# Quantum anomalies in A^{(1)}_r Toda theories with defects

## TL;DR

This paper investigates the quantum integrability of affine Toda theories with defects, revealing that quantum anomalies prevent the conservation of higher-spin currents, thus challenging their quantum integrability.

## Contribution

It provides an exact computation of quantum anomalies in higher-spin currents in affine Toda theories with defects, highlighting limitations of classical integrability at the quantum level.

## Key findings

- Quantum anomalies appear in higher-spin current conservation laws.
- Anomalies in the stress-energy tensor can be canceled by renormalization.
- Higher-spin currents exhibit anomalies that cannot be removed, questioning quantum integrability.

## Abstract

We study quantum integrability of affine Toda theories with a line of defect. In particular, we focus on the problem of constructing quantum higher-spin conserved currents in models defined by two A_r^{(1)} Toda theories separated by a non-trivial type-I defect. For a suitable choice of the defect potential these theories are known to be classically integrable, that is they possess an infinite set of higher-spin conserved charges in involution. Studying the corresponding conservation laws at quantum level we discover that anomalies arise, which we compute exactly at all orders in the coupling constant. While for the stress-energy tensor these anomalies can be cancelled by a finite renormalization of the defect potential, we find that from the first non-trivial higher-spin current this is no longer possible. This opens the question whether these theories are indeed integrable at quantum level.