JT Gravity and the Ensembles of Random Matrix Theory (1907.03363v5)

Abstract: We generalize the recently discovered relationship between JT gravity and double-scaled random matrix theory to the case that the boundary theory may have time-reversal symmetry and may have fermions with or without supersymmetry. The matching between variants of JT gravity and matrix ensembles depends on the assumed symmetries. Time-reversal symmetry in the boundary theory means that unorientable spacetimes must be considered in the bulk. In such a case, the partition function of JT gravity is still related to the volume of the moduli space of conformal structures, but this volume has a quantum correction and has to be computed using Reidemeister-Ray-Singer "torsion." Presence of fermions in the boundary theory (and thus a symmetry $(-1)^F$) means that the bulk has a spin or pin structure. Supersymmetry in the boundary means that the bulk theory is associated to JT supergravity and is related to the volume of the moduli space of super Riemann surfaces rather than of ordinary Riemann surfaces. In all cases we match JT gravity or supergravity with an appropriate random matrix ensemble. All ten standard random matrix ensembles make an appearance -- the three Dyson ensembles and the seven Altland-Zirnbauer ensembles. To facilitate the analysis, we extend to the other ensembles techniques that are most familiar in the case of the original Wigner-Dyson ensemble of hermitian matrices. We also generalize Mirzakhani's recursion for the volumes of ordinary moduli space to the case of super Riemann surfaces.

• The paper extends the JT gravity–RMT duality by including time-reversal symmetry, fermionic fields, and supersymmetry.
• It classifies the ten Altland–Zirnbauer ensembles and applies methods like Mirzakhani recursion to bridge gravitational and matrix integral formulations.
• By incorporating torsion corrections and supergravity aspects, the study unveils new universality classes in quantum chaos and lower-dimensional gravity.

Overview of the Paper: JT Gravity and the Ensembles of Random Matrix Theory

This paper provides an in-depth exploration of the intricate relationship between Jackiw-Teitelboim (JT) gravity and double-scaled random matrix theory (RMT), focusing particularly on how these theories intersect when considering variants of JT gravity that incorporate various symmetries. Specifically, the authors extend the mapping between JT gravity and RMT to include cases with time-reversal symmetry, fermionic fields, and supersymmetry.

Context and Motivation

JT gravity is a notable model for understanding two-dimensional quantum gravity, often utilized in the context of the AdS${}_2$/CFT${}_1$ correspondence. It provides a rich framework for analyzing near-extremal black holes and serves as a useful laboratory for studying holographic principles in lower dimensions. Recent discoveries have highlighted a compelling relationship between JT gravity and RMT, particularly through the lens of the Sachdev-Ye-Kitaev (SYK) model's connection to JT via the Schwarzian theory. This paper extends these insights further by investigating how different symmetry constraints lead to different matrix ensembles, thereby enhancing our understanding of universality classes in quantum chaos and gravity.

Main Contributions

1. Extension of JT/RMT Duality: The authors generalize the known relationship between JT gravity and RMT by incorporating theories with time-reversal symmetry and fermionic fields. The presence of fermions in particular demands symmetry considerations such as $(-1)^$ associated with RMT, leading to complexities such as spin or pin structures in the gravitational theory.
2. Symmetry and Matrix Ensembles: A primary result is the classification of these relationships based on symmetry, revealing that time-reversal symmetry and the presence or absence of supersymmetry in the boundary theory dictate the kind of RMT ensemble one finds on the gravity side. Specifically, they map all ten Altland-Zirnbauer matrix ensembles — including the Dyson ensembles and their extensions — to JT gravity settings.
3. Role of the Torsion: Incorporating time-reversal symmetry makes unorientable spacetimes relevant, necessitating calculations of torsion corrections in the moduli space of conformal structures. These are computed using the Reidemeister-Ray-Singer torsion, a crucial adjustment to account for quantum effects in the gravitational path integral.
4. Supergravity Considerations: When supersymmetry is present, JT gravity transitions into JT supergravity, which is then naturally related to volumes of moduli spaces of super Riemann surfaces. These setups correspond to distinct RMT ensembles where the Hamiltonian has roots or constraints imposed by supersymmetry, yielding new results and predictions about moduli space volumes.
5. Computational Methods: The paper leverages several advanced computational techniques, such as the Mirzakhani recursion for calculating volumes of moduli space, extended here to super Riemann surfaces. Additionally, loop equations for random matrices are adapted to extract genral predictions concerning the genus expansion of matrix integrals—a vital tool for comparing to gravitational path integrals.

Implications and Speculations

This work has profound implications for the theoretical landscape of quantum gravity and its associated mathematical frameworks:

• Practical Outcomes: By aligning JT gravity and RMT symmetries, the research provides a methodical foundation for understanding quantum chaos through gravitational models. It also helps identify universality classes within the dynamics of quantum gravity, potentially influencing quantum information and black hole physics.
• Theoretical Developments: In more abstract aspects, the generalized correspondence opens avenues for further exploration of holographic dualities in other geometric regimes, particularly those involving supersymmetric and fermionic fields.
• Future Directions: This paper poses numerous pathways for future investigation, such as non-perturbative effects in matrix models, higher-dimensional generalizations, and potential connections with other string-inspired models of quantum gravity. The consideration of anomalies and non-trivial topological field theories as controlling parameters adds another layer of depth to future explorations, potentially in connection with string theory and M-theory landscapes.

In summary, the research advances our understanding of JT gravity via matrix models, providing crucial insights into how the microphysics of black holes may be fundamentally tied to matrix ensembles. It bridges important theoretical concepts, uniting gravity and quantum chaos through a symmetry-oriented perspective, and sets a substantial groundwork for further explorations in lower-dimensional quantum gravity and holography.