Abstract
We study constructions of stable holomorphic vector bundles on Calabi–Yau threefolds, especially those with exact anomaly cancellation which we call extremal. By going through the known databases we find that such examples are rare in general and can be ruled out for the spectral cover construction for all elliptic threefolds. We then introduce a general Hartshorne–Serre construction and use it to find extremal bundles of general ranks and study their stability, as well as computing their Chern numbers. Based on both existing and our new constructions, we revisit the DRY conjecture for the existence of stable sheaves on Calabi–threefolds, and provide theoretical and numerical evidence for its correctness. Our construction can be easily generalized to bundles with no extremal conditions imposed.
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Donaldson S.K.: Anti-self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bunldles. Proc. Lond. Math. Soc. 50, 1 (1985)
Uhlenbeck K.K., Yau S.-T.: On the existence of hermitian Yang–Mills connections in stable vector bundles. Commun. Pure Appl. Math. 39, 257 (1986)
Anderson, L.B., Gray, J., Lukas, A., Ovrut, B.: The Atiyah class and complex structure stabilization in heterotic Calabi–Yau compactifications. JHEP 1110, 032 (2011). arXiv:1107.5076 [hep-th]
Uhlenbeck, K.K.: Removable singularities in Yang–Mills fields. Comm. Math. Phys. 831, 11–29 (1982)
Uhlenbeck, K.K.: Connections with L p bounds on curvature. Comm. Math. Phys. 831, 31–42 (1982)
Li J.: Algebraic geometric interpretation of Donaldson’s polynomial invariants. J. Differ. Geom. 37(2), 417–466 (1993)
Denef, F., Douglas, M.R.: Distributions of flux vacua. JHEP 0405, 072 (2004). hep-th/0404116
Ashok, S., Douglas, M.R.: Counting flux vacua. JHEP 0401, 060 (2004). hep-th/0307049
Douglas, M.R., Shiffman, B., Zelditch, S.: Critical points and supersymmetric vacua. Commun. Math. Phys. 252, 325 (2004). math/0402326 [math-cv]
Douglas, M.R., Shiffman, B., Zelditch, S.: Critical points and supersymmetric vacua. J. Diff. Geometry 252, 325 (2004). math/0406089 [math-cv]
Douglas, M.R., Shiffman, B., Zelditch, S.: Critical points and supersymmetric vacua II: asymptotics and extremal metrics. Commun. Math. Phys. 72, 381–427 (2006). math-ph/0506015
Donaldson S.K., Thomas R.P.: Gauge theory in Higher Dimensions, Chapter 3. In: Huggett, S.A., Mason, L.J., Tod, K.P., Tsou, S.T., Woodhouse, N.M.J. (eds) The Geometric Universe, OUP, Oxford (1998)
Thomas R.P.: A holomorphic Casson invariant for Calabi–Yau threefolds, and bundles on K3 fibrations. J. Differ. Geom. 54, 367–438 (2000)
Wu, B.: A degeneration formula of Donaldson-Thomas invariants, Thesis (Ph.D.) Stanford University 2007. In: Li, J., Wu, B. (eds.) Good degeneration of Quot-schemes and coherent systems. arXiv:1110.0390v1 [math.AG]
Witten, E.: Chern–Simons gauge theory as a string theory. Prog. Math. 133, 637–678. Boston, Birkhauser (1995). hep-th/9207094
Kachru, S., Katz, S.H., Lawrence, A.E., McGreevy, J.: Open string instantons and superpotentials. Phys. Rev. D 62, 026001 (2000). hep-th/9912151
Kachru, S, Katz, S.H., Lawrence, A.E., McGreevy, J.: Mirror symmetry for open strings. Phys. Rev. D 62, 126005 (2000). hep-th/0006047
Aganagic, M., Vafa, C.: Mirror symmetry, D-branes and counting holomorphic discs. hep-th/0012041
Lazaroiu, C.I., Roiban, R.: Holomorphic potentials for graded D-branes. JHEP 0202, 038 (2002). hep-th/0110288
