Aharony–Bergman–Jafferis–Maldacena Wilson Loops in the Fermi Gas Approach

Abstract

The matrix model of the Aharony-Bergman-Jafferis-Maldacena theory can be formulated in terms of an ideal Fermi gas with a non-trivial one-particle Hamiltonian. We show that, in this formalism, vacuum expectation values (vevs) of Wilson loops correspond to averages of operators in the statistical-mechanical problem. This makes it possible to calculate these vevs at all orders in 1/N, up to exponentially small corrections, and for arbitrary Chern-Simons coupling, by using the WentzelKramer-Brillouin expansion. We present explicit results for the vevs of 1/6 and the 1/2 BogomolnyiPrasad-Sommerfield Wilson loops, at any winding number, in terms of Airy functions. Our expressions are shown to reproduce the low genus results obtained previously in the 't Hooft expansion.

BPS Wilson loops and Bremsstrahlung function in ABJ(M): a two loop analysis

“…In [33,34] the exact quantum value of this Wilson loop has been obtained by evaluating the matrix model through topological string theory techniques. These results have been further generalized [35] using a powerful Fermi gas approach [36]. The strong coupling limit of the exact expressions matches the predictions from the AdS dual description.…”

“…We have checked this proposal using the weak and strong coupling expansion of the exact expression for W 1/6 n given in [35]. At weak coupling we reproduce exactly…”

“…In particular, since W B (1) = W 1/6 , the second term can be easily computed from the well-known results of localization for the 1/6−BPS bosonic Wilson loop on the maximal circle [31,[33][34][35] and allows to make an interesting prediction for B 1/2 (λ) at three loops…”

“…More generally, since the second term can be easily evaluated both at weak and strong coupling by means of the results already available for the undeformed bosonic Wilson loop [31,35], we can make a prediction for the three loop contribution to B 1/2 (λ), that could be tested by an explicit computation of the cusp anomalous dimension in ABJM theory at that order. Taking into account the series expansion for the 1/6−BPS Wilson loop given in [31,34] we obtain…”

ABJ theory in the higher spin limit

“…Inspired by the seminal work of Drukker, Marino, and Putrov [71,72] and, in good part, with the use of the elegant Fermi gas approach developed by Marino and Putrov [73], a great deal about the ABJ(M) partition function has been uncovered, in particular, at large N , both in perturbative [73,74] and nonperturbative expansions [75][76][77][78][79][80][81][82]. There has also been significant progress in the study of Wilson loops in the ABJ(M) theory [83][84][85][86] as well as the partition functions of more general Chern-Simons-matter theories [87][88][89][90][91]. However, the ABJ partition function in the HS limit (1.1) has not been much investigated in the literature.…”