[Paper Review] Asymptotic Symmetries of Yang-Mills with Theta Term and Monopoles

Carlos Cardona|arXiv (Cornell University)|Apr 21, 2015

Black Holes and Theoretical Physics14 references6 citations

TL;DR

This paper proposes that large singular gauge transformations in Yang-Mills theory with a theta term induce monopoles in the bulk and decouple holomorphic and anti-holomorphic Kac-Moody current algebras at null infinity. By adding a topological theta term, the asymptotic symmetry structure is modified such that the holomorphic and anti-holomorphic currents become independent at specific theta values, resembling two-dimensional WZW models and suggesting a deeper connection to 2D CFTs in the S-matrix.

ABSTRACT

In this short note we suggest that the singular behavior of large gauge transformations preserving the vacuum at null infinity in Yang-Mills theory implies monopoles into the bulk, as well as that the inclusion of a theta term induces a decoupling between holomorphic and anti-holomorphic currents associated to those large gauge transformations

Motivation & Objective

• To investigate the physical consequences of allowing singular large gauge transformations at null infinity in Yang-Mills theory.
• To explore how the inclusion of a theta term modifies the asymptotic symmetry structure of non-Abelian gauge theories.
• To determine whether the holomorphic and anti-holomorphic currents associated with large gauge symmetries decouple under the theta term.
• To connect the resulting asymptotic symmetry algebra to two-dimensional WZW models and understand its implications for the S-matrix.

Proposed method

• Analyzing the asymptotic behavior of Yang-Mills fields in retarded coordinates with a radial gauge fixing, focusing on the boundary at null infinity.
• Applying large singular gauge transformations that preserve the vacuum at spatial infinity, leading to non-trivial field configurations in the bulk.
• Introducing a theta term to the Yang-Mills action, which becomes a Chern-Simons term on the boundary at null infinity.
• Deriving modified asymptotic equations of motion that include the theta term's contribution to the surface term.
• Using the flatness condition $F_{z\bar{z}} = 0$ and current conservation to relate the gauge potentials to punctures on the sphere.
• Showing that at $\theta = 0$, $A_z$ is holomorphic, and at $\theta = -8\pi$, $A_{\bar{z}}$ becomes anti-holomorphic, indicating decoupling of the current algebras.

Experimental results

Research questions

• RQ1 How do singular large gauge transformations at null infinity affect the bulk field configuration in Yang-Mills theory?
• RQ2 What is the role of the theta term in modifying the asymptotic symmetry algebra of non-Abelian gauge theories?
• RQ3 Under what conditions does the holomorphic and anti-holomorphic current algebra decouple in the asymptotic limit?
• RQ4 How does the inclusion of a theta term induce a structure analogous to two-dimensional WZW models in the boundary theory?
• RQ5 What are the implications of CP violation from the theta term for the matching of asymptotic symmetries at past and future null infinity?

Key findings

• Singular large gauge transformations at null infinity induce monopole-like configurations in the bulk of Yang-Mills theory, even in the absence of explicit monopole sources.
• The addition of a theta term modifies the asymptotic equations of motion, introducing a mixing parameter that depends on $\theta/4\pi$.
• At $\theta = 0$, the current algebra is dominated by the holomorphic component $A_z$, while at $\theta = -8\pi$, the anti-holomorphic component $A_{\bar{z}}$ becomes independent.
• The holomorphic and anti-holomorphic currents decouple at specific values of $\theta$, such as $\theta = -8\pi$, leading to two independent Kac-Moody algebras.
• The surface term from the theta term contributes a total derivative that modifies the asymptotic current algebra structure, resembling the WZW model's Chern-Simons term.
• The S-matrix elements are expected to resemble correlation functions in a 2D Euclidean WZW model based on the gauge group $\mathcal{G}$, suggesting a holographic interpretation.