Abstract
In this paper, which is the last of a series including [1, 2] we first verify that the two open-closed effective potentials derived in the previous paper from the WZW theory in the large Hilbert space and the A∞ theory in the small Hilbert space have the same vacuum structure. In particular, we show that mass-term deformations given by the effective (open)2-closed couplings are the same, provided the effective tadpole is vanishing to first order in the closed string deformation. We show that this condition is always realized when the worldsheet BCFT enjoys a global mathcal{N} = 2 superconformal symmetry and the deforming closed string belongs to the chiral ring in both the holomorphic and anti-holomorphic sector. In this case it is possible to explicitly evaluate the mass deformation by localizing the SFT Feynman diagrams to the boundary of world-sheet moduli space, reducing the amplitude to a simple open string two-point function. As a non-trivial check of our construction we couple a constant Kalb-Ramond closed string state to the OSFT on the D3–D(−1) system and we show that half of the bosonic blowing-up moduli become tachyonic, making the system condense to a bound state whose binding energy we compute exactly to second order in the closed string deformation, finding agreement with the literature.
Highlights
Algebraic couplings in the effective action, are an important exception because they receive non-trivial contributions from the boundary of moduli space
We show that this condition is always realized when the worldsheet BCFT enjoys a global N = 2 superconformal symmetry and the deforming closed string belongs to the chiral ring in both the holomorphic and anti-holomorphic sector
In this paper, following the construction started in [28, 29] and continued in [1, 2], we have first established that the effective potential derived from the WZW theory in the large Hilbert space and the one derived from the A∞ theory in the small Hilbert space have the same vacuum structure, they are in general related by a zero momentum field redefinition
Summary
For the cubic effective coupling, the A∞ theory yields. Where V1/2 is a h = 1/2 zero-momentum matter state belonging to the (anti)-chiral ring of a worldsheet N = 2 superconformal algebra. It was shown in [6] that upon assuming the projector condition (2.9), the quartic couplings S3,0(ψ) computed using the WZW-like and A∞ effective actions agree. In line with our expectation that the leading order of the field redefinition relating the effective WZW-like and A∞ theories is trivial, we confirm (at least to the quartic order) that the purely open effective couplings in the two theories at a fixed order in perturbation theory agree, provided that lower order couplings vanish. Since the vanishing of the effective potential order-by-order in perturbation theory is precisely the requirement on massless modes to be exact moduli for any given background, we conclude that both the WZW-like and the A∞ theory yield the same algebraic constraints determining moduli spaces of open string backgrounds
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