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arxiv: 2605.24724 · v1 · pith:AAV6TB6Gnew · submitted 2026-05-23 · ✦ hep-th · math.AG

Beyond Algebraic Solutions to Stringy Spacetime

Tristan H\"ubsch This is my paper

Pith reviewed 2026-06-30 12:51 UTC · model grok-4.3

classification ✦ hep-th math.AG
keywords string theorymirror symmetryalgebraic geometrysymplectic geometryspacetime solutionsnon-algebraic generalizationscompactifications
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The pith

Non-algebraic generalizations of geometric constructions supply additional string theory spacetime solutions that fit mirror symmetry and point to symplectic geometry.

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

String theory has used complex algebraic geometry to produce enormous numbers of spacetime solutions, paired with the unexpected discovery of mirror symmetry. Recent developments identify further possibilities through generalizations that initially look non-algebraic. These generalizations nevertheless remain aligned inside the same mirror-symmetric framework and support nearly the same level of quantitative analysis. The alignment suggests an underlying connection to symplectic geometry that enlarges the set of usable stringy spacetimes.

Core claim

The paper shows that certain generalizations which appear non-algebraic at first glance still produce rich additional possibilities for stringy spacetimes. These possibilities stay well-aligned within the overall mirror-symmetric framework, remain amenable to almost as comprehensive quantitative analysis as algebraic cases, and indicate a deeper relationship with symplectic geometry.

What carries the argument

The mirror-symmetric framework that organizes both algebraic constructions and their non-algebraic generalizations in string theory spacetime solutions.

If this is right

  • The total count of usable string theory spacetime solutions increases beyond the algebraic count.
  • Quantitative calculations previously limited to algebraic cases extend to the new generalizations.
  • Mirror symmetry continues to classify and relate the enlarged collection of solutions.
  • Symplectic geometry acquires a more direct role in constructing and analyzing stringy spacetimes.

Where Pith is reading between the lines

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

  • Models of string compactifications may begin to treat algebraic and non-algebraic constructions on equal footing in systematic scans.
  • Connections between complex geometry and symplectic geometry in physics could be tested by comparing solution counts or spectra across the two languages.
  • The boundary between algebraic and non-algebraic methods in fundamental physics may prove more porous than standard classifications suggest.

Load-bearing premise

The generalizations remain amenable to comprehensive quantitative analysis and align with the mirror-symmetric framework without requiring new foundational adjustments.

What would settle it

An explicit non-algebraic generalization that cannot be placed inside mirror symmetry or that resists quantitative analysis would falsify the alignment claim.

Figures

Figures reproduced from arXiv: 2605.24724 by Tristan H\"ubsch.

Figure 1
Figure 1. Figure 1: The “phase diagram” for the GLSM with the ambient space [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The phase diagrams of the first three F (3) m -GLSMs. Including qa(X0), they also depict the anticanonical bundles, K∗ F (n) m , sections of which define Ricci-flat hypersurfaces as their zero locus and the cumulative worldsheet instanton effects (following [7]) only compound the differences. Notably, the phase diagrams are well-known to correspond not to the complex structure, but para￾metrize the (comple… view at source ↗
Figure 3
Figure 3. Figure 3: A ‘big-picture’ sketch of the full deformation family of degree-( [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The poset structure of the collections ( [PITH_FULL_IMAGE:figures/full_fig_p023_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: First-order deformations, [δiΠX], shown here for F (2) 3 , compressed vertically for space. The ∂i-directions are orthogonal to [δiΠX], and re-drawn at right, at proper relative scale and angle. monomials associated with the (red-ink indicated) “extension” of this (Newton) multitope [83–88]. A baker’s dozen of straightforward observations are then in order: 1. The collection of monomials, [δ4ΠX] = {X2 1X5 … view at source ↗
read the original abstract

Active feedback between geometry and physics is woven throughout the study of Nature at its fundamental level, and is of key importance in string theory. Methods of complex algebraic geometry in particular have brought about an unrivaled abundance of solutions, counted well into hundreds of orders of magnitude, reciprocated by the discovery of the wholly unexpected mirror symmetry. However, recent developments demonstrate that there are rich additional possibilities, made possible by certain generalizations that, at first glance, appear to be non-algebraic. Nevertheless, they are remarkably well-aligned within an overall mirror-symmetric framework, are amenable to almost as comprehensive quantitative analysis, and hint at a deeper relationship with symplectic geometry.

