scholarly journals Non-Degeneracy of 2-Forms and Pfaffian

Symmetry ◽  
2020 ◽  
Vol 12 (2) ◽  
pp. 280
Author(s):  
Jae-Hyouk Lee
Keyword(s):  
Vector Space ◽  
Exterior Power ◽  
Symplectic Vector Space

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.

scholarly journals Isomorphic exponential Weyl algebras

1991 ◽  
Vol 33 (1) ◽  
pp. 7-10 ◽  
Author(s):  
P. L. Robinson
Keyword(s):  
Vector Space ◽  
Weyl Algebra ◽  
Associative Algebras ◽  
Exponential Form ◽  
Canonical Commutation Relations ◽  
Commutation Relations ◽  
Weyl Algebras ◽  
The Subject ◽  
Symplectic Vector Space

Canonically associated to a real symplectic vector space are several associative algebras. The Weyl algebra (generated by the Heisenberg commutation relations) has been the subject of much study; see [1] for example. The exponential Weyl algebra (generated by the canonical commutation relations in exponential form) has been less well studied; see [8].

scholarly journals Equivariant Giambelli formula for the symplectic Grassmannians — Pfaffian Sum Formula

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Takeshi Ikeda ◽  
Tomoo Matsumura
Keyword(s):  
Vector Space ◽  
Closed Formula ◽  
International Audience ◽  
Sum Formula ◽  
Isotropic Subspaces ◽  
Espace Vectoriel ◽  
Symplectic Vector Space ◽  
Symplectic Grassmannians ◽  
Schubert Class

International audience We prove an explicit closed formula, written as a sum of Pfaffians, which describes each equivariant Schubert class for the Grassmannian of isotropic subspaces in a symplectic vector space On démontre une formule close explicite, écrite comme une somme de Pfaffiens, qui décrit toute classe de Schubert équivariante pour la Grassmannienne des sous-espaces isotropes dans un espace vectoriel symplectique.

scholarly journals Crepant resolutions and open strings

2019 ◽  
Vol 2019 (755) ◽  
pp. 191-245 ◽  
Author(s):  
Andrea Brini ◽  
Renzo Cavalieri ◽  
Dustin Ross
Keyword(s):  
Vector Space ◽  
Crepant Resolution ◽  
Complete Proof ◽  
Open Strings ◽  
Witten Theory ◽  
Crepant Resolutions ◽  
Symplectic Vector Space ◽  
Crepant Resolution Conjecture ◽  
Minimal Resolutions ◽  
Type Statement

AbstractIn the present paper, we formulate a Crepant Resolution Correspondence for open Gromov–Witten invariants (OCRC) of toric Lagrangian branes inside Calabi–Yau 3-orbifolds by encoding the open theories into sections of Givental’s symplectic vector space. The correspondence can be phrased as the identification of these sections via a linear morphism of Givental spaces. We deduce from this a Bryan–Graber-type statement for disk invariants, which we extend to arbitrary topologies in the Hard Lefschetz case. Motivated by ideas of Iritani, Coates–Corti–Iritani–Tseng and Ruan, we furthermore propose (1) a general form of the morphism entering the OCRC, which arises from a geometric correspondence between equivariant K-groups, and (2) an all-genus version of the OCRC for Hard Lefschetz targets. We provide a complete proof of both statements in the case of minimal resolutions of threefold {A_{n}}-singularities; as a necessary step of the proof we establish the all-genus closed Crepant Resolution Conjecture with descendents in its strongest form for this class of examples. Our methods rely on a new description of the quantum D-modules underlying the equivariant Gromov–Witten theory of this family of targets.

scholarly journals On the trivectors of a 6-dimensional symplectic vector space. IV

2013 ◽  
Vol 438 (5) ◽  
pp. 2405-2429 ◽  
Author(s):  
B. De Bruyn ◽  
M. Kwiatkowski
Keyword(s):  
Vector Space ◽  
Symplectic Vector Space

Clifford and Heisenberg algebras

2011 ◽  
Author(s):  
Jean-Michel Bismut
Keyword(s):  
Vector Space ◽  
Clifford Algebra ◽  
Bilinear Form ◽  
Scalar Product ◽  
Heisenberg Algebra ◽  
Symmetric Bilinear Form ◽  
Rham Complex ◽  
Heisenberg Algebras ◽  
De Rham ◽  
Symplectic Vector Space

This chapter recalls various results on Clifford algebras and Heisenberg algebras. It first introduces the Clifford algebra of a vector space V equipped with a symmetric bilinear form B and then specializes the construction of the Clifford algebra to the case of V ⊕ V*. Next, the chapter argues that, if (V,ω‎) is a symplectic vector space, then the associated Heisenberg algebra is constructed and then specialized to the case of V ⊕ V*. Hereafter, the chapter considers the combination of the Clifford and Heisenberg algebras for V ⊕ V*, and constructs the complex Λ‎· (V*) ⊗ S· (V*), ̄ƌ) which is the subcomplex of polynomial forms in the de Rham complex. Finally, when V is equipped with a scalar product, this complex is related to a Witten complex over V.

Group-theoretic obstructions to integrability

1995 ◽  
Vol 15 (1) ◽  
pp. 15-48 ◽  
Author(s):  
R. C. Churchill ◽  
D. L. Rod ◽  
M. F. Singer
Keyword(s):  
Hamiltonian System ◽  
Vector Space ◽  
Symplectic Group ◽  
Degree Of Freedom ◽  
Complex Symplectic ◽  
Three Degree Of Freedom ◽  
Symplectic Vector Space

AbstractLet V be a four-dimensional complex symplectic vector space. This paper classifies those connected linear algebraic subgroups of the symplectic group Sp(V) that admit two independent rational invariants. As an application we show the non integrability of a three degree of freedom Hamiltonian system.

scholarly journals HOW LARGE IS THE SHADOW OF A SYMPLECTIC BALL?

2013 ◽  
Vol 05 (01) ◽  
pp. 87-119 ◽  
Author(s):  
ALBERTO ABBONDANDOLO ◽  
ROSTISLAV MATVEYEV
Keyword(s):  
Vector Space ◽  
Unit Ball ◽  
Orthogonal Projection ◽  
Real Dimension ◽  
Symplectic Embedding ◽  
Dimensional Unit ◽  
Dimensional Ball ◽  
Symplectic Vector Space

Consider the image of the 2n-dimensional unit ball by a symplectic embedding into the standard symplectic vector space of dimension 2n. Its 2k-dimensional shadow is its orthogonal projection onto a complex subspace of real dimension 2k. Is it true that the volume of this 2k-dimensional shadow is at least the volume of the unit 2k-dimensional ball? This statement is trivially true when k = n, and when k = 1 it is a reformulation of Gromov's non-squeezing theorem. Therefore, this question can be considered as a middle-dimensional generalization of the non-squeezing theorem. We investigate the validity of this statement in the linear, nonlinear and perturbative setting.

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