Explicit reconstruction in quantum cohomology and K-theory

Abstract

Cohomological genus-0 Gromov-Witten invariants of a given target space can be encoded by the "descendant potential," a generating function defined on the space of power series in one variable with coefficients in the cohomology space of the target. Replacing the coefficient space with the subspace multiplicatively generated by degree-2 classes, we explicitly reconstruct the graph of the differential of the restricted generating function from one point on it. Using the Quantum Hirzebruch--Riemann--Roch Theorem from our joint work with Valentin Tonita, we derive a similar reconstruction formula in genus-0 quantum K-theory. The results amplify the role of the divisor equations, and the structures of $D$-modules and $D_q$-modules in quantum cohomology and quantum K-theory with respect to Novikov's variables.