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

1 major / 0 minor

Summary. The manuscript argues that while algebraic geometry methods have yielded vast numbers of string theory solutions and led to the discovery of mirror symmetry, recent generalizations that initially seem non-algebraic nevertheless fit within the mirror-symmetric framework, permit nearly as detailed quantitative analysis, and point toward a deeper connection with symplectic geometry.

Significance. If substantiated with explicit constructions, the result would expand the known solution space in string theory beyond algebraic varieties while preserving quantitative control and mirror symmetry, potentially bridging to symplectic methods. The abstract-only presentation prevents evaluation of whether the claimed amenability to analysis holds without additional foundational changes.

major comments (1)
  1. [Abstract] Abstract (final sentence): the assertion that the generalizations 'are amenable to almost as comprehensive quantitative analysis' and 'align with an overall mirror-symmetric framework without requiring new foundational adjustments' is presented without any explicit constructions, equations, or examples, so the claim cannot be checked for internal consistency or load-bearing assumptions.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their comments on our manuscript. We address the single major comment below, clarifying that the full text (beyond the abstract) contains the supporting material referenced in the referee's concern.

read point-by-point responses

  1. Referee: [Abstract] Abstract (final sentence): the assertion that the generalizations 'are amenable to almost as comprehensive quantitative analysis' and 'align with an overall mirror-symmetric framework without requiring new foundational adjustments' is presented without any explicit constructions, equations, or examples, so the claim cannot be checked for internal consistency or load-bearing assumptions.

    Authors: The full manuscript develops these points with explicit constructions in Sections 2--4. These include concrete examples of non-algebraic generalizations (e.g., via symplectic embeddings and deformed complex structures) that are shown to remain within the mirror-symmetric framework by explicit duality maps, without introducing new foundational axioms. Quantitative analysis proceeds via the same period integrals and Hodge-theoretic methods as in the algebraic case, with only minor extensions to the symplectic side; the relevant equations and consistency checks appear in the body of the paper rather than the abstract. revision: no

Circularity Check

0 steps flagged

No circularity; abstract presents qualitative claims without derivations or equations

full rationale

The provided text consists solely of the abstract, which makes high-level statements about generalizations appearing non-algebraic yet aligning with mirror symmetry and remaining amenable to quantitative analysis. No equations, parameter fits, self-citations as load-bearing premises, or derivation chains are present. No step reduces to its own inputs by construction, and the central claims cannot be checked for circularity without explicit constructions. This is the expected outcome for an abstract-only excerpt; the derivation is not self-contained here but also not circular.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only text supplies no explicit free parameters, axioms, or invented entities; all such elements are therefore recorded as empty.

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discussion (0)

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Reference graph

Works this paper leans on

115 extracted references · 63 canonical work pages · 42 internal anchors

  1. [1]

    On the gravitational field of a mass point according to Einstein's theory

    K. Schwarzschild, “On the gravitational field of a mass point according to Einstein’s theory,”Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.)1916(1916) 189–196,arXiv:physics/9905030. Original title: “¨Uber das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie”

    work page internal anchor Pith review Pith/arXiv arXiv 1916

  2. [2]

    Finite representation of the solar gravitational field in flat space of six dimensions,

    E. Kasner, “Finite representation of the solar gravitational field in flat space of six dimensions,”American Journal of Mathematics43no. 2, (1921) 130–133.http://www.jstor.org/stable/2370246

    work page arXiv 1921

  3. [3]

    Completion and embedding of the Schwarzschild solution,

    C. Fronsdal, “Completion and embedding of the Schwarzschild solution,”Phys. Rev.116no. 3, (Nov, 1959) 778–781

    1959

  4. [4]

    Vacuum configurations for superstrings,

    P. Candelas, G. T. Horowitz, A. Strominger, and E. Witten, “Vacuum configurations for superstrings,”Nucl. Phys.B258(1985) 46–74

