The KdV Action and Deformed Minimal Models

An action is constructed that gives an arbitrary equation in the KdV or MKdV hierarchies as equation of motion; the second Hamiltonian structure of the KdV equation and the Hamiltonian structure of the MKdV equation appear as Poisson bracket structures derived from this action. Quantization of this theory can be carried out in two different schemes, to obtain either the quantum KdV theory of Kupershmidt and Mathieu or the quantum MKdV theory of Sasaki and Yamanaka. The latter is, for specific values of the coupling constant, related to a generalized deformation of the minimal models, and clarifies the relationship of integrable systems of KdV type and conformal field theories. As a generalization it is shown how to construct an action for the SL(3)-KdV (Boussinesq) hierarchy. An action for the KdV equation should have two basic properties: (a) The associated equation of motion should be the KdV equation (or some equation in the KdV hierarchy). (b) The associated Poisson bracket structure (we will reiterate below how to derive Poisson brackets from an action) should be the second hamiltonian structure of the KdV equation. We should be able to define a quantum theory using our action; given (b) we might expect this to coincide in some sense with the quantum KdV theory as described in [1]. This is clearly desirable, given the correspondence of the conserved quantities of the quantum KdV equation of [1] and the conserved quantities in deformed minimal conformal theories. So we add one further non-essential but desirable property to our list above: (c) The Heisenberg equation of motion associated with our action should be the quantum KdV equation of [1]. In this note I construct an action that has properties (a),(b),(c). The action can also be regarded as an action for the MKdV equation. In this form the action has a kinetic term that describes a theory which is “nearly” free, and an infinite number of potential terms. In quantizing the theory defined by just the kinetic term, we find many of the features of the Feigin-Fuchs construction for the minimal models; in particular it becomes clear that the quantum analogs of the terms in the potential (the quantum MKdV hamiltonians) describe an infinite number of possible integrable deformations of minimal conformal models. These deformations are more general than those considered by Zamolodchikov [2]. Zamolodchikov’s deformations of a conformal field theory are ones that preserve both the integrability and Lorentz invariance of the theory, and there are an infinite number of other perturbations that preserve just the integrability. In the simplest case, the one that we shall consider, the Zamolodchikov deformation gives rise to the integrable, Lorentzinvariant Sine-Gordon theory (as recognized in [3]), whereas the deformations we will consider give rise to theories with equations of motion in the MKdV hierarchy (as is wellknown, all the MKdV flows commute with the Sine-Gordon flow). The correspondence of the conserved quantities of the quantum KdV equation and the conserved quantities of deformed minimal models becomes very clear. 1 In the last part of this note I also show how to construct an action for the SL(3)-KdV equation. Doubtless many physicists, on meeting the KdV equation for the first time, investigate whether it can be derived from an action. It certainly seems that the action has to be nonlocal in the KdV field u, and the simplest actions one might guess appear unenlightening (see for example [4]). Indeed, to satsify condition (b) above we need our action to be non-local in u. An explanation of this is as follows: in a classical mechanical system with phase space coordinates X , i = 1, ..., 2n, a hamiltonian structure is specified either by giving a non-degenerate symplectic form on the phase space Ω = 1 2ωijdX i ∧ dX (1) or the corresponding set of Poisson brackets {X , X} = (ω) (2) The second hamiltonian structure of the KdV equation is given by the Poisson brackets {u(x), u(y)} = − c ( ∂ x + u(x)∂x + ∂xu(x) )