SUSY S-matrix Bootstrap and Friends

TL;DR

This work analyzes the 2D S-matrix bootstrap under supersymmetry and discrete symmetries ($\mathbb{Z}_2$, $\mathbb{Z}_4$), exploring low-dimensional sections of the space of consistent S-matrices by evaluating off-shell couplings at $s_*=2m^2$ and enforcing crossing and unitarity. The authors identify boundary S-matrices corresponding to integrable models, including the supersymmetric sine-Gordon (SSG), restricted sine-Gordon (RSG), and Zamolodchikov's $\mathbb{Z}_4$ S-matrix, and uncover a web of relations among these theories through fusion, limiting procedures, and elliptic deformations. Notably, they exhibit an elliptic deformation preserving integrability but breaking SUSY, and a non-factorizable SUSY deformation that saturates the boundary of the bootstrap space. These results illuminate a rich landscape of exactly solvable 2D QFTs at the boundary of consistency and suggest deep connections between seemingly different integrable models, with potential paths toward Lagrangian realizations and broader generalizations.

Abstract

We consider the 2D S-matrix bootstrap in the presence of supersymmetry, $\mathbb{Z}_2$ and $\mathbb{Z}_4$ symmetry. At the boundary of the allowed S-matrix space we encounter well known integrable models such as the supersymmetric sine-Gordon and restricted sine-Gordon models, novel elliptic deformations thereof, as well as a two parameter family of $\mathbb{Z}_4$ elliptic S-matrices previously proposed by Zamolodchikov. We highlight an intricate web of relations between these various S-matrices.

Figures (7)

• Figure 1: Allowed $\mathcal{N}=1$ S-matrix space with a single bound state of mass $m_{\text{bs}}/m=\{1.73,1.76,1.80,1.85,1.90,1.96 \}$ transforming in the fundamental/anti-fundamental representation represented in red/blue. As we increase the mass of the bound state the allowed space shrinks.
• Figure 2: Coupling space for $\mathbb{Z}_2$ symmetric theories with a single bound state obtained from the S-matrix bootstrap. In this figure $m_\text{bs} =\sqrt{3} m$ and all couplings are measured in units of $m$. The green point is the supersymmetric sine-Gordon theory, while the bold black line corresponds to an integrable elliptic deformation of SSG. The blue region is the fundamental domain: the rest of the 3D space can be obtained from it through trivial reflections corresponding to symmetries of the bootstrap problem.
• Figure 3: S-matrix space for the $\mathbb{Z}_4$ symmetric S-matrix bootstrap at $s_*=2m^2$. The faces are equivalent to the Zamolodchikov's $\mathbb{Z}_4$ S-matrices (appendix \ref{['sec:exactz4matrix']}) and the edges to the sine-Gordon kinks S-matrices (appendix \ref{['sgappendix']}).
• Figure 4: Connections between S-matrices showing up in this work as well as in Paper2Paper4. Green boxes: S-matrices with bound-states. Blue boxes: S-matrices without bound states. $\ast$: Known corresponding Lagrangian field theory (LFT). $\blacksquare$: Unknown corresponding LFT.
• Figure 5: Suppose we scatter two pairs of kinks, the constituents of each pair having the precise relative rapidity $\theta_b$ so as to form a breather. We then let this bound states collide and later decay. That will correspond to a double pole of the $4\to4$ kink S-matrix whose coefficient is proportional to the breather-breather S-matrix of the theory. This is the process described in the left hand side of the picture. From integrability, we can rearrange the incoming wave packets so that the kinks scatter before fusing into bound states, as in the right hand side of the figure. In this way, we relate the breathers S-matrix to a factorised product of four two-to-two kinks S-matrix. For simplicity, we omitted quantum numbers that would be relevant in the sine-Gordon or supersymmetric sine-Gordon theories, such as topological or SUSY charges. The fusing angle is fixed both in SG and SSG to be $\theta_b = i \gamma$.