Topological dualities in the Ising model

• DOI: 10.2140/gt.2022.26.1907
• Publication Date: 2022-12-12T09:18:11Z
• Authors: Daniel S Freed; Constantin Teleman
• Keywords: Lattice gauge theory

We relate two classical dualities in low-dimensional quantum field theory: Kramers-Wannier duality of the Ising and related lattice models 2 dimensions, with electromagnetic for finite gauge theories 3 dimensions.The relation is mediated by notion boundary are pure theory one dimension higher.Thus order/disorder operators endpoints Wilson/'t Hooft defects theory.Symmetry breaking on low-energy states reflects multiplicity topological states.In process we describe as (extended) boundaries domain walls.This allows us to generalize non-abelian groups; finite, semi-simple Hopf algebras; and, a different direction, homotopy arbitrary dimension.In statistical mechanics, 2-dimensional model has earned double distinction being first discrete exhibit, against expectations, phase transitions large volume limit [P], non-trivial be solved explicitly [O].It simplest sigma-model (in apocryphal terminology) only nearest-neighbor interactions, depends single parameter, physically interpreted temperature T , encoded reciprocal β " 1{kT (with Boltzmann's k).Nowadays, detailed treatments can found graduate textbooks [ID, C].The present paper our mathematical attempt understand some features story locate them within algebraic structures (TQFT).The assigns possible (spins valued ˘1) each node lattice.Equal spins nearby nodes probabilistically favored, strongly or weakly, according β.For very lattice, system exhibits phases: ferromagnetic at low temperature, where mostly aligned, sign dominating; paramagnetic high regions both signs co-exist.The also 2D Euclidean (lattice) interpretation, space H assigned any "latticed" circle (subdivided into edges nodes).Specifically, functions set independent spin assignments nodes, acted upon transfer matrix, analogue exponentiated negative-signed Hamiltonian cylindrical space-time.At its top eigenvalue achieved aligned states, δ-functions constant maps t˘1u.At eigenvector function all configurations, matching fact that no particular configuration favored.The global / µ . .2 t˘1u symmetry, acting simultaneously spins, cold symmetry breaking: choosing live near other