Abstract
We show how to compute terms in an expansion of the world-volume superpotential for fairly general D-branes on the quintic Calabi-Yau using linear sigma model techniques, and show in examples that this superpotential captures the geometry and obstruction theory of bundles and sheaves on this Calabi-Yau.
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References
Albano, A., Katz, S.: Lines on the Fermat threefold and the infinitesimal generalized Hodge conjecture. Trans. AMS 324(1), 353–368 (1991)
Aganagic, M., Vafa, C.: Mirror Symmetry, D-Branes and Counting Holomorphic Discs. [arXiv:hep-th/0012041]
Aspinwall, P.S., Douglas, M.R.: D-brane stability and monodromy. JHEP 0205, 031 (2002) [arXiv:hep-th/0110071]
Beilinson, A.A.: Coherent sheaves on ℙn and problems of linear algebra. Funct. Anal. Appl. 12, 214–216 (1978)
Berenstein, D., Douglas, M.R.: Work in progress
Bott, R., Tu, L.W.: Differential Forms in Algebraic Topology. Berin-Heidelberg-New York: Springer, 1982
Brunner, I., Douglas, M.R., Lawrence, A., Römelsberger, C.: D-branes on the quintic. JHEP 0008, 015 (2000) [arXiv:hep-th/9906200]
Brunner, I., Schomerus, V.: On superpotentials for D-branes in Gepner models. JHEP 0010, 016 (2000) [arXiv:hep-th/0008194]
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)
Cattaneo, A.S., Felder, G.: A path integral approach to the Kontsevich quantization formula. Commun. Math. Phys. 212, 591–611 (2000); math.QA/9902090
Denef, F., Greene, B.R., Raugas, M.: Split attractor flows and the spectrum of BPS D-branes on the Quintic. JHEP 0105, 012 (2001); hep-th/0101135
Diaconescu, D.-E.: Enhanced D-branes categories from string field theory. JHEP 06, 016 (2001) [arXiv:hep-th/0104200]
Diaconescu, D.-E., Douglas, M.R.: D-branes on Stringy Calabi-Yau Manifolds. [arXiv:hep-th/0006224]
Donaldson, S.K., Thomas, R.P.: Gauge theory in higher dimensions. In: The Geometric Universe; Science, Geometry and the work of Roger Penrose, (eds.), S.A. Huggest et al, Oxford: Oxford Univ. Press, 1998
Douglas, M.R.: D-branes, categories and supersymmetry. J. Math. Phys. 42, 2818 (2001) [arXiv:hep-th/0011017]
Douglas, M.R., Fiol, B., Römelsberger, C.: The spectrum of BPS branes on a noncompact Calabi–Yau. [arXiv:hep-th/0003263]
Douglas, M.R., Greene, B.R., Morrison, D.R.: Orbifold resolution by D-branes. Nucl. Phys. B 506, 84 (1997) [arXiv:hep-th/9704151]
Douglas, M.R., Moore, G.: D-branes, quivers, and ALE instantons. [arXiv:hep-th/9603167]
Friedman, R., Morgan, J., Witten, E.: Vector bundles and F theory. Commun. Math. Phys. 187, 679 (1997) [arXiv:hep-th/9701162]
Gelfand, S.I., Manin, Yu,I.: Homological algebra. Berin: Springer-Verlag, 1999
Govindarajan, S., Jayaraman, T., Sarkar, T.: World sheet approaches to D-branes on supersymmetric cycles. Nucl. Phys. B 580, 519 (2000) [arXiv:hep-th/9907131]
Govindarajan, S., Jayaraman, T., Sarkar, T.: On D-branes from gauged linear sigma models. Nucl. Phys. B 593, 155 (2001) [arXiv:hep-th/0007075]
Govindarajan, S., Jayaraman, T.: D-branes, exceptional sheaves and quivers on Calabi-Yau manifolds: From Mukai to McKay. Nucl. Phys. B 600, 457 (2001) [arXiv:hep-th/0010196]
Govindarajan, S., Jayaraman, T.: Boundary fermions, coherent sheaves and D-branes on Calabi-Yau manifolds. Nucl. Phys. B 618, 50 (2001) [arXiv:hep-th/0104126]
Govindarajan, S., Jayaraman, T., Sarkar, T.: Disc instantons in linear sigma models. Nucl. Phys. B 646, 498 (2002) [arXiv:hep-th/0108234]
Griffiths, Ph., Harris, J.: Principles of algebraic geometry. New York: Wiley & Sons, 1978
Harris, J.: Galois group of enumerative problems. Duke Math. J. 46, 685–724 (1979)
