Floer Homology: Invariance

Audin, Michèle; Damian, Mihai

doi:10.1007/978-1-4471-5496-9_11

Cited by 5 publications

(6 citation statements)

References 17 publications

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“…This observation allows us to reinterpret variational results of the 1st Schrödinger paper in terms of the gauge-theoretic formalism developed by Floer. A quick introduction to Floer's theory is provided in [71] while the detailed account can be found, for instance, in [72]. Even though Floer theory studies a multitude of closed orbits on symplectic manifolds classically, the extremely sophisticated computational methods developed by Floer exactly parallel those used in nonperturbative (instanton) treatments of the Yang-Mills theory.…”

Section: Progressive Wave Solution Methods By Friedlander From Mechan...mentioning

confidence: 99%

Huygens triviality of the time-independent Schrödinger equation. Applications to atomic and high energy physics

Kholodenko,

Kauffman

2017

Preprint

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Huygens triviality-a concept invented by Jacques Hadamard-describes an equivalence class connecting those 2nd order partial differential equations which are transformable into the wave equation. In this work it is demonstrated, that the Schrödinger equation with the time-independent Hamiltonian belongs to such an equivalence class. The wave equation is the equation for which Huygens' principle (HP) holds. The HP was a subject of confusion in both physics and mathematics literature for a long time. Not surprisingly, the role of this principle was obscured from the beginnings of quantum mechanics causing some theoretical and experimental misunderstandings. The purpose of this work is to bring the full clarity into this topic. By doing so, we obtained a large amount of new results related to uses of Lie sphere geometry, of twistors, of Dupin cyclides, of null electromagnetic fields, of AdS-CFT correspondence, of Penrose limits, of geometric algebra, etc. in physical problems ranging from the atomic to high energy physics and cosmology.

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Section: Progressive Wave Solution Methods By Friedlander From Mechan...mentioning

confidence: 99%

Huygens triviality of the time-independent Schrödinger equation. Applications to atomic and high energy physics

Kholodenko,

Kauffman

2017

Preprint

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“…We begin this section by recalling the data of Hamiltonian Floer theory and symplectic cohomology, fixing conventions along the way. We refer the reader to [4] and [13] for an introduction to Floer theory, and we refer to [20] for details on symplectic cohomology. We then introduce the notion of reduced symplectic cohomology, recently defined and studied by the author in [24], as well as in the independent works of Groman [11], McLean [14], and Varolgunes [22].…”

Section: Floer Theorymentioning

confidence: 99%

“…A closed-string mirror symmetry statement is proved for some cases in [19], and in the full monotone case in [18]. Namely, if E is monotone, then there is an isomorphism (4) SH * (E) Jac(W).…”

Section: Introductionmentioning

confidence: 99%

The quantitative nature of reduced Floer theory

Venkatesh¹

2020

Preprint

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We study the reduced symplectic cohomology of disk subbundles in negative symplectic line bundles. We show that this cohomology theory "sees" the spectrum of a quantum action on quantum cohomology. Precisely, quantum cohomology decomposes into generalized eigenspaces of the action of the first Chern class by quantum cup product. The reduced symplectic cohomology of a disk bundle of radius R sees all eigenspaces whose eigenvalues have size less than R, up to rescaling by a fixed constant. Similarly, we show that the reduced symplectic cohomology of an annulus subbundle between radii R 1 and R 2 captures all eigenspaces whose eigenvalues have size between R 1 and R 2 , up to a rescaling. We show how local closed-string mirror symmetry statements follow from these computations.

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“…for all x, y ∈ CF (O M , Υ : H) (see [15]). We will consider a special class of Morse functions, as in [4] or [18].…”

Section: Pss Isomorphismmentioning

confidence: 99%

Spectral invariants in Lagrangian Floer homology of open subset

Katić¹,

Milinković²,

Nikolić³

2017

Differential Geometry and its Applications

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We define and investigate spectral invariants for Floer homology HF (H, U : M ) of an open subset U ⊂ M in T * M , defined by Kasturirangan and Oh as a direct limit of Floer homologies of approximations. We define a module structure product on HF (H, U : M ) and prove the triangle inequality for invariants with respect to this product. We also prove the continuity of these invariants and compare them with spectral invariants for periodic orbits case in T * M .

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Floer Homology: Invariance

Cited by 5 publications

References 17 publications

Huygens triviality of the time-independent Schrödinger equation. Applications to atomic and high energy physics

Huygens triviality of the time-independent Schrödinger equation. Applications to atomic and high energy physics

The quantitative nature of reduced Floer theory

Spectral invariants in Lagrangian Floer homology of open subset

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