Geometry & Topology

The quantum content of the gluing equations

Tudor Dimofte and Stavros Garoufalidis

Full-text: Open access

Abstract

The gluing equations of a cusped hyperbolic 3–manifold M are a system of polynomial equations in the shapes of an ideal triangulation T of M that describe the complete hyperbolic structure of M and its deformations. Given a Neumann–Zagier datum (comprising the shapes together with the gluing equations in a particular canonical form) we define a formal power series with coefficients in the invariant trace field of M that should (a) agree with the asymptotic expansion of the Kashaev invariant to all orders, and (b) contain the nonabelian Reidemeister–Ray–Singer torsion of M as its first subleading “1–loop” term. As a case study, we prove topological invariance of the 1–loop part of the constructed series and extend it into a formal power series of rational functions on the PSL(2,) character variety of M. We provide a computer implementation of the first three terms of the series using the standard SnapPy toolbox and check numerically the agreement of our torsion with the Reidemeister–Ray–Singer for all 59924 hyperbolic knots with at most 14 crossings. Finally, we explain how the definition of our series follows from the quantization of 3–dimensional hyperbolic geometry, using principles of topological quantum field theory. Our results have a straightforward extension to any 3–manifold M with torus boundary components (not necessarily hyperbolic) that admits a regular ideal triangulation with respect to some PSL(2,) representation.

Article information

Source
Geom. Topol., Volume 17, Number 3 (2013), 1253-1315.

Dates
Received: 6 March 2012
Revised: 25 October 2012
Accepted: 5 January 2013
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513732608

Digital Object Identifier
doi:10.2140/gt.2013.17.1253

Mathematical Reviews number (MathSciNet)
MR3073925

Zentralblatt MATH identifier
1283.57017

Subjects
Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57N10: Topology of general 3-manifolds [See also 57Mxx]

Keywords
volume complex Chern–Simons theory Kashaev invariant gluing equations Neumann–Zagier equations Neumann–Zagier datum hyperbolic geometry ideal triangulations $1$–loop torsion quantum dilogarithm state integral perturbation theory Feynman diagram formal Gaussian integration

Citation

Dimofte, Tudor; Garoufalidis, Stavros. The quantum content of the gluing equations. Geom. Topol. 17 (2013), no. 3, 1253--1315. doi:10.2140/gt.2013.17.1253. https://projecteuclid.org/euclid.gt/1513732608


