Abstract
For compact Calabi-Yau geometries with D5-branes we study N = 1 effective superpotentials depending on both open- and closed-string fields. We develop methods to derive the open/closed Picard-Fuchs differential equations, which control D5-brane deformations as well as complex structure deformations of the compact Calabi-Yau space. Their solutions encode the flat open/closed coordinates and the effective superpotential. For two explicit examples of compact D5-brane Calabi-Yau hypersurface geometries we apply our techniques and express the calculated superpotentials in terms of flat open/closed coordinates. By evaluating these superpotentials at their critical points we reproduce the domain wall tensions that have recently appeared in the literature. Finally we extract orbifold disk invariants from the superpotentials, which, up to overall numerical normalizations, correspond to orbifold disk Gromov-Witten invariants in the mirror geometry.
Similar content being viewed by others
References
Polchinski J.: Dirichlet-Branes and Ramond-Ramond Charges. Phys. Rev. Lett. 75, 4724 (1995)
Kontsevich, M.: Homological Algebra of Mirror Symmetry. http://arXiv.org/abs/alg-geom/9411018v1, 1994
Hori, K., Katz, S., Klemm, A., Pandharipande, R., Thomas, R., Vafa, C., Vakil, R., Zaslow, E.: Mirror Symmetry. Providence RI: Amer. Math. Soc., 2003
Witten, E.: Mirror manifolds and topological field theory. http://arXiv.org/abs/hep-th/9112056v1, 1991
Antoniadis I., Gava E., Narain K.S., Taylor T.R.: Topological amplitudes in string theory. Nucl. Phys. B 413, 162 (1994)
Bershadsky M., Cecotti S., Ooguri H., Vafa C.: Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes. Commun. Math. Phys. 165, 311 (1994)
Douglas M.R.: D-branes, categories and N = 1 supersymmetry. J. Math. Phys. 42, 2818 (2001)
Lazaroiu C.I.: Generalized complexes and string field theory. JHEP 0106, 052 (2001)
Aspinwall P.S., Lawrence A.E.: Derived categories and zero-brane stability. JHEP 0108, 004 (2001)
Lazaroiu C.I.: D-brane categories. Int. J. Mod. Phys. A 18, 5299 (2003)
Aspinwall, P.S.: D-branes on Calabi-Yau manifolds. http://arXiv.org/abs/hep-th/0403166v1, 2004
Witten E.: Chern-Simons Gauge Theory as a String Theory. Prog. Math. 133, 637 (1995)
Kachru S., Katz S.H., Lawrence A.E., McGreevy J.: Open string instantons and superpotentials. Phys. Rev. D 62, 026001 (2000)
Ooguri H., Vafa C.: Knot invariants and topological strings. Nucl. Phys. B 577, 419 (2000)
Labastida J.M.F., Mariño M.: Polynomial invariants for torus knots and topological strings. Commun. Math. Phys. 217, 423 (2001)
Bouchard, V., Klemm, A., Mariño, M., Pasquetti, S.: Remodeling the B-model. http://arXiv.org/abs/0709.1453v1[hep-th], 2007
Brini, A., Tanzini, A.: Exact results for topological strings on resolved Y(p,q) singularities. http://arXiv.org/abs/0804.2598v3[hep-th], 2008
Bouchard, V., Klemm, A., Mariño, M., Pasquetti, S.: Topological open strings on orbifolds. http://arXiv.org/abs/0807.0597v1[hep-th], 2008
Coates, T., Corti, A., Iritani, H., Tseng, H.-H.: Computing Genus-Zero Twisted Gromov-Witten Invariants. http://arXiv.org/abs/math/0702234v3[math.AG], 2007
Bayer, A., Cadman, C.: Quantum cohomology of \({[\mathbb C^n/\mu_r]}\) . http://arXiv.org/abs/0705.2160v1[math.AG], 2007
Bouchard, V., Cavalieri, R.: On the mathematics and physics of high genus invariants of \({\mathbb {C}^3/\mathbb {Z}_3}\) . http://arXiv.org/abs/0709.1453v1[math.AG], 2007
Aganagic, M., Vafa, C.: Mirror symmetry, D-branes and counting holomorphic discs. http://arXiv.org/abs/hep-th/0012041v1, 2000
Aganagic M., Klemm A., Vafa C.: Disk instantons, mirror symmetry and the duality web. Z. Naturforsch. A 57, 1 (2002)
Lerche, W., Mayr, P., Warner, N.: Holomorphic N = 1 special geometry of open-closed type II strings. http://arXiv.org/abs/hep-th/0207259v2, 2002
Lerche, W., Mayr, P., Warner, N.: N = 1 special geometry, mixed Hodge variations and toric geometry. http://arXiv.org/abs/hep-th/0208039v1, 2002
