S-Matrix Bootstrap and Non-Invertible Symmetries

TL;DR

The paper develops a systematic S-matrix bootstrap framework for theories with non-invertible, fusion-category symmetries in 1+1 dimensions by incorporating modified crossing rules dictated by fusion-category data. Using Symmetry Topological Field Theory (SymTFT), it derives how kinks transform under the symmetry, produces Ward identities, and constructs a projector basis that enforces symmetry constraints on S-matrix elements. Applying the formalism to the $ ext{A}_n$ and Fibonacci categories, the authors show that integrable, cusp-like S-matrices corresponding to known RG flows (e.g., $ ext{M}_n- ext{phi}_{1,3}$ and Potts-type flows) sit at extremal points of the allowed space, with additional structure such as bound states controlled by dual-category fusion data. This work demonstrates a concrete path to constrain and identify interacting theories with generalized symmetries, offering a robust route to higher-dimensional generalizations via SymTFT and inviting further exploration of exceptional categories like Haagerup and related form-factor bootstrap approaches. Overall, the integration of categorical symmetries into the S-matrix bootstrap sharpens predictions and clarifies how non-invertible symmetries sculpt scattering, spectrum, and integrability in a principled, energy-scale-spanning manner.

Abstract

We initiate the S-matrix bootstrap analysis of theories with non-invertible symmetries in (1+1) dimensions. Our previous work showed that crossing symmetry of S-matrices in such theories is modified, with modification characterized by the fusion category data. By imposing unitarity, symmetry and the modified crossing, we constrain the space of consistent S-matrices, identifying integrable theories with non-invertible symmetries at the cusps of allowed regions. We also extend the modified crossing rules to cases where vacua transform in non-regular representations of fusion category, utilizing a connection to a dual category $\mathscr C^{*}_{\mathscr{M}}$ and Symmetry Topological Field Theory (SymTFT). This highlights the utility of SymTFT in the analysis of scattering amplitudes.

Figures (7)

• Figure 1: Topological junction defining a module category $\mathscr{M}$ over $\mathcal{C}$ (Left). Boundary $F$-symbols implementing fusion of topological lines on the boundary (Right).
• Figure 2: The kink Hilbert space as an interval Hilbert space (Left). Symmetry action on $\mathcal{H}_{ab}$ by pushing down the $\mathcal{L}$ line. The dots denote the topological junctions of the module category $\mathscr{M}$ of vacua.
• Figure 3: The SymTFT sandwich construction of a theory $X$ (Left) and its representation of symmetries and charged operators (Right).
• Figure 4: SymTFT representation of a boundary condition for $X$ (Left) and its boundary operators (Right).
• Figure 5: (a) Two-body scattering of particles with the same mass $m$ giving the amplitude $S(s)$. (b) Analytic structure of the amplitude in the complex $s$ plane. Singularities of the amplitude $S(s)$ lie on the real axis, including two-particle branch cuts starting at $s=0,4m^2$. Possible bound states appear as poles in $0<s<4m^2$, here we show in blue one bound state of mass $m_b$ and its crossing symmetric image (lighter blue).