ABSTRACT
We give an explicit construction of a class of F-theory models with matter in the three-index symmetric (4) representation of SU(2). This matter is realized at codimen-sion two loci in the F-theory base where the divisor carrying the gauge group is singular; the associated Weierstrass model does not have the form associated with a generic SU(2) Tate model. For 6D theories, the matter is localized at a triple point singularity of arithmetic genus g = 3 in the curve supporting the SU(2) group. This is the first explicit realization of matter in F-theory in a representation corresponding to a genus contribution greater than one. The construction is realized by “unHiggsing” a model with a U(1) gauge factor under which there is matter with charge q = 3. The resulting SU(2) models can be further unHiggsed to realize non-Abelian G 2 × SU(2) models with more conventional matter content or SU(2)3 models with trifundamental matter. The U(1) models used as the basis for this construction do not seem to have a Weierstrass realization in the general form found by Morrison-Park, suggesting that a generalization of that form may be needed to incorporate models with arbitrary matter representations and gauge groups localized on singular divisors.
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References
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. 1, Nucl. Phys. B 473 (1996) 74 [hep-th/9602114] [INSPIRE].
D.R. Morrison and C. Vafa, Compactifications of F-theory on Calabi-Yau threefolds. 2., Nucl. Phys. B 476 (1996) 437 [hep-th/9603161] [INSPIRE].
K. Kodaira, On compact analytic surfaces: Ii, Ann. Math. 77 (1963) 563.
S.H. Katz and C. Vafa, Matter from geometry, Nucl. Phys. B 497 (1997) 146 [hep-th/9606086] [INSPIRE].
M. Bershadsky, K.A. Intriligator, S. Kachru, D.R. Morrison, V. Sadov and C. Vafa, Geometric singularities and enhanced gauge symmetries, Nucl. Phys. B 481 (1996) 215 [hep-th/9605200] [INSPIRE].
D.R. Morrison and W. Taylor, Matter and singularities, JHEP 01 (2012) 022 [arXiv:1106.3563] [INSPIRE].
M. Esole and S.-T. Yau, Small resolutions of SU(5)-models in F-theory, Adv. Theor. Math. Phys. 17 (2013) 1195 [arXiv:1107.0733] [INSPIRE].
H. Hayashi, C. Lawrie, D.R. Morrison and S. Schäfer-Nameki, Box Graphs and Singular Fibers, JHEP 05 (2014) 048 [arXiv:1402.2653] [INSPIRE].
M. Esole, S.-H. Shao and S.-T. Yau, Singularities and Gauge Theory Phases, Adv. Theor. Math. Phys. 19 (2015) 1183 [arXiv:1402.6331] [INSPIRE].
A.P. Braun and S. Schäfer-Nameki, Box Graphs and Resolutions I, Nucl. Phys. B 905 (2016) 447 [arXiv:1407.3520] [INSPIRE].
A. Grassi, J. Halverson and J.L. Shaneson, Matter From Geometry Without Resolution, JHEP 10 (2013) 205 [arXiv:1306.1832] [INSPIRE].
V. Kumar, D.S. Park and W. Taylor, 6D supergravity without tensor multiplets, JHEP 04 (2011) 080 [arXiv:1011.0726] [INSPIRE].
V. Sadov, Generalized Green-Schwarz mechanism in F-theory, Phys. Lett. B 388 (1996) 45 [hep-th/9606008] [INSPIRE].
M. Cvetič, D. Klevers, H. Piragua and W. Taylor, General U(1) × U(1) F-theory compactifications and beyond: geometry of unHiggsings and novel matter structure, JHEP 11 (2015) 204 [arXiv:1507.05954] [INSPIRE].
L.B. Anderson, J. Gray, N. Raghuram and W. Taylor, Matter in transition, JHEP 04 (2016) 080 [arXiv:1512.05791] [INSPIRE].
S. Katz, D.R. Morrison, S. Schäfer-Nameki and J. Sully, Tate’s algorithm and F-theory, JHEP 08 (2011) 094 [arXiv:1106.3854] [INSPIRE].
D. Klevers, D.R. Morrison, N. Raghuram and W. Taylor, to appear.
D. Klevers, D.K. Mayorga Pena, P.-K. Oehlmann, H. Piragua and J. Reuter, F-Theory on all Toric Hypersurface Fibrations and its Higgs Branches, JHEP 01 (2015) 142 [arXiv:1408.4808] [INSPIRE].
A. Grassi and D.R. Morrison, Anomalies and the Euler characteristic of elliptic Calabi-Yau threefolds, Commun. Num. Theor. Phys. 6 (2012) 51 [arXiv:1109.0042] [INSPIRE].
