New structure for orthogonal quantum group invariants
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- by Qingtao Chen and Kefeng Liu
- Proc. Amer. Math. Soc. 143 (2015), 3645-3657
- DOI: https://doi.org/10.1090/proc/12548
- Published electronically: April 28, 2015
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Abstract:
Based on the orthogonal Labastida-Mariño-Ooguri-Vafa conjecture made by L. Chen and Q. Chen (2012), we derive an infinite product formula for Chern-Simons partition functions, which generalizes Liu and Peng’s recent results to the orthogonal case. Symmetry property of this new infinite product structure is also discussed.References
- Anna Beliakova and Christian Blanchet, Skein construction of idempotents in Birman-Murakami-Wenzl algebras, Math. Ann. 321 (2001), no. 2, 347–373. MR 1866492, DOI 10.1007/s002080100233
- Vincent Bouchard, Bogdan Florea, and Marcos Mariño, Topological open string amplitudes on orientifolds, J. High Energy Phys. 2 (2005), 002, 35. MR 2140507, DOI 10.1088/1126-6708/2005/02/002
- R. Brauer, On algebras which are connected with the semisimple continuous groups, Ann. Math. 63 (1937) 854-872.
- Lin Chen, Chern-Simons theory of knots invariants, ProQuest LLC, Ann Arbor, MI, 2009. Thesis (Ph.D.)–University of California, Los Angeles. MR 2713712
- Lin Chen and Qingtao Chen, Orthogonal quantum group invariants of links, Pacific J. Math. 257 (2012), no. 2, 267–318. MR 2972467, DOI 10.2140/pjm.2012.257.267
- Qingtao Chen, Some mathematical aspects of quantum field theory, ProQuest LLC, Ann Arbor, MI, 2009. Thesis (Ph.D.)–University of California, Berkeley. MR 2717855
- P. Freyd, D. Yetter, J. Hoste, W. B. R. Lickorish, K. Millett, and A. Ocneanu, A new polynomial invariant of knots and links, Bull. Amer. Math. Soc. (N.S.) 12 (1985), no. 2, 239–246. MR 776477, DOI 10.1090/S0273-0979-1985-15361-3
- Rajesh Gopakumar and Cumrun Vafa, On the gauge theory/geometry correspondence, Adv. Theor. Math. Phys. 3 (1999), no. 5, 1415–1443. MR 1796682, DOI 10.4310/ATMP.1999.v3.n5.a5
- V. F. R. Jones, A new knot polynomial and von Neumann algebras, Notices Amer. Math. Soc. 33 (1986), no. 2, 219–225. MR 830613
- V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. (2) 126 (1987), no. 2, 335–388. MR 908150, DOI 10.2307/1971403
- Louis H. Kauffman, State models and the Jones polynomial, Topology 26 (1987), no. 3, 395–407. MR 899057, DOI 10.1016/0040-9383(87)90009-7
- Louis H. Kauffman, An invariant of regular isotopy, Trans. Amer. Math. Soc. 318 (1990), no. 2, 417–471. MR 958895, DOI 10.1090/S0002-9947-1990-0958895-7
- José M. F. Labastida and Marcos Mariño, A new point of view in the theory of knot and link invariants, J. Knot Theory Ramifications 11 (2002), no. 2, 173–197. MR 1895369, DOI 10.1142/S0218216502001561
- José M. F. Labastida, Marcos Mariño, and Cumrun Vafa, Knots, links and branes at large $N$, J. High Energy Phys. 11 (2000), Paper 7, 42. MR 1806596, DOI 10.1088/1126-6708/2000/11/007
- W. B. R. Lickorish and Kenneth C. Millett, A polynomial invariant of oriented links, Topology 26 (1987), no. 1, 107–141. MR 880512, DOI 10.1016/0040-9383(87)90025-5
- Chiu-Chu Melissa Liu, Kefeng Liu, and Jian Zhou, A proof of a conjecture of Mariño-Vafa on Hodge integrals, J. Differential Geom. 65 (2003), no. 2, 289–340. MR 2058264
- Kefeng Liu and Pan Peng, On a proof of the Labastida-Mariño-Ooguri-Vafa conjecture, Math. Res. Lett. 17 (2010), no. 3, 493–506. MR 2653684, DOI 10.4310/MRL.2010.v17.n3.a9
- Kefeng Liu and Pan Peng, Proof of the Labastida-Mariño-Ooguri-Vafa conjecture, J. Differential Geom. 85 (2010), no. 3, 479–525. MR 2739811
- Kefeng Liu and Pan Peng, New structures of knot invariants, Commun. Number Theory Phys. 5 (2011), no. 3, 601–615. MR 2864461, DOI 10.4310/CNTP.2011.v5.n3.a1
- Marcos Mariño, String theory and the Kauffman polynomial, Comm. Math. Phys. 298 (2010), no. 3, 613–643. MR 2670922, DOI 10.1007/s00220-010-1088-6
- Hirosi Ooguri and Cumrun Vafa, Knot invariants and topological strings, Nuclear Phys. B 577 (2000), no. 3, 419–438. MR 1765411, DOI 10.1016/S0550-3213(00)00118-8
- Józef H. Przytycki and PawełTraczyk, Invariants of links of Conway type, Kobe J. Math. 4 (1988), no. 2, 115–139. MR 945888
- Arun Ram, A Frobenius formula for the characters of the Hecke algebras, Invent. Math. 106 (1991), no. 3, 461–488. MR 1134480, DOI 10.1007/BF01243921
- Arun Ram, Characters of Brauer’s centralizer algebras, Pacific J. Math. 169 (1995), no. 1, 173–200. MR 1346252
- Arun Ram, A “second orthogonality relation” for characters of Brauer algebras, European J. Combin. 18 (1997), no. 6, 685–706. MR 1468338, DOI 10.1006/eujc.1996.0132
- N. Reshetikhin and V. G. Turaev, Invariants of $3$-manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991), no. 3, 547–597. MR 1091619, DOI 10.1007/BF01239527
- N. Yu. Reshetikhin and V. G. Turaev, Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127 (1990), no. 1, 1–26. MR 1036112
- Hans Wenzl, On the structure of Brauer’s centralizer algebras, Ann. of Math. (2) 128 (1988), no. 1, 173–193. MR 951511, DOI 10.2307/1971466
- Edward Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), no. 3, 351–399. MR 990772
Bibliographic Information
- Qingtao Chen
- Affiliation: Mathematics Section, International Center for Theoretical Physics, Strada Costiera, 11, Trieste, I-34151, Italy
- Email: qchen1@ictp.it
- Kefeng Liu
- Affiliation: Center of Mathematical Sciences, Zhejiang University, Box 310027, Hangzhou, People’s Republic of China — and — Department of Mathematics, University of California at Los Angeles, Box 951555, Los Angeles, California 90095-1555
- MR Author ID: 327618
- Email: liu@math.ucla.edu
- Received by editor(s): October 12, 2013
- Published electronically: April 28, 2015
- Communicated by: Lei Ni
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 3645-3657
- MSC (2010): Primary 57M27; Secondary 81R50
- DOI: https://doi.org/10.1090/proc/12548
- MathSciNet review: 3348806