[Paper Review] BPS Walls and Junctions in ${\cal N}=1$ SUSY Nonlinear Sigma Models

Masashi Naganuma, Muneto Nitta+1 more|ArXiv.org|Oct 22, 2002

Physics of Superconductivity and Magnetism4 citations

TL;DR

This paper constructs exact BPS wall and junction solutions in ${\cal N}=1$ supersymmetric nonlinear sigma models in four dimensions. By analyzing models with chiral superfields and a superpotential, it derives BPS equations that yield stable, finite-energy configurations connecting arbitrary vacua in moduli space, including $Z_N$-symmetric and asymmetric junctions with $N$ discrete vacua, demonstrating exact solutions via complex coordinate mappings and Kähler metric constructions.

ABSTRACT

The ${\cal N}=1$ SUSY nonlinear sigma models in four spacetime dimensions are studied to obtain BPS walls and junctions. A nonlinear sigma model with a single chiral scalar superfield is found which has a moduli space of the topology of $S^1$ and admits BPS walls and junctions connecting arbitrary points in moduli space. New BPS junction solutions connecting $N$ discrete vacua are also found for nonlinear sigma models with several chiral scalar superfields. More detailed exposition of our results can be found in Ref.\cite{NNS}.

Motivation & Objective

• To construct exact BPS solutions—walls and junctions—in ${\cal N}=1$ supersymmetric nonlinear sigma models in four spacetime dimensions.
• To explore the existence of BPS junctions connecting $N$ discrete vacua in models with multiple chiral scalar superfields.
• To generalize previous $Z_3$-symmetric junction solutions to arbitrary $N$ and asymmetric vacua configurations.
• To derive the effective nonlinear sigma model description from the original linear sigma model by eliminating auxiliary fields and expressing the dynamics in terms of a single complex scalar field $T$.
• To analyze the Kähler metric and its singularities, particularly at vacua and along arcs, to understand the geometric structure of the moduli space.

Proposed method

• Derive the $1/4$ BPS equation from the supersymmetric Lagrangian using a superpotential $\mathcal{W}$ and Kähler potential $K$, leading to a first-order differential equation in complex coordinates $z = x^1 + i x^2$.
• Construct exact solutions for junctions by defining auxiliary real-valued functions $f_j$ that satisfy $\prod f_j = 1$, and express the scalar fields $\mathcal{M}_j$ and $\hat{T}$ in terms of these functions.
• Use the BPS equations to eliminate the auxiliary fields $\mathcal{M}_j$ and express the dynamics of $\hat{T}$ solely in terms of $z$, resulting in a nonlinear BPS equation for $\hat{T}$.
• Map the solution to a nonlinear sigma model by inverting the relation between $\hat{T}$ and $z$, and deriving the effective Kähler metric $K_{TT^*}$ from the resulting dynamics.
• Analyze the Kähler metric for $N=4$ and asymmetric $N=3$ cases, identifying singularities along arcs and finite values at discrete vacua, and confirm the presence of negative kinetic energy regions.
• Verify the solution by showing that the energy density is finite and the junction asymptotes to three or four distinct vacua at infinity, forming a stable configuration.

Experimental results

Research questions

• RQ1 Can exact BPS junction solutions be constructed in ${\cal N}=1$ SUSY nonlinear sigma models with multiple chiral superfields?
• RQ2 What is the structure of the moduli space in models that admit BPS walls and junctions, particularly when the moduli space is topologically $S^1$?
• RQ3 How do the BPS equations and Kähler metric behave in models with $Z_N$-symmetric and asymmetric vacua?
• RQ4 Can the dynamics of a linear sigma model with $U(1)\times U(1)$ gauge symmetry be reduced to a nonlinear sigma model with only chiral superfields while preserving BPS solutions?
• RQ5 What are the geometric and physical implications of singularities in the Kähler metric, especially at vacua and along arcs separating discrete vacua?

Key findings

• A single chiral scalar superfield model with a moduli space topologically equivalent to $S^1$ admits exact BPS walls and junctions connecting any two points in moduli space.
• For $N=3$, the model yields a $Z_3$-symmetric junction solution that is an exact BPS solution, with the superpotential and Kähler metric derived from the original linear sigma model.
• The $N=4$ model results in a nonlinear sigma model with a Kähler metric $K_{TT^*}$ that is singular along arcs touching the unit circle at four discrete vacua $T = e^{i2\pi j/4}$, with finite values at the vacua.
• The energy density of the $Z_4$ junction is finite and localized, with the junction forming a stable configuration connecting four vacua at $1, i, -1, -i$ in the complex plane.
• For asymmetric vacua (e.g., $T = 1, i, -i$), the Kähler metric remains finite at the vacua and exhibits negative kinetic energy in regions between $|T|=1$ and an enclosing ellipse, confirming the existence of stable junctions.
• The derived BPS equation for $\hat{T}$ is expressed solely in terms of $\hat{T}$, confirming that the solution corresponds to a nonlinear sigma model with a specific Kähler metric and superpotential.