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Degeneracy and Continuous Deformations of Supersymmetric Domain Walls

M. Shifman

TL;DR

The paper demonstrates that in a broad class of $N=1$ supersymmetric theories there exist continuously degenerate families of BPS-saturated domain walls, labeled by hidden collective coordinates and sharing a common tension. By analyzing generalized two-field Wess-Zumino models, it shows that each wall's internal structure can vary without changing the tension, and that supersymmetry ensures bosonic zero modes on the wall have fermionic partners. A concrete two-field example reveals a continuous AD wall family, with a non-BPS BC wall whose stability depends on the inter-field coupling; integrating out a heavy field can erase this degeneracy, illustrating how effective theories may conceal rich wall structure present in the full theory. These results hint at a potentially important role for wall degeneracy in supersymmetric gauge theories and domain-wall dynamics beyond simple single-field models.

Abstract

In a wide class of supersymmetric theories degenerate families of the BPS-saturated domain walls exist. The internal structure of these walls can continuously vary, without changing the wall tension. This is described by hidden parameters (collective coordinates). Differentiating with respect to the collective coordinates one gets a set of the bosonic zero modes localized on the wall. Neither of them is related to the spontaneous breaking of any symmetry. Through the residual 1/2 of supersymmetry each bosonic zero mode generates a fermionic partner.

Degeneracy and Continuous Deformations of Supersymmetric Domain Walls

TL;DR

The paper demonstrates that in a broad class of supersymmetric theories there exist continuously degenerate families of BPS-saturated domain walls, labeled by hidden collective coordinates and sharing a common tension. By analyzing generalized two-field Wess-Zumino models, it shows that each wall's internal structure can vary without changing the tension, and that supersymmetry ensures bosonic zero modes on the wall have fermionic partners. A concrete two-field example reveals a continuous AD wall family, with a non-BPS BC wall whose stability depends on the inter-field coupling; integrating out a heavy field can erase this degeneracy, illustrating how effective theories may conceal rich wall structure present in the full theory. These results hint at a potentially important role for wall degeneracy in supersymmetric gauge theories and domain-wall dynamics beyond simple single-field models.

Abstract

In a wide class of supersymmetric theories degenerate families of the BPS-saturated domain walls exist. The internal structure of these walls can continuously vary, without changing the wall tension. This is described by hidden parameters (collective coordinates). Differentiating with respect to the collective coordinates one gets a set of the bosonic zero modes localized on the wall. Neither of them is related to the spontaneous breaking of any symmetry. Through the residual 1/2 of supersymmetry each bosonic zero mode generates a fermionic partner.

Paper Structure

This paper contains 5 sections, 46 equations, 6 figures.

Table of Contents

  1. Introduction
  2. General Observations
  3. Elaborating a Specific Example.
  4. Non-BPS wall connecting the saddle points ${\cal M}_2$ and ${\cal M}_3$
  5. Integrating out a heavy field

Figures (6)

  • Figure 1: The potential of the two-field model given in Eq. (\ref{['4']}) for the following values of the parameters: $\lambda = g = 1$, $\mu = 1$, $m = 1.2$. The points $A,B,C,D$ mark four vacua of the model. The four minima $A$ to $D$ correspond to ${\cal M}_1$ to ${\cal M}_4$, see Eq. (\ref{['clamin']}).
  • Figure 2: The range of variation of the fields $\phi$ and $\chi$ on the previous plot is shown by the solid line. The four minima are depicted as closed circles. The dashed lines show the wall trajectories $AB$ and $BD$, while the dotted lines show two (out of infinitely many) possible $AD$ trajectories. $\gamma$ is the injection angle of the creek (at $z\rightarrow -\infty$).
  • Figure 3: The profile of the superpotential $-{\cal W}(\Phi , X)$, Eqs. (\ref{['spone']}), (\ref{['typ']}). The notations are the same as on Figs. 1,2.
  • Figure 4: The profile of the superpotential $-{\cal W}$ in the model (\ref{['shifmod']}) for the following values of the parameters: $\lambda = m = 1$, $\alpha = 0.49$. The points $A,B,C,D$ mark four vacua of the model: $A$ is maximum of $-{\cal W}$ corresponding to ${\cal M}_1$, $D$ is minimum corresponding to ${\cal M}_4$, $B,C$ are saddle points ${\cal M}_{2,3}$.
  • Figure 5: The scalar potential in the same model.
  • ...and 1 more figures