By Recknagel A., Schomerus V.

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**Extra resources for Boundary Conformal Field Theory and the Worldsheet Approach to D-Branes**

**Example text**

Note that boundary correlation functions depend in several ways on the choice of boundary conditions. To begin with, the set of boundary ﬁelds it labelled by d-dimensional momentum vectors k, in other words it depends on the number of bosonic ﬁelds satisfying the Neumann boundary conditions. 48) 20 Free ﬁeld theory with boundaries of the boundary ﬁelds, contain the deformed metric Gij of rank d rather than our original matrix g μν ; in particular, Gij depends on the antisymmetric matrix Bμν . Finally, the couplings possess a non-trivial dependence on the momenta kr through the factor exp(− 2i kr × ks ).

73) m From these formulas one can now determine the commutation relations of Gr and Gn with Ln and among the modes of G. The results read m − n Gm+n Lm , Gn = 2 D 1 {Gm , Gn } = 2Lm+n + m2 − δm+n,0 2 4 in the Ramond sector, and the same for the Neveu–Schwarz sector with m → r, n → s. 23) of a Virasoro 3D algebra with central charge c = D + D 2 = 2 . 71), let us now turn to the boundary theory. We want to impose boundary conditions on the fermions and bosons such that supersymmetry is preserved.

1 Boundary states for a circle compactiﬁcation The compactiﬁcation of a free bosonic ﬁeld X to a circle of radius r is described by a free ﬁeld theory with the additional identiﬁcation X ∼ X + 2πr. The bulk and boundary spectra of this system get slightly more involved due to the appearance of winding modes. We did not incorporate the case of ﬁnite radius r from the very beginning so as to avoid cluttering the exposition in the main sections. But since torus compactiﬁcations play an important role, we should not discard them completely.