Progress towards a new gauge theory for W type algebras

The W₃ algebra is gauged by using the Noether method. This algebra contains spin 2 Virasoro generators L_m and spin 3 generators W_m, and squares of the Virasoro generators appear on the right-hand side of some commutators. The theory contains n scalar matter fields φⁱ ≡ φⁱ(x⁺, x⁻) coupled to spin 2 gauge fields h⁺⁺, h⁻⁻ and spin 3 gauge fields B⁺⁺⁺, B⁻⁻⁻.

The Noether results through third order in the expansion parameter suggest the notion of a “nested covariant derivative”, and by interpreting it as a field equation, we integrate the latter and find the complete, infinitely nonlinear, action in one stroke. The result is very simple, see eq. (20). The transformation laws are then completed by requiring this action to be invariant. The results, again infinitely nonlinear, are given in closed form in eqs. (16) and (23). Recent results concerning W_N and w_∞ gauge theories, as well as some open problems, are mentioned at the end.