×

A TQFT of Turaev-Viro type on shaped triangulations. (English) Zbl 1337.81105

Summary: A shaped triangulation is a finite triangulation of an oriented pseudo-three-manifold where each tetrahedron carries dihedral angles of an ideal hyperbolic tetrahedron. To each shaped triangulation, we associate a quantum partition function in the form of an absolutely convergent state integral which is invariant under shaped 3-2 Pachner moves and invariant with respect to shape gauge transformations generated by total dihedral angles around internal edges through the Neumann-Zagier Poisson bracket. Similarly to Turaev-Viro theory, the state variables live on edges of the triangulation but take their values on the whole real axis. The tetrahedral weight functions are composed of three hyperbolic gamma functions in a way that they enjoy a manifest tetrahedral symmetry. We conjecture that for shaped triangulations of closed three-manifolds, our partition function is twice the absolute value squared of the partition function of Teichmüller TQFT defined by J. E. Andersen and R. Kashaev [Commun. Math. Phys. 330, No. 3, 887–934 (2014; Zbl 1305.57024)]. This is similar to the known relationship between the Turaev-Viro and the Witten-Reshetikhin-Turaev invariants of three-manifolds. We also discuss interpretations of our construction in terms of three-dimensional supersymmetric field theories related to triangulated three-dimensional manifolds.

MSC:

81T45 Topological field theories in quantum mechanics
81T60 Supersymmetric field theories in quantum mechanics
32B25 Triangulation and topological properties of semi-analytic andsubanalytic sets, and related questions
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory

Citations:

