Wilson loop in general representation and RG flow in 1d defect QFT

Abstract

The generalized Wilson loop operator interpolating between the supersymmetric and the ordinary Wilson loop in $\mathcal{N}=4$ SYM theory provides an interesting example of renormalization group flow on a line defect: the scalar coupling parameter $\zeta$ has a non-trivial beta function and may be viewed as a running coupling constant in a 1d defect QFT. In this paper we continue the study of this operator, generalizing previous results for the beta function and Wilson loop expectation value to the case of an arbitrary representation of the gauge group and beyond the planar limit. Focusing on the scalar ladder limit where the generalized Wilson loop reduces to a purely scalar line operator in a free adjoint theory, and specializing to the case of the rank $k$ symmetric representation of $SU(N)$, we also consider a certain semiclassical limit where $k$ is taken to infinity with the product $k{\zeta }^2$ fixed. This limit can be conveniently studied using a 1d defect QFT representation in terms of $N$ commuting bosons. Using this representation, we compute the beta function and the circular loop expectation value in the large $k$ limit, and use it to derive constraints on the structure of the beta function for general representation. We discuss the corresponding 1d RG flow and comment on the consistency of the results with the 1d defect version of the F-theorem.

44 pages. v2: comment below eq.(1.16) and references added

References in corpus (9)

• Holographic Wilson Loops
• Renormalization Group Flows on Line Defects
• Spin Impurities, Wilson Lines and Semiclassics
• Correlators on non-supersymmetric Wilson line in N=4 SYM and AdS${}_2$/CFT${}_1$
• Monodromy Defects in Free Field Theories
• Magnetic impurities at quantum critical points: large-$N$ expansion and SPT connections
• Flag manifold sigma models: spin chains and integrable theories
• Wilson loops in terms of color invariants
• Fixed point annihilation for a spin in a fluctuating field

Cited by in corpus (11)

• Spin Impurities, Wilson Lines and Semiclassics
• Phases of Wilson Lines in Conformal Field Theories
• Defects in scalar field theories, RG flows and Dimensional Disentangling
• Interpolating Wilson loops and enriched RG flows
• Fixed point annihilation for a spin in a fluctuating field
• SU(2)-Symmetric Spin-Boson Model: Quantum Criticality, Fixed-Point Annihilation, and Duality
• Large Charges on the Wilson Loop in $\mathcal{N}=4$ SYM: II. Quantum Fluctuations, OPE, and Spectral Curve
• Defect Localized Entropy: Renormalization Group and Holography
• Wilson loops in Large Symmetric Representations through a Double-Scaling Limit
• Tunable phase transition in a dissipative two-spin system: A renormalization group study
• Fermionic Matrix Models and Bosonization