Abstract
We discuss a categorical approach to the investigation of topological D-branes. Twist functors and their induced action on the cohomology ring of a manifold are studied. A nontrivial spherical object of the derived category of coherent sheaves of a reduced plane singular curve of degree 3 is constructed.
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REFERENCES
E. Zaslow, “Solitons and helices: search for a mathematical physics bridge,” Commun. Math. Phys., 175, 337–347 (1996).
M. Kontsevich, “Homological algebra of mirror symmetry,” in: Proceedings of the International Congress on Mathematics (Zurich, 1994) (1995), pp. 120–139.
J. Polchinski, “Dirichlet branes and Ramond-Ramond charges,” Phys. Rev. Lett., 75, 4724–4727 (1995); Preprint / arxiv: hep-th, No. 9510017 (1995).
E. Witten, “D-branes and K-theory,” J. High Energy Phys., 9812, 19 (1998).
R. Minasian and G. Moore, “K-theory and Ramond-Ramond charges. D-branes and K-theory,” J. High Energy Phys., 11, 102 (1997); Preprint / arxiv: hep-th, No. 9710230 (1997).
D. A. Cox and S. Kaz, “Mirror symmetry and algebraic geometry,” Math. Surveys Monographs, Amer. Math. Soc., 68, 203 (1999).
A. Bondal and D. Orlov, Semiorthogonal Decompositions for Algebraic Varieties, Preprint / arxiv: math. AG, No. 9506006 (1995).
T. Bridgeland, “Flops and derived categories,” Invent. Math., 147, No.3, 613–632 (2002).
P. Seidel and R. Thomas, “Braid group actions on derived categories of coherent sheaves,” Duke Math. J., 108, No.1, 37–108 (2001).
H. Lenzing and H. Meltzer, “Sheaves on weighted projective lines of genus one and representations of a tubular algebra,” Can. Math. Soc. Conf. Proc., 14, 313–337 (1993).
A. Polishchuk, “Yang-Baxter equation and A ∞-constraints,” Adv. Math., 168, 56–95 (2002).
D. Orlov, “Derived categories of coherent sheaves and equivalences between them,” Usp. Mat. Nauk, 58, No.3, 89–172 (2003).
S. I. Gel’fand and Yu. I. Manin, Methods of Homological Algebra [in Russian], Nauka, Moscow (1988).
D. Orlov, “Equivalences of derived categories and K3 surfaces,” J. Math. Sci. (New York), 84, No.5, 1361–1381 (1997).
A. Dold, “Zur Homotopietheorie der Kettenkomplexe,” Math. Ann., 140, 278–298 (1960).
S. Mukai, “On the moduli spaces of bundles on K3 surfaces I,” Vector Bundles on Algebraic Varieties, Stud. Math. Tata Inst. Fundam. Res., 11, 341–413 (1987).
S. Zube, “Exceptional sheaves on Enriques surfaces,” Math. Notes, 61, No.6, 693–699 (1994).
R. Hartshorne, Algebraic Geometry, Springer, New York (1977).
P. Griffiths and J. Harris, Principles of Algebraic Geometry [Russian translation], Mir, Moscow (1982).
I. I. Burban and Yu. A. Drozd, “Coherent sheaves on singular curves with nodal singularities,” Duke Math. J., 121, No.2, 189–229 (2004); Preprint / arxiv: math. AG, No. 0101140 (2001).
I. I. Burban, “Stable bundles on a rational curve with one simple double point,” Ukr. Mat. Zh., 55, No.7, 867–875 (2003).
I. I. Burban, Yu. A. Drozd, and G.-M. Greuel, “Vector bundles on singular projective curves,” in: Application of Algebraic Geometry to Coding Theory, Physics and Computation, Kluwer, New York (2001), pp. 1–15.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 1, pp. 18–31, January, 2005.
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Burban, I.I., Burban, I.M. Twist Functors and D-Branes. Ukr Math J 57, 18–34 (2005). https://doi.org/10.1007/s11253-005-0169-8
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DOI: https://doi.org/10.1007/s11253-005-0169-8