Questions tagged [symplectic-topology]
The symplectic-topology tag has no usage guidance.
244
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Associativity of orientations of determinant bundles in Floer homology
I have been reading the paper "Coherent orientations for periodic orbits problems in symplectic geometry" by Floer and Hofer, trying to understand how we can orient the moduli spaces that ...
2
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0
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120
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Compactness of Moduli spaces in Lagrangian Floer Cohomology
I have been reading Denis Aurox lecture notes on Fukaya Categories https://arxiv.org/pdf/1301.7056.pdf , and in page $9$ he starts to discuss the compactness properties of the moduli spaces and how we ...
3
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0
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96
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Removal of singularities theorem for J-holomorphic curves in non-compact manifolds
Following the book by Mcduff and Salamon, J-holomorphic curves and Symplectic Topology, we know that every $J-$holomorphic curve on the punctured disk with values in a compact symplectic manifold ...
6
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260
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Computation of the Fredhom index in Floer theory
I have been reading Salamon's lecture notes on Floer homology, and to compute the Fredholm index of operators they use the fact that the spectral flow of $A(s)$ is the Fredholm index. Now in the proof ...
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90
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Compactness properties in floer homology of cotangent bundles in the non-periodic case
Following the paper https://arxiv.org/pdf/math/0408280.pdf I have been interested in studying the case of solutions $x:[0,1] \rightarrow T^*M$ such that $x(0)\in T_{q_0}^*M$ and $x(1)\in T_{q_1}^*M$ ...
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Linking number of specific Reeb orbits in a toric domain ($S^3$ diffeomorphic)
Consider a toric domain defined by the region bounded on the first quadrant by a function $f:[0,a]\mapsto [0,b]$ with $a,b>0;f(0)=b,f(a)=0,f(x)>0 \hspace{2mm} \forall x\in [0,a)$. We know that $\...
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71
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Using the removal of singularities theorem in $\mathbb{C}\mathbb{P}^1-\{0,\infty\}$ with lagrangian boundary conditions
Reading the paper "Floer Cohomology of Lagrangian intersections" the authors construct a map $f: \mathbb{R}^n \times [0,2^N]\rightarrow \mathbb{C}\mathbb{P}^n$ such that $f(\tau,0)=f(\tau,2^...
7
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273
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Cotangent bundles of surfaces as varieties
As far as I understand, it is easy to see (and find in the literature) that the affine variety
$$z_1^2+z_2^2+z_3^2=1$$
with the restriction of the standard $\omega_{std}$ of $\mathbb{C}^3$ is ...
2
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0
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79
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Are there paths of non-degenerate $2$-forms joining two symplectic structures on open $4$-manifolds?
There is a Theorem of Conolly, L$\hat{\text{e}}$ and Ono which states that on a closed simply-connected $4$-manifold two cohomologous symplectic forms can be joined by a path of non-degenerate $2$-...
3
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108
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Properties of $I_{\mu}$ for Lagrangian Floer Homology in the Cotangent bundle
Following the notation of the book "Lagrangian intersection Floer theory anomaly and obstruction" suppose we have that our symplectic manifold is a cotangent bundle $T^*M$ with the canonical ...
2
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110
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Higher genus (Hamiltonian perturbed) holomorphic curves in cotangent bundle of S^1
Consider $T^*S^1$ as symplectic manifold, with hamiltonian function $H(x,y) = y^2$ (y is the fiber direction, I know this is morse bott but it can be perturbed). consider the set of maps $u: \Sigma \...
11
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402
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A theorem about the symplectic geometry of projective bundles
I am trying to understand the following theorem about symplectomorphisms of projective bundles. Theorem 1.5 of "Characteristic Classes in Symplectic Topology" A.G. Reznikov. Selecta ...
2
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94
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How does the Maslov index of a loop `project’ to the rotation number?
I’m trying to learn some Legendrian contact homology and the grading of the generators of the DGA are given by computing a fractional rotation number. In the symplectisation, this number is the Conley-...
4
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468
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(Contradiction) All symplectic manifolds are holomorphic
I’m studying symplectic manifolds and almost complex structures. This lead to two propositions:
Proposition 1 (from da Silva’s Lectures on Symplectic Geometry): If $J_0$ and $J_1$ are almost complex ...
2
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0
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Is the forgetful map a submersion away from the nodal points?
Let $\overline{\mathcal{M}}_{0,n}$ be the moduli space of stable curves of genus zero with $n$ marked points. For $n \geq 4$ we have a forgetful map $\pi \colon \overline{\mathcal{M}}_{0,n}\rightarrow ...
3
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91
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Displacing a conormal Lagrangian from the zero section
I was told that the conormal bundle $\nu^*K$ of a knot $K\subset S^3$ can be displaced from the zero section $0_{S^3}$ in $T^*S^3.$ Having no intuition about whether/how often this happens in general, ...
