Fock Spaces and Refined Severi Degrees

In this note we show that, for a large class of pairs |$(S,L)$| of a toric surface with an ample line bundle |L|⁠, containing |P²| and rational-ruled surfaces, the refined Severi degrees can be computed as matrix elements of an operator on a Fock space. The Heisenberg algebra does not depend on the surface considered, only the operator. The result also applies to the Severi degrees counting complex curves and the Welschinger numbers which are a signed count of real curves.