pith. sign in

arxiv: 2507.02037 · v3 · submitted 2025-07-02 · ✦ hep-th

An M-theory dS maximum from Casimir energies on Riemann-flat manifolds

Bruno Valeixo Bento , Miguel Montero This is my paper

Pith reviewed 2026-05-19 05:40 UTC · model grok-4.3

classification ✦ hep-th
keywords Casimir energyde Sitter vacuumM-theoryRiemann-flat manifoldsflux compactificationscale separationCasimir branes
0
0 comments X

The pith

Casimir energies on Riemann-flat manifolds produce an explicit de Sitter maximum in M-theory.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The authors build a concrete five-dimensional de Sitter maximum by compactifying M-theory on a six-dimensional Riemann-flat manifold and balancing fluxes against Casimir energies. These energies localize on special loci called Casimir branes and can cancel exactly in some cases due to a spacetime version of Atkin-Lehner symmetry. The construction evades standard de Sitter no-go theorems because the manifold has no curvature or singular sources, so the entire solution stays inside eleven-dimensional supergravity. The resulting vacuum is scale-separated, with a calculable positive energy density of order 10^{-8} in five-dimensional Planck units and a Hubble radius of 10^4 Planck lengths. Light moduli masses sit at the Hubble scale, and all known corrections remain small.

Core claim

We use Casimir energies to construct an explicit dS5 maximum solution of a flux compactification of M-theory on a specific 6-dimensional RFM. The resulting solution is scale-separated, has a vacuum energy of 10^{-8} in five-dimensional Planck units, the Hubble radius is 10^4 Planck lengths, and the light fields have masses of order H. This is a fully explicit, top-down de Sitter maximum in M-theory, with precisely computable vacuum energy.

What carries the argument

Casimir stress-energy localized on Casimir branes inside the Riemann-flat manifold, whose tension sometimes cancels exactly due to spacetime Atkin-Lehner symmetry.

If this is right

  • The vacuum energy is precisely computable and equals 10^{-8} in five-dimensional Planck units.
  • The solution is scale-separated with Hubble radius 10^4 Planck lengths.
  • Light fields acquire masses of order the Hubble scale H.
  • Higher-derivative and loop corrections are suppressed by δV/V ∼ 10^{-5}.
  • M2- and M5-brane instantons contribute negligibly to the vacuum energy.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The numerical extension of the Ewald method to arbitrary dimensions could be reused for other lattice sums in compactification problems.
  • If the construction survives only perturbative corrections, it motivates checking non-perturbative effects such as virtual black-hole loops in eleven dimensions.
  • The localization of Casimir energy on specific loci suggests similar mechanisms might stabilize moduli in other non-supersymmetric compactifications without singular sources.

Load-bearing premise

Higher-derivative and loop corrections remain suppressed by a small parameter of order 10^{-5} while M2- and M5-brane instantons stay negligible.

What would settle it

An explicit higher-derivative or loop calculation that makes the relative correction δV/V order one instead of 10^{-5}, or shows that M2/M5 instanton contributions are not negligible, would remove control over the vacuum energy.

read the original abstract

We initiate the study of flux compactifications on non-supersymmetric Riemann-flat manifolds (RFM's) with Casimir energy. While curvature and other corrections are suppressed in RFM's, the inclusion of Casimir energies allows one to evade standard dS no-go theorems, and the absence of orientifolds or other singular sources means that the construction is completely captured by ten or eleven-dimensional supergravity. We obtain a fully explicit formula for the Casimir stress-energy in a general RFM, including its ten or eleven-dimensional profile. The Casimir energy localizes in particular loci of the RFM, which we call ``Casimir branes''. The tension of Casimir branes sometimes cancels exactly, due to a spacetime analog of worldsheet Atkin-Lehner symmetry. We use Casimir energies to construct an explicit $dS_5$ maximum solution of a flux compactification of M-theory on a specific 6-dimensional RFM. The resulting solution is scale-separated, has a vacuum energy of $10^{-8}$ in five-dimensional Planck units, the Hubble radius is $10^4$ Planck lengths, and the light fields have masses of order $H$. This is a fully explicit, top-down de Sitter maximum in M-theory, with precisely computable vacuum energy. While the solution is not parametric, it is under very good control: higher derivative and loop corrections to the vacuum energy are suppressed in powers of a small parameter $\delta V/V\sim 10^{-5}$, and M2 and M5-brane instantons are negligible. In short, the solution survives all known corrections. Nevertheless, it might be sensitive to more exotic ones, such as e.g. loops of 11d Planckian virtual black holes if there were a large enough number of them. We also extend the Ewald numerical method for lattice sums to arbitrary dimensions and develop an efficient numerical implementation.}

