D-branes in the Light-cone Gauge and Broken Symmetries

Boundary states for D-branes are constructed using the light cone gauge. The D-brane breaks half the spacetime supersymmetry giving rise to fermionic zero modes living on the brane. The nonlinear realization of the broken supersymmetry on the open string degrees of freedom is analysed and the influence of boundary terms coming from closed string vertex operators is discussed. 1 Boundary states and broken symmetries The introduction of boundaries breaks symmetries of conformal field theories because the boundary relates left-moving and right-moving degrees of freedom. The condition that the conformal invariance on the half plane H = {z|Im(z) ≤ 0} is not completely broken implies the continuity of the holomorphic and antiholomorphic parts of the stress energy tensor T (z) = ¯ T (¯ z) for Im(z) = 0. More generally for a current J, ¯ J of conformal weight (h, h) the condition that the boundary preserves the symmetry is given by the continuity J(z) = ¯ J(¯ z) for z = ¯ z. Mapping the half plane into the semi infinite cylinder via z = exp(τ +iσ) we can translate this condition into a condition on a boundary state. (J n + (−1) h ¯ J −n) | B = 0 (1) We can define two currents acting on the boundary, J ± (σ) = J(z) ± (−1) h ¯ J(¯ z) where J − is the current broken by the boundary conditions. Conformal boundary conditions and boundary states are in one to one correspondence 1. In the case of toroidally compactified bosonic string theory with an enhanced gauge symmetry group G × G it was shown in 2 that the scalars J −a correspond to Goldstone bosons living in the coset G × G/G which is defined trough the breaking of the G × G closed string symmetry to G × G/G by the boundary.