Exact Results in ABJM Theory from Topological Strings

Abstract: Recently, Kapustin, Willett and Yaakov have found, by using localization techniques, that vacuum expectation values of Wilson loops in ABJM theory can be calculated with a matrix model. We show that this matrix model is closely related to Chern-Simons theory on a lens space with a gauge supergroup. This theory has a topological string large N dual, and this makes possible to solve the matrix model exactly in the large N expansion. In particular, we find the exact expression for the vacuum expectation value of a 1/6 BPS Wilson loop in the ABJM theory, as a function of the 't Hooft parameters, and in the planar limit. This expression gives an exact interpolating function between the weak and the strong coupling regimes. The behavior at strong coupling is in precise agreement with the prediction of the AdS string dual. We also give explicit results for the 1/2 BPS Wilson loop recently constructed by Drukker and Trancanelli

• The paper employs localization and matrix model techniques to derive exact expressions for 1/6 and 1/2 BPS Wilson loops in ABJM theory.
• It reveals a duality between ABJM theory and topological strings through a large N expansion that smoothly connects weak and strong coupling regimes.
• The study highlights the role of mirror symmetry and complex dualities in linking three-dimensional gauge theories with higher-dimensional string frameworks.

Overview of "Exact results in ABJM theory from topological strings"

The paper "Exact results in ABJM theory from topological strings" by Marcos Mariño and Pavel Putrov presents an intricate examination of ABJM theory by harnessing topological string theory. The research is significant in its use of localization techniques to provide exact results for Wilson loop vacuum expectation values (vevs) in ABJM theory by examining their relationship with topological strings and matrix models.

Key Contributions

This research meticulously bridges ABJM theory—a superconformal Chern-Simons-matter theory in three dimensions—with topological string theory. The authors leverage the relationship between matrix models connected to ABJM theory and lens space matrix models in Chern-Simons theory. They exploit this duality to arrive at exact results in the large N expansion.

1. Localization Techniques and Matrix Models: The study employs localization techniques advanced by Kapustin, Willett, and Yaakov to reduce the problem of calculating Wilson loop vevs in ABJM theory to a problem in matrix models. The central result indicates that these matrix models are analogous to those arising in Chern-Simons theory with a lens space as a gauge supergroup, particularly in the context of a large N duality with topological strings.
2. Exact Planar Results for Wilson Loops: The authors derive exact expressions for the 1/6 and 1/2 BPS Wilson loops within ABJM theory, constructing an integrable model using the complex relationship between two 't Hooft parameters. These findings not only provide precise interpolating functions between weak and strong coupling regimes but also align the behavior at strong coupling with predictions from the AdS/CFT correspondence.
3. Mirror Symmetry and Dualities: The dual theory invoked is that of the anti-canonical bundle over the Hirzebruch surface within the remit of mirror symmetry. The authors unfold the multi-facet interactions of matrix model characteristics with the spectral curve of topological strings, revealing implicit dualities within ABJM theory through complex structural analyses and the inversion of mirror maps.

Implications and Future Directions

The paper contributes robust analytical tools and numerical results critical for understanding the mathematical structures underlying ABJM theory. It establishes a novel link between high-dimensional string theoretic constructs and three-dimensional gauge theories through matrix model techniques and symmetries.

The implications extend both towards theoretical refinement and practical applications within high-energy physics. The precision of the results strengthens the foundation for future explorations into more intricate dualities in quantum field theories and string theory landscapes. Moreover, the methodologies employed have potential applications in uncovering similar relationships in other gauge theories and exploring new results in the intertwined realms of geometry and physics.

This work opens avenues for investigating non-planar corrections in Wilson loops beyond the planar approximation outlined in this paper. Furthermore, identifying equivalent formulations for these results in various duality frames or employing different choices of flat coordinates, crucial for topological string computations, presents additional avenues for refinement.

Concluding, the paper advances the theoretical landscape of ABJM theory and string theory through rigorous mathematical models and facilitates deeper insights into the geometrical underpinnings of gauge theories, offering substantial grounds for further exploratory and application-oriented research in theoretical physics.