Summing Up All Genus Free Energy of ABJM Matrix Model

Abstract: The localization technique allows us to compute the free energy of the U(N)k x U(N){-k} Chern-Simons-matter theory dual to type IIA strings on AdS_4 x CP^3 from weak to strong 't Hooft coupling \lambda = N / k at finite N, as demonstrated by Drukker, Marino, and Putrov. In this note we study further the free energy at large 't Hooft coupling with the aim of testing AdS/CFT at the quantum gravity level and, in particular, sum up all the 1/N corrections, apart from the worldsheet instanton contributions. The all genus partition function takes a remarkably simple form -- the Airy function, Ai (k^{4/3} \lambda_r), with the renormalized 't Hooft coupling \lambda_r.

• The paper demonstrates advanced localization techniques to compute the full 1/N free energy corrections in the ABJM model.
• It reveals that a renormalized 't Hooft coupling leads to an Airy function form for the free energy and exposes non-planar discrepancies.
• These findings prompt further research into non-planar corrections and alternative gauge group variants to deepen our understanding of quantum gravity.

An Analysis of All Genus Free Energy in ABJM Matrix Models

The study by Hiroyuki Fuji, Shinji Hirano, and Sanefumi Moriyama provides a theoretical examination of the ABJM (Aharony-Bergman-Jafferis-Maldacena) matrix model, aiming to verify the AdS/CFT (Anti-de Sitter/Conformal Field Theory) duality at the quantum gravity level by computing its all-genus free energy. The ABJM theory, which serves as a dual to type IIA string theory on an AdS$_4 \times CP^3$ manifold, constitutes a significant framework in the assessment of M-theory and string dynamics at finite N.

Key Insights on Localization Techniques and Free Energy

A focal point of the paper is the application of advanced localization techniques to compute the free energy associated with the U(N) × U(N)$_{-k}$ Chern-Simons-matter theory, also known as the ABJM theory. Specifically, the paper demonstrates the utility of localization in summing up all 1/N corrections in the free energy at finite N and strong 't Hooft coupling $\lambda = N/k$, thereby contributing to a deeper understanding of non-planar corrections that are typically challenging to compute in quantum gravity.

The free energy, when not accounting for worldsheet instanton contributions, is found to converge to a relatively simple form known as the Airy function, represented as:

$$F_{\text{ABJM}} = \log \left( 2\pi C \cdot \text{Ai} \left( \left( \frac{\pi k}{2} \right)^{2/3} \lambda_{\text{ren}} \right) \right)$$

Then, introducing a renormalized 't Hooft coupling:

$$\lambda_{\text{ren}} = \lambda - \frac{1}{24} - \frac{\lambda^2}{3N}$$

This renormalization effectively incorporates a partial resummation of the free energy, with remaining higher genus free energies adhering to a straightforward recursive relation.

Discrepancies and the Non-Planar Level

While the study confirms the effectiveness of localization techniques in testing the AdS/CFT conjecture, it identifies discrepancies at the non-planar level between predictions made by the matrix model and those of the string theory. Specifically, the matrix model predicts an additional non-planar shift of $-\frac{3\lambda^2}{8N}$, which is absent from the dual string/M-theory forecasts, notably the ones predicting a shift of $\lambda^2/24N$ as linked to higher curvature corrections.

Theoretical and Practical Implications

These findings have notable implications for both theoretical physics and string theory. The renormalized coupling becomes significant for understanding quantum gravity corrections and the nature of AdS/CFT duality beyond the planar limit. The results raise critical questions regarding higher-order corrections in supergravity and their possible implications for the holographic nature of gravitational theories.

Future Research Directions

Given the identified discrepancies between matrix model predictions and established string theory expectations, further research into the SU(N) × SU(N) variant of the theory may elucidate these differences. This exploration could enhance the understanding of non-planar dynamics in M-theory, potentially leading to new insights into the nuances of quantum corrections across different gauge groups.

Conclusion

In summary, this work represents a significant step in understanding the mathematical structure and physical implications of the ABJM matrix models, particularly in the context of AdS/CFT duality. The application of localization techniques provides a sophisticated framework for encapsulating corrections in finite N scenarios, though further exploration will be essential to resolve current discrepancies and to bridge theoretical predictions with broader string theory phenomena.