Covariant Closed String Field Theory
Covariant closed string field theories are considerably more complicated than their open string cousins. Even if one wants to construct a string field theory which only reproduces tree-level interactions between closed strings, the classical action must contain an infinite number of vertices consisting of string polyhedra.
If one demands that on-shell scattering diagrams be reproduced to all orders in the string coupling, one must also include additional vertices arising from higher genus (and hence higher order in ) as well. In general, a manifestly BV invariant, quantizable action takes the form
where denotes an th order vertex arising from a genus surface and is the closed string coupling. The structure of the vertices is in principle determined by a minimal area prescription, although, even for the polyhedral vertices, explicit computations have only been performed to quintic order.
Read more about this topic: String Field Theory
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