AdS vacua of non-supersymmetric strings

Few vacua are known for the three tachyon-free non-supersymmetric string theories. We find new classes of AdS backgrounds by focusing on spaces where the equations of motion reduce to purely algebraic conditions. Our first examples involve non-zero three-form fluxes supported either on direct product internal spaces or on $T_{p,q}$ geometries. For the ${\mathrm{SO}}(16)\times{\mathrm{SO}}(16)$ heterotic string, we then develop a method to engineer vacua with the addition of gauge fields. A formal Kaluza–Klein reduction yields complete solutions on a broad class of coset spaces $G/H$, automatically satisfying the three-form Bianchi identities with $H$-valued gauge fields.

💡 Research Summary

The paper addresses the long‑standing scarcity of explicit anti‑de Sitter (AdS) vacua in the three tachyon‑free, non‑supersymmetric ten‑dimensional string theories: the heterotic SO(16)×SO(16) model, the type 0′B theory, and the Sugimoto orientifold of type IIB. Starting from the low‑energy effective action (2.1) that includes the Einstein–Hilbert term, the dilaton kinetic term, a set of p‑form field strengths and a one‑loop generated scalar potential V(φ)=T e^{γφ}, the authors impose a constant dilaton background. This reduces the equations of motion to algebraic relations between the Ricci tensor, the flux bilinears ι_MF·ι_NF and the cosmological term.

For the heterotic model the only form field is the Kalb–Ramond three‑form H₃ (β₃=−2). The resulting equations (2.4) admit the well‑known AdS₇×X₃ solution with H₃ proportional to the volume form of an internal Einstein three‑manifold X₃ of positive curvature. The authors first review this simple case to fix conventions.

The first new class of solutions generalises the internal sector to a product of two three‑dimensional Einstein spaces, X₃ and Y₃, each threaded by independent magnetic three‑form fluxes (2.13). The external space becomes AdS₄. By normalising the curvatures (2.6) and introducing an angle θ that parametrises the two internal radii, the authors show that for any quantised flux numbers (n_X, n_Y) there exists a θ solving the algebraic constraints (2.18) or (2.21) depending on the sign choices of the internal curvatures. Three distinct curvature assignments are examined: both positive, mixed sign, and one flat. In all cases the dilaton and the AdS radius L grow with the flux quanta, guaranteeing weak coupling and large radii.

A second family of vacua exploits an S¹‑fibration over a product of two Riemann surfaces X₂ and Y₂, the so‑called T_{p,q} spaces. The U(1) connection on the circle carries first Chern numbers (p χ_X, q χ_Y). The harmonic three‑form H₃ is built from the fibre direction wedged with the volume forms of the two bases (2.27). Solving the reduced equations yields AdS₅×X₂×Y₂ solutions with the external AdS₅ radius fixed by the one‑loop tadpole (2.29). Again an angle θ parametrises the two internal radii, and the quantised flux n_H together with (p,q) determines θ via (2.32) or (2.34). The analysis shows that for any (p,q) with non‑vanishing Euler characteristics there is a consistent solution.

The most original contribution concerns the heterotic SO(16)×SO(16) theory with gauge fields. The authors develop a Kaluza–Klein inspired framework in which the internal space is a homogeneous coset G/H. By expanding the Kalb–Ramond three‑form on invariant forms of the coset and allowing the gauge field strength F to take values in the isotropy algebra 𝔥, the Bianchi identity dH=−½ tr F∧F is automatically satisfied. This reduces the problem to solving algebraic Einstein equations on the coset together with flux quantisation conditions.

Explicit examples are worked out for three well‑known six‑dimensional cosets:

• F(1,2;3)=SU(3)/U(1)×U(1) with gauge group H=U(1)×U(1),
• CP³=Sp(2)/Sp(1)×U(1) with H=Sp(1)×U(1),
• S⁶=G₂/SU(3) with H=SU(3).

For each case the authors compute the invariant metric, the structure constants, and the contributions of the gauge fluxes to the stress‑energy tensor. The resulting algebraic system admits solutions that give AdS₄×(coset) backgrounds. All these vacua share a crucial property: there is no scale separation. The internal curvature scale R_int and the AdS radius L scale in the same way with the flux numbers (∼n^{1/2}), as argued in Appendix A. Consequently, the Kaluza–Klein tower is not parametrically heavier than the AdS excitations, mirroring the situation in supersymmetric Freund–Rubin compactifications.

The paper concludes with a discussion of the implications. The presence of a one‑loop tadpole does not improve scale separation, and the non‑supersymmetric nature of the models leaves the issue of perturbative and non‑perturbative stability open. Nevertheless, the systematic method of using homogeneous cosets and gauge‑field induced three‑form fluxes provides a powerful tool to generate a broad landscape of non‑supersymmetric AdS vacua. Future work is suggested on the analysis of potential tachyonic modes, the role of higher‑derivative corrections, and possible holographic applications of these non‑supersymmetric backgrounds.