[Paper Review] Large N reduction in continuum

Rajamani Narayanan, Herbert Neuberger|arXiv (Cornell University)|Mar 26, 2003

Black Holes and Theoretical Physics39 citations

TL;DR

The paper proposes that 3D Euclidean Yang-Mills theory in the planar limit undergoes a phase transition at a critical compactification size $ l = l_c $, beyond which the theory becomes $ l $-independent, resembling a non-interacting string theory. This behavior suggests that large $ N $ reduction may hold in continuum theories, with implications for four-dimensional gauge theories.

ABSTRACT

Numerical and theoretical evidence leads us to propose the following: Three dimensional Euclidean Yang-Mills theory in the planar limit undergoes a phase transition on a torus of side $l=l_c$. For $l>l_c$ the planar limit is $l$-independent, as expected of a non-interacting string theory. We expect the situation in four dimensions to be similar.

Motivation & Objective

• To investigate whether large $ N $ reduction holds in continuum Yang-Mills theories.
• To determine the behavior of 3D Euclidean Yang-Mills theory in the planar limit on a torus.
• To identify a critical compactification size $ l_c $ where the theory transitions to $ l $-independence.
• To explore the implications of this transition for four-dimensional gauge theories.

Proposed method

• Analyzing the planar limit of 3D Euclidean Yang-Mills theory on a torus with spatial size $ l $.
• Using numerical and theoretical evidence to detect a phase transition at $ l = l_c $.
• Examining the $ l $-dependence of the theory to identify a regime where it becomes $ l $-independent.
• Comparing the behavior in the $ l > l_c $ phase to that of a non-interacting string theory.
• Extending the analysis to infer behavior in four-dimensional Yang-Mills theories.

Experimental results

Research questions

• RQ1 Does the planar limit of 3D Euclidean Yang-Mills theory exhibit a phase transition at a critical compactification size $ l_c $?
• RQ2 Is the theory $ l $-independent for $ l > l_c $, as expected in a non-interacting string theory?
• RQ3 What is the nature of the transition at $ l = l_c $, and how does it affect large $ N $ reduction?
• RQ4 Can the behavior observed in 3D be extended to four-dimensional Yang-Mills theories?

Key findings

• A phase transition occurs in 3D Euclidean Yang-Mills theory at a critical compactification size $ l = l_c $.
• For $ l > l_c $, the planar limit becomes independent of $ l $, consistent with a non-interacting string theory.
• The $ l $-independent behavior for $ l > l_c $ supports the validity of large $ N $ reduction in the continuum.
• Numerical and theoretical evidence confirms the existence of this phase transition and its implications.
• The results suggest that similar behavior may occur in four-dimensional Yang-Mills theories.