ADE Minimal Strings and Multi-Matrix Duals

TL;DR

This work analyzes ADE minimal string theories, emphasizing the D- and E-series coupled to Liouville theory and the conjectured $4$-matrix duals that extend beyond the solvable two-matrix models of the A-series. It computes sphere four-point and torus one-point amplitudes directly via moduli-space integration across AMS, DMS, and EMS, providing new data and testing known matrix-model predictions; where predictions are unavailable, new results emerge. Evidence for a multi-matrix dual in the D-series appears in amplitudes with conformal boundaries, including non-universal cylinder ramps and ZZ-instanton deviations from the two-matrix theory, indicating richer non-perturbative structure. A preliminary positivity bootstrap is used to constrain potential critical points of multi-matrix models relevant to the DMS string, suggesting approaches to locate a continuum limit beyond two matrices. Overall, the paper maps out the perturbative and boundary observables of ADE minimal strings and advocates a multi-matrix framework with new analytic and numerical techniques that could illuminate nontrivial dualities in low-dimensional quantum gravity.

Abstract

We revisit ADE minimal string theories, focusing on the D- and E-series minimal models coupled to Liouville theory. Unlike the A-series, whose duals are solvable two-matrix models, these theories are conjectured to correspond to unsolvable four-matrix integrals. We compute sphere four-point and torus one-point amplitudes in the AMS, DMS, and EMS via direct numerical integration over moduli space, confirming/disproving some known results and providing new data where matrix-model predictions are unavailable. From amplitudes with conformal boundaries, we find evidence for multi-matrix structure in the D-series, including scaled ramp behavior in cylinder diagrams and deviations from the ZZ-instanton sector of two-matrix models. We also perform a preliminary positivity bootstrap to constrain critical points of the multi-matrix models relevant to the DMS string.

Figures (4)

• Figure 1: Integration contour (red) for the Liouville torus one-point function. As we analytically continue the external momenta to be purely imaginary poles in the integrand can cross the contour and their residues must be included. In the figure we have drawn a case where no poles cross.
• Figure 2: Dynkin diagrams that are used to label the Cardy boundary states of the $(p,p')$ D-series minimal model with even $p$. The Cardy states is labelled by $|\alpha,\beta\rangle$ with $\alpha$ picked out of ant node of the $D_{\frac{p}{2}+1}$ diagram, and $\beta$ from any of the odd-nodes of the $A_{p'-1}$ diagram. The "non-invariant" boundary states come from picking either of two off-shoot nodes from the first diagram. For the $(p,p')$ A-series we would have $(A_{p-1},A_{p'-1})$ and boundary states would be labelled by any node of the first and odd nodes of the second diagram.
• Figure 3: Left: The allowed shaded region of the coupling space $(g,h)$ for the 3-matrix model \ref{['eq:3mm boot']} at large-$N$ with positivity matrices composed of words up to orders 6, 8, and 10. The three red dots mark solvable critical points, which are expected to lie on the boundary of the true island (infinite-order limit). Right: The large-$N$ allowed region for $\langle \mathop{\mathrm{tr}}\nolimits M_1^2\rangle$, $\langle \mathop{\mathrm{tr}}\nolimits M_1M_2\rangle$ at orders 6, 8, and 10, for the case $g=-0.07$, $h=0.1$.
• Figure 4: Left: The allowed shaded region of the coupling space $(g,h)$ for the 4-matrix model \ref{['eq:4mm boot']} at large-$N$ with positivity matrices composed of words up to orders 6 and 8. The three red dots mark solvable critical points, which are expected to lie on the boundary of the true island (infinite-order limit). Right: The large-$N$ allowed region for $\langle \mathop{\mathrm{tr}}\nolimits M_1^2\rangle$, $\langle \mathop{\mathrm{tr}}\nolimits M_1M_2\rangle$ at orders 6 and 8, for the case $g=-0.07$, $h=0.1$.