Abstract
We study elliptic fibrations that geometrically engineer an SU(2) ×G2 gauge theory realized by a Weierstrass model for the collision III + I * ns0 . We find all the distinct crepant resolutions of such a model and the flops connecting them. We compute the generating function for the Euler characteristic of the SU(2) ×G2-model. In the case of a Calabi-Yau threefold, we consider the compactification of M-theory and F-theory on an SU(2) × G2-model to a five and six-dimensional supergravity theory with eight super-charges. By matching each crepant resolution with each Coulomb chamber of the five-dimensional theory, we determine the number of multiplets and compute the prepotential in each Coulomb chamber. In particular, we discuss the counting numbers of hypermultiplets in the presence of singularities. We discuss in detail the cancellation of anomalies of the six-dimensional theory.
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References
P. Aluffi, Chern classes of blowups, Math. Proc. Cambridge Philos. Soc. 148 (2010) 227.
P. Aluffi and M. Esole, Chern class identities from tadpole matching in type IIB and F-theory, JHEP 03 (2009) 032 [arXiv:0710.2544] [INSPIRE].
J. de Boer, K. Papadodimas and E. Verlinde, Holographic neutron stars, JHEP 10 (2010) 020 [arXiv:0907.2695] [INSPIRE].
L.B. Anderson,M. Esole, L. Fredrickson and L.P. Schaposnik, Singular geometry and higgs bundles in string theory, SIGMA 14 (2018) 037.
L.B. Anderson, J.J. Heckman, S. Katz and L. Schaposnik, T-branes at the limits of geometry, JHEP 10 (2017) 058 [arXiv:1702.06137] [INSPIRE].
F. Apruzzi, J.J. Heckman, D.R. Morrison and L. Tizzano, 4D gauge theories with conformal matter, JHEP 09 (2018) 088 [arXiv:1803.00582] [INSPIRE].
P.S. Aspinwall and M. Gross, The SO(32) heterotic string on a K3 surface, Phys. Lett. B 387 (1996) 735 [hep-th/9605131] [INSPIRE].
P.S. Aspinwall, S.H. Katz and D.R. Morrison, Lie groups, Calabi-Yau threefolds and F-theory, Adv. Theor. Math. Phys. 4 (2000) 95 [hep-th/0002012] [INSPIRE].
S.D. Avramis and A. Kehagias, A systematic search for anomaly-free supergravities in six dimensions, JHEP 10 (2005) 052 [hep-th/0508172] [INSPIRE].
V. V. Batyrev, Birational Calabi-Yau n-folds have equal Betti numbers, in New trends in algebraic geometry (Warwick, 1996), London Mathematical Society Lecture Note Series volume 264, Cambridge University Press, Cambridge U.K. (1999).
F. Baume, M. Cvetič, C. Lawrie and L. Lin, When rational sections become cyclic — Gauge enhancement in F-theory via Mordell-Weil torsion, JHEP 03 (2018) 069 [arXiv:1709.07453] [INSPIRE].
C.W. Bernard, N.H. Christ, A.H. Guth and E.J. Weinberg, Instanton parameters for arbitrary gauge groups, Phys. Rev. D 16 (1977) 2967 [INSPIRE].
M. Bershadsky et al., Geometric singularities and enhanced gauge symmetries, Nucl. Phys. B 481 (1996) 215 [hep-th/9605200] [INSPIRE].
M. Bershadsky and A. Johansen, Colliding singularities in F-theory and phase transitions, Nucl. Phys. B 489 (1997) 122 [hep-th/9610111] [INSPIRE].
L. Bhardwaj, Classification of 6d \( \mathcal{N}=\left(1,0\right) \) gauge theories, JHEP 11 (2015) 002 [arXiv:1502.06594] [INSPIRE].
M. Bies, Cohomologies of coherent sheaves and massless spectra in F-theory, Ph.D. thesis, Heidelberg U., 2018-02. arXiv:1802.08860 [INSPIRE].
F. Bonetti and T.W. Grimm, Six-dimensional (1, 0) effective action of F-theory via M-theory on Calabi-Yau threefolds, JHEP 05 (2012) 019 [arXiv:1112.1082] [INSPIRE].
N. Bourbaki, Groups and Lie algebras. Chapters 7-9 (translated from the 1975 and 1982 French originals by A. Pressley, Elements of mathematics, Berlin, Germany), Springer, Berlin Germany (2005).
A.C. Cadavid, A. Ceresole, R. D’Auria and S. Ferrara, Eleven-dimensional supergravity compactified on Calabi-Yau threefolds, Phys. Lett. B 357 (1995) 76 [hep-th/9506144] [INSPIRE].
P. Candelas and A. Font, Duality between the webs of heterotic and type-II vacua, Nucl. Phys. B 511 (1998) 295 [hep-th/9603170] [INSPIRE].
