Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 016, 21 pages      arXiv:1506.07913

On the Chern-Gauss-Bonnet Theorem and Conformally Twisted Spectral Triples for $C^*$-Dynamical Systems

Farzad Fathizadeh a and Olivier Gabriel b
a) Department of Mathematics, Mail Code 253-37, California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125, USA
b) University of Copenhagen, Universitetsparken 5, 2100 København Ø, Denmark

Received October 26, 2015, in final form February 04, 2016; Published online February 10, 2016

The analog of the Chern-Gauss-Bonnet theorem is studied for a $C^*$-dynamical system consisting of a $C^*$-algebra $A$ equipped with an ergodic action of a compact Lie group $G$. The structure of the Lie algebra $\mathfrak{g}$ of $G$ is used to interpret the Chevalley-Eilenberg complex with coefficients in the smooth subalgebra $\mathcal{A} \subset A$ as noncommutative differential forms on the dynamical system. We conformally perturb the standard metric, which is associated with the unique $G$-invariant state on $A$, by means of a Weyl conformal factor given by a positive invertible element of the algebra, and consider the Hermitian structure that it induces on the complex. A Hodge decomposition theorem is proved, which allows us to relate the Euler characteristic of the complex to the index properties of a Hodge-de Rham operator for the perturbed metric. This operator, which is shown to be selfadjoint, is a key ingredient in our construction of a spectral triple on $\mathcal{A}$ and a twisted spectral triple on its opposite algebra. The conformal invariance of the Euler characteristic is interpreted as an indication of the Chern-Gauss-Bonnet theorem in this setting. The spectral triples encoding the conformally perturbed metrics are shown to enjoy the same spectral summability properties as the unperturbed case.

Key words: $C^*$-dynamical systems; ergodic action; invariant state; conformal factor; Hodge-de Rham operator; noncommutative de Rham complex; Euler characteristic; Chern-Gauss-Bonnet theorem; ordinary and twisted spectral triples; unbounded selfadjoint operators; spectral dimension.

