The complex Liouville string: the matrix integral

TL;DR

This work establishes a controllable holographic duality between the complex Liouville string, built from two Liouville CFTs with central charges $c=13\pm i\lambda$, and a double-scaled two-matrix integral with a non-algebraic spectral curve $\mathsf{x}(z)=-2\cos(\pi b^{-1}\sqrt{z})$, $\mathsf{y}(z)=2\cos(\pi b\sqrt{z})$. The matrix-model side yields a genus expansion computed by topological recursion on the spectral curve, which is then mapped to the worldsheet amplitudes $\mathsf{A}_{g,n}^{(b)}$, with a detailed dictionary linking resolvents to string amplitudes and a sum over stable graphs interpreted as closed-string field-theory vertices. The framework naturally produces a CohFT of infinite rank, whose associated TQFT is $\mathrm{SU}(2)_q$ Yang-Mills, tying the amplitudes to Schur indices of class-$\mathcal{S}$ theories. The paper provides extensive tests (e.g., reproducing bootstrapped sphere4- and torus1 amplitudes, analytic-continuation properties, dilaton equations, and dualities) that strongly support the duality, while non-perturbative aspects and deformations are deferred to subsequent installments. Together, these results illuminate a rich structure connecting matrix-model topological recursion, intersection theory, and holographic descriptions in low dimensions, offering a concrete laboratory for holography beyond traditional minimal-string dualities.

Abstract

We propose a duality between the complex Liouville string and a two-matrix integral. The complex Liouville string is defined by coupling two Liouville theories with complex central charges $c = 13 \pm i λ$ on the worldsheet. The matrix integral is characterized by its spectral curve which allows us to compute the perturbative string amplitudes recursively via topological recursion. This duality constitutes a controllable instance of holographic duality. The leverage on the theory is provided by the rich analytic structure of the string amplitudes that we discussed in arXiv:2409.18759 and allows us to perform numerous tests on the duality.

Figures (6)

• Figure 1: Density of states for the first matrix for $b^2=i$. Close to $E\approx 2$ the density behaves as $\rho_0(E)\approx \sqrt{E-2}$.
• Figure 2: The structure of the spectral curve plotted for $b=\frac{11}{10} \, \mathrm{e}^{\frac{\pi i}{4}}$. The points $z_{(m,n)}^\pm$ are nodal singularities and correspond to the same point on the spectral curve. We denoted this by a dotted line, which can be viewed as a pinched handle of the surface. The red dots correspond to the branch points $z_m^*$ of $\mathsf{x}(z)$. The differently colored regions correspond to the different sheets of $\mathsf{x}(z)$. The yellow and red regions map to $\mathbb{C} \setminus [2,\infty)$ under $\mathsf{x}(z)$, while the green and blue regions map to $\mathbb{C} \setminus (-\infty,-2]$. The yellow region corresponds to the physical sheet. The support of the eigenvalues is the lowest blue parabola which delineates the boundary of the physical sheet. It maps to $[2,\infty)$ under $\mathsf{x}(z)$.
• Figure 3: The three distinct ways of embedding a pair of pants with a distinguished external cuff (labelled by $p_1$ above) into a surface. These correspond to the three classes of terms in (\ref{['eq:Agn simpler residue recursion']}) and (\ref{['eq:Agn recursion relation']}). There is a factor of the sphere three-point amplitude $\mathsf{A}_{0,3}^{(b)}$ corresponding to this pair of pants for each term in the recursion.
• Figure 4: The contour of integration that defines the recursion kernel (\ref{['eq:recursion kernel']}).
• Figure 5: The possible values of the matter central charge for the 2d string theories of table \ref{['tab:minimal string theories']}. The $c\leqslant 1$ region contains a discrete set of points corresponding to the $(2,p)$ minimal string, a dense set of rational points corresponding to the $(p,q)$ minimal string, and a continuum spanned by the Virasoro minimal string.