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Explicit String Vacua and the Geometric Selection Conjecture

A mix of human-written and AI-generated textHuman understanding: some partsmath-ph — Mathematical Physicsmath.AG — Algebraic Geometry

Contributed by Deep Bhattacharjee, Ushashi Bhattacharya ↗

Version 2 / Oct 23, 2026 / CC BY 4.0

Abstract

We combine explicit spectral calculations for the Dwork and weighted Fermat pencils with an audit of the Geometric Selection Conjecture. The residue calculation determines the character labels, cubic couplings and invariant-direction mass maps, while an index argument excludes three standard-embedding families on free quotients of thirteen hypergeometric complete-intersection types. For a nonstandard construction we fix one tetraquadric equation, prove smoothness and a free degree-four action, and calculate equivariant bundle cohomology by an exact determinant certificate. Its projected charged spectrum contains three families and one Higgs doublet pair. At the specified split-bundle point, however, an additional Abelian gauge factor survives, the perturbative Yukawa matrices fail the required rank test, and the tree-level volume remains flat. We then show that the first of these failures is forced: on every Calabi--Yau threefold with at most four K\"ahler parameters, every polystable sum of five line bundles with vanishing first Chern class has a charge matrix of rank at most three and keeps at least one Abelian gauge boson massless at its split locus. On the tetraquadric the polystable K\"ahler classes of the fixed bundle form exactly the volume ray. For the second failure we decide the perturbative selection rules on eleven split data, the fixed bundle and ten further line-bundle sums with three families on the same manifold, by a flow criterion for the singlet insertions: six of the eleven lose a Yukawa matrix identically and five are not excluded. We then follow the extension branch: a single non-split extension never removes the forced gauge boson, while for the fixed bundle and five of the ten further sums a polystable sum of two rank-two extensions and a line bundle, with the same Chern character, exists on one side of a stability wall arbitrarily close to the split point and carries no massless Abelian gauge boson; for three further sums no such bundle exists at any K\"ahler class, and for one it does not exist on the general tetraquadric. We fix the acceptance windows for the stabilization, Yukawa and supersymmetry breaking conditions, so that the conjecture becomes a definite statement, and show that the failures found do not depend on them. No vacuum satisfying all nine conditions is constructed; the Geometric Selection Conjecture remains unproved.

Provenance statement

Human authors: Deep Bhattacharjee, Ushashi Bhattacharya. Submitting contributor: Deep Bhattacharjee, ORCID 0000-0003-0466-750X. Text classification: to be chosen by the submitting author on the form (version 1 of the record, hexagon:2610.00493v1, was submitted as partly checked and understood by a human; this version has been read in full by the submitting author before upload if the stronger declaration is chosen). Basis: the submitting author's own declaration on the Hexagon form, not a claim verified by the assistant. Claude Code (Anthropic) assisted the preparation of the manuscript, the merger of the two earlier Hexagon parts, the proofs and the writing of Section 16 (including the extension-branch results of Section 16.3 and the acceptance windows of Definition 9.1), the Python checks of the finite arithmetic and the submission package. The Anaxia engine (creelie/eBLLM, release 3.1.0, 10.5281/zenodo.23484160) supplied the ten tetraquadric line-bundle classes of Proposition 16.4 and the atlas record that conditions C7 to C9 of the conjecture are fixed by comparison with measurement; it is cited in the paper. Documented tools: Anthropic Claude / Claude Code; Python 3 (exact rational arithmetic, 511 + 45 + 61 checks in the repository, the last with SymPy); LaTeX / pdfLaTeX (amsart); Matplotlib for the figures. Exact historical model versions are not inferred from tool names. Verification scope: every statement of the paper is proved in the text by hand; the programs check finite arithmetic (the matrix certificates, the charge matrices, the Yukawa flow bounds and the slope spaces, extension groups and wall certificates of the extension branch) and no statement rests on them alone. No Lean formalisation is claimed for this paper. The authors retain responsibility for the final mathematics, citations and approval of the manuscript. AI systems are not added as human contributors. The package creation step prepared the source archive and the submission metadata; it did not submit the manuscript to Hexagon. No private prompts, chat history or hidden reasoning traces are included or represented as an exported transcript. Earlier Hexagon versions: https://hexagonmath.org/2610.00493v1 (mirror clause), https://hexagonmath.org/2610.00446v1 (spectral calculations) Research manuscript/source DOI: https://doi.org/10.5281/zenodo.23688897 (release v3.1.0; all versions https://doi.org/10.5281/zenodo.23567114) Repository: https://github.com/creelie/geometric-selection-conjecture
AI provenance filesAI_PROVENANCE.md · 3 KB

