Motives and periods in Bianchi IX gravity models. (English) Zbl 1404.58014
Summary: We show that, when considering the anisotropic scaling factors and their derivatives as affine variables, the coefficients of the heat-kernel expansion of the Dirac-Laplacian on \(SU(2)\) Bianchi IX metrics are algebro-geometric periods of motives of complements in affine spaces of unions of quadrics and hyperplanes. We show that the motives are mixed Tate and we provide an explicit computation of their Grothendieck classes.
MSC:
| 58B34 | Noncommutative geometry (à la Connes) |
| 14C15 | (Equivariant) Chow groups and rings; motives |
| 83F05 | Relativistic cosmology |
Keywords:
Bianchi IX gravity model; spectral action; Dirac-Laplacian; heat kernel expansion; periods; motivesReferences:
| [1] | Babich, MV; Korotkin, DA, Self-dual \(SU(2)\) invariant Einstein metrics and modular dependence of theta-functions, Lett. Math. Phys., 46, 323-337, (1998) · Zbl 0917.53016 · doi:10.1023/A:1007542422413 |
| [2] | Brown, F., Mixed Tate motives over \({\mathbb{Z}}\), Ann. Math. (2), 175, 949-976, (2012) · Zbl 1278.19008 · doi:10.4007/annals.2012.175.2.10 |
| [3] | Chamseddine, AH; Connes, A., Spectral action for Robertson-Walker metrics, J. High Energy Phys., 101, 29, (2012) · Zbl 1397.81413 |
| [4] | Chamseddine, AH; Connes, A., The spectral action principle, Commun. Math. Phys., 186, 731-750, (1997) · Zbl 0894.58007 · doi:10.1007/s002200050126 |
| [5] | Connes, A., Marcolli, M.: Noncommutative Geometry, Quantum Fields and Motives. Colloquium Publications, vol. 55, American Mathematical Society (2008) · Zbl 1159.58004 |
| [6] | Cornish, NJ; Levin, JJ, The mixmaster universe: a chaotic Farey tale, Phys. Rev. D, 55, 7489-7510, (1997) · doi:10.1103/PhysRevD.55.7489 |
| [7] | Fan, W.; Fathizadeh, F.; Marcolli, M., Spectral action for Bianchi type-IX cosmological models, J. High Energy Phys., 10, 085, (2015) · Zbl 1388.83911 · doi:10.1007/JHEP10(2015)085 |
| [8] | Fan, W., Fathizadeh, F., Marcolli, M.: Modular forms in the spectral action of Bianchi IX gravitational instantons. arXiv:1511.05321 · Zbl 1388.83911 |
| [9] | Fang, L., Ruffini, R. (eds.): Quantum Cosmology. World Scientific, Singapore (1987) |
| [10] | Fathizadeh, F.; Ghorbanpour, A.; Khalkhali, M., Rationality of spectral action for Robertson-Walker metrics, J. High Energy Phys., 064, 21, (2014) · Zbl 1333.83130 |
| [11] | Fathizadeh, F.; Marcolli, M., Periods and motives in the spectral action of Robertson-Walker spacetimes, Commun. Math. Phys., 356, 641-671, (2017) · Zbl 1379.83026 · doi:10.1007/s00220-017-2991-x |
| [12] | Gracia-Bondia, J.M., Varilly, J.C., Figueroa, H.: Elements of Noncommutative Geometry. Birkhäuser, Boston (2001) · Zbl 0958.46039 · doi:10.1007/978-1-4612-0005-5 |
| [13] | Khalatnikov, IM; Lifshitz, EM; Khanin, KM; Shchur, LN; Sinai, Ya G., On the stochasticity in relativistic cosmology, J. Stat. Phys., 38, 97-114, (1985) · doi:10.1007/BF01017851 |
