A matrix model for the latitude Wilson loop in ABJM theory

Abstract

In ABJ(M) theory, we propose a matrix model for the exact evaluation of BPS Wilson loops on a latitude circular contour, so providing a new weak-strong interpolation tool. Intriguingly, the matrix model turns out to be a particular case of that computing torus knot invariants in U(N 1 |N 2 ) Chern-Simons theory. At weak coupling we check our proposal against a three-loop computation, performed for generic framing, winding number and representation. The matrix model is amenable of a Fermi gas formulation, which we use to systematically compute the strong coupling and genus expansions. For the fermionic Wilson loop the leading planar behavior agrees with a previous string theory prediction. For the bosonic operator our result provides a clue for finding the corresponding string dual configuration. Our matrix model is consistent with recent proposals for computing Bremsstrahlung functions exactly in terms of latitude Wilson loops. As a by-product, we extend the conjecture for the exact B θ 1/6 Bremsstrahlung function to generic representations and test it with a four-loop perturbative computation. Finally, we propose an exact prediction for B 1/2 at unequal gauge group ranks.

Roadmap on Wilson loops in 3d Chern–Simons-matter theories

“…This result deserves some comments: Firstly, while the framing-dependent contributions seem to exponentiate as in (6.3), the exponent becomes a non trivial function of the coupling, as opposed to the simple linear exponent of pure Chern-Simons theory; secondly, the analysis of [38] shows that while up to two-loops all the framing contributions came from purely gauge contractible propagators, at three-loops vertex-like diagrams with matter also contribute to the framing anomaly. An interesting consequence of the non-triviality of the exponent of (6.4) has to do with the fact [39][40][41][42] that the Bremsstrahlung function (Chapter 10 and 11) associated to 1/2 BPS Wilson loops in ABJM theory (N 1 = N 2 ) can be written as…”

“…This is an awkward choice since f is an integer number whereas q is a real one. This is however supported by the matrix model construction of [41], where a single q parameter is needed in the matrix integral and operator definitions in order to match the known perturbative results with f = q (see Chapter 8).…”

“…However the localization procedure has to yield a matrix model, which is a significant deformation of the one obtained in Chapter 3 and solved in Chapter 4. In fact, an explicit three-loop computation of this observable in the fundamental representation (with q defined in (7.3)) at framing f in ABJM (N 1 = N 2 = N ) gives [41]…”

Toward precision holography in type IIA with Wilson loops

“…Recently, in [13], a matrix model for the exact evaluation of the latitude BPS Wilson loops has been proposed. The expectation value for any genus of the fermionic (in the sense of the superconnection [20]) latitude Wilson loop is given in terms of Airy functions by (see equations (1.3) and (5.44) in [13]),…”

“…We hope that by turning our attention to the AdS 4 /ABJM pair we can gather complementary information to the one already available and ultimately learn about string perturbation theory in curved backgrounds with Ramond-Ramond fluxes. There are, indeed, a number of exact results obtained via localization of the ABJM theory starting with the free energy of the theory on S 3 [2] but most importantly to us there are various exact results for supersymmetric Wilson loops for the 1 2 BPS [12] and, more recently, for the 1 6 BPS configuration [13]. We consider one-loop effective actions of string configurations dual to those supersymmetric Wilson loops in ABJM.…”