined via the cubic products of Chern polynomials, called the cubic Hodge integrals: \[ \int_{\overline{\mathcal{M}}_{g,n}} \psi_1^{i_1}\cdots\psi_n^{i_n} \mathcal{C}_g(-p)\mathcal{C}_g(-q)\mathcal{C}_g(-r),\tag{2} \] where \(p,q,r\) are complex parameters. These integrals are called special if \(p,q,r\) satisfy the following local Calabi-Yau condition: \[ pq+qr+rp=0.\tag{3} \] Let \(\mathcal{H}=\mathcal{H}(\mathbf{t}; p,q,r; \epsilon)\) be the cubic Hodge free energy defined by \begin{align*} \mathcal{H}(\mathbf{t}; p,q,r; \epsilon) &:= \sum_{g\geq0}\epsilon^{2g-2}\mathcal{H}_g(\mathbf{t}; p,q,r), \\ \mathcal{H}_g(\mathbf{t}; p,q,r) &:= \sum_{n\geq 0} \sum_{i_1,\dots,i_n\geq 0} \frac{t_{i_1}\cdots t_{i_n}}{n!} \int_{\overline{\mathcal{M}}_{g,n}} \psi_1^{i_1}\cdots\psi_n^{i_n} \mathcal{C}_g(-p)\mathcal{C}_g(-q)\mathcal{C}_g(-r) . \end{align*} Here \(\mathcal{H}_g(\mathbf{t}; p,q,r)\) is called the genus \(g\) part of the free energy \(\mathcal{H}\). Then the exponential \[ e^{\mathcal{H}(\mathbf{t};p,q,r;\epsilon)} =:Z_{\text{cubic}}(\mathbf{t};p,q,r;\epsilon)=:Z_{\text{cubic}} \] is called the cubic Hodge partition function. Clearly, \(\mathcal{H}_g(\mathbf{t}; p,q,r)\in \mathbb{C}[p,q,r][[\mathbf{t}]]\). The genus zero free energy \(\mathcal{H}_0(\mathbf{t})\) is actually independent of \(p,q,r\) and has the explicit expression \[ \mathcal{H}_0(\mathbf{t}) = \sum_{n\geq 3} \frac{1}{n(n-1)(n-2)} \sum_{i_1+\dots+i_n=n-3} \frac{t_{i_1}}{i_1!} \cdots \frac{t_{i_n}}{i_n!} . \] Define \[ v(\mathbf{t}):=\partial_{t_0}^2 \mathcal{H}_0(\mathbf{t})=\sum_{n\geq 1} \frac1n \sum_{i_1+\dots+i_n=n-1} \frac{t_{i_1}}{i_1!} \cdots \frac{t_{i_n}}{i_n!}.\tag{4} \] It satisfies the following Riemann hierarchy: \[ \frac{\partial v}{\partial t_i} = \frac{v^i}{i!} \frac{\partial v}{\partial t_0} , \quad i\geq 0.\tag{5} \] More generally, if one defines \(w=\epsilon^2 \partial_{t_0}^2 \mathcal{H}(\mathbf{t};p,q,r;\epsilon)\), then \(w\) satisfies an integrable hierarchy of Hamiltonian evolutionary PDEs [A. Buryak et al., J. Differ. Geom. 92, No. 1, 153–185 (2012; Zbl 1259.53079); J. Geom. Phys. 62, No. 7, 1639–1651 (2012; Zbl 1242.53113); B. Dubrovin et al., Adv. Math. 293, 382–435 (2016; Zbl 1335.53114)], called the Hodge hierarchy for the special cubic Hodge integrals, which is a deformation of the Riemann hierarchy. The Hodge-FVH correspondence is given by the following conjecture [S.-Q. Liu et al., Lett. Math. Phys. 108, No. 2, 261–283 (2018; Zbl 1410.37061)].
Conjecture. The Hodge hierarchy for the special cubic Hodge integrals is equivalent, under a certain Miura type transformation, to the fractional Volterra hierarchy (FVH). Furthermore, the special cubic Hodge partition function gives a tau function of the FVH.
