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Nonabelian Lattice Weak Gravity Conjecture and Monopole Confinement

Matthew Reece, Tom Rudelius

Abstract

Within the known landscape of quantum gravity, most theories satisfy the Lattice Weak Gravity Conjecture (LWGC), which requires a superextremal particle at every site in the electric charge lattice $Γ$. However, counterexamples to the LWGC exist, and it was recently hypothesized that such counterexamples necessarily feature fractionally charged confined monopoles. In this work, we verify this hypothesis in toroidal orbifold compactifications of the heterotic string, which notably feature LWGC violation in both the abelian and nonabelian gauge sectors. In all the cases we consider, there exists a discrete subgroup of the center of the gauge group $K \subseteq Z(G)$ such that superextremal particles exist at every site in the charge lattice of the quotient group $G/K$, while (confined) monopoles exist at all sites in the magnetic charge lattice of $G/K$. This suggests that LWGC violation cannot occur for gauge groups with trivial centers, and more generally the degree of LWGC violation in a nonabelian gauge theory is bounded in terms of the maximal order of the center.

Nonabelian Lattice Weak Gravity Conjecture and Monopole Confinement

Abstract

Within the known landscape of quantum gravity, most theories satisfy the Lattice Weak Gravity Conjecture (LWGC), which requires a superextremal particle at every site in the electric charge lattice . However, counterexamples to the LWGC exist, and it was recently hypothesized that such counterexamples necessarily feature fractionally charged confined monopoles. In this work, we verify this hypothesis in toroidal orbifold compactifications of the heterotic string, which notably feature LWGC violation in both the abelian and nonabelian gauge sectors. In all the cases we consider, there exists a discrete subgroup of the center of the gauge group such that superextremal particles exist at every site in the charge lattice of the quotient group , while (confined) monopoles exist at all sites in the magnetic charge lattice of . This suggests that LWGC violation cannot occur for gauge groups with trivial centers, and more generally the degree of LWGC violation in a nonabelian gauge theory is bounded in terms of the maximal order of the center.
Paper Structure (21 sections, 82 equations, 3 figures, 1 table)

This paper contains 21 sections, 82 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Review: Magnetic Monopoles
  3. Magnetic monopoles in nonabelian gauge theories
  4. Example: $SU(n)/\mathbb{Z}_k$
  5. Confined monopoles in compactifications with Wilson lines
  6. Example 1: 9d Heterotic String Theory with a $\mathbb{Z}_2$ Wilson Line
  7. Reminder: the 10d heterotic theory
  8. Compactifying to 9d with a twist
  9. Case 1: ${\vec{q}} \in \Gamma_e$.
  10. Case 2: ${\vec{q}} \in \Gamma_o$.
  11. Example 2: A $T^4/ \mathbb{Z}_3$ Orbifold
  12. Untwisted sector
  13. Twisted sectors
  14. Gauge group
  15. LWGC violation
  16. ...and 6 more sections

Figures (3)

  • Figure 1: An unwrapped twist vortex acts as a domain wall after circle reduction. Here, the wavy line indicates an unphysical branch cut, which plays a role similar to that of the Dirac string. Figure adapted from Etheredge:2025rkn.
  • Figure 2: A magnetic monopole in a domain without a Wilson line may be pushed into the domain with the Wilson line by deforming the domain wall between them. Collapsing the tube surrounding the monopole produces a codimension-2 flux tube, hence the monopole is confined in the theory with the Wilson line. Figure adapted from Etheredge:2025rkn.
  • Figure 3: A collection of $p$ confined monopoles may join to form an unconfined monopole of charge $p$. Figure adapted from Etheredge:2025rkn.