Abstract
We use the remodeling approach to the B-model topological string in terms of recursion relations to study open string amplitudes at orbifold points. To this end, we clarify modular properties of the open amplitudes and rewrite them in a form that makes their transformation properties under the modular group manifest. We exemplify this procedure for the \({{\mathbb C}^3/{\mathbb Z}_3}\) orbifold point of local \({{\mathbb P}^2}\), where we present results for topological string amplitudes for genus zero and up to three holes, and for the one-holed torus. These amplitudes can be understood as generating functions for either open orbifold Gromov–Witten invariants of \({{\mathbb C}^3/{\mathbb Z}_3}\), or correlation functions in the orbifold CFT involving insertions of both bulk and boundary operators.
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Aganagic M., Bouchard V., Klemm A.: Topological strings and (Almost) modular forms. Commun. Math. Phys. 277, 771 (2008)
Aganagic M., Dijkgraaf R., Klemm A., Mariño M., Vafa C.: Topological strings and integrable hierarchies. Commun. Math. Phys. 261, 451 (2006)
Aganagic M., Klemm A., Mariño M., Vafa C.: The topological vertex. Commun. Math. Phys. 254, 425 (2005)
Aganagic M., Klemm A., Vafa C.: Disk instantons, mirror symmetry and the duality web. Z. Naturforsch. A 57, 1 (2002)
Aganagic, M., Vafa, C.: Mirror symmetry, D-branes and counting holomorphic discs. http://arXiv.org/abs/hep-th/0012041v1, 2000
Akemann G.: Higher genus correlators for the Hermitian matrix model with multiple cuts. Nucl. Phys. B 482, 403 (1996)
Akhiezer, N.I.: Elements of Theory of Elliptic Functions, AMS, Providence, RI: Amer. Math.Soc., 1999
Alim M., Lange J.D.: Polynomial structure of the (Open) topological string partition function. JHEP 0710, 045 (2007)
Aspinwall, P.S.: D-branes on Calabi-Yau manifolds. http://arXiv.org/abs/hep-th/0403166v1, 2004
Aspinwall P.S., Greene B.R., Morrison D.R.: Calabi-Yau moduli space, mirror manifolds and spacetime topology change in string theory. Nucl. Phys. B 416, 414 (1994)
Bayer, A., Cadman, C.: Quantum cohomology of \({[{\mathbb C}^n / \mu_r]}\). http://arXiv.org/abs/0705.2160v2[math.AG], 2009
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)
Bershadsky M., Cecotti S., Ooguri H., Vafa C.: Holomorphic anomalies in topological field theories. Nucl. Phys. B 405, 279 (1993)
Bertoldi G., Hollowood T.J.: Large N gauge theories and topological cigars. JHEP 0704, 078 (2007)
Bonelli G., Tanzini A.: The holomorphic anomaly for open string moduli. JHEP 0710, 060 (2007)
Bouchard V., Klemm A., Mariño M., Pasquetti S.: Remodeling the B-model. Commun. Math. Phys. 287, 117–178 (2009)
Bouchard, V., Cavalieri, R.: On the mathematics and physics of high genus invariants of \({{\mathbb C}^3/{\mathbb Z}_3}\). http://arXiv.org/abs/0709.3805v1[math.AG], 2007
Brini, A., Tanzini, A.: Exact results for topological strings on resolved Y(p,q) singularities. http://arXiv.org/abs/0804.2598v4[hep-th], 2008
Bryan, J., Graber, T.: The crepant resolution conjecture. http://arXiv.org/abs/arXiv:math/0610129v2[math.AG], 2007
Cadman, C., Cavalieri, R.: Gerby localization, \({{\mathbb Z}_3}\)-Hodge integrals and the GW theory of \({{\mathbb C}^3/{\mathbb Z}_3}\). http://arXiv.org/abs/0705.2158v3[math.AG], 2007
Cavalieri, R.: Private communication
Chekhov L., Eynard B., Orantin N.: Free energy topological expansion for the 2-matrix model. JHEP 0612, 053 (2006)
Chiang T.M., Klemm A., Yau S.T., Zaslow E.: Local mirror symmetry: Calculations and interpretations. Adv. Theor. Math. Phys. 3, 495 (1999)
Coates, T., Corti, A., Iritani, H., Tseng, H.-H.: Wall-Crossings in toric Gromov-Witten theory I: crepant examples. http://arXiv.org/abs/math/0611550v3[math.AG], 2006
Coates, T., Corti, A., Iritani, H., Tseng, H.-H.: Computing Genus-Zero twisted Gromov-Witten invariants. http://arXiv.org/abs/math/0702234v2[math.AG], 2007
Coates, T.: Wall-Crossings in toric Gromov-Witten theory II: local examples. http://arXiv.org/abs/0804.2592v1[math.AG], 2008
Diaconescu D.E., Florea B.: Large N duality for compact Calabi-Yau threefolds. Adv. Theor. Math. Phys. 9, 31 (2005)
