Quantum Gravity Partition Functions in Three Dimensions

Abstract: We consider pure three-dimensional quantum gravity with a negative cosmological constant. The sum of known contributions to the partition function from classical geometries can be computed exactly, including quantum corrections. However, the result is not physically sensible, and if the model does exist, there are some additional contributions. One possibility is that the theory may have long strings and a continuous spectrum. Another possibility is that complex geometries need to be included, possibly leading to a holomorphically factorized partition function. We analyze the subleading corrections to the Bekenstein-Hawking entropy and show that these can be correctly reproduced in such a holomorphically factorized theory. We also consider the Hawking-Page phase transition between a thermal gas and a black hole and show that it is a phase transition of Lee-Yang type, associated with a condensation of zeros in the complex temperature plane. Finally, we analyze pure three-dimensional supergravity, with similar results.

• The paper computes the AdS3 partition function with quantum corrections, highlighting limitations in pure 3D gravity models.
• It introduces holomorphic factorization via complex saddle points to bridge classical geometries with quantum anomalies.
• The study analyzes subleading corrections to black hole entropy and reinterprets the Hawking-Page transition as a Lee-Yang type phase change.

Overview of Quantum Gravity Partition Functions in Three Dimensions

The paper by Alexander Maloney and Edward Witten explores the theoretical landscape of three-dimensional quantum gravity with a negative cosmological constant. The authors focus on the analytic calculation of partition functions fundamental to understanding the quantum properties of gravity in reduced dimensions. Utilizing known contributions from classical geometries, the study reveals intriguing yet unresolved aspects of quantizing gravity, particularly emphasizing the role of subleading corrections and phase transitions that distinguish three-dimensional configurations from their higher-dimensional counterparts.

Key Contributions

1. Exact Computation in AdS$_3$:
   • The authors compute the partition function in three-dimensional anti-de Sitter (AdS$_3$) space, incorporating quantum corrections to classical geometrical configurations. While the calculations yield explicit results, these results are not physically sensible within the known quantum gravity frameworks, implying potential limitations or missing pieces in current models of pure three-dimensional gravity.
2. Holomorphic Factorization:
   • One of the speculative solutions proposed is the inclusion of complex geometries, potentially leading to holomorphically factorized partition functions. This approach aligns with previous assumptions in lower-dimensional theories where factorization aids in establishing a consistent quantum framework.
3. Corrections to Black Hole Entropy:
   • By analyzing the subleading corrections to the Bekenstein-Hawking entropy, the paper demonstrates that these corrections can be symmetrically incorporated into a holomorphically factorized theory. This contributes to ongoing efforts to bridge classical and quantum descriptions, offering insights into microstates associated with black hole entropy.
4. Hawking-Page Transition:
   • The study revisits the Hawking-Page phase transition, traditionally defined as a thermodynamic transition between a thermal gas and a black hole. The authors identify it as a Lee-Yang type transition, characterized by condensation of the partition function zeros in the complex temperature plane—a perspective providing a different mathematical framework to describe such transitions in lower dimensions.
5. Supergravity Partition Functions:
   • Extending the analysis to supergravity, the authors present analogous results, emphasizing the robustness of their conclusions across varying theoretical constructs. Despite similar challenges and unphysical aspects, supergravity lends additional credibility to the suppositions regarding holographic dualities in lower dimensions.

Implications and Speculations

The findings and hypotheses advanced in this paper have profound implications for the theoretical grounding of quantum gravity. Primarily, the discrepancies and unsolvable aspects in the calculated partition functions suggest the non-existence of a strictly minimal theory for three-dimensional gravity—a notion indicating potential new physics or necessitating additional dimensions, fields, or corrections.

• Theoretical and Mathematical Development:
  • The possibility of a holomorphically factorized approach suggests new avenues for theoreticians to explore symmetry aspects in low-dimensional gravity, potentially enriching our understanding of duality and holography in flat-space quantum theories.
• Impact on Model Building:
  • The paper's exploratory approach to unconventional geometries (complex saddle points) could influence model-building in string-theoretic and gravitational contexts, pressing for models that inherently incorporate quantum corrections and anomalies unaccountable by classical analysis alone.
• Consequences for Higher-dimensional Theories:
  • Given that three-dimensional results often penetrate discussions on higher-dimensional theories, this work might incite further inquiries into modular invariant properties and to what extent they shape theoretical physics across multiple scales.

In conclusion, while "Quantum Gravity Partition Functions In Three Dimensions" illuminates a portion of the theoretical challenges within reduced dimensionality, it also marks a frontier for discovering or postulating missing links within established and new frameworks in quantum gravity, potentially guiding future explorations and mathematical formalizations in the broader context of theoretical physics.