Remodeling the B-model (0709.1453v1)

Abstract: We propose a complete, new formalism to compute unambiguously B-model open and closed amplitudes in local Calabi-Yau geometries, including the mirrors of toric manifolds. The formalism is based on the recursive solution of matrix models recently proposed by Eynard and Orantin. The resulting amplitudes are non-perturbative in both the closed and the open moduli. The formalism can then be used to study stringy phase transitions in the open/closed moduli space. At large radius, this formalism may be seen as a mirror formalism to the topological vertex, but it is also valid in other phases in the moduli space. We develop the formalism in general and provide an extensive number of checks, including a test at the orbifold point of A_p fibrations, where the amplitudes compute the 't Hooft expansion of Wilson loops in lens spaces. We also use our formalism to predict the disk amplitude for the orbifold C^3/Z_3.

• The paper introduces a recursive framework adapted from matrix models to compute open and closed B-model amplitudes in local Calabi-Yau geometries.
• It employs residue calculus on spectral curves, validating computations at orbifold points and linking topological open strings to algebraic spectral data.
• The method provides nonperturbative insights into moduli spaces and string dualities, with potential extensions to compact Calabi-Yau spaces.

Remodeling the B-model: A Formalism for Local Calabi-Yau Geometries

The paper "Remodeling the B-model" by Vincent Bouchard, Albrecht Klemm, Marcos Mariño, and Sara Pasquetti introduces a comprehensive framework for computing open and closed B-model amplitudes within local Calabi-Yau geometries. This framework is particularly centered on the recursive structure discovered by Eynard and Orantin, which was originally applied in the context of matrix models. The authors adapt this recursive solution to address the complexities of B-model strings in local Calabi-Yau spaces and propose a method that facilitates the paper of stringy phase transitions across open/closed moduli spaces.

Structure and Methodology

The paper is dedicated to non-compact Calabi-Yau threefolds, such as those which are mirrors of toric manifolds. The B-model's key advantage lies in its ability to deliver exact results concerning complex moduli—a feature the authors employ to obtain insights into string geometries, especially in non-geometric phases like orbifold and conifold points. The authors' novel approach provides a robust computational technique applicable to the topological vertex at large radii and in other moduli space phases.

The core of the approach is the recursive computation method pioneered by Eynard and Orantin. Initially developed to resolve the loop equations in matrix models, this method involves calculating open and closed string amplitudes through residue calculus on the spectral curves associated with these models. Bouchard et al. argue that this methodology can be suitably adapted to manage the B-model on mirrors of toric backgrounds, effectively establishing this recursive method in the local Calabi-Yau context.

Key Results and Tests

To validate their formalism, the authors conduct extensive tests, exemplified by computations at the orbifold point of A_p fibrations. Here, the amplitudes are instrumental in computing the 't Hooft expansion of Wilson loops within lens spaces. Furthermore, the authors leverage their methodology to predict disk amplitudes for the orbifold $^3 / _3$.

An illustrative case is the relationship between topological open strings on these local geometries and the amplitudes computed in a matrix model. This link effectively abstracts topological amplitudes in terms of purely algebraic quantities on spectral curves, offering significant computational advantages. Additionally, the paper discusses strategies to navigate between various modular spaces by exploiting modularity properties of string amplitudes.

Implications and Future Directions

The implications of this framework are far-reaching both theoretically and practically. By offering a non-perturbative view of the moduli space, it provides a deeper understanding of the geometry and physics of disconnected worldsheets— domains previously elusive in standard perturbative treatments.

The theoretical advancement also suggests practical areas for future research. The application to compact Calabi-Yau spaces remains an open and enticing avenue, as does the further development of connections to physical models such as topological insulators or string-like states in condensed matter physics.

The authors speculate that while the efficacy of the framework in non-compact geometries is evident, extending these concepts into compact scenarios could yield further insights into generalized string phenomena. Moreover, this framework's proposed use in non-geometric phase transitions could lead to groundbreaking revelations in understanding string dualities and the landscape of string theory solutions.

In essence, this paper marks a substantial contribution to the formal understanding of topological strings, laying the groundwork for an expansion beyond perturbative regimes, broadening the applicability of these ideas in both mathematical and physical contexts.