# Constraints for Three-Dimensional Higher-Spin Interactions and Conformal Correlators

## TL;DR

This paper explores the constraints on higher-spin interactions in three dimensions and their correspondence with conformal correlators in two-dimensional theories, revealing shared geometric inequalities.

## Contribution

It establishes that the same triangle and polygon inequalities govern higher-spin interactions and correlators, unifying their structural classification.

## Key findings

- Higher-spin cubic vertices obey triangle inequalities.
- Higher-order correlators satisfy polygon inequalities.
- Structural constraints are consistent across gravity and CFT contexts.

## Abstract

In the context of higher-spin holography, we compare the classification of cubic interaction vertices for higher-spin gravity theories in three dimensions to the possible three-point correlation functions of conserved higher-spin currents in two-dimensional conformal field theories. In both cases, the allowed structures are governed by triangle inequalities for the involved spins. It is established that higher-order correlators satisfy similar polygon inequalities and that the same inequalities are valid for higher-order continuations of cubic vertices in the three-dimensional higher-spin gravity.