A Background-Independent Closed String Action at Tree Level

TL;DR

The paper constructs a background‑independent, off‑shell closed string action at tree level by pairing the planar Zamolodchikov $C$‑function with the sphere partition function into a Planar $cZ$ action $I_0 = c_{ ext{pl}}(r_*) Z_{S^2}$. It proves non‑perturbatively that stationary points of this action coincide with $c=0$ CFTs on the plane for unitary matter sectors flowing from a UV fixed point, and shows equivalence to the Tseytlin $I_0^{\textbf{T2}}$ within the renormalization window via a field redefinition. The analysis identifies higher non‑minimal couplings dressed with powers of the worldsheet curvature as pure gauge and demonstrates the recovery of the dilaton kinetic term from the planar $C$‑function, while also exploring Lorentzian extensions under CTO symmetry. Together, these results advance a consistent, background‑independent description of string dynamics at tree level and suggest connections to open questions in string field theory and quantum gravity.

Abstract

We propose an off-shell bosonic string action that removes the renormalization window constraint of [1]. To all orders in conformal perturbation theory, this action allows for deformations of the worldsheet theory by any primary or descendant irrelevant deformation. Non-perturbatively, this action has no spurious solutions on the space of all worldsheet theories with a unitary matter sector that flows from a UV fixed point. We find that non-minimal couplings dressed with more than one factor of the Ricci curvature behave as gauge redundancies. As part of our investigation of this action, we find non-smooth behavior in the Zamolodchikov $C$-function. Our results mostly apply to Euclidean-signature target spaces, but can be extended to Lorentzian backgrounds which are invariant under time translations and CTO symmetry.

Figures (6)

• Figure 1: (a) is a schematic depiction of the space of 2d QFTs. The dimensions of the space are labeled by worldsheet couplings (i.e. target space fields) $\lambda_{a}$. The marked points are string backgrounds, i.e. $c=0$ CFTs. On the space of QFTs, $c=0$ CFTs may either be isolated (depicted by the isolated points in the diagram) or appear as a continuous family (depicted by the curve). In (b), we have drawn a sketch of a tree-level string effective action $I_0$. $I_0$ is a function on this space of coupling constants. In order for $I_0$ to be a valid string effective action, its set of stationary points must precisely match the set of $c=0$ CFTs (or, equivalently, matter CFTs with $c = 26$). The purpose of this paper is to construct such an $I_0$.
• Figure 2: We emphasize that the sphere effective action we compute from the worldsheet has already summed all contributions of all possible tree-level diagrams, including all possible massive species that may form an internal propagator. For example, in this figure we consider the four-point scattering amplitude of some massless field target space field $\phi$ (represented by the solid lines). The coefficient of the $\phi^4$ term in the effective action we compute is the sum of the $s,t,u$ channels (summed over all possible internal modes, represented by the different styles of dashed lines) and the 4-point contact interaction. This is contrary to the assumption made in Tseytlin-SFTComponents-1987, which argues for a no-go theorem under the assumption that the target space effective action generates the 1PI diagrams at tree-level, not the full tree-level amplitude.
• Figure 3: A pictorial representation of the planar c-function as defined in \ref{['eqn:int-c']}. The region drawn is a subset of the infinite plane. The shaded circle of radius $r_{\star}$ is the region of integration of $|z|^2\langle \Theta(z) \Theta(0) \rangle$ While $\Theta(0)$ is fixed at the origin, $\Theta(z)$ varies over a disk of radius $r_{\star}$.
• Figure 4: A pictorial depiction of the planar cZ action. The $C$-function is drawn as in Fig. \ref{['fig:planar-c-func']}, and the sphere with no insertions represents $Z_{S^2}(r)$.
• Figure 5: In (a) we depict a two-point function $D(z):=\langle\mathcal{O}(z)\mathcal{O}(0)\rangle$ in a QFT deformed from a UV CFT by a relevant operator. A few points, $z_i$, are marked to denote some choices of length scale. In (b), we depict how increasing the length scale of the correlator corresponds to flowing from the UV CFT to the IR fixed point. Near $z_1$, $D(z)$ may be calculated in CPT with an irrelevant deformation of the UV CFT. At $z_2$, we are at large enough length scales that conformal perturbation theory has broken down. By the time we get to $z_3$, we have flowed so close to the IR CFT that the theory may be treated as a deformation by an irrelevant perturbation of the IR CFT. Due to the properties of correlation functions of irrelevant deformations, this implies that at large enough length scales $D(z)$ falls off faster than $1/|z|^4$.