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.

References

The open questions are why the Weyl invariance condition is equivalent to the classical equations of motion of a "critical" theory and whether there exists some generalization of this condition, which is equivalent to the "critical" string theory equations of motion with loop contributions included.

— Sigma model approach to string theory  (2602.10977 - Tseytlin, 11 Feb 2026) in Section 1 (Introduction)

It supplies existence for such string solutions, while leaving open the compatibility of an arbitrary leading family.

— Resolvent reconstruction of minimal strings beyond KdV  (2609.28964 - Ahmed et al., 24 Sep 2026) in Section 3, From string pairs to several boundaries; Section 7, Discussion