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Murat Gunaydin - One of the best experts on this subject based on the ideXlab platform.
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minimal Unitary Representation of 5d superconformal algebra f 4 and ads6 cft5 higher spin super algebras
Nuclear Physics, 2015Co-Authors: Sudarshan Fernando, Murat GunaydinAbstract:Abstract We study the minimal Unitary Representation (minrep) of SO ( 5 , 2 ) , obtained by quantization of its geometric quasiconformal action, its deformations and supersymmetric extensions. The minrep of SO ( 5 , 2 ) describes a massless conformal scalar field in five dimensions and admits a unique “deformation” which describes a massless conformal spinor. Scalar and spinor minreps of SO ( 5 , 2 ) are the 5 d analogs of Dirac's singletons of SO ( 3 , 2 ) . We then construct the minimal Unitary Representation of the unique 5 d superconformal algebra F ( 4 ) with the even subalgebra SO ( 5 , 2 ) × SU ( 2 ) . The minrep of F ( 4 ) describes a massless conformal supermultiplet consisting of two scalar and one spinor fields. We then extend our results to the construction of higher spin AdS 6 / CFT 5 (super)-algebras. The Joseph ideal of the minrep of SO ( 5 , 2 ) vanishes identically as operators and hence its enveloping algebra yields the AdS 6 / CFT 5 bosonic higher spin algebra directly. The enveloping algebra of the spinor minrep defines a “deformed” higher spin algebra for which a deformed Joseph ideal vanishes identically as operators. These results are then extended to the construction of the unique higher spin AdS 6 / CFT 5 superalgebra as the enveloping algebra of the minimal Unitary realization of F ( 4 ) obtained by the quasiconformal methods.
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Minimal Unitary Representation of 5d superconformal algebra F(4) and AdS6/CFT5 higher spin (super)-algebras
Nuclear Physics, 2014Co-Authors: Sudarshan Fernando, Murat GunaydinAbstract:Abstract We study the minimal Unitary Representation (minrep) of SO ( 5 , 2 ) , obtained by quantization of its geometric quasiconformal action, its deformations and supersymmetric extensions. The minrep of SO ( 5 , 2 ) describes a massless conformal scalar field in five dimensions and admits a unique “deformation” which describes a massless conformal spinor. Scalar and spinor minreps of SO ( 5 , 2 ) are the 5 d analogs of Dirac's singletons of SO ( 3 , 2 ) . We then construct the minimal Unitary Representation of the unique 5 d superconformal algebra F ( 4 ) with the even subalgebra SO ( 5 , 2 ) × SU ( 2 ) . The minrep of F ( 4 ) describes a massless conformal supermultiplet consisting of two scalar and one spinor fields. We then extend our results to the construction of higher spin AdS 6 / CFT 5 (super)-algebras. The Joseph ideal of the minrep of SO ( 5 , 2 ) vanishes identically as operators and hence its enveloping algebra yields the AdS 6 / CFT 5 bosonic higher spin algebra directly. The enveloping algebra of the spinor minrep defines a “deformed” higher spin algebra for which a deformed Joseph ideal vanishes identically as operators. These results are then extended to the construction of the unique higher spin AdS 6 / CFT 5 superalgebra as the enveloping algebra of the minimal Unitary realization of F ( 4 ) obtained by the quasiconformal methods.
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su 2 deformations of the minimal Unitary Representation of osp 8 2n as massless 6d conformal supermultiplets
Nuclear Physics, 2011Co-Authors: Sudarshan Fernando, Murat GunaydinAbstract:Abstract Minimal Unitary Representation of SO ⁎ ( 8 ) ≃ SO ( 6 , 2 ) realized over the Hilbert space of functions of five variables and its deformations labeled by the spin t of an SU ( 2 ) subgroup correspond to massless conformal fields in six dimensions as was shown in [S. Fernando, M. Gunaydin, arXiv:1005.3580 ]. In this paper we study the minimal Unitary supermultiplet of OSp ( 8 ⁎ | 2 N ) with the even subgroup SO ⁎ ( 8 ) × USp ( 2 N ) and its deformations using quasiconformal methods. We show that the minimal Unitary supermultiplet of OSp ( 8 ⁎ | 2 N ) admits deformations labeled uniquely by the spin t of an SU ( 2 ) subgroup of the little group SO ( 4 ) of lightlike vectors in six dimensions. We construct the deformed minimal Unitary Representations and show that they correspond to massless 6D conformal supermultiplets. The minimal Unitary supermultiplet of OSp ( 8 ⁎ | 4 ) is the massless supermultiplet of ( 2 , 0 ) conformal field theory that is believed to be dual to M-theory on AdS 7 × S 4 . We study its deformations in further detail and show that they are isomorphic to the doubleton supermultiplets constructed by using twistorial oscillators.