P. Candelas, E. Perevalov and G. Rajesh, Matter from toric geometry, Nucl. Phys. B 519 (1998) 225 [hep-th/9707049] [INSPIRE].
A. Cattaneo, Crepant resolutions of Weierstrass threefolds and non-Kodaira fibres, arXiv:1307.7997 [INSPIRE].
A. Collinucci, F. Denef and M. Esole, D-brane deconstructions in IIB orientifolds, JHEP 02 (2009) 005 [arXiv:0805.1573] [INSPIRE].
A. Collinucci, M. Fazzi and R. Valandro, Geometric engineering on flops of length two, JHEP 04 (2018) 090 [arXiv:1802.00813] [INSPIRE].
M. Del Zotto and G. Lockhart, Universal features of BPS strings in six-dimensional SCFTs, JHEP 08 (2018) 173 [arXiv:1804.09694] [INSPIRE].
M. Del Zotto, J.J. Heckman, A. Tomasiello and C. Vafa, 6d conformal matter, JHEP 02 (2015) 054 [arXiv:1407.6359] [INSPIRE].
D.-E. Diaconescu and R. Entin, Calabi-Yau spaces and five-dimensional field theories with exceptional gauge symmetry, Nucl. Phys. B 538 (1999) 451 [hep-th/9807170] [INSPIRE].
L.J. Dixon, J.A. Harvey, C. Vafa and E. Witten, Strings on orbifolds. 2, Nucl. Phys. B 274 (1986) 285 [INSPIRE].
M. Esole, Introduction to elliptic fibrations, Math. Phys. Stud. 9783319654270 (2017) 247.
M. Esole, J. Fullwood and S.T. Yau, D 5 elliptic fibrations: non-Kodaira fibers and new orientifold limits of F-theory, Commun. Num. Theor. Phys. 09 (2015) 583.
M. Esole, S.G. Jackson, R. Jagadeesan and A.G. Noël, Incidence geometry in a Weyl chamber I: GL n, arXiv:1508.03038 [INSPIRE].
M. Esole, S.G. Jackson, R. Jagadeesan and A.G. Noël, Incidence Geometry in a Weyl chamber II: SL n, arXiv:1601.05070 [INSPIRE].
M. Esole, R. Jagadeesan and M.J. Kang, The geometry of G 2 , Spin(7) and Spin(8)-models, arXiv:1709.04913 [INSPIRE].
M. Esole, P. Jefferson and M.J. Kang, Euler characteristics of crepant resolutions of Weierstrass models, arXiv:1703.00905 [INSPIRE].
M. Esole and M.J. Kang, Flopping and slicing: SO(4) and Spin(4)-models, arXiv:1802.04802 [INSPIRE].
M. Esole, R. Jagadeesan and M.J. Kang, 48 crepant paths to SU(2) × SU(3), to appear.
M. Esole, M.J. Kang and S.-T. Yau, A new model for elliptic fibrations with a rank one Mordell-Weil group: I. Singular fibers and semi-stable degenerations, arXiv:1410.0003 [INSPIRE].
M. Esole, P. Jefferson and M.J. Kang, The geometry of F 4 -models, arXiv:1704.08251 [INSPIRE].
M. Esole, M.J. Kang and S.-T. Yau, Mordell-Weil torsion, anomalies and phase transitions, arXiv:1712.02337 [INSPIRE].
M. Esole and R. Savelli, Tate form and weak coupling limits in F-theory, JHEP 06 (2013) 027 [arXiv:1209.1633] [INSPIRE].
M. Esole and S.-H. Shao, M-theory on elliptic Calabi-Yau threefolds and 6d anomalies, arXiv:1504.01387 [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].
M. Esole, S.-H. Shao and S.-T. Yau, Singularities and gauge theory phases II, Adv. Theor. Math. Phys. 20 (2016) 683 [arXiv:1407.1867] [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].
J. Erler, Anomaly cancellation in six-dimensions, J. Math. Phys. 35 (1994) 1819 [hep-th/9304104] [INSPIRE].
J. Fullwood, On generalized Sethi-Vafa-Witten formulas, J. Math. Phys. 52 (2011) 082304 [arXiv:1103.6066] [INSPIRE].
A. Grassi and D.R. Morrison, Group representations and the Euler characteristic of elliptically fibered Calabi-Yau threefolds, J. Alg. Geom. 12 (2003) 321.
M.B. Green, J.H. Schwarz and P.C. West, Anomaly free chiral theories in six-dimensions, Nucl. Phys. B 254 (1985) 327 [INSPIRE].
T.W. Grimm and H. Hayashi, F-theory fluxes, chirality and Chern-Simons theories, JHEP 03 (2012) 027 [arXiv:1111.1232] [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].
