Abstract
Root bundles appear prominently in studies of vector-like spectra of 4d F-theory compactifications. Of particular importance to phenomenology are the Quadrillion F-theory Standard Models (F-theory QSMs). In this work, we analyze a superset of the physical root bundles whose cohomologies encode the vector-like spectra for the matter representations (3, 2)1/6, (\( \overline{\textbf{3}} \), 1)−2/3 and (1, 1)1. For the family B3(\( {\Delta }_4^{{}^{\circ}} \)) consisting of \( \mathcal{O} \)(1011) F-theory QSM geometries, we argue that more than 99.995% of the roots in this superset have no vector-like exotics. This indicates that absence of vector-like exotics in those representations is a very likely scenario in the \( \mathcal{O} \)(1011) QSM geometries B3(\( {\Delta }_4^{{}^{\circ}} \)).
The QSM geometries come in families of toric 3-folds B3(∆°) obtained from triangulations of certain 3-dimensional polytopes ∆°. The matter curves in XΣ ∈ B3(∆°) can be deformed to nodal curves which are the same for all spaces in B3(∆°). Therefore, one can probe the vector-like spectra on the entire family B3(∆°) from studies of a few nodal curves. We compute the cohomologies of all limit roots on these nodal curves.
In our applications, for the majority of limit roots the cohomologies are determined by line bundle cohomology on rational tree-like curves. For this, we present a computer algorithm. The remaining limit roots, corresponding to circuit-like graphs, are handled by hand. The cohomologies are independent of the relative position of the nodes, except for a few circuits. On these jumping circuits, line bundle cohomologies can jump if nodes are specially aligned. This mirrors classical Brill-Noether jumps. B3(\( {\Delta }_4^{{}^{\circ}} \)) admits a jumping circuit, but the root bundle constraints pick the canonical bundle and no jump happens.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
P. Candelas, G.T. Horowitz, A. Strominger and E. Witten, Vacuum Configurations for Superstrings, Nucl. Phys. B 258 (1985) 46 [INSPIRE].
B.R. Greene, K.H. Kirklin, P.J. Miron and G.G. Ross, A Superstring Inspired Standard Model, Phys. Lett. B 180 (1986) 69 [INSPIRE].
V. Braun, Y.-H. He, B.A. Ovrut and T. Pantev, A Heterotic standard model, Phys. Lett. B 618 (2005) 252 [hep-th/0501070] [INSPIRE].
V. Bouchard and R. Donagi, An SU(5) heterotic standard model, Phys. Lett. B 633 (2006) 783 [hep-th/0512149] [INSPIRE].
V. Bouchard, M. Cvetič and R. Donagi, Tri-linear couplings in an heterotic minimal supersymmetric standard model, Nucl. Phys. B 745 (2006) 62 [hep-th/0602096] [INSPIRE].
L.B. Anderson, J. Gray, Y.-H. He and A. Lukas, Exploring Positive Monad Bundles And A New Heterotic Standard Model, JHEP 02 (2010) 054 [arXiv:0911.1569] [INSPIRE].
L.B. Anderson, J. Gray, A. Lukas and E. Palti, Two Hundred Heterotic Standard Models on Smooth Calabi-Yau Threefolds, Phys. Rev. D 84 (2011) 106005 [arXiv:1106.4804] [INSPIRE].
L.B. Anderson, J. Gray, A. Lukas and E. Palti, Heterotic Line Bundle Standard Models, JHEP 06 (2012) 113 [arXiv:1202.1757] [INSPIRE].
M. Berkooz, M.R. Douglas and R.G. Leigh, Branes intersecting at angles, Nucl. Phys. B 480 (1996) 265 [hep-th/9606139] [INSPIRE].
G. Aldazabal, S. Franco, L.E. Ibáñez, R. Rabadán and A.M. Uranga, D = 4 chiral string compactifications from intersecting branes, J. Math. Phys. 42 (2001) 3103 [hep-th/0011073] [INSPIRE].
