Form factors of descendant operators: $A^{(1)}_{L-1}$ affine Toda theory

TL;DR

This paper develops an exact free field representation for form factors of local operators, including descendant operators, in affine Toda theories, extending previous methods and providing new insights into operator counting and symmetry properties.

Contribution

It generalizes Lukyanov's construction to descendant operators and introduces a free field representation with countably many generators for form factors in affine Toda theories.

Findings

Constructed exact form factors for descendant operators.

Proposed a free field representation with countably many generators.

Validated operator counting against Lagrangian formalism.

Abstract

In the framework of the free field representation we obtain exact form factors of local operators in the two-dimensional affine Toda theories of the $A^{(1)}_{L-1}$ series. The construction generalizes Lukyanov's well-known construction to the case of descendant operators. Besides, we propose a free field representation with a countable number of generators for the `stripped' form factors, which generalizes the recent proposal for the sine/sinh-Gordon model. As a check of the construction we compare numbers of the operators defined by these form factors in level subspaces of the chiral sectors with the corresponding numbers in the Lagrangian formalism. We argue that the construction provides a correct counting for operators with both chiralities. At last we study the properties of the operators with respect to the Weyl group. We show that for generic values of parameters there exist Weyl invariant analytic families of the bases in the level subspaces.