Interpolating String Field Theories

A minimal area problem imposing different length conditions on open and closed curves is shown to define a one parameter family of covariant open-closed quantum string field theories. These interpolate from a recently proposed factorizable open-closed theory up to an extended version of Witten’s open string field theory capable of incorporating on shell closed strings. The string diagrams of the latter define a new decomposition of the moduli spaces of Riemann surfaces with punctures and boundaries based on quadratic differentials with both first order and second order poles. ⋆ Permanent address: Center for Theoretical Physics, MIT, Cambridge, Mass. 02139. Supported in part by D.O.E. contract DE-AC02-76ER03069 and NSF grant PHY91-06210. A covariant quantum theory of open and closed strings was derived recently from string diagrams defined by a minimal area problem [1]. This problem required that the length of any nontrivial open curve be larger or equal to lo = π, and that the length of any nontrivial closed curve be larger or equal to lc = 2π. In the resulting theory one can compute directly all scattering amplitudes involving any possible numbers of open and closed strings on any surface. The Feynman rules of covariant open string theory [2] are simpler, but amplitudes involving external closed strings must be obtained indirectly by factorization. Finding a way of extending the simple string diagrams of open string theory to include external closed strings was the motivation for the present work. The solution is simple. Open string diagrams are now known to arise from minimal area metrics under the condition that all nontrivial open curves be longer or equal to π [3]. We can incorporate external closed strings without changing the minimal area problem at all! Closed string external states correspond to punctures inside the surface, and curves cannot be moved across them. Their presence creates new homotopy types of nontrivial open curves. The minimal area metric will have to adjust itself in order to satisfy the new length conditions arising due to the closed string punctures. The above minimal area problem is recognized to correspond to lo = π, and lc = 0. It is therefore natural to consider the generalized problem where lo = π and lc = a, where a is a constant. We shall see that this defines a one parameter family of string diagrams. Even more interesting is that actually they correspond to a one parameter family of string field theories. This happens simply because the string diagrams can be built by sewing. When a = 2π we recover the open-closed factorizable theory [1], and when a = 0 we get the extended open string theory. A large part of our effort in this paper will be devoted to elucidate the extended open string theory. Since the minimal area problem is showing us a natural way to incorporate external closed strings into the framework of the open string theory we only have to figure out how the string diagrams look, and if they can be built using Feynman rules. The minimal area metrics we shall find always arise from quadratic differentials, and closed strings appear as first order poles. A first order pole corresponds to a finite area conical singularity of the metric with an angle of π at the singularity. We will find the amazingly simple result that a single open-closed interaction is all one needs to add to the Witten open string vertex so