JT gravity as a matrix integral (1903.11115v1)

Abstract: We present exact results for partition functions of Jackiw-Teitelboim (JT) gravity on two-dimensional surfaces of arbitrary genus with an arbitrary number of boundaries. The boundaries are of the type relevant in the NAdS${}_2$/NCFT${}_1$ correspondence. We show that the partition functions correspond to the genus expansion of a certain matrix integral. A key fact is that Mirzakhani's recursion relation for Weil-Petersson volumes maps directly onto the Eynard-Orantin "topological recursion" formulation of the loop equations for this matrix integral. The matrix integral provides a (non-unique) nonperturbative completion of the genus expansion, sensitive to the underlying discreteness of the matrix eigenvalues. In matrix integral descriptions of noncritical strings, such effects are due to an infinite number of disconnected worldsheets connected to D-branes. In JT gravity, these effects can be reproduced by a sum over an infinite number of disconnected geometries -- a type of D-brane logic applied to spacetime.

• The paper demonstrates that JT gravity partition functions can be exactly formulated using matrix integrals, establishing a robust link with random matrix models.
• The paper reveals nonperturbative completions that capture corrections beyond the standard genus expansion, crucial for understanding spectral properties.
• The paper utilizes topological recursion and Mirzakhani's algorithm to relate Weil-Petersson volumes with JT gravity, illuminating deep geometric connections.

Analysis of "JT Gravity as a Matrix Integral"

The paper, "JT gravity as a matrix integral" by Phil Saad, Stephen H. Shenker, and Douglas Stanford, provides a comprehensive examination of Jackiw-Teitelboim (JT) gravity using a matrix integral framework, yielding exact results for partition functions on two-dimensional surfaces of arbitrary genus and with complex boundary configurations. Building on the NAdS${}_2$/NCFT${}_1$ correspondence, the authors effectively bridge connections between JT gravity and random matrix models, supported by rigorous mathematical constructs such as the Eynard-Orantin topological recursion.

The fundamental insight this paper presents is the identification of the genus expansion of JT gravity partition functions with a certain matrix integral. Intriguingly, Mirzakhani's recursion relation for Weil-Petersson volumes translates into the loop equations for the matrix model, revealing deep structural connections between these a priori disparate mathematical frameworks. Notably, the matrix integral provides a nonperturbative completion sensitive to the discreteness inherent in matrix eigenvalues, akin to an infinite series of disconnected worldsheets associated with D-branes in string theory.

Key Contributions and Findings

1. Exact Formulation of JT Gravity via Matrix Models: By mapping JT gravity onto a matrix integral, the authors derive closed-form expressions for partition functions across various topologies and boundaries, particularly emphasizing the significant role of Schwarzian theory in these constructions. This facilitates an efficient exploration of nonperturbative effects and the duality with matrix ensembles.
2. Nonperturbative Completions: The matrix integral offers a non-unique completion beyond genus-expansion, accommodating nonperturbative corrections vital for understanding phenomena such as the plateau in spectral form factors. These corrections hinge upon a "D-brane logic" applied spatially, signifying a profound connection between spacetime and brane dynamics.
3. Topological Recursions and Weil-Petersson Volumes: Utilizing Mirzakhani's algorithms, the work unearths associations between topological recursion—a crucial tool in random matrix theory—and the geometry of moduli spaces, underscoring a shared mathematical underpinning with JT gravity.
4. Connections to Minimal Strings: The paper speculates that JT gravity may epitomize a limit of the $(2,p)$ minimal string models, drawing parallels through the density of states function. This relationship hints at broader applications and interpretations across string theoretical domains.

Theoretical and Practical Implications

The theoretical implications of this work are multifaceted, extending from mathematical physics into potential applications in understanding quantum gravity mechanisms. Specifically, the analytical forms introduced provide fresh perspectives on the nonperturbative phenomena within gravitational models, aligning with matrix integral techniques employed in high-energy physics and condensed matter systems.

This research also whets the appetite for future explorations, such as clarifying the conjectured connections between JT gravity, SYK models, and the broader landscape of AdS/CFT correspondence in higher dimensions. It encloses a hint at bridging seemingly unrelated quantum systems through matrix approaches, thereby potentially yielding a universal language for representing complex gravitational interactions.

The findings may facilitate advanced numerical simulations, particularly those involving long-time dynamics and intricate topologies, serving as critical benchmarks for quantum gravity models undergoing empirical tests in the near future.

Conclusion

"JT gravity as a matrix integral" insightfully integrates JT gravity within a comprehensive matrix integral framework, effectively advancing the dialogue between gravitational physics, topological recursion, and string theory. The pairwise corroboration of analytical results with theoretical constructs underscores the paper's significance within the landscape of theoretical physics, projecting avenues for robust, multifaceted explorations in high-energy phenomenology and mathematics. As the field progresses, this work stands as both a cornerstone and a springboard for researchers striving to decode the complexities of quantum gravity through the elegant machinery of matrix integrals.