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  1. Albeverio S., Høegh-Krohn R., Ergodic actions by compact groups on $C^{\ast} $-algebras, Math. Z. 174 (1980), 1-17.
  2. Atiyah M., Bott R., Patodi V.K., On the heat equation and the index theorem, Invent. Math. 19 (1973), 279-330.
  3. Bhuyain T.A., Marcolli M., The Ricci flow on noncommutative two-tori, Lett. Math. Phys. 101 (2012), 173-194, arXiv:1107.4788.
  4. Branson T.P., Ørsted B., Conformal indices of Riemannian manifolds, Compositio Math. 60 (1986), 261-293.
  5. Bratteli O., Derivations, dissipations and group actions on $C^*$-algebras, Lecture Notes in Math., Vol. 1229, Springer-Verlag, Berlin, 1986.
  6. Carey A.L., Neshveyev S., Nest R., Rennie A., Twisted cyclic theory, equivariant $KK$-theory and KMS states, J. Reine Angew. Math. 650 (2011), 161-191, arXiv:0808.3029.
  7. Carey A.L., Phillips J., Rennie A., Semifinite spectral triples associated with graph $C^\ast$-algebras, in Traces in Number Theory, Geometry and Quantum Fields, Aspects Math., Vol. E38, Friedr. Vieweg, Wiesbaden, 2008, 35-56, arXiv:0707.3853.
  8. Carey A.L., Phillips J., Rennie A., Twisted cyclic theory and an index theory for the gauge invariant KMS state on the Cuntz algebra $O_n$, J. K-Theory 6 (2010), 339-380, arXiv:0801.4605.
  9. Chamseddine A.H., Connes A., The spectral action principle, Comm. Math. Phys. 186 (1997), 731-750, hep-th/9606001.
  10. Chamseddine A.H., Connes A., Scale invariance in the spectral action, J. Math. Phys. 47 (2006), 063504, 19 pages, hep-th/0512169.
  11. Cohen P.B., Connes A., Conformal geometry of the irrational rotation algebra, Preprint MPI, 1992.
  12. Connes A., $C^{\ast}$ algèbres et géométrie différentielle, C. R. Acad. Sci. Paris Sér. A-B 290 (1980), A599-A604, hep-th/0101093.
  13. Connes A., Noncommutative differential geometry, Inst. Hautes Études Sci. Publ. Math. 62 (1985), 257-360.
  14. Connes A., Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994.
  15. Connes A., Moscovici H., The local index formula in noncommutative geometry, Geom. Funct. Anal. 5 (1995), 174-243.
  16. Connes A., Moscovici H., Type III and spectral triples, in Traces in number theory, geometry and quantum fields, Aspects Math., Vol. E38, Friedr. Vieweg, Wiesbaden, 2008, 57-71, math.OA/0609703.
  17. Connes A., Moscovici H., Modular curvature for noncommutative two-tori, J. Amer. Math. Soc. 27 (2014), 639-684, arXiv:1110.3500.
  18. Connes A., Tretkoff P., The Gauss-Bonnet theorem for the noncommutative two torus, in Noncommutative Geometry, Arithmetic, and Related Topics, Johns Hopkins Univ. Press, Baltimore, MD, 2011, 141-158, arXiv:0910.0188.
  19. Cuntz J., Simple $C^*$-algebras generated by isometries, Comm. Math. Phys. 57 (1977), 173-185.
  20. Dąbrowski L., Sitarz A., An asymmetric noncommutative torus, SIGMA 11 (2015), 075, 11 pages, arXiv:1406.4645.
  21. Devastato A., Martinetti P., Twisted spectral triple for the standard model and spontaneous breaking of the grand symmetry, arXiv:1411.1320.
  22. Fathizadeh F., On the scalar curvature for the noncommutative four torus, J. Math. Phys. 56 (2015), 062303, 14 pages, arXiv:1410.8705.
  23. Fathizadeh F., Khalkhali M., The algebra of formal twisted pseudodifferential symbols and a noncommutative residue, Lett. Math. Phys. 94 (2010), 41-61, arXiv:0810.0484.
  24. Fathizadeh F., Khalkhali M., Twisted spectral triples and Connes' character formula, in Perspectives on Noncommutative Geometry, Fields Inst. Commun., Vol. 61, Amer. Math. Soc., Providence, RI, 2011, 79-101, arXiv:1106.6127.
  25. Fathizadeh F., Khalkhali M., The Gauss-Bonnet theorem for noncommutative two tori with a general conformal structure, J. Noncommut. Geom. 6 (2012), 457-480, arXiv:1005.4947.
  26. Fathizadeh F., Khalkhali M., Scalar curvature for the noncommutative two torus, J. Noncommut. Geom. 7 (2013), 1145-1183, arXiv:1110.3511.
  27. Fathizadeh F., Khalkhali M., Weyl's law and Connes' trace theorem for noncommutative two tori, Lett. Math. Phys. 103 (2013), 1-18, arXiv:1111.1358.
  28. Fathizadeh F., Khalkhali M., Scalar curvature for noncommutative four-tori, J. Noncommut. Geom. 9 (2015), 473-503, arXiv:1301.6135.
  29. Fathizadeh F., Wong M.W., Noncommutative residues for pseudo-differential operators on the noncommutative two-torus, J. Pseudo-Differ. Oper. Appl. 2 (2011), 289-302.
