Geometry & Topology

Classical and quantum dilogarithmic invariants of flat $PSL(2,\mathbb{C})$–bundles over 3–manifolds

Stephane Baseilhac and Riccardo Benedetti

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We introduce a family of matrix dilogarithms, which are automorphisms of NN, N being any odd positive integer, associated to hyperbolic ideal tetrahedra equipped with an additional decoration. The matrix dilogarithms satisfy fundamental five-term identities that correspond to decorated versions of the 23 move on 3–dimensional triangulations. Together with the decoration, they arise from the solution we give of a symmetrization problem for a specific family of basic matrix dilogarithms, the classical (N=1) one being the Rogers dilogarithm, which only satisfy one special instance of five-term identity. We use the matrix dilogarithms to construct invariant state sums for closed oriented 3–manifolds W endowed with a flat principal PSL(2,)–bundle ρ, and a fixed non empty link L if N>1, and for (possibly “marked”) cusped hyperbolic 3–manifolds M. When N=1 the state sums recover known simplicial formulas for the volume and the Chern–Simons invariant. When N>1, the invariants for M are new; those for triples (W,L,ρ) coincide with the quantum hyperbolic invariants defined by the first author, though our present approach clarifies substantially their nature. We analyse the structural coincidences versus discrepancies between the cases N=1 and N>1, and we formulate “Volume Conjectures”, having geometric motivations, about the asymptotic behaviour of the invariants when N.

Article information

Geom. Topol., Volume 9, Number 1 (2005), 493-569.

Received: 2 August 2003
Revised: 5 April 2005
Accepted: 5 April 2005
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M27: Invariants of knots and 3-manifolds 57Q15: Triangulating manifolds
Secondary: 57R20: Characteristic classes and numbers 20G42: Quantum groups (quantized function algebras) and their representations [See also 16T20, 17B37, 81R50]

dilogarithms state sum invariants quantum field theory Cheeger–Chern–Simons invariants scissors congruences hyperbolic 3–manifolds.


Baseilhac, Stephane; Benedetti, Riccardo. Classical and quantum dilogarithmic invariants of flat $PSL(2,\mathbb{C})$–bundles over 3–manifolds. Geom. Topol. 9 (2005), no. 1, 493--569. doi:10.2140/gt.2005.9.493.

