Classical strings and the double copy

TL;DR

The paper investigates extending the classical double copy to the realm of string theory by studying an open string in a constant background gauge field and constructing its double copy as a closed string in a gravity background obtained via the Kerr–Schild framework, revealing a non-inertial reference frame and Rindler-horizon structure that encode gauge-boundary data. The analysis shows that the double copy of a constant electric field yields a gravity background that can be transformed away to Minkowski space, while boundary conditions persist as observable non-inertial effects; for transverse constant fields, a similar memory persists in the gravity description. In the string context, open strings exhibit a DBI stability bound $f^2<1$ and modified zero modes, whereas closed strings in the double-copy metric retain a mode structure that reflects the gauge origin, demonstrating that the double copy remains meaningful in classical string theory beyond flat-space amplitudes. Overall, the work lays groundwork for a systematic study of double copy phenomena in classical string settings and clarifies how gauge data and boundary conditions translate into gravitational backgrounds, potentially enriching our understanding of KLT-type relations in nontrivial backgrounds.

Abstract

The double copy is by now a well-established relationship between scattering amplitudes and classical solutions in gauge and gravity (field) theories, and is itself inspired by amplitude relations in string theory. In this paper, we generalise the classical double copy to the motion of strings, taking as a case study the motion of an open string in a background abelian gauge field. We argue that the double copy of this situation is a closed string moving in a spacetime background arising as the double copy of the gauge theory background. The gauge theory background we consider is that of a constant electric field, which has a critical value beyond which the open string motion is pathological. We find no counterpart of this behaviour in the double copy, and interpret this result. We then examine how the closed string nevertheless still knows about the single copy gauge theory. Our results pave the way for more systematic study of the double copy in a classical string context, thus going beyond the KLT relations for amplitudes in flat space.

Figures (4)

• Figure 1: Trajectories of objects at rest in the double copy metric of eq. (\ref{['ds2E']}), as seen in the Minkowski coordinates of eqs. (\ref{['dsE2']}, \ref{['UVdef2']}). Results are shown for: (a) $f>0$ and fixed values of $h_0<0$; (b) $f<0$ and fixed values of $h_0>0$.
• Figure 2: (a) A constant electric field can be obtained by placing positively and negatively charged plates at the boundary of space, where the fields due to each plate (shown in red and blue) combine to make the black arrows; (b) the double copy maps the charged plates to masses, whose gravitational fields (shown in red) cancel out.
• Figure 3: Trajectories in the $(X,T)$ plane associated with stationary observers in the non-inertial frame of eq. (\ref{['ds2EB']}). Results are shown for different initial values of $x=x_0$, with $f^2=1$ and $x^i_0=0$.
• Figure 4: Trajectories in the $(X^2,T)$ plane associated stationary observers in the non-inertial frame of eq. (\ref{['ds2EB']}). Results are shown for different initial values $x^2=x^2_0$, and $x_0=0$, with: (a) $f^2=1$; (b) $f^2=-1$.