Sphere Packings, Lattices and Groups

Several related operator-algebraic constructions for quantum field theory models on Minkowski spacetime are reviewed. The common theme of these constructions is that of a Borchers triple, capturing the structure of observables localized in a Rindler wedge. After reviewing the abstract setting, we discuss in this framework i) the construction of free field theories from standard pairs, ii) the inverse scattering construction of integrable QFT models on two-dimensional Minkowski space, and iii) the warped convolution deformation of QFT models in arbitrary dimension, inspired from non-commutative Minkowski space.

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Algebraic Constructive Quantum Field Theory: Integrable Models and Deformation Techniques

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, Poincare 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 non-linear 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.

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Towards an Operator-Algebraic Construction of Integrable Global Gauge Theories

Our previous constructions of Borchers triples are extended to massless scattering with nontrivial left and right components. A massless Borchers triple is constructed from a set of left–left, right–right and left–right scattering functions. We find a correspondence between massless left–right scattering S-matrices and massive block diagonal S-matrices. We point out a simple class of S-matrices with examples. We study also the restriction of two-dimensional models to the lightray. Several arguments for constructing strictly local two-dimensional nets are presented and possible scenarios are discussed.

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Integrable QFT and Longo–Witten Endomorphisms

The anomaly for the Monster group $${\mathbb{M}}$$ acting on its natural (aka moonshine) representation $${V^\natural}$$ is a particular cohomology class $${\omega^\natural \in {\rm H}^3(\mathbb{M},{\rm U}(1))}$$ that arises as a conformal field theoretic generalization of the second Chern class of a representation. This paper shows that $${\omega^\natural}$$ has order exactly 24 and is not a Chern class. In order to perform this computation, this paper introduces a finite-group version of T-duality, which is used to relate $${\omega^\natural}$$ to the anomaly for the Leech lattice CFT.

The Moonshine Anomaly

Part II. Construction of Longo–Witten unitaries and models in BQFT 47 Chapter 4. Longo–Witten unitaries for conformal nets associated with lattices and loop group models. . Introduction In algebraic quantum field theory (AQFT) [Haa96] one studies local nets (e.g. von Neumann algebras) that assign to a space-time region the algebra of observables localized in it. These nets are asked to fulfill certain axioms coming from basic physical principles; we mention as examples the locality principle—which asks that the algebras assigned to causally disjoint regions should commute (local nets)—and the covariant assignment with respect to some " symmetry group " of the space-time. This approach brought many conceptional and model independent results, but recently it seems to be useful for construction and classification of models. In the AQFT approach also conformal quantum field theory (CQFT) has been treated successfully by considering nets on two dimensional Minkowski space and its chiral parts, which can be regarded as nets on the real line or as nets on the circle. Some important achievements are classification results [KL04, Xu05] and new constructions [Xu07], which were not obtained in other approaches. An important tool is the the relation to subfactors: inclusions of von Neumann algebras with trivial centers. For example there is a notion of index of subfactors giving relations to the statistical dimensions of representations. For finite index subfactors and there are several invariants with combinatorial nature making classifications possible. A conformal net on the real line (or the circle) is a net (precosheaf), which assigns to each proper interval a von Neumann algebra on a fixed Hilbert space H. Locality is encoded by asking that the algebras of disjoint intervals commute, covariance that there is a positive energy representation of the Möbius group (PSL(2, R) PSU(1, 1)) such that A(gI) = U(g)A(I)U(g) *. Often these nets turn out to be also covariant with respect to Diff + (S 1) (the orientation preserving diffeomorphisms of the circle) and there is a subnet Vir giving a representation of the Virasoro algebra and one can associates a central charge c. The von Neumann algebras turn out to be type III 1 factor in Connes classification. Superselection sectors or Doplicher–Haag–Roberts (DHR) representations can be described by localized endomorphisms giving a connection to Jones theory subfactors. In contrast to the result of higher dimensions, where just Fermions and Bosons can exist, in low dimensions more general so-called anyons exist …

Marcel Bischo

Papers

Construction of Models in low-dimensional Quantum Field Theory using Operator Algebraic Methods

Semigroup elements associated to conformal nets and boundary quantum eld theory