Classical Versus Quantum Symmetries for Toda Theories with a Nontrivial Boundary Perturbation

TL;DR

This paper investigates quantum conservation laws in Toda field theories with boundaries, revealing that quantum effects alter classical symmetries and that boundary potential redefinitions are crucial for maintaining certain conserved charges.

Contribution

It demonstrates how boundary perturbations affect quantum conservation laws in Toda theories and identifies conditions under which classical symmetries are preserved or broken at the quantum level.

Findings

Quantum boundary conservation laws are sensitive to boundary potential redefinitions.

Spin-3 currents require finite renormalization for quantum conservation.

Higher-spin symmetries, such as spin four, generally do not survive quantization.

Abstract

In this paper we present a detailed study of the quantum conservation laws for Toda field theories defined on the half plane in the presence of a boundary perturbation. We show that total derivative terms added to the currents, while irrelevant at the classical level, become important at the quantum level and in general modify significantly the quantum boundary conservation. We consider the first nontrivial higher--spin currents for the simply laced $a^{(1)}_n $ Toda theories: we find that the spin--three current leads to a quantum conserved charge only if the boundary potential is appropriately redefined through a finite renormalization. Contrary to the expectation we demonstrate instead that at spin four the classical symmetry does not survive quantization and we suspect that this feature will persist at higher--spin levels. Finally we examine the first nontrivial conservations at spin four for the $d^{(2)}_3$ and $c^{(1)}_2$ nonsimply laced Toda theories. In these cases the addition of total derivative terms to the bulk currents is necessary but sufficient to ensure the existence of corresponding quantum exact conserved charges.