Wall Crossing of BPS States on the Conifold from Seiberg Duality and Pyramid Partitions

TL;DR

This paper establishes a precise correspondence between pyramid partitions with specific configurations and BPS state counts of D-branes on the resolved conifold, revealing new insights into wall crossing phenomena via Seiberg duality.

Contribution

It introduces a novel connection between pyramid partitions and BPS state counting, utilizing Seiberg duality and quiver mutations to analyze wall crossing in the conifold setting.

Findings

Pyramid partition generating functions match D6/D2/D0 BPS partition functions in certain chambers.

Defined new pyramid partitions with finite ERC for counting BPS degeneracies in other chambers.

Confirmed quiver structures and superpotential rules through Ext group computations.

Abstract

In this paper we study the relation between pyramid partitions with a general empty room configuration (ERC) and the BPS states of D-branes on the resolved conifold. We find that the generating function for pyramid partitions with a length n ERC is exactly the same as the D6/D2/D0 BPS partition function on the resolved conifold in particular Kaehler chambers. We define a new type of pyramid partition with a finite ERC that counts the BPS degeneracies in certain other chambers. The D6/D2/D0 partition functions in different chambers were obtained by applying the wall crossing formula. On the other hand, the pyramid partitions describe T^3 fixed points of the moduli space of a quiver quantum mechanics. This quiver arises after we apply Seiberg dualities to the D6/D2/D0 system on the conifold and choose a particular set of FI parameters. The arrow structure of the dual quiver is confirmed by computation of the Ext group between the sheaves. We show that the superpotential and the stability condition of the dual quiver with this choice of the FI parameters give rise to the rules specifying pyramid partitions with length n ERC.