Objective B-fields and a Hitchin-Kobayashi correspondence

Wang, Shuguang

doi:10.1090/S0002-9947-2011-05413-9

Objective B-fields and a Hitchin-Kobayashi correspondence

Author: Shuguang Wang
Journal: Trans. Amer. Math. Soc. 364 (2012), 2087-2107
MSC (2010): Primary 53C07, 53C08, 57R20
DOI: https://doi.org/10.1090/S0002-9947-2011-05413-9
Published electronically: December 1, 2011
MathSciNet review: 2869199
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Abstract: A simple trick invoking objective B-fields is employed to refine the concept of characteristic classes for twisted bundles. Then the objective stability and objective Einstein metrics are introduced and a new Hitchin-Kobayashi correspondence is established between them.

[1] Michael Atiyah, 𝐾-theory past and present, Sitzungsberichte der Berliner Mathematischen Gesellschaft, Berliner Math. Ges., Berlin, 2001, pp. 411–417. MR 2091892
[2] Peter Bouwknegt, Alan L. Carey, Varghese Mathai, Michael K. Murray, and Danny Stevenson, Twisted 𝐾-theory and 𝐾-theory of bundle gerbes, Comm. Math. Phys. 228 (2002), no. 1, 17–45. MR 1911247, https://doi.org/10.1007/s002200200646
[3] Peter Bouwknegt and Varghese Mathai, D-branes, 𝐵-fields and twisted 𝐾-theory, J. High Energy Phys. 3 (2000), Paper 7, 11. MR 1756434, https://doi.org/10.1088/1126-6708/2000/03/007
[4] Jean-Luc Brylinski, Loop spaces, characteristic classes and geometric quantization, Progress in Mathematics, vol. 107, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1197353
[5] Andrei Căldăraru, Nonfine moduli spaces of sheaves on 𝐾3 surfaces, Int. Math. Res. Not. 20 (2002), 1027–1056. MR 1902629, https://doi.org/10.1155/S1073792802109093
[6] D. S. Chatterjee, On Gerbes, Ph.D. thesis, Cambridge University, 1998.
[7] Ron Donagi, Tony Pantev, Burt A. Ovrut, and René Reinbacher, 𝑆𝑈(4) instantons on Calabi-Yau threefolds with ℤ₂×ℤ₂ fundamental group, J. High Energy Phys. 1 (2004), 022, 58. MR 2045884, https://doi.org/10.1088/1126-6708/2004/01/022
[8] S. K. Donaldson, Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proc. London Math. Soc. (3) 50 (1985), no. 1, 1–26. MR 765366, https://doi.org/10.1112/plms/s3-50.1.1
[9] S. K. Donaldson, Infinite determinants, stable bundles and curvature, Duke Math. J. 54 (1987), no. 1, 231–247. MR 885784, https://doi.org/10.1215/S0012-7094-87-05414-7
[10] S. K. Donaldson, The orientation of Yang-Mills moduli spaces and 4-manifold topology, J. Differential Geom. 26 (1987), no. 3, 397–428. MR 910015
[11] S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1990. Oxford Science Publications. MR 1079726
[12] Nigel Hitchin, Lectures on special Lagrangian submanifolds, Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds (Cambridge, MA, 1999) AMS/IP Stud. Adv. Math., vol. 23, Amer. Math. Soc., Providence, RI, 2001, pp. 151–182. MR 1876068
[13] Daniel Huybrechts and Paolo Stellari, Proof of Căldăraru’s conjecture. Appendix: “Moduli spaces of twisted sheaves on a projective variety” [in Moduli spaces and arithmetic geometry, 1–30, Math. Soc. Japan, Tokyo, 2006; MR2306170] by K. Yoshioka, Moduli spaces and arithmetic geometry, Adv. Stud. Pure Math., vol. 45, Math. Soc. Japan, Tokyo, 2006, pp. 31–42. MR 2310257
