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Sine-Gordon solitons in AdS, dS and other hyperbolic spaces

E. T. Akhmedov, D. V. Diakonov

Abstract

We find infinitely many soliton-like solutions in a deformation of the sine-Gordon theory in $(d+1)$-dimensional $AdS_{d+1}$ (anti-de Sitter) spacetime for $d \geq 2$, as well as single solitonic solutions in $dS_{d+1}$ (de Sitter) and $\mathrm{H}{d+1}$ (Lobachevsky) spaces for $d \geq 1$ and in $AdS_2$. We also find a deformation of the kink solution in scalar field theory with a polynomial potential in $AdS_2$. The deformation of the sine-Gordon theory strikingly resembles the bosonic part of the flat-space supersymmetric sine-Gordon theory. In the infinite radius limit, single soliton solutions reduce to solitons in flat space. Meanwhile, the multisoliton solution of $AdS{d+1}$, $d\geq 2$ for certain values of the parameters reduces in the same limit to a single soliton solution boosted in the normal direction. However, there are also multisoliton solutions in $AdS_{d+1}$, $d \geq 2$ that do not have a flat space limit.

Sine-Gordon solitons in AdS, dS and other hyperbolic spaces

Abstract

We find infinitely many soliton-like solutions in a deformation of the sine-Gordon theory in -dimensional (anti-de Sitter) spacetime for , as well as single solitonic solutions in (de Sitter) and (Lobachevsky) spaces for and in . We also find a deformation of the kink solution in scalar field theory with a polynomial potential in . The deformation of the sine-Gordon theory strikingly resembles the bosonic part of the flat-space supersymmetric sine-Gordon theory. In the infinite radius limit, single soliton solutions reduce to solitons in flat space. Meanwhile, the multisoliton solution of , for certain values of the parameters reduces in the same limit to a single soliton solution boosted in the normal direction. However, there are also multisoliton solutions in , that do not have a flat space limit.

Paper Structure

This paper contains 11 sections, 89 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Geometry of hyperbolic spaces and hyperbolic plane waves
  3. Solitons in $AdS$ spacetime
  4. Polynomial potential
  5. Visualizing single soliton solution in $AdS_{1+1}$
  6. Visualizing multiple soliton solutions in $AdS_{2+1}$
  7. Energy and Stability
  8. Soliton in $dS$ spacetime
  9. Visualizing the soliton in $dS_{1+1}$ spacetime
  10. Soliton in Lobachevsky space
  11. Conclusion and acknowledgments

Figures (8)

  • Figure 1: The value of the $\phi$ field of the soliton in $AdS$ for $m=1$ (left) and $m=2$ (right) with the same null vector $\xi=(1,0,1)$ is shown. On the left picture, the field value changes from $0$ (red) to $2\pi$ (green). On the right picture, the field value changes from $-2\pi$ (red) to $2\pi$ (green).
  • Figure 2: A set of graphs of $\phi$ in $AdS_{1+2}$ shown on the Poincaré disk for different moments in time $\tau\in(0,2\pi)$. The value of the $\phi$ field of the soliton for $m=1$ and null vector $\eta=(0,1,1,0)$. The value of the field changes from $-2\pi$ (purple) to $2\pi$ (yellow).
  • Figure 3: The value of the $\phi$ field in Poincaré coordinates of $AdS$ for $m=1$ with the null vector $\xi=(-1,0,-1)$.
  • Figure 4: The graph of the potential $V(z)$. Blue line: $0 < m \le \frac{d-1}{2}$, orange line: $\frac{d-1}{2} < m < \frac{d+1}{2}$, and green line: $m \ge \frac{d+1}{2}$.
  • Figure 5: The value of the $\phi$ field of the soliton in $AdS$ for $m=1$ and $m=2$ with the same null vector $\xi=(1,0,1)$. On the left picture, the field value changes from $-2\pi$ (red) to $2\pi$ (green). On the right picture, the field value changes from $0$ (red) to $2\pi$ (green).
  • ...and 3 more figures