Questions tagged [sg.symplectic-geometry]
Hamiltonian systems, symplectic flows, classical integrable systems
1,436
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Volume of manifolds embedded in $\mathbb{R}^n$
Let $N$ be a closed, connected, oriented hypersurface of $\mathbb{R}^n$. Such a manifold inherits a volume form from the usual volume from on $\mathbb{R}^n$ and has an associated volume given by ...
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Why hasn't anyone proved that the two standard approaches to quantizing Chern-Simons theory are equivalent?
The two standard approaches to the quantization of Chern-Simons theory are geometric quantization of character varieties, and quantum groups plus skein theory. These two approaches were both first ...
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0
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Limitations of the splitting construction and SFT
I am trying to understand the so-called symplectic field theory (SFT) machinery used in symplectic topology. As I understand it, one of the applications of SFT (or rather, of the splitting ...
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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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Transitivity of Diff on the space of embeddings of balls
Given 2 symplectic embeddings $g_0$ and $g_1$ of a 4-ball of radius $r \leq 1$ into the 4-ball of radius 1 (all equipped with the standard symplectic form coming from $\mathbb{R}^4$), does there exist ...
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The mirror of the Landau--Ginzburg model given by elliptically fibered K3
Let $f:X\rightarrow \mathbb{P}^1$ be an elliptically fibered K3 surface. Choose a coordinate on $\mathbb{P}^1$ and consider $X\backslash f^{-1}(\infty)\rightarrow \mathbb{C}$ as a Landau--Ginzburg ...
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Pairs of J-holomorphic curves
Let $(M, \omega)$ be a symplectic 4-manifold and let $A$ and $B$ be symplectic submanifolds on M such that $A \cap B = p \in M$.Under what conditions can I find a $\omega$-compatible almost complex ...
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An inequality for symplectic manifolds
Question: Is there a closed symplectic manifold satisfying the following? $$|Td(M)| > \sum_{i \in \mathbb{Z}} b_{i}(M) $$
here $Td(X)$ is the Todd genus of $M$ i.e. the integral of the Todd class ...
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4
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1
answer
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The singular cohomology embeds into the symplectic cohomology
Viterbo's theorem on cotangent bundles $M=T^*N$ tells you in particular that singular cohomology $H^*(M)$ gets embedded in $SH^*(M)$ via the $c^*$ map. Having a Weinstein manifold (or more generally ...
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Symplectic Chern class of holomorphic symplectic manifold
I've posted this question already on math.stackexchage. Unfortunately, it did not receive any answers even though there was a bounty on it. So maybe somebody of you could help me. I apologize if this ...
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Diffeomorphism of $ \mathbb{C}P^2 \# ~\overline{\mathbb{C}P^2}$
I am currently reading Dusa McDuff's paper "Blow ups and symplectic embedding in dimension 4" and had a few questions regarding the paper.
In the paper McDuff uses the following notation. $X = \...
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1
answer
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An elliptic operator whose corresponding symbol Hamiltonian vector field has an isolated periodic orbit
Let $D$ be a differential operator on the space of smooth functions on a manifold $M$. The symbol of $D$ can be considered as a Hamiltonian on the cotangent bundle $T^*M$. We call ...
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Closed symplectic manifold with Euler characteristic 2
I am working about an article. In this article, author said that if close symplectic manifold $M$ has two fixed points implies that either $M$ is 2-sphere or $\dim M=6$.
The closed symplectic manifold ...
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Compatibility of Kirillov-Kostant-Souriau form and Killing form
Let $\mathfrak{g}$ be a real semisimple Lie algebra. We know that a coadjoint orbit $\mathcal{O} \hookrightarrow \mathfrak{g}^*$ carries a natural symplectic form $\omega$, namely the Kirillov-Kostant-...
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1
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Gromov-Witten invariants and the mod 2 spectral flow
I'm reading Lee-Parker, “A structure theorem for the Gromov-Witten invariants of Kähler
surfaces”, which studies Gromov-Witten invariants within symplectic
geometry. Lee-Parker write (&...
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4
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Abstract stationary phase
I have been reading Semi-classical analysis by Guillemin and Sternberg. At the end of Chapter 8, they gave an abstract version of the stationary phase method. I have a hard time figuring out what $a_i$...