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  • S Baseilhac, Dilogarithme Quantique et Invariants de 3–Variétés, PhD Thesis, Université Paul Sabatier, Toulouse, France (October 2001)
  • S Baseilhac, R Benedetti, QHI, 3–manifold scissors congruence classes and the volume conjecture, from: “Invariants of Knots and 3–manifolds (Kyoto 2001)”, \gtmref4200221328
  • S Baseilhac, R Benedetti, Quantum hyperbolic invariants of 3–manifolds with $PSL(2,\C)$–characters, Topology 43 (2004) 1373–1423
  • S Baseilhac, R Benedetti, 3D Quantum Hyperbolic Field Theory.
  • R Benedetti C. Petronio, Lectures on Hyperbolic Geometry, Universitext, Springer (1992)
  • R Benedetti, C Petronio, Branched Standard Spines of $3$–manifolds, Lecture Notes in Mathematics 1653, Springer (1997)
  • V V Bazhanov, N Yu Reshetikhin, Remarks on the quantum dilogarithm, Journal of Physics A: Mathematics & General 28 (1995) 2217–2226
  • S Baaj, G Skandalis, Unitaires multiplicatifs et dualité pour les produits croisés de $C^*$–algèbres, Ann. Sci. l'Ecole Norm. Sup. 26 (1993) 425–488
  • B G Casler, An embedding theorem for connected $3$–manifolds with boundary, Proc. Amer. Math. Soc. 16 (1965) 559–566
  • F Costantino, A calculus for branched spines of $3$–manifolds, Math. Zeit. to appear
  • V Chari, A Pressley, A Guide To Quantum Groups, Cambridge University Press (1994)
  • A Davydov, Pentagon equation and matrix bialgebras, Comm. Algebra 29 (2001) 2627–2650
  • N Dunfield, Cyclic surgery, degree of maps of character curves, and volume rigidity for hyperbolic manifolds, Invent. Math. 136 (1999) 623–657
  • J L Dupont, C-H Sah, Scissors congruences II, J. Pure Appl. Algebra 25 (1982) 159–195
  • J L Dupont, The dilogarithm as a characteristic class for flat bundles, J. Pure Appl. Algebra 44 (1987) 137–164
  • J L Dupont, Scissors Congruences, Group Homology and Characteristic Classes, Nankai Tracts in Mathematics Volume 1, World Scientific (2001)
  • D B A Epstein, R Penner, Euclidian decompositions of non-compact hyperbolic manifolds, J. Differential Geom. 27 (1988) 67–80
  • L D Faddeev, R. M. Kashaev, Quantum dilogarithm, Modern Phys. Lett. A 9 (1994) 427–434
  • S Francaviglia, Hyperbolic volume of representations of fundamental groups of cusped $3$–manifolds, Internat. Math. Res. Not. 9 (2004) 425–459
  • R M Kashaev, A link invariant from quantum dilogarithm, Modern Phys. Lett. A 10 (1995) 1409–1418
  • R M Kashaev, The algebraic nature of quantum dilogarithm, Geometry and integrable models (Dubna 1994), World Scientific Publishing, River Edge NJ (1996) 32–51
  • R M Kashaev, The hyperbolic volume of knots from the quantum dilogarithm, Lett. Math. Phys. 39 (1997) 269–275
  • C Kassel, Quantum Groups, Graduate Texts in Mathematics 155, Springer (1995)
  • L Lewin, Dilogarithm and Associated Functions, Macdonald (1958)
  • S V Matveev, Transformations of special spines and the Zeeman conjecture, Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987) 1104–1116, 1119 (Russion); translation in Math USSR-Izv. 31 (1988) 423–434
  • R Meyerhoff, Hyperbolic $3$–manifolds with equal volumes but different Chern-Simons invariants, from: “Low-dimensional topology and Kleinian groups”, (D B A Epstein, editor), LMS Lecture Notes Series, 112 (1986) 209–215
  • H Murakami, J Murakami, The colored Jones polynomials and the simplicial volume of a knot, Acta Math. 186 (2001) 85–104
  • W D Neumann, Combinatorics of triangulations and the Chern-Simons invariant for hyperbolic $3$–manifolds, from: “Topology '90 (Columbus, OH, 1990)”, Ohio State Univ. Math. Res. Inst. Publ. 1, De Gruyter, Berlin (1992) 243–271
  • W D Neumann, Hilbert's 3rd problem and invariants of 3–manifolds, from: “The Epstein Birthday Schrift”, \gtmref1199919393411
  • W D Neumann, Extended Bloch group and the Chern-Simons class, incomplete working version.
  • W D Neumann, Extended Bloch group and the Cheeger–Chern–Simons class, \gtref8200410413474
  • W D Neumann, J Yang, Bloch invariants of hyperbolic $3$–manifolds, Duke Math. J. 96 (1) (1999) 29–59
  • W D Neumann, D Zagier, Volumes of hyperbolic $3$–manifolds, Topology 24 (1985) 307–332
  • T Ohtsuki (Editor), Invariants of Knots and $3$–Manifolds (Kyoto 2001), a collection of papers and problems, Geom. Topol. Monogr. 4 (2002–4)
  • R Piergallini, Standard moves for standard polyhedra and spines, from “Third National Conference on Topology (Trieste, 1986)”, Rend. Cir. Mat. Palermo (2) Suppl. 18 (1988) 391–414
  • M A Semenov-Tian-Shansky, Poisson-Lie groups, quantum duality principle, and the quantum double, from: “Mathematical aspects of conformal and topological field theories and quantum groups (South Hadley, MA, 1992)”, Contemp. Math. 175 (1994) 219–248
  • V Turaev, Quantum Invariants of $3$–Manifolds, Studies in Mathematics 18, De Gruyter, Berlin (1994)
  • D Zagier, The remarkable dilogarithm, J. Math. Phys. Sci. 22 (1988) 131–145