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Closed string field theory without the level-matching condition

In formulating covariant closed string field theories, we have always used closed string fields with the level-matching condition. Recently, open superstring field theories including the Ramond sector were constructed, and one approach was to use open superstring fields with a constraint analogous to the level-matching condition for closed string fields. There is another approach developed by Sen, where no constraints are imposed on open superstring fields but additional free string fields are introduced so that covariant kinetic terms for the Ramond sector were constructed. Motivated by this development, we construct closed string field theory without imposing the level-matching condition on the closed string field, while we introduce an additional free string field. This is the first implementation of general covariance in the context of string theory without using the level-matching condition. We explicitly expand the string fields up to the level relevant to massless fields, and we confirm the equivalence to the theory with the level-matching condition up to decoupled free fields.