Ivancevic_Applied-Diff-Geom

1236 Applied Differential Geometry: A Modern Introductionopen string problem is to omit or not specify the boundary term with B inthe action (6.347) and simply impose the boundary conditions (6.348).For B = 0, the boundary conditions in (6.348) are Neumann boundaryconditions. When B has rank r = p and B −→ ∞, or equivalently g ij → 0along the spatial directions of the brane, the boundary conditions becomeDirichlet; indeed, in this limit, the second term in (6.348) dominates, and,with B being invertible, (6.348) reduces to ∂ t x j = 0. This interpolationfrom Neumann to Dirichlet boundary conditions will be important, since wewill eventually take B −→ ∞ or g ij −→ 0. For B very large or g very small,each boundary of the string world–sheet is attached to a single point in theDp−brane, as if the string is attached to a zero–brane in the Dp−brane.Intuitively, these zero–branes are roughly the constituent zero-branes ofthe Dp−brane as in the matrix model of M−theory [Seiberg and Witten(1999)], an interpretation that is supported by the fact that in the matrixmodel the construction of Dp−branes requires a nonzero B−field.Our main focus is the case that Σ is a disc, corresponding to the classicalapproximation to open string theory. The disc can be conformally mappedto the upper half plane; in this description, the boundary conditions (6.348)areg ij (∂ − ¯∂)x j + 2πα ′ B ij (∂ + ¯∂)x j | z=¯z = 0,where ∂ = ∂/∂z, ¯∂ = ∂/∂¯z, and Im z ≥ 0.boundary conditions isThe propagator with these〈x i (z)x j (z ′ )〉 = −α ′ [g ij log |z − z ′ | − g ij log |z − ¯z ′ | (6.349)+ G ij log |z − ¯z ′ | 2 + 12πα ′ θij log z − ¯z′¯z − z ′ + Dij ].Here(G ij 1=g + 2πα ′ B) ijS() ij1=g + 2πα ′ B g 1g − 2πα ′ ,BG ij = g ij − (2πα ′ ) 2 (Bg −1 B) ij ,( ) ij () ijθ ij = 2πα ′ 1g + 2πα ′ = −(2πα ′ ) 2 1BAg + 2πα ′ B B 1g − 2πα ′ ,Bwhere ( ) S and ( ) A denote the symmetric and antisymmetric part of thematrix. The constants D ij in (6.349) can depend on B but are independentof z and z ′ ; they play no essential role and can be set to a convenient value.The first three terms in (6.349) are manifestly single–valued. The fourth