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper constructs an explicit dS5 maximum solution in M-theory via flux compactification on a specific 6-dimensional Riemann-flat manifold (RFM), using Casimir energies to generate positive vacuum energy. It derives a general explicit formula for the Casimir stress-energy tensor (including its 11D profile and localization on 'Casimir branes'), claims exact tension cancellation in some cases via a spacetime analog of Atkin-Lehner symmetry, and presents a concrete example with vacuum energy V ∼ 10^{-8} in 5D Planck units, scale separation, Hubble radius ∼ 10^4 Planck lengths, and light fields of mass ∼ H. Higher-derivative/loop corrections are asserted to be suppressed by δV/V ∼ 10^{-5} and M2/M5 instantons negligible, so the solution survives all known corrections; the paper also extends the Ewald method for lattice sums to arbitrary dimensions.

Significance. If the explicit construction and control over corrections hold, this would be a notable result: a fully top-down, explicit de Sitter maximum in M-theory with precisely computable vacuum energy, achieved without orientifolds or singular sources and captured entirely within 11D supergravity. It offers a concrete counterexample to dS no-go theorems via Casimir effects on non-supersymmetric RFMs and provides a scale-separated solution with light fields of mass order H. The extension of the Ewald summation technique to higher dimensions is a useful technical contribution for lattice computations in compactifications.

major comments (2)
  1. Abstract and final paragraph: The assertion that higher-derivative and loop corrections (e.g., R^4 terms in the M-theory effective action) are suppressed by δV/V ∼ 10^{-5} is load-bearing for the claim that the solution survives all known corrections and remains a maximum. The manuscript states this suppression factor but does not provide an explicit power-counting evaluation or action computation of the correction terms on the specific non-supersymmetric 6D RFM background with the flux quanta and radius chosen to realize the quoted V ∼ 10^{-8} and Hubble radius of 10^4 Planck lengths; without this, it is unclear whether numerical prefactors could allow corrections to flip the sign of V.
  2. Section deriving the explicit dS5 solution: The construction requires that the 11D supergravity equations of motion are solved with the computed Casimir stress-energy source (including its localized profile). The manuscript should specify where the full verification is carried out for the chosen RFM and fluxes, particularly confirming consistency with the claimed scale separation and the precise numerical value of the vacuum energy.
minor comments (2)
  1. Numerical methods section: The extension of the Ewald method to arbitrary dimensions is a positive addition; including a short comparison to prior higher-dimensional lattice sum techniques or a brief description of the implementation's convergence properties would improve clarity.
  2. Abstract and introduction: The specific 6D RFM geometry and flux quanta are presented as a concrete choice yielding the quoted results; a short table or explicit listing of these parameters (e.g., radii, flux integers) would facilitate independent numerical checks of the Casimir energy computation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and insightful comments on our work. We address each major comment in detail below, providing clarifications and indicating where revisions will strengthen the manuscript. Our responses focus on the technical substance of the points raised.

read point-by-point responses

  1. Referee: Abstract and final paragraph: The assertion that higher-derivative and loop corrections (e.g., R^4 terms in the M-theory effective action) are suppressed by δV/V ∼ 10^{-5} is load-bearing for the claim that the solution survives all known corrections and remains a maximum. The manuscript states this suppression factor but does not provide an explicit power-counting evaluation or action computation of the correction terms on the specific non-supersymmetric 6D RFM background with the flux quanta and radius chosen to realize the quoted V ∼ 10^{-8} and Hubble radius of 10^4 Planck lengths; without this, it is unclear whether numerical prefactors could allow corrections to flip the sign of V.

    Authors: We agree that an explicit evaluation on the chosen background would remove any ambiguity regarding prefactors. The quoted suppression δV/V ∼ 10^{-5} follows from the compactification scale (Hubble radius ∼ 10^4 Planck lengths) and the structure of the M-theory effective action, where each higher-derivative term is accompanied by additional powers of the Planck length over the radius; for the leading R^4 correction this yields a relative shift well below the quoted figure. Nevertheless, to address the referee's concern directly we will add a dedicated paragraph in the revised manuscript performing the power-counting explicitly for the selected RFM, fluxes, and radius, confirming that the sign of the vacuum energy remains positive. revision: yes

  2. Referee: Section deriving the explicit dS5 solution: The construction requires that the 11D supergravity equations of motion are solved with the computed Casimir stress-energy source (including its localized profile). The manuscript should specify where the full verification is carried out for the chosen RFM and fluxes, particularly confirming consistency with the claimed scale separation and the precise numerical value of the vacuum energy.

