Justification and regime of validity of the factorized-scattering criterion for (non)integrability of AdS/CFT string backgrounds

Wulff (2019) showed that tree-level factorization of the light-cone-gauge worldsheet S-matrix expanded about the GKP null-cusp vacuum constrains the warp factor of any putative integrable supersymmetric string background, ruling out integrability for all supersymmetric $AdS_7$ and $AdS_6$ backgrounds under the assumption that the warp factor admits a regular critical point. The paper itself flags four unresolved issues bundled in this problem: (a) how to treat physically interesting backgrounds whose warp factor lacks a regular critical point, so that the GKP-type expansion is unavailable; (b) how to characterize integrability for backgrounds known only locally or perturbatively; (c) the lack of a rigorous justification for using factorized tree-level scattering as the diagnostic of integrability in a worldsheet theory that is not manifestly Lorentz invariant and contains massless modes; and (d,e) the extension to backgrounds that break AdS isometries through NSNS $B$-fields (such as omega-deformations) or that require supergravity sources (such as D8-brane-solved $AdS_7$ solutions). These issues have not been settled in the literature since 2019: partial evidence both for and against the naive integrability-to-factorization link exists, the omega-deformed background was proven nonintegrable by an independent analytic method, and the local warp-factor test has been applied to further marginal deformations, but the criterion itself remains an assumption and its domain of applicability remains open.

Background

A central lesson of the AdS/CFT correspondence (the conjectured duality between string theory on anti-de Sitter (AdS) spacetimes and conformal field theories living on the boundary) is that certain string sigma models, i.e. the two-dimensional field theories describing a string worldsheet embedded in the target spacetime, are integrable, meaning they possess infinitely many conserved charges. The paradigm is type IIB string theory on $AdS_5 \times S^5$, where integrability explains the exact spectral data of $\mathcal{N}=4$ super-Yang-Mills theory (a four-dimensional maximally supersymmetric gauge theory). This raises the classification question: which other supersymmetric AdS backgrounds support an integrable string sigma model? A powerful obstruction was developed in Wulff (2019), 'Constraining integrable AdS/CFT with factorized scattering' (JHEP 04 (2019) 133). The setup: in light-cone gauge (a parametrization of the string worldsheet in which the worldsheet coordinates are traded for the target-space light-cone variables), the Green-Schwarz superstring (the standard formulation of string theory with manifest target-space supersymmetry) expanded about the GKP null-cusp solution (a classical string solution found by Gubser-Klebanov-Polyakov, about which the worldsheet theory simplifies enough that the fluctuation spectrum and interactions can be computed) defines a worldsheet theory whose tree-level two-to-two scattering matrix can be extracted from the quartic interaction vertex. In a Lorentz-invariant two-dimensional theory with a strictly massive spectrum, integrability implies that this S-matrix factorizes, i.e. multi-particle scattering decomposes into products of two-body scattering with no particle production. Standard proofs of this implication, however, rely on Lorentz invariance and on the absence of massless particles, and the light-cone-gauge worldsheet theory about the GKP vacuum satisfies neither: Lorentz invariance is broken by the interactions and the spectrum contains massless modes. Wulff therefore treats factorization of tree-level scattering about the GKP vacuum as an assumption-laden diagnostic of integrability, and derives from it a constraint on the warp factor $A(x)$ (the function controlling how the geometry of the transverse directions varies with position) via its local expansion $e^{2A} = 1 + 2 a_{ij} x^i x^j + \cdots$: the coefficients $a_{ij}$ must satisfy a second-order algebraic condition that no supersymmetric $AdS_7$ or $AdS_6$ background satisfies, so those backgrounds are nonintegrable conditionally. The conclusions are thus conditional on the background admitting a regular critical point of the warp factor (the point about which the GKP-type solution is constructed), and the paper explicitly leaves open: (a) how to treat physically interesting backgrounds without such a regular critical point; (b) how to analyze backgrounds known only locally or in perturbation theory, where existence of a regular critical point is not guaranteed; (c) a more rigorous justification of the applicability and limitations of the factorization criterion in this non-relativistic, massless-extended context; and (d) backgrounds breaking AdS isometries via the NSNS $B$-field (the antisymmetric two-form field of string theory), such as the omega-deformed backgrounds of Bobev-Gautason-Hristov (2019), and (e) backgrounds requiring supergravity sources, such as the massive type IIA (a one-parameter extension of type IIA supergravity with a nonzero Romans-mass flux term) $AdS_7$ solution with space-filling D8-branes (extended solitonic objects carrying D8-brane charge) of Filippas-Nunez-van Gorsel (2019), whose integrable NS-NS sector sits in tension with the fact that kappa symmetry (the local fermionic gauge symmetry of the Green-Schwarz formulation) implies source-free supergravity equations (Tseytlin-Wulff 2016).

Problem Statement

The problem is to put the factorized-scattering diagnostic of string-theoretic integrability on a justified footing and to extend it beyond its current regime. The factorized-scattering diagnostic is the criterion, introduced by Wulff (2019), that integrability of a string background implies tree-level factorization of the light-cone-gauge worldsheet S-matrix about the GKP null-cusp vacuum, from which an algebraic obstruction on the transverse warp factor $A(x)$, through its expansion $e^{2A} = 1 + 2 a_{ij} x^i x^j + \cdots$ about a regular critical point, is derived. Specifically: (1) Rigorously justify, or delimit, the use of factorization of the tree-level light-cone-gauge worldsheet S-matrix about the GKP null-cusp vacuum as a criterion for (non)integrability of a string background, given that the worldsheet theory is not manifestly Lorentz invariant and contains massless modes, so that the standard massive-relativistic proof that integrability implies factorized scattering does not apply; that is, identify the precise hypotheses under which the implication holds, or exhibit a background where it fails. (2) Determine to what extent physically interesting supersymmetric AdS backgrounds lack the regular warp-factor critical point assumed in the GKP-type expansion, and develop a treatment of those backgrounds, so that nonintegrability statements (e.g. for supersymmetric $AdS_7$ and $AdS_6$ backgrounds) no longer need to be conditional on that assumption. (3) Extend the warp-factor obstruction analysis to backgrounds known only locally or perturbatively, where the existence of a regular critical point is not guaranteed. (4) Extend the analysis to backgrounds that break AdS isometries via NSNS $B$-fields (such as omega-deformed backgrounds) and to backgrounds with supergravity sources (such as D8-brane-solved massive type IIA $AdS_7$ solutions), clarifying the superstring interpretation of sourced solutions given that kappa symmetry of the Green-Schwarz string implies source-free supergravity equations. The four parts above together constitute the single objective of justifying the diagnostic and extending it beyond its current regime; they are independent of one another, and resolving any subset alone does not settle the objective. In particular, (2) determines which of the relevant supersymmetric $AdS_7$ and $AdS_6$ backgrounds lack the regular warp-factor critical point and provides an analysis of those backgrounds, so that their nonintegrability no longer depends on that assumption.

References

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