Abstract
This is the second paper of a series of three. We construct effective open-closed superstring couplings by classically integrating out massive fields from open superstring field theories coupled to an elementary gauge invariant tadpole proportional to an on-shell closed string state in both large and small Hilbert spaces, in the NS sector. This source term is well known in the WZW formulation and by explicitly performing a novel large Hilbert space perturbation theory we are able to characterize the first orders of the vacuum shift solution, its obstructions and the non-trivial open-closed effective couplings in closed form. With the aim of getting all order results, we also construct a new observable in the A∞ theory in the small Hilbert space which correctly provides a gauge invariant coupling to physical closed strings and which descends from the WZW open-closed coupling upon partial gauge fixing and field redefinition. Armed with this new A∞ observable we use tensor co-algebra techniques to efficiently package the whole perturbation theory necessary for computing the effective action and we give all order results for the open-closed effective couplings in the small Hilbert space.
Highlights
This source term is well known in the WZW formulation and by explicitly performing a novel large Hilbert space perturbation theory we are able to characterize the first orders of the vacuum shift solution, its obstructions and the non-trivial open-closed effective couplings in closed form
With the aim of getting all order results, we construct a new observable in the A∞ theory in the small Hilbert space which correctly provides a gauge invariant coupling to physical closed strings and which descends from the WZW open-closed coupling upon partial gauge fixing and field redefinition
Before attacking the explicit evaluation of the first non-trivial open-closed couplings we address the same problem in the A∞ open superstring field theory in the small Hilbert space [8], which can be obtained from the WZW theory in the large Hilbert space by gauge fixing the η gauge invariance and performing a field redefinition [19,20,21]
Summary
The WZW theory coupled to the superstring version of the Ellwood-invariant in the large Hilbert space [9, 25]1 has the following action. Notice that the shifted kinetic operator Qμ is nilpotent even without assuming that Φμ solves the tadpole-sourced equation of motion This may look odd but the need for a proper solution is contained in [Qμ, η] = adη(e−Φμ QeΦμ ) = adμe = 0,. As in the bosonic case these conditions should capture the vanishing of S-matrix elements between the deforming bulk closed strings and a single massless open string Notice that these amplitudes are written explicitly in the large Hilbert space and make use of both propagators h and hin a completely symmetric way. The corresponding holomorphic bilocal field will depend on the gluing condition for the spin field which we will generically take to be Sβ(z) → F ββSβ(z∗), where the chirality of βdepends on the boundary conditions In this case we find that (assuming zero momentum in the open string sector) the only possible outcome is. We see that in both cases of NS-NS or R-R deformations, the first order obstruction to the vacuum shift is associated to the creation of a physical NS open string field via the bulk-boundary OPE
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