Observables of Non-Commutative Gauge Theories

TL;DR

This work constructs gauge-invariant, momentum-carrying observables in non-commutative Yang–Mills by attaching straight open Wilson lines to local operators, reducing to ordinary local operators in the IR. Perturbative analysis reveals a universal exponential growth in UV two-point functions due to the long Wilson line, which is echoed by the AdS/CFT dual via a NC deformation of the AdS$_5\times S^5$ background and a corresponding exponential suppression in the absorption cross section. For higher-point functions, the ratio to the product of two-point functions is exponentially suppressed in the UV, and this behavior mirrors, yet is distinct from, high-energy fixed-angle string scattering. The results establish a coherent field-theory–gravity correspondence for NCYM observables and highlight the non-decoupling of the $U(1)$ sector and the nonlocal, Wilson-line–driven nature of the observables.

Abstract

We construct gauge invariant operators in non-commutative gauge theories which in the IR reduce to the usual operators of ordinary field theories (e.g. F^2). We show that in the deep UV the two-point functions of these operators admit a universal exponential behavior which fits neatly with the dual supergravity results. We also consider the ratio between n-point functions and two-point functions to find exponential suppression in the UV which we compare to the high energy fixed angle scattering of string theory.

Figures (6)

• Figure 1: Moving the location of the local operator on the Wilson line does not change the definition of the gauge invariant operator only if the line is straight (a). For a generic line one gets a different gauge invariant operator (b).
• Figure 2: Leading contributions to the two-point functions. (a) is the $g^0$ order which agrees with the ordinary gauge theory results. (b), (c) and (d) are the non-commutative corrections at order $g^2$.
• Figure 3: Ladder diagram contributions to the rectangular Wilson loop in ordinary field theories and to the open Wilson lines in non-commutative gauge theories.
• Figure 4: WKB approximation to the absorption cross section. (a) In AdS the potential falls at infinity like $1/U^2$ and hence the absorption cross section is suppressed like a power of the cutoff. (b) In the NC version of AdS the potential goes to a constant at infinity and so the absorption cross section is suppressed like an exponential of the cutoff.
• Figure 5: The ladder diagrams dominate at large momentum due to the integration over $\zeta_i$ which grows linearly with the size of the line.