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517
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On prequantization bundles over integral symplectic manifolds
I am trying to clarify certain subtleties regarding prequantization bundles over symplectic manifolds, for which I haven't found any clear explanation so far.
Let me fix some definitions first.
...
3
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0
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99
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Continuation map interpolating two quadratic Hamiltonians with respect to different contact boundaries
Let $(M,\lambda)$ be a Liouville manifold. Consider two different contact boundaries $\partial_{\infty}^1M$ and $\partial_{\infty}^2M$ with respect to the same Liouville flow $Z$. Each of them ...
6
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490
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Hamiltonian dynamics on cotangent bundle
I'm stuck with the following claim made in Section 13.1 of Y-G. Oh's book "Symplectic topology and Floer homology". Assume that $N$ is a differential manifold and $S_0 ,S_1\subseteq N$ two ...
3
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1
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391
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On some prerequisites for J-holomorphic curves and Gromov-Witten invariants
I'm currently reading through J-holomorphic curves and Quantum cohomology by McDuff and Salamon, and I've been facing some unfamiliarity issues with respect to the PDE and functional analysis tools ...
2
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0
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83
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Two closed lagrangians embedded in $\mathbb{R}^{2n}$ can be shifted and perturbed to intersect at two points
In Polterovich paper "The surgery of Lagrange manifolds". It is stated in page 204 - in proof of Theorem 7a - the following,
If $L_1$ and $L_2$ are two Lagrangian submaifolds of $\mathbb{C}^...
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0
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64
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Definition of signs of isomorphisms $c_u : o(x_1) \to o(x_0)$ in the definition of Floer cohomology via Seidel's book
I'm reading Paul Seidel's book "Fukaya Categories and Picard-Lefschetz Theory", chapter 12, and I'm currently trying to understand the differential on Floer cohomology in terms of ...
3
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0
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127
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Moment map of $\mathrm{O}(n)$-action on $\mathbb{C}^n$
Let $(\mathbb{C}^n, \omega_0)$ be the complex Euclidean space of dimension $n$ with the standard Kähler structure $\omega_0$. I am looking for a Hamiltonian $\mathrm{O}(n)$-action on $(\mathbb{C}^n, \...
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2
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558
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Which curves are boundary of pseudoholomorphic curves?
I have posted it on Mathstackexchange but nobody replied.
Consider a loop $\gamma:\mathbb{S}^1\to M^{2n}$ in a symplectic manifold $(M^{2n},\omega)$. Let $J$ be an $\omega$-compatible almost complex ...
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764
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Roadmap to Floer homotopy theory?
I am a young postdoc working in symplectic topology.
Recently I became intrigued by Floer homotopy, especially after seeing it had been applied to classical questions in symplectic topology. (e.g. ...
1
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1
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339
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Terminology for exact symplectomorphism
Let $(M,\omega = d\alpha)$ be an exact symplectic manifold. Then a symplectomorphism $\varphi \colon M \to M$ is said to be exact, iff $\varphi^*\alpha - \alpha$ is exact. Is there a terminology for ...
13
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1
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423
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How not to use J-holomorphic curves [closed]
The field of symplectic topology is filled with subtle traps for the unwary, particularly when it comes to the analysis of $J$-holomorphic curves. So that the next generation of symplectic topologists ...
2
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92
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Local contractibility of group of symplectomorphisms for open manifolds
It is well know that for a closed symplectic manifold $(M, \omega)$ the group of symplectomorphisms in locally contractible. The gist of this proof goes as follows.
Given a $\psi \in \operatorname{...
6
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299
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Is the symplectic quotient $\mu^{-1}(0)/G$ unique up to something?
Given a Hamiltonian action of a compact Lie group $G$ on a symplectic manifold $(M,\omega)$, we may choose a moment map $\mu \colon M\to \mathfrak{g}^* $ and obtain the symplectic reduction $M/\!\!/G =...
2
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0
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101
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Augmentations of wrapped Floer cochains
Let $M$ be a closed, simply-connected spin manifold and let $F_b \subset T^*M$ be the cotangent fiber over a point $b \in M$. Let $CW^*(L,L)$ be the $A_{\infty}$-algebra of wrapped Floer cochains over ...
3
votes
1
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360
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Viterbo restriction map surjective on Weinstein neighbourhood
In a Liouville manifold $M$ having a Liouville subdomain $i: N \hookrightarrow M$, there is the so-called Viterbo restriction map in symplectic cohomology $$SH^*(i): SH^*(M)\rightarrow SH^*(N).$$
In ...
2
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1
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188
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Superlevel sets of a parametrized quadratic forms
Let $N$ be an odd integer, $n\in\mathbb{N}$, and $-\frac{2T}{NR^2}\leq a\leq0$ for given $R,T\in\mathbb{R}$ with $\frac{T}{NR^2}\leq\frac{\pi}{2}$.
Now consider the quadratic form $\Omega(a)=\sum_{l\...
