Universal Box Operator: $\mathbf{O}(D,D)$-Symmetry and $α^{\prime}$-Corrections

Kawon Lee; Jeong-Hyuck Park

Paper

Universal Box Operator: $\mathbf{O}(D,D)$-Symmetry and $α^{\prime}$-Corrections

Abstract

We construct a fully covariant,

-symmetric d'Alembertian -- or box operator -- that acts on tensor fields of arbitrary rank and provides a universal kinetic term for all bosonic closed-string states. In its Riemannian parametrization, the operator packages the Riemann curvature,

-flux, and dilaton gradient into a single duality-covariant object. This yields

-symmetric gravitational-wave equations for the massless sector, governs the tachyon and all massive modes, and clarifies how higher excitations contribute to

-corrections. The box operator thus supplies a unified description of closed-string dynamics across the entire spectrum. Our analysis shows that any apparent breaking of

symmetry arises only after integrating out massive modes in a Wilsonian sense, where loop momentum integrals obscure half of the doubled momenta. We stand on the view that

symmetry and doubled diffeomorphisms remain exact and undeformed at the fundamental level of string theory.