Ivancevic_Applied-Diff-Geom

1224 Applied Differential Geometry: A Modern Introductionabove question rigorously, we would have to study boundary conditions onworld–sheets with boundaries which preserve the Q−symmetry.In the A−model, one can only construct 3D–branes wrapping so–called‘Lagrangian’ submanifolds of M. Here, ‘Lagrangian’ means that the Kählerform ω vanishes on this submanifold. In the B−model, one can constructD–branes of any even dimension, as long as these branes wrap holomorphicsubmanifolds of M.Just like in ordinary string theory, when we consider open topologicalstrings ending on a D–brane, there should be a field theory on the braneworld–volume describing the low–energy physics of the open strings. Moreover,since we are studying topological theories, one may expect such atheory to inherit the property that it only depends on a restricted amountof data of the manifolds involved. A key example is the case of the A−modelon the deformed conifold, M = T ∗ S 3 , where we wrap ND–branes on the S 3in the base. (One can show that this is indeed a Lagrangian submanifold.)In ordinary string theory, the world–volume theory on ND–branes has aU(N) gauge symmetry, so putting the ingredients together we can makethe guess that the world–volume theory is a 3D topological field theorywith U(N) gauge symmetry. There is really only one candidate for such atheory: the Chern–Simons gauge theory. Recall that it consists of a singleU(N) gauge field, and has the actionS = k4π∫S 3 Tr(A ∧ dA + 2 3 A ∧ A ∧ A ). (6.334)Before the invention of D–branes, E. Witten showed that this is indeedthe theory one gets. In fact, he showed even more: this theory actuallydescribes the full topological string–field theory on the D–branes, evenwithout going to a low–energy limit [Witten (1995a)].Let us briefly outline the argument that gives this result. In his paper,Witten derived the open string–field theory action for the open A−modeltopological string; it reads∫S = Tr(A ∗ Q A A + 2 )3 A ∗ A ∗ A .The form of this action is very similar to Chern–Simons theory, but itsinterpretation is completely different: A is a string–field (a wave functionon the space of all maps from an open string to the space–time manifold),Q A is the topological symmetry generator, which has a natural action on thestring–field, and ∗ is a certain noncommutative product. Witten shows that