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arxiv: 2605.23900 · v1 · pith:WIESTCQWnew · submitted 2026-05-22 · ✦ hep-th

What to do with a Ricci-flat Calabi--Yau metric?

Per Berglund , Tristan H\"ubsch , Vishnu Jejjala This is my paper

Pith reviewed 2026-05-25 03:27 UTC · model grok-4.3

classification ✦ hep-th
keywords Calabi-Yau metricsRicci-flat metricsstring compactificationsheterotic stringsYukawa couplingsmoduli stabilizationmirror symmetrynumerical geometry
0
0 comments X

The pith

Numerical approximations to Ricci-flat Calabi-Yau metrics allow computation of non-holomorphic quantities in string compactifications.

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

The paper argues that accurate numerical Ricci-flat metrics on Calabi-Yau manifolds shift explicit string compactifications from reliance on topology and holomorphy alone to the full geometric data. In heterotic models this supplies the inputs needed for matter Kähler metrics, canonically normalized Yukawa couplings, Kaluza-Klein spectra, threshold corrections, and soft terms. The same data also supply quantitative handles on moduli stabilization, alpha-prime corrections, de Sitter constructions, axion physics, swampland tests, and variable internal geometry, while opening computational routes to mathematical questions about special Lagrangians, SYZ fibrations, mirror symmetry, and metric degeneration.

Core claim

Numerical approximations to Ricci-flat Calabi-Yau metrics make it possible to move beyond the topological and holomorphic data that have traditionally dominated explicit string compactifications. In heterotic compactifications such data are needed to determine matter Kähler metrics, canonically normalized Yukawa couplings, Kaluza-Klein spectra, threshold effects, soft terms, and other non-holomorphic ingredients of the four-dimensional effective action. More broadly, numerical Calabi-Yau geometry provides quantitative input for moduli stabilization, alpha-prime-corrected backgrounds, de Sitter model building, axion physics, swampland distance tests, and compactifications in which theinternal

What carries the argument

Numerical approximations to Ricci-flat Calabi-Yau metrics together with the associated Hermitian Yang-Mills bundle data.

If this is right

  • Matter Kähler metrics and canonically normalized Yukawa couplings become determinable in heterotic compactifications.
  • Kaluza-Klein spectra, threshold effects, and soft supersymmetry-breaking terms can be calculated from the metric.
  • Quantitative input becomes available for moduli stabilization, alpha-prime corrections, and de Sitter model building.
  • Swampland distance conjectures and axion physics can be tested with explicit geometric data.
  • Computational approaches open to questions about special Lagrangian submanifolds, SYZ fibrations, and mirror symmetry.

Where Pith is reading between the lines

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

  • If the numerical metrics reach the stated accuracy, direct numerical comparison between different compactification schemes on the same manifold becomes possible.
  • The same data could be used to study compactifications in which the internal geometry varies over the external spacetime in a controlled way.
  • Mathematical conjectures about calibrated submanifolds and metric restrictions to fibers could be checked on explicit examples rather than abstractly.

Load-bearing premise

Numerical metrics of sufficient accuracy and the associated Hermitian Yang-Mills bundle data can be computed in practice.

What would settle it

A concrete heterotic example in which the required numerical accuracy cannot be reached for computing canonically normalized Yukawa couplings or matter Kähler metrics would show the listed applications remain out of reach.

read the original abstract

Numerical approximations to Ricci-flat Calabi--Yau metrics make it possible to move beyond the topological and holomorphic data that have traditionally dominated explicit string compactifications. This article explains what new physics and mathematics become accessible once the metric, and eventually the associated Hermitian Yang--Mills bundle data, can be computed. In heterotic compactifications, such data are needed to determine matter K\"ahler metrics, canonically normalized Yukawa couplings, Kaluza--Klein spectra, threshold effects, soft terms, and other non-holomorphic ingredients of the four-dimensional effective action. More broadly, numerical Calabi--Yau geometry provides quantitative input for moduli stabilization, $\alpha'$-corrected backgrounds, de~Sitter model building, axion physics, swampland distance tests, and compactifications in which the internal geometry varies over spacetime. Geometric data permit a computational approach to long-standing mathematical questions involving special Lagrangian submanifolds, SYZ fibrations, mirror symmetry, calibrated geometry, metric degeneration, restrictions of Ricci-flat metrics to fibers, and the search for analytic or semi-analytic structures. We present these directions as a roadmap for future work.

