# Holographic four-point functions in Toda field theories in $AdS_2$

## TL;DR

This paper computes four-point functions of boundary operators in Toda field theories on Euclidean AdS2 and suggests a correspondence with $ ext{W}$-algebra generators, revealing a deep connection between boundary correlators and algebraic structures.

## Contribution

It provides explicit four-point function calculations in Toda theories on AdS2 and proposes a conjecture linking boundary operators to $ ext{W}$-algebra generators.

## Key findings

- Four-point functions match $ ext{W}$-algebra generator correlators
- Boundary operators are conjectured to correspond to $ ext{W}$-algebra generators
- Results support a holographic relation between Toda theories and $ ext{W}$-algebras

## Abstract

We consider Toda field theories in a classical Euclidean $AdS_2$ background. We compute the four-point functions of boundary operators in the $a_1$, $a_2$ and $b_2$ Toda field theories. They take the same form as the four-point functions of generators in the corresponding $\mathcal{W}$-algebras. Therefore we conjecture that the boundary operators are in one-to-one correspondence with the generators in the $\mathcal{W}$-algebras.