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Quiver Invariants from Intrinsic Higgs States

Seung-Joo Lee, Zhao-Long Wang, Piljin Yi

TL;DR

This work investigates BPS states in quiver quantum mechanics by separating Higgs-phase ground states into those arising from pull-backs of ambient D-term cohomology and intrinsic Higgs states. The authors conjecture two key ideas: (1) Coulomb-phase states correspond bijectively to the pulled-back cohomology $i^*_M(H(X))$ and (2) the intrinsic Higgs index $\chi(M)-\chi(i^*_M(H(X)))$ is a quiver-invariant quantity independent of wall-crossing branches. They analytically verify the conjectures for 3-gon quivers and provide extensive numerical checks for 4-, 5-, and 6-gon quivers, finding that intrinsic states lie in the middle cohomology and are angular-momentum singlets for simple loops. The results link Higgs-phase counting to a robust geometric invariant of the quiver, with implications for single-center black hole microstate interpretations and wall-crossing physics.

Abstract

In study of four-dimensional BPS states, quiver quantum mechanics plays a central role. The Coulomb phases capture the multi-centered nature of such states, and are well understood in the context of wall-crossing. The Higgs phases are given typically by F-term-induced complete intersections in the ambient D-term-induced toric varieties, and the ground states can be far more numerous than the Coulomb phase counterparts. We observe that the Higgs phase BPS states are naturally and geometrically grouped into two parts, with one part given by the pulled-back cohomology from the D-term-induced ambient space. We propose that these pulled-back states are in one-to-one correspondence with the Coulomb phase states. This also leads us to conjecture that the index associated with the rest, intrinsic to the Higgs phase,is a fundamental invariant of quivers, independent of branches. For simple circular quivers, these intrinsic Higgs states belong to the middle cohomology and thus are all angular momentum singlets, supporting the single-center black hole interpretation.

Quiver Invariants from Intrinsic Higgs States

TL;DR

This work investigates BPS states in quiver quantum mechanics by separating Higgs-phase ground states into those arising from pull-backs of ambient D-term cohomology and intrinsic Higgs states. The authors conjecture two key ideas: (1) Coulomb-phase states correspond bijectively to the pulled-back cohomology and (2) the intrinsic Higgs index is a quiver-invariant quantity independent of wall-crossing branches. They analytically verify the conjectures for 3-gon quivers and provide extensive numerical checks for 4-, 5-, and 6-gon quivers, finding that intrinsic states lie in the middle cohomology and are angular-momentum singlets for simple loops. The results link Higgs-phase counting to a robust geometric invariant of the quiver, with implications for single-center black hole microstate interpretations and wall-crossing physics.

Abstract

In study of four-dimensional BPS states, quiver quantum mechanics plays a central role. The Coulomb phases capture the multi-centered nature of such states, and are well understood in the context of wall-crossing. The Higgs phases are given typically by F-term-induced complete intersections in the ambient D-term-induced toric varieties, and the ground states can be far more numerous than the Coulomb phase counterparts. We observe that the Higgs phase BPS states are naturally and geometrically grouped into two parts, with one part given by the pulled-back cohomology from the D-term-induced ambient space. We propose that these pulled-back states are in one-to-one correspondence with the Coulomb phase states. This also leads us to conjecture that the index associated with the rest, intrinsic to the Higgs phase,is a fundamental invariant of quivers, independent of branches. For simple circular quivers, these intrinsic Higgs states belong to the middle cohomology and thus are all angular momentum singlets, supporting the single-center black hole interpretation.

Paper Structure

This paper contains 8 sections, 62 equations, 1 figure, 3 tables.

Table of Contents

  1. Conjectures
  2. Higgs Phases of Quivers with a Loop
  3. Basics on $H(M)$ and $i^*_M(H(X))$ for Cyclic $(n+1)$-Gons
  4. Analytical Check: 3-Gons
  5. Numerical Evidences: 4-Gons, 5-Gons, and 6-Gons
  6. Comments
  7. Lefschetz Hyperplane Theorem
  8. An Exercise in Cohomology Counting

Figures (1)

  • Figure 2.1: A cyclic $(n+1)$-gon quiver consists of $n+1$ nodes, cyclically connected by directed arrows. Associated to each node is a $U(1)$ gauge group, whose FI constant is denoted by $\zeta_i$, and the $a_i$ arrows from the $i$-th node to the $(i+1)$-th node correspond to the $a_i$ bifundamental fields $Z_i = (Z_i^{(1)}, \cdots, Z_i^{(a_i)})$.