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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.


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.


Hasebe K.,Takuma National College of Technology | Totsuka K.,Kyoto University
Physical Review B - Condensed Matter and Materials Physics | Year: 2013

We present a detailed analysis of topological properties of the valence-bond solid (VBS) states doped with fermionic holes. As concrete examples, we consider the supersymmetric extension of the SU(2) and the SO(5) VBS states, dubbed UOSp(1|2) and UOSp(1|4) supersymmetric VBS states, respectively. Specifically, we investigate the string-order parameters and the entanglement spectra of these states to find that, even when the parent states (bosonic VBS states) do not support the string order, they recover it when holes are doped and the fermionic sector appears in the entanglement spectrum. These peculiar properties are discussed in light of the symmetry-protected topological order. To this end, we characterize a few typical classes of symmetry-protected topological orders in terms of supermatrix product states (SMPS). From this, we see that the topological order in the bulk manifests itself in the transformation properties of the SMPS in question and thereby affects the structure of the entanglement spectrum. Then, we explicitly relate the existence of the string order and the structure of the entanglement spectrum to explain the recovery and the stabilization of the string order in the supersymmetric systems. © 2013 American Physical Society.


Hasebe K.,Takuma National College of Technology
Physical Review D - Particles, Fields, Gravitation and Cosmology | Year: 2010

Introducing a noncompact version of the Hopf map, we demonstrate remarkable close relations between quantum Hall effect and twistor theory. We first construct quantum Hall effect on a hyperboloid based on the noncompact 2nd Hopf map of split-quaternions. We analyze a hyperbolic one-particle mechanics, and explore many-body problem, where a many-body ground state wave function and membrane-like excitations are derived explicitly. In the lowest Landau level, the symmetry is enhanced from SO(3,2) to the SU(2,2) conformal symmetry. We point out that the quantum Hall effect naturally realizes the philosophy of twistor theory. In particular, emergence mechanism of fuzzy space-time is discussed somehow in detail. © 2010 The American Physical Society.


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.


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.


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

Chiral topological insulator (AIII-class) with Landau levels is constructed based on the Nambu 3-algebraic geometry. We clarify the geometric origin of the chiral symmetry of the AIII-class topological insulator in the context of non-commutative geometry of 4D quantum Hall effect. The many-body groundstate wavefunction is explicitly derived as a (l, l, l - 1) Laughlin-Halperin type wavefunction with unique K-matrix structure. Fundamental excitation is identified with anyonic string-like object with fractional charge 1/(2(l -1)2 + 1). The Hall effect of the chiral topological insulators turns out be a color version of Hall effect, which exhibits a dual property of the Hall and spin-Hall effects. © 2014 The Author.


Hasebe K.,Takuma National College of Technology | Totsuka K.,Kyoto University
Physical Review B - Condensed Matter and Materials Physics | Year: 2011

Supersymmetric valence-bond solid models are extensions of the VBS model, a paradigmatic model of "solvable" gapped quantum antiferromagnets, to the case with doped fermionic holes. In this paper, we present a detailed analysis of physical properties of the models. For systematic studies, a supersymmetric version of the matrix-product formalism is developed. On 1D chains, we exactly evaluate the hole-doping behavior of various physical quantities, such as the spin or the charge excitation spectrum, the superconducting order parameter. A generalized hidden order is proposed, and the corresponding string nonlocal order parameter is also calculated. The behavior of the string-order parameter is discussed in the light of the entanglement spectrum. © 2011 American Physical Society.


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.


Taniguchi H.,Takuma National College of Technology
European Journal of Combinatorics | Year: 2010

Let d ≥ 3. Let H be a d + 1-dimensional vector space over G F (2) and {e0, ..., ed} be a specified basis of H. We define S u p p (t) {colon equals} {et1, ..., etl}, a subset of a specified base for a non-zero vector t = et1 + ⋯ + etl of H, and S u p p (0) {colon equals} 0{combining long solidus overlay}. We also define J (t) {colon equals} S u p p (t) if | S u p p (t) | is odd, and J (t) {colon equals} S u p p (t) ∪ {0} if | S u p p (t) | is even. For s, t ∈ H, let {a (s, t)} be elements of H ⊕ (H ∧ H) which satisfy the following conditions: (1) a (s, s) = (0, 0), (2) a (s, t) = a (t, s), (3) a (s, t) ≠ (0, 0) if s ≠ t, (4) a (s, t) = a (s′, t′) if and only if {s, t} = {s′, t′}, (5) {a (s, t) | t ∈ H} is a vector space over G F (2), (6) {a (s, t) | s, t ∈ H} generate H ⊕ (H ∧ H). Then, it is known that S {colon equals} {X (s) | s ∈ H}, where X (s) {colon equals} {a (s, t) | t ∈ H {set minus} {s}}, is a dual hyperoval in P G (d (d + 3) / 2, 2) = (H ⊕ (H ∧ H)) {set minus} {(0, 0)}. In this note, we assume that, for s, t ∈ H, there exists some xs, t in G F (2) such that a (s, t) satisfies the following equation: a (s, t) = under(∑, w ∈ J (t)) a (s, w) + xs, t (a (s, 0) + a (s, e0)) . Then, we prove that the dual hyperoval constructed by {a (s, t)} is isomorphic to either the Huybrechts' dual hyperoval, or the Buratti and Del Fra's dual hyperoval. © 2009 Elsevier Ltd. All rights reserved.

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