scholarly journals Computation of Quantum Cohomology From Fukaya Categories

2020 ◽  
Author(s):  
Fumihiko Sanda
Keyword(s):  
Symplectic Manifold ◽  
Symplectic Structure ◽  
Blow Up ◽  
Quantum Cohomology ◽  
Fukaya Category ◽  
Fukaya Categories ◽  
Compact Symplectic Manifold

Abstract Assume the existence of a Fukaya category $\textrm{Fuk}(X)$ of a compact symplectic manifold $X$ with some expected properties. In this paper, we show $\mathscr{A} \subset \textrm{Fuk}(X)$ split generates a summand $\textrm{Fuk}(X)_e \subset \textrm{Fuk}(X)$ corresponding to an idempotent $e \in QH^{\bullet }(X)$ if the Mukai pairing of $\mathscr{A}$ is perfect. Moreover, we show $HH^{\bullet }(\mathscr{A}) \cong QH^{\bullet }(X) e$. As an application, we compute the quantum cohomology and the Fukaya category of a blow-up of $\mathbb{C} P^2$ at four points with a monotone symplectic structure.

scholarly journals Geometric prequantization on the path space of a prequantized manifold

2015 ◽  
Vol 12 (03) ◽  
pp. 1550030
Author(s):  
Indranil Biswas ◽  
Saikat Chatterjee ◽  
Rukmini Dey
Keyword(s):  
Symplectic Manifold ◽  
Symplectic Form ◽  
Symplectic Structure ◽  
Gordon Equation ◽  
Solution Space ◽  
Path Space ◽  
Klein Gordon ◽  
Class Of Functions ◽  
Compact Symplectic Manifold ◽  
Klein Gordon Equation

Given a compact symplectic manifold M, with integral symplectic form, we prequantize a certain class of functions on the path space for M. The functions in question are induced by functions on M. We apply our construction to study the symplectic structure on the solution space of Klein–Gordon equation.

scholarly journals Homological mirror symmetry for higher-dimensional pairs of pants

2020 ◽  
Vol 156 (7) ◽  
pp. 1310-1347
Author(s):  
Yankı Lekili ◽  
Alexander Polishchuk
Keyword(s):  
Mirror Symmetry ◽  
Derived Category ◽  
Affine Variety ◽  
Fukaya Category ◽  
Homological Mirror Symmetry ◽  
Endomorphism Algebra ◽  
Generating Set ◽  
Fukaya Categories ◽  
Higher Dimensional ◽  
Abelian Covers

Using Auroux’s description of Fukaya categories of symmetric products of punctured surfaces, we compute the partially wrapped Fukaya category of the complement of $k+1$ generic hyperplanes in $\mathbb{CP}^{n}$, for $k\geqslant n$, with respect to certain stops in terms of the endomorphism algebra of a generating set of objects. The stops are chosen so that the resulting algebra is formal. In the case of the complement of $n+2$ generic hyperplanes in $\mathbb{C}P^{n}$ ($n$-dimensional pair of pants), we show that our partial wrapped Fukaya category is equivalent to a certain categorical resolution of the derived category of the singular affine variety $x_{1}x_{2}\ldots x_{n+1}=0$. By localizing, we deduce that the (fully) wrapped Fukaya category of the $n$-dimensional pair of pants is equivalent to the derived category of $x_{1}x_{2}\ldots x_{n+1}=0$. We also prove similar equivalences for finite abelian covers of the $n$-dimensional pair of pants.

Deformation quantization and boundary value problems

2016 ◽  
Vol 13 (02) ◽  
pp. 1650007
Author(s):  
Nikolai Tarkhanov
Keyword(s):  
Harmonic Oscillator ◽  
Boundary Value Problems ◽  
Symplectic Manifold ◽  
Deformation Quantization ◽  
Boundary Value ◽  
Index Theorem ◽  
Manifold With Boundary ◽  
Quantum Observables ◽  
Compact Symplectic Manifold

We describe a natural construction of deformation quantization on a compact symplectic manifold with boundary. On the algebra of quantum observables a trace functional is defined which as usual annihilates the commutators. This gives rise to an index as the trace of the unity element. We formulate the index theorem as a conjecture and examine it by the classical harmonic oscillator.