    Authors: The explicit dS5 maximum is obtained by substituting the derived Casimir stress-energy tensor (with its 11D profile and localization on Casimir branes) into the 11D supergravity equations together with the chosen fluxes on the RFM. The equations are satisfied by construction once the effective 5D potential is minimized at the quoted vacuum energy; scale separation follows from the large radius relative to the Planck length and the resulting mass spectrum. We will revise the relevant section to include a concise step-by-step outline of this verification, explicitly referencing the Casimir tensor formula, the flux quantization conditions, and the numerical evaluation of the vacuum energy. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via explicit Casimir formula

full rationale

The paper provides an explicit formula for the Casimir stress-energy on general RFMs, including its 11D profile and localization to Casimir branes, then applies it to a specific 6D RFM flux compactification to obtain the dS5 maximum with quoted vacuum energy 10^{-8}. This computation is presented as independent of the target result rather than fitted or self-defined; the δV/V ∼ 10^{-5} suppression is stated as a derived bound on corrections without reducing to a self-citation chain or ansatz smuggling. No load-bearing step equates the output vacuum energy or scale separation to an input by construction, and the construction is captured entirely within 11D supergravity without external uniqueness theorems from the same authors.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The construction rests on the validity of 11D supergravity, the definition of Riemann-flat manifolds, and the localization of Casimir energy; the specific numerical value depends on a chosen manifold whose parameters are not shown to be derived from first principles.

free parameters (1)
  • Specific 6D RFM geometry and flux quanta
    The explicit dS5 solution is obtained on a particular manifold whose size and fluxes are chosen to produce the reported vacuum energy and scale separation.
axioms (1)
  • domain assumption Eleven-dimensional supergravity captures the full dynamics of the compactification without singular sources.
    The abstract states that the construction is completely captured by ten or eleven-dimensional supergravity.
invented entities (1)
  • Casimir branes no independent evidence
    purpose: Loci where Casimir energy localizes inside the RFM
    New term introduced to describe the ten- or eleven-dimensional profile of the Casimir stress-energy.

pith-pipeline@v0.9.0 · 5879 in / 1534 out tokens · 59644 ms · 2026-05-19T05:40:58.473106+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 5 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Isolated Critical Points for Scherk-Schwarz Compactifications of M-theory

    hep-th 2026-05 unverdicted novelty 6.0

    Isolated critical points of the potential are located in Scherk-Schwarz M-theory compactifications to d=8,6,5,4 by fixed points under unbroken duality groups, plus a conjectured duality-covariant anomaly cancellation ...

  2. de Sitter in String Theory vs. Gibbons & Hawking

    hep-th 2026-04 conditional novelty 5.0

    Perturbative string theory forbids de Sitter times closed manifold solutions under the assumption that the Euclidean sphere partition function receives a nonzero 1/G_N contribution.

  3. Instabilities in scale-separated Casimir vacua

    hep-th 2025-07 unverdicted novelty 5.0

    Casimir-stabilized AdS vacua with parametric scale separation in supergravity exhibit perturbative and non-perturbative instabilities under deformations.

  4. Dark energy from string theory: an introductory review

    hep-th 2026-03 unverdicted novelty 2.0

    String theory imposes constraints on dark energy but permits various construction attempts for de Sitter vacua and single-field exponential quintessence models despite obstructions.

  5. Aspects of strings without spacetime supersymmetry

    hep-th 2025-09 unverdicted novelty 2.0

    A survey of tachyons and tadpoles in non-supersymmetric closed and orientifold strings, including ten-dimensional models and landscape attempts.

Reference graph

Works this paper leans on

167 extracted references · 167 canonical work pages · cited by 5 Pith papers · 59 internal anchors

  1. [1]

    Measurements of Omega and Lambda from 42 High-Redshift Supernovae

    Supernova Cosmology Project Collaboration, S. Perlmutter et al., “Measurements of Ω and Λ from 42 High Redshift Supernovae”, Astrophys. J. 517 (1999) 565–586, arXiv:astro-ph/9812133

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  2. [2]

    Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant

    Supernova Search Team Collaboration, A. G. Riess et al., “Observational evidence from supernovae for an accelerating universe and a cosmological constant”, Astron. J. 116 (1998) 1009–1038, arXiv:astro-ph/9805201

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  3. [3]

    Is the Superstring Weakly Coupled?