From now on, we assume that \(p,q,r\) satisfy the local Calabi-Yau condition (3). The case with \(p,q,r\in \mathbb{Q}\) is called rational. Let us first consider the rational case. Due to the symmetry property of the cubic Hodge integrals with respect to \(p,q,r\), and the homogeneity property (deduced from (1)) \[ \mathcal{H}_g( \mathbf{t}; \lambda p, \lambda q, \lambda r)|_{t_i\mapsto t_i \lambda^{i-1}} = \lambda^{3g-3} \mathcal{H}_g( \mathbf{t}; p, q, r), \] we can assume that \[ p=\frac{1}{K_1},\quad q=\frac{1}{K_2}, \quad r=-\frac{1}{h},\tag{6} \] where \(K_1, K_2 \in \mathbb{N}\), \((K_1,K_2)=1\) and \(h:=K_1+K_2\). We denote, for \(\ell\geq 0\), \begin{align*} b_{\alpha+h\ell} &:= \frac{\alpha}{K_1}+\ell, \quad c_{\alpha+h\ell} := \binom{b_{\alpha+h\ell} h}{b_{\alpha+h\ell} K_1} , \qquad \alpha=0,\dots,K_1-1,\tag{7} \\ b_{\alpha+h\ell} &:= \frac{-\alpha}{K_2}+\ell, \quad c_{\alpha+h\ell} := \binom{b_{\alpha+h\ell} h}{b_{\alpha+h\ell} K_2}, \qquad \alpha=-(K_2-1),\dots,-1,\tag{8} \end{align*} and \[ \mathbb{N}_*=(\mathbb{N}-K_2) \backslash \left(\{0\} \cup (h\mathbb{N}-K_2)\right), \quad \text{where } a\mathbb{N}-K_2:=\{ak-K_2|k\in \mathbb{N}\}. \] Define \[ Z(x,\mathbf{s};\epsilon):=\exp\left(\frac{A(x,\tilde{\mathbf{s}})}{\epsilon^2}\right) Z_{\text{cubic}}\left(\mathbf{t}(x,\mathbf{s}); \frac{1}{K_1}, \frac{1}{K_2}, -\frac{1}{h};\epsilon \right), \tag{9} \] where \(\mathbf{s}:=(s_k)_{k\in \mathbb{N}_*}\) is an infinite vector of indeterminates, \(\tilde{s}_k=s_k-c_h^{-1}\delta_{k,h}\) (\(k\in \mathbb{N}_*\)), \[ t_i=t_i(x,\mathbf{s})=\sum_{k\in \mathbb{N}_*} b_k^{i+1} c_k \tilde s_k+\delta_{i,1}+x \delta_{i,0},\quad i\geq 0,\tag{10} \] and \(A\) is the quadratic series \[ A:=A(x,\mathbf{s})=\frac{1}{2}\sum_{k_1,k_2\in \mathbb{N}_*} \frac{b_{k_1} b_{k_2}}{b_{k_1} + b_{k_2}} c_{k_1} c_{k_2} s_{k_1} s_{k_2} + x \sum_{k\in \mathbb{N}_*} c_k s_k. \] Note that for \(g\geq0\), \(\mathcal{H}_g(\mathbf{t}(x,\mathbf{s});\frac{1}{K_1}, \frac{1}{K_2}, -\frac{1}{h})\) is a well-defined formal power series in \(\mathbb{C}[[x-1]][[\mathbf{s}]]\).
Denote \(I=\{-(K_2-1),\dots,K_1-1\}\) and \(I_*=I\backslash \{0\}\), and define a family of linear operators \(L_m=L_m\left(\epsilon^{-1}x,\epsilon^{-1}\mathbf{s},\epsilon \partial/\partial \mathbf{s}\right)\), \(m\geq 0\) by \begin{align*} L_0=&\sum_{k\in \mathbb{N}_*} b_k s_k \frac{\partial}{\partial s_k}+\frac{x^2}{2\epsilon^2}+\frac{1}{24}\left(\frac1h-\frac1{K_1}-\frac1{K_2}\right), \tag{11} \\ L_m=&\sum_{k\in \mathbb{N}_*} b_k s_k \frac{\partial}{\partial s_{k+h m}}+x\frac{\partial}{\partial s_{hm}} \\ & +\frac{\epsilon^2}{2} \sum_{\ell=1}^{m-1}\frac{\partial^2}{\partial s_{h \ell} \partial s_{h(m-\ell)}} + \frac{\epsilon^2}{2} \sum_{\alpha,\beta \in I_*} \sum_{\ell=0}^{m-1} G^{\alpha\beta} \frac{\partial^2}{\partial s_{\alpha+h\ell} \partial s_{\beta+h(m-1-\ell)}}, \tag{12} \end{align*} where \(\left(G^{\alpha\beta}\right)_{\alpha,\beta\in I}\) is a symmetric nondegenerate constant matrix defined by \[ G^{\alpha\beta}= \begin{cases} \frac{K_1}{h}\delta^{\alpha+\beta,-K_2}, &\alpha,\beta<0;\\ 1, &\alpha=\beta=0;\\ \frac{K_2}{h}\delta^{\alpha+\beta, K_1}, &\alpha,\beta>0;\\ 0 &\text{elsewhere}. \end{cases}\tag{13} \] It is easy to check that the operators \(L_m\) satisfy the following Virasoro commutation relations: \[ \left[ L_m,L_n \right]= \left(m-n\right) L_{m+n}, \quad \forall\, m,n\geq 0. \] Theorem 1. For the rational numbers \(p,q,r\) given by (6), the series \(Z(x,\mathbf{s};\epsilon)\) defined by (9) satisfies the following Virasoro constraints: \[ L_m\left(\epsilon^{-1}x,\epsilon^{-1}\tilde{\mathbf{s}},\epsilon \partial/\partial \mathbf{s}\right) Z(x,\mathbf{s};\epsilon)=0, \quad m\geq 0.\tag{14} \]
We proceed to the general case. Denote \[ \sigma_1 = -(p+q+r), \quad \sigma_3 = -2 \left(p^3+q^3+r^3\right).\tag{15} \] From the local Calabi-Yau condition (3) and the fact that the integral in (2) is symmetric in \(p,q,r\), we know that \[ \mathcal{H}_g:=\mathcal{H}_g(\mathbf{t};p,q,r)\in \mathbb{C}[\sigma_1,\sigma_3][[\mathbf{t}]], \quad g\ge 0.\tag{16} \] The following theorem is the main result of the present paper.
Theorem 2. The equation \[ \begin{split} \sum_{i\geq0 }\left( \partial^i \Theta+\sum_{j=1}^i \binom{i}{j} P_{j-1,i-j+1}\right) \frac{\partial \Delta H }{\partial z_i} \\ = \frac{\Theta^2}{16}-\left(\frac{1}{16}-\frac{\sigma_1}{24}\right)\Theta+ \epsilon^2\sum_{i\geq0}\partial^{i+2}\left(\frac{\Theta^2}{16}-\left(\frac{1}{16}-\frac{\sigma_1}{24}\right)\Theta\right) \frac{\partial \Delta H}{\partial z_i}\\ + \frac{\epsilon^2}{2}\sum_{i,j\geq0} P_{i+1,j+1} \left(\frac{\partial^2\Delta H}{\partial z_i \partial z_j} +\frac{\partial \Delta H}{\partial z_i}\frac{\partial \Delta H}{\partial z_j}\right), \end{split} \tag{17} \] which is called the Dubrovin-Zhang loop equation for the special cubic Hodge integrals, has a unique solution of the form \[ \Delta H:=\sum_{g\geq 1} \epsilon^{2g-2} H_g,\quad H_g:=H_g(z_0,\dots,z_{3g-2};\sigma_1,\sigma_3) \] up to the addition of a constant to each \(H_g\), \(g\geq 1\). These constants can be uniquely determined by the following conditions: \[ H_1= \frac1{24} \log z_1 + \frac{\sigma_1}{24} z_0, \quad \sum_{j=1}^{3g-2} j z_j \frac{\partial H_g}{\partial z_j} = (2g-2)H_g, \quad g\geq 2. \tag{18} \] Here \[ \partial:=\sum_{k\geq 0} z_{k+1} \frac{\partial}{\partial z_k},\quad \Theta:= \frac1{1-e^{z_0}/\mu}, \] the coefficients \(P_{i,j}\) are certain polynomials in \(\Theta, \sigma_1,\sigma_3, z_1,z_2,\dots\) whose explicit expressions are given in Section 4, and \(\mu\) is an arbitrary parameter. Moreover, let \(v(\mathbf{t})\) be defined in (4), then the genus \(g\) (\(g\geq 1\)) special cubic Hodge free energy has the expression \[ \mathcal{H}_g =H_g\left(v(\mathbf{t}), \frac{\partial v(\mathbf{t})}{\partial t_0},\cdots,\frac{\partial^{3g-2} v(\mathbf{t})}{\partial t_0^{3g-2}};\sigma_1,\sigma_3\right).\tag{19} \]
One can recursively solve the loop equation to obtain the free energies \(H_g\), \(g\geq 1\).