Abstract
Spectral triples and quantum statistical mechanical systems are two important constructions in noncommutative geometry. In particular, both lead to interesting reconstruction theorems for a broad range of geometric objects, including number fields, spin manifolds, graphs. There are similarities between the two structures, and we show that the notion of twisted spectral triple, introduced recently by Connes and Moscovici, provides a natural bridge between them. We investigate explicit examples, related to the Bost-Connes quantum statistical mechanical system and to Riemann surfaces and graphs.
Similar content being viewed by others
References
R. Akhoury and A. Comtet, “Anomalous behavior of the Witten index — exactly soluble models,” Nucl. Phys. B 246, 253–278 (1984).
J. B. Bost and A. Connes, “Hecke algebras, Type III factors and phase transitions with spontaneous symmetry breaking in number theory,” Selecta Math. 1(3), 411–457 (1995).
O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics 2, second ed., Texts and Monographs in Physics (Springer-Verlag, 1997).
D. Buchholz and R. Longo, “Graded KMS-functionals and the breakdown of supersymmetry,” Adv. Theor. Math. Phys. 3(3), 615–626 (1999).
B. Ćaćić, “A reconstruction theorem for almost-commutative spectral triples,” Lett. Math. Phys. 100(2), 181–202 (2012).
S. Cecotti and L. Girardello, “Functional measure, topology, and dynamical supersymmetry breaking,” Phys. Lett. B 110, 39–43 (1982).
A. Chamseddine and A. Connes, “The spectral action principle,” Comm. Math. Phys. 186(3), 731–750 (1997).
A. Chamseddine, A. Connes and M. Marcolli, “Gravity and the standard model with neutrino mixing,” Adv. Theor. Math. Phys. 11, 991–1090 (2007).
A. Connes, “Compact metric spaces, Fredholm modules, and hyperfiniteness,” Ergod. Th. Dynam. Sys. 9, 207–220 (1989).
A. Connes, Noncommutative Geometry (Academic Press, 1994).
A. Connes, “Geometry from the spectral point of view,” Lett. Math. Phys. 34(3), 203–238 (1995).
A. Connes, “A unitary invariant in Riemannian geometry,” Int. J. Geom. Meth. Mod. Phys. 5(8), 1215–1242 (2008).
A. Connes, “On the spectral characterization of manifolds,” J. Noncomm. Geom. 7(1), 1–82 (2013).
A. Connes and M. Marcolli, “A walk in the noncommutative garden,” in An Invitation to Noncommutative Geometry, pp. 1–128 (World Scientific, 2008).
A. Connes and M. Marcolli, Noncommutative Geometry, Quantum Fields and Motives, Colloquium Publications 55 (American Math. Society, 2008).
A. Connes, M. Marcolli and N. Ramachandran, “KMS states and complex multiplication,” Selecta Math. (N.S.) 11(3–4), 325–347 (2005).
A. Connes and H. Moscovici, “The local index formula in noncommutative geometry,” Geom. Funct. Anal. 5(2), 174–243 (1995).
A. Connes and H. Moscovici, “Type III and spectral triples,” in Traces in Number Theory, Geometry and Quantum Fields, pp. 57–71, Aspects Math. E 38 (Vieweg, 2008).
M. Coornaert, “Mesures de Patterson-Sullivan sur le bord d’un espace hyperbolique au sens de Gromov,” Pacific J. Math. 159(2), 241–270 (1993).
G. Cornelissen and J. W. de Jong, “The spectral length of a map between Riemannian manifolds,” J. Noncomm. Geom. 6(4), 721–748 (2012).
G. Cornelissen and M. Marcolli, “Zeta functions that hear the shape of a Riemann surface,” J. Geom. Phys. 58(1), 57–69 (2008).
G. Cornelissen and M. Marcolli, “Quantum statistical mechanics, L-series and anabelian geometry,” [arXiv:1009.0736].
G. Cornelissen and M. Marcolli, “Graph reconstruction and quantum statistical mechanics,” to appear in J. Geom. Phys. [arXiv:1209.5783].
P. B. Gilkey, Asymptotic Formulae in Spectral Geometry, Studies in Advanced Mathematics (Chapman Hall/CRC, 2004).
E. Ha and F. Paugam, “Bost-Connes-Marcolli systems for Shimura varieties. I. Definitions and formal analytic properties,” IMRP Int. Math. Res. Pap. 5, 237–286 (2005).
N. Higson and J. Roe, Analytic K-Homology (Oxford Univ. Press, 2000).
A. Jaffe, “Mathematics motivated by physics,” in The Legacy of John von Neumann (Hempstead, NY, 1988), pp. 137–150, Proc. Sympos. Pure Math. 50 (Amer. Math. Soc., Providence, RI, 1990).
J. W. de Jong, “Graphs, spectral triples and Dirac zeta functions,” p-Adic Numbers Ultrametric Anal. Appl. 1(4), 286–296 (2009).
B. Julia, “Statistical theory of numbers,” in Number Theory and Physics (Springer, 1990).
D. Kastler, “Cyclic cocycles from graded KMS functionals,” Commun. Math. Phys. 121, 345–350 (1989).
T. Kimura, S. Koyama and N. Kurokawa, “Euler products beyond the boundary,” [arXiv:1210.1216].
M. Laca and I. Raeburn, “A semigroup crossed product arising in number theory,” J. London Math. Soc. (2) 59(1), 330–344 (1999).
M. Laca, N. Larsen and S. Neshveyev, “On Bost-Connes types systems for number fields,” J. Number Theory 129(2), 325–338 (2009).
S. Lord, A. Rennie and J. C. Varilly, “Riemannian manifolds in noncommutative geometry,” J. Geom. Phys. 62(7), 1611–1638 (2012).
J. Lott, “Limit sets as examples in noncommutative geometry,” K-Theory 34(4), 283–326 (2005).
S. J. Patterson, “The limit set of a Fuchsian group,” Acta Math. 136(3–4), 241–273 (1976).
M. A. Rieffel, “Compact quantum metric spaces,” in Operator Algebras, Quantization, and Noncommutative Geometry, Contemp. Math. 365, pp. 315–330 (Amer. Math. Soc., 2004).
D. Spector, “Supersymmetry and the Möbius inversion function,” Commun. Math. Phys. 127, 239–252 (1990).
D. Spector, “Duality, partial supsersymmetry, and arithmetic number theory,” J. Math. Phys. 39(4), 1919–1927 (1998).
O. Stoytchev, “The modular group and super-KMS functionals,” Lett. Math. Phys. 27, 43–50 (1993).
D. Sullivan, “The density at infinity of a discrete group of hyperbolic motions,” Inst. Hautes Études Sci. Publ. Math. 50, 171–202 (1979).
E. Witten, “Constraints on supersymmetry breaking,” Nucl. Phys. B 202, 253–316 (1982).
B. Yalkinoglu, “On arithmetic models and functoriality of Bost-Connes systems” [arXiv:1105.5022].
D. Zhang, “Projective Dirac operators, twisted K-theory and local index formula,” to appear in J. Noncomm. Geom. [arXiv:1008.0707].
Author information
Authors and Affiliations
Corresponding author
Additional information
The text was submitted by the authors in English.
Rights and permissions
About this article
Cite this article
Greenfield, M., Marcolli, M. & Teh, K. Twisted spectral triples and quantum statistical mechanical systems. P-Adic Num Ultrametr Anal Appl 6, 81–104 (2014). https://doi.org/10.1134/S2070046614020010
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S2070046614020010