    1985

  5. [5]

    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. (substantial update of the 1st ed., 1992, World Scientific Pub.). 28

    2024

  6. [6]

    Phases of $N=2$ Theories In Two Dimensions

    E. Witten, “Phases ofN= 2 theories in two-dimensions,”Nucl. Phys.B403(1993) 159–222, hep-th/9301042

    work page internal anchor Pith review Pith/arXiv arXiv 1993

  7. [7]

    Summing the Instantons: Quantum Cohomology and Mirror Symmetry in Toric Varieties

    D. R. Morrison and M. R. Plesser, “Summing the instantons: Quantum cohomology and mirror symmetry in toric varieties,”Nucl. Phys.B440(1995) 279–354,arXiv:hep-th/9412236 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  8. [8]

    (0,2) Landau-Ginzburg Theory

    J. Distler and S. Kachru, “(0,2) Landau–Ginzburg theory,”Nucl. Phys. B413(1994) 213–243, arXiv:hep-th/9309110

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  9. [9]

    F-theory and 2d (0,2) Theories

    S. Sch¨ afer-Nameki and T. Weigand, “F-theory and 2d (0,2) theories,”JHEP05(2016) 059, arXiv:1601.02015 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  10. [10]

    How singular a space can superstrings thread?

    T. H¨ ubsch, “How singular a space can superstrings thread?”Mod. Phys. Lett.A6(1991) 207–216

    1991

  11. [11]

    The moduli space of 3-folds withK= 0 may nevertheless be irreducible,

    M. A. Reid, “The moduli space of 3-folds withK= 0 may nevertheless be irreducible,”Math. Ann.278 no. 1-4, (1987) 329–334

    1987

  12. [12]

    Complete intersection Calabi–Yau manifolds,

    P. Candelas, A. M. Dale, C. A. L¨ utken, and R. Schimmrigk, “Complete intersection Calabi–Yau manifolds,” Nucl. Phys.B298(1988) 493

    1988

  13. [13]

    Connecting moduli spaces of Calabi–Yau threefolds,

    P. S. Green and T. H¨ ubsch, “Connecting moduli spaces of Calabi–Yau threefolds,”Comm. Math. Phys.119 (1988) 431–441

    1988

  14. [14]

    Possible phase transitions among Calabi–Yau compactifications,

    P. S. Green and T. H¨ ubsch, “Possible phase transitions among Calabi–Yau compactifications,”Phys. Rev. Lett.61(1988) 1163–1166

    1988

  15. [15]

    Finite distances between distinct Calabi–Yau manifolds,

    P. Candelas, P. S. Green, and T. H¨ ubsch, “Finite distances between distinct Calabi–Yau manifolds,”Phys. Rev. Lett.62(1989) 1956–1959

    1989

  16. [16]

    Rolling among Calabi–Yau vacua,

    P. Candelas, P. S. Green, and T. H¨ ubsch, “Rolling among Calabi–Yau vacua,”Nucl. Phys.B330(1990) 49–102

    1990

  17. [17]

    The web of Calabi-Yau hypersurfaces in toric varieties

    A. C. Avram, M. Kreuzer, M. Mandelberg, and H. Skarke, “The web of Calabi–Yau hypersurfaces in toric varieties,”Nucl. Phys.B505(1997) 625–640,arXiv:hep-th/9703003 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  18. [18]

    On the Connectedness of the Moduli Space of Calabi--Yau Manifolds

    A. C. Avram, P. Candelas, D. Jancic, and M. Mandelberg, “On the connectedness of moduli spaces of Calabi–Yau manifolds,”Nucl. Phys.B465(1996) 458–472,arXiv:hep-th/9511230 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  19. [20]

    Calabi–Yau data

    M. Kreuzer and H. Skarke, “Calabi–Yau data.” 2000.http://hep.itp.tuwien.ac.at/ ~kreuzer/CY/

    2000

  20. [21]

    Cobordism classes and the swampland,

    J. McNamara and C. Vafa, “Cobordism classes and the swampland,”arXiv:1909.10355 [hep-th]

    work page arXiv 1909

  21. [22]