    M. Dine and N. Seiberg, “Is the Superstring Weakly Coupled?”, Phys. Lett. B 162 (1985) 299–302

    work page 1985

  4. [4]

    On the Geometry of the String Landscape and the Swampland

    H. Ooguri and C. Vafa, “On the Geometry of the String Landscape and the Swampland”, Nucl. Phys. B 766 (2007) 21–33, arXiv:hep-th/0605264

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  5. [5]

    Emergent strings from infinite distance limits

    S.-J. Lee, W. Lerche, and T. Weigand, “Emergent strings from infinite distance limits”, JHEP 02 (2022) 190, arXiv:1910.01135 [hep-th]

    work page arXiv 2022

  6. [6]

    The Swampland: Introduction and Review

    E. Palti, “The Swampland: Introduction and Review”, Fortsch. Phys. 67 (2019) no. 6, 1900037, arXiv:1903.06239 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  7. [7]

    van Beest, J

    M. van Beest, J. Calder´ on-Infante, D. Mirfendereski, and I. Valenzuela, “Lectures on the Swampland Program in String Compactifications”, Phys. Rept. 989 (2022) 1–50, arXiv:2102.01111 [hep-th]

    work page arXiv 2022

  8. [8]

    What if string theory has no de Sitter vacua?

    U. H. Danielsson and T. Van Riet, “What if string theory has no de Sitter vacua?”, Int. J. Mod. Phys. D 27 (2018) no. 12, 1830007, arXiv:1804.01120 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  9. [9]

    De Sitter Space and the Swampland

    G. Obied, H. Ooguri, L. Spodyneiko, and C. Vafa, “De Sitter Space and the Swampland”, arXiv:1806.08362 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv

  10. [10]

    Distance and de Sitter Conjectures on the Swampland

    H. Ooguri, E. Palti, G. Shiu, and C. Vafa, “Distance and de Sitter Conjectures on the Swampland”, Phys. Lett. B 788 (2019) 180–184, arXiv:1810.05506 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  11. [11]

    On the de Sitter swampland criterion

    D. Andriot, “On the de Sitter swampland criterion”, Phys. Lett. B 785 (2018) 570–573, arXiv:1806.10999 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  12. [12]

    Further refining the de Sitter swampland conjecture

    D. Andriot and C. Roupec, “Further refining the de Sitter swampland conjecture”, Fortsch. Phys. 67 (2019) no. 1-2, 1800105, arXiv:1811.08889 [hep-th]

    work page arXiv 2019

  13. [13]

    Bedroya and C

    A. Bedroya and C. Vafa, “Trans-Planckian Censorship and the Swampland”, JHEP 09 (2020) 123, arXiv:1909.11063 [hep-th]

    work page arXiv 2020

  14. [14]

    Asymptotic Flux Compactifications and the Swampland

    T. W. Grimm, C. Li, and I. Valenzuela, “Asymptotic Flux Compactifications and the Swampland”, JHEP 06 (2020) 009, arXiv:1910.09549 [hep-th] . [Erratum: JHEP 01, 007 (2021)]

    work page arXiv 2020

  15. [15]

    Calder´ on-Infante, I

    J. Calder´ on-Infante, I. Ruiz, and I. Valenzuela, “Asymptotic accelerated expansion in string theory and the Swampland”, JHEP 06 (2023) 129, arXiv:2209.11821 [hep-th]

    work page arXiv 2023

  16. [16]

    Etheredge, B

    M. Etheredge, B. Heidenreich, T. Rudelius, I. Ruiz, and I. Valenzuela, “Taxonomy of infinite distance limits”, JHEP 03 (2025) 213, arXiv:2405.20332 [hep-th]

    work page arXiv 2025

  17. [17]

    On Moduli Stabilisation and de Sitter Vacua in MSSM Heterotic Orbifolds

    S. L. Parameswaran, S. Ramos-Sanchez, and I. Zavala, “On Moduli Stabilisation and de Sitter Vacua in MSSM Heterotic Orbifolds”, JHEP 01 (2011) 071, arXiv:1009.3931 [hep-th] . – 112 –

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  18. [18]

    de Sitter Vacua in String Theory

    S. Kachru, R. Kallosh, A. D. Linde, and S. P. Trivedi, “De Sitter vacua in string theory”, Phys. Rev. D 68 (2003) 046005, arXiv:hep-th/0301240

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  19. [19]

