Abstract
In this note we describe a method to calculate the action of a particular Fourier-Mukai transformation on a basis of brane charges on elliptically fibered Calabi-Yau threefolds with and without a section. The Fourier-Mukai kernel is the ideal sheaf of the relative diagonal and for fibrations that admit a section this is essentially the Poincaré sheaf. We find that in this case it induces an action of the modular group on the charges of 2-branes.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
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].
J. Gray, A.S. Haupt and A. Lukas, Topological Invariants and Fibration Structure of Complete Intersection Calabi-Yau Four-Folds, JHEP 09 (2014) 093 [arXiv:1405.2073] [INSPIRE].
L.B. Anderson, X. Gao, J. Gray and S.-J. Lee, Fibrations in CICY Threefolds, JHEP 10 (2017) 077 [arXiv:1708.07907] [INSPIRE].
Y.-C. Huang and W. Taylor, On the prevalence of elliptic and genus one fibrations among toric hypersurface Calabi-Yau threefolds, JHEP 03 (2019) 014 [arXiv:1809.05160] [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].
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.N. Schellekens and N.P. Warner, Anomalies, Characters and Strings, Nucl. Phys. B 287 (1987) 317 [INSPIRE].
A.N. Schellekens and N.P. Warner, Anomalies and modular invariance in string theory, Phys. Lett. B 177 (1986) 317 CERN-TH-4464-86 (1986).
A. Klemm, P. Mayr and C. Vafa, BPS states of exceptional noncritical strings, Nucl. Phys. Proc. Suppl. 58 (1997) 177 [hep-th/9607139] [INSPIRE].
B. Haghighat, A. Iqbal, C. Kozçaz, G. Lockhart and C. Vafa, M-Strings, Commun. Math. Phys. 334 (2015) 779 [arXiv:1305.6322] [INSPIRE].
S.-J. Lee, W. Lerche and T. Weigand, Tensionless Strings and the Weak Gravity Conjecture, JHEP 10 (2018) 164 [arXiv:1808.05958] [INSPIRE].
S.-J. Lee, W. Lerche and T. Weigand, A Stringy Test of the Scalar Weak Gravity Conjecture, Nucl. Phys. B 938 (2019) 321 [arXiv:1810.05169] [INSPIRE].
P. Candelas, A. Font, S.H. Katz and D.R. Morrison, Mirror symmetry for two parameter models. 2., Nucl. Phys. B 429 (1994) 626 [hep-th/9403187] [INSPIRE].
S. Hosono, M.H. Saito and A. Takahashi, Holomorphic anomaly equation and BPS state counting of rational elliptic surface, Adv. Theor. Math. Phys. 3 (1999) 177 [hep-th/9901151] [INSPIRE].
S. Hosono, Counting BPS states via holomorphic anomaly equations, Fields Inst. Commun. 38 (2013) 57 [hep-th/0206206] [INSPIRE].
A. Klemm, J. Manschot and T. Wotschke, Quantum geometry of elliptic Calabi-Yau manifolds, arXiv:1205.1795 [INSPIRE].
M. Alim and E. Scheidegger, Topological Strings on Elliptic Fibrations, Commun. Num. Theor. Phys. 08 (2014) 729 [arXiv:1205.1784] [INSPIRE].
M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes, Commun. Math. Phys. 165 (1994) 311 [hep-th/9309140] [INSPIRE].
M.-x. Huang, S. Katz and A. Klemm, Topological String on elliptic CY 3-folds and the ring of Jacobi forms, JHEP 10 (2015) 125 [arXiv:1501.04891] [INSPIRE].
G. Oberdieck and J. Shen, Curve counting on elliptic Calabi-Yau threefolds via derived categories, arXiv:1608.07073 [INSPIRE].
G. Oberdieck and A. Pixton, Holomorphic anomaly equations and the Igusa cusp form conjecture, Invent. Math. 213 (2018) 507 [arXiv:1706.10100] [INSPIRE].
G. Oberdieck and A. Pixton, Gromov-Witten theory of elliptic fibrations: Jacobi forms and holomorphic anomaly equations, arXiv:1709.01481 [INSPIRE].
S. Katz, Elliptically fibered Calabi-Yau threefolds: mirror symmetry and Jacobi forms, in String-Math Conference, 27 June 2016 – 2 July 2016, https://indico.cern.ch/event/375104/contributions/2153264/.
M. Del Zotto, J. Gu, M.-X. Huang, A.-K. Kashani-Poor, A. Klemm and G. Lockhart, Topological Strings on Singular Elliptic Calabi-Yau 3-folds and Minimal 6d SCFTs, JHEP 03 (2018) 156 [arXiv:1712.07017] [INSPIRE].
