Inverse problem for the classical spectral curve of the $\mathrm{AdS}_5\times S^5$ $\lambda$-superstring

The classical spectral curve of the AdS5 x S5 lambda-superstring — an integrable deformation of the type IIB superstring whose quantum-group parameter is a root of unity — was constructed by Hollowood, Miramontes and Price (JHEP 05 (2020) 010) as a Riemann–Hilbert problem: densities satisfying integral equations determine resolvents and, through them, the quasi-momenta of the monodromy of the Lax connection. That work explicitly records as an outstanding problem the inverse construction: given the spectral data of a curve, construct the classical worldsheet solution — the group-valued field and fermions of the gauged WZW formulation — that produces it. Both the inverse reconstruction problem and the lambda-string quantum spectral curve remain open.

Background

Classically integrable two-dimensional sigma models are characterized by their spectral data. The equations of motion are equivalent to the flatness of a Lax connection — a worldsheet gauge connection depending on an auxiliary complex spectral parameter $z$ — and the monodromy of this connection around the closed string, a path-ordered integral $T(z)$, then has a conserved spectrum. Its eigenvalues $e^{ip_A(z)}$ define quasi-momenta $p_A(z)$: branches of a multi-valued function whose branch cuts, poles and gluing conditions constitute the classical spectral curve. The conserved charges, including (after gauge fixing) the energy, are read off from the behaviour of the quasi-momenta at distinguished points of the $z$-plane. In the general theory of finite-gap integrable systems the moduli of the curve are the action variables and, in favourable circumstances, an inverse algorithm reconstructs the solution from the curve. For the string on $\mathrm{AdS}_5\times S^5$, however, only the forward direction is workable in generality: classical solutions such as spinning and folded strings or giant magnons (large worldsheet excitations with magnon dispersion) are found by ansatz and then verified through their curves, and a general reconstruction of the solution from curve data is not available.

The $\lambda$-deformation introduced by Sfetsos is an integrable deformation of the Wess–Zumino–Witten (WZW) model — a sigma model with a topological level-$k$ term — that interpolates between the WZW point and the non-abelian T-dual (a duality trading isometry variables for Lagrange-multiplier vectors) of the principal chiral model. Hollowood, Miramontes and Schmidtt extended the construction to the semi-symmetric supercoset (the supermanifold whose bosonic body is $\mathrm{AdS}_5\times S^5$, enlarged by fermionic directions) underlying the type IIB superstring on $\mathrm{AdS}_5\times S^5$, producing the $\lambda$-string: a gauged WZW-type model for the supergroup $PSU(2,2|4)$ (the superalgebra whose bosonic subalgebra combines the $\mathrm{AdS}_5$ and $S^5$ isometry algebras) with deformation parameter $\lambda$ and level $k$, which preserves integrability and $\kappa$-symmetry (the local fermionic gauge symmetry of superstrings) and is one-loop finite. Its quantum-group parameter is a root of unity, in contrast with the $\eta$-deformed string, where it is real. Giant-magnon solutions in the $\lambda$ background were constructed by direct ansatz by Appadu, Hollowood, Miramontes, Price and Schmidtt.

For the undeformed string the exact spectral problem is governed by the quantum spectral curve (QSC) of Gromov, Kazakov, Leurent and Volin — a finite system of functional equations for Q-functions computing the full finite-coupling spectrum — and the classical spectral curve arises as its semiclassical, WKB-type limit, where logarithms of Q-functions become resolvents whose cuts carry Bethe-root densities. For the $\eta$-deformed string a QSC was constructed by Klabbers and van Tongeren. Hollowood, Miramontes and Price then constructed the classical spectral curve of the $\lambda$-string: the quasi-momenta are determined by densities solving Riemann–Hilbert-type integral equations, the level-matching condition appears as equality of the monodromy spectra at the two special points $z=\lambda^{\pm 1/2}$, and the curve is the semiclassical limit of XXZ-type (trigonometric, rather than the undeformed rational XXX-type) nested Bethe ansatz equations — the algebraic equations for Bethe roots that diagonalize integrable spin chains — for $PSU(2,2|4)$. They presented the curve explicitly as a test bed for a QSC of the $\lambda$-string to be given in a companion paper, and recorded the inverse problem — the construction of the classical solution corresponding to a given curve — as an outstanding problem. The present problem is that inverse reconstruction.

Problem Statement

Given the spectral data of the classical spectral curve of the $\mathrm{AdS}_5\times S^5$ $\lambda$-superstring — the quasi-momenta $p_A(z)$, $A=1,\ldots,8$, or equivalently the resolvents and densities $\rho_r(x)$ satisfying the curve's Riemann–Hilbert-type integral equations, with their prescribed pole and asymptotic structure at the distinguished points of the spectral-parameter plane (including the level-matching condition that the monodromy spectra at $z=\lambda^{1/2}$ and $z=\lambda^{-1/2}$ coincide) and with the conserved charges read off from the quasi-momentum asymptotics at the special points and the filling fractions (the integrality conditions on the resolvents) — construct the corresponding classical solution of the $\lambda$-string worldsheet theory: the $PSU(2,2|4)$-valued group field and its fermionic partners of the gauged WZW formulation, as functions of worldsheet time and space, whose Lax-connection monodromy reproduces precisely this spectral data. A complete solution consists of (i) an explicit reconstruction algorithm, (ii) its demonstration on nontrivial curve data beyond the vacuum (point-like string) configuration, obtained by passing the forward map back over the reconstructed fields, and (iii) a characterization of the class of admissible curves covered together with the non-uniqueness modulo symmetries. The quantum spectral curve of the $\lambda$-string and the open-string extension requiring a modified monodromy lie outside the scope of the present problem, whose subject is the classical inverse reconstruction.

References

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