    Systematics of Moduli Stabilisation in Calabi-Yau Flux Compactifications

    V. Balasubramanian, P. Berglund, J. P. Conlon, and F. Quevedo, “Systematics of moduli stabilisation in Calabi-Yau flux compactifications”, JHEP 03 (2005) 007, arXiv:hep-th/0502058

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  20. [20]

    Control issues of KKLT,

    X. Gao, A. Hebecker, and D. Junghans, “Control issues of KKLT”, Fortsch. Phys. 68 (2020) 2000089, arXiv:2009.03914 [hep-th]

    work page arXiv 2020

  21. [21]

    Small cosmological constants in string theory

    M. Demirtas, M. Kim, L. McAllister, J. Moritz, and A. Rios-Tascon, “Small cosmological constants in string theory”, JHEP 12 (2021) 136, arXiv:2107.09064 [hep-th]

    work page arXiv 2021

  22. [22]

    Junghans, LVS de Sitter vacua are probably in the swampland , Nucl

    D. Junghans, “LVS de Sitter vacua are probably in the swampland”, Nucl. Phys. B 990 (2023) 116179, arXiv:2201.03572 [hep-th]

    work page arXiv 2023

  23. [23]

    Bare-bones de Sitter vacua

    I. Bena, E. Dudas, M. Gra˜ na, G. Lo Monaco, and D. Toulikas, “Bare-bones de Sitter vacua”, Phys. Rev. D 108 (2023) no. 2, L021901, arXiv:2202.02327 [hep-th]

    work page arXiv 2023

  24. [24]

    The LVS parametric tadpole constraint

    X. Gao, A. Hebecker, S. Schreyer, and V. Venken, “The LVS parametric tadpole constraint”, JHEP 07 (2022) 056, arXiv:2202.04087 [hep-th]

    work page arXiv 2022

  25. [25]

    Holography and the KKLT scenario

    S. L¨ ust, C. Vafa, M. Wiesner, and K. Xu, “Holography and the KKLT scenario”, JHEP 10 (2022) 188, arXiv:2204.07171 [hep-th]

    work page arXiv 2022

  26. [26]

    L¨ ust and L

    S. L¨ ust and L. Randall, “Effective Theory of Warped Compactifications and the Implications for KKLT”, Fortsch. Phys. 70 (2022) no. 7-8, 2200103, arXiv:2206.04708 [hep-th]

    work page arXiv 2022

  27. [27]

    Hebecker, S

    A. Hebecker, S. Schreyer, and V. Venken, “Curvature corrections to KPV: do we need deep throats?”, JHEP 10 (2022) 166, arXiv:2208.02826 [hep-th]

    work page arXiv 2022

  28. [28]

    D3-branes and gaugino condensation

    I. Bena, E. Dudas, M. Gra˜ na, G. Lo Monaco, and D. Toulikas, “D3-branes and gaugino condensation”, JHEP 12 (2023) 019, arXiv:2211.14381 [hep-th]

    work page arXiv 2023

  29. [29]

    De Sitter vacua — when are ‘subleading corrections’ really subleading?

    B. Valeixo Bento, D. Chakraborty, S. Parameswaran, and I. Zavala, “De Sitter vacua — when are ‘subleading corrections’ really subleading?”, JHEP 11 (2023) 075, arXiv:2306.07332 [hep-th]

    work page arXiv 2023

  30. [30]

    McAllister, J

    L. McAllister, J. Moritz, R. Nally, and A. Schachner, “Candidate de Sitter vacua”, Phys. Rev. D 111 (2025) no. 8, 086015, arXiv:2406.13751 [hep-th]

    work page arXiv 2025

  31. [31]

    Kim,String perturbation theory of Klebanov-Strassler throat,JHEP05(2025) 234, [2409.19048]

    M. Kim, “String perturbation theory of Klebanov-Strassler throat”, arXiv:2409.19048 [hep-th]

    work page arXiv

  32. [32]

    G2-manifolds from Diophantine equations

    J. Moritz, “ G2-manifolds from Diophantine equations”, arXiv:2505.15883 [hep-th]

    work page arXiv

  33. [33]

    ASPECTS OF SUPERGRAVITY THEORIES

    G. W. Gibbons, “ASPECTS OF SUPERGRAVITY THEORIES”, in XV GIFT Seminar on Supersymmetry and Supergravity. 6, 1984

    work page 1984

  34. [34]

    Supergravity description of field theories on curved manifolds and a no go theorem

    J. M. Maldacena and C. Nunez, “Supergravity description of field theories on curved manifolds and a no go theorem”, Int. J. Mod. Phys. A 16 (2001) 822–855, arXiv:hep-th/0007018

