Jean Avan

Classical integrability and higher symmetries of collective string field theory

Abstract We demonstrate the complete integrability of classical collective field theory. An exact equivalence to the classical N -body problem of Calogero type is described. A Lax pair is constructed for the general continuum collective equations. A form of background independence is pointed out. An infinite set of commuting currents is constructed in the general case. A large symmetry algebra is discovered and shown to take the form of a W ∞ algebra.

Quantum integrability and exact eigenstates of the collective string field theory

Abstract The quantum w∞ algebra associated with the collective string theory is used to construct exact discrete eigenstates of the cubic hamiltonian. Imaginary-eigenvalued creation/annihilation operators are constructed which build a F ⊗ F - graded algebra . Real-eigenvalued operators are formally obtained as combinations of discrete eigenvalued ones and vice versa.

The classical r-matrix for the relativistic Ruijsenaars-Schneider system

Abstract We compute the classical r -matrix for the relativistic generalization of the Calogero-Moser model, or Ruijsenaars-Schneider model, at all values of the speed-of-light parameter λ. We connect it with the nonrelativistic Calogero-Moser r -matrix ( λ → −1) and the λ = 1 sine-Gordon soliton limit.

STRING FIELD ACTIONS FROM W

Starting from W∞ as a fundamental symmetry and using the coadjoint orbit method, we derive an action for one-dimensional strings. It is shown that on the simplest non-trivial orbit this gives the single scalar collective field theory. On higher orbits one finds generalized KdV type field theories with increasing number of components.

Collective field theory of the matrix-vector models

We construct collective field theories associated with one-matrix plus r vector models. Such field theories describe the continuum limit of spin Calogero Moser models. The invariant collective fields consist of a scalar density coupled to a set of fields in the adjoint representation of U(r). Hermiticity conditions for the general quadratic Hamiltonians lead to a new type of extended non-linear algebra of differential operators acting on the Jacobian. It includes both Virasoro and SU(r) (included in sl(r, C) × sl(r, C)) current algebras. A systematic construction of exact eigenstates for the coupled field theory is given and exemplified.

Algebraic structures and eigenstates for integrable collective field theories

Conditions for the construction of polynomial eigen-operators for the Hamiltonian of collective string field theories are explored. Such eigen-operators arise for only one monomial potentialv(x)=μx2 in the collective field theory. They form aw∞-algebra isomorphic to the algebra of vertex operators in 2d gravity. Polynomial potentials of orders only strictly larger or smaller than 2 have no non-zero-energy polynomial eigen-operators. This analysis leads us to consider a particular potentialv(x)=μx2+g/x2. A Lie algebra of polynomial eigen-operators is then constructed for this potential. It is a symmetric 2-index Lie algebra, also represented as a subalgebra ofU(sl(2)).

Alternative Lax structures for the classical and quantum Neumann model

Abstract A Lax representation by sl(2, C ) matrices, following from the geometrical formulation of Moser, is introduced for the Neumann model. Its Poisson bracket structure is given by the rational antisymmetric sl(2, C ) R -matrix. The quantum version of its determinant, suitably ordered, gives rise to commuting operators, exact counterparts of the classical Uhlenbeck conserved quantities.

Interacting theory of collective and topological fields in 2 dimensions

Abstract We propose a generalization of the collective field theory hamiltonian, including interactions between the original bosonic collective field w0(z) and supplementary fields w j (z) realizing classically a w∞ algebra. The latter are then shown to represent a 3-dimensional topological field theory. This generalization follows from a conjectured representation of the W1+∞ algebra of bilinear fermion operators underlying the original matrix model. It provides an improved bosonization scheme for (1+1)-dimensional fermion theories.

Collective hamiltonians with Kac-Moody algebraic conditions

Abstract We describe the general framework for constructing collective-theory hamiltonians whose hermiticity requirements imply a Kac-Moody algebra of constraints on the associated jacobian. We give explicit examples for the algebras sl(2)k and sl(3)k. The reduction to Wn-constraints, relevant to n-matrix models, is described for the jacobians.

W∞ currents in three-dimensional Toda theory

Chiral densities obeying a