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2d affine XY-spin model/ 4d gauge theory duality and deconfinement

Mohamed M. Anber, Erich Poppitz, Mithat Unsal

TL;DR

This work establishes a long-distance duality between 2d affine XY-spin models with symmetry-breaking perturbations and 4d SU(2) and SU(2)/Z_2 gauge theories compactified on small circles, using an electric-magnetic Coulomb gas framework to bridge spin and gauge dynamics. It shows that near deconfinement, vortices in the spin models map to W-bosons in the gauge theory while magnetic defects correspond to monopole-instantons and magnetic bions, enabling analytic control over confinement and chiral symmetry realization. For SU(2), the deconfinement transition aligns with a Z4 clock-model criticality at Tc ≈ g4^2/(8πL), with a finite but large correlation-length divergence governed by fugacities; for SU(3), a vector XY description on the root lattice captures the transition as a coupled-spin system, with e-m duality subtleties and a richer phase structure. Overall, the paper provides a unifying, analytically tractable connection between 2d spin systems and 4d gauge theories at finite temperature, offering new avenues to study deconfinement, chiral transitions, and symmetry realization in QCD(adj) and related theories.

Abstract

We introduce a duality between two-dimensional XY-spin models with symmetry-breaking perturbations and certain four-dimensional SU(2)and SU(2)/Z_2 gauge theories, compactified on a small spatial circle R^(1,2) x S^1, and considered at temperatures near the deconfinement transition. In a Euclidean set up, the theory is defined on R^2 x T^2. Similarly, thermal gauge theories of higher rank are dual to new families of "affine" XY-spin models with perturbations. For rank two, these are related to models used to describe the melting of a 2d crystal with a triangular lattice. The connection is made through a multi-component electric-magnetic Coulomb gas representation for both systems. Perturbations in the spin system map to topological defects in the gauge theory, such as monopole-instantons or magnetic bions, and the vortices in the spin system map to the electrically charged W-bosons in field theory (or vice versa, depending on the duality frame). The duality permits one to use the two-dimensional technology of spin systems to study the thermal deconfinement and discrete chiral transitions in four-dimensional SU(N) gauge theories with n_f >=1 adjoint Weyl fermions.

2d affine XY-spin model/ 4d gauge theory duality and deconfinement

TL;DR

This work establishes a long-distance duality between 2d affine XY-spin models with symmetry-breaking perturbations and 4d SU(2) and SU(2)/Z_2 gauge theories compactified on small circles, using an electric-magnetic Coulomb gas framework to bridge spin and gauge dynamics. It shows that near deconfinement, vortices in the spin models map to W-bosons in the gauge theory while magnetic defects correspond to monopole-instantons and magnetic bions, enabling analytic control over confinement and chiral symmetry realization. For SU(2), the deconfinement transition aligns with a Z4 clock-model criticality at Tc ≈ g4^2/(8πL), with a finite but large correlation-length divergence governed by fugacities; for SU(3), a vector XY description on the root lattice captures the transition as a coupled-spin system, with e-m duality subtleties and a richer phase structure. Overall, the paper provides a unifying, analytically tractable connection between 2d spin systems and 4d gauge theories at finite temperature, offering new avenues to study deconfinement, chiral transitions, and symmetry realization in QCD(adj) and related theories.

Abstract

We introduce a duality between two-dimensional XY-spin models with symmetry-breaking perturbations and certain four-dimensional SU(2)and SU(2)/Z_2 gauge theories, compactified on a small spatial circle R^(1,2) x S^1, and considered at temperatures near the deconfinement transition. In a Euclidean set up, the theory is defined on R^2 x T^2. Similarly, thermal gauge theories of higher rank are dual to new families of "affine" XY-spin models with perturbations. For rank two, these are related to models used to describe the melting of a 2d crystal with a triangular lattice. The connection is made through a multi-component electric-magnetic Coulomb gas representation for both systems. Perturbations in the spin system map to topological defects in the gauge theory, such as monopole-instantons or magnetic bions, and the vortices in the spin system map to the electrically charged W-bosons in field theory (or vice versa, depending on the duality frame). The duality permits one to use the two-dimensional technology of spin systems to study the thermal deconfinement and discrete chiral transitions in four-dimensional SU(N) gauge theories with n_f >=1 adjoint Weyl fermions.

Paper Structure

This paper contains 28 sections, 138 equations.

Table of Contents

  1. Introduction, summary, and outline
  2. Deconfinement in $\mathbf{SU(2)}$ QCD(adj)
  3. The theory of melting of 2d-crystals and $\mathbf{SU(3)}$ QCD(adj)
  4. Remarks on the affine XY-spin model and $\mathbf{SU(N_c > 3)}$
  5. The zero temperature dynamics of QCD(adj) on $\mathbf{{\mathbb R}^{1,2} \times {\mathbb S}^1_L}$
  6. Review of perturbative dynamics at zero temperature
  7. Review of the $\mathbf{SU(2)}$ non-perturbative dynamics at zero temperature
  8. Hamiltonian quantization of compact $\mathbf{U(1)}$, $\mathbf{SO(3)}$ vs. $\mathbf{SU(2)}$, discrete fluxes, and the topological $\mathbf{{\mathbb Z}_2}$ symmetry.
  9. Nonperturbative effects in the $\mathbf{SU(2)}$ theory on $\mathbf{{\mathbb R}^{1,2} \times {\mathbb S}^1_L}$.
  10. The $\mathbf{SU(2)}$(adj) theory at finite temperature and the $\mathbf{Z_p \; \; XY}$ model
  11. The thermal partition function as an electric-magnetic Coulomb gas
  12. The renormalization group equations for the magnetic bion/W-boson plasma and the approach to ${\mathbf T_c}$
  13. Dual descriptions and symmetry realizations above and below ${\mathbf T_c}$
  14. $\mathbf{{\mathbb Z}_p}$ XY models as Coulomb gases
  15. $\mathbf{{\mathbb Z}_p}$ XY-models as duals of the $\mathbf{W}$-boson/bion gas
  16. ...and 13 more sections