# A Lie theoretical construction of a Landau–Ginzburg model without projective mirrors

@article{Ballico2019ALT, title={A Lie theoretical construction of a Landau–Ginzburg model without projective mirrors}, author={Edoardo Ballico and Severin Barmeier and Elizabeth Gasparim and Lino Grama and Luiz A. B. San Martin}, journal={manuscripta mathematica}, year={2019}, volume={158}, pages={85-101} }

We describe the Fukaya–Seidel category of a Landau–Ginzburg model $$\mathrm {LG}(2)$$LG(2) for the semisimple adjoint orbit of $$\mathfrak {sl}(2, {\mathbb {C}})$$sl(2,C). We prove that this category is equivalent to a full triangulated subcategory of the category of coherent sheaves on the second Hirzebruch surface. We show that no projective variety can be mirror to $$\mathrm {LG}(2)$$LG(2), and that this remains so after compactification.

## 8 Citations

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## References

SHOWING 1-10 OF 23 REFERENCES

Classical deformations of local surfaces and their moduli of instantons

- Mathematics
- 2016

We describe the semiuniversal deformation spaces for the noncompact surfaces $Z_k := \operatorname{Tot} (\mathcal O_{\mathbb P^1} (-k))$ and prove that any nontrivial deformation $\mathcal Z_k$ of…

Some Landau--Ginzburg models viewed as rational maps

- Mathematics
- 2016

[GGSM2] showed that height functions give adjoint orbits of semisimple Lie algebras the structure of symplectic Lefschetz fibrations (superpotential of the LG model in the language of mirror…

More about vanishing cycles and mutation

- Mathematics
- 2000

The paper continues the discussion of symplectic aspects of Picard-Lefschetz theory begun in "Vanishing cycles and mutation" (this archive). There we explained how to associate to a suitable…

Mirror symmetry for weighted projective planes and their noncommutative deformations

- Mathematics
- 2004

We study the derived categories of coherent sheaves of weighted projective spaces and their noncommutative deformations, and the derived categories of Lagrangian vanishing cycles of their mirror…

Tilting Bundles on Rational Surfaces and Quasi-Hereditary Algebras

- Mathematics
- 2011

Let $X$ be any rational surface. We construct a tilting bundle $T$ on $X$. Moreover, we can choose $T$ in such way that its endomorphism algebra is quasi-hereditary. In particular, the bounded…

Quivers, Floer cohomology, and braid group actions

- Mathematics
- 2000

We consider the derived categories of modules over a certain family A_m of graded
rings, and Floer cohomology of Lagrangian intersections in the symplectic manifolds which
are the Milnor fibres of…

A Beginner’s Introduction to Fukaya Categories

- Mathematics
- 2014

The goal of these notes is to give a short introduction to Fukaya categories and some of their applications. The first half of the text is devoted to a brief review of Lagrangian Floer (co)homology…

Lagrangian intersection floer theory : anomaly and obstruction

- Mathematics
- 2009

Part I Introduction Review: Floer cohomology The $A_\infty$ algebra associated to a Lagrangian submanifold Homotopy equivalence of $A_\infty$ algebras Homotopy equivalence of $A_\infty$ bimodules…

Classical deformations of noncompact surfaces and their moduli of instantons

- MathematicsJournal of Pure and Applied Algebra
- 2019

Abstract We describe semiuniversal deformation spaces for the noncompact surfaces Z k : = Tot ( O P 1 ( − k ) ) and prove that any nontrivial deformation Z k ( τ ) of Z k is affine. It is known that…

The full exceptional collections of categorical resolutions of curves

- Mathematics
- 2016

Abstract This paper gives a complete answer of the following question: which (singular, projective) curves have a categorical resolution of singularities which admits a full exceptional collection?…