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.