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References

  • M Aganagic, V Bouchard, A Klemm, Topological strings and (almost) modular forms, Comm. Math. Phys. 277 (2008) 771–819
  • J E Andersen, R Kashaev, A TQFT from quantum Teichmüller theory
  • D Bar-Natan, E Witten, Perturbative expansion of Chern–Simons theory with noncompact gauge group, Comm. Math. Phys. 141 (1991) 423–440
  • E W Barnes, The genesis of the double gamma functions, Proc. London Math. Soc. 31 (1899) 358
  • R Benedetti, C Petronio, Branched standard spines of $3$–manifolds, Lecture Notes in Mathematics 1653, Springer, Berlin (1997)
  • D Bessis, C Itzykson, J B Zuber, Quantum field theory techniques in graphical enumeration, Adv. in Appl. Math. 1 (1980) 109–157
  • D W Boyd, N M Dunfield, F Rodriguez-Villegas, Mahler's measure and the dilogarithm (II)
  • B A Burton, Regina: Normal surface and $3$–manifold topology software Available at \setbox0\makeatletter\@url http://regina.sourceforge.net {\unhbox0
  • D Calegari, Real places and torus bundles, Geom. Dedicata 118 (2006) 209–227
  • B G Casler, An imbedding theorem for connected $3$–manifolds with boundary, Proc. Amer. Math. Soc. 16 (1965) 559–566
  • A A Champanerkar, $A$–polynomial and Bloch invariants of hyperbolic $3$–manifolds, PhD thesis, Columbia University (2003) Available at \setbox0\makeatletter\@url http://search.proquest.com/docview/305332823 {\unhbox0
  • D Cooper, M Culler, H Gillet, D D Long, P B Shalen, Plane curves associated to character varieties of $3$–manifolds, Invent. Math. 118 (1994) 47–84
  • M Culler, N M Dunfield, J R Weeks, SnapPy, a computer program for studying the geometry and topology of $3$–manifolds (2011) Available at \setbox0\makeatletter\@url http://snappy.computop.org {\unhbox0
  • J A De Loera, J Rambau, F Santos, Triangulations, Algorithms and Computation in Mathematics 25, Springer, Berlin (2010)
  • T Dimofte, Quantum Riemann surfaces in Chern–Simons theory
  • T Dimofte, D Gaiotto, S Gukov, Gauge theories labelled by three-manifolds
  • T Dimofte, S Gukov, J Lenells, D Zagier, Exact results for perturbative Chern–Simons theory with complex gauge group, Commun. Number Theory Phys. 3 (2009) 363–443
  • J Dubois, Non abelian twisted Reidemeister torsion for fibered knots, Canad. Math. Bull. 49 (2006) 55–71
  • J Dubois, S Garoufalidis, Rationality of the $\mathrm{SL}(2,\mathbb{C})$–Reidemeister torsion in dimension $3$
  • N M Dunfield, Cyclic surgery, degrees of maps of character curves, and volume rigidity for hyperbolic manifolds, Invent. Math. 136 (1999) 623–657
  • N M Dunfield, S Friedl, N Jackson, Twisted Alexander polynomials of hyperbolic knots, Exp. Math. 21 (2012) 329–352
  • N M Dunfield, S Garoufalidis, Incompressibility criteria for spun-normal surfaces, Trans. Amer. Math. Soc. 364 (2012) 6109–6137
  • J L Dupont, C H Sah, Scissors congruences, II, J. Pure Appl. Algebra 25 (1982) 159–195
  • J L Dupont, C K Zickert, A dilogarithmic formula for the Cheeger–Chern–Simons class, Geom. Topol. 10 (2006) 1347–1372
  • D B A Epstein, R C Penner, Euclidean decompositions of noncompact hyperbolic manifolds, J. Differential Geom. 27 (1988) 67–80
  • B Eynard, Invariants of spectral curves and intersection theory of moduli spaces of complex curves
  • L D Faddeev, Discrete Heisenberg–Weyl group and modular group, Lett. Math. Phys. 34 (1995) 249–254
  • L D Faddeev, R M Kashaev, Quantum dilogarithm, Modern Phys. Lett. A 9 (1994) 427–434
  • L D Faddeev, R M Kashaev, A Y Volkov, Strongly coupled quantum discrete Liouville theory. I. Algebraic approach and duality, Comm. Math. Phys. 219 (2001) 199–219
  • V V Fok, L O Chekhov, Quantum Teichmüller spaces, Teoret. Mat. Fiz. 120 (1999) 511–528
  • S Garoufalides, D Zagier, Empirical relations betwen $q$–series and Kashaev's invariant of knots, Preprint (2013)
  • S Garoufalidis, Quantum knot invariants, Mathematische Arbeitstagung talk 2011
  • S Garoufalidis, On the characteristic and deformation varieties of a knot, from: “Proceedings of the Casson Fest”, Geom. Topol. Monogr. 7 (2004) 291–309
  • S Garoufalidis, Chern–Simons theory, analytic continuation and arithmetic, Acta Math. Vietnam. 33 (2008) 335–362
  • S Garoufalidis, C D Hodgson, J H Rubinstein, H Segerman, $1$–efficient triangulations and the index of a cusped hyperbolic $3$–manifold
  • S Garoufalidis, T T Q Lê, Asymptotics of the colored Jones function of a knot, Geom. Topol. 15 (2011) 2135–2180
  • S Garoufalidis, D Zagier, Asymptotics of quantum knot invariants, Preprint (2013)
  • S Goette, C K Zickert, The extended Bloch group and the Cheeger–Chern–Simons class, Geom. Topol. 11 (2007) 1623–1635
