Gauge-invariant operators of open bosonic string field theory in the low-energy limit

In the AdS/CFT correspondence we consider correlation functions of gauge-invariant operators on the gauge theory side, which we obtain in the low-energy limit of the open string sector. To investigate this low-energy limit we consider the action of open bosonic string field theory including source terms for gauge-invariant operators and classically integrate out massive fields to obtain the effective action for massless fields. While the gauge-invariant operators depend linearly on the open string field and do not resemble the corresponding operators such as the energy-momentum tensor in the low-energy limit, we find that nonlinear dependence is generated in the process of integrating out massive fields. We also find that the gauge transformation is modified in such a way that the effective action and the modified gauge transformation can be written in terms of the same set of multi-string products which satisfy weak $A_\infty$ relations, and we present explicit expressions for the multi-string products.