G. Aldazabal, S. Franco, L.E. Ibáñez, R. Rabadán and A.M. Uranga, Intersecting brane worlds, JHEP 02 (2001) 047 [hep-ph/0011132] [INSPIRE].
L.E. Ibáñez, F. Marchesano and R. Rabadan, Getting just the standard model at intersecting branes, JHEP 11 (2001) 002.
R. Blumenhagen, B. Körs, D. Lüst and T. Ott, The standard model from stable intersecting brane world orbifolds, Nucl. Phys. B 616 (2001) 3 [hep-th/0107138] [INSPIRE].
M. Cvetič, G. Shiu and A.M. Uranga, Three-family supersymmetric standardlike models from intersecting brane worlds, Phys. Rev. Lett. 87 (2001) 201801 [hep-th/0107143] [INSPIRE].
M. Cvetič, G. Shiu and A.M. Uranga, Chiral four-dimensional N = 1 supersymmetric type 2A orientifolds from intersecting D6 branes, Nucl. Phys. B 615 (2001) 3 [hep-th/0107166] [INSPIRE].
R. Blumenhagen, M. Cvetič, P. Langacker and G. Shiu, Toward realistic intersecting D-brane models, Ann. Rev. Nucl. Part. Sci. 55 (2005) 71 [hep-th/0502005] [INSPIRE].
T.L. Gomez, S. Lukic and I. Sols, Constraining the Kähler moduli in the heterotic standard model, Commun. Math. Phys. 276 (2007) 1 [hep-th/0512205] [INSPIRE].
V. Bouchard and R. Donagi, On heterotic model constraints, JHEP 08 (2008) 060.
C. Vafa, Evidence for F-theory, Nucl. Phys. B 469 (1996) 403 [hep-th/9602022] [INSPIRE].
D.R. Morrison and C. Vafa, Compactifications of F-theory on Calabi-Yau threefolds. (II), Nucl. Phys. B 476 (1996) 437 [hep-th/9603161] [INSPIRE].
D.R. Morrison and C. Vafa, Compactifications of F-theory on Calabi-Yau threefolds. (I), Nucl. Phys. B 473 (1996) 74 [hep-th/9602114] [INSPIRE].
M. Cvetič, J. Halverson, G. Shiu and W. Taylor, Snowmass White Paper: String Theory and Particle Physics, arXiv:2204.01742 [INSPIRE].
T.W. Grimm and H. Hayashi, F-theory fluxes, Chirality and Chern-Simons theories, JHEP 03 (2012) 027 [arXiv:1111.1232] [INSPIRE].
S. Krause, C. Mayrhofer and T. Weigand, Gauge Fluxes in F-theory and Type IIB Orientifolds, JHEP 08 (2012) 119 [arXiv:1202.3138] [INSPIRE].
V. Braun, T.W. Grimm and J. Keitel, Geometric Engineering in Toric F-theory and GUTs with U(1) Gauge Factors, JHEP 12 (2013) 069 [arXiv:1306.0577] [INSPIRE].
M. Cvetič, A. Grassi, D. Klevers and H. Piragua, Chiral Four-Dimensional F-theory Compactifications With SU(5) and Multiple U(1)-Factors, JHEP 04 (2014) 010 [arXiv:1306.3987] [INSPIRE].
M. Cvetič, D. Klevers, D.K.M. Peña, P.-K. Oehlmann and J. Reuter, Three-Family Particle Physics Models from Global F-theory Compactifications, JHEP 08 (2015) 087 [arXiv:1503.02068] [INSPIRE].
L. Lin, C. Mayrhofer, O. Till and T. Weigand, Fluxes in F-theory Compactifications on Genus-One Fibrations, JHEP 01 (2016) 098 [arXiv:1508.00162] [INSPIRE].
L. Lin and T. Weigand, G4-flux and standard model vacua in F-theory, Nucl. Phys. B 913 (2016) 209 [arXiv:1604.04292] [INSPIRE].
