Wall-crossing of D4-branes using flow trees

TL;DR

This paper investigates how D4-branes' stability depends on moduli in Calabi-Yau spaces using flow trees, establishing positivity criteria and implications for BPS partition functions and S-duality invariants.

Contribution

It introduces an iterative method to determine flow tree positivity without explicit flow calculations and proves the positivity of a key quadratic form for trees with up to three endpoints.

Findings

Flow parameters' signs can be determined iteratively from initial moduli.

The quadratic form in the D4-D2-D0 mass expression is positive definite for trees with ≤3 endpoints.

The BPS partition function converges and is a generating function of rational invariants.

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.