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Homological Berglund-Hübsch-Henningson mirror symmetry for curve singularities

Matthew Habermann

TL;DR

The paper proves homological Berglund–Hübsch–Henningson mirror symmetry for invertible curve singularities with equivariant A–models, confirming Futaki–Ueda’s conjecture by lifting the transpose potential to a crepant resolution and matching B–model matrix factorisations with an A–model Fukaya–Seidel category. It introduces a tilting object on the B–model side and computes its endomorphism algebra as a quiver, establishing generation via Polishchuk–Vaintrob results. The A–model is built using resonant Morsifications that respect dual symmetry, including a non–exact total space framework and careful vanishing–cycle analysis, yielding a generated directed category that matches the B–model quivers. The work also develops a deformation theory via Hochschild cohomology, showing all Γ–equivariant deformations correspond to B–fields on the A–side, thereby realizing a LG Dubrovin–type semisimplicity criterion in non–maximally graded settings and extending mirror symmetry to deformed categories with practical implications for FJRW theory.

Abstract

In this article, we establish homological Berglund--Hübsch mirror symmetry for curve singularities where the A--model incorporates equivariance, otherwise known as homological Berglund--Hübsch--Henningson mirror symmetry, including for certain deformations of categories. More precisely, we prove a conjecture of Futaki and Ueda in arXiv:1004.0078 which posits that the equivariance in the A-model can be incorporated by pulling back the superpotential to the total space of the corresponding crepant resolution. Along the way, we show that the B--model category of matrix factorisations has a tilting object whose length is the dimension of the state space of the FJRW A--model, a result which might be of independent interest for its implications in the Landau--Ginzburg analogue of Dubrovin's conjecture.

Homological Berglund-Hübsch-Henningson mirror symmetry for curve singularities

TL;DR

The paper proves homological Berglund–Hübsch–Henningson mirror symmetry for invertible curve singularities with equivariant A–models, confirming Futaki–Ueda’s conjecture by lifting the transpose potential to a crepant resolution and matching B–model matrix factorisations with an A–model Fukaya–Seidel category. It introduces a tilting object on the B–model side and computes its endomorphism algebra as a quiver, establishing generation via Polishchuk–Vaintrob results. The A–model is built using resonant Morsifications that respect dual symmetry, including a non–exact total space framework and careful vanishing–cycle analysis, yielding a generated directed category that matches the B–model quivers. The work also develops a deformation theory via Hochschild cohomology, showing all Γ–equivariant deformations correspond to B–fields on the A–side, thereby realizing a LG Dubrovin–type semisimplicity criterion in non–maximally graded settings and extending mirror symmetry to deformed categories with practical implications for FJRW theory.

Abstract

In this article, we establish homological Berglund--Hübsch mirror symmetry for curve singularities where the A--model incorporates equivariance, otherwise known as homological Berglund--Hübsch--Henningson mirror symmetry, including for certain deformations of categories. More precisely, we prove a conjecture of Futaki and Ueda in arXiv:1004.0078 which posits that the equivariance in the A-model can be incorporated by pulling back the superpotential to the total space of the corresponding crepant resolution. Along the way, we show that the B--model category of matrix factorisations has a tilting object whose length is the dimension of the state space of the FJRW A--model, a result which might be of independent interest for its implications in the Landau--Ginzburg analogue of Dubrovin's conjecture.
Paper Structure (38 sections, 26 theorems, 162 equations, 12 figures)

This paper contains 38 sections, 26 theorems, 162 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Strategy of proof
  3. Structure of the paper
  4. Acknowledgements
  5. Loop B-model
  6. Recollections on maximal Cohen--Macaulay modules
  7. Graded matrix factorisations for loop polynomials
  8. Morphisms between the ${{}^{}K_{x}}$'s, ${{}^{}K_{y}}$'s and ${{}^{}K_{w_r}}$'s
  9. Morphisms between the ${{}^{}K_{w_r}}$'s and ${{}^{}K_{0}}$'s
  10. Morphisms between ${{}^{}K_{x}}$'s and ${{}^{}K_{y}}$'s and ${{}^{}K_{0}}$'s
  11. Morphisms between ${{}^{}K_{0}}$'s
  12. The total endomorphism algebra of the basic objects.
  13. Generation
  14. Loop A-model
  15. The crepant resolution of the $A_{\ell-1}$ singularity
  16. ...and 23 more sections

Key Result

Theorem 1

FUHomologicalBHHMSConjecture holds for all invertible curve singularities and any admissible symmetry group. Moreover, for any $\Gamma$-equivariant deformation $\mathbf{w}_{\vec{\varepsilon}}$ of $\mathbf{w}$ which has an isolated singularity at the origin, there exists a B--field $B\in H^2(\widetil

Figures (12)

  • Figure 1: The quiver describing the category $\mathcal{B}$ for loop polynomials. There are $\ell$ blocks of size $\frac{q-1}{\ell}\times\frac{p-1}{\ell}$.
  • Figure 2: Quiver corresponding to an alternative collection of $I_{i,j}$ for $\ell=2$. Whilst we have considered different objects, it is clear that the resulting category is the same, since the quiver is manifestly a rearrangement of \ref{['LoopEndAlgebra']} for $\ell=2$.
  • Figure 3: Quiver corresponding to $x^py+y^px$.
  • Figure 4: Sketch of the fibre of $\widecheck{\widetilde{\mathbf{w}}}_\varepsilon$ above the origin. This does not represent any specific polynomial, and is only meant to convey the general shape. See \ref{['fig:loopsingularfibrespecificexample']} for a specific example.
  • Figure 5: Red Lagrangian (alternating dashes and dots) corresponds to the real vanishing cycle. Blue (dashed) Lagrangian is another vanishing cycle -- away from the neck regions in the exceptional locus, it is just the shift of the real vanishing cycle by a fixed argument. Light dashes indicate the Lagrangian is on the back side of the surface.
  • ...and 7 more figures

Theorems & Definitions (58)

  • Example 1.1
  • Conjecture 1: UedaSimpleTakahashiLekiliUeda
  • Conjecture 2: FutakiUedaD
  • Theorem 1
  • Remark 1.2
  • Remark 1.3
  • Corollary 1
  • Conjecture 3: Landau--Ginzburg Dubrovin conjecture
  • Remark 2.1
  • Remark 2.2
  • ...and 48 more