Questions tagged [sg.symplectic-geometry]
Hamiltonian systems, symplectic flows, classical integrable systems
496
questions with no upvoted or accepted answers
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Are there connected closed 4-manifolds admitting a regular Almost Lagrangian distribution, and which are not Lorentzian?
In the category of real differential manifolds, connected (of $ C ^ {\infty} $ class in the sequel), closed of dimension 4, is there any manifold admitting a regular Almost Lagrangian distribution and ...
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Lagragian floer homology vs homology of $\Omega(L_0,L_1)$
I'm very new to this subject, so apologies for a very naive question and probably many mistakes. Let $M$ be some compact sympletic manifold with $L_0,L_1$ Lagrangian submanifolds which intersects ...
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Area preserving diffeomorphisms of surfaces with only hyperbolic periodic points
This is a (probably very naive question) about area-preserving maps of surfaces.
Does there exist a Hamiltonian diffeomorphism
$$ f: \Sigma \to \Sigma $$
of a symplectic surface (real dimension $2$), ...
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166
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Derivative of a pullback by a $2$-parameter family
Let $(M,\omega = d\alpha)$ be an exact symplectic manifold and $\gamma \in C^\infty(I,M)$ a path in $M$. Given $X_1,X_2 \in \mathfrak{X}(\gamma)$ I would like to compute
$$\frac{\partial^2}{\partial\...
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Mirror symmetry for $C^*$
The Liouville manifold $T^*S^1$ is said to be "mirror" to the complex variety $C^*$. (see for instance lecture 7 here: http://math.columbia.edu/~topology/Eilenberg_lectures_fall_2016)
This is ...
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Legendrian surgery and invertible elements in zeroth degree symplectic cohomology
Is there anything known about the relation between Legendrian handle attachment and invertible elements in $\mathit{SH}^0(M)$?
As the simplest interesting case, take $M_0$ to be the cotangent bundle $...
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341
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Lagrangian subgroup of a nonabelian Lie group
My post here concerns the concept of Lagrangian subgroup for a non-abelian Lie group, such as a semi-simple non-abelian Lie group for gauge theory.
See a previous post for other background ...
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symplectic sum of two copies of $Bl_{p}(\mathbb{CP}^{2})$
Let $M^{4}= \mathbb{P}(\mathcal{O} \oplus \mathcal{O}(1))$ and $\omega$ the symplectic form on $M^{4}$ given by the anticanonical polarisation.
Suppose we form a symplectic (Gompf) sum of two copies ...
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Is there any known relationship between sutured contact homology and Legendrian contact homology?
On one hand, Colin-Ghiggini-Honda-Hutchings' construction (https://arxiv.org/abs/1004.2942) provides an invariant of a Legendrian $L$ in a closed contact manifold $(M,\xi)$ via the sutured contact ...
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GSO projection and $H^d(M, \mathbb{Z}_2)$
This follows up the comment which suggests that asking the later 2nd part of subquestion in "GSO (Gliozzi-Scherk-Olive) projection and its Mathematics" as a new different question
GSO (...
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Is there a simply connected contact manifold, "non-exactly" fillable, cappable, such that the whole thing is symplectically aspherical?
Is there an example of a simply connected contact manifold W, with a non exact symplectic filling $M_1$, (that is, $M_1$ is a symplectic manifold, with contact boundary $W$ and a Liouville vector ...
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Isn't the quantomorphism group really just the "WKB-quantomorphism" group?
Introduction
In his second-most upvoted post, called "Why quantum mechanics?" (second only to his post on fibre bundles & gauge theory) in the physics SE community, Urs Schreiber, in the setting ...
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Lagrangian foliation for a holomorphic symplectic manifold
I am interested in gathering as many examples as possible for Lagrangian foliations of holomorphically symplectic manifolds $(X, \omega)$, where $X$ is a $2n$-dimensional complex manifold equipped ...
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Does there exist a candidate for 'holomorphic' instanton Floer homology?
The Euler characteristic of instanton Floer homology agrees with the Casson invariant. Thomas introduced the notion of holomorphic Casson invariant, defined using the holomorphic Chern-Simons ...
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The structure of Banach manifolds in symplectic geometry
Let $M$ be a symplectic manifold, and let $L_0$ and $L_1$ be Lagrangian submanifolds which transverse to each other. In Floer theory, we need to consider a Banach manifold $\mathcal B$ of maps $u:\...
