The FPS model — the relativistic limit of the mixed-flux AdS3×S3×T4\mathrm{AdS}_3\times S^3\times T^4 superstring worldsheet theory, with 'squared' N=(2,2)\mathcal{N}=(2,2) supersymmetry — has a solved thermodynamic Bethe ansatz whose ultraviolet (UV) limit decouples into massive, massless-chiral and massless-antichiral sectors with central charges cmassive=6(1−1/k)c_{\text{massive}}=6(1-1/k) and cmassless=N∘+2c_{\text{massless}}=N_\circ+2, where k≥2k\ge2 is the integer flux parameter and N∘N_\circ is the number of massless momentum-carrying flavours. The value of N∘N_\circ is ambiguous — the Bethe–Yang equations suggest two flavours while semiclassical comparisons of the string ground-state energy suggest one — so the total UV central charge is undetermined by one unit, and the UV conformal field theory itself is unidentified: the massive-sector central charge matches no superconformal minimal model, and the Y-system structure instead suggests, at least for k≥3k\ge3, a non-supersymmetric split into a c=4c=4 sector plus Zk−2\mathbb{Z}_{k-2} parafermions. The problem is to fix N∘N_\circ from first principles, establish the exact total UV central charge c(k)c(k), and identify the UV conformal field theory (or theories), including the fate of the squared supersymmetry algebra at the fixed point.

Background

Two-dimensional integrable quantum field theories (QFTs) are special in that the exact two-particle S-matrix, once known, determines the finite-volume and finite-temperature thermodynamics through the thermodynamic Bethe ansatz (TBA) — a set of coupled integral equations for pseudo-energies, commonly repackaged as functional relations among Y-functions (the 'Y-system'). In the ultraviolet (UV) limit where the spatial circle of size LL shrinks, the TBA ground-state energy satisfies c=−6LE/πc=-6L\mathcal{E}/\pi, where cc is the central charge of the conformal field theory (CFT) describing the model's UV fixed point; the constant-Y-function solutions of the UV Y-system, together with Rogers dilogarithm identities, turn the extraction of cc into an exact algebraic computation. This is the standard laboratory for identifying which CFT a massive integrable model flows to in the UV.

The model of interest here arises from string theory. Type IIB superstrings on AdS3×S3×T4\mathrm{AdS}_3\times S^3\times T^4 supported by a mixture of Ramond–Ramond (RR) and Neveu–Schwarz–Neveu–Schwarz (NSNS) three-form flux ('mixed flux') have an integrable worldsheet theory, with an integer kk measuring the relative NSNS flux strength. In the relativistic limit of small effective string tension, the worldsheet S-matrix reduces to that of a genuine two-dimensional relativistic QFT with massive and massless particles; Frolov, Polvara and Sfondrini constructed this exact S-matrix including the scalar 'dressing phases' required by crossing symmetry (the consistency condition relating the S-matrix to its analytic continuation in crossed scattering channels), and the resulting model is called the FPS model. Its particle multiplets are built by tensoring (not summing) pairs of short representations, so the model carries twice the supercharges of the N=(2,2)\mathcal{N}=(2,2) supersymmetric integrable model of Fendley and Intriligator: it realises two copies of the N=(2,2)\mathcal{N}=(2,2) supersymmetry algebra sharing the same central (bosonic) charges — 'N=2\mathcal{N}=2 squared' — which is not the standard N=(4,4)\mathcal{N}=(4,4) algebra.

Kervyn, Polvara and Sfondrini derived the TBA and Y-system of the FPS model. Massive and massless modes decouple, giving three sectors: massive, massless right-moving (chiral), and massless left-moving (antichiral). For the physical case (labelled Zα=2Z_\alpha=2, i.e. the squared supersymmetry) the massive sector contributes cmassive=6(1−1/k)c_{\text{massive}}=6(1-1/k); in the UV the massive Y-system itself splits into free-like functions (Y1Y_1, Yk−1Y_{k-1}, Y↑Y_\uparrow, Y↓Y_\downarrow) and a remainder identical to the Y-systems of minimal-model S-matrices of Dynkin type Ak−3A_{k-3}, which is the structure of Zk−2\mathbb{Z}_{k-2} parafermionic CFTs (Fateev–Zamolodchikov; conformal field theories whose symmetry currents carry fractional spin, generalising fermions), whose central charge 2−6/k2-6/k accounts for the interacting part. The massless sectors contribute cmassless=N∘+2c_{\text{massless}}=N_\circ+2, where N∘N_\circ is the number of massless momentum-carrying flavours: on the worldsheet the massless excitations carry an su(2)∘su(2)_\circ index α˙=1,2\dot{\alpha}=1,2, but the relativistic Bethe–Yang equations alone do not settle whether both flavours survive as independent thermodynamic species.

