Advertisement

Invariants of families of flat connections using fiber integration of differential characters

  • Ishan MataEmail author
Article
First Online:
  • 24 Downloads

Abstract

Let \(E\rightarrow B\) be a smooth vector bundle of rank n, and let \(P \in I^p(GL(n,{\mathbb {R}}))\) be a \(GL(n,{\mathbb {R}})\)-invariant polynomial of degree p compatible with a universal integral characteristic class \( u \in H^{2p}(BGL(n,{\mathbb {R}}),{\mathbb {Z}})\). Cheeger–Simons theory associates a rigid invariant in \(H^{2p-1}(B,{\mathbb {R}}/{\mathbb {Z}})\) to any flat connection on this bundle. Generalizing this result, Jaya Iyer (Lett Math Phys 106 (1):131–146, 2016) constructed maps \(H_r({\mathcal {D}}(E)) \rightarrow H^{2p-r-1}(B,{\mathbb {R}}/{\mathbb {Z}})\) for \(p>r+1\). Here, \({\mathcal {D}}(E)\) is the simplicial abelian group whose group of r-simplices is freely generated by \((r+1)\)-tuples of relatively flat connections. In this article, we construct such maps for the cases \(p<r\) and \(p>r+1\) using fiber integration of differential characters. We find that for \(p>r+1\) case, the invariants constructed here coincide with those obtained by Jaya Iyer, and that in the \(p<r\) case the invariants are trivial. We further compare our construction with other results in the literature.

Keywords

Differential characters Flat connections Fiber integration 

Mathematics Subject Classification

53C07 53C08 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgements

I am thankful to my supervisor Dr. Rishikesh Vaidya for discussions and support. I wish to profusely thank the anonymous reviewer whose feedback significantly helped improve the presentation. I am financially supported by the Council of Scientific & Industrial Research-Human Resource Development Group (CSIR-HRDG) under the CSIR-SRF(NET) scheme. I am grateful to CSIR-HRDG for the same.

References

  1. 1.
    Bär, C., Becker, C.: Differential Characters and Geometric Chains, pp. 1–90. Springer International Publishing, Cham (2014).  https://doi.org/10.1007/978-3-319-07034-6_1 CrossRefzbMATHGoogle Scholar
  2. 2.
    Becker, C.: Cheeger–Chern–Simons theory and differential string classes. Ann. Henri Poincaré 17(6), 1529–1594 (2016).  https://doi.org/10.1007/s00023-016-0485-6 MathSciNetCrossRefzbMATHADSGoogle Scholar
  3. 3.
    Biswas, I., López, M.C.: Flat connections and cohomology invariants. Math. Nachr. 290(14–15), 2170–2184 (2017)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Borel, A., Hirzebruch, F.: Characteristic classes and homogeneous spaces, i. Am. J. Math. 80(2), 458–538 (1958)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bott, R., Tu, L.W.: Differential Forms in Algebraic Topology, Graduate Texts in Mathematics, vol. 82. Springer, Berlin (1982)CrossRefGoogle Scholar
  6. 6.
    Castrillón López, M., Ferreiro Pérez, R.: Differential characters and cohomology of the moduli of flat connections. Lett. Math. Phys. (2018).  https://doi.org/10.1007/s11005-018-1095-7 CrossRefzbMATHGoogle Scholar
  7. 7.
    Cheeger, J., Simons, J.: Differential characters and geometric invariants. In: Geometry and Topology, pp. 50–80. Springer, Berlin (1985)Google Scholar
  8. 8.
    Chern, S.S.: On the characteristic classes of complex sphere bundles and algebraic varieties. Am. J. Math. 75(3), 565–597 (1953)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chern, S.S., Simons, J.: Characteristic forms and geometric invariants. Ann. Math. 2(99), 48–69 (1974).  https://doi.org/10.2307/1971013 MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dupont, J.L., Kamber, F.W.: Gerbes, simplicial forms and invariants for families of foliated bundles. Commun. Math. Phys. 253(2), 253–282 (2005).  https://doi.org/10.1007/s00220-004-1193-5 MathSciNetCrossRefzbMATHADSGoogle Scholar
  11. 11.
    Dupont, J.L., Ljungmann, R.: Integration of simplicial forms and deligne cohomology. Math. Scand. 97(1), 11–39 (2005).  https://doi.org/10.7146/math.scand.a-14961 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ewald, C.: Hochschild homology, and de rham cohomology of stratifolds. Ph.D. thesis, Universitä Heidelberg (2002)Google Scholar
  13. 13.
    Ewald, C.: A de rham isomorphism in singular cohomology and stokes theorem for stratifolds. Int. J. Geom. Methods Mod. Phys. 2(1), 63–81 (2005)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Freed, D.S.: Classical Chern–Simons theory. ii. Houst. J. Math. 28(2), 293–310 (2002)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Gomi, K., Terashima, Y.: A fiber integration formula for the smooth deligne cohomology. Int. Math. Res. Not. 2000(13), 699–708 (2000).  https://doi.org/10.1155/S1073792800000386 MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Guruprasad, K., Kumar, S.: A new geometric invariant associated to the space of flat connections. Compos. Math. 73(2), 199–222 (1990)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Iyer, J.N.N.: Cohomological invariants of a variation of flat connections. Lett. Math. Phys. 106(1), 131–146 (2016).  https://doi.org/10.1007/s11005-015-0807-5 MathSciNetCrossRefzbMATHADSGoogle Scholar
  18. 18.
    Kreck, M.: Differential Algebraic Topology, From Stratifolds to Exotic Spheres, Graduate Studies in Mathematics, vol. 110. American Mathematical Society, Providence (2010).  https://doi.org/10.1090/gsm/110 CrossRefzbMATHGoogle Scholar
  19. 19.
    Ljungmann, R.: Secondary invariants for families of bundles. Ph.D. thesis (2006)Google Scholar
  20. 20.
    Narasimhan, M.S., Ramanan, S.: Existence of universal connections. Am. J. Math. 83(3), 563–572 (1961)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Tu, L.W.: Differential Geometry. Springer, Berlin (2017).  https://doi.org/10.1007/978-3-319-55084-8 CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of PhysicsBirla Institute of Technology and Science- PilaniPilaniIndia

Personalised recommendations

Cite article

Buy options