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Abstract

We continue our study of effective field theory via homotopy transfer of
$L_\infty$-algebras, and apply it to tree-level non-Wilsonian effective actions
of the kind discussed by Sen in which the modes integrated out are comparable
in mass to the modes that are kept. We focus on the construction of effective
actions for string states at fixed levels and in particular on the construction
of weakly constrained double field theory. With these examples in mind, we
discuss closed string theory on toroidal backgrounds and resolve some subtle
issues involving vertex operators, including the proper form of cocycle factors
and of the reflector state. This resolves outstanding issues concerning the
construction of covariant closed string field theory on toroidal backgrounds.
The weakly constrained double field theory is formally obtained from closed
string field theory on a toroidal background by integrating out all but the \lq
doubly massless' states and homotopy transfer then gives a prescription for
determining the theory's vertices and symmetries. We also discuss consistent
truncation in the context of homotopy transfer.