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

0 major / 2 minor

Summary. The manuscript is a forward-looking review and roadmap that argues numerical approximations to Ricci-flat Calabi-Yau metrics (and eventually associated Hermitian Yang-Mills data) will enable access to non-holomorphic quantities in string compactifications. It enumerates concrete applications in heterotic models (matter Kähler metrics, canonically normalized Yukawa couplings, Kaluza-Klein spectra, threshold corrections, soft terms) and broader topics (moduli stabilization, α'-corrections, de Sitter constructions, axion physics, swampland tests, spacetime-dependent geometry). On the mathematical side it lists questions involving special Lagrangians, SYZ fibrations, mirror symmetry, calibrated geometry, metric degeneration, and restrictions to fibers. All claims are explicitly conditioned on future availability of sufficiently accurate numerical data.

Significance. If the computational prerequisites are met, the identified directions would allow quantitative string phenomenology and geometric investigations that have been inaccessible with purely topological or holomorphic data. The paper performs a useful service by compiling these possibilities into a coherent agenda while avoiding over-claims about current numerical capabilities; this framing is appropriate for a discussion article and could usefully guide subsequent work.

minor comments (2)
  1. [Abstract] Abstract, final sentence: the phrase 'we present these directions as a roadmap' is slightly redundant with the preceding sentence; a single concluding sentence would suffice.
  2. The manuscript would benefit from a short dedicated paragraph (perhaps near the end) that explicitly lists the main computational bottlenecks that must still be overcome before the listed applications become feasible.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their thoughtful review and for recommending acceptance. The report accurately captures the scope and intent of the manuscript as a forward-looking roadmap.

Circularity Check

0 steps flagged

No significant circularity; forward-looking roadmap with no derivations or predictions

full rationale

The paper is a discussion of future directions conditional on the availability of accurate numerical Ricci-flat metrics and HYM data. It offers no equations, derivations, fitted parameters, or predictions that could reduce to inputs by construction. No load-bearing self-citations, uniqueness theorems, or ansatze are invoked. The central claim is explicitly hypothetical, making the text self-contained against external benchmarks with no circular steps.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

As a review paper the central claim rests on the existence and usability of numerical methods developed elsewhere; no new free parameters, axioms, or invented entities are introduced in this work.

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discussion (0)

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Reference graph

Works this paper leans on

174 extracted references · 174 canonical work pages · 80 internal anchors

  1. [1]

    Yau,Calabi’s conjecture and some new results in algebraic geometry,Proc

    S.-T. Yau,Calabi’s conjecture and some new results in algebraic geometry,Proc. Nat. Acad. Sci.74(1977) 1798–1799

    work page 1977

  2. [2]

    Yau,On the Ricci curvature of a compact K¨ ahler manifold and the complex Monge–Amp` ere equation

    S.-T. Yau,On the Ricci curvature of a compact K¨ ahler manifold and the complex Monge–Amp` ere equation. I,Commun. Pure Appl. Math.31(1978) 339–411

    work page 1978

  3. [3]

    Numerical Ricci-flat metrics on K3

    M. Headrick and T. Wiseman,Numerical Ricci-flat metrics on K3,Class. Quant. Grav.22 (2005) 4931–4960, [hep-th/0506129]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  4. [4]

    S. K. Donaldson,Some numerical results in complex differential geometry,Pure Appl. Math. Quart.5(12, 2009) 571–618, [math/0512625]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  5. [5]

    M. R. Douglas, R. L. Karp, S. Lukic and R. Reinbacher,Numerical Calabi–Yau metrics,J. Math. Phys.49(2008) 032302, [hep-th/0612075]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  6. [6]

    Energy functionals for Calabi-Yau metrics

    M. Headrick and A. Nassar,Energy functionals for Calabi–Yau metrics,Adv. Theor. Math. Phys.17(2013) 867–902, [0908.2635]. – 40 –

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  7. [7]

    Ashmore, Y.-H

    A. Ashmore, Y.-H. He and B. A. Ovrut,Machine learning Calabi–Yau metrics,Fortsch. Phys.68(2020) 2000068, [1910.08605]

    work page arXiv 2020

  8. [8]

    L. B. Anderson, M. Gerdes, J. Gray, S. Krippendorf, N. Raghuram and F. Ruehle, Moduli-dependent Calabi–Yau andSU(3)-structure metrics from machine learning,JHEP 05(2021) 013, [2012.04656]

    work page arXiv 2021

  9. [9]