| [14] | Kontsevich, M., Zagier, D.: Periods. In: Mathematics unlimited—2001 and beyond. Springer, pp. 771-808 (2001) · Zbl 1039.11002 |
| [15] | Li, D., The Ponzano-Regge model and parametric representation, Commun. Math. Phys., 327, 243-260, (2014) · Zbl 1287.81111 · doi:10.1007/s00220-014-1945-9 |
| [16] | Manin, YuI; Marcolli, M., Continued fractions, modular symbols, and non-commutative geometry., Sel. Math. New Ser., 8, 475-521, (2002) · Zbl 1116.11033 · doi:10.1007/s00029-002-8113-3 |
| [17] | Manin, Yu.I., Marcolli, M.: Big Bang, blowup, and modular curves: algebraic geometry in cosmology. SIGMA Symmetry Integrability Geom. Methods Appl. 10 Paper 073 (2014) · Zbl 1296.85004 |
| [18] | Manin, YuI; Marcolli, M., Symbolic dynamics, modular curves, and Bianchi IX cosmologies, Ann. Fac. Sci. Toulouse Math. (6), 25, 517-542, (2016) · Zbl 1380.37071 · doi:10.5802/afst.1503 |
| [19] | Marcolli, M.: Modular curves, \(C^*\)-algebras, and chaotic cosmology. In: Frontiers in Number Theory, Physics, and Geometry. II. Springer, pp. 361-372 (2007) · Zbl 1176.83162 |
| [20] | Marcolli, M.: Feynman Motives. World Scientific, Singapore (2010) · Zbl 1192.14001 |
| [21] | Marcolli, M.: Noncommutative Cosmology. World Scientific (2018) · Zbl 1401.81005 |
| [22] | Mayer, D., Relaxation properties of the mixmaster universe, Phys. Lett. A, 121, 390-394, (1987) · doi:10.1016/0375-9601(87)90483-X |
| [23] | McKeag, P., Safarov, Yu.: Pseudodifferential operators on manifolds: a coordinate-free approach. In: Partial Differential Equations and Spectral Theory. Oper. Theory Adv. Appl., vol. 211. Birkhäuser, pp. 321-341 (2011) · Zbl 1254.58009 |
| [24] | Misner, CW, Mixmaster universe, Phys. Rev. Lett., 22, 1071-1074, (1969) · Zbl 0177.28701 · doi:10.1103/PhysRevLett.22.1071 |
| [25] | Rost, M.: The motive of a Pfister form. Preprint (1998). http://www.physik.uni-regensburg.de/ rom03516/motive.html |
| [26] | Shubin, M.: Pseudodifferential operators and spectral theory. Springer, Berlin (1987) · Zbl 0616.47040 · doi:10.1007/978-3-642-96854-9 |
| [27] | Vishik, A.: Integral motives of quadrics. Max-Planck-Institut für Mathematik Bonn, Preprint MPI-1998-13, 1-82 (1998) |
| [28] | Vishik, A.: Motives of quadrics with applications to the theory of quadratic forms. In: Geometric Methods in the Algebraic Theory of Quadratic Forms. Lecture Notes in Math., vol. 1835. Springer, pp. 25-101 (2004) · Zbl 1047.11033 |
| [29] | van Suijlekom, W.: Noncommutative Geometry and Particle Physics. Springer, Berlin (2014) · Zbl 1305.81007 |
| [30] | Voevodsly, V., Suslin, A., Friedlander, E.M.: Cyles, Transfers, and Motivic Homology Theories. Princeton University Press, Princeton (2000) · Zbl 1021.14006 |
| [31] | Wodzicki, M., Local invariants of spectral asymmetry, Invent. Math., 75, 143-177, (1984) · Zbl 0538.58038 · doi:10.1007/BF01403095 |
| [32] | Wodzicki, Mariusz, Noncommutative residue Chapter I. Fundamentals, 320-399, (1987), Berlin, Heidelberg · Zbl 0649.58033 · doi:10.1007/BFb0078372 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