4
votes
2
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396
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Contactomorphisms have in general no fixed points
Let $(V, \xi = \ker \alpha)$ be a cooriented contact manifold. A contactomorphism of $(V, \xi)$ is a diffeomorphism $\phi$ which preserves the contact structure $\xi$ and its coorientation. In other ...
2
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1
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178
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Comparing the minimal Chern number and the cup-length of a symplectic manifold
Let $(M,\omega)$ be a symplectic manifold. One can define its minimal Chern number $N_M$ as:
$$
N_M := \text{inf} \lbrace k > 0 \ |\ \exists A \in H_2(M; \mathbb{Z}), \langle c_1, A \rangle = k \...
7
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1
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645
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Lagrangian intersection Floer homology: understanding some assumptions
Let $(X,\omega)$ be a symplectic manifold and $L\subset X$ be a Lagrangian subspace.
Let $\mu_L:H_2(X,L;\mathbb{Z})\to \mathbb{Z}$ be the Maslov index
homomorphism.
Usual hypothesis
Recall that $L$...
2
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1
answer
482
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Two Lagrangian submanifolds with clean intersections
Having two closed exact Lagrangian submanifolds $L_1$ and $L_2$ that intersect cleanly inside a Liouville manifold $M$ with $c_1(M)=0,$ is there (with possibly some other conditions) any relation ...
10
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0
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1k
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Roadmap to understanding Gromov's Non-squeezing theorem
I'm a graduate student starting out to venture into the areas of Symplectic Geometry/Topology, and was somewhat motivated by the essence of Gromov's non-squeezing theorem which in a sense made me feel ...
10
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394
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Words and ranks
Let me state two problems that look very much alike. The first one can be solved putting together answers that different people have given to some questions I asked here a few weeks ago. The second ...
5
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1
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448
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Diffeomorphic but not isotopic symplectic forms
Do we know of any closed symplectic manifold $M$ with 2 cohomologous symplectic forms $\omega_1$ and $\omega_2$ such that there exist $\psi \in \text{Diff}(M)$ and $\psi^* \omega_1 = \omega_2$ but $\...
3
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1
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185
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Star-shaped domain in $\mathbb{C}P^2$
Consider $(\mathbb{C}P^2,\omega_{FS})$ where $\omega_{FS}$ is the standard Fubini-Study form. Let $L$ denote a sphere in $\mathbb{C}P^2$ in the class $\mathbb{C}P^1$. Further let $\int_{L} \omega_{FS} ...
5
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1
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480
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Lagrangian torus fibrations and Arnol'd-Liouville theorem
Let $(X, \omega)$ be a closed symplectic manifold of dimension $2n$ and $\pi: X \rightarrow Q$ a Lagrangian torus fibration. Let $F_q$ denote the fiber at $q \in Q$. It is claimed in a paper of ...
4
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1
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133
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Orientable surface bundle
Is it true that every orientable surface bundle can be made into a symplectic fibration?If yes, why?
What about the particular case that $M$ is a connected compact 4-manifold?
3
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1
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227
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Compactly supported symplectomorphisms of $D^2$
I'm trying to understand a more basic version about Gromov's theorem about the compactly supported symplectomorphisms of $D^2 \times D^2$ being contractible.
Consider the dimensional disk $D^2 \...
2
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0
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359
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The norm squared of a moment map
I am studying the paper by E. Lerman:
https://arxiv.org/abs/math/0410568
Let $(M,\sigma)$ be a connected symplectic manifold with a Hamiltonian action of a compact Lie group $G$, so that there exist ...
4
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1
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324
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Every symplectic submanifold is J-holomorphic
I am trying to show that every symplectic submanifold $N$ of a 2n- dimensional symplectic manifold $(M,\omega)$ is J-holomorphic for some compatible almost complex structure $J$.
The way I am ...
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71
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Invariance under tame almost complex structure of the fibre tangent space of the symplectic normal bundle
I am trying to understand the construction of symplectic inflation and I am stuck in the following point.
Suppose we have a 4 dimensional symplectic manifold $(M, \omega)$. Also suppose that $N \...
5
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241
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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 $...
2
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0
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187
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What is the relation between the different generating functions thought as finite approximations of action functionals
In the book Introduction to symplectic topology by MC Duff and Salamon, a discrete analogue of the action functional is defined on $\mathbb{R}^{2n}$. The idea is that a Hamiltonian isotopy can be ...
4
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143
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Mixed characteristic in symplectic geometry
Are there any mixed-characteristic phenomena in symplectic geometry/mirror symmetry?
There are papers on symplectic geometry by Abouzaid (inspired by Kontsevich--Soibelman, I believe) in which there ...
6
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1
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580
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Relationship between Gromov-Witten and Taubes' Gromov invariant
Fix a compact, symplectic four-manifold ($X$, $\omega$).
Recall Taubes' Gromov invariant is a certain integer-valued function on $H^2(X; \mathbb{Z})$ defined by weighted counts of pseudoholomorphic ...