zbMATH — the first resource for mathematics

Knot invariants and topological strings. (English) Zbl 1036.81515
Summary: We find further evidence for the conjecture relating large $$N$$ Chern-Simons theory on $$S^3$$ with topological string on the resolved conifold geometry by showing that the Wilson loop observable of a simple knot on $$S^3$$ (for any representation) agrees to all orders in $$N$$ with the corresponding quantity on the topological string side. For a general knot, we find a reformulation of the knot invariant in terms of new integral invariants, which capture the spectrum (and spin) of M2 branes ending on M5 branes embedded in the resolved conifold geometry. We also find an intriguing link between knot invariants and superpotential terms generated by worldsheet instantons in $$N=1$$ supersymmetric theories in 4 dimensions.

MSC:
 81T45 Topological field theories in quantum mechanics 57M27 Invariants of knots and $$3$$-manifolds (MSC2010) 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
Full Text:
References:
 [1] Periwal, V., Topological closed-string interpretation of chern – simons theory, Phys. rev. lett., Vol. 71, 1295, (1993) · Zbl 0972.81596 [2] Douglas, M.R., Chern – simons – witten theory as a topological Fermi liquid [3] Gopakumar, R.; Vafa, C., On the gauge theory/geometry correspondence · Zbl 1026.81029 [4] Witten, E., Chern – simons gauge theory as a string theory · Zbl 0844.58018 [5] Faber, C.; Pandharipande, R., Hodge integrals and gromov – witten theory · Zbl 0960.14031 [6] Candelas, P.; De La Ossa, X.C.; Green, P.S.; Parkes, L., A pair of calabi – yau manifolds as an exactly soluble superconformal theory, Nucl. phys. B, Vol. 359, 21, (1991) · Zbl 1098.32506 [7] Hosono, S.; Klemm, A.; Theisen, S.; Yau, S.-T., Mirror symmetry, mirror map and applications to complete intersection calabi – yau spaces, Nucl. phys. B, Vol. 433, 501, (1995) · Zbl 1020.32508 [8] Bershadsky, M.; Cecotti, S.; Ooguri, H.; Vafa, C., Holomorphic anomalies in topological field theories, Nucl. phys. B, Vol. 405, 279, (1993) · Zbl 0908.58074 [9] Bershadsky, M.; Cecotti, S.; Ooguri, H.; Vafa, C., Kodaira – spencer theory of gravity and exact results for quantum string theory, Commun. math. phys., Vol. 165, 311, (1994) · Zbl 0815.53082 [10] Gopakumar, R.; Vafa, C., M-theory and topological strings I, II [11] Witten, E., Quantum field theory and the Jones polynomial, Commun. math. phys., Vol. 121, 351, (1989) · Zbl 0667.57005 [12] Gross, D.J.; Taylor, W., Twists and Wilson loops in the string theory of two-dimensional QCD, Nucl. phys. B, Vol. 403, 395, (1993) · Zbl 1030.81518 [13] Antoniadis, I.; Gava, E.; Narain, K.S.; Taylor, T.R., Topological amplitudes in superstring theory, Nucl. phys. B, Vol. 413, 162, (1994) · Zbl 1007.81522 [14] Antoniadis, I.; Gava, E.; Narain, K.S.; Taylor, T.R., N=2 type II — heterotic duality and higher derivative F-terms, Nucl. phys. B, Vol. 455, 109, (1995) · Zbl 0925.81158 [15] Strominger, A.; Yau, S.-T.; Zaslow, E., Mirror symmetry is T-duality, Nucl. phys. B, Vol. 479, 243, (1996) · Zbl 0896.14024 [16] N. Berkovits, private communication [17] Berkovits, N.; Vafa, C., N=4 topological strings, Nucl. phys. B, Vol. 433, 123, (1995) · Zbl 1020.81761 [18] Hanany, A.; Hori, K., Branes and N=2 theories in two dimensions, Nucl. phys. B, Vol. 513, 119, (1998) · Zbl 1052.81625 [19] Brunner, I.; Douglas, M.R.; Lawrence, A.; Romelsberger, C., D-branes on the quintic · Zbl 0989.81100 [20] S. Kachru, S. Katz, A. Lawrence, J. McGreevy, Open string instantons and superpotentials, to appear [21] K. Hori, C. Vafa, Mirror symmetry, to appear [22] Gukov, S.; Vafa, C.; Witten, E., CFT’s from calabi – yau four-folds · Zbl 0984.81143 [23] Vafa, C., Extending mirror conjecture to calabi – yau with bundles · Zbl 0986.14500 [24] J. Labastida, M. Mariño, Polynomial invariants for torus knots and topological strings, to appear
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.