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Alim, Murad (2009): Mirror Symmetry, Toric Branes and Topological String Amplitudes as Polynomials. Dissertation, LMU München: Fakultät für Physik



The central theme of this thesis is the extension and application of mirror symmetry of topological string theory. Mirror symmetry is an equivalence between the topological string A-model on a manifold X and the B-model on a mirror manifold Y, together with their deformation spaces. Deformations of the target space on the A-side are Kaehler deformations which change the volume of the manifold. Deformations on the B-side change the complex structure. The power of mirror symmetry is due to its simple physical origin which connects two different areas of mathematics, symplectic- and complex geometry. This connection has far reaching and unexpected consequences both on the mathematical and on the physical side. Physical problems can be given a precise mathematical meaning and can be solved. In this regard, quantum corrected superpotentials are computed in this work on the one hand. On the other hand the mathematical understanding of the background dependence is used to reorganize a perturbative Feynman diagram expansion in terms of a more efficient polynomial expansion. The contribution of this work on the mathematical side is given by interpreting the calculated partition functions as generating functions for mathematical invariants which are extracted in various examples. The main idea of mirror symmetry is to map the solution of simple problems to the solution of equivalent difficult problems. To do so, a mirror map is needed, which is at the heart of mirror symmetry. The computation of this map is possible thanks to a physical structure which occurs in both mathematical realizations. This structure is the vacuum bundle, together with a grading which varies over the space of deformations. In the context of the B-model the study of the variation of this grading leads to differential equations which allow the computation of the mirror map as well as other quantities which describe quantum geometry when translated to the A-side. In this thesis, the extension of the variation of the vacuum bundle to include D-branes on compact geometries is studied. Based on previous work for non-compact geometries a system of differential equations is derived which allows to extend the mirror map to the deformation spaces of the D-Branes. Furthermore, these equations allow the computation of the full quantum corrected superpotentials which are induced by the D-branes. Based on the holomorphic anomaly equation, which describes the background dependence of topological string theory relating recursively loop amplitudes, this work generalizes a polynomial construction of the loop amplitudes, which was found for manifolds with a one dimensional space of deformations, to arbitrary target manifolds with arbitrary dimension of the deformation space. The polynomial generators are determined and it is proven that the higher loop amplitudes are polynomials of a certain degree in the generators. Furthermore, the polynomial construction is generalized to solve the extension of the holomorphic anomaly equation to D-branes without deformation space. This method is applied to calculate higher loop amplitudes in numerous examples and the mathematical invariants are extracted.