Zbl 1305.57024
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Andersen, J.E., Kashaev, R.: A TQFT from quantum Teichmüller theory. Commun. Math. Phys. 330(3), 887-934 (2014). arXiv:1109.6295 [math.QA] · Zbl 1305.57024
[2] Atiyah M.: Topological quantum field theories. Inst. Hautes Études Sci. Publ. Math. 68, 175-186 (1988) · Zbl 0692.53053 · doi:10.1007/BF02698547
[3] Baseilhac S., Benedetti R.: Quantum hyperbolic geometry. Algebraic Geom. Topol. 7, 845-917 (2007) · Zbl 1139.57008 · doi:10.2140/agt.2007.7.845
[4] Barnes E.W.: A transformation of generalized hypergeometric series. Q. J. Math. 41, 136-140 (1910) · JFM 41.0503.01
[5] Bazhanov V.V., Sergeev S.M.: Elliptic gamma-function and multi-spin solutions of the Yang-Baxter equation. Nucl. Phys. B 856, 475-496 (2012) · Zbl 1246.81076 · doi:10.1016/j.nuclphysb.2011.10.032
[6] Barrett J.W., Westbury B.W.: Invariants of piecewise-linear 3-manifolds. Trans. Am. Math. Soc. 348(10), 3997-4022 (1996) · Zbl 0865.57013 · doi:10.1090/S0002-9947-96-01660-1
[7] Blanchet C., Habegger N., Masbaum G., Vogel P.: Three-manifold invariants derived from the Kauffman bracket. Topology 31, 685-699 (1992) · Zbl 0771.57004 · doi:10.1016/0040-9383(92)90002-Y
[8] Blanchet C., Habegger N., Masbaum G., Vogel P.: Topological quantum field theories derived from the Kauffman bracket. Topology 34, 883-927 (1995) · Zbl 0887.57009 · doi:10.1016/0040-9383(94)00051-4
[9] Chekhov L.O., Fock V.V.: Quantum Teichmüller spaces. Theor. Math. Phys. 120, 1245-1259 (1999) · Zbl 0986.32007 · doi:10.1007/BF02557246
[10] Dijkgraaf R., Fuji H., Manabe M.: The volume conjecture, perturbative knot invariants, and recursion relations for topological strings. Nucl. Phys. B 849, 166-211 (2011) · Zbl 1215.81082 · doi:10.1016/j.nuclphysb.2011.03.014
[11] Dimofte, T.: Quantum Riemann surfaces in Chern-Simons theory. Adv. Theor. Math. Phys. 17(3), 479-599 (2013) · Zbl 1304.81143
[12] Dimofte, T., Gaiotto, D., Gukov, S.: 3-Manifolds and 3d indices. Adv. Theor. Math. Phys. 17(5), 975-1076 (2013) · Zbl 1297.81149
[13] Dimofte, T., Gaiotto, D., Gukov, S.: Gauge theories labelled by three-manifolds. Commun. Math. Phys. 325(2), 367-419 (2014) · Zbl 1292.57012
[14] Dimofte T., Gukov S., Lenells J., Zagier D.: Exact results for perturbative Chern-Simons theory with complex gauge group. Commun. Num. Theor. Phys. 3, 363-443 (2009) · Zbl 1214.81151 · doi:10.4310/CNTP.2009.v3.n2.a4
[15] Dolan F.A., Osborn H.: Applications of the superconformal index for protected operators and q-hypergeometric identities to \[{\mathcal{N}=1}N=1\] dual theories. Nucl. Phys. B 818, 137-178 (2009) · Zbl 1194.81220 · doi:10.1016/j.nuclphysb.2009.01.028
[16] Dolan F.A.H., Spiridonov V.P., Vartanov G.S.: From 4d superconformal indices to 3d partition functions. Phys. Lett. B 704(3), 234-241 (2011) · doi:10.1016/j.physletb.2011.09.007
[17] Faddeev L.D.: Discrete Heisenberg-Weyl group and modular group. Lett. Math. Phys. 34(3), 249-254 (1995) · Zbl 0836.47012 · doi:10.1007/BF01872779
[18] Faddeev L.D., Kashaev R.M., Volkov A.Y.: Strongly coupled quantum discrete Liouville theory. 1. Algebraic approach and duality. Commun. Math. Phys. 219, 199-219 (2001) · Zbl 0981.81052 · doi:10.1007/s002200100412
[19] Futer, D., Guéritaud, F.: From angled triangulations to hyperbolic structures. In: Interactions Between Hyperbolic Geometry, Quantum Topology and Number Theory, Contemporary Mathematics, vol. 541, pp. 159-182. American Mathematical Society, Providence (2011) · Zbl 1236.57002
[20] Gasper, G., Rahman, M.: Basic hypergeometric series. In: Encyclopedia of Mathematics and its Applications, 2nd edn, vol. 96. Cambridge University Press, Cambridge (2004) (with a foreword by Richard Askey) · Zbl 1129.33005
[21] Geer, N., Kashaev, R., Turaev, V.: Tetrahedral forms in monoidal categories and 3-manifold invariants. J. Reine Angew. Math. 673, 69-123 (2012) · Zbl 1261.57013
[22] Hama N., Hosomichi K., Lee S.: Notes on SUSY gauge theories on three-sphere. JHEP 1103, 127 (2011) · Zbl 1301.81133 · doi:10.1007/JHEP03(2011)127
[23] Hama N., Hosomichi K., Lee S.: SUSY gauge theories on squashed three-spheres. JHEP 1105, 014 (2011) · Zbl 1296.81061 · doi:10.1007/JHEP05(2011)014
[24] Hikami K.: Hyperbolicity of Partition Function and Quantum Gravity. Nucl. Phys. B 616, 537-548 (2001) · Zbl 1020.83015 · doi:10.1016/S0550-3213(01)00464-3
[25] Hikami K.: Generalized volume conjecture and the A-polynomials—the Neumann-Zagier potential function as a classical limit of quantum invariant. J. Geom. Phys. 57, 1895-1940 (2007) · Zbl 1139.57013 · doi:10.1016/j.geomphys.2007.03.008
[26] Jafferis D.L.: The exact superconformal R-symmetry extremizes Z. JHEP 1205, 159 (2012) · Zbl 1348.81420 · doi:10.1007/JHEP05(2012)159
[27] Kapustin A., Willett B., Yaakov I.: Exact results for Wilson loops in superconformal Chern-Simons theories with matter. JHEP 1003, 089 (2010) · Zbl 1271.81110 · doi:10.1007/JHEP03(2010)089
