[Paper Review] Generating the superpotential on a D-brane category: I

Calin Iuliu Lazaroiu|ArXiv.org|Oct 10, 2006

Black Holes and Theoretical Physics34 references9 citations

TL;DR

This paper constructs a formal topological string field theory action that generates the D-brane superpotential for algebraic Calabi-Yau triangulated categories with a triangle generator, using cyclic $A_{∞}$ enhancements. It establishes that the Serre pairing on such categories is equivalent to a cyclic pairing on the generator’s enhancement, enabling a systematic construction of the superpotential from a finite set of generating D-branes.

ABSTRACT

I study A-infinity enhancements of algebraic Calabi-Yau triangulated categories admitting a (triangle) generator, showing that the Serre pairing on such categories determines and is determined by a cyclic pairing on an enhancement of the generator. Using this result, I construct a formal topological string field action inducing an extended D-brane superpotential for such categories. I also give a procedure for lifting certain 2d boundary topological field theories to open topological string theories generated by a single D-brane.

Motivation & Objective

• To develop a method for generating the full D-brane superpotential from a finite set of generating D-branes in a Calabi-Yau triangulated category.
• To establish a correspondence between the Serre pairing on a triangulated category and a cyclic pairing on an $A_{\infty}$ enhancement of a triangle generator.
• To construct a formal open string field theory action that induces the extended superpotential for such categories.
• To provide a procedure for lifting 2D boundary topological field theories to open topological string theories generated by a single D-brane.
• To demonstrate that homologically non-degenerate cyclic pairings on $A_{\infty}$ algebras admit flat symmetric minimal models, ensuring the existence of cyclic minimal models.

Proposed method

• Uses $A_{\infty}$ enhancements of algebraic Calabi-Yau triangulated categories that admit a triangle generator to relate the Serre pairing to a cyclic pairing on the generator’s enhancement.
• Applies the minimal model theorem for $A_{\infty}$ algebras to construct a cyclic minimal model from a homologically non-degenerate cyclic pairing.
• Employs the noncommutative Darboux theorem to ensure the existence of a flat symplectic form on the minimal model, preserving the cyclic structure.
• Derives a formal topological string field action from the cyclic $A_{\infty}$ structure on the generator, inducing the superpotential via higher-order correlation functions.
• Utilizes the cohomological splitting of the $A_{\infty}$ algebra to induce minimal models that preserve cyclic pairings and descent to cohomology.
• Constructs a prolongation of the open string field action using suspended morphism spaces and graded cyclic symmetry, ensuring compatibility with boundary observables.

Experimental results

Research questions

• RQ1 How can the superpotential of a D-brane category be systematically generated from a finite set of generating D-branes?
• RQ2 What is the precise relationship between the Serre pairing on a triangulated category and a cyclic pairing on an $A_{\infty}$ enhancement of a generator?
• RQ3 Can a formal topological string field theory action be constructed that induces the extended superpotential for such categories?
• RQ4 Under what conditions does a homologically non-degenerate cyclic pairing on an $A_{\infty}$ algebra admit a flat symmetric minimal model?
• RQ5 How can 2D boundary topological field theories be lifted to open topological string theories generated by a single D-brane?

Key findings

• The Serre pairing on a Calabi-Yau triangulated category is equivalent to a cyclic pairing on an $A_{\infty}$ enhancement of a triangle generator, establishing a duality between global and local structures.
• A formal topological string field action is constructed that induces the extended D-brane superpotential from a cyclic minimal $A_{\infty}$ generator, generalizing the physical superpotential construction.
• The induced open string field action is compatible with the $A_{\infty}$ relations and graded cyclicity, ensuring consistency with topological string theory axioms.
• Homologically non-degenerate cyclic pairings on $A_{\infty}$ algebras admit flat symmetric minimal models, which are unique up to isomorphism of cyclic $A_{\infty}$ algebras.
• The minimal model theorem ensures that any symmetric $A_{\infty}$ algebra with a homologically non-degenerate cyclic pairing admits a cyclic minimal model, preserving the pairing under quasi-isomorphism.
• The pullback of the cyclic pairing via an $A_{\infty}$ quasi-isomorphism preserves non-degeneracy on cohomology, ensuring the existence of a non-degenerate pairing on the minimal model.