Global structure and physical content of the Longo-Witten deformed boundary quantum field theory nets: relation to Lechner's factorizing-scattering models, minimal regularity, covariance, and global classification

Longo and Witten (Commun. Math. Phys. 303 (2011) 213-232) constructed local, time-translation covariant nets of von Neumann algebras on the Minkowski half-plane from a chiral conformal net and a unitary $V$ in a semigroup $E(\mathcal{A})$ of half-line endomorphisms, parameterized (for the $U(1)$-current net) by symmetric inner functions or matrix-valued analytic data. Under the split property these deformed boundary nets are locally isomorphic to the conformal boundary net, but their physical content was left undetermined: the paper explicitly asks whether the construction is related to Lechner's factorizing-scattering models of two-dimensional QFT ("at the moment there is no link between the two constructions"), assumes a Hoelder-type regularity of the inner function at zero without identifying the minimal requirement, produces nets that are only time-translation covariant, and does not analyze the global (non-local) structure of the deformed nets. Later work (Tanimoto 2012-2014, Bischoff-Tanimoto 2013-2015, Lechner-Schlemmer-Tanimoto 2013) built a substantial bridge between Longo-Witten endomorphisms and factorizing S-matrix constructions in the two-dimensional wedge-local, mostly massless setting, and proved the two deformation schemes equivalent there; however, the relation between the boundary nets themselves and Lechner's massive models was still explicitly "unclear" as of 2015, strict locality of the Longo-Witten-based massive family is open, and the minimal regularity, covariance characterization, and global classification of the boundary nets (superselection structure, modular invariants, global equivalence to the undeformed net) have no found solution. The problem asks to determine the physical content of the deformed boundary nets along these four jointly linked axes.

Background

In the operator-algebraic approach to quantum field theory (also called algebraic QFT, AQFT), a model is a Haag-Kastler net: an assignment $\mathcal{A}(O)$ of von Neumann algebras (algebras of bounded operators on a Hilbert space, closed under taking adjoints and under weak operator limits) of observables to open spacetime regions $O$, subject to isotony (bigger regions have bigger algebras), locality (algebras of spacelike separated regions commute), and covariance under a unitary representation of the spacetime symmetry group with a vacuum vector. A chiral conformal field theory is described in this framework by a local Moebius covariant net on the circle $S^1$ (or the real line $\mathbb{R}$, obtained by deleting a point): the Moebius group $PSL(2,\mathbb{R})$ acts projectively, with positive energy and a unique vacuum. Boundary quantum field theory (BQFT) studies theories on the Minkowski half-plane $M_+=\{(t,x):x>0\}$, whose timelike boundary plays the role of a reflecting boundary condition; the conformal case was developed by Longo and Rehren via the net $\mathcal{A}_+$ obtained by folding the two lightlike components of a chiral net.

Longo and Witten (2011) gave a general algebraic method to produce boundary nets beyond the conformal ones. Starting from a local Moebius covariant net $\mathcal{A}$ on $\mathbb{R}$ (viewed as the time axis of $M_+$) and a unitary $V$ on the chiral Hilbert space that commutes with time translations, preserves the vacuum, and maps the algebras of future lightlike intervals into themselves compatibly with locality, they define, for a double cone (rectangle) $O=I_-\times I_+$ in $M_+$,

$$\mathcal{A}_V(O)=\mathcal{A}(I_-)\vee V\mathcal{A}(I_+)V^*,$$

where $\vee$ denotes the von Neumann algebra generated by the two. The admissible unitaries form a semigroup $E(\mathcal{A})$; its elements implement endomorphisms of the half-line algebra $\mathcal{A}(0,\infty)$ and are now called Longo-Witten endomorphisms. The case $V=1$ recovers the Moebius covariant net $\mathcal{A}_+$; general $V$ give nets that are local and time-translation covariant but not conformally covariant. A key structural result of the paper is an abstract analog of the Beurling-Lax theorem from Hardy-space theory (the Hardy spaces being the spaces of bounded analytic functions on a disk or half-plane): at the one-particle level, i.e. for a non-degenerate standard pair (a closed real subspace $H$ of a complex Hilbert space that is standard, i.e. $H+iH$ is dense and $H\cap iH=\{0\}$, together with a positive-energy unitary translation group $T$), the semigroup $E(H,T)$ consists, in the irreducible case, exactly of the unitaries $\varphi(P)$, where $P$ is the generator of $T$ and $\varphi$ is a symmetric inner function on the upper half-plane $S_\infty$ (a bounded analytic function with boundary values of modulus one a.e. satisfying $\overline{\varphi(p)}=\varphi(-p)$); in the reducible case the unitaries are matrices $(\varphi_{hk}(P))$ of such boundary values that are unitary matrices for a.e. $p>0$. Via second quantization (lifting a one-particle unitary to the bosonic Fock space, the direct sum of the symmetric tensor powers of the one-particle space) this yields large families of boundary nets for the $U(1)$-current net (the chiral net generated by a free massless bosonic current), for any Buchholz-Mach-Todorov (BMT) local extension of it (coset models $SO(4N)_1/SO(2N)_2$), and for the Ising net. Two restrictions appear in the construction: the extension of the endomorphism to BMT extensions assumes that $|\varphi(p)-1|^2/|p|$ is locally integrable at $p=0$ (called Hoelder continuity at zero), and for a chiral net with the split property (the property that algebras of intervals with disjoint closures generate a tensor product, implied by a summable vacuum character) each $\mathcal{A}_V$ is locally isomorphic to $\mathcal{A}_+$ (Proposition 3.5 of the paper).