Tools used

Anthropic
Claude CodeVersion Fable 5.1 Medium

References

  1. P. Candelas, G. T. Horowitz, A. Strominger and E. Witten, Vacuum configurations for superstrings, Nucl. Phys. B 258 (1985), 46--74, doi:10.1016/0550-3213(85)90602-9 (https://doi.org/10.1016/0550-3213(85)90602-9).DOI
  2. E. Witten, Symmetry breaking patterns in superstring models, Nucl. Phys. B 258 (1985), 75--100, doi:10.1016/0550-3213(85)90603-0 (https://doi.org/10.1016/0550-3213(85)90603-0).DOI
  3. A. Strominger, Yukawa couplings in superstring compactification, Phys. Rev. Lett. 55 (1985), 2547--2550, doi:10.1103/PhysRevLett.55.2547 (https://doi.org/10.1103/PhysRevLett.55.2547).DOI
  4. P. Candelas, Yukawa couplings between (2,1)-forms, Nucl. Phys. B 298 (1988), 458--492, doi:10.1016/0550-3213(88)90351-3 (https://doi.org/10.1016/0550-3213(88)90351-3).DOI
  5. 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. B 359 (1991), 21--74, doi:10.1016/0550-3213(91)90292-6 (https://doi.org/10.1016/0550-3213(91)90292-6).DOI
  6. L. J. Dixon, V. Kaplunovsky and J. Louis, On effective field theories describing (2,2) vacua of the heterotic string, Nucl. Phys. B 329 (1990), 27--82, doi:10.1016/0550-3213(90)90057-K (https://doi.org/10.1016/0550-3213(90)90057-K).DOI
  7. P. Candelas and X. C. de la Ossa, Moduli space of Calabi--Yau manifolds, Nucl. Phys. B 355 (1991), 455--481, doi:10.1016/0550-3213(91)90122-E (https://doi.org/10.1016/0550-3213(91)90122-E).DOI
  8. P. A. Griffiths, On the periods of certain rational integrals, I, II, Ann. of Math. 90 (1969), 460--541, doi:10.2307/1970746 (https://doi.org/10.2307/1970746).DOI
  9. J. A. Carlson and P. A. Griffiths, Infinitesimal variations of Hodge structure and the global Torelli problem, in Journées de géométrie algébrique d'Angers 1979 , Sijthoff and Noordhoff, Alphen aan den Rijn, 1980, 51--76.
  10. N. Hitchin, Harmonic spinors, Adv. Math. 14 (1974), 1--55, doi:10.1016/0001-8708(74)90021-8 (https://doi.org/10.1016/0001-8708(74)90021-8).DOI
Show all 41 referencesShow fewer references
  1. A. J. Hanson, A construction for computer visualization of certain complex curves, Notices Amer. Math. Soc. 41 (1994), 1156--1163.
  2. G. Butbaia, D. Mayorga Peña, J. Tan, P. Berglund, T. Hübsch, V. Jejjala and C. Mishra, Physical Yukawa couplings in heterotic string compactifications, Adv. Theor. Math. Phys. 28 (2024), 2783--2822, doi:10.4310/atmp.241119041341 (https://doi.org/10.4310/atmp.241119041341).DOI
  3. J. Distler and B. Greene, Some exact results on the superpotential from Calabi--Yau compactifications, Nucl. Phys. B 309 (1988), 295--316, doi:10.1016/0550-3213(88)90084-3 (https://doi.org/10.1016/0550-3213(88)90084-3).DOI
  4. F. Pham, Formules de Picard--Lefschetz généralisées et ramification des intégrales, Bull. Soc. Math. France 93 (1965), 333--367, doi:10.24033/bsmf.1628 (https://doi.org/10.24033/bsmf.1628).DOI
  5. M. Sebastiani and R. Thom, Un résultat sur la monodromie, Invent. Math. 13 (1971), 90--96, doi:10.1007/BF01390095 (https://doi.org/10.1007/BF01390095).DOI
