A "Bootstrap-like" Approach for the Loop Equations

Peter D Anderson, Purdue University


Non-Abelian gauge theories are instrumental in formulating the standard model. The lattice has been the main viable option for obtaining results in the strong coupling regime. However, AdS/CFT correspondence is considered to be an alternative method of doing strongly coupled calculations via simpler string theory computations. Recent advancements in the formulation of certain supersymmetric theories on the lattice [1– 5] allow one to check results from AdS/CFT and, in principle, to interpolate between QCD and supersymmetric theories where AdS/CFT can be applied. Another method for understanding this connection is through the loop equations of N = 4 SYM [6–9]. It is known that pure gauge theories can be formulated solely in terms of Wilson loops with the loop equation. In the Large-N limit this equation closes in the expectation value of single loops. In particular, using the lattice as a regulator, it becomes a well defined equation for a discrete set of loops. An equivalent description of the N = 4 SYM lattice loop equations have not yet been formulated, but we outline here some first steps using the lattice description developed by Catterall et al [5]. Here we investigate different numerical approaches to solving the pure gauge loop equation and we find that previous methods gave good results in the strong coupling region but were found wanting in the more interesting weak coupling regime. We propose an alternative method based on the observation that certain matrices, ρ(L), of Wilson loop expectation values are positive definite. They also have unit trace (ρ(L) ≥ 0, Trρ(L) = 1), in fact they can be defined as density matrices in the space of open loops after tracing over color indices and can be used to define an entropy associated with the loss of information due to such trace SWL = –Tr(ρ (L) logL ρ(L)). The condition that such matrices are positive definite allows us to study the weak coupling region which is relevant for the continuum limit. In the exactly solvable case of two dimensions, this approach gives very good results by considering just a few loops. In higher dimensions, it gives good results in the weak coupling region and therefore is complementary to the strong coupling expansion. We compare the results with standard Monte Carlo simulations.




Kruczenski, Purdue University.

Subject Area

Theoretical physics|Particle physics

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