Topological Field Theory With The Exact Solvable Model

This thesis is concerned with the relations between topological string theory and some exactly solvable models. The first example is the so-called Clabi-Yau crystal, which turns out making a remarkable simplicity in calculation of the amplitude of topological A-model on toric Calabi-Yau manifold. In more specific, the A-model am-plitude can be exactly reproduced by generating functions of simple models of crystal melting knowing from statistical mechanics. Moreover, the crystal partition can be interpreted as sums of quantum foam of Calabi-Yau geometry, on where some effective gravitational theory live. In this sense, Calabi-Yau crystal provide a illustrate example of open/closed duality in topological string theory. In the first part of the thesis, we are mainly interested in constructing new Calabi-Yau crystal model via introducing some deformed algebra. After a two-chapter's introduction on topological field/string theory, we investigate the model in [100] which developed a q-deformed Calabi-Yau crystal model. We confirm that that to introduce a q-deformed commutator is equiv-alent to constrain the shape of crystal with a wall. The position of the wall can be taken as Kahler moduli parameter of resolved conifold on where A-model open string lives. By the same procedure, we calculate the partition function of q-deformed CY crystal with one or two wall. Especially interesting case is the configuration restricted by two wall, we prove in that case the model is equivalent to the so-called cubic model whose partition calculation is highly nontrivial.As an integrable system, Calabi-Yau crystal can be mapped into other integrable models in some proper limits [25]. The highlight case is quantum spin XXZ chain which is the central issue of our second part of thesis. We mainly concern about the computation of correlated function (or scalar product of Bethe states). For the open chain with non-diagonal boundary term, instead of having one reference state in diagonal case, we need one more set of pseudo-vacuum state, therefore two set of reflection K-matrix, two set of Bethe states et.al. Another keypoint in calculation, we should make Drinfeld twist in quantum space of spin chain in order to get a proper F- basis, on which the K-matrix become diagonal. In chapter 5, we construct the proper F-basis in quantum space of open XXZ spin chain with non-diagonal boundary term, and obtain the determinant representations of the scalar products of the corresponding Bethe states. Moreover, we also study the second reference state of the open XYZ spin chain with non-diagonal boundary term, and the associated Bethe states exactly yield the second set of eigenvalues.