  6. K. Sakamoto, The Seifert matrices of Milnor fiberings defined by holomorphic functions, J. Math. Soc. Japan 26 (1974), 714--721, doi:10.2969/jmsj/02640714 (https://doi.org/10.2969/jmsj/02640714).DOI
  7. D. R. Morrison, Picard--Fuchs equations and mirror maps for hypersurfaces, in Essays on Mirror Manifolds , International Press, Hong Kong, 1992, 241--264, arXiv:hep-th/9111025 (https://arxiv.org/abs/hep-th/9111025).arXiv
  8. A. Klemm and S. Theisen, Considerations of one-modulus Calabi--Yau compactifications: Picard--Fuchs equations, Kähler potentials and mirror maps, Nucl. Phys. B 389 (1993), 153--180, doi:10.1016/0550-3213(93)90289-2 (https://doi.org/10.1016/0550-3213(93)90289-2).DOI
  9. A. Font, Periods and duality symmetries in Calabi--Yau compactifications, Nucl. Phys. B 391 (1993), 358--388, doi:10.1016/0550-3213(93)90152-F (https://doi.org/10.1016/0550-3213(93)90152-F).DOI
  10. J. H. M. Steenbrink, Intersection form for quasi-homogeneous singularities, Compositio Math. 34 (1977), 211--223.
  11. I. Dolgachev, Weighted projective varieties, in Group Actions and Vector Fields , Lecture Notes in Math. 956 , Springer, Berlin, 1982, 34--71, doi:10.1007/BFb0101508 (https://doi.org/10.1007/BFb0101508).DOI
  12. M. F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes. II. Applications, Ann. of Math. (2) 88 (1968), 451--491, doi:10.2307/1970721 (https://doi.org/10.2307/1970721).DOI
  13. H. A. Hamm, Lokale topologische Eigenschaften komplexer Räume, Math. Ann. 191 (1971), 235--252, doi:10.1007/BF01578709 (https://doi.org/10.1007/BF01578709).DOI
  14. C. F. Doran and J. W. Morgan, Mirror symmetry and integral variations of Hodge structure underlying one-parameter families of Calabi--Yau threefolds, in Mirror Symmetry V , AMS/IP Stud. Adv. Math. 38 , Amer. Math. Soc., Providence, RI, 2006, 517--537, doi:10.1090/amsip/038/22 (https://doi.org/10.1090/amsip/038/22).DOI
  15. V. Braun, On free quotients of complete intersection Calabi--Yau manifolds, J. High Energy Phys. 2011 (2011), no. 4, 005, doi:10.1007/JHEP04(2011)005 (https://doi.org/10.1007/JHEP04(2011)005).DOI
  16. D. Bhattacharjee, U. Bhattacharya and S. Ghosh, One missing axiom for the complete closure of the heterotic landscape , Preprints.org (2026), version 2, doi:10.20944/preprints202603.0792.v2 (https://doi.org/10.20944/preprints202603.0792.v2). Cited for the formulation of the nine conditions, not as a proved existence theorem.DOI
  17. D. Bhattacharjee and U. Bhattacharya, Which string vibrations become which generations on the Dwork quintic , Hexagon 2610.00446v1 (https://hexagonmath.org/2610.00446v1); code archive creelie/strings v1.2.0 (https://github.com/creelie/strings/releases/tag/v1.2.0).hexagon
  18. D. Bhattacharjee and U. Bhattacharya, The mirror clause of the Geometric Selection Conjecture is empty , Hexagon 2610.00493v1 (https://hexagonmath.org/2610.00493v1); doi:10.5281/zenodo.23567115 (https://doi.org/10.5281/zenodo.23567115).hexagonDOI
