On Topological Strings, Chern-Simons Theory and the Topological Vertex
Vebjørn Træet Gilberg · NORA - Norwegian Open Research Archives · 2020
Topological string theory was constructed as a toy model of the full physical string theory that has proved to be fruitful in its physical interpretation as well as bridging parts of contemporary mathematics with physics.The basic building block of the amplitudes from topological string theory is the open string amplitude for C 3 , which is represented by a trivalent vertex and is called the topological vertex.Closed vertex amplitudes can be calculated by the gluing of patches of C 3 to obtain more complicated toric Calabi-Yau manifolds.The topological vertex is derived from the duality between Chern-Simons theory as an open string field theory on T * S 3 and the A-model of topological string theory on the resolved conifold.A single vertex is given by the Schur polynomials which are written as products of homogeneous symmetric functions that are found through the Jacobi-Trudi formula in order to ultimately compute the open topological vertex amplitudes from C R•• to C R 1 R 2 R 3 .We calculate open and closed topological string amplitudes with the goal of reproducing the results given in Aganagic et al. (2005).From the gluing of two open vertex amplitudes as patches of C 3 the closed amplitudes of the resolved conifold are obtained in the full closed topological string partition function, Z P 1 , up to order Q 5 .