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Hasebe K.,Takuma National College of Technology
Nuclear Physics B | Year: 2012

Fuzzy hyperboloids naturally emerge in the geometries of D-branes, twistor theory, and higher spin theories. In this work, we perform a systematic study of higher dimensional fuzzy hyperboloids (ultra-hyperboloids) based on non-compact Hopf maps. Two types of non-compact Hopf maps; split-type and hybrid-type, are introduced from the cousins of division algebras. We construct arbitrary even-dimensional fuzzy ultra-hyperboloids by applying the Schwinger operator formalism and indefinite Clifford algebras. It is shown that fuzzy hyperboloids, . HF2p,2q, are represented by the coset, . HF2p,2q≃SO(2p,2q+1)/U(p,q), and exhibit two types of generalized dimensional hierarchy; hyperbolic-type (for . q≠. 0) and hybrid-type (for . q=. 0). Fuzzy hyperboloids can be expressed as fibre-bundle of fuzzy fibre over hyperbolic basemanifold. Such bundle structure of fuzzy hyperboloid gives rise to non-compact monopole gauge field. Physical realization of fuzzy hyperboloids is argued in the context of lowest Landau level physics. © 2012 Elsevier B.V. Source


Hasebe K.,Takuma National College of Technology
Nuclear Physics B | Year: 2011

We argue supersymmetric generalizations of fuzzy two- and four-spheres based on the unitary-orthosymplectic algebras, uosp(N|2) and uosp(N|4), respectively. Supersymmetric version of Schwinger construction is applied to derive graded fully symmetric representation for fuzzy superspheres. As a classical counterpart of fuzzy superspheres, graded versions of 1st and 2nd Hopf maps are introduced, and their basic geometrical structures are studied. It is shown that fuzzy superspheres are represented as a "superposition" of fuzzy superspheres with lower supersymmetries. We also investigate algebraic structures of fuzzy two- and four-superspheres to identify su(2|N) and su(4|N) as their enhanced algebraic structures, respectively. Evaluation of correlation functions manifests such enhanced structure as quantum fluctuations of fuzzy supersphere. © 2011 Elsevier B.V. Source


Taniguchi H.,Takuma National College of Technology
Finite Fields and their Applications | Year: 2013

The concept of dimensional dual hyperovals was introduced by Huybrechts and Pasini [4] in 1999. Let d≥3. It is conjectured in Yoshiara (2004) [13] that, if d-dimensional dual hyperoval S generates V(n,2), n-dimensional vector space over GF(2), then 2d-1≤n≤d(d+1)/2. Simply connected d-dimensional dual hyperovals are known only for n=2d-1, n=2d and n=d(d+1)/2. In this note, we will present simply connected d-dimensional dual hyperovals for n=3d-3 with d≥4, n=4d-6 with d≥5, and n=3d-2 with 4≤d≤14 satisfying some conditions. © 2013 Elsevier Inc. Source


Taniguchi H.,Takuma National College of Technology
Finite Fields and their Applications | Year: 2012

Using a quadratic APN function f on GF(2 d+1), Yoshiara (2009) [15] constructed a d-dimensional dual hyperoval S f in PG(2d+1,2). In Taniguchi and Yoshiara (2005) [13], we prove that the dual of S f, which we denote by S f ⊥, is also a d-dimensional dual hyperoval if and only if d is even. In this note, for a quadratic APN function f(x)= x3+Tr( x9) on GF(2 d+1) by Budaghyan, Carlet and Leander (2009) [2], we show that the dual S f ⊥ and the transpose of the dual S f ⊥T are not isomorphic to the known bilinear dual hyperovals if d is even and d≥6. © 2011 Elsevier Inc. All rights reserved. Source


Hasebe K.,Stanford University | Hasebe K.,Takuma National College of Technology
Nuclear Physics B | Year: 2014

We perform a detail study of higher dimensional quantum Hall effects and A-class topological insulators with emphasis on their relations to non-commutative geometry. There are two different formulations of non-commutative geometry for higher dimensional fuzzy spheres: the ordinary commutator formulation and quantum Nambu bracket formulation. Corresponding to these formulations, we introduce two kinds of monopole gauge fields: non-abelian gauge field and antisymmetric tensor gauge field, which respectively realize the non-commutative geometry of fuzzy sphere in the lowest Landau level. We establish connection between the two types of monopole gauge fields through Chern-Simons term, and derive explicit form of tensor monopole gauge fields with higher string-like singularity. The connection between two types of monopole is applied to generalize the concept of flux attachment in quantum Hall effect to A-class topological insulator. We propose tensor type Chern-Simons theory as the effective field theory for membranes in A-class topological insulators. Membranes turn out to be fractionally charged objects and the phase entanglement mediated by tensor gauge field transforms the membrane statistics to be anyonic. The index theorem supports the dimensional hierarchy of A-class topological insulator. Analogies to D-brane physics of string theory are discussed too. © 2014 The Author. Source

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