BPS States, Refined Indices, and Quiver Invariants

TL;DR

This work proves two refined-index conjectures for cyclic Abelian quivers: first that the Coulomb phase equivariant index Omega_Coulomb^(k)(y) equals (-y)^{-d_k}D_k(-y), i.e. the pulled-back ambient cohomology in the Higgs phase reproduces the Coulomb-phase counting, and second that the intrinsic Higgs sector yields a quiver-invariant refined index, with the full Higgs index Omega_Higgs^(k)(y) decomposing into a pulled-back part and a k-independent intrinsic part. The authors develop a complete Higgs phase computation using complete-intersection moduli spaces M_k, Lefschetz symmetry arguments, and a refined Euler character chi_xi(M_k), and they prove the second conjecture by showing the intrinsic part is invariant under wall-crossing via contour-integral techniques. They further provide explicit lattice-based counts for D_k(-1) and illustrate the refined Hodge structures through numerical examples, demonstrating large intrinsic degeneracies and detailed R-symmetry organization of Intrinsic Higgs states. Collectively, the results offer a concrete, refined classification of BPS states in quiver quantum mechanics, link Coulomb and Higgs phase data, and suggest deeper connections to wall-crossing formalisms and black hole microstate interpretations.

Abstract

For D=4 BPS state construction, counting, and wall-crossing thereof, quiver quantum mechanics offers two alternative approaches, the Coulomb phase and the Higgs phase, which sometimes produce inequivalent counting. The authors have proposed, in arXiv:1205.6511, two conjectures on the precise relationship between the two, with some supporting evidences. Higgs phase ground states are naturally divided into the Intrinsic Higgs sector, which is insensitive to wall-crossings and thus an invariant of quiver, plus a pulled-back ambient cohomology, conjectured to be an one-to-one image of Coulomb phase ground states. In this note, we show that these conjectures hold for all cyclic quivers with Abelian nodes, and further explore angular momentum and R-charge content of individual states. Along the way, we clarify how the protected spin character of BPS states should be computed in the Higgs phase, and further determine the entire Hodge structure of the Higgs phase cohomology. This shows that, while the Coulomb phase states are classified by angular momentum, the Intrinsic Higgs states are classified by R-symmetry.

Figures (5)

• Figure 3.1: The figure, borrowed from Ref. Lee:2012sc, shows a cyclic Abelian quiver with $n+1$ nodes. Associated with each node is a FI constant $\zeta_i$ and a $U(1)$ gauge field. Arrows between $i$-th and $(i+1)$-th nodes represent $a_i$ chiral multiplets of charge $(-1,1)$, say $Z_i = (Z_i^{(1)}, \cdots, Z_i^{(a_i)})$.
• Figure 5.1: A pictorial representation of the Poincaré polynomials for a 3-gon quiver when $d_i \geq -1$ for $i=1,2,3$. Two mutually-inverted equilateral triangles $O_1 O_2 O_3$ and $\tilde{O}_1 \tilde{O}_2 \tilde{O}_3$ are placed in a triangular lattice, overlapping with each other at a shaded hexagonal region. In each branch $k$, $E_k = \frac{ a_1 a_2 a_3}{a_k} - D_k(-1)$ counts the number of lattice points inside this hexagon.
• Figure 5.2: A pictorial representation of the Poincaré polynomials for a 3-gon quiver when $d_3 < -1$. Two mutually-inverted equilateral triangles $O_1 O_2 O_3$ and $\tilde{O}_1 \tilde{O}_2 \tilde{O}_3$ are placed in a triangular lattice, overlapping with each other at the shaded parallelogram. In each branch $k$, $E_k = \frac{ a_1 a_2 a_3}{a_k} - D_k(-1)$ counts the number of lattice points inside this parallelogram. In particular, $D_3(-1)=0$ and $E_3=a_1 a_2$ in branch 3.
• Figure 5.3: A pictorial representation of the Poincaré polynomials for a 4-gon quiver when $d_i \geq -1$ for $i=1,2,3,4$.
• Figure 5.4: A pictorial representation of the Poincaré polynomials for a 4-gon quiver when $d_i \geq -1$ for $i=1,2,3,4$. A hyper-parallelogram with sides of lengths $a_i -1$ has been added to Figure \ref{['fig:tetrahedron']}. Counting of the lattice points inside the region with thick edges gives $E_k = \frac{ a_1 a_2 a_3}{a_k} - D_k(-1)$ in any branches $k$.