Observables of String Field Theory

TL;DR

Key problem: identify off-shell gauge invariant observables in open string field theory that encode closed-string physics. The authors construct gauge-invariant operators by inserting an on-shell closed-string vertex at the open-string midpoint, denoting them as $O_V = ⟨I| V(π/2)|A⟩$ with $V(σ) = c_+(σ) c_-(σ) O(σ)$, establishing a one-to-one correspondence with closed-string vertex operators. They prove linear and nonlinear gauge invariance using BRST properties and midpoint regularization, and interpret correlators as disk amplitudes for closed-string scattering with open-string boundaries. An explicit oscillator construction confirms the formal results (including a tachyon example), and the discussion of vacuum string field theory reveals a mismatch that challenges a purely ghost-based kinetic operator, pointing to the need for matter dependence to recover the correct on-shell closed-string sector.

Abstract

We study gauge invariant operators of open string field theory and find a precise correspondence with on-shell closed strings. We provide a detailed proof of the gauge invariance of the operators and a heuristic interpretation of their correlation functions in terms of on-shell scattering amplitudes of closed strings. We also comment on the implications of these operators to vacuum string field theory.

Figures (4)

• Figure 1: The closed string is inserted at $\tau =0$ at some $\sigma$. The open string is inserted as usual at $\tau- -\infty$. The dashed line represents the midpoint.
• Figure 2: World sheet description of the gauge invariant operator (\ref{['oneform']}) in the coordinate where the metric is flat everywhere but at the insertion of the closed string vertex.
• Figure 3: World sheet description of closed string vertex operator $V$ acting on a state which is a product of the form $A*\Lambda$. The geometry is such that the metric is flat away from the midpoint.
• Figure 4: Open string field theory description of one closed two open string disk amplitude. The closed string is described by a gauge invariant operator which propagate to the interaction point.