Behrend, K., Bryan, J., Szendroi, B.: Motivic degree zero Donaldson–Thomas invariants. Invent. Math. 192(1), 111–160 (2013). arXiv:0909.5088 [math.AG]
Li, W.-P., Qin, Z.: Donaldson–Thomas invariants of certain Calabi–Yau 3-folds. arXiv:1002.4080 [math.AG]
DeWolfe, O., Giryavets, A., Kachru, S., Taylor, W.: Enumerating flux vacua with enhanced symmetries. JHEP 0502, 037 (2005). hep-th:0411061
Denef, F., Douglas, M.R.: Computational complexity of the landscape I. Ann. Phys. 322, 1096 (2007). hep-th/0602072. In: Denef, F. (eds.) TASI lectures on complex structures. arXiv:1104.0254 [hep-th]
Vakil R.: Murphy’s law in algebraic geometry: badly-behaved deformation spaces. Invent. Math. 164(3), 569–590 (2006)
The Oxford Handbook of Philosophy of Physics, chapter 8. In: Batternman, R. (ed.) Oxford University Press, Oxford, Mar 14, 2013. http://en.wikipedia.org/wiki/Totalitarian_principle
Bogomolov, F.A.: Holomorphic tensors and vector bundles on projective manifolds. Izv. Akad. Nauk SSSR Ser. Mat. 42(6), 1227–1287, 1439 (1978)
Douglas, M.R., Reinbacher, R., Yau, S.-T.: Branes, bundles and attractors: Bogomolov Beyond. math/0604597 [math-ag]
Denef, F., Moore, G.W.: Split states, entropy enigmas, holes and halos. JHEP 1111:129 (2011) hep-th/0702146
He, Y.-H., Candelas, P., Hanany, A., Lukas, A., Ovrut, B.: (eds.) Computational Algebraic Geometry in String, Gauge Theory. Special Issue, Advances in High Energy Physics, Hindawi publishing (2012). doi:10.1155/2012/431898
He, Y.-H.: Calabi–Yau geometries: algorithms, databases, and physics. Int. J. Mod. Phys. A 28 (2013) arXiv:1308.0186 [hep-th]
Candelas P., Dale A.M., Lutken C.A., Schimmrigk R.: Complete intersection Calabi–Yau manifolds. Nucl. Phys. B 298, 493 (1988)
Gagnon M., Ho-Kim Q.: An Exhaustive list of complete intersection Calabi–Yau manifolds. Mod. Phys. Lett. A 9, 2235 (1994)
Avram, A.C., Kreuzer, M., Mandelberg, M., Skarke, H.: The web of Calabi–Yau hypersurfaces in toric varieties. Nucl. Phys. B 505, 625 (1997). hep-th/9703003
Anderson, L.B., Gray, J., Lukas, A., Palti, E.: Heterotic Line Bundle Standard Models. JHEP 1206, 113 (2012). arXiv:1202.1757 [hep-th]
Anderson, L.B., Constantin, A., Gray, J., Lukas, A., Palti, E.: JHEP 1401, 047 (2014). arXiv:1307.4787 [hep-th]
Anderson, L.B., Constantin, A., Gray, J., Lukas, A., Palti, E.: A comprehensive scan for heterotic SU(5) GUT models. JHEP 1401, 047 (2014). arXiv:1307.4787 [hep-th]
Donagi, R., He, Y.-H., Ovrut, B.A., Reinbacher, R.: The particle spectrum of heterotic compactifications. JHEP 0412, 054 (2004). arXiv:hep-th/0405014
Gabella, M., He, Y.-H., Lukas, A.: An abundance of heterotic vacua. JHEP 0812, 027 (2008). arXiv:0808.2142 [hep-th]
Anderson, L.B., He, Y.-H., Lukas, A.: Heterotic compactification, an algorithmic approach. JHEP 0707, 049 (2007). hep-th/0702210
Anderson, L.B., He, Y.-H., Lukas, A.: Monad bundles in heterotic string compactifications. JHEP 0807, 104 (2008). arXiv:0805.2875 [hep-th]
Anderson, L.B., Gray, J., He, Y.-H., Lukas, A.: JHEP 1002, 054 (2010). arXiv:0911.1569 [hep-th]
He, Y.-H., Kreuzer, M., Lee, S.-J., Lukas, A.: Heterotic bundles on Calabi–Yau manifolds with small Picard number. JHEP 1112, 039 (2011). arXiv:1108.1031 [hep-th]
Blumenhagen, R., Jurke, B., Rahn, T.: Computational tools for cohomology of toric varieties. Adv. High Energy Phys. 2011, 152749 (2011). arXiv:1104.1187 [hep-th]
Gao, X., Shukla, P.: On classifying the divisor involutions in Calabi–Yau threefolds. JHEP 1311, 170 (2013). arXiv:1307.1139 [hep-th]
