Refined, Motivic, and Quantum

TL;DR

This paper explores the behavior of refined BPS invariants under wall crossing in string theory, introduces a combinatorial model for their counting, and compares them with motivic Donaldson-Thomas invariants, revealing their deep interrelation.

Contribution

It provides a refined wall crossing formula for BPS invariants, introduces a new statistical model for refined pyramid partitions, and clarifies the relationship between refined, motivic, and quantum invariants.

Findings

Refined BPS degeneracies for the conifold are computed in various chambers.

A new combinatorial model for counting refined pyramid partitions is proposed.

The wall crossing behavior of refined BPS invariants aligns with that of motivic Donaldson-Thomas invariants.

Abstract

It is well known that in string compactifications on toric Calabi-Yau manifolds one can introduce refined BPS invariants that carry information not only about the charge of the BPS state but also about the spin content. In this paper we study how these invariants behave under wall crossing. In particular, by applying a refined wall crossing formula, we obtain the refined BPS degeneracies for the conifold in different chambers. The result can be interpreted in terms of a new statistical model that counts `refined' pyramid partitions; the model provides a combinatorial realization of wall crossing and clarifies the relation between refined pyramid partitions and the refined topological vertex. We also compare the wall crossing behavior of the refined BPS invariants with that of the motivic Donaldson-Thomas invariants introduced by Kontsevich-Soibelman. In particular, we argue that, in the context of BPS state counting, the three adjectives in the title of this paper are essentially synonymous.