Intertwiner existence and modular nuclearity for multi-component factorizing S-matrix models (including the O(N) nonlinear sigma models)

In the operator-algebraic inverse-scattering program for two-dimensional integrable quantum field theories, wedge-local models can be constructed from any regular factorizing S-matrix, including multi-component ones with matrix-valued S. Proving that their double-cone (local) observable algebras are nontrivial, however, rests on the modular nuclearity condition, whose verification for genuinely matrix-valued S depends on an unproven intertwiner existence conjecture. The conjecture asks for multiplication operators that are holomorphic on shrinking tube domains, intertwine the S-symmetrized n-particle representation of the permutation group with the antisymmetric one, and have operator norms (and those of their inverses) growing at most exponentially in the particle number n. It is known for scalar and for diagonal S-matrices, and for the O(N) sigma-model S-matrices only up to n = 2. The open problem — posed in S. Alazzawi's PhD thesis (2015) and elaborated by Alazzawi and Lechner (2017) — is to establish or refute the intertwiner property for regular matrix-valued factorizing S-matrices, which by the nuclearity theorems of the sources implies modular nuclearity for the associated wedge-local models (for double cones above a minimal size $s_{\min}$; only this one-way implication is proven), or to establish or refute that modular nuclearity by any other method, with the massive O(N) nonlinear sigma models as the central test case. The problem remains open.

Background

In two space-time dimensions, relativistic quantum field theories with infinitely many conservation laws ("integrable" models) have elastic scattering without particle production: the n-particle S-matrix factorizes into two-particle factors. Each one-particle state is parametrized by a rapidity $\theta$, related to energy and momentum by $E = m\cosh\theta$, $p = m\sinh\theta$, and the two-particle scattering operator is a function $S(\theta_1 - \theta_2)$ acting on the internal degrees of freedom of the two-particle tensor product. Zamolodchikov and Zamolodchikov determined exact factorizing S-matrices for many such models, including the O(N)-invariant nonlinear sigma models. The inverse scattering problem asks to reverse this: given a factorizing S-matrix satisfying the Yang–Baxter equation (the consistency condition for factorized multi-particle scattering), unitarity, hermitian analyticity, crossing symmetry, PCT invariance, translational and gauge invariance, and standard strip analyticity, construct an interacting quantum field theory whose scattering operator is the given S.

The most successful rigorous route is operator-algebraic. In algebraic quantum field theory (AQFT), a theory is a net of von Neumann algebras of observables localized in spacetime regions, and constructing the theory means producing this net. Lechner (2008) solved the inverse scattering problem for scalar (single-species) regular S-matrices: starting from the Zamolodchikov–Faddeev (ZF) algebra — the quadratic exchange relations $z^{\dagger}(\theta_1) z^{\dagger}(\theta_2) = z^{\dagger}(\theta_2) z^{\dagger}(\theta_1) S(\theta_2 - \theta_1)$ obeyed by creation operators, generalizing the canonical commutation and anticommutation relations — one obtains fields localized only in Rindler wedges $W_R$, $W_L$ (the wedge-local fields). Lechner and Schützenhofer (2014) extended this wedge-local construction to multi-component particle spectra: finitely many massive particle species transforming under an arbitrary compact global gauge group, so the one-particle space $\mathcal{K}$ is finite-dimensional with $\dim\mathcal{K} > 1$ and $S(\theta)$ is a genuinely matrix-valued function. The vacuum Hilbert space is an S-symmetrized Fock space: on the n-particle tensor power $\mathcal{H}_1^{\otimes n}$ the permutation group $\mathfrak{S}_n$ acts through an S-dependent representation $D_n$ generated by the tensor flip combined with S-factors, replacing the Bose and Fermi symmetrizers.

Wedge locality is not yet a local theory: observables of direct physical interest are localized in bounded regions (double cones $\mathcal{O} = W_R \cap (W_L + x)$). The double-cone algebra is defined as the intersection of wedge algebras $\mathcal{A}(W_R) \cap \mathcal{A}(W_L + x)$, but it could a priori be trivial (only multiples of the identity). The tool ruling this out is the modular nuclearity condition of Buchholz, D'Antoni and Longo. A linear map between Banach spaces is nuclear if it is a sum of rank-one maps with summable norms (an operator analogue of trace-class maps); nuclearity of the map $\mathcal{A}(W_R) \ni A \mapsto \Delta^{1/4} U(x) A \Omega$ — where $\Delta$ is the Tomita–Takesaki modular operator of the wedge algebra, whose action coincides with Lorentz boosts by the Bisognano–Wichmann property, $U(x)$ are translations, and $\Omega$ is the vacuum — implies that the vacuum is cyclic for the double-cone algebras (a Reeh–Schlieder property) and that the net has the split property (algebras of spacelike separated regions factorize through an intermediate tensor product), so in particular the local algebras are nontrivial.

