[Paper Review] The moment graph for Bott-Samelson varieties and applications to quantum cohomology

Camron Withrow|arXiv (Cornell University)|Aug 28, 2018

Algebraic Geometry and Number Theory11 references4 citations

TL;DR

This paper presents a complete description of the moment graph for Bott-Samelson varieties in arbitrary Lie type, using curve neighborhoods and moduli space computations to derive a presentation of the small quantum cohomology ring for a specific Type A Bott-Samelson variety $Z = Z(\alpha_1, \alpha_2, \alpha_1)$. It proves the conjecture $\mathcal{O}$ of Galkin, Golyshev, and Iritani for this variety, confirming its eigenvalue structure under the first Chern class operator.

ABSTRACT

We give a description of the moment graph for Bott-Samelson varieties in arbitrary Lie type. We use this, along with curve neighborhoods and explicit moduli space computations, to compute a presentation for the small quantum cohomology ring of a particular Bott-Samelson variety in Type $A$. We also show the conjecture $\mathcal{O}$ of Galkin, Golyshev, and Iritani holds for that Bott-Samelson variety.

Motivation & Objective

• To provide a complete description of the moment graph for Bott-Samelson varieties in arbitrary Lie type.
• To compute a presentation of the small quantum cohomology ring for a specific Bott-Samelson variety in Type $A_2$.
• To verify Conjecture $\mathcal{O}$ of Galkin, Golyshev, and Iritani for this variety.
• To establish a framework for studying quantum cohomology in non-convex, non-toric varieties using torus actions and stable maps.

Proposed method

• Constructs the moment graph using $T$-fixed points and $T$-stable curves, parameterized by binary sequences $\varepsilon \in \{0,1\}^n$.
• Applies an inductive characterization of $T$-stable curves based on the first differing index in binary tuples.
• Uses the moduli space of stable maps $\overline{M}_{0,1}(Z,\beta)$ to compute Gromov-Witten invariants via fibered Cartesian diagrams.
• Employs Chevalley matrices to encode quantum product relations and derive the quantum cohomology presentation.
• Computes the matrix of the first Chern class operator $\hat{c}_1$ by specializing quantum parameters to 1.
• Verifies Conjecture $\mathcal{O}$ by analyzing eigenvalues of $\hat{c}_1$, confirming the maximal modulus eigenvalue is unique and roots of unity structure holds.

Experimental results

Research questions

• RQ1 How can the moment graph of a Bott-Samelson variety be fully described in arbitrary Lie type?
• RQ2 What is the explicit presentation of the small quantum cohomology ring for the Bott-Samelson variety $Z(\alpha_1, \alpha_2, \alpha_1)$ in Type $A_2$?
• RQ3 Do the Gromov-Witten invariants required for the quantum cohomology computation admit a geometric interpretation via moduli spaces of stable maps?
• RQ4 Does Conjecture $\mathcal{O}$, relating eigenvalues of the first Chern class operator to the Fano index, hold for this non-toric, non-convex variety?
• RQ5 Can the Giambelli formula be explicitly computed in the quantum cohomology of this Bott-Samelson variety?

Key findings

• The small quantum cohomology ring $QH^*(Z)$ for $Z = Z(\alpha_1, \alpha_2, \alpha_1)$ is generated by $\sigma_{100}, \sigma_{010}, \sigma_{001}, q_1, q_2, q_3$ with explicit relations derived from Chevalley matrices.
• The Gromov-Witten invariant $I_{\beta_1}([Z_{001}]) = y_3 = 1$, computed via an isomorphism of moduli spaces $\overline{M}_{0,1}(Z,\beta_1) \to \overline{M}_{0,1}(Z',[Z'_{01}])$.
• The Giambelli formulae express all Schubert classes $\sigma_{110}, \sigma_{101}, \sigma_{011}, \sigma_{111}$ as polynomials in the algebra generators with quantum corrections.
• The quantum cohomology ring is associative, as verified by the consistency of the derived relations and Giambelli formulae.
• The matrix of the first Chern class operator $\hat{c}_1$ has eight distinct eigenvalues, with one real eigenvalue of maximal modulus, and all other eigenvalues strictly smaller in modulus.
• Conjecture $\mathcal{O}$ holds for $Z = Z(\alpha_1, \alpha_2, \alpha_1)$: the maximal eigenvalue is unique and all eigenvalues of equal modulus are roots of unity scaled by the Fano index.