Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 12356.8.1.2 Open Strings in the Presence of Constant B−FieldBosonic StringsFollowing [Seiberg and Witten (1999)], we now study strings in flat space,with metric g ij , in the presence of a constant Neveu–Schwarz B–field andwith Dp−branes. The B−field is equivalent to a constant magnetic fieldon the brane.We denote the rank of the matrix B ij as r; r is of course even. Since thecomponents of B not along the brane can be gauged away, we can assumethat r ≤ p + 1. When our target space has Lorentzian signature, we willassume that B 0i = 0, with “0” the time direction. With a Euclidean targetspace we will not impose such a restriction. Our discussion applies equallywell if space is R 10 or if some directions are toroidally compactified withx i ∼ x i + 2πr i . (One could pick a coordinate system with g ij = δ ij , inwhich case the identification of the compactified coordinates may not besimply x i ∼ x i + 2πr i , but we will not do that.) If our space is R 10 , we canpick coordinates so that B ij is nonzero only for i, j = 1, . . . , r and that g ijvanishes for i = 1, . . . , r, j ≠ 1, . . . , r. If some of the coordinates are on atorus, we cannot pick such coordinates without affecting the identificationx i ∼ x i + 2πr i . For simplicity, we will still consider the case B ij ≠ 0 onlyfor i, j = 1, . . . , r and g ij = 0 for i = 1, . . . , r, j ≠ 1, . . . , r.The world–sheet action isS = 1 (gij4πα∫Σ′ ∂ a x i ∂ a x j − 2πiα ′ B ij ɛ ab ∂ a x i ∂ b x j)= 14πα∫Σ ′ g ij ∂ a x i ∂ a x j − i ∫B ij x i ∂ t x j , (6.347)2where Σ is the string world–sheet, which we take to be with Euclideansignature. ∂ t is a tangential derivative along the world–sheet boundary ∂Σ.The equations of motion determine the boundary conditions. For i alongthe Dp−branes they are∂Σg ij ∂ n x j + 2πiα ′ B ij ∂ t x j | ∂Σ = 0, (6.348)where ∂ n is a normal derivative to ∂Σ. These boundary conditions are notcompatible with real x, though with a Lorentzian world–sheet the analogousboundary conditions would be real. Nonetheless, the open string theory canbe analyzed by determining the propagator and computing the correlationfunctions with these boundary conditions. In fact, another approach to the