Fonctionne avec Indico
v3.2.9
The real Gromov-Witten theory is an enumerative theory of pseudo-holomorphic maps from Riemann surfaces to a symplectic manifold, in which the maps are compatible with an anti-holomorphic involution of the domain and an anti-symplectic involution of the target.
In this talk, I will discuss the case of Riemann surface targets. The invariants involving only conjugated points constraints are called stationnary. In the complex Gromov-Witten theory, the stationnary invariants can be computed as an explicit weighted count of maps. The corresponding statement for the real invariants requires to count the maps with signs that I will describe. This combinatorial perspective yields explicit formulas for the stationnary invariants.
In the case of the target $\mathbb{P}^1$, the real invariants satisfy a familly of equations that define a representation of a sub-algebra of the Virasoro algebra. They allow to express the non-stationnary invariants in terms of the stationnary ones. The two results fully describe the real Gromov-Witten invariants of $\mathbb{P}^1$.