Symplectic Topology, Mirror Symmetry and Integrable systems

The plan of the work is the following: ² In Chapter 1 we recall, basically from [16] and [14], the ideas and methods of Symplectic Field Theory. Our review will focus on the algebraic structure arising from topology, more than on the geometry underlying it. In particular we de¯ne the SFT analogue of the Gromov-Witten potential as an element in some graded Weyl algebra and consider its properties (grading, master equations, semiclassical limit). We then stress (following [18]) how this algebraic structure allows the appearence of a system of commuting di®erential operators (on the homology of the Weyl algebra) which can be thought of as a system of quantum Hamiltonian PDEs with symmetries. Sometimes this symmetries are many enough to give rise to a complete integrable system (at least at the semiclassical level) and we examine the main examples where this happens. Eventually we review some results of [16] which turn out to be very useful in computations and which we actually employ in the next chapters. ² In Chapter 2 we apply the methods of Symplectic Field Theory to the computation of the Gromov-Witten invariants of target Riemann surfaces. Our computations reproduce the results of [27], [28] which, in principle, solve the theory of target curves, but are fairly more explicit and, above all, clarify the role of the KdV hierarchy in this topological theory. More precisely we are able to describe the full descendant Gromov-Witten potential as the solution to SchrÄodinger equation for a quantum dispersionless KdV system. This quantization of KdV, already appearing in [31], can be easily dealt with in the fermionic formalism to give extremely explicit results, like closed formulae for the Gromov-Witten potential at all genera and given degree. These results where published by the author in [32]. ² In Chapter 3 we use basically the same techniques of Chapter 2 to compute the Gromov- Witten theory of target curves with orbifold points (orbicurves). As in the smooth case, the coe±cient for the Gromov-Witten potential are written in terms of Hurwitz numbers. It turns out that we can even classify those target orbicurves whose potential involves only a ¯nite number of these a priori unknown Hurwitz coe±cients, so that they can be determined using WDVV equations. These polynomial P1-orbifolds are the object of our study for the ¯nal part of this work. Moreover, we extend the theorem by Bourgeois ([4]) about Hamiltonian structures of ¯bration type to allow singular ¯bers (Seifert ¯brations), so that we can use our result on Gromov-Witten invariants of polynomial P1-orbifolds to deduce the SFT-Hamiltonians of the ¯bration. ² In Chapter 4 we completely solve the rational Gromov-Witten problem for polynomial P1-orbifolds. Namely we ¯nd a Landau-Ginzburg model which is mirror symmetric to these spaces. This model consists in a Frobenius manifold structure on the space of what we call for brevity tri-polynomials, i.e. polynomials of three variables of the form ¡xyz + P1(x) + P2(y) + P3(z). The main results here are the explicit construction of the Frobenius manifold structure with closed expressions for °at coordinates and the mirror theorem 4.0.3, i.e. the isomorphism of this Frobenius structure with the one on the quantum cohomology of polynomial P1-orbifolds. From the polynomiality property of the Frobenius potentials involved, one is able to show that there is also a third mirror symmetric partner in the picture, namely the Frobenius manifold associated to extended a±ne Weyl groups of type A, D, E ([12]). The results of these last two chapters appeared in [33]. ² In the Conclusions we summarize our results and analyze possible further developments and directions to be explored.