Aganagic M., Klemm A., Mariño M., Vafa C.: The topological vertex. Commun. Math. Phys. 254, 425 (2005)
Walcher J.: Opening mirror symmetry on the quintic. Commun. Math. Phys. 276, 671 (2007)
Morrison, D.R., Walcher, J.: D-branes and Normal Functions. http://arXiv.org/abs/0709.4028v1[hep-th], 2007
Knapp, J., Scheidegger, E.: Towards Open String Mirror Symmetry for One-Parameter Calabi-Yau Hypersurfaces. http://arXiv.org/abs/0805.1013v2[hep-th], 2008
Krefl D., Walcher J.: Real Mirror Symmetry for One-parameter Hypersurfaces. JHEP 0809, 031 (2008)
Taylor T.R., Vafa C.: RR flux on Calabi-Yau and partial supersymmetry breaking. Phys. Lett. B 474, 130 (2000)
Gukov, S., Vafa, C., Witten, E.: CFT’s from Calabi-Yau four-folds. Nucl. Phys. B 584, 69 (2000) [Erratum-ibid. B 608, 477 (2001)]
Witten E.: Branes and the dynamics of QCD. Nucl. Phys. B 507, 658 (1997)
Mayr P.: N = 1 mirror symmetry and open/closed string duality. Adv. Theor. Math. Phys. 5, 213 (2002)
Cachazo F., Intriligator K.A., Vafa C.: A large N duality via a geometric transition. Nucl. Phys. B 603, 3 (2001)
Kachru S., Katz S.H., Lawrence A.E., McGreevy J.: Mirror symmetry for open strings. Phys. Rev. D 62, 126005 (2000)
Karoubi M., Leruste C.: Algebraic topology via differential geometry. Cambridge University Press, Cambridge (1987)
Deligne P.: Théorie de Hodge II. Publ. Math. I.H.E.S. 40, 5 (1971)
Voisin C.: Hodge Theory and Complex Algebraic Geometry II. Cambridge University Press, Cambridge (2003)
Griffiths P.: On the periods of certain rational integrals: I. Ann. Math. 90, 460 (1969)
Candelas P.: Yukawa couplings between (2,1) forms. Nucl. Phys. B 298, 458 (1988)
Lerche W., Smit D.J., Warner N.P.: Differential equations for periods and flat coordinates in two-dimensional topological matter theories. Nucl. Phys. B 372, 87 (1992)
Libgober A., Teitelbaum J.: Lines on Calabi-Yau complete intersections, mirror symmetry, and Picard Fuchs equations. Int. Math. Res. Not. 1993, 13 (1993)
Griffiths P.: A theorem concerning the differential equations satisfied by normal functions associated to algebraic cycles. Am. J. Math. 101, 94 (1979)
Deligne P.: Théorie de Hodge III. Publ. Math. I.H.E.S. 55, 5 (1974)
Cecotti S., Vafa C.: Topological antitopological fusion. Nucl. Phys. B 367, 359 (1991)
Lerche W., Vafa C., Warner N.P.: Chiral Rings in N = 2 Superconformal Theories. Nucl. Phys. B 324, 427 (1989)
Strominger A.: Special Geometry. Commun. Math. Phys. 133, 163 (1990)
Aganagic M., Bouchard V., Klemm A.: Topological Strings and (Almost) Modular Forms. Commun. Math. Phys. 277, 771 (2008)
Klemm A., Theisen S.: Considerations of one modulus Calabi-Yau compactifications: Picard-Fuchs equations, Kahler potentials and mirror maps. Nucl. Phys. B 389, 153 (1993)
Greene B.R., Plesser M.R.: Duality in Calabi-Yau moduli space. Nucl. Phys. B 338, 15 (1990)
Rainville E.: Special Functions. The Macmillan Company, New York (1960)
Witten E.: Phases of N = 2 theories in two dimensions. Nucl. Phys. B 403, 159 (1993)
Intriligator K.A., Vafa C.: Landau-Ginzburg Orbifolds. Nucl. Phys. B 339, 95 (1990)
Dijkgraaf R., Verlinde H.L., Verlinde E.P.: Topological Strings in D < 1. Nucl. Phys. B 352, 59 (1991)
Candelas P., De La Ossa X., Font A., Katz S.H., Morrison D.R.: Mirror symmetry for two parameter models. I. Nucl. Phys. B 416, 481 (1994)
Candelas P., Font A., Katz S.H., Morrison D.R.: Mirror symmetry for two parameter models. 2. Nucl. Phys. B 429, 626 (1994)
Candelas P., De La Ossa X.C., Green P.S., Parkes L.: A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory. Nucl. Phys. B 359, 21 (1991)
Hori K., Walcher J.: F-term equations near Gepner points. JHEP 0501, 008 (2005)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A. Kapustin
Rights and permissions
About this article
Cite this article
Jockers, H., Soroush, M. Effective Superpotentials for Compact D5-Brane Calabi-Yau Geometries. Commun. Math. Phys. 290, 249–290 (2009). https://doi.org/10.1007/s00220-008-0727-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-008-0727-7