    M. R. Douglas, S. Lakshminarasimhan and Y. Qi,Numerical Calabi–Yau metrics from holomorphic networks, inProceedings of the 2nd Mathematical and Scientific Machine Learning Conference(J. Bruna, J. Hesthaven and L. Zdeborova, eds.), vol. 107, pp. 223–252, 2021.2012.04797

    work page arXiv 2021

  10. [10]

    Jejjala, D

    V. Jejjala, D. K. Mayorga Pe˜ na and C. Mishra,Neural network approximations for Calabi–Yau metrics,JHEP08(2022) 105, [2012.15821]

    work page arXiv 2022

  11. [11]

    M. R. Douglas,Holomorphic feedforward networks,Pure Appl. Math. Quart.18(2022) 251–268, [2105.03991]

    work page arXiv 2022

  12. [12]

    Larfors, A

    M. Larfors, A. Lukas, F. Ruehle and R. Schneider,Numerical metrics for complete intersection and Kreuzer–Skarke Calabi–Yau manifolds,Mach. Learn. Sci. Tech.3(2022) 035014, [2205.13408]

    work page arXiv 2022

  13. [13]

    Berglund, G

    P. Berglund, G. Butbaia, T. H¨ ubsch, V. Jejjala, D. Mayorga Pe˜ na, C. Mishra and J. Tan, Machine learned Calabi–Yau metrics and curvature,Adv. Theor. Math. Phys.27(2023) 1107–1158, [2211.09801]

    work page arXiv 2023

  14. [14]

    Hendi, M

    Y. Hendi, M. Larfors and M. Walden,Learning group invariant Calabi–Yau metrics by fundamental domain projections,Mach. Learn. Sci. Tech.6(2025) 015050, [2407.06914]

    work page arXiv 2025

  15. [15]

    M. R. Douglas, S. Lakshminarasimhan and Y. Qi,MLGeometry, (2020) https://github.com/yidiq7/MLGeometry

    work page 2020

  16. [16]

    Larfors, A

    M. Larfors, A. Lukas, F. Ruehle and R. Schneider,Learning size and shape of Calabi–Yau spaces, inMachine Learning and the Physical Sciences, Workshop at 35th NeurIPS, 11, 2021.2111.01436

    work page arXiv 2021

  17. [17]

    Gerdes and S

    M. Gerdes and S. Krippendorf,CYJAX: A package for Calabi–Yau metrics with JAX, Mach. Learn. Sci. Tech.4(2023) 025031, [2211.12520]

    work page arXiv 2023

  18. [18]

    Butbaia, D

    G. Butbaia, D. Mayorga Pe˜ na, J. Tan, P. Berglund, T. H¨ ubsch, V. Jejjala and C. Mishra, cymyc: Calabi–Yau metrics, Yukawas, and curvature,JHEP03(2025) 28, [2410.19728]

    work page arXiv 2025

  19. [19]

    S. K. Donaldson,Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles,Proc. Lond. Math. Soc.50(1985) 1–26

    work page 1985

  20. [20]

    Uhlenbeck and S

    K. Uhlenbeck and S. T. Yau,On the existence of Hermitian-Yang-Mills connections in stable vector bundles,Commun. Pure Appl. Math.39(1986) S257–S293

    work page 1986

  21. [21]

    Distribution of zeros of random and quantum chaotic sections of positive line bundles

    B. Shiffman and S. Zelditch,Distribution of zeros of random and quantum chaotic sections of positive line bundles,Commun. Math. Phys.200(1999) 661–683, [math/9803052]

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  22. [22]

    Calabi-Yau Metrics for Quotients and Complete Intersections

    V. Braun, T. Brelidze, M. R. Douglas and B. A. Ovrut,Calabi–Yau metrics for quotients and complete intersections,JHEP05(2008) 080, [0712.3563]. – 41 –

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  23. [23]

    L. B. Anderson, V. Braun, R. L. Karp and B. A. Ovrut,Numerical hermitian Yang–Mills connections and vector bundle stability in heterotic theories,JHEP06(2010) 107, [1004.4399]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  24. [24]

    Bertrand,Calcul des probabilit´ es

    J. Bertrand,Calcul des probabilit´ es. Gauthier-Villars, Paris, 1889

    work page

  25. [25]

    E. T. Jaynes,The Well-Posed Problem,Found. Phys.3(1973) 477–493

    work page 1973

  26. [26]

