Supersymmetric heterotic flux compactifications with non-zero three-form field strength H, most prominently the N=2 Minkowski vacua on principal T^2-fibrations over K3 surfaces of Fu–Yau type, can be constructed so that the supersymmetry conditions and the heterotic Bianchi identity are solved exactly at first order in the string expansion parameter α'. Melnikov, Minasian and Sethi (JHEP 2014) emphasized that this success is formal: because the flux is quantized, these backgrounds have string-scale cycles and no ten-dimensional large-radius limit, so the supergravity system truncated at O(α') cannot be assumed to be closed, and whether the constructed solutions have physical significance is unknown. The open problem is to determine the fate of these vacua at O(α'^2) and beyond: either exhibit explicit second-order-corrected solutions of the full α'^2 supersymmetry conditions, Bianchi identity (with the composite Hull connection), and equations of motion, showing that the O(α') solution deforms consistently and whether the corrections can be absorbed by field redefinitions such as a redefinition of the dilaton function; or prove that the second-order terms obstruct the solutions. The general α'^2 supersymmetry framework was only derived in 2025, and no flux background has yet been pushed through it, so the question remains open.

Background

The heterotic string is the ten-dimensional superstring whose gauge fields live on a rank-16 bundle; its massless bosonic fields include the metric, the two-form B with three-form field strength H = dB, the dilaton, and a Yang–Mills gauge field with curvature F. Consistency of the world-sheet theory forces the anomaly-cancellation (Bianchi) identity dH=α′(tr⁡R∧R−tr⁡F∧F)dH=\alpha'(\operatorname{tr}R\wedge R-\operatorname{tr}F\wedge F), where R is the curvature of a spacetime connection that itself differs from the Levi-Civita connection by terms of order α' proportional to H. Here α' is the inverse string tension, so the identity couples geometry and gauge fields order by order in the α' expansion.

Supersymmetric compactifications to four-dimensional Minkowski space on a complex threefold X with non-zero H were characterized by Strominger: the geometry is complex but non-Kähler, conformally balanced (the Lee form, measuring the failure of the hermitian volume form to be closed, must be exact and proportional to the dilaton gradient), and the gauge field must solve the hermitian Yang–Mills equations (the gauge curvature must be of type (1,1) and primitive, i.e. it vanishes when wedged with the square of the hermitian form); together with the Bianchi identity these conditions form the Hull–Strominger system. The non-Kähler requirement follows because H is of type (2,1)+(1,2) and non-zero, which is incompatible with a Kähler metric.

The best-understood compact solutions of this system are the Fu–Yau backgrounds: principal T^2-fibrations over a K3 surface (a simply-connected complex surface with trivial canonical bundle) with Abelian gauge fields, where the Strominger system reduces to a non-linear Monge–Ampère-type equation for the dilaton and metric data whose solvability was proven by Fu and Yau. Melnikov, Minasian and Sethi (arXiv:1403.4298) re-examined these and related constructions, keeping careful track of α' factors in both the equations and the solution. Their central observations are: (i) the O(α') corrections can be treated self-consistently, and one can find solutions of the O(α') supersymmetry and Bianchi equations to all orders in the expansion of the leading-order fields — both for Calabi–Yau/bundle backgrounds with small flux and for the genuinely non-Kähler flux vacua; (ii) nevertheless, since the flux parameter (e.g. the first Chern number of the T^2-bundle curvature) is quantized, these flux vacua have no ten-dimensional large-radius limit in which higher α' corrections are parametrically small, so treating the truncated system as a closed system of equations is not physically justified — essential changes to the equations enter at O(α'^2) and beyond.

A further subtlety clarified by later work is that at O(α'^2) the connection entering the Bianchi identity is the composite Hull connection (the hermitian connection plus a torsion correction), and the second-order-corrected supersymmetry transformations of the Bergshoeff–de Roo effective action have only recently been analyzed in this context (McOrist–Picard, arXiv:2507.02388). For backgrounds preserving N=2 spacetime supersymmetry (8 supercharges), extended supersymmetry strongly constrains any quantum corrections, and dual M/F-theory arguments make the existence of the underlying vacua very plausible — which sharpens rather than removes the question of whether the α'-expansion solution matches them. The situation is to be contrasted with Calabi–Yau compactifications, where Nemeschansky–Sen and more recently Jardine–Quigley (arXiv:1801.04336) showed the Ricci-flat metric can be corrected order by order to give a conformally invariant world-sheet theory: for the flux vacua no analogous statement exists.

Problem Statement

Do the heterotic flux vacua constructed as all-order solutions of the O(α') supersymmetry conditions and Bianchi identity — principally the N=2 Minkowski vacua on principal T^2-fibrations over K3 surfaces with quantized flux (Fu–Yau type), together with the α'-corrected Calabi–Yau/bundle backgrounds — persist as solutions of the full heterotic string equations at second order in α'? Concretely, using the α'^2-corrected supersymmetry transformations and equations of motion of the heterotic effective action (with the composite Hull connection entering the Bianchi identity, as required at this order): either (a) construct the explicit O(α'^2)-corrected background fields (metric, H-flux, dilaton, gauge bundle) for these classes of vacua, demonstrating that the O(α') solution deforms consistently and determining whether the second-order terms can be absorbed by field redefinitions — for example a redefinition of the function entering the non-linear PDE for the dilaton — so that the leading Fu–Yau-type solution is exact or near-exact in string theory; or (b) prove that the O(α'^2) terms obstruct these solutions, i.e. that no deformation within the supersymmetric field content can absorb them. For a class of backgrounds that must include the principal N=2 Minkowski vacua on T^2-fibrations over K3 surfaces with quantized flux, does the formal O(α') construction reflect bona fide vacua of the heterotic string? The further, explicitly conditional stage identified by the source — making sense of world-sheet instanton corrections in the presence of H-flux, where the instanton action acquires a non-topological imaginary part, presumably through more general saddles in a continued field space of the non-linear sigma model — is the downstream successor of this question and is not required for its resolution.

References

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