S. Krause, C. Mayrhofer and T. Weigand, G4-flux, chiral matter and singularity resolution in F-theory compactifications, Nucl. Phys. B 858 (2012) 1 [arXiv:1109.3454] [INSPIRE].
M. Cvetič, L. Lin, M. Liu and P.-K. Oehlmann, An F-theory Realization of the Chiral MSSM with ℤ2-Parity, JHEP 09 (2018) 089 [arXiv:1807.01320] [INSPIRE].
M. Cvetič, J. Halverson, L. Lin, M. Liu and J. Tian, Quadrillion F -Theory Compactifications with the Exact Chiral Spectrum of the Standard Model, Phys. Rev. Lett. 123 (2019) 101601 [arXiv:1903.00009] [INSPIRE].
M. Bies, C. Mayrhofer, C. Pehle and T. Weigand, Chow groups, Deligne cohomology and massless matter in F-theory, arXiv:1402.5144 [INSPIRE].
M. Bies, C. Mayrhofer and T. Weigand, Gauge Backgrounds and Zero-Mode Counting in F-theory, JHEP 11 (2017) 081 [arXiv:1706.04616] [INSPIRE].
M. Bies, Cohomologies of coherent sheaves and massless spectra in F-theory, Ph.D. Thesis, Faculty of Physics and Astronomy, Heidelberg University, Heidelberg, Germany (2018) [DOI] [arXiv:1802.08860] [INSPIRE].
The Toric Varieties project authors, The ToricVarieties project, https://github.com/homalg-project/ToricVarieties_project, (2019).
D.R. Grayson and M.E. Stillman, Macaulay2, a software system for research in algebraic geometry, http://www.math.uiuc.edu/Macaulay2/.
The Sage Developers, Sagemath, the Sage Mathematics Software System (Version 8.5), http://www.sagemath.org, (2018).
J. Carifio, J. Halverson, D. Krioukov and B.D. Nelson, Machine Learning in the String Landscape, JHEP 09 (2017) 157 [arXiv:1707.00655] [INSPIRE].
J. Halverson, B. Nelson and F. Ruehle, Branes with Brains: Exploring String Vacua with Deep Reinforcement Learning, JHEP 06 (2019) 003 [arXiv:1903.11616] [INSPIRE].
S. Abel, A. Constantin, T.R. Harvey and A. Lukas, String Model Building, Reinforcement Learning and Genetic Algorithms, in Nankai Symposium on Mathematical Dialogues: In celebration of S.S.Chern’s 110th anniversary, Online China, August 2–13 2021 [arXiv:2111.07333] [INSPIRE].
F. Ruehle, Data science applications to string theory, Phys. Rept. 839 (2020) 1 [INSPIRE].
M. Bies, M. Cvetič, R. Donagi, L. Lin, M. Liu and F. Ruehle, Machine Learning and Algebraic Approaches towards Complete Matter Spectra in 4d F-theory, JHEP 01 (2021) 196 [arXiv:2007.00009] [INSPIRE].
M. Bies, M. Cvetič, R. Donagi, L. Lin, M. Liu and F. Ruehle, Database, https://github.com/Learning-line-bundle-cohomology, (2020).
Brill, Alexander and Noether, Max, Ueber die algebraischen Functionen und ihre Anwendung in der Geometrie, Math. Ann. 7 (1874) 269.
D. Eisenbud, M. Green and J. Harris, Cayley-Bacharach theorems and conjectures, Bull. Am. Math. Soc. 33 (1996) 295.
T. Watari, Vector-like pairs and Brill-Noether theory, Phys. Lett. B 762 (2016) 145 [arXiv:1608.00248] [INSPIRE].
P. Jefferson, W. Taylor and A.P. Turner, Chiral matter multiplicities and resolution-independent structure in 4D F-theory models, arXiv:2108.07810 [INSPIRE].
M. Bies, M. Cvetič, R. Donagi, M. Liu and M. Ong, Root bundles and towards exact matter spectra of F-theory MSSMs, JHEP 09 (2021) 076 [arXiv:2102.10115] [INSPIRE].
