Open string field theory in lightcone gauge

TL;DR

This paper shows that covariant open bosonic SFT in a well-defined lightcone gauge yields gauge-fixed vertices that are Mandelstam diagrams with attached stubs, and that the moduli-space coverage of these vertices differs from covariant vertices due to longitudinal exchanges. The author proves an equivalence theorem equating covariant off-shell amplitudes of DDF states with lightcone off-shell amplitudes of corresponding lightcone states, up to stub-length data, and introduces longitudinal freezing to explain how longitudinal dynamics decouple inside propagator strips. The construction provides a consistent lightcone measure and a framework to compute quartic and higher vertices by transversely projecting covariant Siegel-gauge amplitudes while accounting for longitudinal subgraphs that fill gaps in moduli space. The results extend to higher-order vertices, establish graphical compatibility conditions, and discuss implications for closed and superstring field theories, the soft string problem, and potential finite, stubbed formulations of lightcone string interactions.

Abstract

We study covariant open bosonic string field theory in lightcone gauge. When lightcone gauge is well-defined, we find two results. First, the vertices of the gauge-fixed action consist of Mandelstam diagrams with stubs covering specific portions of the moduli spaces of Riemann surfaces. This is true regardless of how the vertices of the original covariant string field theory are constructed (e.g. through minimal area metrics, hyperbolic geometry, and so on). Second, the portions of moduli space covered by gauge-fixed vertices are changed relative to those covered by the original covariant vertices. The extra portions are supplied through the exchange of longitudinal degrees of freedom in scattering processes.

Figures (23)

• Figure 3.1: A Mandelstam diagram is given by gluing together rectangular strip domains $\rho_i$. The strip domains $\rho_1,...,\rho_5$ in this figure represent external states, and should be imagined as extending to plus or minus infinity. The strip domains $\rho_{23}$ and $\rho_{234}$ represent propagators. We have chosen to label the propagators by the list of punctures which are separated from the first puncture at degeneration.
• Figure 3.2: This figure shows how to construct the vertical displacement $\sigma_{34}$ of the strip domain $\rho_{34}$. First we note that $\rho_{34}$ does not extend to $+\infty$, so we mark a point on the lower right hand corner. Second, we note that $u_1$ is the rightmost puncture in the upper half plane. Since $\rho_1$ does not extend to $+\infty$, we also mark a point on the lower right hand corner. Then we trace a counterclockwise path connecting the marked points. Every time the path crosses a puncture, there is a corresponding contribution to the vertical displacement $\sigma_{34}$.
• Figure 3.3: Conformal transformations between the upper half plane $u$, the Mandelstam diagram $\rho$, and the local coordinates $\xi_i$ on the strip domains $\rho_i$ within the Mandelstam diagram.
• Figure 3.4: Relation between the covariant fiber bundle, lightcone fiber bundle, and the moduli space.
• Figure 3.5: Assuming by convention that $\alpha_3$ has largest magnitude, the inequalities (\ref{['eq:s1l']})-(\ref{['eq:s3l']}) are plotted here as a function of the dimensionless ratio $U = |\alpha_2/\alpha_3|\in [0,1]$. This ratio is equal to the position of the cubic interaction point on the real qxis of the upper half plane. The line $r_1=r_2=r_3= \frac{4}{3\sqrt{3}}$ corresponds to the Witten vertex. In this case there is no interval of $U$ where all three inequalities are obeyed. Below $r_1=r_2=r_3=1/4$, there is an interval containing $U=1/2$ where all stub lengths are positive. The point $U=1/2$ represents the Mandelstam diagram where strings 1 and 2 have equal minus momenta.