[14] Shoshichi Kobayashi, Differential geometry of complex vector bundles, Publications of the Mathematical Society of Japan, vol. 15, Princeton University Press, Princeton, NJ; Princeton University Press, Princeton, NJ, 1987. Kanô Memorial Lectures, 5. MR 909698
[15] J. Li, Anti-self-dual connections and stable vector bundles, Gauge theory and the topology of four-manifolds (Park City, UT, 1994) IAS/Park City Math. Ser., vol. 4, Amer. Math. Soc., Providence, RI, 1998, pp. 23–49. MR 1611446
[16] Martin Lübke, Stability of Einstein-Hermitian vector bundles, Manuscripta Math. 42 (1983), no. 2-3, 245–257. MR 701206, https://doi.org/10.1007/BF01169586
[17] Ernesto Lupercio and Bernardo Uribe, Gerbes over orbifolds and twisted 𝐾-theory, Comm. Math. Phys. 245 (2004), no. 3, 449–489. MR 2045679, https://doi.org/10.1007/s00220-003-1035-x
[18] Marco Mackaay, A note on the holonomy of connections in twisted bundles, Cah. Topol. Géom. Différ. Catég. 44 (2003), no. 1, 39–62 (English, with French summary). MR 1961525
[19] Marco Mackaay and Roger Picken, Holonomy and parallel transport for abelian gerbes, Adv. Math. 170 (2002), no. 2, 287–339. MR 1932333, https://doi.org/10.1006/aima.2002.2085
[20] Varghese Mathai, Richard B. Melrose, and Isadore M. Singer, The index of projective families of elliptic operators, Geom. Topol. 9 (2005), 341–373. MR 2140985, https://doi.org/10.2140/gt.2005.9.341
[21] M. K. Murray, Bundle gerbes, J. London Math. Soc. (2) 54 (1996), no. 2, 403–416. MR 1405064, https://doi.org/10.1093/acprof:oso/9780199534920.003.0012
[22] Michael K. Murray and Michael A. Singer, Gerbes, Clifford modules and the index theorem, Ann. Global Anal. Geom. 26 (2004), no. 4, 355–367. MR 2103405, https://doi.org/10.1023/B:AGAG.0000047514.71785.96
[23] J. Pan, Y. Ruan, X. Yin, Gerbes and twisted orbifold quantum cohomology, math.AG/0504369.
[24] Carlos T. Simpson, Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization, J. Amer. Math. Soc. 1 (1988), no. 4, 867–918. MR 944577, https://doi.org/10.1090/S0894-0347-1988-0944577-9
[25] K. Uhlenbeck and S.-T. Yau, On the existence of Hermitian-Yang-Mills connections in stable vector bundles, Comm. Pure Appl. Math. 39 (1986), no. S, suppl., S257–S293. Frontiers of the mathematical sciences: 1985 (New York, 1985). MR 861491, https://doi.org/10.1002/cpa.3160390714
[26] Shuguang Wang, A Narasimhan-Seshadri-Donaldson correspondence over non-orientable surfaces, Forum Math. 8 (1996), no. 4, 461–474. MR 1393324, https://doi.org/10.1515/form.1996.8.461
[27] Shuguang Wang, Gerbes, holonomy forms and real structures, Commun. Contemp. Math. 11 (2009), no. 1, 109–130. MR 2498389, https://doi.org/10.1142/S0219199709003302
[28] Alan Weinstein, The Maslov gerbe, Lett. Math. Phys. 69 (2004), 3–9. MR 2104435, https://doi.org/10.1007/s11005-004-0342-2
[29] R. O. Wells Jr., Differential analysis on complex manifolds, 2nd ed., Graduate Texts in Mathematics, vol. 65, Springer-Verlag, New York-Berlin, 1980. MR 608414
[30] Kōta Yoshioka, Moduli spaces of twisted sheaves on a projective variety, Moduli spaces and arithmetic geometry, Adv. Stud. Pure Math., vol. 45, Math. Soc. Japan, Tokyo, 2006, pp. 1–30. MR 2306170