# Charged Quantum Fields in AdS$_2$

## TL;DR

This paper studies quantum charged fields in AdS$_2$ near extreme Kerr black holes, revealing novel boundary behaviors, constructing a flux S-matrix, and exploring representations of $SL(2, ext{R})$ for both fermionic and bosonic fields.

## Contribution

It introduces a new framework for quantum fields in AdS$_2$ with background electric fields, including a UV completion and a flux S-matrix, and analyzes highly charged particles via $SL(2, ext{R})$ representations.

## Key findings

- Charged fields oscillate near AdS boundary at large charge.
- A novel flux S-matrix is constructed for these modes.
- The free massless fermion case is exactly solvable.

## Abstract

We consider quantum field theory near the horizon of an extreme Kerr black hole. In this limit, the dynamics is well approximated by a tower of electrically charged fields propagating in an $SL(2,\mathbb{R})$ invariant AdS$_2$ geometry endowed with a constant, symmetry preserving background electric field. At large charge the fields oscillate near the AdS$_2$ boundary and no longer admit a standard Dirichlet treatment. From the Kerr black hole perspective, this phenomenon is related to the presence of an ergosphere. We discuss a definition for the quantum field theory whereby we 'UV' complete AdS$_2$ by appending an asymptotically two dimensional Minkowski region. This allows the construction of a novel observable for the flux-carrying modes that resembles the standard flat space S-matrix. We relate various features displayed by the highly charged particles to the principal series representations of $SL(2,\mathbb{R})$. These representations are unitary and also appear for massive quantum fields in dS$_2$. Both fermionic and bosonic fields are studied. We find that the free charged massless fermion is exactly solvable for general background, providing an interesting arena for the problem at hand.