In the factorizing-S setting this nuclearity question becomes a problem in the complex analysis of several variables. The n-particle vacuum coefficient functions $(A\Omega)_n(\theta_1, \dots, \theta_n)$ of wedge-algebra elements in the ZF basis continue analytically to tube domains in $\mathbb{C}^n$ with suitable Hardy-type bounds (Hardy spaces on tubes; this is called property (H)), which Alazzawi and Lechner proved for all regular S-matrices. Property (H) yields nuclearity of each n-particle component $\Xi_n(s)$ of the nuclearity map, but with bounds growing like $n^n$, which are not summable over n. The intertwiner property is the device that repairs summability: it demands multiplication operators $I_n$, acting on the internal tensor factor $\mathcal{K}^{\otimes n}$, that intertwine the S-dependent permutation representation $D_n$ with the representation $D^-_n$ in which S is replaced by the negative tensor flip $-F$ (for the class considered here a regular S takes the value $S(0) = -F$ at zero rapidity transfer, so antisymmetric statistics is reached by analytic continuation). Passing to antisymmetric functions improves the nuclearity bounds by a factor of order $1/n!$, a Pauli-type improvement, making the bounds summable over n and yielding nuclearity of the full map — for double cones above a minimal size $s_{\min} > 0$.

The status of the intertwiner property defines the open problem. For scalar S ($\mathcal{K} = \mathbb{C}$) it is known from Lechner's original work via a product-of-square-roots formula, although that nuclearity proof contained a mistake (repaired in Refs. 1 and 2), and the repaired arguments cover localization regions only above a minimal size $s_{\min}$. Alazzawi and Lechner proved the intertwiner property for regular diagonal S-matrices with symmetric coefficients. For the O(N)-invariant nonlinear sigma-model S-matrices (a θ-dependent linear combination of three commuting O(N)-invariant operators on $\mathbb{C}^N \otimes \mathbb{C}^N$ — the identity, the tensor flip $F$, and the rank-one projection $P$ onto the $O(N)$-invariant singlet vector — with scalar coefficient functions $\sigma_1, \sigma_2, \sigma_3$ built from Gamma functions; see eq. (5.14) of Alazzawi–Lechner 2017; it is regular and satisfies $S_N(0) = -F$), intertwiners were constructed only up to particle number $n = 2$: the natural ansatz expands $I_n$ in the explicitly known basis of the invariant algebra $\mathrm{End}_{O(N)}((\mathbb{C}^N)^{\otimes n})$, which reduces the problem to bounded holomorphic scalar coefficient functions, but their analytic existence for $n \ge 3$ is open — already $(\mathbb{C}^N)^{\otimes 3}$ fails to be multiplicity-free under O(N) (some irreducible O(N)-representations occur with multiplicity, so the invariant algebra is non-commutative), complicating the diagonalization of the $\mathfrak{S}_n$-action. "Regular" here means: S satisfies the full S-matrix axioms listed above (unitarity, hermitian analyticity, Yang–Baxter equation, crossing symmetry, PCT invariance, translational and gauge invariance) and has a bounded analytic continuation to the enlarged strip $-\kappa < \mathrm{Im}\,\theta < \pi + \kappa$ for some $0 < \kappa < \pi/2$, hence no poles there; the class $\mathcal{S}^-_0$ is the subclass of regular S-matrices with $S(0) = -F$ ($F$ the tensor flip), to which the O(N) models belong.

Problem Statement

The problem is to complete — or refute — the verification of the modular nuclearity condition for the wedge-local quantum field theories constructed from regular matrix-valued factorizing S-matrices, by settling the intertwiner existence conjecture on which that verification rests.

Concretely (Conjecture 1 of the source thesis; the 'intertwiner property', Definition 5.2 of Alazzawi–Lechner 2017): let $S \in \mathcal{S}^-_0$ be a factorizing S-matrix satisfying the full S-matrix axioms above and regular in the sense defined in the Background, acting on a finite-dimensional one-particle space $\mathcal{K}$ with $\dim\mathcal{K} > 1$, and satisfying $S(0) = -F$ ($F$ the tensor flip) — the class containing the O(N)-invariant nonlinear sigma-model S-matrices $S_N$, $N \ge 3$. Do there exist, for every particle number $n \in \mathbb{N}$, multiplication intertwiners $I_n$ — holomorphic functions on the shrinking tube domains (cubes of width proportional to $1/n$ around the symmetry point) to which the Hardy-space estimates extend, with values in $\mathrm{End}(\mathcal{K}^{\otimes n})$ — that intertwine the S-dependent representation $D_n$ of the permutation group $\mathfrak{S}_n$ with the antisymmetric representation $D^-_n$, such that $I_n$ is bounded on the corresponding S-symmetric Hardy space with operator norm at most $\gamma^n$, and the inverse $I_n^{-1}$ maps the antisymmetric Hardy space boundedly back with norm at most $\gamma'^n$, for constants $\gamma, \gamma' > 0$ independent of $n$?

A positive answer implies, by the nuclearity theorems of the sources (a one-way implication: the converse is not established there), that the modular nuclearity condition holds for the associated wedge-local ZF-algebra models for all double cones above a minimal size $s_{\min}$, so that the local observable algebras are nontrivial with the vacuum as cyclic vector — completing the operator-algebraic inverse-scattering construction of these interacting theories, most prominently the massive O(N) nonlinear sigma models. The problem is answered equally by any other method that establishes (or disproves) modular nuclearity for this class without the intertwiner hypothesis — for instance, by improving the n-dependence of the nuclearity estimates on the S-symmetric Hardy spaces, a route explicitly contemplated in the sources — or by a refutation: an explicit regular $S \in \mathcal{S}^-_0$ for which intertwiners with the required bounds cannot exist, or for which nuclearity of the map $\Xi(s)$ fails for arbitrarily large splitting distances $s$ (i.e., for $s$ above every candidate minimal size $s_{\min}$); failure of nuclearity only at small $s$ would be compatible with the intertwiner property holding for that S and would not refute it.

References

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