[Paper Review] Wall-crossing of D4-branes using flow trees

Jan Manschot|arXiv (Cornell University)|Mar 8, 2010

Black Holes and Theoretical Physics4 citations

TL;DR

This paper establishes that the BPS partition function for D4-D2-D0 branes is convergent for flow trees with up to three endpoints in the large volume limit of a Calabi-Yau threefold, by proving that an indefinite quadratic form governing the BPS mass is positive definite under stability conditions. It further shows that S-duality invariance requires the partition function to be a generating function of rational invariants, not integer invariants, resolving a key consistency condition in wall-crossing mechanisms.

ABSTRACT

The moduli dependence of D4-branes on a Calabi-Yau manifold is studied using attractor flow trees, in the large volume limit of the Kahler cone. One of the moduli dependent existence criteria of flow trees is the positivity of the flow parameters along its edges. It is shown that the sign of the flow parameters can be determined iteratively as function of the initial moduli, without explicit calculation of the flow of the moduli in the tree. Using this result, an indefinite quadratic form, which appears in the expression for the D4-D2-D0 BPS mass in the large volume limit, is proven to be positive definite for flow trees with 3 or less endpoints. The contribution of these flow trees to the BPS partition function is therefore convergent. From non-primitive wall-crossing is deduced that the S-duality invariant partition function must be a generating function of rational, multi-covering invariants instead of integer invariants.

Motivation & Objective

• To resolve the convergence of the BPS partition function for D4-D2-D0 branes in the large volume limit using attractor flow trees.
• To determine the moduli dependence of flow tree existence by analyzing the sign of flow parameters without solving the full moduli flow.
• To clarify the role of rational invariants in S-duality-invariant partition functions, showing they are necessary for consistency with non-primitive wall-crossing.
• To extend the framework of flow trees beyond two endpoints, addressing the complexity of three-endpoint configurations.
• To lay the groundwork for generalizing the flow tree approach to higher-order trees and non-primitive charges.

Proposed method

• Uses iterative determination of flow parameter signs from initial moduli, avoiding explicit solution of the full moduli flow along the tree.
• Applies the Kontsevich-Soibelman wall-crossing formula to derive index jumps in terms of nested flow tree structures.
• Analyzes the BPS mass formula in the large volume limit, focusing on the indefinite quadratic form (Q−B)+2−∑(Qi−B)i2.
• Proves that this quadratic form is positive definite for stable three-endpoint flow trees, ensuring convergence of the partition function contribution.
• Derives that S-duality invariance requires the partition function to be a generating function of rational invariants Ω̄(Γ) = ∑_{m|Γ} Ω(Γ/m)/m².
• Constructs the partition function contribution for three-endpoint trees, including cases with equal charges, and shows compatibility with modular forms and mock Siegel theta functions.

Experimental results

Research questions

• RQ1 Can the sign of flow parameters along edges of a D4-brane flow tree be determined iteratively from initial moduli without solving the full moduli flow?
• RQ2 Is the indefinite quadratic form in the BPS mass formula positive definite for stable three-endpoint flow trees in the large volume limit?
• RQ3 Does the S-duality invariance of the partition function require the use of rational invariants rather than integer invariants?
• RQ4 How do contributions from non-primitive and primitive flow trees combine in the partition function, and do they yield modular objects?
• RQ5 Can the condition S(T(12)3,t) ≠ 0 be interpreted more deeply in a mathematical or physical context?

Key findings

• The sign of the flow parameter along each edge of a flow tree can be determined iteratively from the initial moduli, without solving the full moduli flow, providing a practical criterion for tree existence.
• The indefinite quadratic form (Q−B)+2−∑(Qi−B)i2 is proven to be positive definite for all stable three-endpoint flow trees in the large volume limit, ensuring convergence of the partition function contribution.
• The contribution of three-endpoint flow trees to the BPS partition function is convergent, extending the convergence result previously known only for two-endpoint trees.
• The S-duality invariant partition function must be a generating function of rational invariants Ω̄(Γ) = ∑_{m|Γ} Ω(Γ/m)/m², not integer invariants Ω(Γ), to be compatible with non-primitive wall-crossing.
• The partition function contributions from flow trees with equal charges for two endpoints combine consistently with vector-valued modular forms and mock Siegel theta functions, supporting modular invariance.
• The results suggest that the positive definiteness of the quadratic form and convergence of the partition function likely extend to flow trees with any number of endpoints, though this remains to be proven.