M. Kontsevich, Homological Algebra of Mirror Symmetry, alg-geom/9411018 [INSPIRE].
D. Orlov, Equivalences of derived categories and K3 surfaces, alg-geom/9606006.
D. Huybrechts, M. Huybrechts and O.U. Press, Fourier-Mukai Transforms in Algebraic Geometry, Oxford Mathematical Monographs, Clarendon Press, (2006), https://books.google.at/books?id=9HQTDAAAQBAJ.
R.P. Horja, Hypergeometric functions and mirror symmetry in toric varieties, math/9912109.
D.R. Morrison, Geometric aspects of mirror symmetry, math/0007090.
B. Andreas, G. Curio, D.H. Ruipérez and S.-T. Yau, Fourier-Mukai transform and mirror symmetry for D-branes on elliptic Calabi-Yau, math/0012196.
B. Andreas and D. Hernandez Ruipérez, Fourier Mukai transforms and applications to string theory, math/0412328.
R. Friedman, J. Morgan and E. Witten, Vector bundles and F-theory, Commun. Math. Phys. 187 (1997) 679 [hep-th/9701162] [INSPIRE].
P. Corvilain, T.W. Grimm and I. Valenzuela, The Swampland Distance Conjecture for Kähler moduli, arXiv:1812.07548 [INSPIRE].
D.H. Ruipérez, A. Martín and F.S. de Salas, Relative integral functors for singular fibrations and singular partners, math/0610319.
W. Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, (1984), https://books.google.at/books?id=cmoPAQAAMAAJ.
V. Braun, T.W. Grimm and J. Keitel, Complete Intersection Fibers in F-theory, JHEP 03 (2015) 125 [arXiv:1411.2615] [INSPIRE].
P.-K. Oehlmann, J. Reuter and T. Schimannek, Mordell-Weil Torsion in the Mirror of Multi-Sections, JHEP 12 (2016) 031 [arXiv:1604.00011] [INSPIRE].
C.F. Cota, A. Klemm and T. Schimannek, work in progress.
P.S. Aspinwall and M.R. Douglas, D-brane stability and monodromy, JHEP 05 (2002) 031 [hep-th/0110071] [INSPIRE].
P.S. Aspinwall, Some navigation rules for D-brane monodromy, J. Math. Phys. 42 (2001) 5534 [hep-th/0102198] [INSPIRE].
D. Erkinger and J. Knapp, Hemisphere Partition Function and Monodromy, JHEP 05 (2017) 150 [arXiv:1704.00901] [INSPIRE].
T. Weigand, TASI Lectures on F-theory, arXiv:1806.01854 [INSPIRE].
D.A. Cox and S. Katz, Mirror symmetry and algebraic geometry, AMS, (2000).
P.S. Aspinwall, D-branes on Calabi-Yau manifolds, in Progress in string theory. Proceedings, Summer School, TASI 2003, Boulder, U.S.A., June 2–27, 2003, pp. 1–152, hep-th/0403166 [INSPIRE].
H. Iritani, An integral structure in quantum cohomology and mirror symmetry for toric orbifolds, arXiv:0903.1463.
P.S. Aspinwall and D.R. Morrison, Nonsimply connected gauge groups and rational points on elliptic curves, JHEP 07 (1998) 012 [hep-th/9805206] [INSPIRE].
C. Mayrhofer, D.R. Morrison, O. Till and T. Weigand, Mordell-Weil Torsion and the Global Structure of Gauge Groups in F-theory, JHEP 10 (2014) 16 [arXiv:1405.3656] [INSPIRE].
R. Wazir, Arithmetic on Elliptic Threefolds, math/0112259.
A. Gerhardus and H. Jockers, Quantum periods of Calabi-Yau fourfolds, Nucl. Phys. B 913 (2016) 425 [arXiv:1604.05325] [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].
M. Esole and M.J. Kang, Characteristic numbers of elliptic fibrations with non-trivial Mordell-Weil groups, arXiv:1808.07054 [INSPIRE].
X. De la Ossa, B. Florea and H. Skarke, D-branes on noncompact Calabi-Yau manifolds: k-theory and monodromy, Nucl. Phys. B 644 (2002) 170 [hep-th/0104254] [INSPIRE].
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1902.08215
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Schimannek, T. Modularity from monodromy. J. High Energ. Phys. 2019, 24 (2019). https://doi.org/10.1007/JHEP05(2019)024
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2019)024