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  35. [35]

    The Quantum Dynamics of Kaluza-Klein Theories

    T. Appelquist and A. Chodos, “The Quantum Dynamics of Kaluza-Klein Theories”, Phys. Rev. D 28 (1983) 772

    work page 1983

  36. [36]

    Quantum Horizons of the Standard Model Landscape

    N. Arkani-Hamed, S. Dubovsky, A. Nicolis, and G. Villadoro, “Quantum Horizons of the Standard Model Landscape”, JHEP 06 (2007) 078, arXiv:hep-th/0703067. – 113 –

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  37. [37]

    Hyperbolic compactification of M-theory and de Sitter quantum gravity

    G. B. De Luca, E. Silverstein, and G. Torroba, “Hyperbolic compactification of M-theory and de Sitter quantum gravity”, SciPost Phys. 12 (2022) no. 3, 083, arXiv:2104.13380 [hep-th]

    work page arXiv 2022

  38. [38]

    N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space . Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge, UK, 1982

    work page 1982

  39. [39]

    Minimal Simple de Sitter Solutions

    S. S. Haque, G. Shiu, B. Underwood, and T. Van Riet, “Minimal simple de Sitter solutions”, Phys. Rev. D 79 (2009) 086005, arXiv:0810.5328 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  40. [40]

    Towards Classical de Sitter Solutions in String Theory

    U. H. Danielsson, S. S. Haque, G. Shiu, and T. Van Riet, “Towards Classical de Sitter Solutions in String Theory”, JHEP 09 (2009) 114, arXiv:0907.2041 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  41. [41]

    Fayet–Iliopoulos D-term in non-supersymmetric heterotic string orbifolds

    A. R. D. Avalos, A. E. Faraggi, V. G. Matyas, and B. Percival, “Fayet–Iliopoulos D-term in non-supersymmetric heterotic string orbifolds”, Eur. Phys. J. C 83 (2023) no. 10, 926, arXiv:2302.10075 [hep-th]

    work page arXiv 2023

  42. [42]

    D-term uplifts in nonsupersymmetric heterotic string models

    A. R. D. Avalos, A. E. Faraggi, V. G. Matyas, and B. Percival, “D-term uplifts in nonsupersymmetric heterotic string models”, Phys. Rev. D 108 (2023) no. 8, 086007, arXiv:2306.16878 [hep-th]

    work page arXiv 2023

  43. [43]

    Vacuum energy of nonsupersymmetric S˜ heterotic string models

    L. A. Detraux, A. R. D. Avalos, A. E. Faraggi, and B. Percival, “Vacuum energy of nonsupersymmetric S˜ heterotic string models”, Phys. Rev. D 110 (2024) no. 8, 086006, arXiv:2407.19980 [hep-th]

    work page arXiv 2024

  44. [44]

    Fixing Moduli in Exact Type IIA Flux Vacua

    B. S. Acharya, F. Benini, and R. Valandro, “Fixing moduli in exact type IIA flux vacua”, JHEP 02 (2007) 018, arXiv:hep-th/0607223

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  45. [45]

    Smearing orientifolds in flux compactifications can be OK

    S. Baines and T. Van Riet, “Smearing orientifolds in flux compactifications can be OK”, Class. Quant. Grav. 37 (2020) no. 19, 195015, arXiv:2005.09501 [hep-th]

    work page arXiv 2020

  46. [46]

    Localized O6-plane solutions with Romans mass

    F. Saracco and A. Tomasiello, “Localized O6-plane solutions with Romans mass”, JHEP 07 (2012) 077, arXiv:1201.5378 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  47. [47]

    O-Plane Backreaction and Scale Separation in Type IIA Flux Vacua

    D. Junghans, “O-Plane Backreaction and Scale Separation in Type IIA Flux Vacua”, Fortsch. Phys. 68 (2020) no. 6, 2000040, arXiv:2003.06274 [hep-th]

    work page arXiv 2020

  48. [48]

    On supersymmetric AdS 4 orientifold vacua

    F. Marchesano, E. Palti, J. Quirant, and A. Tomasiello, “On supersymmetric AdS 4 orientifold vacua”, JHEP 08 (2020) 087, arXiv:2003.13578 [hep-th]

    work page arXiv 2020

  49. [49]

    Type IIA Moduli Stabilization

    O. DeWolfe, A. Giryavets, S. Kachru, and W. Taylor, “Type IIA moduli stabilization”, JHEP 07 (2005) 066, arXiv:hep-th/0505160

