Exact Results in ABJM Theory from Topological Strings

Journal of High Energy Physics (Impact Factor: 6.11). 12/2009; 2010(6). DOI: 10.1007/JHEP06(2010)011
Source: arXiv


Recently, Kapustin, Willett and Yaakov have found, by using localization techniques, that vacuum expectation values of Wilson loops in ABJM theory can be calculated with a matrix model. We show that this matrix model is closely related to Chern-Simons theory on a lens space with a gauge supergroup. This theory has a topological string large N dual, and this makes possible to solve the matrix model exactly in the large N expansion. In particular, we find the exact expression for the vacuum expectation value of a 1/6 BPS Wilson loop in the ABJM theory, as a function of the 't Hooft parameters, and in the planar limit. This expression gives an exact interpolating function between the weak and the strong coupling regimes. The behavior at strong coupling is in precise agreement with the prediction of the AdS string dual. We also give explicit results for the 1/2 BPS Wilson loop recently constructed by Drukker and Trancanelli Comment: 18 pages, two figures, small misprints corrected and references added, final version to appear in JHEP

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    • "Membrane instantons are already visible in the semi-classical small -expansion of the Fermi gas, while worldsheet instanton effects are non-perturbative in . One can study such worldsheet instanton effects from the opposite large regime, by 1/-expansion of the matrix model associated with the Fermi gas system, which corresponds to the ordinary genus expansion of the matrix model with the string coupling given by g s = 1/ [8] [9] [10]. From the viewpoint of this g s -expansion, the membrane instantons appear as nonperturbative effects in g s . "
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    ABSTRACT: We investigate the large $N$ instanton effects of partition functions in a class of $\mathcal{N}=4$ circular quiver Chern-Simons theories on a three-sphere. Our analysis is based on the supersymmetry localization and the Fermi-gas formalism. The resulting matrix model can be regarded as a two-parameter deformation of the ABJM matrix model, and has richer non-perturbative structures. Based on a systematic semi-classical analysis, we find analytic expressions of membrane instanton corrections. We also exactly compute the partition function for various cases and find some exact forms of worldsheet instanton corrections, which appear as quantum mechanical non-perturbative corrections in the Fermi-gas system.
    Journal of High Energy Physics 05/2015; 2015(9). DOI:10.1007/JHEP09(2015)046 · 6.11 Impact Factor
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    • "This still remains non-trivial after the localization. In the 't Hooft limit, this was done in [8] [9] in great detail. After overcoming several technical issues, a recursion relation to determine the all-genus free energy was found [8]. "
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    ABSTRACT: The exact partition function in ABJM theory on three-sphere can be regarded as a canonical partition function of a non-interacting Fermi-gas with an unconventional Hamiltonian. All the information on the partition function is encoded in the discrete spectrum of this Hamiltonian. We explain how (quantum mechanical) non-perturbative corrections in the Fermi-gas system appear from a spectral consideration. Basic tools in our analysis are a Mellin-Barnes type integral representation and a spectral zeta function. From a consistency with known results, we conjecture that the spectral zeta function in the ABJM Fermi-gas has an infinite number of "non-perturbative" poles, which are invisible in the semi-classical expansion of the Planck constant. We observe that these poles indeed appear after summing up perturbative corrections. As a consequence, the perturbative resummation of the spectral zeta function causes non-perturbative corrections to the grand canonical partition function. We also present another example associated with a spectral problem in topological string theory. A conjectured non-perturbative free energy on the resolved conifold is successfully reproduced in this framework.
    Journal of High Energy Physics 03/2015; 2015(11). DOI:10.1007/JHEP11(2015)086 · 6.11 Impact Factor
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    • "These matrix integrals can be studied in the conventional 't Hooft expansion [4] [5]. In [7] it was pointed out that, to fully understand the non-perturbative effects, one has to go beyond the 't Hooft 1/N expansion and study the M-theory regime of (2.1). "
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    ABSTRACT: The partition function of ABJ(M) theories on the three-sphere can be regarded as the canonical partition function of an ideal Fermi gas with a non-trivial Hamiltonian. We propose an exact expression for the spectral determinant of this Hamiltonian, which generalizes recent results obtained in the maximally supersymmetric case. As a consequence, we find an exact WKB quantization condition determining the spectrum which is in agreement with numerical results. In addition, we investigate the factorization properties and functional equations for our conjectured spectral determinants. These functional equations relate the spectral determinants of ABJ theories with consecutive ranks of gauge groups but the same Chern-Simons coupling.
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