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Quantum Schubert calculus. (English) Zbl 0945.14031

Denote by \(W_{\vec{a}}\) Schubert varieties associated to \(\vec{a}\) and by \(\sigma_{\vec{a}}\in\text{H}^{2|\vec{a}|}(G,\mathbb{C})\) the corresponding elements in cohomology where \(G\) is the Grassmannian. The symbol \(W_a\) stands for a special Schubert variety associated to \((a,0,\dots,0)\). Choose general points \(p_1,\dots,p_N\in\mathbb{P}^1\) and general translates of the \(W_{\vec{a}}\). The Gromov-Witten intersection number \(\langle W_{\vec{a}_1},\dots,W_{\vec{a}_N}\rangle_d\) is, by a naive definition, the number of holomorphic maps \(f:\mathbb{P}^1\to G\) of degree \(d\) with the property that \(f(p_i)\in W_{\vec{a}_i}\) for all \(i=1,\dots,N\). For \(d=0\) one gets the original intersection number. The small quantum ring is the vector space \(\text{H}^\ast(G,\mathbb{C})[q]\) over \(C[q]\) with an associative product which obeys \[ \sigma_{\vec{a}_1}\ast\dots\ast\sigma_{\vec{a}_N}= \sum_{d\geq 0}q^d\left(\sum_{\vec{a}} \langle W_{\vec{a}},W_{\vec{a}_1},\dots,W_{\vec{a}_N}\rangle_d \sigma_{\vec{a}}\right). \] The paper generalizes Giambelli’s formula and Pieri’s formula to the small quantum ring. In the quantum Giambelli formula that reads \(\sigma_{\vec{a}}=\Delta_{\vec{a}}(\sigma_\ast)\) no higher terms in \(q\) arise. The Giambelli determinant in cohomology classes corresponding to special Schubert varieties is evaluated in \(\text{H}^\ast(G,\mathbb{C})[q]\). On the other hand, the quantum Pieri formula has a correction term, \[ \sigma_a\ast\sigma_{\vec{a}}= p_{a,\vec{a}}(\sigma_{\vec\ast})+ q\left(\sum_{\vec c}\sigma_{\vec c}\right), \] with an appropriate range of \(\vec c\). Before giving the proofs the Gromov-Witten number is defined rigorously by considering intersections of Schubert varieties on the moduli space \(\mathcal M_d\) of holomorphic maps of degree \(d\) from \(\mathbb{P}^1\) to \(G\) with \(\mathcal M_d\) being an open subscheme in the Grothendieck quoted scheme. As a corollary of the quantum Giambelli’s formula the author also shows a Vafa and Intriligator formula for the Gromov-Witten intersection number of special Schubert varieties.

MSC:

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
14M15 Grassmannians, Schubert varieties, flag manifolds
14J81 Relationships between surfaces, higher-dimensional varieties, and physics
81T20 Quantum field theory on curved space or space-time backgrounds
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