Two conflicting indications exist for N∘N_\circ. The Bethe–Yang equations (the asymptotic, large-volume Bethe equations obtained from the S-matrix) naturally suggest N∘=2N_\circ=2. On the other hand, the semiclassical comparison of Frolov, Pribytok and Sfondrini for the twisted pure-RR AdS3×S3×T4\mathrm{AdS}_3\times S^3\times T^4 string — whose mirror-TBA proposal (the TBA written for the analytically continued, 'mirror' worldsheet theory, where the spatial circle is finite and the TBA applies directly) originally contained two massless Y0Y_0-functions — showed agreement with the semiclassical ground-state energy only when a single massless Y0Y_0-function is kept, i.e. N∘=1N_\circ=1. The FPS thermodynamics paper therefore leaves N∘N_\circ free, making the total UV central charge c(k)=6(1−1/k)+(N∘+2)c(k)=6(1-1/k)+(N_\circ+2) ambiguous by one unit. The ambiguity has a visible precedent in earlier work on the relativistic massless sector, where a central charge c=3c=3 — the value corresponding to N∘=1N_\circ=1 — was doubled to c=6c=6 by an argument that appears to disregard the mutual interaction of the massless momentum-carrying modes, and which the FPS authors explicitly question. Beyond the central charge, no candidate UV CFT has been established: 6(1−1/k)6(1-1/k) matches no minimal model (the discrete unitary series of rational CFTs) with N=(2,2)\mathcal{N}=(2,2) or N=(4,4)\mathcal{N}=(4,4) supersymmetry, the Y-system periodicity analysis (which normally reveals the conformal dimension of the perturbing operator) is obstructed because the Zα=2Z_\alpha=2 Y-system is not periodic, and it is unknown whether the squared N=(2,2)\mathcal{N}=(2,2) algebra closes into a consistent superconformal algebra at the fixed point.

Problem Statement

Determine the ultraviolet fixed point of the FPS integrable quantum field theory (the relativistic limit of the mixed-flux AdS3×S3×T4\mathrm{AdS}_3\times S^3\times T^4 worldsheet theory, with integer flux parameter k≥2k\ge2 and squared N=(2,2)\mathcal{N}=(2,2) supersymmetry). The single objective is the UV classification of the model, with three jointly necessary components:

  1. Fix the number of massless momentum-carrying flavours N∘N_\circ by derivation from the model's defining data (the mixed-flux worldsheet construction, the exact S-matrix with dressing factors, or the Bethe–Yang/TBA equations), resolving the conflict between the Bethe–Yang count N∘=2N_\circ=2 and the semiclassical indication N∘=1N_\circ=1.

  2. Establish the exact UV central charges of the decoupled sectors — cmassless=N∘+2c_{\text{massless}}=N_\circ+2 and cmassive=6(1−1/k)c_{\text{massive}}=6(1-1/k) — and hence the exact total central charge c(k)=6(1−1/k)+(N∘+2)c(k)=6(1-1/k)+(N_\circ+2) as a function of the integer k≥2k\ge2.

  3. Identify the ultraviolet conformal field theory (or theories): a conventional superconformal theory (and which one), or a decoupled sum of sectors — for instance a c=4c=4 theory together with Zk−2\mathbb{Z}_{k-2} parafermions of central charge 2−6/k2-6/k, the split conjectured at least for k≥3k\ge3 — and determine whether the model's supersymmetry (two copies of N=(2,2)\mathcal{N}=(2,2) with identical central elements) gives rise to a consistent superconformal algebra at the fixed point, or exactly how it fails to.

These components form one objective: without (1) the central charge in (2) is undetermined, and (3) must reproduce the sector structure and central charges of (1)–(2). A partial result — for example a numerical study confirming the UV decoupling for an arbitrarily chosen N∘N_\circ, or a central-charge match without CFT identification — is not a complete answer.

References

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