Welschinger--Witt invariants

Welschinger invariants are signed counts of real rational curves satisfying contraints. Quadratic Gromov--Witten invariants give such counts over general fields of characteristic different from 2 and 3. For rational del Pezzo surfaces over a field, we propose a conjectural relationship between Welschinger and quadratic Gromov--Witten invariants. We construct multivariable unramified Witt invariants, in the sense of Serre, from Welschinger invariants and call them Welschinger--Witt invariants. We show that quadratic Gromov--Witten invariants are also Witt invariants and control their ramification. We then conjecture an equality between these Witt invariants, in particular giving a conjectural computation of all the quadratic Gromov--Witten invariants of $k$-rational surfaces. We prove this conjecture for $k$-rational del Pezzo surfaces of degree at least 6.