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Twisted smooth Deligne cohomology

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Abstract

Deligne cohomology can be viewed as a differential refinement of integral cohomology, hence captures both topological and geometric information. On the other hand, it can be viewed as the simplest nontrivial version of a differential cohomology theory. While more involved differential cohomology theories have been explicitly twisted, the same has not been done to Deligne cohomology, although existence is known at a general abstract level. We work out what it means to twist Deligne cohomology, by taking degree one twists of both integral cohomology and de Rham cohomology. We present the main properties of the new theory and illustrate its use with examples and applications. Given how versatile Deligne cohomology has proven to be, we believe that this explicit and utilizable treatment of its twisted version will be useful.

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Notes

  1. Note that we are working over the small site of Cartesian spaces, so stackification of this prestack is not necessary.

  2. Note that this is different than the smooth stack \(\mathbb {R}^{\times }:=C^{\infty }(-;\mathbb {R}^{\times })\). It is obtained by regarding \(\mathbb {R}^{\times }\) as a discrete group.

  3. The sheafification is computed as a limit over refinements of covers, and here we know that the sequences are exact.

  4. The associated map here is provided via the adjunction \([M,\mathbf {B}\mathbb {Z}/2]\cong [\Pi (M),B\mathbb {Z}/2]\cong \hom (\pi _1(M),\mathbb {Z}/2)\).

  5. It is a straightforward exercise to show that this follows in general for any such web of short exact sequences.

References

  1. Adolphson, A., Sperber, S.: On twisted de Rham cohomology. Nagoya Math. J. 146, 55–81 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ando, M., Blumberg, A.J., Gepner, D.J.: Twists of K-theory and TMF, superstrings, geometry, topology, and \(C^\ast \)-algebras. In: Proceedings of Symposia in Pure Mathematics, vol. 81, pp. 27–63. American Mathematical Society, Providence, RI (2010). [arXiv:1002.3004] [math.AT]

  3. Ando, M., Blumberg, A.J., Gepner, D., Hopkins, M.J., Rezk, C.: Units of ring spectra, orientations and Thom spectra via rigid infinite loop space theory. J. Topol. 7(4), 1077–1117 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Aomoto, K., Kita, M., Orlik, P., Terao, H.: Twisted de Rham cohomology groups of logarithmic forms. Adv. Math. 128(1), 119–152 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bär, C., Becker, C.: Differential Characters. Lecture Notes in Mathematics, vol. 2112. Springer, Cham (2014)

    MATH  Google Scholar 

  6. Beilinson, A.: Higher regulators and values of L-functions. J. Sov. Math. 30, 2036–2070 (1985)

    Article  MATH  Google Scholar 

  7. Bott, R., Tu, L.W.: Differential Forms in Algebraic Topology. Springer, New York (1982)

    Book  MATH  Google Scholar 

  8. Brylinski, J.-L.: Loop Spaces, Characteristic Classes and Geometric Quantization. Progress in Mathematics, vol. 107. Birkhäuser, Boston (1993)

  9. Bunke, U.: Differential cohomology. [arXiv:1208.3961] [math.AT]

  10. Bunke, U., Kreck, M., Schick, T.: A geometric description of differential cohomology. Ann. Math. Blaise Pascal 17(1), 1–16 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bunke, U., Nikolaus, T.: Twisted differential cohomology. arXiv:1406.3231

  12. Bunke, U., Nikolaus, T., Völkl, M.: Differential cohomology theories as sheaves of spectra. J. Homotopy Relat. Struct. 11(1), 1–66 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bunke, U., Schick, T.: Uniqueness of smooth extensions of generalized cohomology theories. J. Topol. 3, 110–156 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Carlson, J., Hain, R.: Extensions of variations of mixed Hodge structure. Astérisque 179–180(9), 39–65 (1989)

    MathSciNet  MATH  Google Scholar 

  15. Cheeger, J., Simons, J.: Differential Characters and Geometric Invariants. Lecture Notes in Mathematics, vol. 1167, pp. 50–80. Springer, Berlin (1985)

