Gromov-Witten
invariants of a symplectic manifold are defined via intersection theory in
moduli spaces of
pseudo-holomorphic curves in these manifolds. The invariants satisfy numerous identities
universal in the sense that
their form is independent on the
choice of the target symplectic manifold. The peculiar structure formed by the
invariants and the universal
identities has been subject to
extensive study and has lead to the theory of Frobenius manifolds and other concepts of
axiomatic Gromov-Witten theory. Some recent work shows that the axiomatic
structure posesses a certain
loop group of hidden symmetries.
In the first lecture we plan to outline the approach to axiomatic Gromov-Witten theory which
emphasizes the role of this
symmetry group. In the second lecture we intend to discuss the place of the
so-called Virasoro constraints -
some conjectural indentities which play a key role in Gromov-Witten theory at its
current state.