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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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1
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On the existence and classification of prequantization spaces over a closed symplectic manifold
Let $(M,\omega)$ be a closed symplectic manifold. If the cohomology class $[\omega]$ is rational, that is if it lies in the image of the natural homomorphism $H^2(M,\mathbb{Z}) \to H^2(M,\mathbb{R})$, ...
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Cohomological indices VS cup-length, Lusternik-Schnirelmann category, betti sum in critical point set theory
Let $M$ be a (closed) manifold acted on by a compact Lie group $G$. For any characteristic class $\alpha \in H^*(BG)$, one can define a so-called cohomological index of $M$ as follows:
$$\text{ind}_{\...
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Is $\Bbb S^2 \times \Bbb S^4$ symplectic?
I'm playing around with products $M = \Bbb S^{n_1} \times \Bbb S^{n_2}$, and a quick computation using the Künneth formula tells us that if $(n_1,n_2)$ is not $(1,1)$ or $(2,4)$, $M$ is not symplectic ...
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Analogy between BV formalism and integration by residues
Domenico Fiorenza begins his description of the Batalin–Vilkovisky formalism by pointing out an analogy with integration by residues:
Take a top form (density) on $\mathbf R$ resp. space of fields;
...
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1
answer
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Normal coordinates for isotropic submanifolds
Let $(M,\omega)$ be a symplectic manifold and $N$ an isotropic submanifold. For a point $p\in N$, can we always find coordinates $(x_1,\ldots,x_n,y_1,\ldots,y_n)$ in a neighbourhood $U$ of $p$ such ...
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First Chern Class of Contact Structure which is not Torsion
Let $(M,\xi)$ be a closed connected $3-$dimensional contact manifold with contact structure $\xi$. It is known that the first Chern class $c_{1}(\xi)$ defines an element in $H^{2}(M;\mathbb{Z})$ and ...
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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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On different definitions of a prequantization space
Geometric quantization associates to a symplectic manifold $(M,\omega)$ a hermitian line bundle $L \to M$ with connection $\nabla$ whose curvature is $\omega$ (up to some constant).
Without talking ...
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2
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Moduli space of curves
Let $(M,\omega)$ be a symplectic manifold, and let $\mathscr{J}$ be the set of compatible almost complex structures on $M$.Finally let $A \in H^2(M,\mathbb{Z})$. Then we can consider the moduli space ...
- 17.2k
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Half-dimensional torus fibration vs Lagrangian torus fibration
Assume we have a closed symplectic manifold $M$ which is the total space of a smooth fibration by half-dimensional tori. Can we infer that $M$ is the total space of a smooth fibration by Lagrangian ...
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The automorphism group of a symplectic symmetric space
Why is the automorphism group of a sympelctic symmetric space a Lie group?
$\\$
A symplectic symmetric space is a triple $(M, \omega, s)$, where $(M, \omega)$ is a symplectic manifold and $ s \; \...
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2
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Uniqueness of a compatible Kahler-Einstein structure on a symplectic manifold?
$\require{AMScd}$
Preliminaries: Let $(X,\omega,J)$ be a closed Kahler manifold. That is, $X$ is a closed $2n$-manifold, $\omega$ is a symplectic form and $J$ is a compatible (integrable) complex ...
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2
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969
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Relation between mirror symmetry, homological mirror symmetry, and SYZ conjecture
I'm very new to mirror symmetry, and have a hard time establishing a broad overview of the subject. In particular I do not understand the precise relation between the following three conjectures:
...
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The $\frak{sl}_2$-representation on a symplectic manifold
Any symplectic manifold $(M,\omega)$ carries a representation of $\frak{sl}_2$: Define the maps
$$
L: \Omega^\bullet \to \Omega^{\bullet}, ~~~~~~~~~ \Lambda: \Omega^\bullet \to \Omega^{\bullet}, ~~~~~~...
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690
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Two homeomorphic non-diffeomorphic complex manifolds
Does there exist a closed topological manifold supporting two non-diffeomorphic smooth structures both of which admit a compatible complex structure? Also the same question, but for symplectic ...
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2
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Manifolds distinguished by Gromov-Witten invariants?
What is a simplest example of a manifold $M^{2n}$ that admits two symplectic structures with isotopic almost complex structures, and such that Gromov-Witten invariants of these symplectic structures ...
