All the D-Branes of Resurgence

Abstract

It was recently shown how to account for all instantons of hermitian matrix models via (anti-) eigenvalue-tunneling -- including both exponentially-suppressed and exponentially-enhanced transseries-transmonomials which are predicted by resurgence. Matrix-model eigenvalue-tunneling corresponds to ZZ-branes. The present work shows how matrix-model anti-eigenvalues correspond to negative-tension ZZ-branes; and how to compute generic nonperturbative sectors -- with both ZZ and negative-tension-ZZ branes -- in the minimal-string free-energy. Negative-tension D-branes are herein a requirement of resurgence. This results in the construction of minimal-string free-energy transseries and the analytic computation of their resurgent Stokes data. Calculations are presented via Liouville boundary conformal field theory and via (matching) matrix model analysis. Minimal-string results are extended to Jackiw-Teitelboim gravity. Building on the matrix model analysis, one extension towards topological string theory is obtained via the remodeling-conjecture -- which allows for addressing one-cut, toric Calabi-Yau geometries. Building on the Liouville theory calculation, one other extension towards critical string theory is obtained via the H3+ - Liouville correspondence -- which allows for addressing negative-tension D-instantons in AdS spacetime. Throughout, checks of the construction and formulae are made in several examples, against both Borel resurgent analysis and string-equation transseries data.

Figures (11)

• Figure 1: Example of the $(2,5)$ minimal-string FZZT Riemann surface. Its two sheets are separated by the cut (in green) starting at $a$ and ending at $\infty$ (from the matrix model point-of-view to be discussed later, this is a double-scaled spectral curve). Furthermore, we have labeled the two sheets with $\alpha=0$ (physical sheet in blue) and with $\alpha=1$ (orange); following \ref{['eq:sheet-labels']}. Throughout this section we shall return to similar figures to illustrate our discussion.
• Figure 2: Visualization of the disk contribution ${\mathcal{A}}_{\alpha \beta}^{[-1]}(x)$ in \ref{['eq:1instFZZTcontribution']} as a contour integral over the FZZT Riemann surface. Here we illustrate again the $(2,5)$ example as in the previous figure \ref{['fig:minimalstringspectralcurve-start']}.
• Figure 3: The $(2,5)$ FZZT surface revisited. On the left-plot we show the integration cycle associated to the configuration ${\mathscr Z}_{10}$ (this is \ref{['eq:1instFZZTcontribution']} with $\alpha=1$, $\beta=0$, which culminates in \ref{['eq:1-instanton-result']}). On the right-plot we show the exactly switched configuration${\mathscr Z}_{01}$ (again \ref{['eq:1instFZZTcontribution']}, this time around with $\alpha=0$, $\beta=1$, or, equivalently, as visualized above, \ref{['eq:negative-1-instanton-result']} with $\alpha=1$, $\beta=0$).
• Figure 7: Schematic plot of the real-part of the matrix-model holomorphic effective-potential (on the physical sheet) with two real saddle-points $x_1^{\star}$ and $x_2^{\star}$. The saddle $x_1^{\star}$ is closer to the cut with endpoints $a$ and $b$. The corresponding instanton actions $A_1$ (to $x_1^{\star}$) and $A_2$ (to $x_2^{\star}$) are real and resonant. Compare this plot to the double-scaled $(2,5)$ minimal-string plot in figure \ref{['fig:minimal-string-resonance-check']}.
• Figure 8: Schematic visualization of tunneling one eigenvalue from the perturbative-cut to the saddle $x_2^{\star}$, using the matrix-model potential in figure \ref{['fig:Matrix-Potential-Configuration']}. First (upper row), undergoing a forward discontinuity with the perturbative configuration tunnels an eigenvalue from the perturbative-cut to $x_1^{\star}$ (here the cut of the effective potential is visualized in green; the nonperturbative saddles are shown in red; the eigenvalue steepest-descent contours are schematically plotted in blue; and $\epsilon$ denotes the offset of the argument of $g_{\text{s}}$ coming from the discontinuity). The corresponding Borel plane picture is shown on the right, where the residue attached to the action $A_1$ is picked up (we only show the singularity relevant for our transition). Second (lower row), we undergo the backwards discontinuity starting from the $(1|0)(0|0)$ configuration. Because we are employing the backwards discontinuity the phase of $g_{\text{s}}$ has shifted by $\pi$. We observe how the backwards discontinuity indeed populates the second saddle $x_2^{\star}$ in this way. Notice also the appearance of an additive perturbative term in this transition which has been studied in mss22. Again on the right-hand side we show the Borel plane with the singularity shown in red.