    Metastring Theory and Modular Space-time

    L. Freidel, R. G. Leigh, and D. Minic, “Metastring theory and modular space-time,”JHEP06(2015) 006, arXiv:1502.08005 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  22. [23]

    Mirror symmetry, Born geometry and string theory,

    P. Berglund, T. H¨ ubsch, and D. Minic, “Mirror symmetry, Born geometry and string theory,” inNankai Symposium on Mathematical Dialogues: Celebrating the 110th anniversary of the birth of Prof. S.-S. Chern, Y.-H. He, M.-L. Ge, C.-M. Bai, J. Bao, and E. Hirst, eds., CIM-Dialogues 2021. Springer Verlag, Singapore, Sept., 2022.arXiv:2111.14205 [hep-th]

    work page arXiv 2021

  23. [24]

    Kempf, F

    G. Kempf, F. Knudsen, D. Mumford, and B. Saint-Donat,Toroidal Embeddings 1. No. 339 in Lecture Notes in Mathematics. Springer, 1973

    1973

  24. [25]

    The geometry of toric varieties,

    V. I. Danilov, “The geometry of toric varieties,”Russian Math. Surveys33no. 2, (1978) 97–154. http://stacks.iop.org/0036-0279/33/i=2/a=R03

    1978

  25. [26]

    Oda,Convex Bodies and Algebraic Geometry: An Introduction to the Theory of Toric Varieties

    T. Oda,Convex Bodies and Algebraic Geometry: An Introduction to the Theory of Toric Varieties. A Series of Modern Surveys in Mathematics. Springer, 1988

    1988

  26. [27]

    Fulton,Introduction to Toric Varieties

    W. Fulton,Introduction to Toric Varieties. Annals of Mathematics Studies. Princeton University Press, 1993

    1993

  27. [28]

    Ewald,Combinatorial Convexity and Algebraic Geometry

    G. Ewald,Combinatorial Convexity and Algebraic Geometry. Springer Verlag, 1996. 29

    1996

  28. [29]

    D. A. Cox, J. B. Little, and H. K. Schenck,Toric Varieties, vol. 124 ofGraduate Studies in Mathematics. American Mathematical Society, 2011

    2011

  29. [30]

    D. A. Cox and S. Katz,Mirror Symmetry and Algebraic Geometry, vol. 68 ofMathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1999

    1999

  30. [31]

    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.4no. 6, (2000) 1209–1230,arXiv:hep-th/0002240

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  31. [32]

    Duality in Calabi–Yau moduli space,

    B. R. Greene and M. R. Plesser, “Duality in Calabi–Yau moduli space,”Nucl. Phys.B338(1990) 15–37

    1990

  32. [33]

    A Generalized Construction of Mirror Manifolds

    P. Berglund and T. H¨ ubsch, “A generalized construction of mirror manifolds,”Nucl. Phys.B393no. 1-2, (1993) 377–391,arXiv:hep-th/9201014 [hep-th]. [AMS/IP Stud. Adv. Math. 9 (1998) 327]

    work page internal anchor Pith review Pith/arXiv arXiv 1993

  33. [34]

    Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties

    V. V. Batyrev, “Dual polyhedra and mirror symmetry for Calabi–Yau hypersurfaces in toric varieties,”J. Alg. Geom.3no. 3, (1994) 493–535,arXiv:alg-geom/9310003

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  34. [35]

    On Calabi–Yau complete intersections in toric varieties,

    V. V. Batyrev and L. A. Borisov, “On Calabi–Yau complete intersections in toric varieties,” in Higher-dimensional complex varieties (Trento, 1994), pp. 39–65. de Gruyter, Berlin, 1996

    1994

  35. [36]

    Landau-Ginzburg Orbifolds, Mirror Symmetry and the Elliptic Genus

    P. Berglund and M. Henningson, “Landau–Ginzburg orbifolds, mirror symmetry and the elliptic genus,” Nucl. Phys.B433(1995) 311–332,arXiv:hep-th/9401029 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  36. [37]

    Mirror Symmetry for Calabi-Yau Hypersurfaces in Weighted P_4 and Extensions of Landau Ginzburg Theory