D.R. Morrison and D.S. Park, F-Theory and the Mordell-Weil Group of Elliptically-Fibered Calabi-Yau Threefolds, JHEP 10 (2012) 128 [arXiv:1208.2695] [INSPIRE].
D.R. Morrison and W. Taylor, Classifying bases for 6D F-theory models, Central Eur. J. Phys. 10 (2012) 1072 [arXiv:1201.1943] [INSPIRE].
D.R. Morrison and W. Taylor, Non-Higgsable clusters for 4D F-theory models, JHEP 05 (2015) 080 [arXiv:1412.6112] [INSPIRE].
D.S. Park, Anomaly Equations and Intersection Theory, JHEP 01 (2012) 093 [arXiv:1111.2351] [INSPIRE].
L.B. Anderson, I. Garc´ıa-Etxebarria, T.W. Grimm and J. Keitel, Physics of F-theory compactifications without section, JHEP 12 (2014) 156 [arXiv:1406.5180] [INSPIRE].
M. Cvetič, D. Klevers and H. Piragua, F-Theory Compactifications with Multiple U(1)-Factors: Constructing Elliptic Fibrations with Rational Sections, JHEP 06 (2013) 067 [arXiv:1303.6970] [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].
J. Erler, Anomaly cancellation in six-dimensions, J. Math. Phys. 35 (1994) 1819 [hep-th/9304104] [INSPIRE].
D.S. Park and W. Taylor, Constraints on 6D Supergravity Theories with Abelian Gauge Symmetry, JHEP 01 (2012) 141 [arXiv:1110.5916] [INSPIRE].
D.R. Morrison and W. Taylor, Sections, multisections and U(1) fields in F-theory, arXiv:1404.1527 [INSPIRE].
T.W. Grimm, A. Kapfer and D. Klevers, The Arithmetic of Elliptic Fibrations in Gauge Theories on a Circle, JHEP 06 (2016) 112 [arXiv:1510.04281] [INSPIRE].
C. Lawrie and D. Sacco, Tate’s algorithm for F-theory GUTs with two U(1)s, JHEP 03 (2015) 055 [arXiv:1412.4125] [INSPIRE].
J. Tate, Algorithm for determining the type of a singular fiber in an elliptic pencil, in Lecture Notes in Mathematics. Vol. 476: Modular functions of one variable IV, Springer, Berlin Germany (1975), pg. 33.
E. Witten, Phase transitions in M-theory and F-theory, Nucl. Phys. B 471 (1996) 195 [hep-th/9603150] [INSPIRE].
L. Bhardwaj, M. Del Zotto, J.J. Heckman, D.R. Morrison, T. Rudelius and C. Vafa, F-theory and the Classification of Little Strings, Phys. Rev. D 93 (2016) 086002 [arXiv:1511.05565] [INSPIRE].
W. Taylor, TASI Lectures on Supergravity and String Vacua in Various Dimensions, arXiv:1104.2051 [INSPIRE].
P. Griffiths and J. Harris, Principles of algebraic geometry, John Wiley & Sons, New York U.S.A. (2014).
M. Cvetič, D. Klevers, H. Piragua and P. Song, Elliptic fibrations with rank three Mordell-Weil group: F-theory with U(1) × U(1) × U(1) gauge symmetry, JHEP 03 (2014) 021 [arXiv:1310.0463] [INSPIRE].
D.R. Morrison and W. Taylor, Toric bases for 6D F-theory models, Fortsch. Phys. 60 (2012) 1187 [arXiv:1204.0283] [INSPIRE].
W. Taylor, On the Hodge structure of elliptically fibered Calabi-Yau threefolds, JHEP 08 (2012) 032 [arXiv:1205.0952] [INSPIRE].
G. Martini and W. Taylor, 6D F-theory models and elliptically fibered Calabi-Yau threefolds over semi-toric base surfaces, JHEP 06 (2015) 061 [arXiv:1404.6300] [INSPIRE].
S.B. Johnson and W. Taylor, Calabi-Yau threefolds with large h 2,1, JHEP 10 (2014) 23 [arXiv:1406.0514] [INSPIRE].
W. Taylor and Y.-N. Wang, Non-toric Bases for Elliptic Calabi-Yau Threefolds and 6D F-theory Vacua, arXiv:1504.07689 [INSPIRE].
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Klevers, D., Taylor, W. Three-index symmetric matter representations of SU(2) in F-theory from non-Tate form Weierstrass models. J. High Energ. Phys. 2016, 171 (2016). https://doi.org/10.1007/JHEP06(2016)171
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DOI: https://doi.org/10.1007/JHEP06(2016)171