- 28.8k
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2
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Symplectic connections are (locally) Levi-Civita connections
I was wondering... Is every symplectic connection $\nabla$
on some symplectic manifold $(M,ω)$ the Levi-Civita connection of some metric $g$ on $M$? What about the local statement?
- 8,157
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1
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Beilinson-Drinfeld quantization and stable bundles
To motivate this question, I'm going to try and explain some background notions. This won't be absolutely necessary for experts, but I want to be vaguely honest about where this question comes from. ...
2
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1
answer
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Cotangent bundle of coadjoint orbit is stein manifold?
Let me first define stein manifolds and coadjoint orbits.
A complex manifold $X$ of complex dimension $n$ is called a Stein manifold if the following conditions hold:
$X$ is holomorphically convex, ...
- 151k
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Compact simply-connected homogeneous symplectic manifold
I was reading a paper in which the authors use the fact that any compact simply-connected homogeneous symplectic manifold has non-zero Euler characteristic. They prove it by quoting a theorem by ...
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2
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Linearization of hamiltonian torus action
Let $(M,\omega)$ be a symplectic manifold with a hamiltonian effective torus action. Suppose it has an isolated fixed point $p$. Is it true that there exists an invariant neighborhood $U$ of $p$ such ...
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1
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Existence of symplectic basis
Let $R$ be a PID and $M$ a free, finite rank $R$-module with a perfect billinear form $\omega$ such that $\omega(v,v)=0$ for all $v \in M$. Does anyone know a reference for the fact that a symplectic ...
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Grassmannians of planes isotropic with respect to general tensors
In symplectic geometry, the Grassmannian of isotropic planes for a symplectic vector space is a well known and well studied object; for example, one can realize it as a homogeneous space with a known ...
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1
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Global symplectic reduction
Let $M$ be a symplectic manifold equipped with a hamiltonian action of a compact Lie group $G$ with moment map $\mu\colon M\to \mathfrak g^*$. Assume $c\in \mathfrak g^*$. Then the symplectic ...
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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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GSO (Gliozzi-Scherk-Olive) projection and its Mathematics?
GSO (Gliozzi-Scherk-Olive) projection is an ingredient used in constructing a consistent model in superstring theory. The projection is a selection of a subset of possible vertex operators in the ...
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615
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The Fukaya category of a simple singularity (reference request)
I have heard that for an ADE singularity $f$,
$ D^b\mathrm{Fuk}(f) \simeq D^b(\mathrm{Rep}\ Q)$
where $Q$ is the corresponding Dynkin quiver. (As one would hope, if $\mathrm{Fuk}$ is some kind of ...
- 3,507
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1
answer
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Morse theory in infinite dimensions
It seems that people often talk of "doing Morse theory" on loop spaces in two quite different contexts.
Case 1: When one does Morse theory on a loop space $\Omega(M; p,q)$ using the energy ...
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Bound on critical points of Lefschetz fibration over the disk with prescribed monodromy
Let $\phi$ be a right-veering diffeomorphism of a surface $\Sigma$ of genus $g$ and $r$ boundary components. Suppose that the diffeomorphism is freely periodic so if $M$ is the associated open book ...
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523
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What is the mirror of an algebraic group?
Background: Kontsevich's homological mirror symmetry conjecture posits the existence of pairs $(X,\check X)$ with an equivalence of dg/$A_\infty$-categories
$$\mathcal F(X)=\mathcal D^b(\check X)$$
...
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1
answer
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Symplectic Lefschetz fibrations in terms of factorization in symplectic mapping class group
There is a well-known theorem stating that there is a bijection between diffeomorphism classes of Lefschetz fibrations over $S^2$ whose general fiber is a closed orientable surface $\Sigma_g$ of genus ...
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0
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Effective classes in toric Kähler manifolds
In an article about toric manifolds, I have seen the following notions, which I don't understand. We view a symplectic toric manifold $(M,\omega)$ as a Kähler manifold with Kähler form $\omega$, and ...
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Existence of compact leaf for certain foliation of a symplectic manifold
Is there a symplectic manifold $(M,\omega)$ equipped with a Riemannian metric such that $d^* \omega \wedge dd^*\omega=0$ and there is a compact leaf for the foliation of $M\setminus S$ defined by $d^*...
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