    MATH  Google Scholar 

  16. Deligne, P.: Equations defférentielles a points singuliers réguliers. Lecture Notes in Mathematics. Springer, Berlin (1970)

    Book  MATH  Google Scholar 

  17. Deligne, P.: Théorie de Hodge : II. Publ. Math. IHES 40, 5–57 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  18. Deligne, P., Mostow, G.D.: Monodromy of hypergeometric functions and non-lattice integral monodromy groups. Publ. Math. IHES 63, 5–90 (1986)

    Article  MATH  Google Scholar 

  19. Dugger, D.: Sheaves and homotopy theory, 1999 draft. http://pages.uoregon.edu/ddugger/

  20. Dugger, D., Hollander, S., Isaksen, D.: Hypercovers and simplicial presheaves. Math. Proc. Camb. Philos. Soc. 136(1), 9–51 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  21. Dupont, J.L., Ljungmann, R.: Integration of simplicial forms and Deligne cohomology. Math. Scand. 97(1), 11–39 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Esnault, H., Viehweg, E.: Deligne–Beilinson cohomology, Beilinson’s conjectures on special values of L-functions. In: Rapoport, M., Schappacher, N., Schneider, P. (eds.) Perspectives in Mathematics, pp. 43–91. Academic Press, Boston (1988)

  23. Farber, M.: Topology of Closed One-forms. American Mathematical Society, Providence (2004)

    Book  MATH  Google Scholar 

  24. Fiorenza, D., Sati, H., Schreiber, U.: Extended higher cup-product Chern–Simons theory. J. Geom. Phys. 74, 130–163 (2013). [arXiv:1207.5449] [hep-th]

  25. Fiorenza, D., Sati, H., Schreiber, U.: A Higher stacky perspective on Chern–Simons theory. In: Calaque, D., Strobl, T. (eds.) Mathematical Aspects of Quantum Field Theories. Springer, Berlin (2015). [arXiv:1301.2580] [hep-th]

  26. Fiorenza, D., Sati, H., Schreiber, U.: The \(E_8\) moduli 3-stack of the C-field in M-theory. Commun. Math. Phys. 333, 117–151 (2015). [arXiv:1202.2455] [hep-th]

  27. Fiorenza, D., Schreiber, U., Stasheff, J.: Čech cocycles for differential characteristic classes—an infinity-Lie theoretic construction. Adv. Theor. Math. Phys. 16, 149–250 (2012). [arXiv:1011.4735] [math.AT]

  28. Freed, D.S.: The Verlinde algebra is twisted equivariant K-theory. Turk. J. Math. 25, 159–167 (2001)

    MathSciNet  MATH  Google Scholar 

  29. Gajer, P.: Geometry of Deligne cohomology. Invent. Math. 127, 155–207 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  30. Gawedzki, K., Suszek, R.R., Waldorf, K.: Bundle gerbes for orientifold sigma models. Adv. Theor. Math. Phys. 15, 621–688 (2011). [arXiv:0809.5125] [math-ph]

  31. Gillet, H.: Deligne homology and Abel–Jacobi maps. Bull. Am. Math. Soc. 10, 285–288 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  32. Gomi, K.: Equivariant smooth Deligne cohomology. Osaka J. Math. 42(2), 309–337 (2005)

    MathSciNet  MATH  Google Scholar 

  33. Grady, D., Sati, H.: Massey products in differential cohomology via stacks. J. Homotopy Relat. Struct. (2017). [arXiv:1510.06366] [math.AT]

  34. Grady, D., Sati, H.: Primary operations in differential cohomology. [arXiv:1604.05988] [math.AT]

  35. Grady, D., Sati, H.: Spectral sequences in smooth generalized cohomology. Algebr. Geom. Topol. 17(4), 2357–2412 (2017). [arXiv:1605.03444] [math.AT]

  36. Hain, R.: Deligne–Beilinson cohomology of affine groups. [arXiv:1507.03144] [math.AG]

  37. Harvey, R., Zweck, J.: Steifel–Whitney currents. J. Geom. Anal. 8, 809–844 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  38. Hekmati, P., Murray, M.K., Szabo, R.J., Vozzo, R.F.: Real bundle gerbes, orientifolds and twisted KR-homology. [arXiv:1608.06466] [hep-th]

