Off-Shell Strings I: S-matrix and Action

TL;DR

<3-5 sentence high-level summary> This work defends the viability of off-shell string theory by articulating Tseytlin's nonlinear sigma model framework and showing that the Weyl-frame choice is a renormalization-scheme freedom absorbed by target-space field redefinitions. It provides a detailed account of how Tseytlin's sphere prescriptions T1 and T2 yield the correct tree-level S-matrix and equations of motion across perturbation theory, while clarifying the connections to conformal perturbation theory and c-theorems. By examining the role of the worldsheet UV cutoff ε and the iε prescription, it explains how locality interpolates between a local action and the conventional S-matrix, and how to extract Lorentzian physics from off-shell amplitudes. The paper also discusses extensions to noncompact CFTs and supersymmetry, and sets the stage for applications to black hole entropy in a Part II, highlighting the broader significance for understanding off-shell string dynamics and RG flows in string theory.

Abstract

We explain why Tseytlin's off-shell formulation of string theory is well-defined. Although quantizing strings on an off-shell background requires an arbitrary choice of Weyl frame, this choice is not physically significant since it can be absorbed into a field redefinition of the target space fields. The off-shell formalism is particularly subtle at tree-level, due to the treatment of the noncompact conformal Killing group SL(2,$\mathbb{C}$) of the sphere. We prove that Tseytlin's sphere prescriptions recover the standard tree-level Lorentzian S-matrix, and show how to extract the stringy $i\varepsilon$ prescription from the UV cutoff on the worldsheet. We also demonstrate that the correct tree-level equations of motion are obtained to all orders in perturbation theory in $g_s$ and $α^{\prime}$, and illuminate the close connection between the string action and the c-theorem.

Figures (9)

• Figure 1: Hard disks of radius ${\epsilon}$ around each of four vertex operator insertions. These disks are not allowed to intersect, so the two disks on top are close to the boundary of the space of allowed positions.
• Figure 2: A conformal transformation which replaces the sphere regulated with a hard disk cutoff, with an open Riemann surface. The vertex operator insertions become state insertions on the boundaries. The length of the external and internal tubes is determined by the relative positioning of $z_1 \ldots z_4$ on the worldsheet, relative to our choice of Weyl frame $\omega$. In fact this is the sole effect of $\omega$ and ${\epsilon}$, as everything else is conformally invariant. In principle this conformal transformation allows one to re-express Tseytlin's off-shell formalism in the language of string field theory, but with an rather exotic rule for determining where to truncate the external propagators.
• Figure 3: The sliding scale of worldsheet locality vs. target space locality, as controlled by the UV cutoff length ${\epsilon}$, with specific numerical values for illustrative purposes. (We take the Weyl frame to be a unit sphere.) Smaller values of ${\epsilon}$ make the worldsheet theory more local, but the target space effective action $I^\text{eff}_0$ becomes less local, ultimately culminating in the S-matrix regime where strings can make it out to asymptotic infinity. Since the nonlocality in target space grows very slowly as ${\epsilon} \to 0$, there is a wide range of values with good approximate locality on both sides. The large ${\epsilon}$ regime is confusing, but might be related to attempts to discretize the string worldsheet Ginsparg:1993Polchinski:1994Thorn:2014Thorn:2015.
• Figure 4: A commutative diagram of the moduli spaces discussed in this section. The downward arrows represent UV regulation by the hard disk ${\epsilon}$, while the rightward arrows represent quotienting by the action of SL(2,$\mathbb{C}$). The arrow from ${\cal M}_{0,n}$ to ${\cal M}_{0,n}^{(\epsilon)}$ is implicitly defined by the other arrows.
• Figure 5: (i) A visualization of the regulated gauge orbit for the case $n=2$. Two red vertex operators are shown on the $S_2$ spherical worldsheet, connected by a brown geodesic through the interior of hyperbolic space $H_3$, each point of which represents a conformal frame of $S_2$. The locus of light blue points, which is within a fixed $O(\log {\epsilon}^{-1})$ proper distance of the geodesic, are those points in the SL(2,$\mathbb{C}$) gauge orbit for which both points are at least ${\epsilon}$ apart on the sphere. Symmetry ensures that the surface of this locus is a fixed proper distance from the geodesic; hence the hyperbolic volume of the regulated gauge orbit is infinite. (ii) The regulated gauge orbit for three points ($n = 3$), i.e. ${\cal M}^{(\epsilon)}_{0,3}\!{}\;$. This is the set of points in the intersection (shaded light blue) of the three different $n = 2$ loci associated with each pair of points. (Only one dark blue edge is shown for each pair of vertex operators, as the edge on the other side is too far away to contribute to the boundary the regulated gauge orbit.) The volume of this gauge orbit (and any other regulated gauge orbit with $n\ge 3$) is finite, and can be integrated by picking a point $p$ somewhere in the interior, and shooting out rays $r$ in all possible directions. Although the boundary of the regulated orbit is not spherically symmetric or even smooth, all directions have the same universal $\log {\epsilon}$ contribution.