[Paper Review] Color superconductivity, BPS strings and monopole confinement in N=2 and N=4 super Yang-Mills theories

Marco A. C. Kneipp|ArXiv.org|Jan 29, 2004

Black Holes and Theoretical Physics4 citations

TL;DR

This paper investigates monopole confinement via BPS strings in deformed N=2 and N=4 super Yang-Mills theories with SU(N) gauge groups, showing that monopole magnetic fluxes are always integer multiples of string fluxes and that BPS string tensions in the Higgs phase satisfy Casimir scaling. The analysis reveals a confining three-monopole system in SU(3) and establishes a duality between chromomagnetic flux tubes and chromoelectric strings in the confining phase.

ABSTRACT

We review some recent developments on BPS string solutions and monopole confinement in the Higgs or (color) superconducting phase of deformed N=2 and N=4 super Yang-Mills theories. In particular, the monopole magnetic fluxes are shown to be always integer linear combinations of string fluxes. Moreover, a bound for the threshold length of the string breaking is obtained. When the gauge group SU(N) is broken to Z_N, the BPS string tension satisfies the Casimir scaling law. Furthermore in the SU(3) case the string solutions are such that they allow the formation of a confining system with three monopoles.

Motivation & Objective

• To understand monopole confinement in weakly coupled, supersymmetric Yang-Mills theories as a dual to strong-coupling quark confinement.
• To analyze BPS string solutions in the Higgs or (color) superconducting phase of deformed N=2 and N=4 SYM theories.
• To determine whether monopole magnetic fluxes are quantized in integer multiples of string fluxes.
• To investigate whether BPS string tensions in SU(N) gauge theories satisfy the Casimir scaling law.
• To explore the formation of three-monopole confining systems in SU(3), analogous to baryonic states.

Proposed method

• Constructs a two-step symmetry breaking mechanism: first to a Coulomb phase with solitonic monopoles, then to a Higgs or (color) superconducting phase with flux tubes.
• Uses a potential derived from the bosonic part of N=4 or N=2 SYM theories with mass deformations, ensuring BPS conditions are satisfied.
• Analyzes string solutions for two distinct representations of scalar fields: adjoint and symmetric product of fundamental representations.
• Applies electromagnetic duality to map monopole-antimonopole confinement in a superconductor to string formation.
• Derives a lower bound on string tension using the BPS condition and magnetic flux quantization.
• Computes string tensions for arbitrary fundamental weights λₖ in SU(N), showing agreement with Casimir scaling.

Experimental results

Research questions

• RQ1 Are monopole magnetic fluxes always integer multiples of the fluxes carried by BPS strings in N=2 and N=4 SYM theories with SU(N) gauge groups?
• RQ2 Do BPS string tensions in the Higgs phase of SU(N) gauge theories satisfy the Casimir scaling law?
• RQ3 Can a confining system of three monopoles form in SU(3), and if so, what are the topological conditions for such a system?
• RQ4 How does the choice of scalar representation (adjoint vs. symmetric product) affect the residual gauge group and confinement mechanism?
• RQ5 What is the threshold length for string breaking in these BPS systems, and how is it bounded?

Key findings

• Monopole magnetic fluxes are always integer linear combinations of string fluxes, ensuring topological consistency in monopole confinement.
• The BPS string tension for a weight ω = λₖ − β_ω satisfies the Casimir scaling law: Tₖ^BPS = T₁^BPS × k(N−k)/(N−1) for SU(N).
• For SU(3), a confining system with three monopoles (α₁, α₂, −α₁−α₂) can form via attached strings, analogous to a baryonic state.
• When φ₁ and φ₂ are in the symmetric product representation R^sym_{kλ_φ}, only a U(1) subgroup is broken, leading to monopole-antimonopole confinement.
• When φ₁ and φ₂ are in the adjoint representation, the full SU(N) gauge group is broken to a discrete Z_N subgroup, forming a color superconductor.
• The lower bound on string tension is saturated by BPS solutions, and the BPS string tension for fundamental weight λ₁ is T₁^BPS = (mμπ)/(2e) × (N−1)²/(2N).