Resurgence in the Virasoro Minimal String and 3d Gravity

Abstract

We compute non-perturbative, resurgent contributions to the Virasoro minimal string and 3d gravity using techniques from hermitian matrix models. In particular, we construct a fully non-perturbative partition function for the Virasoro minimal string in terms of a Zak transform. In this context, negative tension D-branes appear naturally, which in the matrix model correspond to anti-eigenvalues, or instantons on the involuted sheet of the spectral curve. We further extend this analysis to resolvents and observe resurgent wall crossing phenomena between ZZ- and FZZT-branes. Using recent results that relate the Virasoro minimal string to 3d gravity with end-of-the-world branes we proceed to study the resurgent consequences of summing over the genus in 3d gravity, where we find non-perturbative contributions of doubly exponential type. These statements are then tested using resurgent large-order asymptotics. Lastly, we compute the non-perturbative eigenvalue density for generic hermitian matrix models and identify the change of asymptotic behavior at the edge of the eigenvalue distribution with a Stokes transition. This allows us to identify oscillations in the eigenvalue density with anti-Stokes behavior of FZZT-branes. In the case of 3d gravity with end-of-the-world branes we comment how this Stokes transition coincides with the onset of black hole behaviour and compute the non-perturbative primary density. Furthermore, we apply these techniques to the eigenvalue density of JT gravity to compute higher genus corrections.

Figures (14)

• Figure 1: Pictorial description of the Borel resummation of a divergent series with genus expansion. Start with the left-downward arrow by performing a Borel transform $\mathcal{B}_{\gamma}$, which yields a power-series convergent in a disk around the origin ($\gamma \in\mathbb{R}$ see m12s14abs18 for details). Moreover, because we started with a genus expansion the Borel transform and its analytic continuation will be parity fixed -- the resulting Borel-plane singularities will be symmetrically arranged. The final step with the right-upward arrow is resummation via Laplace transform $\mathcal{L}_{\theta}^{\gamma}$ along the angle $\theta$ (inverse Borel transform); such transformation will turn the Borel plane singularities into exponential contributions where the location of the singularity determines the exponential weight. Therefore, because of the symmetry on the Borel plane that we inherited from the genus expansion, such exponentials appear always in pairs $(A, -A)$ -- a phenomenon often referred to as resonance. Notice how this argument is independent of the choice of parameter $\gamma$. This figure illustrates and builds on a graphic from mss22.
• Figure 2: The spectral curve of the VMS. We show the branch cut (green) starting at $0$ and ending at $\infty$. Furthermore we visualize the two sheets: the physical sheet on top (light blue) and the non-physical sheet at the bottom (light orange). Those two sheets meet at the cut and at the saddle points \ref{['eq:saddlesVMS']} (red). In addition we show the integration cycle (blue) of the supressed instanton action \ref{['eq:action']} associated to the saddle point $E^{\star}_{1, +}$ in the physical sheet. On the other hand, for the same saddle-point $E^{\star}_{1, +}$, we also show the integration cycle (orange) associated to the exponentially growing instanton \ref{['eq:naiveminussignforaction']} on the non-physical sheet. Both contributions are seen by the asymptotics of the free energy (see section \ref{['subsec:largeorderchecks']}). This plot builds on previous graphics from mss22sst23.
• Figure 3: Depiction of the effective potential $V_{\text{eff}}(E) = \int\text{d}E y(E)$ as a function of $E$ at $b=1/2$. Same as the spectral curve \ref{['eq:spectralCurveVMS']} it has two sheets: the physical sheet (light blue) and the non-physical one (light orange). Compare also to figure \ref{['fig:MSspectralcurve']}. The saddles (red) of the potential on the physical sheet are exactly the resurgent instanton actions \ref{['eq:action']} and we can indeed see that they oscillate between positive and negative values. In addition, taking into account the non-physical sheet we find additional saddle points (red) (On the uniformization cover each saddle point splits into two, see for example formula \ref{['eq:naiveminussignforaction']}.), exactly with a relative minus sign in comparison with the physical sheet.
• Figure 4: Plot of the sequence $M_g(b)$ and its Richardson transforms of order $n$$\text{RT}_n$ as a function of $g$ for the values $b=1/3$ (left) and $b=1/2$ (right). We also show the theoretical prediction in formula \ref{['eq:asymptoticscheck']} in gray. Even though we do not have many terms in the asymptotic series we achieve an agreement of 5 digits with the predicted value.
• Figure 5: Plot of the numerical extrapolation $M_{\infty}(b)$ as a function of $b$. We also show the theoretical prediction in formula \ref{['eq:asymptoticscheck']} in blue. The precision is 4 to 5 digits for each point, but seems to drop a little bit close to $b=1$ exactly as expected in cemr23. In fact a very similar plot has already appeared in cemr23: The difference here is that here we got this plot from the corrected asymptotics \ref{['eq:asymptoticscheck']} that take into account that the free energy is even.