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 $\barΩ^Q(γ)$ 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 $γ$ 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.

Figures (14)

• Figure 1: The Kronecker quivers $K_n$ for $n=2,3$. The notation $\gamma_j$ is explained in the main text.
• Figure 2: Examples of Galois covers of the Kronecker quiver $Q=K_2$. Here, the nodes $j_\alpha \in \tilde{Q}_0$ are indexed by $\gamma_{j, \alpha}$. The arrow gradings $d_a$ are indicated in each case.
• Figure 3: Galois pair of the BPS quivers of the 4d $N_f=2$ (Left) and $N_f=0$ theories (Right). Note that we sometimes use the obvious notation of writing multiple arrows on top of each other.
• Figure 4: Galois pair of the BPS quivers of the 5d $D_{S^1} E_6$ (Left) and $D_{S^1} E_0$ (Right) theories. Note that the $E_6$ quiver is written in the block notation, which means that each arrow between from a block of $m_1$ nodes to a block of $m_2$ nodes means that there are $m_1 m_2$ arrows that connect each node in the first block to each node in the second block, with no arrows within a block --- hence this $E_6$ quiver contains 9 nodes and 18 arrows.
• Figure 7: Left:$\mathbb{Z}_2$-cover of the $\mathbb{F}_0$ quiver, giving us the quiver for 5d $\text{SU}(4)_0$ SYM. Right:$\mathbb{Z}_3$-cover of the $\mathbb{F}_0$ quiver, giving us the quiver for 5d $\text{SU}(6)_0$ SYM.