J.J. Heckman, D.R. Morrison and C. Vafa, On the classification of 6D SCFTs and generalized ADE orbifolds, JHEP 05 (2014) 028 [Erratum ibid. 1506 (2015) 017] [arXiv:1312.5746] [INSPIRE].
J.J. Heckman and T. Rudelius, Top down approach to 6D SCFTs, arXiv:1805.06467 [INSPIRE].
K.A. Intriligator, D.R. Morrison and N. Seiberg, Five-dimensional supersymmetric gauge theories and degenerations of Calabi-Yau spaces, Nucl. Phys. B 497 (1997) 56 [hep-th/9702198] [INSPIRE].
P. Jefferson, S. Katz, H.-C. Kim and C. Vafa, On geometric classification of 5d SCFTs, JHEP 04 (2018) 103 [arXiv:1801.04036] [INSPIRE].
V.G. Kac. Infinite-dimensional Lie algebras, 3rd edition, Cambridge University Press, Cambridge U.K. (1990).
S.H. Katz and C. Vafa, Matter from geometry, Nucl. Phys. B 497 (1997) 146 [hep-th/9606086] [INSPIRE].
D. Klevers, D.R. Morrison, N. Raghuram and W. Taylor, Exotic matter on singular divisors in F-theory, JHEP 11 (2017) 124 [arXiv:1706.08194] [INSPIRE].
V. Kumar, D.R. Morrison and W. Taylor, Global aspects of the space of 6D N = 1 supergravities, JHEP 11 (2010) 118 [arXiv:1008.1062] [INSPIRE].
S.-J. Lee, D. Regalado and T. Weigand, 6d SCFTs and U(1) flavour symmetries, JHEP 11 (2018) 147 [arXiv:1803.07998] [INSPIRE].
J. Marsano and S. Schäfer-Nameki, Yukawas, G-flux and Spectral Covers from Resolved Calabi-Yau’s, JHEP 11 (2011) 098 [arXiv:1108.1794] [INSPIRE].
J.W.G. McKay and J. Patera, Tables of dimensions, indices, and branching rules for representations of simple Lie algebras, M. Dekker, New York U.S.A. (1981).
R. Miranda. Smooth models for elliptic threefolds, in The birational geometry of degenerations, R. Friedman and D.R. Morrison ed., Birkhäuser, Boston U.S.A. (1983).
S. Monnier, G.W. Moore and D.S. Park, Quantization of anomaly coefficients in 6D \( \mathcal{N}=\left(1,0\right) \) supergravity, JHEP 02 (2018) 020 [arXiv:1711.04777] [INSPIRE].
D.R. Morrison and W. Taylor, Matter and singularities, JHEP 01 (2012) 022 [arXiv:1106.3563] [INSPIRE].
D.R. Morrison and W. Taylor, Classifying bases for 6D F-theory models, Central Eur. J. Phys. 10 (2012) 1072.
D.S. Park, Anomaly equations and intersection theory, JHEP 01 (2012) 093 [arXiv:1111.2351] [INSPIRE].
S. Randjbar-Daemi, A. Salam, E. Sezgin and J.A. Strathdee, An anomaly free model in six-dimensions, Phys. Lett. 151B (1985) 351 [INSPIRE].
V. Sadov, Generalized Green-Schwarz mechanism in F-theory, Phys. Lett. B 388 (1996) 45 [hep-th/9606008] [INSPIRE].
A. Sagnotti, A note on the Green-Schwarz mechanism in open string theories, Phys. Lett. B 294 (1992) 196 [hep-th/9210127] [INSPIRE].
J.H. Schwarz, Anomaly-free supersymmetric models in six-dimensions, Phys. Lett. B 371 (1996) 223 [hep-th/9512053] [INSPIRE].
M.G. Szydlo, Flat regular models of elliptic schemes, ProQuest LLC, Ann Arbor, U.S.A. (1999).
T. van Ritbergen, A.N. Schellekens and J.A.M. Vermaseren, Group theory factors for Feynman diagrams, Int. J. Mod. Phys. A 14 (1999) 41 [hep-ph/9802376] [INSPIRE].
T. Weigand, TASI Lectures on F-theory, arXiv:1806.01854 [INSPIRE].
R. Wazir, Arithmetic on elliptic threefolds, Comp. Math. 140 (2004) 567.
E. Witten, Phase transitions in M-theory and F-theory, Nucl. Phys. B 471 (1996) 195 [hep-th/9603150] [INSPIRE].
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Esole, M., Kang, M.J. The Geometry of the SU(2) × G2-model. J. High Energ. Phys. 2019, 91 (2019). https://doi.org/10.1007/JHEP02(2019)091
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DOI: https://doi.org/10.1007/JHEP02(2019)091