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Compactifying the Parameter Space for the Quantum Multiplication for Hypertoric Varieties

Jeremy Peters

TL;DR

The paper develops a rigorous framework for compactifying the parameter space of quantum multiplication on hypertoric varieties. It first encodes the quantum product via a holonomy-Lie-algebra structure and proves injectivity of the key map that connects the modified holonomy algebra to endomorphisms acting on equivariant cohomology. It then constructs a deConcini–Gaiffi–style compactification of toric arrangements by passing to a toric variety $X_\Sigma$, blowing up layers to obtain $Y_\Phi$, and finally extending the quantum-parameter map to $Q_\Phi: Y_\Phi \to Gr(k, \mathfrak{u}_\Phi^1)$; this extension also extends to the boundary divisors of $X_\Sigma$. The results provide a global, well-behaved parameter space for hypertoric quantum multiplication and connect the quantum deformation to stable-basis techniques and toric/hyperplane-arrangement geometry, opening avenues for studying the full quantum cohomology algebra in this setting.

Abstract

In this paper, we will be studying the parameter space for the quantum multiplication for hypertoric varieties. The operation of quantum multiplication for hypertoric varieties has an explicit formulation which is given by McBreen and Shenfeld. In particular, this multiplication depends on a parameter which lives in the complement of a toric arrangement. Following a paper of deConcini and Gaiffi, I will define a compactification of this parameter space and show how the quantum multiplication can be extended to this compactification.

Compactifying the Parameter Space for the Quantum Multiplication for Hypertoric Varieties

TL;DR

The paper develops a rigorous framework for compactifying the parameter space of quantum multiplication on hypertoric varieties. It first encodes the quantum product via a holonomy-Lie-algebra structure and proves injectivity of the key map that connects the modified holonomy algebra to endomorphisms acting on equivariant cohomology. It then constructs a deConcini–Gaiffi–style compactification of toric arrangements by passing to a toric variety , blowing up layers to obtain , and finally extending the quantum-parameter map to ; this extension also extends to the boundary divisors of . The results provide a global, well-behaved parameter space for hypertoric quantum multiplication and connect the quantum deformation to stable-basis techniques and toric/hyperplane-arrangement geometry, opening avenues for studying the full quantum cohomology algebra in this setting.

Abstract

In this paper, we will be studying the parameter space for the quantum multiplication for hypertoric varieties. The operation of quantum multiplication for hypertoric varieties has an explicit formulation which is given by McBreen and Shenfeld. In particular, this multiplication depends on a parameter which lives in the complement of a toric arrangement. Following a paper of deConcini and Gaiffi, I will define a compactification of this parameter space and show how the quantum multiplication can be extended to this compactification.
Paper Structure (11 sections, 50 theorems, 285 equations)

This paper contains 11 sections, 50 theorems, 285 equations.

Table of Contents

  1. Introduction
  2. Algebraic Structure of Quantum Multiplication
  3. Hypertoric Varieties and Circuits
  4. Defining the Algebraic Structure of the Quantum Multiplication
  5. Injectivity of $\gamma$
  6. Codomain of $Q$
  7. The deConcini--Gaiffi Compactification
  8. The Toric Variety
  9. Compactifying Toric Arrangements
  10. Extending to a Compactification of the Torus $T^\text{reg}$
  11. Extending to the Boundary Divisors of $X_\Sigma$

Key Result

Theorem 1.1

Let $X = T^*\mathbb{C}^n///_{0,\chi}T^k$ be a Hypertoric variety with an action of $G = T^d\times\mathbb{C}^\mathsf{x}$, where $d = n-k$ and define the following map: Then $\gamma$ is well-defined on the level of Lie algebras and is injective.

Theorems & Definitions (165)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Remark 1
  • ...and 155 more