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su 2 su 2 deformations of the minimal Unitary Representation of osp 8 2n osp 8 2n as massless 6d conformal supermultiplets
Nuclear Physics, 2011Co-Authors: Sudarshan Fernando, Murat GunaydinAbstract:Minimal Unitary Representation of SO⁎(8)≃SO(6,2)SO⁎(8)≃SO(6,2) realized over the Hilbert space of functions of five variables and its deformations labeled by the spin t of an SU(2)SU(2) subgroup correspond to massless conformal fields in six dimensions as was shown in [S. Fernando, M. Gunaydin, arXiv:1005.3580]. In this paper we study the minimal Unitary supermultiplet of OSp(8⁎|2N)OSp(8⁎|2N) with the even subgroup SO⁎(8)×USp(2N)SO⁎(8)×USp(2N) and its deformations using quasiconformal methods. We show that the minimal Unitary supermultiplet of OSp(8⁎|2N)OSp(8⁎|2N) admits deformations labeled uniquely by the spin t of an SU(2)SU(2) subgroup of the little group SO(4)SO(4) of lightlike vectors in six dimensions. We construct the deformed minimal Unitary Representations and show that they correspond to massless 6D conformal supermultiplets. The minimal Unitary supermultiplet of OSp(8⁎|4)OSp(8⁎|4) is the massless supermultiplet of (2,0)(2,0) conformal field theory that is believed to be dual to M-theory on AdS7×S4AdS7×S4. We study its deformations in further detail and show that they are isomorphic to the doubleton supermultiplets constructed by using twistorial oscillators.
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deformations of the minimal Unitary Representation of as massless 6D conformal supermultiplets
Nuclear Physics, 2011Co-Authors: Sudarshan Fernando, Murat GunaydinAbstract:Abstract Minimal Unitary Representation of SO ⁎ ( 8 ) ≃ SO ( 6 , 2 ) realized over the Hilbert space of functions of five variables and its deformations labeled by the spin t of an SU ( 2 ) subgroup correspond to massless conformal fields in six dimensions as was shown in [S. Fernando, M. Gunaydin, arXiv:1005.3580 ]. In this paper we study the minimal Unitary supermultiplet of OSp ( 8 ⁎ | 2 N ) with the even subgroup SO ⁎ ( 8 ) × USp ( 2 N ) and its deformations using quasiconformal methods. We show that the minimal Unitary supermultiplet of OSp ( 8 ⁎ | 2 N ) admits deformations labeled uniquely by the spin t of an SU ( 2 ) subgroup of the little group SO ( 4 ) of lightlike vectors in six dimensions. We construct the deformed minimal Unitary Representations and show that they correspond to massless 6D conformal supermultiplets. The minimal Unitary supermultiplet of OSp ( 8 ⁎ | 4 ) is the massless supermultiplet of ( 2 , 0 ) conformal field theory that is believed to be dual to M-theory on AdS 7 × S 4 . We study its deformations in further detail and show that they are isomorphic to the doubleton supermultiplets constructed by using twistorial oscillators.
Sudarshan Fernando - One of the best experts on this subject based on the ideXlab platform.
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minimal Unitary Representation of 5d superconformal algebra f 4 and ads6 cft5 higher spin super algebras
Nuclear Physics, 2015Co-Authors: Sudarshan Fernando, Murat GunaydinAbstract:Abstract We study the minimal Unitary Representation (minrep) of SO ( 5 , 2 ) , obtained by quantization of its geometric quasiconformal action, its deformations and supersymmetric extensions. The minrep of SO ( 5 , 2 ) describes a massless conformal scalar field in five dimensions and admits a unique “deformation” which describes a massless conformal spinor. Scalar and spinor minreps of SO ( 5 , 2 ) are the 5 d analogs of Dirac's singletons of SO ( 3 , 2 ) . We then construct the minimal Unitary Representation of the unique 5 d superconformal algebra F ( 4 ) with the even subalgebra SO ( 5 , 2 ) × SU ( 2 ) . The minrep of F ( 4 ) describes a massless conformal supermultiplet consisting of two scalar and one spinor fields. We then extend our results to the construction of higher spin AdS 6 / CFT 5 (super)-algebras. The Joseph ideal of the minrep of SO ( 5 , 2 ) vanishes identically as operators and hence its enveloping algebra yields the AdS 6 / CFT 5 bosonic higher spin algebra directly. The enveloping algebra of the spinor minrep defines a “deformed” higher spin algebra for which a deformed Joseph ideal vanishes identically as operators. These results are then extended to the construction of the unique higher spin AdS 6 / CFT 5 superalgebra as the enveloping algebra of the minimal Unitary realization of F ( 4 ) obtained by the quasiconformal methods.