scholarly journals SCALING ASYMPTOTICS FOR QUANTIZED HAMILTONIAN FLOWS

2012 ◽  
Vol 23 (10) ◽  
pp. 1250102 ◽  
Author(s):  
ROBERTO PAOLETTI
Keyword(s):  
Line Bundle ◽  
Symplectic Manifold ◽  
Natural Generalization ◽  
Local Scaling ◽  
Hamiltonian Flows ◽  
Szegő Kernel ◽  
Szego Kernel ◽  
Positive Line Bundle ◽  
Positive Line ◽  
Compact Symplectic Manifold

In recent years, the near diagonal asymptotics of the equivariant components of the Szegö kernel of a positive line bundle on a compact symplectic manifold have been studied extensively by many authors. As a natural generalization of this theme, here we consider the local scaling asymptotics of the Toeplitz quantization of a Hamiltonian symplectomorphism, and specifically how they concentrate on the graph of the underlying classical map.

scholarly journals FLOER HOMOLOGY FOR SYMPLECTOMORPHISM

2009 ◽  
Vol 11 (06) ◽  
pp. 895-936 ◽  
Author(s):  
HAI-LONG HER
Keyword(s):  
Fixed Points ◽  
Symplectic Manifold ◽  
Floer Homology ◽  
Symplectic Manifolds ◽  
Homology Groups ◽  
Symplectic Diffeomorphism ◽  
Novikov Ring ◽  
Compact Symplectic Manifold

Let (M,ω) be a compact symplectic manifold, and ϕ be a symplectic diffeomorphism on M, we define a Floer-type homology FH*(ϕ) which is a generalization of Floer homology for symplectic fixed points defined by Dostoglou and Salamon for monotone symplectic manifolds. These homology groups are modules over a suitable Novikov ring and depend only on ϕ up to a Hamiltonian isotopy.

scholarly journals Floer-Novikov cohomology and symplectic fixed points, revisited

2020 ◽  
Vol 13 (4) ◽  
pp. 89-115
Author(s):  
Kaoru Ono ◽  
Hong Van Le
Keyword(s):  
Lower Bound ◽  
Fixed Points ◽  
Symplectic Manifold ◽  
Compact Symplectic Manifold

This note is mostly an exposition of a few versions of Floer-Novikov cohomology with a few new observations. For example, we state a lower bound for the number of symplectic fixed points of a non-degenerate symplectomorphism, which is symplectomorphic isotopic to the identity, on a compact symplectic manifold, more precisely than previous statements in [14,10].

scholarly journals UNBOUNDEDNESS OF THE FIRST EIGENVALUE OF THE LAPLACIAN IN THE SYMPLECTIC CATEGORY

2013 ◽  
Vol 05 (01) ◽  
pp. 13-56
Author(s):  
LEV BUHOVSKY
Keyword(s):  
Symplectic Manifold ◽  
Symplectic Structure ◽  
Riemannian Metrics ◽  
First Eigenvalue ◽  
The First Eigenvalue

Given a closed symplectic manifold (M2n, ω) of dimension 2n ≥ 4, we consider all Riemannian metrics on M, which are compatible with the symplectic structure ω. For each such metric g, we look at the first eigenvalue λ1 of the Laplacian associated with it. We show that λ1 can be made arbitrarily large, when we vary g. This generalizes previous results of Polterovich, and of Mangoubi.

scholarly journals Contact blow up and cylindrical contact homology of toric contact manifolds of Reeb type

2017 ◽  
Vol 102 (116) ◽  
pp. 61-71
Author(s):  
Aleksandra Marinkovic
Keyword(s):  
Symplectic Manifold ◽  
Blow Up ◽  
Contact Manifold ◽  
Contact Homology ◽  
Contact Manifolds ◽  
Special Cases ◽  
Toric Contact Manifolds

Let (V,?) be a toric contact manifold of Reeb type that is a prequantization of a toric symplectic manifold (M,?). A contact blow up of (V,?) is the prequantization of a symplectic blow up of (M,?). Thus, a contact blow up of (V,?) is a new toric contact manifold of Reeb type. In some special cases we are able to compute the cylindrical contact homology for the contact blowup using only the cylindrical contact homology of the contact manifold we started with.

A quantum modification of relative cohomology

2015 ◽  
Vol 12 (02) ◽  
pp. 1550021
Author(s):  
Dong Zhang ◽  
Bohui Chen ◽  
Cheng-Yong Du
Keyword(s):  
Symplectic Manifold ◽  
Cup Product ◽  
Relative Cohomology ◽  
Compact Symplectic Manifold

In this paper, we give a quantum modification of the relative cup product on H*(X, S;ℝ) by using Gromov–Witten invariants when S is a compact codimension 2k symplectic submanifold of the compact symplectic manifold (X, ω).

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