de Sitter Constraints and Vacuum-Structure Claims

Classical no-go theorems and swampland de Sitter conjectures are different objects. A no-go result excludes positive stationary points only under explicit compactness, source, curvature, flux, and correction hypotheses. Gradient and Hessian inequalities are broader conjectures whose constants and domains must be specified.

Required background. Quantum-Gravity Consistency Claims and Comparison Contract fixes logical status; Vacuum Energy and the Cosmological Constant supplies the EFT observable.

Helpful background. de Sitter Holography: Dictionary Completeness, Obstructions, and Status supplies holographic limitations; Euclidean Gravitational Saddles and Boundary Terms supplies metastability.

Evidence cutoff: 25 July 2026.

The three questions at a candidate point

For scalar metric $G_{ij}$ and potential $V$, a de Sitter critical point has

$$V_*>0,\qquad \nabla_iV|_*=0.$$

Metastability additionally requires nonnegative eigenvalues of the covariant Hessian $M^i{}_j=M_{\rm Pl}^2G^{ik}\nabla_k\nabla_jV/V$, up to controlled decay channels. The refined de Sitter conjecture instead proposes that at least one branch holds,

$$\frac{M_{\rm Pl}\lVert\nabla V\rVert}{V}\ge c \quad\text{or}\quad \frac{M_{\rm Pl}^2\lambda_{\min}(\nabla\nabla V)}{V}\le-c',$$

with positive constants conjectured to be order one in a specified asymptotic regime.

First application: classify one proposed vacuum

Suppose a four-dimensional compactification yields

$$V_*=V_0>0,\qquad \nabla V|_*=0,\qquad \frac{M_{\rm Pl}^2\lambda_{\min}}{V_0}=-0.20.$$

The point violates the gradient-only conjecture for every $c>0$, satisfies the Hessian branch only when $c'\le0.20$, and is perturbatively unstable. A Maldacena–Nuñez-type classical no-go theorem excludes it only if its compact smooth internal space, two-derivative equations, source content, and energy conditions match the theorem Maldacena and Nuñez 2001. If the construction uses orientifold negative tension or quantum corrections, that particular theorem is inapplicable rather than disproved.

The refined criteria were proposed in Obied et al. 2018 and Garg and Krishnan 2019. They remain conjectural and variant-dependent.

Control and metastability

A claimed vacuum must exhibit small string-loop and $\alpha'$ corrections, stabilized moduli, flux quantization, tadpole cancellation, and separation from KK and string scales. A positive Hessian in the truncated field set is insufficient if an omitted modulus or brane nucleation channel is unstable.

Adversarial control: add the first omitted term

Let $V=V^{(0)}+\epsilon V^{(1)}$. If the positive value or smallest Hessian eigenvalue is $O(\epsilon)$, the leading omitted correction can create or remove the critical point. Varying into a region where $g_s$ or curvature is order one likewise destroys the calculation’s control without proving that no exact vacuum exists.

The strongest conclusion must name its hypotheses: a theorem may rigorously exclude a restricted class; a controlled construction may evade that class; a conjectural inequality can then be tested but not assumed as its own proof.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Garg, Sumit K., and Chethan Krishnan. “Bounds on Slow Roll and the de Sitter Swampland.” Journal of High Energy Physics 2019, 11 (2019): 075. DOI. Open PDF.
• Maldacena, Juan, and Carlos Nuñez. “Supergravity Description of Field Theories on Curved Manifolds and a No Go Theorem.” International Journal of Modern Physics A 16, 822–855 (2001). DOI. Open PDF.
• Obied, Georges, Hirosi Ooguri, Lev Spodyneiko, and Cumrun Vafa. “De Sitter Space and the Swampland.” arXiv:1806.08362 [hep-th] (2018). arXiv.