One-to-one correspondence between defect form-factor solutions and local defect operators in the scaling Lee–Yang model

The scaling Lee–Yang model with an integrable topological defect is the simplest interacting integrable quantum field theory with a purely transmitting line defect. Its transmission matrix is known, all defect form factors of primary fields have been constructed, the ultraviolet defect conformal field theory has been solved, and finite-volume form factors have been validated numerically. One item of the original bootstrap program, however, remains unfinished: proving (or refuting) that the space of polynomial solutions of the defect form-factor equations stands in one-to-one correspondence with the complete local operator content of the defect Lee–Yang conformal field theory, including all Virasoro descendants. The counting was completed for the free boson and for the lowest levels of the Lee–Yang primaries, but the left/right operator mixing and dimension cancellations of the non-unitary defect conformal field theory prevented the general counting. The surviving open core is the all-levels, descendant-inclusive correspondence.

Background

In a two-dimensional integrable quantum field theory, scattering is factorized: multi-particle scattering decomposes into products of two-body scattering amplitudes $S(\theta)$ depending on the rapidity difference $\theta$ , where the rapidity of a particle of mass $m$ parametrizes its energy and momentum as $E=m\cosh\theta$ , $p=m\sinh\theta$ . Local operators can then be characterized by their form factors, the matrix elements $\langle 0|\mathcal{O}(0)|\theta_1,\dots,\theta_n\rangle$ between the vacuum and $n$ -particle states. The form-factor bootstrap determines these matrix elements recursively from the requirements of factorization (Watson's equations) and crossing symmetry. The scaling Lee–Yang model is the simplest interacting example: it is the unique non-unitary minimal conformal field theory (a solvable family of two-dimensional conformal field theories with finitely many primary fields, i.e. fields annihilated by the lowering generators of the chiral Virasoro algebras; its central charge, the number characterizing the conformal anomaly, is $c=-22/5$ ) perturbed by its single relevant field, giving a massive theory with one particle species. A topological (purely transmitting) defect is a line defect that transmits particles with rapidity-dependent transmission amplitudes $T_\pm(\theta)$ without reflection, carries no energy momentum of its own, and preserves integrability. Bajnok and Simon fixed the defect transmission matrix of the Lee–Yang model via a fusion construction (building the defect amplitudes out of products of bulk scattering amplitudes), obtaining a one-parameter family supporting a defect bound state. Bajnok and el Deeb then set up the defect form-factor bootstrap: form factors of operators living on the defect line satisfy recursive equations whose nontrivial content is carried by polynomials $Q_n$ in the rapidity variables, one independent set per particle number $n$ . In the short-distance (ultraviolet) limit the massive theory flows to a defect conformal field theory (DCFT), i.e. the Lee–Yang conformal field theory modified by the topological defect line. Local defect operators are organized in tensor products of Virasoro modules — irreducible representations of the chiral Virasoro algebras of the left- and right-moving degrees of freedom; a Virasoro descendant is an operator obtained from a primary by acting with the chiral raising generators. Bajnok, Holló and Watts solved this DCFT completely, finding that for the nontrivial topological defect the operators living on the defect span $H(1,1)=(V_0\otimes\bar V_0)\oplus(V_1\otimes\bar V_0)\oplus(V_0\otimes\bar V_1)\oplus 2\,(V_1\otimes\bar V_1)$ , where $V_0$ and $V_1$ are the two Lee–Yang Virasoro modules (the identity and the perturbing primary), with five primary defect fields $I,\phi,\bar\phi,\Phi^-,\Phi^+$ , and computed all its structure constants. The structure of the solution side is known: each solution is a symmetric polynomial $Q_n$ in the variables $x_i=e^{\theta_i}$ multiplied by a universal kinematical and dynamical prefactor $K_n$ (the kernel), and every solution is generated from a top representative of the form $\sigma_1^{a_1}\cdots\sigma_n^{a_n}K_n$ , where $\sigma_k$ is the $k$ -th elementary symmetric polynomial in the $x_i$ , by repeated application of level-raising operators that add one particle — mirroring the primary/descendant organization of the conformal side. The counting question is whether every local defect operator of this DCFT corresponds to exactly one independent form-factor solution $Q_n$ at some particle number, and conversely. For the free boson such a one-to-one correspondence was proven. For the interacting Lee–Yang case, solutions at the lowest levels were matched to the operator content, but the general counting was never completed: in a non-unitary theory the left/right operator mixing and cancellations among conformal dimensions make the enumeration subtle, and the number of operators and descendants is large. The boundary analogue of this question — counting boundary local operators via the boundary form-factor bootstrap in the boundary Lee–Yang model (the model restricted to a half-plane with a conformally invariant boundary condition) — was completed by Szőcs and Takács, and serves as the methodological template. Bajnok, Buccheri, Holló, Konczer and Takács later constructed all defect form factors of primary fields, computed exact vacuum expectation values, extended the defect form factors to finite volume with all polynomial $1/L$ corrections, and validated them against truncated conformal space approach (TCSA, a numerical method that diagonalizes the Hamiltonian in a finite conformal-space basis) with excellent agreement; they also confirmed the operator identification at the level of primaries through two-point functions. What their analysis explicitly did not do is the descendant-level counting: relating the form-factor solution space to the full space of local defect operators remained, in their words, a highly nontrivial task.

Problem Statement

Determine whether the space of polynomial solutions $Q_n$ of the defect form-factor equations of the scaling Lee–Yang model with the integrable topological defect is in one-to-one correspondence with the complete local operator content of the ultraviolet defect Lee–Yang conformal field theory (the operators living on the nontrivial defect, spanning $H(1,1)$ with its five primary defect fields and all their Virasoro descendants), and establish the correspondence or exhibit the mismatch. Concretely: (1) enumerate the local defect operators of the defect Lee–Yang DCFT, including all primaries and their Virasoro descendants with their left/right module structure and conformal dimensions; (2) determine the space of defect form-factor solutions at every particle number $n$ , identifying independent top representatives and eliminating descendants of lower-level solutions; and (3) construct the explicit map between the two spaces and prove it is a bijection, or exhibit a specific defect operator with no corresponding form-factor solution or a specific solution with no corresponding operator, together with a proof of the mismatch.

References

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