[Paper Review] Gromov-Witten invariants for G/B and Pontryagin product for ΩK

Naichung Conan Leung, Changzheng Li|arXiv (Cornell University)|Oct 27, 2008

Advanced Combinatorial Mathematics5 references3 citations

TL;DR

This paper provides an explicit formula for equivariant quantum Schubert structure constants in the quantum cohomology ring of flag varieties $G/B$ by deriving an explicit expression for the Pontryagin product on the equivariant homology of the based loop group $\Omega K$. The key result is a finite, combinatorially defined formula involving rational functions $c_{x,[y]}$ and $d_{x,[y]}$, which computes the 3-pointed genus zero Gromov-Witten invariants via a sum over a finite set of coweight pairs, with the result being a constant when the degree condition $\langle\lambda,2\rho\rangle = \ell(u)+\ell(v)-\ell(w)$ is satisfied.

ABSTRACT

We give an explicit formula for (T-equivariant) 3-pointed genus zero Gromov-Witten invariants for G/B. We derive it by finding an explicit formula for the equivariant Pontryagin product on the homology of the based loop group ΩK.

Motivation & Objective

• To provide an explicit formula for the (equivariant) quantum Schubert structure constants in the quantum cohomology ring $QH^*_{T}(G/B)$, which generalize classical Schubert structure constants by incorporating genus zero Gromov-Witten invariants.
• To establish a connection between the equivariant quantum cohomology of $G/B$ and the Pontryagin product on the equivariant homology of the based loop group $\Omega K$, leveraging the homotopy equivalence $\Omega_{\text{an}}K \simeq \Omega K$.
• To derive a finite, combinatorial expression for the 3-pointed genus zero Gromov-Witten invariants $N_{u,v}^{w,\lambda}$ using rational functions $c_{x,[y]}$ and $d_{x,[y]}$, valid under the degree condition $\langle\lambda,2\rho\rangle = \ell(u)+\ell(v)-\ell(w)$.

Proposed method

• The authors define rational functions $c_{x,[y]}$ and $d_{x,[y]}$ combinatorially for elements $x,y$ in the affine Weyl group $W_{\text{af}} = W \ltimes Q^\vee$, based on the structure of the based loop group $\Omega K$ and its equivariant homology.
• They express the Pontryagin product on $H_*^{T}(\Omega K)$ using the formula $p_{x,y}^z = \sum_{v \in W_{\text{af}}^-} d_{x,v} d_{y,v} c_{z,v}$, which is then used to compute the quantum Schubert structure constants.
• The main formula for the quantum Schubert structure constant is derived as $N_{u,v}^{w,\lambda} = \sum_{\lambda_1,\lambda_2 \in Q^\vee} c_{ut_A,[t_{\lambda_1}} c_{vt_A,[t_{\lambda_2}} d_{wt_{2A+\lambda},[t_{\lambda_1+\lambda_2}}]$, with $A = -\frac{1}{2}n(n+1)\sum_{i=1}^n w_i^\vee$, and the sum is reduced to a finite set $\Gamma \times W$.
• The finite summation is achieved by restricting $\lambda_1, \lambda_2 \succcurlyeq A$ and $\lambda_1 + \lambda_2 \preccurlyeq 2A + \lambda$, with $\lambda_1, \lambda_2 \in \tilde{Q}^\vee$, ensuring only finitely many nonzero terms.
• The formula is extended to the $T$-equivariant setting by using evaluation maps $\text{ev}: \hat{S} \to S$, which send the null root $\delta = \alpha_0 + \theta$ to zero, and by expressing $T$-equivariant structure constants as $\tilde{p}_{x,y}^z = \sum_{v \in W_{\text{af}}^-} d_{x,[v]} d_{y,[v]} c_{z,[v]}$.
• The method relies on the Bruhat decomposition of $\mathcal{G}/\mathcal{P}_0 \simeq \Omega K$, which induces a Schubert basis on $H_*(\Omega K, \mathbb{Z})$, and uses the $\hat{T}$-equivariant structure of $\mathcal{G}/\mathcal{B}$ to relate cohomology classes to those in $\Omega K$.

Experimental results

Research questions

• RQ1 How can the equivariant quantum Schubert structure constants $N_{u,v}^{w,\lambda}$ for $G/B$ be computed explicitly in terms of combinatorial data from the affine Weyl group?
• RQ2 What is the precise relationship between the Pontryagin product on the equivariant homology of $\Omega K$ and the quantum cohomology ring $QH^*_{T}(G/B)$?
• RQ3 Can the infinite sum over coweights in the Gromov-Witten invariants be reduced to a finite sum using combinatorial constraints on the coweight parameters?
• RQ4 What is the role of the parameter $A = -\frac{1}{2}n(n+1)\sum_{i=1}^n w_i^\vee$ in simplifying the structure constants, and can it be replaced by a smaller one in specific cases?
• RQ5 How do the rational functions $c_{x,[y]}$ and $d_{x,[y]}$ encode the geometry of the based loop group and allow for explicit evaluation of Gromov-Witten invariants?

Key findings

• The quantum Schubert structure constant $N_{u,v}^{w,\lambda}$ is given by a finite sum over $\Gamma \times W$, where $\Gamma$ is a finite set of coweight pairs satisfying $\lambda_1, \lambda_2 \succcurlyeq A$, $\lambda_1 + \lambda_2 \preccurlyeq 2A + \lambda$, and $\lambda_1, \lambda_2 \in \tilde{Q}^\vee$, with the formula $N_{u,v}^{w,\lambda} = \sum_{(\lambda_1,\lambda_2,v_1) \in \Gamma \times W} c_{ut_A,[v_1 t_{\lambda_1}} c_{vt_A,[v_1 t_{\lambda_2}} d_{wt_{2A+\lambda},[v_1 t_{\lambda_1+\lambda_2}}]$.
• For $G = SL(3,\mathbb{C})$, with $u = v = s_1 s_2 s_1$, $w = s_1 s_2$, and $\lambda = \theta^\vee$, the formula yields $N_{u,v}^{w,\lambda} = 1$, computed as $(-1/\theta)^2 \cdot \theta^2 = 1$, confirming agreement with Mihalcea’s algorithm.
• The formula is valid only when $\langle\lambda, 2\rho\rangle = \ell(u) + \ell(v) - \ell(w)$; otherwise, $N_{u,v}^{w,\lambda} = 0$, ensuring degree constraints are respected.
• The parameter $A$ is not unique; in many cases, it can be replaced by a smaller one, such as $A = -\theta^\vee$ for $SL(3,\mathbb{C})$, simplifying computation.
• The rational functions $c_{x,[y]}$ and $d_{x,[y]}$ are defined combinatorially and are used to express the Pontryagin product on $H_*^T(\Omega K)$, which in turn computes the quantum product on $QH^*_{T}(G/B)$.
• The $T$-equivariant Schubert structure constants are recovered via evaluation maps $\text{ev}: \hat{S} \to S$, which set $\alpha_0 = -\theta$, and the resulting polynomials in $S$ match the $T$-equivariant structure constants $\tilde{p}_{x,y}^z = \sum_v d_{x,[v]} d_{y,[v]} c_{z,[v]}$.