Showing papers on "Symplectic vector space published in 2020"
TL;DR: In this article, the authors studied moduli spaces of meromorphic connections over Riemann surfaces together with corresponding spaces of monodromy data (involving Stokes matrices) and described natural symplectic structures both explicitly and from an infinite dimensional viewpoint.
Abstract: We study moduli spaces of meromorphic connections (with arbitrary order poles) over Riemann surfaces together with the corresponding spaces of monodromy data (involving Stokes matrices). Natural symplectic structures are found and described both explicitly and from an infinite dimensional viewpoint (generalising the Atiyah-Bott approach). This enables us to give an intrinsic symplectic description of the isomonodromic deformation equations of Jimbo, Miwa and Ueno, thereby putting the existing results for the six Painleve equations and Schlesinger's equations into a uniform framework.
82 citations
TL;DR: In this paper, the authors study the Berezin-Toeplitz quantization using as quantum space the space of eigenstates of the renormalized Bochner Laplacian corresponding to eigenvalues localized near the origin on a symplectic manifold.
Abstract: We study the Berezin–Toeplitz quantization using as quantum space the space of eigenstates of the renormalized Bochner Laplacian corresponding to eigenvalues localized near the origin on a symplectic manifold. We show that this quantization has the correct semiclassical behavior and construct the corresponding star-product.
24 citations
TL;DR: In this article, the authors use virtual fundamental chains of moduli spaces of nonconstant pseudo-holomorphic disks to define a Maurer-Cartan element of a Lie bracket operation in string topology.
Abstract: Let $L$ be a Lagrangian submanifold in a symplectic vector space which is closed, oriented and spin. Using virtual fundamental chains of moduli spaces of nonconstant pseudo-holomorphic disks with boundaries on $L$, one can define a Maurer-Cartan element of a Lie bracket operation in string topology (the loop bracket) defined at chain level. This observation is due to Fukaya, who also pointed out its important consequences in symplectic topology. The goal of this paper is to work out details of this observation. Our argument is based on a string topology chain model previously introduced by the author, and the theory of Kuranishi structures on moduli spaces of pseudo-holomorphic disks, which has been developed by Fukaya-Oh-Ohta-Ono.
5 citations
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TL;DR: In this paper, the authors define an analogue of the resolvent algebra on the cotangent bundle of an $n$-torus by first generalizing the classical analogue of this algebra defined by the first author of this paper in earlier work, and subsequently applying Weyl quantisation.
Abstract: Buchholz and Grundling (Comm. Math. Phys., 272, 699--750, 2007) introduced a C$^\ast$-algebra called the resolvent algebra as a canonical quantisation of a symplectic vector space, and demonstrated that this algebra has several desirable features. We define an analogue of their resolvent algebra on the cotangent bundle $T^*\mathbb{T}^n$ of an $n$-torus by first generalizing the classical analogue of the resolvent algebra defined by the first author of this paper in earlier work (J. Funct. Anal., 277, 2815--2838, 2019), and subsequently applying Weyl quantisation. We prove that this quantisation is almost strict in the sense of Rieffel and show that our resolvent algebra shares many features with the original resolvent algebra. We demonstrate that both our classical and quantised algebras are closed under the time evolutions corresponding to large classes of potentials. Finally, we discuss their relevance to lattice gauge theory.
3 citations
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TL;DR: In this article, the Schur multiplier was used to associate a π-representation with the Weyl Weyl calculus, and the properties of π -representation were systematically studied.
Abstract: Let $\left( W,\sigma \right) $ be a symplectic vector space and let $% T:W\rightarrow W$ be a linear map that satisfies a certain condition of non-degeneracy. We define the Schur multiplier $\omega _{\sigma ,T}$ on $W$. To this multiplier we associate a $\omega _{\sigma ,T}$-representation and and we build the $T$-Weyl calculus, $\mathrm{Op}_{\sigma ,T}$, whose properties are are systematically studied further.
2 citations
TL;DR: In this article, it was shown that enumerative counts of holomophic disks give useful information about linking of Lagrangian tori in the symplectic vector space, and that any two Clifford tori are unlinked in a strong sense.
Abstract: We prove some results about linking of Lagrangian tori in the symplectic vector space $(\mathbb{R}^4, \omega)$. We show that certain enumerative counts of holomophic disks give useful information about linking. This enables us to prove, for example, that any two Clifford tori are unlinked in a strong sense. We extend work of Dimitroglou Rizell and Evans on linking of monotone Lagrangian tori to a class of non-monotone tori in $\mathbb{R}^4$ and also strengthen their conclusions in the monotone case in $\mathbb{R}^4$.
2 citations
Abstract: We study particular classes of states on the Weyl algebra $\mathcal{W}$ associated with a symplectic vector space $S$ and on the von Neumann algebras generated in representations of $\mathcal{W}$. Applications in quantum physics require an implementation of constraint equations, e.g., due to gauge conditions, and can be based on so-called Dirac states. The states can be characterized by nonlinear functions on $S$ and it turns out that those corresponding to non-trivial Dirac states are typically discontinuous. We discuss general aspects of this interplay between functions on $S$ and states, but also develop an analysis for a particular example class of non-trivial Dirac states. In the last part, we focus on the specific situation with $S = L^2(\mathbb{R}^n)$ or test functions on $\mathbb{R}^n$ and relate properties of states on $\mathcal{W}$ with those of generalized functions on $\mathbb{R}^n$ or with harmonic analysis aspects of corresponding Borel measures on Schwartz functions and on temperate distributions.
1 citations
TL;DR: As an application, it is shown the non-degenerate skew symmetric ( 0 , 2 ) -tensor ω has a property of rigidity that it is determined by its exterior power.
Abstract: In this article, we study the non-degeneracy of 2-forms (skew symmetric ( 0 , 2 ) -tensor) α along the Pfaffian of α . We consider a symplectic vector space V with a non-degenerate skew symmetric ( 0 , 2 ) -tensor ω , and derive various properties of the Pfaffian of α . As an application we show the non-degenerate skew symmetric ( 0 , 2 ) -tensor ω has a property of rigidity that it is determined by its exterior power.