ﻻ يوجد ملخص باللغة العربية
This thesis is concerned with a realisation of topological theories in terms of statistical models known as Calabi-Yau crystals. The thesis starts with an introduction and review of topological field and string theories. Subsequently several new results are presented. The main focus of the thesis is on the topological string theory. In this case crystal models correspond to three-dimensional partitions and their relations with the topological vertex theory and knot invariants are studied. Two-dimensional crystal models corresponding to topological gauge theories on ALE spaces are also introduced and analysed. Essential mathematical tools are summarised in appendices.
We present the most complete list of mirror pairs of Calabi-Yau complete intersections in toric ambient varieties and develop the methods to solve the topological string and to calculate higher genus amplitudes on these compact Calabi-Yau spaces. The
We show that Calabi-Yau crystals generate certain Chern-Simons knot invariants, with Lagrangian brane insertions generating the unknot and Hopf link invariants. Further, we make the connection of the crystal brane amplitudes to the topological vertex
We study the geometry of the scalar manifolds emerging in the no-scale sector of Kahler moduli and matter fields in generic Calabi-Yau string compactifications, and describe its implications on scalar masses. We consider both heterotic and orientifol
We calculate the generating functions of BPS indices using their modular properties in Type II and M-theory compactifications on compact genus one fibered CY 3-folds with singular fibers and additional rational sections or just $N$-sections, in order
The disk partition function of certain 3d N=2 supersymmetric gauge theories computes a quantum K-theoretic ring for Kahler manifolds X. We study the 3d gauge theory/quantum K-theory correspondence for global and local Calabi-Yau manifolds with severa