欢迎访问中国科学院大学学报,今天是
论文

高阶丛束的K-理论

  • 吴正尧 ,
  • 高玉彬 ,
  • 唐国平
展开
  • 1. 中国科学院研究生院数学科学学院,北京 100049;
    2. 北京航空航天大学,北京 100191

收稿日期: 2010-04-08

  修回日期: 2010-04-28

  网络出版日期: 2010-11-15

基金资助

Supported by the National Natural Science Foundation of China (10671202) and "Hundred Talent" Program of the Chinese Academy of Sciences

K-theory of higher bundle gerbes

  • WU Zheng-Yao ,
  • GAO Yu-Bin ,
  • TANG Guo-Ping
Expand
  • 1. School of Mathematical Sciences, Graduate University, Chinese Academy of Sciences, Beijing 100049, China;
    2. Beijing University of Aeronautics and Astronautics, Beijing 100191, China

Received date: 2010-04-08

  Revised date: 2010-04-28

  Online published: 2010-11-15

摘要

旨在将束的概念推广到高阶束.回顾了D-膜电荷的分类作为研究束的依据,并按照高阶层-堆的过程建立高阶束的结构.考虑了纤维丛上的例子,给出了稳定同构的3个等价条件。

本文引用格式

吴正尧 , 高玉彬 , 唐国平 . 高阶丛束的K-理论[J]. 中国科学院大学学报, 2010 , 27(6) : 721 -738 . DOI: 10.7523/j.issn.2095-6134.2010.6.001

Abstract

We aim to generalize gerbes to higher gerbes. This paper reviews the history of classification of D-brane charges as evidence for gerbes, then present a higher sheaf-stack procedure to investigate the n-gerbe structure.This study also considers examples on a fibred bundle. Our conclusion offers three equivalent conditions for stable isomorpoism.

参考文献


[1] Giraud J. Cohomologie non abelienne
[M]. Die Grundlehren der mathematischen Wissenschaften 179. Berlin/Heidelberg: Springer, 1971.

[2] Brylinski J-L. Loop spaces, characteristic class and geometric quantinization
[M]. Progress in Mathematics 107. Boston: Birkhuser, 1993.

[3] Weil A. Sur les theoremes de de Rham
[J]. Comm Math Helv, 1952, 26: 17- 43.

[4] Kostant B. Quantization and unitary representations //Lectures in Modern Analysis and Applications III, Lecture Notes in Mathematics 170. Berlin/Heidelberg: Springer, 1970: 87-208.

[5] Murray M. Bundle gerbes
[J]. London Mathematical Society, 1996, 2(54): 403- 416.

[6] Bouwknegt P, Carey A, Mathai V, et al. Twisted K-theory and K-theory of bundle gerbes
[J]. Commun Math Phys, 2002, 228: 17- 45.

[7] Minasian R, Moore G. K-theory and Ramond-Ramond charge
[J]. J High Energy Phys, 1997 (11): 002.

[8] Witten E. D-branes and K-theory
[J]. J High Energy Phys, 1998 (12): 019.

[9] Ho ava P. Type-IIA D-branes, K-theory and matrix theory
[J]. Adv Theor Math Phys, 1998 (2): 1373-1404.

[10] Olsena K, Szabo R. Constructing D-branes from K-theory
[J]. Adv Theor Math Phys, 1999, 3: 889-1025.

[11] Freed D, Witten E. Anomalies in string theory with D-branes . arXiv: hep-th/9907189v2, 2000-03-06 . http://arxiv.org/PS cache/hep-th/pdf/9907/9907189v2.pdf.

[12] Dixmier J, Douady A. Champs continus d’espaces hilbertiens et de C*-algébres
[J]. Bull Soc Math France, 1963, 91: 227-284.

[13] Maldacena J, Moore G, Seiberg N. D-brane instantons and K-theory charges
[J]. J High Energy Phys, 2001 (11): 062.

[14] Freed D, Hopkins M, Teleman C. Twisted equivariant K-theory with complex coefficients . arXiv: math/0206257v4, 2006- 07-23 . http://arxiv.org/PS cache/math/pdf/0206/0206257v4.pdf.

[15] Alekseev A, Schomerus V. Quantum moduli spaces of flat connections . arXiv: q-alg/9612037, 1996-12-31 . http://arxiv.org/PS cache/q-alg/pdf/9612/9612037v1.pdf.

[16] Mathai V, Stevenson D. Chern character in twisted K-theory: equivariant and holomorphic cases
[J]. Commun Math Phys, 2003, 236:161-186.

[17] Breen L. On the classification of 2-gerbes and 2-stacks
[M]. Astérisque 225. Paris: Société Mathématique de France, 1994.

