[Paper Review] Notes on BPS Wilson Loops and the Cusp Anomalous Dimension in ABJM theory

Daniele Marmiroli|arXiv (Cornell University)|Dec 10, 2013

Black Holes and Theoretical Physics59 references4 citations

TL;DR

This paper introduces a new 1/6 BPS Wilson loop in ABJM theory on S³ that couples to scalar fields at an angle θ on ℂP³, enabling non-perturbative computation of the bremsstrahlung function B(λ) via localization. It relates the small-cusp-angle expansion of the cusp anomalous dimension to the logarithmic derivative of the Wilson loop, matching known weak-coupling results and revealing a leading-order strong-coupling agreement but a mismatch in the constant term compared to h(λ), highlighting a surprising similarity between two a priori unrelated functions.

ABSTRACT

We introduce a new purely bosonic, $\frac{1}{6}$ BPS Wilson loop for ABJM theory on $S^3$ that couples scalar fields to a latitude at an angle $θ$ on $S^2\in\mathbb{C}P^3$. Through localization of this operator, we relate the expansion of the cusp anomalous dimension at small cusp angles to the logarithmic derivative of the ABJM Wilson loop. This defines, non-perturbatively in the 't Hooft coupling, the bremsstrahlung function $B(λ)$ describing in three dimensions the soft radiation of a $W$-boson undergoing a sudden change in trajectory. We compare our results for $B(λ)$ to the known weak/strong coupling expansions of the function $h(λ)$ that enters integrability. At weak coupling we precisely match the previously known two-loop result. At strong coupling we find agreement at leading order in $\sqrtλ$, but a mismatch of the constant coefficient. We comment on the striking similarity that we observe between these two, in principle, unrelated functions.

Motivation & Objective

• To construct a new purely bosonic, 1/6 BPS Wilson loop in ABJM theory on S³ that couples to scalar fields at an angle θ on ℂP³.
• To use localization of this operator to relate the small-cusp-angle expansion of the cusp anomalous dimension to the logarithmic derivative of the Wilson loop expectation value.
• To compute the bremsstrahlung function B(λ) non-perturbatively in the 't Hooft coupling λ.
• To compare the resulting B(λ) with the known weak- and strong-coupling expansions of h(λ), a function central to integrability in ABJM theory.
• To investigate the striking similarity between B(λ) and h(λ), despite their different physical origins.

Proposed method

• Construct a Wilson loop operator that couples gauge fields to a latitude at angle φ on S² ⊂ S³ and scalar fields to a latitude at angle θ on S² ⊂ ℂP³, preserving 1/6 of the supersymmetry.
• Identify the conditions under which this operator is BPS by explicitly solving for the Killing spinors and verifying the preserved supercharges.
• Apply localization techniques to compute the expectation value of this Wilson loop exactly in the 't Hooft coupling λ.
• Relate the small-θ expansion of the cusp anomalous dimension to the logarithmic derivative of the Wilson loop, thereby extracting B(λ) non-perturbatively.
• Use the matrix model formulation of ABJM theory, including the lens space matrix model and Fermi gas reformulation, to compute the Wilson loop at weak and strong coupling.
• Compare the resulting B(λ) with the known expansions of h(λ) from integrability and perturbative field theory.

Experimental results

Research questions

• RQ1 How can a new 1/6 BPS Wilson loop be constructed in ABJM theory that couples to scalar fields at an arbitrary angle on ℂP³?
• RQ2 What is the non-perturbative relation between the cusp anomalous dimension at small cusp angles and the logarithmic derivative of the Wilson loop expectation value?
• RQ3 Does the bremsstrahlung function B(λ) computed via this method match known weak-coupling results and integrability predictions at strong coupling?
• RQ4 Why does B(λ) exhibit a striking similarity to h(λ), despite being derived from different physical principles?
• RQ5 What explains the mismatch in the constant term of the strong-coupling expansion between B(λ) and h(λ), despite agreement at leading order in √λ?

Key findings

• The paper constructs a new 1/6 BPS Wilson loop in ABJM theory that couples to scalar fields at angle θ on ℂP³ and is preserved by 2 out of 12 superconformal charges.
• The small-cusp-angle expansion of the cusp anomalous dimension is related to the logarithmic derivative of the Wilson loop, enabling non-perturbative computation of the bremsstrahlung function B(λ).
• At weak coupling, the computed B(λ) precisely matches the known two-loop perturbative result.
• At strong coupling, B(λ) agrees with h(λ) at leading order in √λ, but the constant coefficient in the expansion differs.
• The authors observe a striking similarity between B(λ) and h(λ), despite their distinct physical origins in soft radiation and integrability, respectively.
• The mismatch in the constant term at strong coupling suggests a deeper, yet-to-be-understood connection between the bremsstrahlung function and the integrability function h(λ).