Z. Naturforsch. **68a,** 178 – 209
(2013)

doi:10.5560/ZNA.2012-0118

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

Albrecht
Klemm^{1},
Marcos
Mari~no^{2},
Marc
Schiereck^{1}, and
Masoud
Soroush^{1}

^{1 }Bethe Center for Theoretical Physics, Physikalisches Institut der Universität Bonn, Nussallee 12, D-53315 Bonn, Germany

^{2 }Département de Physique Théorique et Section de Mathématiques, Université de Genève, Genève, CH-1211 Switzerland

Received November 29, 2012 / published online February 15, 2013

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 Wentzel–Kramer–Brillouin expansion. We present explicit results for the vevs of 1/6 and the 1/2 Bogomolnyi–Prasad–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.

*Key words:* Sring Gauge Theory, Duality, Strongly Coupled Systems.