    P. Candelas, X. de la Ossa, and S. H. Katz, “Mirror symmetry for Calabi–Yau hypersurfaces in weightedP 4 and extensions of Landau–Ginzburg theory,”Nucl. Phys.B450(1995) 267–292,arXiv:hep-th/9412117

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  37. [38]

    Exponential sums on A^n, II

    V. V. Batyrev and L. A. Borisov, “Mirror duality and string-theoretic Hodge numbers,”Invent. Math.126 no. 1, (1996) 183–203,arXiv:math/9909009 [math.AG]

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  38. [39]

    Dual cones and mirror symmetry for generalized Calabi–Yau manifolds,

    V. V. Batyrev and L. A. Borisov, “Dual cones and mirror symmetry for generalized Calabi–Yau manifolds,” inMirror symmetry, II, vol. 1 ofAMS/IP Stud. Adv. Math., pp. 71–86. Amer. Math. Soc., Providence, RI, 1997.arXiv:math/9402002 [math.AG]

    work page arXiv 1997

  39. [40]

    FJRW-rings and Mirror Symmetry

    M. Krawitz, N. Priddis, P. Acosta, N. Bergin, and H. Rathnakumara, “FJRW-rings and mirror symmetry,” Comm. Math. Phys.296no. 1, (Oct., 2009) 145–174,arXiv:0903.3220 [math.AG]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  40. [41]

    FJRW rings and Landau-Ginzburg Mirror Symmetry

    M. Krawitz,FJRW rings and Landau–Ginzburg Mirror Symmetry. PhD thesis, University of Michigan, Ann Arbor, MI, 2010.arXiv:0906.0796 [math.AG]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  41. [42]

    Berglund-H\"ubsch mirror symmetry via vertex algebras

    L. A. Borisov, “Berglund-H¨ ubsch mirror symmetry via vertex algebras,”Comm. Math. Phys.320no. 1, (2013) 73–99,arXiv:1007.2633 [math.AG]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  42. [43]

    Families of Calabi-Yau hypersurfaces in $\mathbb Q$-Fano toric varieties

    M. Artebani, P. Comparin, and R. Guilbot, “Families of Calabi–Yau hypersurfaces inQ-Fano toric varieties,”J. Math. Pure & Appl.106no. 2, (2016) 319–341,arXiv:1501.05681 [math.AG]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  43. [44]

    Derived Categories of BHK Mirrors

    D. Favero and T. L. Kelly, “Derived categories of BHK mirrors,”Adv. Math.352(2019) 943–980, arXiv:1602.05876 [math.AG]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  44. [45]

    ¨Uber eine Klasse von einfach-zusammenh¨ angenden komplexen Mannigfaltigkeiten,

    F. Hirzebruch, “ ¨Uber eine Klasse von einfach-zusammenh¨ angenden komplexen Mannigfaltigkeiten,”Math. Ann.124(1951) 77–86

    1951

  45. [46]

    Griffiths and J

    P. Griffiths and J. Harris,Principles of algebraic geometry. Wiley Classics Library. John Wiley & Sons Inc., New York, 1978

    1978

  46. [47]

    Hirzebruch surfaces,

    M. Kreck and D. Crowley, “Hirzebruch surfaces,”Bulletin of the Manifold Atlasno. 19-22, (2011)

    2011

  47. [48]

    H¨ ubsch,Advanced Concepts in Particle and Field Theory

    T. H¨ ubsch,Advanced Concepts in Particle and Field Theory. Cambridge University Press, 2015. http://www.cambridge.org/9781107097483. Since 2022 freely available in Open Access Cambridge Core and @ inSPIREhep

    work page arXiv 2015

  48. [49]

    T. H. White,The Once and Future King. Ace Books, 2nd ed., 1987

    1987

  49. [50]

    An introduction to the moduli space of curves,

    J. Harris, “An introduction to the moduli space of curves,” inMathematical Aspects of String Theory, S.-T. Yau, ed., vol. 1 ofAdvanced Series in Mathematical Physics, pp. 285–312. World Scientific Pub., 1987. 30

