## 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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