Infinitesimally tight Lagrangian orbits
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Mathematische Zeitschrift
Infinitesimally tight Lagrangian orbits Elizabeth Gasparim1 · Luiz A. B. San Martin2 · Fabricio Valencia3 Received: 16 March 2019 / Accepted: 24 June 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We describe isotropic orbits for the restricted action of a subgroup of a Lie group acting on a symplectic manifold by Hamiltonian symplectomorphisms and admitting an Ad*-equivariant moment map. We obtain examples of Lagrangian orbits of complex flag manifolds, of cotangent bundles of orthogonal Lie groups, and of products of flags. We introduce the notion of infinitesimally tight and study the intersection theory of such Lagrangian orbits, giving many examples. Keywords Lagrangian orbits · Product of flags · Infinitesimally tight · Orthogonal Lie group Mathematics Subject Classification 53D12 · 14M15
1 Introduction Let (M, ω) be a connected symplectic manifold, G a Lie group with Lie algebra g, and L a Lie subgroup of G. Assume that there exists a Hamiltonian action of G on M which admits an Ad∗ -equivariant moment map μ : M → g∗ . The purpose of this paper is to study those orbits L x with x ∈ M that are Lagrangian submanifolds of (M, ω), or more generally, isotropic submanifolds. We also discuss some essential features of the intersection theory of such Lagrangian orbits, namely the concepts of locally tight and infinitesimally tight Lagrangians. The famous Arnold–Givental conjecture, proved in many cases, predicts that the number of intersection points of a Lagrangian L and its image ϕ(L) by the flow of a Hamiltonian vector field can be estimated from below by the sum of its Z2 Betti numbers:
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Fabricio Valencia [email protected] Elizabeth Gasparim [email protected] Luiz A. B. San Martin [email protected]
1
Depto. Matemáticas, Univ. Católica del Norte, Antofagasta, Chile
2
Imecc-Unicamp, Depto. de Matemática, Campinas, Brazil
3
Inst. Matemáticas, Univ. de Antioquia, Medellín, Colombia
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|L ∩ ϕ(L)| ≥
bk (L; Z2 ).
The concepts of tightness address those Lagrangians which attain the lower bound, and are therefore of general interest in symplectic geometry. For us additional motivation to study Lagrangians and their intersection theory comes from questions related to the Homological Mirror Symmetry conjecture and in particular from concepts of objects and morphisms in the so called Fukaya–Seidel categories, which are generated by Lagrangian vanishing cycles (and their thimbles) with prescribed behavior inside of symplectic fibrations. In [7, Thm. 2.2] it was shown that the usual height function from Lie theory gives adjoint orbits of semisimple Lie groups the structure of symplectic Lefschetz fibrations. These give rise to what is known as Landau–Ginzburg (LG) models. We wish to study the Fukaya–Seidel category of these LG models. Finding Lagrangian submanifolds and understanding their intersection theory inside a compactification is an initial tool to investigate possible thimbles. The Fukaya–Seidel category of th
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