A scattering function is a symmetric inner function on the strip of width $\pi$ with the extra symmetry $S_2(-q)=S_2(q)$; such functions are the two-particle scattering data of two-dimensional integrable models with factorizing S-matrices (no particle production, scattering factorized into two-particle phases). Lechner (2008) rigorously constructed interacting Haag-Kastler nets on the full two-dimensional Minkowski plane from a scattering function satisfying regularity and $S_2(0)=-1$ (fermionic) conditions, realizing the inverse-scattering program: the wedge-local observable algebra is built from Zamolodchikov-type deformed fields (deformations of free fields modeled on the Zamolodchikov-Faddeev algebra that organizes factorized scattering states), and strict locality (existence of enough double-cone-localized observables) follows from a modular nuclearity criterion (a bound, formulated with Tomita-Takesaki modular operators, on the size of the set of states localized in a bounded region). In this framework a two-dimensional model can equivalently be encoded by a Borchers triple (a single wedge algebra with a positive-energy translation representation and a vacuum vector), by Borchers' theorem; passing from wedge-local to strictly local nets via intersections is the difficult step.

The two constructions share striking analytic data: the same inner functions $\varphi$ (or their matrix-valued generalizations) that parameterize Longo-Witten endomorphisms and the deformed boundary nets also include, as a special symmetric subclass, the scattering functions driving Lechner's models. Longo and Witten explicitly left open whether this is a coincidence or a structural correspondence: "One may wonder whether our construction is related to Lechner models on the 2-dimensional Minkowski spacetime associated with a scattering function, yet at the moment there is no link between the two constructions." Three further structural questions about the deformed nets arise directly from the paper's results and assumptions: the minimal regularity of $\varphi$ at zero needed for the extension constructions (the Hoelder condition is assumed, not shown necessary); the characterization of the special choices of $V$ (if any beyond phases) for which $\mathcal{A}_V$ is Moebius or fully conformally covariant; and the global classification of the nets $\mathcal{A}_V$ beyond their proved local isomorphism to $\mathcal{A}_+$, i.e. their superselection structure (the inequivalent irreducible representations, or charge sectors, of the observable algebra), modular invariants (invariants of how chiral components combine, classically encoded as torus partition functions invariant under the modular group $SL(2,\mathbb{Z})$), spectra of conserved quantities, and global equivalence classes - in particular whether the deformation is visible at all at the global level.

Problem Statement

Determine the physical content of the Longo-Witten deformed boundary quantum field theory nets $\mathcal{A}_V$ on the Minkowski half-plane, constructed from a local Moebius covariant net $\mathcal{A}$ with split property and a unitary $V\in E(\mathcal{A})$ (parameterized, for the $U(1)$-current net, at the one-particle level by symmetric inner functions or matrices of symmetric analytic functions), beyond the local isomorphism with the conformal boundary net $\mathcal{A}_+$ proved in the source paper. This comprises four jointly linked questions:

(1) Relation to factorizing-scattering models: establish, or refute, a precise correspondence between the Longo-Witten construction (the boundary nets $\mathcal{A}_V$ and the Longo-Witten endomorphisms parameterized by inner-function data) and Lechner's models of two-dimensional interacting QFT associated with scattering functions - including whether the boundary nets arise as boundary restrictions/boundary conditions of Lechner-type bulk models, and which S-matrix data (if any) the deformation of the boundary net encodes. The source paper explicitly states that no link between the two constructions was known; later work has proved equivalence of the associated two-dimensional deformation schemes in the massless wedge-local setting, but the boundary-level and massive-level correspondence remains open. The constituent claims of this question are ordered by implication, the existence of a correspondence map being the weakest: the realization of the boundary nets as boundary restrictions/boundary conditions of Lechner-type bulk models implies it, and the identification of the S-matrix data (if any) encoded by the deformation presupposes it. Settling a constituent therefore settles every constituent it entails: refuting the existence of any correspondence map settles the question in the negative at once, while an affirmative answer requires the two substantive constituents - the boundary-restriction realization and the encoded S-matrix data - which are independent of each other.

(2) Minimal regularity: identify the precise minimal regularity of the symmetric inner function $\varphi$ at $p=0$ needed for the extension constructions of the source paper (which assume that $|\varphi(p)-1|^2/|p|$ is locally integrable at zero) and for the resulting net to be local, time-translation covariant and to have the split property - that is, the weakest condition on $\varphi$ under which the extension constructions go through and the resulting net has all of these properties.

(3) Covariance: characterize the choices of $V$ (inner functions or matrix-valued data) for which the deformed net $\mathcal{A}_V$ is Moebius covariant, and determine the additional constraints required for full conformal covariance.

(4) Global classification: since $\mathcal{A}_V$ is locally isomorphic to $\mathcal{A}_+$ under the split property, classify the nets $\mathcal{A}_V$ up to global (non-local) equivalence - their superselection structure, modular invariants, spectra of conserved quantities, and other global invariants - deciding in particular whether the deformation is globally visible at all (whether some or all $\mathcal{A}_V$ are globally isomorphic to $\mathcal{A}_+$).

The common objective of the four questions (1)-(4) is whether and how the Longo-Witten deformation produces genuinely distinct (interacting, or at least globally non-equivalent) boundary QFTs, and where these models sit relative to the known constructions of interacting two-dimensional quantum field theories.

References

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