Hartshorne, R.: Algebraic geometry. Berin-Heidelberg-New York: Springer-Verlag, 1979
Hellerman, S., McGreevy, J.: Linear sigma model toolshed for D-brane physics. JHEP 0110, 002 (2001) [arXiv:hep-th/0104100]
Hellerman, S., Kachru, S., Lawrence, A.E., McGreevy, J.: Linear sigma models for open strings. JHEP 0207, 002 (2002) [arXiv:hep-th/0109069]
Hori, K., Vafa, C.: Mirror symmetry. [arXiv:hep-th/0002222]
Hori, K., Iqbal, A., Vafa, C.: D-branes and mirror symmetry. [arXiv:hep-th/0005247]
Kac, V.: Infinite root systems, representations of graphs and Invariant Theory. Invent. Math. 56, 57 (1980)
Kontsevich, M., Soibelman, Y.: Homological mirror symmetry and torus fibrations. In: Symplectic geometry and mirror symmetry (Seoul, 2000), River Edge, NJ: World Sci. Publishing, 2001, pp. 203–263 [arXiv:math.sg/0011041]
Lazaroiu, C.I.: String field theory and brane superpotentials. JHEP 0110, 018 (2001) [arXiv:hep-th/0107162]
Mac Cleary, J.: A user’s guide to spectral sequences. Cambridge: Cambridge University Press, 2001
Mayr, P.: Phases of supersymmetric D-branes on Kaehler manifolds and the McKay correspondence. JHEP 0101, 018 (2001) [arXiv:hep-th/0010223]
Mayr, P.: N = 1 mirror symmetry and open/closed string duality. Adv. Theor. Math. Phys. 5, 213 (2002) [arXiv:hep-th/0108229]; Lerche, W., Mayr, P.: On N = 1 mirror symmetry for open type II strings. [arXiv:hep-th/0111113]
Merkulov, S.: Strong homotopy algebras of a Kähler manifold: Internat. Math. Res. Notices 3, 153–164 (1999) [arXiv:math.AG/9809172]
Mitchell, B.: Rings with several objects. Adv. Math. 8, 1–161 (1972)
Morrison, D.R., Ronen Plesser, M.: Towards mirror symmetry as duality for two dimensional abelian gauge theories. Nucl. Phys. Proc. Suppl. 46, 177 (1996) [arXiv:hep-th/9508107].
Polishchuk, A.: Homological Mirror Symmetry with Higher Products. In: Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds (Cambridge, MA, 1999), AMS/IP Stud. Adv. Math. 23, Providence, RI: Am. Math. Soc., 2001, pp. 247–259 [arXiv:math.AG/9901025]
Recknagel, A., Schomerus, V.: D-branes in Gepner models. Nucl. Phys. B531, 185 (1998) [arXiv:hep-th/9712186].
Reid, M.: McKay correspondence. [arXiv:alg-geom/9702016]
Reid, M.: La correspondance de McKay. Séminaire Bourbaki (Novembre 1999) 867, Astérisque No. 276, 53–72 (2002) [arXiv:math.AG/9911165]
Seidel, P.: Private communication
Seidel, P., Thomas, R.: Braid group actions on derived categories of sheaves. Duke Math. J. 108(1), 37–108 (2001) [arXiv:math.AG/0001043].
Strominger, A., Yau, S.T., Zaslow, E.: Nucl. Phys. B 479, 243 (1996) [arXiv:hep-th/9606040].
Tomasiello, A.: A-infinity structure and superpotentials. JHEP 0109, 030 (2001) [arXiv:hep-th/0107195]
Tomasiello, A.: D-branes on Calabi-Yau manifolds and helices. JHEP 0102, 008 (2001) [arXiv:hep-th/0010217]
Witten, E.: Chern-Simons Gauge Theory as a String Theory. Prog. Math. 133, 637 (1995) [arXiv:hep-th/9207094]
Witten, E.: Phases of N = 2 theories in two dimensions. Nucl. Phys. B 403, 159 (1993) [arXiv:hep-th/9301042]
Witten, E.: Branes and the Dynamics of QCD. Nucl. Phys. B507, 658 (1997) [arXiv:hep-th/9706109]
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Communicated by R.H. Dijkgraaf
Louis Michel Professor
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Douglas, M., Govindarajan, S., Jayaraman, T. et al. D-branes on Calabi–Yau Manifolds and Superpotentials. Commun. Math. Phys. 248, 85–118 (2004). https://doi.org/10.1007/s00220-004-1091-x
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DOI: https://doi.org/10.1007/s00220-004-1091-x