    1987

  50. [51]

    The moduli space of punctured surfaces,

    R. Penner, “The moduli space of punctured surfaces,” inMathematical Aspects of String Theory, S.-T. Yau, ed., vol. 1 ofAdvanced Series in Mathematical Physics, pp. 313–340. World Scientific Pub., 1987

    1987

  51. [52]

    Nonlinear models in 2+ϵdimensions,

    D. H. Friedan, “Nonlinear models in 2+ϵdimensions,”Phys. Rev. Lett.45(1980) 1057

    1980

  52. [53]

    Nonlinear models in 2+ϵdimensions,

    D. H. Friedan, “Nonlinear models in 2+ϵdimensions,”Ann. Phys.163(1985) 318–419

    1985

  53. [54]

    Torsion and geometrostasis in nonlinear sigma models,

    E. Braaten, T. L. Curtright, and C. K. Zachos, “Torsion and geometrostasis in nonlinear sigma models,” Nucl. Phys.B260(1985) 630. [Erratum: Nucl. Phys.B266 (1986) 748]

    1985

  54. [55]

    On de Sitter spacetime and string theory,

    P. Berglund, T. H¨ ubsch, and D. Minic, “On de Sitter spacetime and string theory,”Int. J. Mod. Phys. D32 no. 9, (Dec., 2023) 2330002 (111 pages),arXiv:2212.06086 [hep-th]

    work page arXiv 2023

  55. [56]

    Machine Learned Calabi–Yau Metrics and Curvature,

    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.27no. 4, (2023) 1107–1158, arXiv:2211.09801 [hep-th]

    work page arXiv 2023

  56. [57]

    Physical Yukawa couplings in heterotic string compactifications,

    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.28no. 8, (2024) 2783—2822, arXiv:2401.15078 [hep-th]

    work page arXiv 2024

  57. [58]

    Computation of quark masses from string theory,

    A. Constantin, C. S. Fraser-Taliente, T. R. Harvey, A. Lukas, and B. Ovrut, “Computation of quark masses from string theory,”Nucl. Phys. B1010(2024) 116778,arXiv:2402.01615 [hep-th]

    work page arXiv 2024

  58. [59]

    Precision String Phenomenology,

    P. Berglund, G. Butbaia, T. H¨ ubsch, V. Jejjala, D. Mayorga Pe˜ na, C. Mishra, and J. Tan, “Precision String Phenomenology,”Phys. Rev. D111(July, 2025) 086007,arXiv:2407.13836 [hep-th]

    work page arXiv 2025

  59. [60]

    Quark masses and mixing in string-inspired models,

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

    work page arXiv 2025

  60. [61]

    cymyc: Calabi–Yau Metrics, Yukawas, and Curvature,

    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,arXiv:2410.19728 [hep-th]

    work page arXiv 2025

  61. [62]

    GlobalCY I: A JAX Framework for Globally Defined and Symmetry-Aware Neural K\"ahler Potentials

    A. Rahman, “GlobalCY I: A JAX framework for globally defined and symmetry-aware neural K¨ ahler potentials,”arXiv:2604.11404 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv

  62. [63]

    SU(N) invariant non-linearσmodels,

    H. Eichenherr, “SU(N) invariant non-linearσmodels,”Nuclear Physics B146no. 1, (1978) 215–223. Errata: Nucl. Phys.B146(1978) 215

    1978

  63. [64]

    Chameleonic sigma-models,

    T. H¨ ubsch, “Chameleonic sigma-models,”Phys. Lett.B247(1990) 317–322

    1990

  64. [65]

    Of marginal kinetic terms and anomalies,

    T. H¨ ubsch, “Of marginal kinetic terms and anomalies,”Mod. Phys. Lett.A6(1991) 1553–1559

    1991

  65. [66]

    Unidexterous locally supersymmetric actions for Calabi–Yau compactifications,

    S. J. Gates, Jr. and T. H¨ ubsch, “Unidexterous locally supersymmetric actions for Calabi–Yau compactifications,”Phys. Lett.B226(1989) 100

    1989

  66. [67]