[18] Brylinski J-L, McLaughlin D. The geometry of degree-four characteristic classes and of line bundles on loop spaces. I
[J]. Duke Math J, 1994, 75(3): 603-638.

[19] Brylinski J-L, McLaughlin D. A geometric construction of the first Pontryagin class // Quantum Topology in Ser. Knots Everything 3. River Edge, New Jersey: World Sci Publishing, 1993: 209-220.

[20] Carey A, Murray M, Wang B. Higher bundle gerbes and cohomology classes in gauge theory
[J]. J Geom Phys, 1997, 21: 183-197.

[21] Stevenson D. Bundle 2-gerbes
[J]. Proc London Math Soc, 2004, 88(2): 405- 435.

[22] Yang C N, Mills R. Conservation of isotopic spin and isotopic gauge invariance
[J]. Physical Review, 1954, 96(1): 191-195.

[23] Henneaux M, Teitelboim C. p-form electrodynamics
[J]. Foundations of Physics, 1986, 16(7): 593-617.

[24] Hitchin N. What is a gerbe?
[J]. Notices of the AMS, 2003, 50(2): 218-219.

[25] Hitchin N. Lectures on special lagrangian submanifolds . arXiv: math/9907034, 1999- 07- 06 . http://arxiv.org/PS cache/math/pdf/9907/9907034v1.pdf.

[26] Ekstrand C. k-gerbes, line bundles and anomalies
[J]. J High Energy Phys, 2000 (10): 038.

[27] Gomi K. Central extensions of gauge transformation groups of higher abelian gerbes
[J]. J Geom Phys, 2006, 56: 1767-1781.

[28] Gomi K. Projective unitary representations of smooth Deligne cohomology groups . arXiv: math/0510087, 2006. http://arxiv.org/PS cache/math/pdf/0510/0510087v1.pdf.

[29] Zunger Y. p-gerbes and extended objects in string theory . arXiv: hep-th/0002074, 2000- 02-10 . http://arxiv.org/PS cache/hep-th/pdf/0002/0002074v2.pdf.

[30] Keurentjes A. Classifying orientifolds by flat n-gerbes
[J]. J High Energy Phys, 2001 (7): 010.

[31] Bachas C. Lectures on D-branes . arXiv: hepth/9806199, 1999-01-17 . http://arxiv.org/PS cache/hep-th/pdf/9806/9806199v2.pdf.

[32] Milnor J, Stasheff J. Characteristic classes
[M]. Annals of Mathematical Studies 76. Princeton: Princeton University Express, 1974.19

[33] Husemler D, Echterhoff S, Fredenhagen S, et al . Basic bundle theory and K-cohomology invariants
[M]. Lect Notes Phys 726. Berlin/Heidelberg: Springer, 2008.

[34] Atiyah M. K-theory
[M]. 2nd ed. Advanced Book Classics, Addison-Wesley: 1989.

[35] Demazure M, Grothendieck A. Séminaire de Géométrie Algébrique du Bois Marie - 1962~1964 - Schémas en groupes (SGA 3)
[M]. Lecture Notes in Mathematics 151 Berlin/Heidelberg: Springer, 1970.

[36] Artin M, Grothendieck A, Verdier J-L. Séminaire de Géométrie Algébrique du Bois Marie - 1963~1964 - Théorie des topos et cohomologie étale des schémas - (SGA 4)
[M]. Lecture notes in Mathematics 269. Berlin/Heidelberg: Springer, 1972.

[37] MacLane S. Categories for the working mathematician
[M]. 2nd ed. Grad Texts Math 5, Berlin/Heidelberg: Springer, 1971.

[38] Husemler D. Fibred Bundles
[M]. 3rd ed. Grad Texts Math 20, Berlin/Heidelberg: Springer, 1994.

[39] Atiyah M, Segal G. Twisted K-theory . arXiv: math/0407054, 2005-10-31 . http://arxiv.org/PS cache/math/pdf/0407/0407054v2.pdf.

[40] Hartshorne R. Algebraic geometry
[M]. Grad Texts Math 52, Berlin/Heidelberg: Springer, 1977.

[41] Vistoli A. Notes on grothendieck topologies, fibered categories and descent theory . arXiv:math/0412512v4, 2007- 05-24 .http://arxiv.org/PS cache/math/pdf/0412/0412512v4.pdf.

[42] Basu S. What is an Eilenberg-MacLane space? . Graduate Student Seminar Talks at State University of New York, Stony Brook, 2009 . http://www.math.sunysb.edu/ basu/notes/GSS4.pdf.

文章导航

/