Algebraic Approaches To Landau-Ginzburg Orbifolds

The research of Landau-Ginzburg models has always been a major focus of both mathematicians and physicists.It connects the singularity theory to several other research branches in mathematics,like non-commutative geometry,Hodge theory,deformation theory as well as quantum cohomology,and it provides many essential topics in modern mathematical physics.As the most important one among them,the mirror symmetry phenomena between Landau-Ginzburg models is rarely investigated,due to the lack of a mathematical construction of B-model theory.In this thesis,we construct the B-model Frobenius manifolds,the genus zero data of a complete B-model theory,via the deformation theory of a certain type of curved algebras.To be explicit,it includes two parts.Firstly,we use the Hochschild cohomology to define the B-model state spaces,on which we show that there exists a Frobenius algebraic structure.Secondly,analogue to Saito's singularity theory,we construct flat structures,including the flat connections on compact type of periodic cyclic homology as well as the higher residue pairing,and the consequential Frobenius manifold structures on the deformation spaces of those curved algebras.However,we will calculate via G-twisted Hochschild(co)chains instead,because of the existence of a special homotopy retraction from the G-twisted(co)chains to the much smaller Koszul(co)chains,and generalize the braces structure on cochains as well as higher operations on chains to their G-twisted versions.Via this homotopy retraction,we can associate the calculation with the study of different types of quantum differential operators we defined in this thesis and compute the explicit cup product formula for general invertible polynomials for the first time.Simultaneously,we write down the deformed algebraic structures on those curved algebras and the extended Getzler-Gauss-Manin connections explicitly.As an application,we calculate certain examples of Landau-Ginzburg orbifolds related to ADE singularities,whose Frobenius manifolds are isomorphic to those coming from the Saito's theory for some other ADE singularities constructed by associated crepant resolutions,indicated by the crepant resolution conjecture between LG models.