Generalize sigma-model Weyl invariance to include string loop corrections

Determine a generalization of the two-dimensional sigma-model Weyl invariance condition that yields equations of motion equivalent to those of a critical string theory when loop (higher-genus) contributions are included, thereby extending the tree-level equivalence between Weyl invariance and classical string field equations to the full quantum (loop-corrected) regime.

Background

The paper reviews how sigma-model Weyl invariance at genus zero reproduces the classical equations of motion for massless string fields, and argues for an effective action representation using the central-charge functional. While tree-level equivalence is elaborated and supported, the authors emphasize that a non-perturbative dynamical principle and its loop-level generalization remain to be formulated.

In particular, they highlight the need for a condition that, beyond genus zero, would be tantamount to full string equations of motion including loop effects, potentially expressed via vanishing of loop-corrected tadpoles or a generalized Weyl invariance. This problem targets a background-independent formulation that consistently incorporates moduli and ghost contributions without integrating over the conformal factor.