  • S Gukov, Three-dimensional quantum gravity, Chern–Simons theory, and the $A$–polynomial, Comm. Math. Phys. 255 (2005) 577–627
  • S Gukov, H Murakami, $\mathrm{SL}(2,\mathbb{C})$ Chern–Simons theory and the asymptotic behavior of the colored Jones polynomial, Lett. Math. Phys. 86 (2008) 79–98
  • S Gukov, P Sułkowski, $A$–polynomial, $B$–model, and quantization, J. High Energy Phys. (2012) 070, front matter+56
  • K Hikami, Hyperbolic structure arising from a knot invariant, Internat. J. Modern Phys. A 16 (2001) 3309–3333
  • K Hikami, Generalized volume conjecture and the $A$–polynomials: the Neumann–Zagier potential function as a classical limit of the partition function, J. Geom. Phys. 57 (2007) 1895–1940
  • C D Hodgson, J H Rubinstein, H Segerman, Triangulations of hyperbolic $3$–manifolds admitting strict angle structures
  • K Hori, S Katz, A Klemm, R Pandharipande, R Thomas, C Vafa, R Vakil, E Zaslow, Mirror symmetry, Clay Mathematics Monographs 1, Amer. Math. Soc. (2003)
  • V F R Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. 126 (1987) 335–388
  • E Kang, Normal surfaces in non-compact $3$–manifolds, J. Aust. Math. Soc. 78 (2005) 305–321
  • E Kang, J H Rubinstein, Ideal triangulations of $3$–manifolds, I: Spun normal surface theory, from: “Proceedings of the Casson Fest”, (C Gordon, Y Rieck, editors), Geom. Topol. Monogr. 7 (2004) 235–265
  • R M Kashaev, Quantum dilogarithm as a $6j$–symbol, Modern Phys. Lett. A 9 (1994) 3757–3768
  • R M Kashaev, A link invariant from quantum dilogarithm, Modern Phys. Lett. A 10 (1995) 1409–1418
  • R M Kashaev, Quantization of Teichmüller spaces and the quantum dilogarithm, Lett. Math. Phys. 43 (1998) 105–115
  • S V Matveev, Transformations of special spines, and the Zeeman conjecture, Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987) 1104–1116, 1119
  • S Matveev, Algorithmic topology and classification of $3$–manifolds, 2nd edition, Algorithms and Computation in Mathematics 9, Springer, Berlin (2007)
  • W Müller, Analytic torsion and $R$–torsion for unimodular representations, J. Amer. Math. Soc. 6 (1993) 721–753
  • 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”, (B Apanasov, W D Neumann, A W Reid, L Siebenmann, editors), Ohio State Univ. Math. Res. Inst. Publ. 1, de Gruyter, Berlin (1992) 243–271
  • W D Neumann, Extended Bloch group and the Cheeger–Chern–Simons class, Geom. Topol. 8 (2004) 413–474
  • W D Neumann, D Zagier, Volumes of hyperbolic three-manifolds, Topology 24 (1985) 307–332
  • R Piergallini, Standard moves for standard polyhedra and spines, Rend. Circ. Mat. Palermo Suppl. (1988) 391–414
  • M Polyak, Feynman diagrams for pedestrians and mathematicians, from: “Graphs and patterns in mathematics and theoretical physics”, (M Lyubich, L Takhtajan, editors), Proc. Sympos. Pure Math. 73, Amer. Math. Soc. (2005) 15–42
  • B Ponsot, J Teschner, Clebsch-Gordan and Racah–Wigner coefficients for a continuous series of representations of $\mathcal U\sb q(\mathrm{sl}(2,\mathbb{R}))$, Comm. Math. Phys. 224 (2001) 613–655
  • J Porti, Torsion de Reidemeister pour les variétés hyperboliques, Mem. Amer. Math. Soc. 128 (1997) x+139
  • D B Ray, I M Singer, $R$–torsion and the Laplacian on Riemannian manifolds, Advances in Math. 7 (1971) 145–210
  • D Shale, Linear symmetries of free boson fields, Trans. Amer. Math. Soc. 103 (1962) 149–167
  • V P Spiridonov, G S Vartanov, Elliptic hypergeometry of supersymmetric dualities II: orthogonal groups, knots, and vortices
  • W P Thurston, The geometry and topology of three-manifolds, Princeton Univ. Math. Dept. Lecture Notes (1979) Available at \setbox0\makeatletter\@url http://msri.org/publications/books/gt3m/ {\unhbox0
  • W P Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. 6 (1982) 357–381
  • S Tillmann, Normal surfaces in topologically finite $3$–manifolds, Enseign. Math. 54 (2008) 329–380
  • S Tillmann, Degenerations of ideal hyperbolic triangulations, Math. Z. 272 (2012) 793–823
  • V G Turaev, The Yang–Baxter equation and invariants of links, Invent. Math. 92 (1988) 527–553
  • A Weil, Sur certains groupes d'opérateurs unitaires, Acta Math. 111 (1964) 143–211
  • E Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989) 351–399
  • E Witten, Quantization of Chern–Simons gauge theory with complex gauge group, Comm. Math. Phys. 137 (1991) 29–66
  • Y Yamaguchi, A relationship between the non-acyclic Reidemeister torsion and a zero of the acyclic Reidemeister torsion, Ann. Inst. Fourier (Grenoble) 58 (2008) 337–362
  • C K Zickert, The volume and Chern–Simons invariant of a representation, Duke Math. J. 150 (2009) 489–532