Abstract
Abstract We introduce a duality between two-dimensional XY-spin models with symmetry-breaking perturbations and certain four-dimensional SU(2) and SU(2)/ $ {\mathbb{Z}_2} $ gauge theories, compactified on a small spatial circle $ {\mathbb{R}^{{^{{{1},{2}}}}}} $ × $ {\mathbb{S}^{{^{{1}}}}} $ , and considered at temperatures near the deconfinement transition. In a Euclidean set up, the theory is defined on $ {\mathbb{R}^{{^{{2}}}}} $ × $ {\mathbb{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 c ) gauge theories with n f ≥1 adjoint Weyl fermions.
Highlights
It is well-known since the late 70’s that two-dimensional (2d) XY-spin models with appropriate symmetry-breaking perturbations map to a 2d Coulomb gas with electric and magnetic charges [1]
We introduce a duality between two-dimensional XY-spin models with symmetry-breaking perturbations and certain four-dimensional SU(2) and SU(2)/Z2 gauge theories, compactified on a small spatial circle R1,2 × S1, and considered at temperatures near the deconfinement transition
We introduce a long distance duality between 2d XYspin models and certain four-dimensional (4d) gauge theories compactified on R2 × T2, with boundary conditions as specified below
Summary
We consider four-dimensional (4d) SU(Nc) Yang-Mills theory with nf massless Weyl fermions in the adjoint representation, a class of QCD-like (vector) theories usually denoted QCD(adj). Tunneling events between states of zero fermion number that change the flux sectors as |n → |n + 2 , do occur This effect is a descendant of magnetic bions, a certain type of topological molecule which changes the magnetic flux by two units At the T mW temperatures of interest, the W -boson partition function is that of a 2d gas of electrically charged non-relativistic particles, which interact via Coulomb forces with themselves (as well as with the magnetically charged objects, the bions, as described below).
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