Galois Covers of Calabi-Yau Quivers and BPS State Counting

Abstract

BPS quivers are central to our understanding of BPS states in 4d $\mathcal{N}=2$ supersymmetric field theories and of D-branes at Calabi-Yau threefold singularities. The two subjects are deeply interrelated through geometric engineering in Type II string theory, where a CY$_3$ quiver, also known as a 5d BPS quiver, describes fractional branes at a threefold singularity ${\bf X}$. We study the Galois cover ${\skew{2}\tilde Q}\rightarrow Q$ of any BPS quiver $Q$ by a finite abelian group $\mathbb{G}$, leading to a covering quiver ${\skew{2}\tilde Q}$. The Galois cover is determined by a $\mathbb{G}$-grading of the arrows of the quiver $Q$, which can be understood as an orbifolding procedure. In particular, if $Q$ is a CY$_3$ quiver for ${\bf X}$, then the Galois cover $\skew{2}\tilde Q$ is the CY$_3$ quiver for the orbifold singularity ${\bf X}/\mathbb{G}$. We explore such Galois covering procedures in the language of supersymmetric quiver quantum mechanics, in terms of fixed loci under $\mathbb{G}$ actions on moduli spaces of quiver representations, and in terms of homomorphisms between the Kontsevich-Soibelman algebras of $Q$ and ${\skew{2}\tilde Q}$. Our main result is an explicit covering formula for the BPS invariants of 4d $\mathcal{N}=2$ field theories, wherein the rational BPS invariant $\bar{\Omega}^Q(\gamma)$ of $Q$ is expressed as a sum of BPS invariants of $\skew{2}\tilde Q$. We derive this formula in various special cases, which include the case when $\gamma$ is a primitive charge vector, the case of general charge vectors for quivers without loops, and the case of CY$_3$ quivers for some simple geometries such as the conifold or local del Pezzo surfaces. The general formula is presented as a conjecture that can be verified in many examples.