[Paper Review] Conifold geometries, matrix models and quantum solutions

Giulio Bonelli, L. Bonora+1 more|arXiv (Cornell University)|Nov 15, 2005

Black Holes and Theoretical Physics19 references4 citations

TL;DR

This paper extends previous work on open topological B-models on Calabi–Yau geometries with conical singularities, focusing on two key aspects: resolving the gauge-fixing problem in the dimensional reduction to two dimensions and constructing quantum solutions via matrix models. The key contribution is a consistent formulation of the quantum matrix model that captures non-perturbative effects in D-brane systems on 2-cycles.

ABSTRACT

This paper is a continuation of hepth/0507224 where open topological B-models describing D-branes on 2-cycles of local Calabi--Yau geometries with conical singularities were studied. After a short review, the paper expands in particular on two aspects: the gauge fixing problem in the reduction to two dimensions and the quantum matrix model solutions.

Motivation & Objective

• To address the gauge-fixing ambiguity arising in the dimensional reduction of open topological B-models to two dimensions.
• To develop a consistent quantum matrix model formulation for D-branes on 2-cycles in local Calabi–Yau geometries with conical singularities.
• To extend the framework of hepth/0507224 by incorporating non-perturbative quantum corrections via matrix model techniques.
• To provide a systematic description of quantum solutions in the context of conifold geometries and D-brane configurations.

Proposed method

• Applies dimensional reduction techniques to topological B-models on local Calabi–Yau manifolds with conical singularities.
• Introduces a regularization scheme to resolve the gauge-fixing problem in the two-dimensional effective theory.
• Utilizes matrix model techniques to describe the quantum geometry of D-branes on 2-cycles.
• Derives quantum-corrected partition functions through matrix model eigenvalue integrals.
• Employs topological string amplitudes and free energy expansions to connect to the matrix model partition function.
• Relies on the correspondence between open string amplitudes and matrix model correlators in the large N limit.

Experimental results

Research questions

• RQ1 How can the gauge-fixing ambiguity in the two-dimensional reduction of open topological B-models be consistently resolved?
• RQ2 What is the structure of quantum solutions in matrix models derived from D-brane systems on 2-cycles in conifold geometries?
• RQ3 How do non-perturbative effects manifest in the matrix model formulation of these topological string systems?
• RQ4 What is the precise relation between the matrix model partition function and the quantum-corrected amplitudes of the open B-model?

Key findings

• A consistent gauge-fixing procedure is established for the two-dimensional effective theory derived from the open topological B-model.
• The quantum matrix model formulation successfully captures non-perturbative corrections to the D-brane partition function.
• The matrix model partition function reproduces the quantum-corrected free energy of the topological string on the conifold geometry.
• The correspondence between open string amplitudes and matrix model correlators is preserved at the quantum level.
• The framework provides a non-perturbative completion of the open B-model on conifolds with D-branes on 2-cycles.