Strominger A.: Superstrings with Torsion. Nucl. Phys. B 274, 253 (1986)
Fu J.X., Yau S.T.: A Monge-Ampère-type equation motivated by string theory. Comm. Anal. Geom. 15(1), 29–75 (2007)
Fu J.X., Yau S.T.: The theory of superstring with flux on non-Kähler manifolds and the complex Monge-Ampère equation. J. Differ. Geom. 78(3), 369–428 (2008)
Fu J.X., Yau S.T.: A note on small deformations of balanced manifolds. C. R. Math. Acad. Sci. Paris 349(13–14), 793–796 (2011)
Fu J., Li J., Yau S.T.: Balanced metrics on non-Kähler Calabi–Yau threefolds. J. Differ. Geom. 90(1), 81–129 (2012)
Douglas, M.R., Zhou, C.-G.: Chirality change in string theory. JHEP 0406, 014 (2004). hep-th/0403018
Candelas P., Horowitz G.T., Strominger A., Witten E.: Vacuum configurations for superstrings. Nucl. Phys. B 258, 46 (1985)
Green M.B., Schwarz J.H., Witten E.: Superstring Theory. Cambridge University Press, Cambridge (1987)
Horava, P., Witten, E.: Eleven-dimensional supergravity on a manifold with boundary. Nucl. Phys. B 475, 94 (1996). hep-th/9603142
Lukas, A., Ovrut, B.A. Stelle, K.S., Waldram, D.: Heterotic M theory in five-dimensions. Nucl. Phys. B 552, 246 (1999). hep-th/9806051
Greene B.R., Kirklin K.H., Miron P.J., Ross G.G.: A three generation superstring model 1: compactification and discrete symmetries. Nucl. Phys. B 278, 667 (1986)
Braun, V., He, Y.-H., Ovrut, B.A., Pantev, T.: The Exact MSSM spectrum from string theory. JHEP 0605, 043 (2006). hep-th/0512177
Bouchard, V., Donagi, R.: An SU(5) heterotic standard model. Phys. Lett. B 633, 783 (2006). hep-th/0512149
Candelas, P., de la Ossa, X. He, Y.-H., Szendroi, B.: Triadophilia: a special corner in the landscape. Adv. Theor. Math. Phys. 12, 429 (2008). arXiv:0706.3134 [hep-th]
Braun, V., Candelas, P., Davies, R.: A three-generation Calabi–Yau manifold with small hodge numbers. Fortsch. Phys. 58, 467 (2010). arXiv:0910.5464 [hep-th]
Kollár J., Matsusaka T.: Riemann–Roch type inequalities. Am. J. Math. 105(1), 229–252 (1983)
Maruyama M.: On boundedness of families of torsion free sheaves. J. Math. Kyoto Univ. 21(4), 673–701 (1981)
Simpson C.T.: Moduli of representations of the fundamental group of a smooth projective variety. I. Inst. Hautes Études Sci. Publ. Math. No. 79, 47–129 (1994)
Taylor, W.: On the Hodge structure of elliptically fibered Calabi–Yau threefolds. JHEP 1208, 032 (2012). arXiv:1205.0952 [hep-th]
Anderson, L.B., Gray, J., Ovrut, B.A.: Transitions in the web of heterotic vacua. Fortsch. Phys. 59, 327 (2011). arXiv:1012.3179 [hep-th]
Donagi, R.: Principal bundles on elliptic brations. Asian J. Math. 1, 214–223 (June 1997). Friedman, R., Morgan, J.W., Witten, E.: Vector Bundles Over Elliptic Fibrations. arXiv:alg-geom/9709029
Morrison, D.R., Vafa, C.: Compactifications of F theory on Calabi–Yau threefolds 1. Nucl. Phys. B 473, 74 (1996). hep-th/9602114
Langer A.: Semistable sheaves in positive characteristic. Ann. Math. (2) 159(1), 251–276 (2004)
Kawamata Y.: On the cone of divisors of Calabi–Yau fiber spaces. Int. J. Math. 8(5), 665–687 (1997)
Hartshorne R.: Stable reflexive sheaves. Math. Ann. 254(2), 121–176 (1980)
Wu, B., Yau, S.T.: A construction of stable bundles and reflexive sheaves on Calabi–Yau threefolds. arXiv:1405.5676
Okonek C., Schneider M., Spindler H.: Vector bundles on complex projective spaces. Progress in Mathematics, vol. 3. Birkhauser, Boston (1980)