  19. E. I. Buchbinder, A. Constantin and A. Lukas, The moduli space of heterotic line bundle models: a case study for the tetra-quadric , J. High Energy Phys. 03 (2014), 025, doi:10.1007/JHEP03(2014)025 (https://doi.org/10.1007/JHEP03(2014)025). The bundle in eq:bundle is the published example of Section 5.1.DOI
  20. 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, doi:10.1103/fbdz-t73z (https://doi.org/10.1103/fbdz-t73z).DOI
  21. J. M. Lee, Introduction to Smooth Manifolds , second ed., Graduate Texts in Mathematics 218, Springer, 2013.
  22. D. Bhattacharjee and U. Bhattacharya, Explicit String Vacua and the Geometric Selection Conjecture , version 2.1.0 (2026), doi:10.5281/zenodo.23647986 (https://doi.org/10.5281/zenodo.23647986).DOI
  23. L. B. Anderson, J. Gray, A. Lukas and E. Palti, Heterotic line bundle standard models , J. High Energy Phys. 06 (2012), 113, doi:10.1007/JHEP06(2012)113 (https://doi.org/10.1007/JHEP06(2012)113).DOI
  24. L. R. Ford and D. R. Fulkerson, Maximal flow through a network , Canad. J. Math. 8 (1956), 399--404, doi:10.4153/CJM-1956-045-5 (https://doi.org/10.4153/CJM-1956-045-5).DOI
  25. L. B. Anderson, J. Gray, A. Lukas and B. Ovrut, The edge of supersymmetry: stability walls in heterotic theory , Phys. Lett. B 677 (2009), 190--194, doi:10.1016/j.physletb.2009.05.025 (https://doi.org/10.1016/j.physletb.2009.05.025).DOI
  26. L. B. Anderson, J. Gray, A. Lukas and B. Ovrut, Stability walls in heterotic theories , J. High Energy Phys. 09 (2009), 026, doi:10.1088/1126-6708/2009/09/026 (https://doi.org/10.1088/1126-6708/2009/09/026).DOI
  27. D. Huybrechts and M. Lehn, The Geometry of Moduli Spaces of Sheaves , second ed., Cambridge Mathematical Library, Cambridge University Press, 2010, doi:10.1017/CBO9780511711985 (https://doi.org/10.1017/CBO9780511711985).DOI
  28. Z.-z. Xing, H. Zhang and S. Zhou, Updated values of running quark and lepton masses , Phys. Rev. D 77 (2008), 113016, doi:10.1103/PhysRevD.77.113016 (https://doi.org/10.1103/PhysRevD.77.113016).DOI
  29. T. Banks, D. B. Kaplan and A. E. Nelson, Cosmological implications of dynamical supersymmetry breaking , Phys. Rev. D 49 (1994), 779--787, doi:10.1103/PhysRevD.49.779 (https://doi.org/10.1103/PhysRevD.49.779).DOI
  30. B. S. Acharya, G. Kane and P. Kumar, Compactified string theories---generic predictions for particle physics , Int. J. Mod. Phys. A 27 (2012), 1230012, doi:10.1142/S0217751X12300128 (https://doi.org/10.1142/S0217751X12300128).DOI
  31. D. Bhattacharjee, Anaxia: the BLLM engine , https://github.com/creelie/eBLLM, module construct/lbs_tetraquadric.py at commit a85ae17 ; engine release 3.1.0, doi:10.5281/zenodo.23484160 (https://doi.org/10.5281/zenodo.23484160).DOI

Version history

  1. v1 · 5 pagesSubmitted by Deep BhattacharjeeInitial depositSupersededOct 23, 2026
  2. v2 · 49 pagesSubmitted by Deep BhattacharjeeSubmitted as a new versionCurrentOct 27, 2026