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Emergent Loop Geometry in Gauge-Equivariant Graph Neural Networks

arXiv 2610.18742 ↗First posted 13 Oct 2026cond-mat.stat-mech cond-mat.str-el cs.LGcited by 0

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Abstract (arXiv)

Gauge-equivariant graph neural networks generate ordered products of gauge links through local message passing, but the geometry of these internal structures is not captured by the receptive field alone. We investigate the statistical ensemble of closed lattice walks generated by recursive node- and edge-feature operations, independently of network training. Analytical and numerical results reveal a separation between rapid growth of contour length and much slower spatial expansion: deeper compositions generate increasingly long paths through repeated traversal of relatively compact regions. Loop expansions of fermionic observables provide a physical interpretation of this hierarchy, showing how spatial locality remains compatible with high-order dependence on nearby gauge fields. These findings identify complementary roles of network depth in extending spatial reach and enriching the representations constructed within that reach, providing a geometrical perspective on learning lattice gauge observables.

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References · 20

  1. 2017Inductive Representation Learning on Large Graphscited by 6,320
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  12. 2020Sampling using SU(N) gauge equivariant flowscited by 84
8 more
  1. 2022Gauge-equivariant flow models for sampling in lattice field theories with pseudofermionscited by 31
  2. 2020Gauge equivariant neural networks for quantum lattice gauge theoriescited by 28
  3. 2021Gauge Invariant and Anyonic Symmetric Transformer and RNN Quantum States for Quantum Lattice Modelscited by 18
  4. 2023Gauge-equivariant neural networks as preconditioners in lattice QCDcited by 9
  5. 2023Gauge-equivariant pooling layers for preconditioners in lattice QCDcited by 6
  6. 2024Deep learning lattice gauge theoriescited by 3
  7. 2026Gauge-Equivariant Graph Neural Networks for Lattice Gauge Theoriescited by 3
  8. 2026Graph Neural Networks in the Wilson Loop Representation of Abelian Lattice Gauge Theoriescited by 2

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