We will look at the Soliton solutions in the sine-Gordon equation (which also shares correspondence with the (massive) Thirring model in perturbation theory). Let us first see a standard example of soliton in field theory. We take a non-linear scalar field theory $\phi$ with Lagrangian

$$ \mathcal{L} = -\frac{1}{2}\partial_\mu \phi \partial^\mu \phi - U(\phi) $$

where potential $U(\phi)$ is described by

$$ U(\phi)=\lambda\left(\phi^2-m^2/\lambda\right)^2/4 $$

and the dimensionless coupling constant is $g=\lambda/m^2$. Here, $m$ is the mass of the elementary solutions of $\phi$. Then we define the  topological current

$$ j_u = \frac{\sqrt{g}}{2}\epsilon_{\mu \nu}\partial^\nu \phi$$

$$ Q = \int_{-\infty}^{\infty}dx\ j_0$$

integrating it becomes

$$ Q = \frac{\sqrt{g}}{2}\left(\phi(\infty)-\phi(-\infty)\right) $$

where the $\infty$ is for a kink solution and $-\infty$ is for an anti-kink solution. These kinks deserve our attention here. $\phi$ varies from the minimum of $U(\phi)$ at $\phi = \mp 1/\sqrt{g}$ at $x=\infty$ to the minimum of $U(\phi)$ at $\phi = \pm 1/\sqrt{g}$ at $x=-\infty$. We can write a solution to this equation, which follows

$$ \phi^{''}=\frac{\partial U}{\partial \phi} $$

integrating this with $U$ with $\phi'$ vanishing at infinity we get

$$ \frac{1}{2}(\phi^{'})^{2} = U(\phi).$$

Integrating this now over our choice of $U$ will give us the kink (k) and anti-kink (k') solution

$$\phi(x)_{k(k')} = \pm \frac{m}{\sqrt{\lambda}} tanh\left[m(x-x_0)/\sqrt{2}\right].$$

The rest mass for the soliton is given by

$$ E = \int dx \frac{1}{2}\left(\phi^{'}\right)^{2} + U(\phi) = \frac{2\sqrt{2}}{3}\frac{m}{g}$$

which clearly states that kink (rest) mass divided by the $m$ is proportional to $1/g$. This is also an indication that solitons are non-perturbative physics.

Anyway, the previous example was about solutions of just one theory, where kink and elementary solutions shared a relation. The nature of these kinks will be apparent in the next post. Now, what about a duality between two sectors of different theories. For this, we will turn to the massive Thirring model, which shares a correspondence with the sine-Gordon theory, in a next post. 

With V. Kalvakota, we wrote an essay pointing out the traditional points of holography where we have contrasted the case for AdS and de Sitter. The latter has points that are non-trivial in these traditional senses, so people have looked out for answers in different holographic settings. This paper was written for GRF 2024.

Vakil's note on algebraic geometry has a lot of emphasis on, of course, the duality between geometry and algebra.