Calabi-Yau crystals in topological string theory

TL;DR

This thesis explores Calabi-Yau crystal models as statistical representations of topological string theories, revealing new connections with topological vertex theory, knot invariants, and gauge theories on ALE spaces.

Contribution

It introduces novel crystal models for topological string theories and analyzes their relations to vertex theory, knot invariants, and gauge theories on ALE spaces.

Findings

Calabi-Yau crystal models correspond to three-dimensional partitions.

Relations between crystal models and topological vertex theory are established.

Two-dimensional crystal models are linked to topological gauge theories on ALE spaces.

Abstract

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.