Formulate a background-independent dynamical equation expressing generalized Weyl invariance

Construct a precise, background-independent dynamical equation that expresses a generalized Weyl invariance condition for the closed bosonic string, and ascertain its equivalence to the vanishing of full (loop-corrected) massless tadpole amplitudes on arbitrary backgrounds.

Background

The authors argue that a non-perturbative dynamical principle is missing in closed string theory and suggest that a generalized Weyl invariance condition should be the backbone of such a principle. They conjecture that this condition would match a vacuum stability criterion—vanishing of all massless tadpoles including loop corrections.

However, they explicitly note that the formulation of the basic dynamical equation embodying generalized Weyl invariance is not yet known, motivating an explicit construction that is background-independent and compatible with string field theory structures.