Non-perturbative effects and the refined topological string

Abstract: The partition function of ABJM theory on the three-sphere has non-perturbative corrections due to membrane instantons in the M-theory dual. We show that the full series of membrane instanton corrections is completely determined by the refined topological string on the Calabi-Yau manifold known as local P1xP1, in the Nekrasov-Shatashvili limit. Our result can be interpreted as a first-principles derivation of the full series of non-perturbative effects for the closed topological string on this Calabi-Yau background. Based on this, we make a proposal for the non-perturbative free energy of topological strings on general, local Calabi-Yau manifolds.

• The paper derives complete non-perturbative corrections to the ABJM free energy using refined topological string techniques in the NS limit.
• It rigorously computes membrane instanton effects via a Fermi gas approach and TBA, linking them to local Calabi–Yau geometries.
• The authors demonstrate the HMO cancellation mechanism and quantum period methods to ensure analytic control in the dual M-theory framework.

Non-Perturbative Effects and the Refined Topological String

The paper "Non-perturbative effects and the refined topological string" by Yasuyuki Hatsuda, Marcos Mariño, Sanefumi Moriyama, and Kazumi Okuyama provides a comprehensive examination of the non-perturbative structure of the free energy in M-theory on $AdS_4 \times S^7/\mathbb{Z}_k$. This work stands as a notable contribution to the understanding of large $N$ dualities, particularly focusing on the intricate interplay between gauge theories and string theories. The authors meticulously derive the complete series of non-perturbative corrections to the partition function of ABJM theory on the three-sphere by leveraging the refined topological string in the Nekrasov–Shatashvili (NS) limit.

Summary of Findings

1. ABJM Theory and M-Theory Duality: The paper revisits the large $N$ duality between ABJM theory and M-theory, highlighting the analytic determination of the non-perturbative structure of ABJM's partition function. The research underscores the correspondence between M-theory expansions and quantum corrections in the refined topological string.
2. Membrane Instanton Corrections: Central to the study is the role of membrane instantons, which are identified through an equivalence with the refined topological string theory on a local Calabi–Yau manifold, specifically local $^1 \times ^1$. These membrane instanton effects are rigorously determined using the Fermi gas approach combined with Thermodynamic Bethe Ansatz (TBA) techniques.
3. Refined Topological String in the NS Limit: The analysis posits that the non-perturbative ABJM partition function is intrinsically linked to the refined topological string in the NS limit. This connection paves the way for explicating various grand potential terms using quantum mirror maps and quantum periods, offering an elegant mathematical framework to capture these topological properties.
4. Implications of the HMO Cancellation Mechanism: The paper discusses the HMO cancellation mechanism, ensuring cancellation of singularities in the grand potential at integer values of the Chern–Simons level $k$. This mechanism guarantees the regularity of the partition function through a precise relationship among the pole structures of different contributions, including worldsheet instantons and membrane instanton bound states.
5. Theoretical Innovations and Quantum Periods: The researchers elucidate how quantum periods associated with the spectral curve of the Calabi–Yau manifold naturally emerge from the NS limit of the topological string. This advance provides a methodological basis for computing non-perturbative effects systematically across large $N$ expansions, going beyond perturbative regimes.

Implications and Future Directions

The elegant synthesis of refined topological string theory and ABJM theory presented in this research could have significant ramifications for both mathematical physics and string theory. The formulation not only resolves practical computational challenges associated with M-theory but also proposes a refined lattice of dualities that connect seemingly disparate physical phenomena.

Future studies may further explore the broader applicability of this framework to other gauge and string theories, possibly extending these insights to globally non-perturbative regimes and other local Calabi–Yau geometries. Moreover, the study invites further exploration into the non-perturbative refinements of topological strings, potentially enriching the spectrum of solutions available under these theoretical constructs. Understanding how these results may illuminate similar structures in related physical systems could offer profound advancements in theoretical physics.