Fuzzy Complex Grassmannian Spaces and Their Star Products

Dolan, Brian P.; Jahn, O.

doi:10.1142/s0217751x03014113

Cited by 32 publications

(35 citation statements)

References 30 publications

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“…In Section 2, we generalize and simplify the above description of the fuzzy 2-sphere (S 2 ∼ = P 1 (C)), analogous results of H. Grosse and A. Strohmaier for P 2 (C) (see [10]) and similar, independently obtained results of B.P. Dolan and co-workers (see [2] and [8]) on projective spaces and Grassmannians. We obtain by purely representationtheoretic methods that all complex Grassmannians M = Gr n (C n+m ) allow for a sequence of linear subspaces E N N ≥1 in their function algebra C ∞ (M ) such that E N ⊂ E N +1 and N ≥1 E N is dense in C ∞ (M ), and such that E N is isomorphic to a matrix algebra A N .…”

Section: Introductionmentioning

confidence: 63%

Fuzzy complex Grassmannians and quantization of line bundles

Halima¹,

Wurzbacher²

2009

Abh. Math. Semin. Univ. Hambg.

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We construct by purely representation-theoretic methods fuzzy versions of an arbitrary complex Grassmannian M = Gr n (C n+m ), i.e., a sequence of matrix algebras tending SU (n + m)-equivariantly to the algebra of smooth functions on M . We also show that this approximation can be interpreted in terms of the Berezin-Toeplitz quantization of M . Furthermore, we use branching rules to prove that the quantization of every complex line bundle over M is given by a SU (n + m)-equivariant truncation of the space of its L 2 -sections.

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Section: Introductionmentioning

confidence: 63%

Fuzzy complex Grassmannians and quantization of line bundles

Halima¹,

Wurzbacher²

2009

Abh. Math. Semin. Univ. Hambg.

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“…It is known that the LLL wave functions for the QHE on CP N has a correspondence with the algebra of functions on fuzzy CP N [15]. Extending these results to the LLL wave functions on Gr 2 (C N ) and fuzzy Grassmannians as discussed in [20,24,26], would provide additional insights. It may also be possible to develop Chern-Simons type effective field theories along the lines of [6] to shed more light on the structure of the QHE on Gr 2 (C 4 ) in particular.…”

Section: Discussionmentioning

confidence: 84%

“…It is possible to interchange the Young tableaux of the two SU (2)'s in (3.11). This flips the sign of the U (1) charge, n → −n; in the formulas for the energy and degeneracy, etc, this fact can be compensated by substituting |n| for n. 3 It may be useful to state that this volume is computed with the help of the repeated iteration of (special) unitary group manifolds in terms of the odd dimensional spheres, 20) (for N ≥ 3) where ≈ means "locally equal to" and ∼ = indicates isomorphism. Considering this local expression we can expand all the special unitary groups in (3.1) and employ the volume formula for spheres to obtain an approximation for the volume of the Grassmannians [23], namely,…”

Section: U(1) Gauge Field Backgroundmentioning

confidence: 99%

Constructing the quantum Hall system on the Grassmannians Gr₂(C^N)

Ballı

Behtash

Kürkçüoğlu

et al. 2015

J. Phys.: Conf. Ser.

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Abstract. In this talk, we give the formulation of Quantum Hall Effects (QHEs) on the complex Grassmann manifolds Gr2(C N ). We set up the Landau problem in Gr2(C N ), solve it using group theoretical techniques and provide the energy spectrum and the eigenstates in terms of the SU (N ) Wigner D-functions for charged particles on Gr2(C N ) under the influence of abelian and non-abelian background magnetic monopoles or a combination of these thereof. For the simplest case of Gr2(C 4 ) we provide explicit constructions of the single and manyparticle wavefunctions by introducing the Plücker coordinates and show by calculating the two-point correlation function that the lowest Landau level (LLL) at filling factor ν = 1 forms an incompressible fluid. Finally, we heuristically identify a relation between the U (1) Hall effect on Gr2(C 4 ) and the Hall effect on the odd sphere S 5 , which is yet to be investigated in detail, by appealing to the already known analogous relations between the Hall effects on CP 3 and CP 7 and those on the spheres S 4 and S 8 , respectively. The talk is given by S. Kürkçüoglu at the Group 30 meeting at Ghent University, Ghent, Belgium in July 2014 and based on the article by F.Ballı, A.Behtash, S.Kürkçüoglu, G.Ünal [1]. IntroductionA 4-dimensional generalization of the quantum Hall effect (QHE) was introduced by Hu and Zhang in [2]. They treat the Landau problem on S 4 for charged particles carrying an additional SU (2) degree of freedom which are under the influence of an SU (2) background gauge field. In the thermodynamic limit, the multi-particle problem in the lowest Landau level (LLL) with filling factor ν = 1 may be seen as an incompressible 4-dimensional quantum Hall liquid as demonstrated by these authors. Appearance of massless chiral bosons at the edge of a 2-dimensional quantum Hall droplet [3,4,5,6] generalizes to this setting. It is found that among the edge excitations of this 4-dimensional quantum Hall droplet not only photons and gravitons but also other massless higher spin states occur.Further developments took place after the work of Hu and Zhang. Other higher-dimensional generalizations of QHE to a variety of manifolds including complex projective spaces CP N , S 8 , S 3 , the Flag manifold

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“…There is a way of expressing (22) in terms of d = a ∧ b and d * = r * ∧ s * . First, we unify the notation by defining e 5 := E 5 and e * 5 := E * 5 .…”

Section: The Plücker Embedding Of the Super Flag Fl(2|0 2|1; 4|1)mentioning

confidence: 99%

“…[18,19,20] the property of being coadjoint orbits is also exploited using the so-called Shapovalov pairing of Verma modules. Finally we can mention the possibility of quantizing these spaces as fuzzy spaces [21,22].…”

Section: Introductionmentioning

confidence: 99%

The Segre embedding of the quantum conformal superspace

Fioresi

Latini

Lledó

et al. 2018

Advances in Theoretical and Mathematical Physics

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In this paper study the quantum deformation of the superflag Fl(2|0, 2|1, 4|1), and its big cell, describing the complex conformal and Minkowski superspaces respectively. In particular, we realize their projective embedding via a generalization to the super world of the Segre map and we use it to construct a quantum deformation of the super line bundle realizing this embedding. This strategy allows us to obtain a description of the quantum coordinate superring of the superflag that is then naturally equipped with a coaction of the quantum complex conformal supergroup SL q (4|1).

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Fuzzy Complex Grassmannian Spaces and Their Star Products

Cited by 32 publications

References 30 publications

Fuzzy complex Grassmannians and quantization of line bundles

Fuzzy complex Grassmannians and quantization of line bundles

Constructing the quantum Hall system on the Grassmannians Gr₂(C^N)

The Segre embedding of the quantum conformal superspace

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Fuzzy Complex Grassmannian Spaces and Their Star Products

Cited by 32 publications

References 30 publications

Fuzzy complex Grassmannians and quantization of line bundles

Fuzzy complex Grassmannians and quantization of line bundles

Constructing the quantum Hall system on the Grassmannians Gr2(CN)

The Segre embedding of the quantum conformal superspace

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Constructing the quantum Hall system on the Grassmannians Gr₂(C^N)