# From 3d dualities to 2d free field correlators and back

## TL;DR

This paper explores the deep connections between 3d supersymmetric theories and 2d conformal field theory correlators, revealing how dualities in one domain translate into identities in the other, and vice versa.

## Contribution

It establishes a correspondence between 3d N=2 theories and 2d free field correlators, deriving new dualities and interpreting analytic continuations as geometric transitions.

## Key findings

- 3d partition functions reduce to DF integral identities in 2d limit
- New dualities in 3d theories correspond to known CFT identities
- Analytic continuation in screening charges interpreted as geometric transition

## Abstract

We investigate the relation between 3d N=2 theories and 2d free field correlators or Dotsenko-Fateev (DF) integrals for Liouville CFT. We show that the S^2xS^1 partition functions of some known 3d Seiberg-like dualities reduce, in a suitable 2d limit, to known basic duality identities for DF correlators. These identities are applied in a variety of contexts in CFT, as for example in the derivation of the DOZZ 3-point function. Reversing the logic, we can try to guess new 3d IR dualities which reduce to more intricate duality relations for the DF correlators. For example, we show that a recently proposed duality relating the U(N) theory with one flavor and one adjoint to a WZ model can be regarded as the 3d ancestor of the evaluation formula for the DF integral representation of the 3-point correlator. We are also able to interpret the analytic continuation in the number of screening charges, which is performed on the CFT side to reconstruct the DOZZ 3-point function, as the geometric transition relating the 3d U(N) theory to the 5d T_2 theory.