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Darboux's Theorem
I read proofs of Darboux's theorem that a symplectic form $\omega$ on M locally is symplectomorphic to a standard symplectic form $\omega_{0}= \sum dp \wedge dq$ in Rolf Berndt's An Introduction to ...
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What is the fundamental group of Kontsevich's space of stable maps?
... at least in the case where the target is a rationally connected variety.
This question is a follow-up to question
Constructing embedded families of curves with general moduli
and Jason Starr's ...
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Virasoro constraints for parametrized GW invariants
Gromov-Witten invariants count isolated stable maps from Riemann surfaces to a fixed symplectic manifold $(M,\omega)$ subject to some incidence conditions. If we instead replace the target manifold ...
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connectedness of moduli space of Calabi-Yau 3-folds by symplectic surgery theory
"Motto" Moduli space of Calabi-Yau varieties can be connected by using Symplectic surgery theory.
Miles Reid’s Fantasy:“There is only one Calabi-Yau space”
i.e "All CY connected through conifold ...
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The Hofer topology
Let $(M,\omega)$ be a symplectic manifold and let $\operatorname{Ham}(M,\omega)$ be the group of compactly supported Hamiltonian diffeomorphisms equipped with the Hofer metric. I would like to collect ...
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Deformation quantization of Poisson bracket without star-product
Kontsevich's formality theorem implies in particular that star-products on a $C^\infty$-manifold $M$,
$$f\star g = fg + \sum_{k\geq1} \hbar^k B_k(f,g),\qquad f,g\in C^\infty(M),$$ where $B_k$ are ...
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Symplectic leaves in positive characteristic
I am currently considering a family of filtered algebras over an algebraically closed field of positive characteristic with the property that the associated graded algebra is a finitely generated ...
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Grading in Lagrangian Floer homology
What are the conditions on a symplectic manifold (M,w) and on a Lagrangian submanifold L so that Lagrangian Floer complex CF(L, f(L)) is Z-graded? Here f is a compactly supported Hamiltonian isotopy. ...
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The autonomous diameter of the group of Hamiltonian diffeomorphisms of the standard symplectic space
The autonomous norm of a Hamiltonian diffeomorphism $h$ of a symplectic manifold $(M,\omega)$ is the smallest number $n\in \mathbf N$ such that $h=a_1\dots a_n$, where $a_i$ are autonomous ...
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Interpolating from a Hard Lefschetz class to a Kaehler class
Let $X$ be a compact smooth manifold that admits symplectic and Kaehler structures.
There is a paper by Ugarte, Rudyak, Tralle, and Ibanez, showing how the Lefschetz rank can vary along a path of ...
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997
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Complex but not Symplectic
For every even integer $n>2$, does there exist a smooth $n$ dimensional manifold $M$ that admits a complex structure but not a symplectic one?
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Symplectic sum and Symplectic cut
The symplectic sum of Gompf and the symplectic cut of Lerman are known to be inverse of each other, in the sense that if you apply one of these first and the other one afterward, you obtain the ...
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Classification of Compact Symplectic Homogeneous Spaces
Let $M=G/H$ be a compact homogeneous space, $G$ a compact Lie Group and $H$ a closed subgroup. Is there some classification, akin to the Kaehler case, for which such manifolds admit a symplectic ...
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Sophus Lie on the symplectic foliation theorem
Given a Poisson manifold $(P,\{\cdot,\cdot\})$, its characteristic distribution $\mathcal C$ is the singular tangent distribution on $M$ generated by the Hamiltonian vector fields,i.e.
$$\mathcal C=\...
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Lie-infinity structure in Lagrangian Floer theory ?
Is there (besides the A-infinity structure) also a L-infinity structure in Lagrangian Floer theory (forming together a G-infinity structure) - like in Hochschild cohomology ?
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Quantum dynamics on varieties: asymptotic quantum trace distance on SHL varieties
The Question Asked
Definition: the Second-Hand Lion trace distance $D_k$
Let $\mathcal{M}^{(kk)}_r$ be the set of $k\times k$ complex matrices of rank $\le r$ having unit trace norm. Then the ...
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Degenerate moduli spaces in Floer homology
Let $(W,\omega)$ be a closed symplectially aspherical symplectic
manifold, and fix a Hamiltonian $H\in C^{\infty}(W\times S^{1};\mathbb{R})$
and a compatible almost complex structure $J$ on $W$. Given ...
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603
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Almost-Kahler Einstein four manifolds
Are the odd-degree Betti numbers of a compact Almost-Kahler Einstein four manifold necessarily even ?
