Classical aspects of quantum cohomology of generalized flag varieties
Let \(P\) be a parabolic subgroup of a simply connected complex simple Lie group \(G\). The presentation of the ring structure of the quantum cohomology of the flag varieties, denoted by \(QH^*(G/P)\), have been studied by many mathematicians. The structure constants of the product in \(QH^*(G/P)\) are given by the 3-point genus 0 Gromov-Witten invariants of \(G/P\). By the Peterson-Woodward comparison formula these GW invariants of \(G/P\) can be recovered from the GW invariants of the special case of the complete flag variety \(G/B\) where \(B\) is a Borel subgroup.NEWLINENEWLINEThe main result of the paper under review proves vanishing and relations among some of the GW invariants of \(G/B\). As an application of this result, it is proven that in the \(A_n\) cases, certain GW invariants of \(G/B\) are the classical intersection numbers, in other words these invariants satisfy the ``quantum to classical'' principle. The proof of the main result uses the functorial relations resulted from the construction of the natural filtered algebra structure on \(QH^*(G/B)\) in the earlier work of the authors of the paper under review.
scientific article; zbMATH DE number 6078260