## 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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