Non-perturbative topological string theory on compact Calabi-Yau manifolds from M-theory (2408.09255v2)

Abstract: We show that the full non-perturbative topological string free energy, in the holomorphic limit, follows simply from a target space integrating out calculation of M2 states. Qualitatively, this is the same as the calculation performed by Gopakumar and Vafa, but we find that the final expression must be modified due to a subtlety with poles induced by non-perturbative physics. Accounting for this modification leads to a Gopakumar-Vafa-like formula, which we propose as the exact formulation of the integrating out procedure. Evaluating the formula necessarily requires a contour integral in a complexified Schwinger proper time parameter. We show that this evaluation yields the full non-perturbative topological string free energy, and can be applied to a compact, or non-compact, Calabi-Yau threefold. The explicit formula presented holds as long as the two-cycles wrapped by the branes are rigid and smooth, but the methodology can be used to study also more general Calabi-Yau geometries.

• The paper presents a novel formula for non-perturbative topological string free energy by integrating out M2-brane states via a contour integral approach.
• It refines the traditional Gopakumar-Vafa framework by capturing both holomorphic aspects and non-perturbative poles in the string spectrum.
• The study bridges M-theory and topological string theory, paving the way for new insights into string dualities and generalized Calabi-Yau geometries.

Non-Perturbative Topological String Theory on Compact Calabi-Yau Manifolds from M-theory

The paper by Jarod Hattab and Eran Palti explores non-perturbative aspects of topological string theory on compact Calabi-Yau manifolds, approached through M-theory considerations. The paper builds upon the foundational Gopakumar-Vafa (GV) framework, offering a revised and arguably more complete formulation for calculating the full non-perturbative topological string free energy. This paper makes significant strides in formalizing the intricacies involved in integrating out M-brane states, with notable attention to the introduced complexities due to non-perturbative physics.

Overview

The cornerstone of Hattab and Palti's proposal is a novel formula for non-perturbative topological string free energy, derived from a careful examination of M2-brane states in M-theory. The free energy is evaluated via a contour integral approach within the field of Schwinger proper time parameters, differing from GV’s original formulation by accounting for poles induced by non-perturbative effects. The formulation is applied to both compact and non-compact Calabi-Yau threefolds, contingent on the rigidity and smoothness of the two-cycles wrapped by branes.

Methodology and Results

The methodology employed involves deriving the non-perturbative contributions using an integration-out calculation over complexified paths. This differs from the standard perturbative expansions and captures both holomorphic properties as well as the challenging poles that emerge from non-perturbative dynamics. The new expression for the free energy, denoted as $F$, is shown to provide an exact non-perturbative spectrum, enhancing the traditional genus-expansion perturbative series into an exact closed form that includes both perturbative and non-perturbative data.

For the resolved conifold case—a prominent non-compact Calabi-Yau manifold used as a test bed—the authors demonstrate consistency between their proposed formula and expectations from existing literature. Specifically, they find that perturbative and non-perturbative components can be segregated by examining the contour integrals' contribution from various types of poles. The analysis highlights the presence of non-perturbative poles located at specific quantized intervals when viewed in the complex plane—leading to insights consistent with Gopakumar-Vafa derived data.

Implications and Future Research Directions

The implications of this work are twofold. Practically, the proposed formulation broadens the landscape for computing string free energies beyond perturbative limits, promising enhancements in understanding string theory's path integral at high precision. Theoretically, it deepens the connection between M-theory's compactification physics and topological string theory, offering a bridge to explore more considerable dualities in theoretical physics.

Future research could explore extending this formalism to more generalized Calabi-Yau geometries, especially cases entailing less rigid cycle structures. Furthermore, examining the applicability of these results in broader string duality contexts, such as mirror symmetry and geometric transitions, might reveal new layers of mathematical and physical insight. The treatment of complexified string couplings vis-à-vis real couplings could spur further developments in resurgence theory and related non-perturbative physics.

In summary, Hattab and Palti's paper provides a substantial extension of topological string theory in non-perturbative regimes, asserting modifications to classical approaches for accuracy in accounting for intricate physical interactions in the quantum field. This represents a noteworthy contribution towards refining the mathematical apparatus that underpins string theory, promising new explorations in theoretical high-energy physics and complex geometry.