Abstract
We consider the link average of the half-BPS Wilson loop operators in \( \mathcal{N}=6 \) superconformal Chern-Simons-matter theory, which is called ABJM theory. We show that this loop average is reduced to a (super)matrix integral by the localization method, in a similar way to the bosonic U(N) Chern-Simons theory. Using this matrix integral, we compute the two- and three-link averages with an operator formalism inspired by a three-dimensional topological field theory. We obtain a factorization of the link average, and the Verlinde formula in a sector of supergroup representations. We also propose a refined version of ABJM theory, and compute some refined link averages.
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References
E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121 (1989) 351 [INSPIRE].
O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, N = 6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [INSPIRE].
O. Aharony, O. Bergman and D.L. Jafferis, Fractional M2-branes, JHEP 11 (2008) 043 [arXiv:0807.4924] [INSPIRE].
V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].
A. Kapustin, B. Willett and I. Yaakov, Exact results for Wilson loops in superconformal Chern-Simons theories with matter, JHEP 03 (2010) 089 [arXiv:0909.4559] [INSPIRE].
N. Drukker and D. Trancanelli, A supermatrix model for N = 6 super Chern-Simons-matter theory, JHEP 02 (2010) 058 [arXiv:0912.3006] [INSPIRE].
M. Mariño and P. Putrov, Exact results in ABJM theory from topological strings, JHEP 06 (2010) 011 [arXiv:0912.3074] [INSPIRE].
D. Gaiotto and E. Witten, Janus configurations, Chern-Simons couplings, and the theta-angle in N = 4 super Yang-Mills theory, JHEP 06 (2010) 097 [arXiv:0804.2907] [INSPIRE].
M. Aganagic and S. Shakirov, Knot homology and refined Chern-Simons index, Commun. Math. Phys. 333 (2015) 187 [arXiv:1105.5117] [INSPIRE].
D. Gaiotto and X. Yin, Notes on superconformal Chern-Simons-Matter theories, JHEP 08 (2007) 056 [arXiv:0704.3740] [INSPIRE].
M. Mariño, Chern-Simons theory, matrix integrals and perturbative three manifold invariants, Commun. Math. Phys. 253 (2004) 25 [hep-th/0207096] [INSPIRE].
A. Berele and A. Regev, Hook Young-diagrams with applications to combinatorics and to representations of Lie-superalgebras, Adv. Math. 64 (1987) 118 [INSPIRE].
M. Aganagic, A. Klemm, M. Mariño and C. Vafa, Matrix model as a mirror of Chern-Simons theory, JHEP 02 (2004) 010 [hep-th/0211098] [INSPIRE].
I.G. Macdonald, Symmetric functions and Hall polynomials, 2nd edition, Oxford University Press, Oxford U.K. (1997).
M. Mariño and P. Putrov, ABJM theory as a Fermi gas, J. Stat. Mech. 1203 (2012) P03001 [arXiv:1110.4066] [INSPIRE].
E. Moens and J. Van der Jeugt, A determinantal formula for supersymmetric Schur polynomials, J. Alg. Comb. 17 (2003) 283.
B. Eynard and T. Kimura, Towards U(N|M) knot invariant from ABJM theory, arXiv:1408.0010 [INSPIRE].
E.P. Verlinde, Fusion rules and modular transformations in 2D conformal field theory, Nucl. Phys. B 300 (1988) 360 [INSPIRE].
R. Dijkgraaf, S. Gukov, V.A. Kazakov and C. Vafa, Perturbative analysis of gauged matrix models, Phys. Rev. D 68 (2003) 045007 [hep-th/0210238] [INSPIRE].
P. Etingof and A. Kirillov Jr., On Cherednik-Macdonald-Mehta identities, Electron. Res. Announc. Amer. Math. Soc. 4 (1998) 43 [q-alg/9712051].
T. Quella and V. Schomerus, Superspace conformal field theory, J. Phys. A 46 (2013) 494010 [arXiv:1307.7724] [INSPIRE].
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Kimura, T. Linking loops in ABJM and refined theory. J. High Energ. Phys. 2015, 30 (2015). https://doi.org/10.1007/JHEP07(2015)030
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DOI: https://doi.org/10.1007/JHEP07(2015)030