Lucia Caporaso, Cinzia Casagrande and Maurizio Cornalba, Moduli of Roots of Line Bundles on Curves, Trans. Am. Math. Soc. 359 (2007) 3733.
M. Bies, M. Cvetič and M. Liu, Statistics of limit root bundles relevant for exact matter spectra of F-theory MSSMs, Phys. Rev. D 104 (2021) L061903 [arXiv:2104.08297] [INSPIRE].
M. Kreuzer and H. Skarke, Classification of reflexive polyhedra in three-dimensions, Adv. Theor. Math. Phys. 2 (1998) 853 [hep-th/9805190] [INSPIRE].
J. Halverson and J. Tian, Cost of seven-brane gauge symmetry in a quadrillion F-theory compactifications, Phys. Rev. D 95 (2017) 026005 [arXiv:1610.08864] [INSPIRE].
V.V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Alg. Geom. 3 (1994) 493 [alg-geom/9310003] [INSPIRE].
E. Perevalov and H. Skarke, Enhanced gauged symmetry in type-II and F theory compactifications: Dynkin diagrams from polyhedra, Nucl. Phys. B 505 (1997) 679 [hep-th/9704129] [INSPIRE].
D.A. Cox and S. Katz, Mirror Symmetry and Algebraic Geometry, Mathematical surveys and monographs, American Mathematical Society (1999).
F. Rohsiepe, Lattice polarized toric K3 surfaces, hep-th/0409290 [INSPIRE].
A.P. Braun, C. Long, L. McAllister, M. Stillman and B. Sung, The Hodge Numbers of Divisors of Calabi-Yau Threefold Hypersurfaces, arXiv:1712.04946 [INSPIRE].
E. Witten, On flux quantization in M-theory and the effective action, J. Geom. Phys. 22 (1997) 1 [hep-th/9609122] [INSPIRE].
A. Collinucci and R. Savelli, On Flux Quantization in F-theory, JHEP 02 (2012) 015 [arXiv:1011.6388] [INSPIRE].
T. Weigand, F-theory, PoS TASI2017 (2018) 016 [arXiv:1806.01854] [INSPIRE].
Jesús A. De Loera, Jörg Rambau and Francisco Santos, Triangulations, Graduate studies in mathematics, Springer, Heidelberg Dordrecht London New York (2010) [DOI].
D. Cox, J. Little and H. Schenck, Toric Varieties, Graduate studies in mathematics, American Mathematical Society (2011) [DOI].
M. Kreuzer, Toric geometry and Calabi-Yau compactifications, Ukr. J. Phys. 55 (2010) 613 [hep-th/0612307] [INSPIRE].
J. Harris and I. Morrison, Moduli of Curves, Graduate Texts in Mathematics, Springer New York (2006) [ISBN: 9780387227375].
Rambau, Jörg, TOPCOM: Triangulations of Point Configurations and Oriented Matroids, in Proceedings of the International Congress of Mathematical Software, World Scientific (2002) [DOI].
Jordan, Charles and Joswig, Michael and Kastner, Lars, Parallel Enumeration of Triangulations, The Electronic Journal of Combinatorics 25 (2018) 3.
OSCAR — Open Source Computer Algebra Research system, Version 0.8.0-DEV, https://oscar.computeralgebra.de, (2022).
Eder, Christian and Decker, Wolfram and Fieker, Claus and Horn, Max and Joswig, Michael, ed., The OSCAR book, (2024).
Bezanson, Jeff and Edelman, Alan and Karpinski, Stefan and Shah, Viral B, Julia: A fresh approach to numerical computing, SIAM Rev. 59 (2017) 65.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2205.00008
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Bies, M., Cvetič, M., Donagi, R. et al. Brill-Noether-general limit root bundles: absence of vector-like exotics in F-theory Standard Models. J. High Energ. Phys. 2022, 4 (2022). https://doi.org/10.1007/JHEP11(2022)004
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP11(2022)004