  30. Gabriel O., Grensing M., Ergodic actions and spectral triples, arXiv:1302.0426.
  31. Gilkey P.B., Invariance theory, the heat equation, and the Atiyah-Singer index theorem, Mathematics Lecture Series, Vol. 11, Publish or Perish, Inc., Wilmington, DE, 1984.
  32. Gracia-Bondía J.M., Várilly J.C., Figueroa H., Elements of noncommutative geometry, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Boston, Inc., Boston, MA, 2001.
  33. Greenfield M., Marcolli M., Teh K., Twisted spectral triples and quantum statistical mechanical systems, p-Adic Numbers Ultrametric Anal. Appl. 6 (2014), 81-104.
  34. Higson N., The residue index theorem of Connes and Moscovici, in Surveys in noncommutative geometry, Clay Math. Proc., Vol. 6, Amer. Math. Soc., Providence, RI, 2006, 71-126.
  35. Høegh-Krohn R., Landstad M.B., Størmer E., Compact ergodic groups of automorphisms, Ann. of Math. 114 (1981), 75-86.
  36. Iochum B., Masson T., Crossed product extensions of spectral triples, J. Noncommut. Geom., to appear, arXiv:1406.4642.
  37. Kaad J., On modular semifinite index theory, arXiv:1111.6546.
  38. Kaad J., Senior R., A twisted spectral triple for quantum ${\rm SU}(2)$, J. Geom. Phys. 62 (2012), 731-739, arXiv:1109.2326.
  39. Kato T., Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995.
  40. Khalkhali M., Landi G., van Suijlekom W.D., Holomorphic structures on the quantum projective line, Int. Math. Res. Not. 2011 (2011), 851-884, arXiv:0907.0154.
  41. Knapp A.W., Lie groups, Lie algebras, and cohomology, Mathematical Notes, Vol. 34, Princeton University Press, Princeton, NJ, 1988.
  42. Krýsl S., Hodge theory for elliptic complexes over unital $C^*$-algebras, Ann. Global Anal. Geom. 45 (2014), 197-210, arXiv:1309.4560.
  43. Krýsl S., Hodge theory for complexes over $C^*$-algebras with an application to $A$-ellipticity, Ann. Global Anal. Geom. 47 (2015), 359-372, arXiv:1309.4560.
  44. Krýsl S., Elliptic complexes over $C^*$-algebras of compact operators, J. Geom. Phys. 101 (2016), 27-37, arXiv:1506.06244.
  45. Lesch M., Moscovici H., Modular curvature and Morita equivalence, arXiv:1505.00964.
  46. Lord S., Sukochev F., Zanin D., Singular traces. Theory and applications, De Gruyter Studies in Mathematics, Vol. 46, De Gruyter, Berlin, 2013.
  47. Moscovici H., Local index formula and twisted spectral triples, in Quanta of Maths, Clay Math. Proc., Vol. 11, Amer. Math. Soc., Providence, RI, 2010, 465-500, arXiv:0902.0835.
  48. Paterson A.L.T., Contractive spectral triples for crossed products, Math. Scand. 114 (2014), 275-298, arXiv:1204.4404.
  49. Ponge R., Wang H., Noncommutative geometry and conformal geometry. III. Vafa-Witten inequality and Poincaré duality, Adv. Math. 272 (2015), 761-819, arXiv:1310.6138.
  50. Reed M., Simon B., Methods of modern mathematical physics. I. Functional analysis, Academic Press, Inc., New York, 1980.
  51. Reed M., Simon B., Methods of modern mathematical physics. II. Fourier analysis, selfadjointness, Academic Press, Inc., New York, 1975.
  52. Rieffel M.A., Metrics on states from actions of compact groups, Doc. Math. 3 (1998), 215-229, math.OA/9807084.
  53. Simon B., Trace ideals and their applications, Mathematical Surveys and Monographs, Vol. 120, 2nd ed., Amer. Math. Soc., Providence, RI, 2005.
  54. Størmer E., Spectra of ergodic transformations, J. Funct. Anal. 15 (1974), 202-215.
  55. Wahl C., Index theory for actions of compact Lie groups on $C^*$-algebras, J. Operator Theory 63 (2010), 217-242, arXiv:0707.3207.
  56. Wassermann A., Ergodic actions of compact groups on operator algebras. I. General theory, Ann. of Math. 130 (1989), 273-319.
  57. Wassermann A., Ergodic actions of compact groups on operator algebras. II. Classification of full multiplicity ergodic actions, Canad. J. Math. 40 (1988), 1482-1527.
  58. Wassermann A., Ergodic actions of compact groups on operator algebras. III. Classification for ${\rm SU}(2)$, Invent. Math. 93 (1988), 309-354.
  59. Wodzicki M., Local invariants of spectral asymmetry, Invent. Math. 75 (1984), 143-177.
  60. Wodzicki M., Noncommutative residue. I. Fundamentals, in $K$-Theory, Arithmetic and Geometry (Moscow, 1984-1986), Lecture Notes in Math., Vol. 1289, Springer, Berlin, 1987, 320-399.

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