    Berglund, G

    P. Berglund, G. Butbaia, T. H¨ ubsch, V. Jejjala, D. Mayorga Pe˜ na, C. Mishra and J. Tan, Point Selection for Spectral Network Approximations to Ricci-flat Calabi–Yau Metrics, Unpublished results presented at String Data, CalTech(2023)

    work page 2023

  27. [27]

    Ahmed and F

    H. Ahmed and F. Ruehle,Level crossings, attractor points and complex multiplication, JHEP06(2023) 164, [2304.00027]

    work page arXiv 2023

  28. [28]

    Calabi–Yau Manifolds

    F. Ruehle,Backreaction of Fluxes on Calabi–Yau Metrics,Talk at the Pollica Workshop “Calabi–Yau Manifolds”, June 2–6, 2025(2025)

    work page 2025

  29. [29]

    Numerical Weil-Petersson metrics on moduli spaces of Calabi-Yau manifolds

    J. Keller and S. Lukic,Numerical Weil-Petersson Metrics on Moduli Spaces of Calabi-Yau Manifolds,J. Geom. Phys.92(2015) 252–270, [0907.1387]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  30. [30]

    Candelas, G

    P. Candelas, G. T. Horowitz, A. Strominger and E. Witten,Vacuum Configurations for Superstrings,Nucl. Phys. B258(1985) 46–74

    work page 1985

  31. [31]

    Strominger,Yukawa Couplings in Superstring Compactification,Phys

    A. Strominger,Yukawa Couplings in Superstring Compactification,Phys. Rev. Lett.55 (1985) 2547

    work page 1985

  32. [32]

    Candelas and X

    P. Candelas and X. de la Ossa,Moduli Space of Calabi-Yau Manifolds,Nucl. Phys. B355 (1991) 455–481

    work page 1991

  33. [33]

    Candelas, X

    P. Candelas, X. C. de la Ossa, P. S. Green and L. Parkes,A pair of Calabi–Yau manifolds as an exactly soluble superconformal theory,Nucl. Phys. B359(1991) 21–74

    work page 1991

  34. [34]

    Butbaia, D

    G. Butbaia, D. Mayorga Pe˜ na, J. Tan, P. Berglund, T. H¨ ubsch, V. Jejjala and C. Mishra, Physical Yukawa couplings in heterotic string compactifications,Adv. Theor. Math. Phys. 28(2024) 2783–2822, [2401.15078]

    work page arXiv 2024

  35. [35]

    Berglund, G

    P. Berglund, G. Butbaia, T. H¨ ubsch, V. Jejjala, D. Mayorga Pe˜ na, C. Mishra and J. Tan, Precision string phenomenology,Phys. Rev. D111(2025) 086007, [2407.13836]

    work page arXiv 2025

  36. [36]

    Constantin, C

    A. Constantin, C. S. Fraser-Taliente, T. R. Harvey, A. Lukas and B. Ovrut,Computation of quark masses from string theory,Nucl. Phys. B1010(2025) 116778, [2402.01615]

    work page arXiv 2025

  37. [37]

    Constantin, C

    A. Constantin, C. S. Fraser-Taliente, T. R. Harvey, L. T. Leung and A. Lukas,Quark masses and mixing in string-inspired models,JHEP06(2025) 175, [2410.17704]

    work page arXiv 2025

  38. [38]

    Constantin, L

    A. Constantin, L. T.-Y. Leung, A. Lukas and L. A. Nutricati,Reproducing Standard Model fermion masses and mixing in string theory: A heterotic line bundle study,Phys. Rev. D 113(2026) 046005, [2507.03076]

    work page arXiv 2026

  39. [39]

    Mishra and J

    C. Mishra and J. Tan,Hermitian Yang–Mills connections on general vector bundles: geometry and physical Yukawa couplings,2512.10907

    work page arXiv

  40. [40]

    Constantin, A

    A. Constantin, A. Lukas and L. A. Nutricati,Calabi–Yau metrics with K¨ ahler moduli dependence,2603.12384. – 42 –

    work page arXiv

  41. [41]

    The Exact MSSM Spectrum from String Theory

    V. Braun, Y.-H. He, B. A. Ovrut and T. Pantev,The exact MSSM spectrum from string theory,JHEP05(2006) 043, [hep-th/0512177]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  42. [42]

    An SU(5) Heterotic Standard Model

    V. Bouchard and R. Donagi,AnSU(5)heterotic standard model,Phys. Lett. B633(2006) 783–791, [hep-th/0512149]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  43. [43]

