Topological gravity for arbitrary Dyson index

TL;DR

This work extends the duality between topological gravity/JT gravity and Airy matrix models to arbitrary Dyson index $\beta$, defining $\beta$-topological gravity and introducing $\beta$-Airy Weil–Petersson volumes that interpolate between orientable and unorientable sectors while revealing non-Wigner–Dyson contributions. Using loop equations and a double-cover coordinate formulation, the authors derive a universal structure for $\beta$-Airy resolvents and demonstrate a Mirzakhani-like recursion in the $\beta$-dependent setting, giving geometric interpretation via generalized Kontsevich diagrams. They study quantum chaos through the canonical and microcanonical spectral form factors, obtaining perturbative agreement with universal RMT in the symplectic class ($\beta=4$) and providing a framework to extend chaos diagnostics to general $\beta$ with numerical validation. The results indicate robust chaotic behavior across bosonic topological gravity variants, supported by constraints on $\beta$-dependent WP volumes and by a constructive approach to the universal microcanonical SFF for arbitrary $\beta$. The work sets the stage for deeper analytic control of the universal regime for general Dyson index and for JT gravity variants beyond the bosonic classes.

Abstract

We use the well established duality of topological gravity to a double scaled matrix model with the Airy spectral curve to define what we refer to as topological gravity with arbitrary Dyson index $\upbeta$ ($\upbeta$ topological gravity). On the matrix model side this is an interpolation in the Dyson index between the Wigner-Dyson universality classes, on the gravity side it can be thought of as interpolating between orientable and unorientable manifolds in the gravitational path integral, opening up the possibility to study moduli space volumes of manifolds ``in between''. Using the perturbative loop equations we study correlation functions of this theory and prove several structural properties, having clear implications for the generalised moduli space volumes. Additionally we give a geometric interpretation of these properties using the generalisation to arbitrary Dyson index of the recently found Mirzakhani-like recursion for unorientable surfaces. Using these properties, we investigate whether $\upbeta$-topological gravity is quantum chaotic in the sense of the Bohigas-Giannoni-Schmit conjecture. Along the way we answer this question for the symplectic Wigner-Dyson class, not studied in the literature yet, and establish strong evidence for quantum chaos for this version of the theory, and thus for all bosonic varieties of topological gravity. We further argue for quantum chaoticity in the general $\upbeta$ case, based on novel constraints we find to be obeyed by genuinely non-Wigner-Dyson parts of the moduli space volumes. As for the general $\upbeta$ case the universal behaviour expected from a chaotic system is not known fully analytically we give a novel way to approach it, starting with the result of $\upbeta$ topological gravity and compare the results to a numerical evaluation of the universal result.

Figures (8)

• Figure 2.1: Manifold contributing at genus $g=3$ to the correlation function of three partition functions with (complex) inverse temperatures $\beta_1,\beta_2,\beta_3$. In grey are the "trumpets", cut off along geodesic boundaries of lengths $b_1,b_2,b_3$.
• Figure 3.1: Depiction of the different "glueings" i.e. the separation of a surface of constant negative curvature of genus $g$ and $n$ geodesic boundaries of lengths $L_1,\dots,L_n$ into a 3-holed sphere and another such surface. These separations are the same as for the case of unorientable surfaces in Stanford2023. The only difference for our setting, \ref{['eq:Mirz_extrBdry', 'eq:Mirz_Cont_DiscGlue', 'eq:Mirz_CrosscapGlue']}, is, that there is an additional factor of $\frac{1}{\upbeta}$ for the glueing in case a) and an additional factor of $\frac{\qty(2-\upbeta)}{\upbeta}$ for case b).
• Figure 3.2: Possible decomposition of a surface of genus $g$ and $n$ geodesic boundaries into 3-holed spheres and crosscaps with the maximal number of crosscaps possible. The numbers below the first and second part of this construction indicate how many 3-holed spheres/3-holed spheres with an attached crosscap are needed.
• Figure 3.3: Diagrams contributing to $V_{\frac{1}{2},2}$, i.e. the elements of $\Gamma_{\frac{1}{2},2}$.
• Figure 4.1: Comparison of the numerical evaluation of the normalised microcanonical SFF (cf.\ref{['eq:normalised_microSFF']}) for the general $\upbeta$ Gaussian matrix model (blue line) with our prediction up to second order (green line), the prediction of Bianchi2024 to all orders (orange line) and up to second order (red line) for different values of $\upbeta\in\qty[1,4]$. We used matrices of size $N_m=200$, averaging over $N_r=8000$ realisations.