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Quantum \(K\)-theory on flag manifolds, finite-difference Toda lattices and quantum groups. (English) Zbl 1051.14063

Let \(X \hookrightarrow \Pi\) be the Plücker embedding of the manifold \(X\) of complete flags in \(\mathbb C^{r+1}\) into the product of projective spaces \(\Pi = \prod_{i=1}^{r} \mathbb C\mathbb P^{n_{i}-1}\). The space of degree \(d\) holomorphic maps \(\mathbb C\mathbb P^1 \to \mathbb C\mathbb P^n\) is compactified to a complex projective space of dimension \((N+1)(d+1)-1\), which is denoted by \(\mathbb C\mathbb P^N_d\). This construction defines the compactification \(\Pi_d = \prod_{i=1}^r \mathbb C\mathbb P_{d_i}^{n_i-1}\) of the space of degree \(d\) maps from \(\mathbb C\mathbb P^1\) to \(\Pi\). Composing degree \(d\) holomorphic maps from \(\mathbb C\mathbb P^1\) to \(X\) with the Plücker embedding, we can embed the space of such maps into \(\Pi_d\). The closure \(QM_d\) of the image of this embedding is called the Drinfel’d compactification of the space of degree \(d\) maps from \(\mathbb C\mathbb P^1\) to \(X\), or the space of quasi maps.
Since \(X\) is a homogeneous space and there is a natural action of \(S^1\) on \(\mathbb C\mathbb P^1\) we have an action of \(G=S^1 \times SU_{r+1}\) on \(QM_d\). We denote by \(P = (P_1, \dots , P_r)\) the \(G\)-equivariant line bundles obtained by pulling back the Hopf bundles over the factor of \(\Pi_d\), and by \(P^z = P^{z_1}_1 \otimes \cdots \otimes P^{z_r}_r\) its tensor product. Next, we define the function \[ {\mathcal G}(Q, z, q, \Lambda) = \sum_d \; Q^d \chi_G (H^*\widetilde{QM_d}, P^z)), \] where \(q\) and \(\Lambda_0, \dots, \Lambda_r\) are multiplicative coordinates on \(S^1\) and on the maximal torus \(T^r\) of \(SU_{r+1}\), respectively, and \(Q = (Q_1, \dots, Q_r)\) are formal variables. Here, \(\widetilde{QM_d}\) is a \(G\)-equivariant desingularization of \(QM_d\).
The main result of the paper states that the function \[ G(Q,Q') = {\mathcal G}(Q, \frac{\ln Q'- \ln Q}{\ln q}, q, \Lambda) \] is the eigenfunction of the Toda operator \[ \widehat{H}_{Q', q}G = (\Lambda_0 + \dots + \Lambda_r)G, \] where \[ \widehat{H}_{Q,q} = q ^{\frac{\partial}{\partial q_0}} + q ^{\frac{\partial}{\partial q_1}} (1-e^{t_0-t_1}) + \cdots + q ^{\frac{\partial}{\partial q_r}}(1-e^{t_{r-1}-t_r}). \] The authors also explore the possibility to extend the above theorem for a general case of flag manifolds \(X=G/B\) where \(G\) is an arbitrary semi-simple complex Lie group \(G\).

MSC:

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14M15 Grassmannians, Schubert varieties, flag manifolds
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
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