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Are there cohomology classes on a hyperkähler manifolds which pull back to the Stiefel-Whitney classes on every Lagrangian submanifold?
This is a bit of a stab in the dark but I was wondering if anyone has defined cohomology classes on a hyperkähler manifold which pull back to the Stiefel-Whitney classes on any submanifold which is ...
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About a non-degeneracy of Hodge-Riemann form..
Let $(M^{2n},\omega)$ be a closed connected symplectic manifold and let
$HR : H^2(M;R) \times H^2(M;R) \rightarrow R$ be the Hodge-Riemann form defined by
$HR(\alpha,\beta) = \int_M \alpha \beta \...
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symplectic matrices
If $( A B | C D)$ is a symplectic matrix with entries in the finite field with two elements, is it necessarily the case that
$\sum_{i,j,k} a_{ij}b_{ij}c_{ik}d_{ik} = 0$?
This arose in connection ...
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Pre-Symplectic Mapping
I have been studying symplectic integrators and their pre-symplectic extensions for dissipative systems. According to França, Jordan, and Vidal - On dissipative symplectic integration with ...
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Coulomb branches which are not of cotangent type
To each $3d \, N=4$ supersymmetric quantum field theory $\mathcal{T}$, there is a related space called the Coulomb branch of this theory, $\mathcal{M}_C(\mathcal{T})$ (it is a piece of the moduli ...
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A couple of questions about the moduli space of annuli with some marked points on the boundary components
I'm trying to work out an answer for my previous question and I'm stuck with the following issue:
In the paper Deformations of Bordered Riemann surfaces and associahedral polytopes by Devadoss, Heath ...
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Examples of non-equivariant momentum maps
What are examples of non-equivariant momentum maps?
Off the top of my hat, I know about the following examples:
the action of translations of a symplectic vector space (yielding the Heisenberg group ...
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132
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Deformation quantization of an integrable system
What is known about lifting n Poisson commuting functions on a 2n-dimensional symplectic manifolds (say R^2n) to Moyal-Weyl commuting functions?
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What are known properties of the boundary curves of J-holomorphic curve with boundary
Suppose $\Sigma$ be a punctured Riemann surface with punctured boundary, and $(M, J)$ be a $2n$-manifold with almost complex structure $J$. Let $L$ be a totally real submanifolds of $M$ in smooth ...
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The homotopy type of the space of symplectic structures
While reading the book Introduction to the $h$-Principle by Y. Eliashberg and N. Mishachev, I noticed that the authors state, at the end of section 9.1.A, that the space of all symplectic structures ...
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Overtwisted contact structures on $S^3$
All the isotopy classes of overtwisted contact structures are classified by the Hopf invariant. Are any of these contact structures contactomorphic?
Suppose $d_{3}(\xi_{n}) = n$, then my guess is that ...
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Size of Hilbert space in geometric quantization from index theorem
In these notes on geometric quantization by Nair, on page 24, the Bohr-Sommerfeld rule in quantum mechanics is interpreted in terms of the Atiyah-Singer index theorem.
To be precise, the polarization ...
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Writting the Floer map in local coordinates using the exponential chart
Following Salamon's Notes in Floer Homology , consider the Floer equation
$$\mathcal{F}(u):=\partial_su+J_t(\partial_tu+\nabla H_t(u))=0$$
Then we can write in local coordinates
$$\mathcal \Phi_u^{-1}(...
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Some clarifications on the PSS isomorphism in Hamiltonian Floer cohomology
I'm looking for some help in understanding the PSS isomorphism map in the context of Hamiltonian Floer cohomology and Morse cohomology with universal Novikov coefficients $\Lambda_{\omega}$ (à la ...
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Relationship between exact symplectomorphisms and Hamiltonian diffeomorphisms (in the context of Milnor fibers)
Here is a preamble/setup. Suppose we have a symplectic manifold $(M,d\lambda)$ with an exact symplectic form. By Stokes theorem, $M$ must have nonempty boundary. An exact symplectomorphism $\phi:M \to ...
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Choice of almost complex structure in Seidel's Symplectic Floer Homology of a Dehn twist
I'm looking for a clarification of a construction done in Seidel's Symplectic Floer Homology of a Dehn twist: I don't get why his choice of almost complex structure on $\Sigma$ is a valid one for ...
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Example of a computation of the volume of a subvariety in projective space $\mathbb{P}^n$
Let us consider the projective space $\mathbb{P}^n$ with the standard Fubini Study metric.
I searched all over the internet but I can't find an example of a calculation of the volume for a projective ...