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  50. [50]

    Hecke operators on γ0(m)

    A. O. Atkin and J. Lehner, “Hecke operators on γ0(m)”, Mathematische Annalen 185 (1970) 134–160

    work page 1970

  51. [51]

    ATKIN-LEHNER SYMMETRY

    G. W. Moore, “ATKIN-LEHNER SYMMETRY”, Nucl. Phys. B 293 (1987) 139. [Erratum: Nucl.Phys.B 299, 847 (1988)]

    work page 1987

  52. [52]

    Generalized Atkin-Lehner symmetry

    K. R. Dienes, “Generalized Atkin-Lehner symmetry”, Phys. Rev. D 42 (1990) 2004–2021

    work page 1990

  53. [53]

    Notes on a vanishing cosmological constant without Bose–Fermi cancellation

    Y. Satoh and Y. Sugawara, “Notes on a vanishing cosmological constant without Bose–Fermi cancellation”, PTEP 2022 (2022) no. 5, 053B04, arXiv:2111.09663 [hep-th]

    work page arXiv 2022

  54. [54]

    A No-go on dS minima from Casimir energies on Riemann-flat manifolds

    B. Valeixo Bento and M. Montero, “A No-go on dS minima from Casimir energies on Riemann-flat manifolds”. In preparation, 2025. – 114 –

    work page 2025

  55. [55]

    De Sitter hunting in a classical landscape

    U. H. Danielsson, S. S. Haque, P. Koerber, G. Shiu, T. Van Riet, and T. Wrase, “De Sitter hunting in a classical landscape”, Fortsch. Phys. 59 (2011) 897–933, arXiv:1103.4858 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  56. [56]

    A note on obstinate tachyons in classical dS solutions

    U. H. Danielsson, G. Shiu, T. Van Riet, and T. Wrase, “A note on obstinate tachyons in classical dS solutions”, JHEP 03 (2013) 138, arXiv:1212.5178 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  57. [57]

    Stability Constraints on Classical de Sitter Vacua

    G. Shiu and Y. Sumitomo, “Stability Constraints on Classical de Sitter Vacua”, JHEP 09 (2011) 052, arXiv:1107.2925 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  58. [58]

    A Global View on The Search for de-Sitter Vacua in (type IIA) String Theory

    X. Chen, G. Shiu, Y. Sumitomo, and S. H. H. Tye, “A Global View on The Search for de-Sitter Vacua in (type IIA) String Theory”, JHEP 04 (2012) 026, arXiv:1112.3338 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  59. [59]

    DESI 2024 VI: Cosmological Constraints from the Measurements of Baryon Acoustic Oscillations

    DESI Collaboration, A. G. Adame et al., “DESI 2024 VI: cosmological constraints from the measurements of baryon acoustic oscillations”, JCAP 02 (2025) 021, arXiv:2404.03002 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  60. [60]

    Lodha et al

    DESI Collaboration, K. Lodha et al., “DESI 2024: Constraints on physics-focused aspects of dark energy using DESI DR1 BAO data”, Phys. Rev. D 111 (2025) no. 2, 023532, arXiv:2405.13588 [astro-ph.CO]

    work page arXiv 2024

  61. [61]

    DESI DR2 Results II: Measurements of Baryon Acoustic Oscillations and Cosmological Constraints

    DESI Collaboration, M. Abdul Karim et al., “DESI DR2 Results II: Measurements of Baryon Acoustic Oscillations and Cosmological Constraints”, arXiv:2503.14738 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv

  62. [62]

    Extended Dark Energy analysis using DESI DR2 BAO measurements

    DESI Collaboration, K. Lodha et al., “Extended Dark Energy analysis using DESI DR2 BAO measurements”, arXiv:2503.14743 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv

  63. [63]

    Dark Energy and the Refined de Sitter Conjecture

    P. Agrawal and G. Obied, “Dark Energy and the Refined de Sitter Conjecture”, JHEP 06 (2019) 103, arXiv:1811.00554 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  64. [64]

    Symmetries and M-theory-like Vacua in Four Dimensions

    S. Chen, D. van de Heisteeg, and C. Vafa, “Symmetries and M-theory-like Vacua in Four Dimensions”, arXiv:2503.16599 [hep-th]

    work page arXiv

  65. [65]

    Tachyonic de Sitter Solutions of 10d Type II Supergravities

    D. Andriot, “Tachyonic de Sitter Solutions of 10d Type II Supergravities”, Fortsch. Phys. 69 (2021) no. 7, 2100063, arXiv:2101.06251 [hep-th]

    work page arXiv 2021

  66. [66]