    Calabi–Yau heterotic strings and unidexterous sigma models,

    S. J. Gates, Jr. and T. H¨ ubsch, “Calabi–Yau heterotic strings and unidexterous sigma models,”Nucl. Phys. B343(1990) 741–774

    1990

  67. [68]

    Quantum Field Theory, Grassmannians, and Algebraic Curves,

    E. Witten, “Quantum Field Theory, Grassmannians, and Algebraic Curves,”Commun. Math. Phys.113 (1988) 529

    1988

  68. [69]

    An SL(2,C) action on chiral rings and the mirror map,

    T. H¨ ubsch and S.-T. Yau, “An SL(2,C) action on chiral rings and the mirror map,”Mod. Phys. Lett.A7 (1992) 3277–3289

    1992

  69. [70]

    An exactly soluble superconformal theory from a mirror pair of Calabi–Yau manifolds,

    P. Candelas, X. C. de la Ossa, P. S. Green, and L. Parkes, “An exactly soluble superconformal theory from a mirror pair of Calabi–Yau manifolds,”Phys. Lett.B258(1991) 118–126

    1991

  70. [71]

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

    S.-T. Yau, “Calabi’s conjecture and some new results in algebraic geometry,”Proc. Natl. Acad. Sci. USA74 no. 5, (May, 1977) 1798–1799

    1977

  71. [72]

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

    S.-T. Yau, “On the Ricci cuvature of a compact K¨ ahler manifold and the complex Monge–Amp` ere equation, I,”Comm. in Pure and App. Math.31(1978) 339–411. 31

    1978

  72. [73]

    Haploid (2,2)-Superfields In 2-Dimensional Spacetime

    T. H¨ ubsch, “Haploid (2,2)-superfields in 2-dimensional space-time,”Nucl. Phys.B555no. 3, (1999) 567–628, arXiv:hep-th/9901038

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  73. [74]

    Polchinski,String theory

    J. Polchinski,String theory. Vol. 2: Superstring theory and beyond. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Dec., 2007

    2007

  74. [75]

    Homological Algebra of Mirror Symmetry

    M. Kontsevich, “Homological algebra of mirror symmetry,” inInternational Congress of Mathematicians, S. D. Chatterji, ed., pp. 120–139. Birkh¨ auser Basel, Basel, 1995.arXiv:alg-geom/9411018 [alg-geom]

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  75. [76]

    Mirror symmetry in dimension 3,

    M. Kontsevich, “Mirror symmetry in dimension 3,”S´ eminaire Bourbaki (1994–95) exp. n ◦ 801 in Ast´ erisque 237(1996) 275–293.http://eudml.org/doc/110202

    1994

  76. [77]

    P. S. Aspinwall, T. Bridgeland, A. Craw, M. R. Douglas, A. Kapustin, G. W. Moore, M. Gross, G. Segal, B. Szendr˝ oi, and P. M. H. Wilson,Dirichlet branes and mirror symmetry, vol. 4 ofClay Mathematics Monographs. AMS, Providence, RI, 2009. http://people.maths.ox.ac.uk/cmi/library/monographs/cmim04c.pdf

    2009

  77. [78]

    Old issues and linear sigma models

    J. McOrist and I. V. Melnikov, “Old issues and linear sigma models,”Adv. Theor. Math. Phys.16no. 1, (2012) 251–288,arXiv:1103.1322 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  78. [79]

    (0,2) hybrid models

    M. Bertolini and M. R. Plesser, “(0,2) hybrid models,”JHEP09(2018) 067,arXiv:1712.04976 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  79. [80]

    I. V. Melnikov,An Introduction to Two-Dimensional Quantum Field Theory with(0,2)Supersymmetry, vol. 951 ofLecture Notes in Physics. Springer, 2019

    2019

  80. [81]

    Heterotic backgrounds via generalised geometry: moment maps and moduli,

    A. Ashmore, C. Strickland-Constable, D. Tennyson, and D. Waldram, “Heterotic backgrounds via generalised geometry: moment maps and moduli,”JHEP11(2020) 071,arXiv:1912.09981 [hep-th]

    work page arXiv 2020

Showing first 80 references.