[28] Kashaev R.M.: Quantization of Teichmüller spaces and the quantum dilogarithm. Lett. Math. Phys. 43, 105-115 (1998) · Zbl 0897.57014 · doi:10.1023/A:1007460128279
[29] Kashaev R.M.: Quantum dilogarithm as a 6j-symbol. Mod. Phys. Lett. A 9(40), 3757-3768 (1994) · Zbl 1015.17500 · doi:10.1142/S0217732394003610
[30] Kirillov, A. Jr., Balsam, B.: Turaev-Viro invariants as an extended TQFT. (2010, preprint). arXiv:1004.1533 · Zbl 1178.33019
[31] Korepanov I.G.: Euclidean 4-simplices and invariants of four-dimensional manifolds: I. Moves \[{3\rightarrow 3}3\]→3. Theor. Math. Phys. 131(3), 765-774 (2002) · Zbl 1042.57011 · doi:10.1023/A:1015971322591
[32] Korepanov I.G.: Pachner move \[{3\rightarrow3}3\]→3 and affine volume-preserving geometry in R3. SIGMA 1, 021 (2005) · Zbl 1101.57011
[33] Faddeev L.D., Popov V.N.: Feynman diagrams for the Yang-Mills field. Phys. Lett. B 25, 29 (1967) · doi:10.1016/0370-2693(67)90067-6
[34] Luo, F.: Solving Thurston equation in a commutative ring (2012, preprint). arXiv:1201.2228 · Zbl 1298.81332
[35] Luo, F.: Volume optimization, normal surfaces, and Thurston’s equation on triangulated 3-manifolds. J. Differ. Geom. 93(2), 299-326 (2013) · Zbl 1292.57018
[36] Pestun V.: Localization of gauge theory on a four-sphere and supersymmetric Wilson loops. Commun. Math. Phys. 313, 71-129 (2012) · Zbl 1257.81056 · doi:10.1007/s00220-012-1485-0
[37] Ponzano, G., Regge, T.: Semiclassical limit of Racah coefficients. In: Bloch, F. (ed.) Spectroscopic and Group Theoretical Methods in Physics, pp. 1-58. North-Holland Publ. Co., Amsterdam (1968) · Zbl 0692.53053
[38] Rivin I.: Euclidean structures on simplicial surfaces and hyperbolic volume. Ann. Math. (2) 139(3), 553-580 (1994) · Zbl 0823.52009 · doi:10.2307/2118572
[39] Reshetikhin N., Turaev V.: Ribbon graphs and their invariants derived fron quantum groups. Commun. Math. Phys. 127, 1-26 (1990) · Zbl 0768.57003 · doi:10.1007/BF02096491
[40] Reshetikhin N., Turaev V.: Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math. 103, 547-597 (1991) · Zbl 0725.57007 · doi:10.1007/BF01239527
[41] Ruijsenaars S.N.M.: First order analytic difference equations and integrable quantum systems. J. Math. Phys. 38, 1069-1146 (1997) · Zbl 0877.39002 · doi:10.1063/1.531809
[42] Segal, G.B.: The definition of conformal field theory. In: Differential Geometrical Methods in Theoretical Physics (Como, 1987), NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, vol. 250, pp. 165-171. Kluwer Academic Publishers, Dordrecht (1988)
[43] Spiridonov V.P.: On the elliptic beta function. Uspekhi Mat. Nauk 56(1), 181-182 (2001) · Zbl 0997.33009 · doi:10.4213/rm374
[44] Spiridonov V.P.: On the elliptic beta function. Russ. Math. Surv. 56(1), 185-186 (2001) · Zbl 0997.33009 · doi:10.1070/RM2001v056n01ABEH000374
[45] Spiridonov V.P.: A Bailey tree for integrals. Theor. Math. Phys. 139(1), 536-541 (2004) · Zbl 1178.33019 · doi:10.1023/B:TAMP.0000022745.45082.18
[46] Spiridonov V.P.: Elliptic beta integrals and solvable models of statistical mechanics. Contemp. Math. 563, 181-211 (2012) · Zbl 1244.82019 · doi:10.1090/conm/563/11170
[47] Spiridonov V.P., Vartanov G.S.: Elliptic hypergeometry of supersymmetric dualities. Commun. Math. Phys. 304, 797-874 (2011) · Zbl 1225.81137 · doi:10.1007/s00220-011-1218-9
[48] Spiridonov, V.P., Vartanov, G.S.: Elliptic hypergeometry of supersymmetric dualities II. Orthogonal groups, knots, and vortices. Commun. Math. Phys. 325(2), 421-486 (2014) · Zbl 1285.81064
[49] Terashima Y., Yamazaki M.: SL(2,R) Chern-Simons, Liouville, and gauge theory on duality walls. JHEP 1108, 135 (2011) · Zbl 1298.81332 · doi:10.1007/JHEP08(2011)135
[50] Teschner, J., Vartanov, G.: 6j symbols for the modular double, quantum hyperbolic geometry, and supersymmetric gauge theories. Lett. Math. Phys. 104(5), 527-551 (2014) · Zbl 1296.81038
[51] Tillmann S.: Normal surfaces in topologically finite 3-manifolds. Enseign. Math. (2) 54(3-4), 329-380 (2008) · Zbl 1214.57022
[52] Turaev V.G.: Quantum Invariants of Knots and 3-Manifolds. De Gruyter Studies in Mathematics, vol. 18. Walter de Gruyter & Co., Berlin (1994) · Zbl 0812.57003
[53] Turaev V.G., Viro O.Ya.: State sum invariants of 3-manifolds and quantum 6j-symbols. Topology 31(4), 865-902 (1992) · Zbl 0779.57009 · doi:10.1016/0040-9383(92)90015-A
[54] Turaev, V., Virelizier, A.: On two approaches to 3-dimensional TQFTs (2010, preprint). arXiv:1006.3501 · Zbl 1254.57012
[55] Willett, B., Yaakov, \[I.: {\mathcal{N}=2}N=2\] dualities and Z extremization in three dimensions (2011, preprint). arXiv:1104.0487 · Zbl 1178.33019
[56] Witten E.: Topological quantum field theory. Commun. Math. Phys. 117(3), 353-386 (1988) · Zbl 0656.53078 · doi:10.1007/BF01223371
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.