    Automated consistent truncations and stability of flux compactifications

    D. Andriot, P. Marconnet, M. Rajaguru, and T. Wrase, “Automated consistent truncations and stability of flux compactifications”, JHEP 12 (2022) 026, arXiv:2209.08015 [hep-th] . [Addendum: JHEP 04, 044 (2023)]

    work page arXiv 2022

  67. [67]

    Charting the landscape of (anti-) de Sitter and Minkowski solutions of 10d supergravities

    D. Andriot, L. Horer, and P. Marconnet, “Charting the landscape of (anti-) de Sitter and Minkowski solutions of 10d supergravities”, JHEP 06 (2022) 131, arXiv:2201.04152 [hep-th]

    work page arXiv 2022

  68. [68]

    On classical de Sitter solutions and parametric control

    D. Andriot and F. Ruehle, “On classical de Sitter solutions and parametric control”, JHEP 06 (2024) 101, arXiv:2403.07065 [hep-th]

    work page arXiv 2024

  69. [69]

    The dS/dS Correspondence

    M. Alishahiha, A. Karch, E. Silverstein, and D. Tong, “The dS/dS correspondence”, AIP Conf. Proc. 743 (2004) no. 1, 393–409, arXiv:hep-th/0407125

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  70. [70]

    dS/dS and $T\bar T$

    V. Gorbenko, E. Silverstein, and G. Torroba, “dS/dS and T T ”, JHEP 03 (2019) 085, arXiv:1811.07965 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  71. [71]

    Supersymmetry Breaking by Fluxes

    S. Sethi, “Supersymmetry Breaking by Fluxes”, JHEP 10 (2018) 022, arXiv:1709.03554 [hep-th]. – 115 –

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  72. [72]

    Supersymmetry-breaking compactifications on Riemann-flat manifolds

    G. Dall’Agata and F. Zwirner, “Supersymmetry-breaking compactifications on Riemann-flat manifolds”, 2025

    work page 2025

  73. [73]

    Flux compactifications in string theory: a comprehensive review

    M. Grana, “Flux compactifications in string theory: A Comprehensive review”, Phys. Rept. 423 (2006) 91–158, arXiv:hep-th/0509003

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  74. [74]

    Physics of String Flux Compactifications

    F. Denef, M. R. Douglas, and S. Kachru, “Physics of String Flux Compactifications”, Ann. Rev. Nucl. Part. Sci. 57 (2007) 119–144, arXiv:hep-th/0701050

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  75. [75]

    Bl˚ ab¨ ack, U

    J. Bl˚ ab¨ ack, U. Danielsson, G. Dibitetto, and S. Giri, “Constructing stable de Sitter in M-theory from higher curvature corrections”, JHEP 09 (2019) 042, arXiv:1902.04053 [hep-th]

    work page arXiv 2019

  76. [76]

    Constraining Neutrino Masses, the Cosmological Constant and BSM Physics from the Weak Gravity Conjecture

    L. E. Ibanez, V. Martin-Lozano, and I. Valenzuela, “Constraining Neutrino Masses, the Cosmological Constant and BSM Physics from the Weak Gravity Conjecture”, JHEP 11 (2017) 066, arXiv:1706.05392 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  77. [77]

    Weak Gravity Conjecture, Multiple Point Principle and the Standard Model Landscape

    Y. Hamada and G. Shiu, “Weak Gravity Conjecture, Multiple Point Principle and the Standard Model Landscape”, JHEP 11 (2017) 043, arXiv:1707.06326 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  78. [78]

    Gilbarg and N

    D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order . Classics in Mathematics. Springer Berlin Heidelberg, 2001. https://books.google.com/books?id=eoiGTf4cmhwC

    work page 2001

  79. [79]

    Evans, Partial Differential Equations

    L. Evans, Partial Differential Equations . Graduate studies in mathematics. American Mathematical Society, 2010. https://books.google.com/books?id=Xnu0o_EJrCQC

    work page 2010

  80. [80]

    Gravity from thermodynamics: Optimal transport and negative effective dimensions

    G. B. D. Luca, N. De Ponti, A. Mondino, and A. Tomasiello, “Gravity from thermodynamics: Optimal transport and negative effective dimensions”, SciPost Phys. 15 (2023) no. 2, 039, arXiv:2212.02511 [hep-th]

    work page arXiv 2023

Showing first 80 references.