Classical algebraic structures in string theory effective actions

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We study generic properties of string theory effective actions obtained by classically integrating out massive excitations from string field theories based on cyclic homotopy algebras of $A_\infty$ or $L_\infty$ type.

• We construct observables in the UV theory and we discuss their fate after integration-out.
• Furthermore, we discuss how to compose two subsequent integrations of degrees of freedom (horizontal composition) and how to integrate out degrees of freedom after deforming the UV theory with a new consistent interaction (vertical decomposition).
• We then apply our general results to the open bosonic string using Witten's open string field theory.