[Paper Review] Homological mirror symmetry for hypertoric varieties II (with an Appendix written jointly with Laurent Côté and Justin Hilburn)

Benjamin Gammage, Michael McBreen+1 more|arXiv (Cornell University)|Mar 19, 2019

Algebraic structures and combinatorial models48 references16 citations

TL;DR

This paper establishes a homological mirror symmetry equivalence for multiplicative hypertoric varieties by matching holomorphic Lagrangian skeleta on the A-model with noncommutative resolutions on the B-model, using hyperkähler t-structures to clarify SYZ duality. The key result is a derived equivalence of categories that extends to monodromy autoequivalences via perverse schobers, providing a prototype for K-theoretic Coulomb branch mirror symmetry.

ABSTRACT

In this paper, we prove a homological mirror symmetry equivalence for pairs of multiplicative hypertoric varieties, and we calculate monodromy autoequivalences of these categories by promoting our result to an equivalence of perverse schobers. We prove our equivalence by matching holomorphic Lagrangian skeleta, on the A-model side, with non-commutative resolutions on the B-model side. The hyperkähler geometry of these spaces provides each category with a natural t-structure, which helps clarify SYZ duality in a hyperkähler context. Our results are a prototype for mirror symmetry statements relating pairs of K-theoretic Coulomb branches.

Motivation & Objective

• To establish a homological mirror symmetry equivalence for pairs of multiplicative hypertoric varieties, extending previous results on additive hypertoric varieties.
• To clarify the role of hyperkähler geometry in SYZ duality by equipping both A- and B-model categories with natural t-structures.
• To calculate monodromy autoequivalences of the derived categories by promoting the mirror symmetry equivalence to an equivalence of perverse schobers.
• To provide a prototype for mirror symmetry statements relating K-theoretic Coulomb branches in 3d and 4d supersymmetric gauge theories.
• To formalize the duality between complex moment map parameters β and Kähler-like parameters α = γ·exp(δ), with γ a B-field in H²(U; R/Z).

Proposed method

• Construct multiplicative hypertoric varieties as hyperhamiltonian reductions of (T*Cⁿ)◦ by unimodular subtori T ⊂ (C×)ⁿ.
• Use holomorphic Lagrangian skeleta on the A-model side to model the Fukaya category, leveraging the integrable system structure and nodal torus fibers.
• On the B-model side, realize the mirror as a noncommutative resolution of the algebraic variety, using the parameter γ ∈ T∨R to define a class in H²(U; R/Z) via the Kirwan map.
• Establish a derived equivalence between the Fukaya category of the A-model and the derived category of coherent sheaves on the B-model via matching of t-structures and microlocal sheaf theory.
• Apply the theory of perverse schobers to lift the derived equivalence to an equivalence of categories that captures monodromy actions, particularly around singular fibers.
• Use microlocal sheaf theory, including specialization functors and Fourier-Sato transforms, to analyze singular supports and prove that the cohomology of the sheaf H is isomorphic to its stalk at the chamber ∆.

Experimental results

Research questions

• RQ1 How can homological mirror symmetry be extended from additive to multiplicative hypertoric varieties?
• RQ2 What is the precise relationship between the complex moment map parameter β and the Kähler-like parameter α = γ·exp(δ) under mirror symmetry?
• RQ3 How do monodromy autoequivalences arise in the derived category of a multiplicative hypertoric variety, and how can they be captured categorically?
• RQ4 In what way does the hyperkähler structure endow both A- and B-model categories with a natural t-structure, and how does this clarify SYZ duality?
• RQ5 Can the mirror symmetry equivalence be promoted to an equivalence of perverse schobers to encode monodromy data?

Key findings

• The paper proves a homological mirror symmetry equivalence between the Fukaya category of a multiplicative hypertoric variety U(β,α) and the derived category of coherent sheaves on its mirror, which is a noncommutative resolution depending on the B-field γ.
• The equivalence is established by matching holomorphic Lagrangian skeleta on the A-model with noncommutative resolutions on the B-model, using the hyperkähler t-structure to align the categories.
• Monodromy autoequivalences are calculated by lifting the mirror symmetry equivalence to an equivalence of perverse schobers, which encode the action of monodromy around singular fibers.
• The singular support of the sheaf H is shown to lie in the closure of ∆(Λ), a union of polar dual cones, via microlocal analysis and specialization functors.
• The cohomology H∗(V, H) is isomorphic to the stalk H∆, which follows from the fact that H recedes from the chamber ∆ and the microlocal Morse lemma applies to the restriction map.
• The parameter α = γ·exp(δ) governs the complexified symplectic form on the A-model, while β controls the algebraic structure of the B-model, and they are exchanged under mirror symmetry.