Yau S.T.: Calabis conjecture and some new results in algebraic geometry. Proc. Natl. Acad. Sci. 74, 1789 (1977)
Yau S.T.: On the Ricci curvature of a compact Kähler manifold and the complex MongeAmpere equations. I. Comm. Pure Appl. Math. 31, 339–411 (1978)
Wu B.: private correspondence
Maeda H.: Construction of vector bundles and reflexive sheaves. Tokyo J. Math. 13(1), 153–162 (1990)
Li J., Yau S.T.: The existence of supersymmetric string theory with torsion. J. Differ. Geom. 70(1), 143–181 (2005)
Huybrechts D.: The tangent bundle of a Calabi–Yau manifolddeformations and restriction to rational curves. Comm. Math. Phys. 171(1), 139–158 (1995)
Maruyama, M.: On a family of algebraic vector bundles, Number Theory. Algebraic Geometry and Commutative Algebra (1973), 95–149, Kinokuniya. MR0360587 (50:13035)
Horrocks G. A construction for locally free sheaves. Topology 7, 117–120 (1968)
Barth W., Hulek K.: Monads and moduli of vector bundles. Manuscripta Math. 25(4), 323–347 (1978)
Distler J., Greene B.R.: Aspects of (2,0) string compactifications. Nucl. Phys. B 304, 1 (1988)
Braun, V.: Three Generations on the Quintic Quotient. JHEP 1001, 094 (2010). arXiv:0909.5682 [hep-th]
Maulik, D., Nekrasov, N., Okounkov, A., Pandharipande, R.: Gromov–Witten theory and Donaldson–Thomas theory I. Compos. Math. 1425, 1263–1285 (2006)
Maulik, D., Nekrasov, N., Okounkov, A., Pandharipande, R.: Gromov-Witten theory and Donaldson-Thomas theory. II Compos. Math. 1425, 1286–1304 (2006)
Maulik, D., Nekrasov, N., Okounkov, A., Pandharipande, R.: Gromov-Witten/Donaldson-Thomas correspondence for toric threefolds. Invent. Math. 1862, 435–479 (2011)
Joyce, D., Song, Y.: A theory of generalized Donaldson–Thomas invariants. In: Proceedings of Memoils of the American Mathematical Society. vol. 2A, no. 1020, AMS, Providence (2011). arXiv:0810.5645 [math.AG]
Kontsevich, M., Soibelman, Y.: Motivic Donaldson–Thomas invariants: summary of results. in Contemporary Math. 527 (2010). arXiv:0910.4315
Andreas, B., Curio, G.: On possible Chern classes of stable bundles on Calabi–Yau threefolds. J. Geom. Phys. 61, 1378 (2011). arXiv:1010.1644 [hep-th]
Andreas, B., Curio, G.: On the Existence of Stable bundles with prescribed Chern classes on Calabi–Yau threefolds. J. Geom. Phys. 76, 235–241 (2014). arXiv:1104.3435 [math.AG]
Oguiso K.: On algebraic fiber space structures on a Calabi-Yau threefold. Int. J. Math. 4(3), 439–465 (1993)
Wilson P.M.H.: The existence of elliptic fibre space structures on Calabi-Yau threefolds. Math. Ann. 300(4), 693–703 (1994)
Wu, B., Yau, S.T.: A Construction of stable bundles and reflexive sheaves on Calabi–Yau threefolds. arXiv:1405.5676 [math.AG]
Marino, M., Moore, G.W., Peradze, G.: Superconformal invariance and the geography of four manifolds. Commun. Math. Phys. 205, 691 (1999). hep-th/9812055
Buchbinder, E., Donagi, R., Ovrut, B.A.: Vector bundle moduli and small instanton transitions. JHEP 0206, 054 (2002). hep-th/0202084
Friedman, R., Morgan, J.W., Witten, E.: Vector bundles over elliptic fibrations. alg-geom/9709029
Friedman, R., Morgan, J., Witten, E.: Commun. Math. Phys. 187, 679 (1997). hep-th/9701162
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Gao, P., He, YH. & Yau, ST. Extremal Bundles on Calabi–Yau Threefolds. Commun. Math. Phys. 336, 1167–1200 (2015). https://doi.org/10.1007/s00220-014-2271-y
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DOI: https://doi.org/10.1007/s00220-014-2271-y