Equivalence of loop-corrected tadpole equations to generalized Weyl invariance (consistency conjecture)

Prove that the vacuum stability conditions obtained by summing massless tadpole amplitudes across all genera, \(\sum_{n=0}^{\infty}\langle V_i \rangle_n = 0\), are equivalent to a generalized Weyl invariance condition in the two-dimensional sigma-model, such that any background satisfying these equations is independent of the chosen Weyl gauge.

Background

In developing a background-independent perspective, the paper connects string field theory equations of motion with sigma-model correlators computed in a fixed Weyl gauge and integrated over moduli. The authors propose that the full set of tadpole equations across genera encodes a generalized vacuum stability criterion.

They explicitly state a consistency conjecture that these equations are equivalent to generalized Weyl invariance, implying gauge (Weyl) independence of the resulting backgrounds. Establishing this equivalence would bridge sigma-model renormalization structures with non-perturbative string field theory dynamics.

References

The basic consistency conjecture is that (2.24) is equivalent to a generalized Weyl invariance condition, i.e. that a background which satisfies (2.24) should be independent of a Weyl gauge chosen.

— Sigma model approach to string theory  (2602.10977 - Tseytlin, 11 Feb 2026) in Section 2 (Sigma model and second quantized string theory)