    H¨ ubsch,Calabi–Yau Manifolds: a Bestiary for Physicists

    T. H¨ ubsch,Calabi–Yau Manifolds: a Bestiary for Physicists. World Scientific Pub. Europe Ltd., London, UK, 2nd ed., 2024 (1st ed., 1992, World Scientific Pub., Singapore)

    work page 2024

  44. [44]

    V. V. Batyrev,Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties,J. Alg. Geom.3(1994) 493–545, [alg-geom/9310003]

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  45. [45]

    Complete classification of reflexive polyhedra in four dimensions

    M. Kreuzer and H. Skarke,Complete classification of reflexive polyhedra in four-dimensions, Adv. Theor. Math. Phys.4(2000) 1209–1230, [hep-th/0002240]

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  46. [46]

    MacFadden, A

    N. MacFadden, A. Schachner and E. Sheridan,The DNA of Calabi–Yau hypersurfaces, 2405.08871

    work page arXiv

  47. [47]

    Berglund, G

    P. Berglund, G. Butbaia, Y.-H. He, E. Heyes, E. Hirst and V. Jejjala,Generating triangulations and fibrations with reinforcement learning,Phys. Lett. B860(2025) 139158, [2405.21017]

    work page arXiv 2025

  48. [48]

    MacFadden and E

    N. MacFadden and E. Sheridan,Calabi–Yau threefolds from vex triangulations,2512.14817

    work page arXiv

  49. [49]

    Sch¨ oller and H

    F. Sch¨ oller and H. Skarke,All Weight Systems for Calabi–Yau Fourfolds from Reflexive Polyhedra,Commun. Math. Phys.372(2019) 657–678, [1808.02422]

    work page arXiv 2019

  50. [50]

    Berglund, Y.-H

    P. Berglund, Y.-H. He, E. Heyes, E. Hirst, V. Jejjala and A. Lukas,New Calabi–Yau manifolds from genetic algorithms,Phys. Lett. B850(2024) 138504, [2306.06159]

    work page arXiv 2024

  51. [51]

    Nemeschansky and A

    D. Nemeschansky and A. Sen,Conformal invariance of supersymmetricσmodels on Calabi–Yau manifolds,Phys. Lett. B178(1986) 365–369

    work page 1986

  52. [52]

    String Corrected Spacetimes and SU(N)-Structure Manifolds

    K. Becker, M. Becker and D. Robbins,String corrected spacetimes andSU(N)-structure manifolds,Nucl. Phys. B898(2015) 715–735, [1503.04237]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  53. [53]

    C. M. Hull,Anomalies, Ambiguities and Superstrings,Phys. Lett. B167(1986) 51–55

    work page 1986

  54. [54]

    C. M. Hull,Compactifications of the Heterotic Superstring,Phys. Lett. B178(1986) 357–364

    work page 1986

  55. [55]

    Strominger,Superstrings with torsion,Nucl

    A. Strominger,Superstrings with torsion,Nucl. Phys. B274(1986) 253–284

    work page 1986

  56. [56]

    Li and S.-T

    J. Li and S.-T. Yau,Hermitian-Yang-Mills Connection on non-Kahler Manifolds, in Mathematical Aspects of String Theory, (Singapore), pp. 560–573, World Scientific, 1987

    work page 1987

  57. [57]

    Geometric Model for Complex Non-Kaehler Manifolds with SU(3) Structure

    E. Goldstein and S. Prokushkin,Geometric model for complex nonKahler manifolds with SU(3) structure,Commun. Math. Phys.251(2004) 65–78, [hep-th/0212307]

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  58. [58]

    The theory of superstring with flux on non-Kahler manifolds and the complex Monge-Ampere equation

    J.-X. Fu and S.-T. Yau,The theory of superstring with flux on non-Kahler manifolds and the complex Monge-Ampere equation,J. Diff. Geom.78(2008) 369–428, [hep-th/0604063]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  59. [59]

    Heterotic Flux Compactifications and Their Moduli

    K. Becker and L.-S. Tseng,Heterotic flux compactifications and their moduli,Nucl. Phys. B 741(2006) 162–179, [hep-th/0509131]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  60. [60]

    A Metric for Heterotic Moduli

    P. Candelas, X. de la Ossa and J. McOrist,A metric for heterotic moduli,Commun. Math. Phys.356(2017) 567–612, [1605.05256]. – 43 –

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  61. [61]

    C. S. Fraser-Taliente, T. R. Harvey and M. Kim,Not so flat metrics,2411.00962

    work page arXiv

  62. [62]