Dijkgraaf R., Gukov S., Kazakov V.A., Vafa C.: Perturbative analysis of gauged matrix models. Phys. Rev. D 68, 045007 (2003)
Dijkgraaf R., Vafa C.: Matrix models, topological strings, and supersymmetric gauge theories. Nucl. Phys. B 644, 3 (2002)
Dijkgraaf, R., Vafa, C.: Two dimensional Kodaira-Spencer theory and three dimensional chern-simons gravity. http://arXiv.org/abs/0711.1932v1[hep-th], 2007
Dubrovin B., Zhang Y.: Bihamiltonian hierarchies in 2D topological field theory at one-loop approximation. Commun. Math. Phys. 198, 311 (1998)
Eynard B.: Topological expansion for the 1-hermitian matrix model correlation functions. JHEP 0411, 031 (2004)
Eynard B., Mariño M., Orantin N.: Holomorphic anomaly and matrix models. JHEP 0706, 058 (2007)
Eynard, B., Orantin, N.: Invariants of algebraic curves and topological expansion. http://arXiv.org/abs/math-ph/0702045v4, 2007
Ghoshal D., Vafa C.: c = 1 String as the topological theory of the conifold. Nucl. Phys. B 453, 121 (1995)
Givental, A.: Elliptic Gromov-Witten invariants and the generalized mirror conjecture. In: Integrable Systems and Algebraic Geometry (Kobe/Kyoto, 1997), River Edge, NJ: World Sci. Publ., 1998, pp. 107–155
Graber, T., Zaslow, E.: Open-String Gromov-Witten invariants: calculations and a mirror “Theorem”. http://arXiv.org/abs/hep-th/0109075v1, 2001
Grimm T.W., Klemm A., Mariño M., Weiss M.: Direct integration of the topological string. JHEP 0708, 058 (2007)
Harvey R., Lawson H.B.: Calibrated geometries. Acta Mathematica 148, 47–157 (1982)
Hori, K., Iqbal, A., Vafa, C.: D-branes and mirror symmetry. http://arXiv.org/abs/hep-th/0005247v2, 2000
Hori, K., Vafa, C.: Mirror symmetry. http://arXiv.org/abs/hep-th/0002222v3, 2000
Huang M.x., Klemm A.: Holomorphic anomaly in gauge theories and matrix models. JHEP 0709, 054 (2007)
Huang, M.x., Klemm, A., Quackenbush, S.: Topological string theory on compact Calabi-Yau: modularity and boundary conditions. In: Homological Mirror Symmetry: New Dev. and Perspectives, A. Kapustin (ed.), Lect. Notes in Phys. 757, Berlin-Heidelberg-New York: Springer, 2009, pp. 45–102
Kaneko, M., Zagier, D.B.: A generalized Jacobi theta function and quasimodular forms. In: The Moduli Space of Curves. Progr. Math. 129, Boston, MA: Birkhauser, 1995, pp. 165–172
Katz S., Liu C.-C.M.: Enumerative geometry of stable maps with Lagrangian boundary conditions and Multiple Covers of the Disc. Adv. Theor. Math. Phys. 5, 1–49 (2002)
Konishi, Y., Minabe, S.: On solutions to Walcher’s extended holomorphic anomaly equation. http://arXiv.org/abs/0708.2898v2[math.AG], 2007
Lerche, W., Mayr, P.: On N = 1 mirror symmetry for open type II strings. http://arXiv.org/abs/hep-th/0111113v2, 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
Mariño M.: Open string amplitudes and large order behavior in topological string theory. JHEP 0803, 060 (2008)
Morrison, D.R., Walcher, J.: D-branes and normal functions. http://arXiv.org/abs/0709.4028v1[hep-th], 2007
Orlov D.: Triangulated categories of singularities and D-branes in Landau-Ginzburg models. Proc. Steklov Inst. Math. 246(3), 227–248 (2004)
Orlov, D.: Derived categories of coherent sheaves and triangulated categories of singularities. http://arXiv.org/abs/math.AG/0503632v3, 2005
Ruan, Y.: The cohomology ring of crepant resolutions of orbifolds. In: Gromov-Witten Theory of Spin Curves and Orbifolds, Vol. 403 of Contemp. Math., Providence, RI: Amer. Math. Soc., 2006, pp. 117–126
Walcher J.: Extended holomorphic anomaly and loop amplitudes in open topological string. Nucl. Phys. B 817(3), 167–207 (2009)
Witten E.: Chern-Simons Gauge theory as a string theory. Prog. Math. 133, 637 (1995)
Witten E.: Phases of N = 2 theories in two dimensions. Nucl. Phys. 403, 159 (1993)
Yamaguchi S., Yau S.T.: Topological string partition functions as polynomials. JHEP 0407, 047 (2004)
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Bouchard, V., Klemm, A., Mariño, M. et al. Topological Open Strings on Orbifolds. Commun. Math. Phys. 296, 589–623 (2010). https://doi.org/10.1007/s00220-010-1020-0
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DOI: https://doi.org/10.1007/s00220-010-1020-0