Towards an Operator-Algebraic Construction of Integrable Global Gauge Theories

Abstract

The recent construction of integrable quantum field theories on two-dimensional Minkowski space by operator-algebraic methods is extended to models with a richer particle spectrum, including finitely many massive particle species transforming under a global gauge group. Starting from a two-particle S-matrix satisfying the usual requirements (unitarity, Yang-Baxter equation, Poincaré and gauge invariance, crossing symmetry, ...), a pair of relatively wedge-local quantum fields is constructed which determines the field net of the model. Although the verification of the modular nuclearity condition as a criterion for the existence of local fields is not carried out in this paper, arguments are presented that suggest it holds in typical examples such as nonlinear O(N ) σ-models. It is also shown that for all models complying with this condition, the presented construction solves the inverse scattering problem by recovering the S-matrix from the model via Haag-Ruelle scattering theory, and a proof of asymptotic completeness is given.

$${{SO(d,1)}}$$ S O ( d , 1 ) -Invariant Yang–Baxter Operators and the dS/CFT Correspondence

“…Before presenting examples, we recall how an invariant Yang-Baxter operator R gives rise to an R-symmetric Fock space, following [LM95,Lec03,LS14]. In this Fock space construction, we consider an invariant Yang-Baxter operator R and call its group representation, conjugation, and Hilbert space V 1 , 1 , and H 1 , as these data enter on the one particle level.…”

“…It is clear that˜ n is a conjugation on H ⊗n 1 , and thanks to (R3), it commutes with P R n and thus restricts to H R n [LS14]. We call this restriction n :=˜ n | H R n .…”

“…Both examples can be combined by taking the inner symmetry group as G = O(N ), as one would do for describing the O(N )-models [LS14,Ala14].…”

“…2.1), and later restrict to G = SO ↑ (d, 1). Our exposition is related to [BLM11], where a scalar version of such models was presented, and [LS14], where a massive version with finite-dimensional representation V 1 was analyzed.…”

“…Furthermore, φ R,k ( f ) * = φ R, k ( f ) on an appropriate domain. We do not repeat the calculations from [LS14] here (see, however, Sect. 4 for a concrete version of these properties), but rather focus on the locality aspects.…”