  39. Hopkins, M.J., Quick, G.: Hodge filtered complex bordism. J. Topol. 8, 147–183 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  40. Hopkins, M.J., Singer, I.M.: Quadratic functions in geometry, topology, and M-theory. J. Differ. Geom. 70(3), 329–452 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  41. Jannsen, U.: Deligne Homology, Hodge-D-Conjecture, and Motives. Beilinson’s Conjectures on Special Values of L-Functions, Perspectives in Mathematics, vol. 4, pp. 305–372. Academic Press, Boston (1988)

    MATH  Google Scholar 

  42. Jardine, J.F.: Local Homotopy Theory. Springer, New York (2015)

    Book  MATH  Google Scholar 

  43. Kapranov, M.: Real mixed Hodge structures. J. Noncommut. Geom. 6, 321–342 (2012). [arXiv:0802.0215] [math.AG]

  44. Kita, M.: On hypergeometric functions in several variables II. The Wronskian of the hypergeometric functions. J. Math. Soc. Jpn. 45, 645–669 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  45. Kita, M.: On vanishing of the twisted rational de Rham cohomology associated with hypergeometric functions. Nagoya Math. J. 135, 55–85 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  46. Lind, J.A., Sati, H., Westerland, C.: Twisted iterated algebraic K-theory and topological T-duality for sphere bundles. [arXiv:1601.06285] [math.AT]

  47. Lurie, J.: Higher algebra, prepublication book draft. http://www.math.harvard.edu/~lurie/ (2011)

  48. May, J.P.: \(E_\infty \) ring spaces and \(E_\infty \) ring spectra, with contributions by Quinn, F., Ray, N., Tornehave, J. Lecture Notes in Mathematics 577. Springer, Berlin (1977)

  49. May, J.P., Sigurdsson, J.: Parametrized Homotopy Theory. American Mathematical Society, Providence (2006)

    Book  MATH  Google Scholar 

  50. Nikolaus, T., Schreiber, U., Stevenson, D.: Principal \(\infty \)-bundles: general theory. J. Homotopy Relat. Struct. 10(4), 749–801 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  51. Sati, H., Westerland, C., Twisted Morava K-theory and E-theory. J. Topol. 8(4), 887–916 (2015). [arXiv:1109.3867] [math.AT]

  52. Schreiber, U.: Differential cohomology in a cohesive infinity-topos. [arXiv:1310.7930] [math-ph]

  53. Schreiber, U., Schweigert, C., Waldorf, K.: Unoriented WZW models and holonomy of bundle gerbes. Commun. Math. Phys. 274, 31–64 (2007). [arXiv:hep-th/0512283]

  54. Shipley, B.: \(H{\mathbb{Z}}\)-algebra spectra are differential graded algebras. Am. J. Math. 129(2), 351–379 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  55. Simons, J., Sullivan, D.: Axiomatic characterization of ordinary differential cohomology. J. Topol. 1(1), 45–56 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  56. Witten, E.: Supersymmetry and Morse theory. J. Differ. Geom. 17(4), 661–692 (1982)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the organizers and participants of the Geometric Analysis and Topology Seminar at the Courant Institute for Mathematical Sciences for asking about twisting Deligne cohomology, during a talk by H.S., which encouraged the authors to revisit and carry out this project. D.G. would like to thank the Mathematics Department at the University of Pittsburgh for hospitality during the final writing of this paper. The authors thank Richard Hain for bringing to their attention related works in algebraic geometry and the referee for useful remarks.

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  1. Division of Science and Mathematics, New York University NYUAD, Saadiyat Island, Abu Dhabi, UAE

    Daniel Grady & Hisham Sati

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Grady, D., Sati, H. Twisted smooth Deligne cohomology. Ann Glob Anal Geom 53, 445–466 (2018). https://doi.org/10.1007/s10455-017-9583-z

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