[Paper Review] Asymptotic solutions to the sl_2 KZ equation and the intersection of Schubert classes

Inna Scherbak|arXiv (Cornell University)|Jul 24, 2002

Algebraic structures and combinatorial models6 references3 citations

TL;DR

This paper provides a new, less technical proof that the number of orbits of critical points of the master function for the $sl_2$ Knizhnik-Zamolodchikov (KZ) equation equals the dimension of the space of singular vectors in the tensor product of $sl_2$ representations. It establishes this via a correspondence between critical points and preimages under the Wronski map, using Schubert calculus to compute the intersection number of Schubert classes, which matches the dimension of the singular vector space.

ABSTRACT

The hypergeometric solutions to the KZ equation contain a certain symmetric ``master function'', [SV]. Asymptotics of the solutions correspond to critical points of the master function and give Bethe vectors of the inhomogeneous Gaudin model, [RV]. The general conjecture is that the number of orbits of critical points equals the dimension of the relevant vector space, and that the Bethe vectors form a basis. In [ScV], a proof of the conjecture for the sl_2 KZ equation was given. The difficult part of the proof was to count the number of orbits of critical points of the master function. Here we present another, ``less technical'', proof based on a relation between the master function and the map sending a pair of polynomials into the Wronski determinant. Within these frameworks, the number of orbits becomes the intersection number of appropriate special Schubert classes. Application of the Schubert calculus to the sl_p KZ equation is discussed.

Motivation & Objective

• To provide an alternative, less technical proof of the conjecture that the number of orbits of critical points of the $sl_2$ KZ master function equals the dimension of the space of singular vectors.
• To establish a correspondence between critical points of the master function and preimages under the Wronski map in the Grassmannian of 2-planes.
• To use Schubert calculus to compute the intersection number of Schubert classes as an upper bound for the number of critical point orbits.
• To extend the method to $sl_p$ KZ equations using a generalized master function and the Wronski map for $p$-dimensional planes.
• To demonstrate that the upper bound from Schubert calculus matches the dimension of the singular vector space, confirming the Bethe vectors form a basis.

Proposed method

• The master function $\Phi(t)$ is defined on $\mathcal{C}_{k,n}(t,z)$, with critical points corresponding to solutions of the Bethe equations for the $sl_2$ Gaudin model.
• A one-to-one correspondence is established between orbits of critical points and 2-planes in the preimage of the Wronski map for the polynomial $W(x) = \prod_{i=1}^n (x - z_i)^{m_i}$.
• The number of such 2-planes is computed as the intersection number of special Schubert classes in the Grassmannian $Gr(2, \text{Poly}_d)$.
• This intersection number provides an upper bound for the number of critical point orbits, which is shown to equal the dimension of $\text{Sing}_k$.
• The method is extended to $sl_p$ by using a generalized master function related to the Wronski map for $p$-dimensional planes in the Grassmannian.
• The rational curve in the Grassmannian corresponding to the Wronski map is identified via the matrix of derivatives of a monomial row $F(\xi) = (\xi^d, \dots, 1)$.

Experimental results

Research questions

• RQ1 Does the number of orbits of critical points of the $sl_2$ master function equal the dimension of the space of singular vectors in the tensor product of $sl_2$ representations?
• RQ2 Can the number of critical point orbits be computed via Schubert calculus using the Wronski map?
• RQ3 How does the Wronski map relate to the master function of the $sl_p$ KZ equation for tensor products of symmetric powers?
• RQ4 Is the intersection number of Schubert classes a valid upper bound for the number of critical point orbits in the $sl_p$ case?
• RQ5 What rational curves in the Grassmannian correspond to master functions of $sl_p$ KZ equations for other tensor products?

Key findings

• The number of orbits of nondegenerate critical points of the $sl_2$ master function $\Phi(t)$ equals the dimension of the space $\text{Sing}_k$ of singular vectors of weight $|M| - 2k$.
• The upper bound for the number of orbits, derived from the Schubert calculus intersection number, matches the dimension of $\text{Sing}_k$, confirming the conjecture for $sl_2$.
• For the $sl_p$ KZ equation, the number of orbits of critical points of the master function is shown to be equal to the dimension of the corresponding singular vector space $\text{Sing}_{\mathbf{k}}L$.
• The master function for $sl_p$ is linked to the Wronski map for $p$-dimensional planes, and the number of critical point orbits is computed as an intersection number of Schubert classes.
• In the $sl_p$ case with two weights, the number of critical points is 1 when the highest weight is in the tensor product and 0 otherwise, matching the dimension of the singular vector space.
• The Wronski map corresponds to a rational curve in the Grassmannian defined by the matrix of derivatives of the monomial row $F(\xi)$, providing a geometric interpretation of the master function.