    The Euler characteristic correction to the Kaehler potential - revisited

    F. Bonetti and M. Weissenbacher,The Euler characteristic correction to the K¨ ahler potential — revisited,JHEP01(2017) 003, [1608.01300]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  63. [63]

    Supersymmetry Breaking and alpha'-Corrections to Flux Induced Potentials

    K. Becker, M. Becker, M. Haack and J. Louis,Supersymmetry breaking and alpha′-corrections to flux induced potentials,JHEP06(2002) 060, [hep-th/0204254]

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  64. [64]

    Soft supersymmetry-breaking terms from supergravity and superstring models

    A. Brignole, L. E. Ib´ a˜ nez and C. Mu˜ noz,Soft supersymmetry-breaking terms from supergravity and superstring models, inPerspectives on Supersymmetry, vol. 18 ofAdvanced Series on Directions in High Energy Physics, pp. 125–148. World Scientific, 1998. hep-ph/9707209. DOI

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  65. [65]

    Matter Field Kahler Metric in Heterotic String Theory from Localisation

    S ¸. Blesneag, E. I. Buchbinder, A. Constantin, A. Lukas and E. Palti,Matter field k¨ ahler metric in heterotic string theory from localisation,JHEP04(2018) 139, [1801.09645]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  66. [66]

    Scalar geometry and masses in Calabi-Yau string models

    D. Farquet and C. A. Scrucca,Scalar geometry and masses in Calabi–Yau string models, JHEP09(2012) 025, [1205.5728]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  67. [67]

    M. Berg, D. Marsh, L. McAllister and E. Pajer,Sequestering in string compactifications, JHEP06(2011) 134, [1012.1858]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  68. [68]

    The Leading Quantum Corrections to Stringy Kahler Potentials

    L. Anguelova, C. Quigley and S. Sethi,The leading quantum corrections to stringy K¨ ahler potentials,JHEP10(2010) 065, [1007.4793]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  69. [69]

    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. D68(2003) 046005, [hep-th/0301240]

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  70. [70]

    S. B. Giddings, S. Kachru and J. Polchinski,Hierarchies from fluxes in string compactifications,Phys. Rev. D66(2002) 106006, [hep-th/0105097]

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  71. [71]

    McAllister, J

    L. McAllister, J. Moritz, R. Nally and A. Schachner,Candidate de Sitter vacua,Phys. Rev. D111(2025) 086015, [2406.13751]

    work page arXiv 2025

  72. [72]

    I. R. Klebanov and M. J. Strassler,Supergravity and a confining gauge theory: Duality cascades andχSB-resolution of naked singularities,JHEP08(2000) 052, [hep-th/0007191]

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  73. [73]

    Brane/Flux Annihilation and the String Dual of a Non-Supersymmetric Field Theory

    S. Kachru, J. Pearson and H. Verlinde,Brane/flux annihilation and the string dual of a non-supersymmetric field theory,JHEP06(2002) 021, [hep-th/0112197]

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  74. [74]

    The Ubiquitous Throat

    A. Hebecker and J. March-Russell,The ubiquitous throat,Nucl. Phys. B781(2007) 99–111, [hep-th/0607120]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  75. [75]

    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,JHEP03(2005) 007, [hep-th/0502058]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  76. [76]

    D3-brane Potentials from Fluxes in AdS/CFT

    D. Baumann, A. Dymarsky, S. Kachru, I. R. Klebanov and L. McAllister,D3-brane potentials from fluxes in AdS/CFT,JHEP06(2010) 072, [1001.5028]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  77. [77]

    A Toolkit for Perturbing Flux Compactifications

    S. Gandhi, L. McAllister and S. Sjors,A toolkit for perturbing flux compactifications,JHEP 12(2011) 053, [1106.0002]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  78. [78]

    M. Dine, J. A. P. Law-Smith, S. Sun, D. Wood and Y. Yu,Obstacles to constructing de Sitter space in string theory,JHEP02(2021) 050, [2008.12399]. – 44 –

    work page arXiv 2021

  79. [79]

    U. H. Danielsson and T. Van Riet,What if string theory has no de Sitter vacua?,Int. J. Mod. Phys. D27(2018) 1830007, [1804.01120]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  80. [80]

    I. Bena, G. Giecold, M. Grana, N. Halmagyi and S. Massai,The backreaction of anti-D3 branes on the Klebanov-Strassler geometry,JHEP06(2013) 060, [1106.6165]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

Showing first 80 references.