W-Constraints And Genus Zero Virasoro-like Constraints For Frobenius Manifolds

The notion of Frobenius manifold is a geometric abstraction of the genus zero partsof2D topological field theories (TFT). It connects2D TFT with several diferent researchbranches of mathematics, including the theory of Gromov-Witten invariants, singularitytheory and the theory of integrable systems, and the development of the theory of Frobe-nius manifolds is an important research subject in modern mathematical physics.One of the most important research activities, that are connected to the theory ofFrobenius manifolds, is to reconstruct the full genera free energy of a2D TFT from itsgenus zero primary free energy. In this thesis we construct, in terms of the geomet-ric structures of Frobenius manifolds, two families of constraints for the free energiesassociated to the Frobenius manifolds. These constraints, obtained from certain lineardiferential operators, are calledW-constraints and genus zero Virasoro-like constraints.We show, for certain classes of Frobenius manifolds, that these constraints themselves ortogether with other known constraints uniquely determine the associated full genera freeenergies.More specifically, for a semisimple Frobenius manifold with a non-degenerate G-matrix, we construct the natural twisted module over the Heisenberg vertex algebra as-sociated to it by employing the natural quantization of the gradients of periods. Then byusing a vertex algebra approach, we define a vertex subalgebra of the Heisenberg vertexalgebra as the intersection of kernels of certain screening operators. We call this vertexsubalgebra theW-algebra. One of our main results is that the image of theW-algebraunder the natural twisted state-field map provides constraints for the total descendantpotential (the exponential of the full genera free energy) given by Givental in terms ofcertain quantization formulas. These constraints are then called theW-constraints. Thedirect corollary of this result is that the total descendant potential satisfies the Virasoroconstraints, where the associated Virasoro operators coincide with those introduced inthe Dubrovin-Zhang theory on the relationship between Frobenius manifolds and inte-grable systems. In particular, for the class of semisimple Frobenius manifolds associatedto simple singularities, we prove that theW-constraints uniquely determine the total de-scendant potential up to a constant factor. We also calculate in several examples someintersection numbers by applying this uniqueness theorem. Another main result of the thesis is the construction of a new family of constraintsfor the genus zero free energy of an arbitrary Frobenius manifold. These constraintscontain the genus zero Virasoro constraints as a subset and are called the genus zeroVirasoro-like constraints. For the topologicalσ-model coupled to gravity, Virasoro-likeconstraints can be considered as an analogue of theW-constraints. In particular, fortheP1topologicalσ-model coupled to gravity, we show that the genus zero Virasoro-like constraints together with the genus zero dilaton equation uniquely determine thecorresponding genus zero free energy. As a result, according to the Dubrovin-Zhangtheory, these constraints together with the higher genus Virasoro constraints uniquelydetermine the Gromov-Witten invariants withP1as the target.