Covariant w∞ gravity and its reduction to WN gravity

Covariant w∞ gravity

“…In ref. [7 ] this was done by gauging two copies of the w~ algebra ( 1 ), for m 1> -(i + 1 ), setting all curvatures to zero, and shifting away further gauge fields (by algebraic transformations) such that the only remaining independent fields are those given in (7). Here, we shall obtain the required transformation rules from the non-chiral rules (3) in the following way.…”

“…Writing this action out in + light-cone notation yields the action ofref. [ 7 ]. The action (25) is invariant under both k and 2 transformations.…”

“…[ 7 ], we find it convenient to use this parametrisation of the transformations. ) Note that if we now use the 2 + transformations to gauge away the e+ + o-l ) dummy fields, then solving for the compensating 2 + transformations needed to maintain Be+ + ~-~) =0 under k + transformations implies that (8) produces the e_V+k + terms in (3), and so in the covariant transformation rules we simply omit those terms from the k + transformations.…”

“…This is not yet enough because the transformation rule of e_ contains e_ itself. However a covariant expression, which is needed for closure of the algebra, can be obtained by an iterative procedure, via the recursive definition [ 7 ] A+O-l)=V+e_+O-l) V_e++O-i) …”

“…[ 6,7]. This work proceeded by introducing gauge fields for all of the generators of the algebra but then requiring them all to be pure gauge by setting all the curvatures to zero.…”