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Minimal Unitary Representation of 5d superconformal algebra F(4) and AdS6/CFT5 higher spin (super)-algebras
Nuclear Physics, 2014Co-Authors: Sudarshan Fernando, Murat GunaydinAbstract:Abstract We study the minimal Unitary Representation (minrep) of SO ( 5 , 2 ) , obtained by quantization of its geometric quasiconformal action, its deformations and supersymmetric extensions. The minrep of SO ( 5 , 2 ) describes a massless conformal scalar field in five dimensions and admits a unique “deformation” which describes a massless conformal spinor. Scalar and spinor minreps of SO ( 5 , 2 ) are the 5 d analogs of Dirac's singletons of SO ( 3 , 2 ) . We then construct the minimal Unitary Representation of the unique 5 d superconformal algebra F ( 4 ) with the even subalgebra SO ( 5 , 2 ) × SU ( 2 ) . The minrep of F ( 4 ) describes a massless conformal supermultiplet consisting of two scalar and one spinor fields. We then extend our results to the construction of higher spin AdS 6 / CFT 5 (super)-algebras. The Joseph ideal of the minrep of SO ( 5 , 2 ) vanishes identically as operators and hence its enveloping algebra yields the AdS 6 / CFT 5 bosonic higher spin algebra directly. The enveloping algebra of the spinor minrep defines a “deformed” higher spin algebra for which a deformed Joseph ideal vanishes identically as operators. These results are then extended to the construction of the unique higher spin AdS 6 / CFT 5 superalgebra as the enveloping algebra of the minimal Unitary realization of F ( 4 ) obtained by the quasiconformal methods.
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su 2 deformations of the minimal Unitary Representation of osp 8 2n as massless 6d conformal supermultiplets
Nuclear Physics, 2011Co-Authors: Sudarshan Fernando, Murat GunaydinAbstract:Abstract Minimal Unitary Representation of SO ⁎ ( 8 ) ≃ SO ( 6 , 2 ) realized over the Hilbert space of functions of five variables and its deformations labeled by the spin t of an SU ( 2 ) subgroup correspond to massless conformal fields in six dimensions as was shown in [S. Fernando, M. Gunaydin, arXiv:1005.3580 ]. In this paper we study the minimal Unitary supermultiplet of OSp ( 8 ⁎ | 2 N ) with the even subgroup SO ⁎ ( 8 ) × USp ( 2 N ) and its deformations using quasiconformal methods. We show that the minimal Unitary supermultiplet of OSp ( 8 ⁎ | 2 N ) admits deformations labeled uniquely by the spin t of an SU ( 2 ) subgroup of the little group SO ( 4 ) of lightlike vectors in six dimensions. We construct the deformed minimal Unitary Representations and show that they correspond to massless 6D conformal supermultiplets. The minimal Unitary supermultiplet of OSp ( 8 ⁎ | 4 ) is the massless supermultiplet of ( 2 , 0 ) conformal field theory that is believed to be dual to M-theory on AdS 7 × S 4 . We study its deformations in further detail and show that they are isomorphic to the doubleton supermultiplets constructed by using twistorial oscillators.
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su 2 su 2 deformations of the minimal Unitary Representation of osp 8 2n osp 8 2n as massless 6d conformal supermultiplets
Nuclear Physics, 2011Co-Authors: Sudarshan Fernando, Murat GunaydinAbstract:Minimal Unitary Representation of SO⁎(8)≃SO(6,2)SO⁎(8)≃SO(6,2) realized over the Hilbert space of functions of five variables and its deformations labeled by the spin t of an SU(2)SU(2) subgroup correspond to massless conformal fields in six dimensions as was shown in [S. Fernando, M. Gunaydin, arXiv:1005.3580]. In this paper we study the minimal Unitary supermultiplet of OSp(8⁎|2N)OSp(8⁎|2N) with the even subgroup SO⁎(8)×USp(2N)SO⁎(8)×USp(2N) and its deformations using quasiconformal methods. We show that the minimal Unitary supermultiplet of OSp(8⁎|2N)OSp(8⁎|2N) admits deformations labeled uniquely by the spin t of an SU(2)SU(2) subgroup of the little group SO(4)SO(4) of lightlike vectors in six dimensions. We construct the deformed minimal Unitary Representations and show that they correspond to massless 6D conformal supermultiplets. The minimal Unitary supermultiplet of OSp(8⁎|4)OSp(8⁎|4) is the massless supermultiplet of (2,0)(2,0) conformal field theory that is believed to be dual to M-theory on AdS7×S4AdS7×S4. We study its deformations in further detail and show that they are isomorphic to the doubleton supermultiplets constructed by using twistorial oscillators.
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deformations of the minimal Unitary Representation of as massless 6D conformal supermultiplets
Nuclear Physics, 2011Co-Authors: Sudarshan Fernando, Murat GunaydinAbstract:Abstract Minimal Unitary Representation of SO ⁎ ( 8 ) ≃ SO ( 6 , 2 ) realized over the Hilbert space of functions of five variables and its deformations labeled by the spin t of an SU ( 2 ) subgroup correspond to massless conformal fields in six dimensions as was shown in [S. Fernando, M. Gunaydin, arXiv:1005.3580 ]. In this paper we study the minimal Unitary supermultiplet of OSp ( 8 ⁎ | 2 N ) with the even subgroup SO ⁎ ( 8 ) × USp ( 2 N ) and its deformations using quasiconformal methods. We show that the minimal Unitary supermultiplet of OSp ( 8 ⁎ | 2 N ) admits deformations labeled uniquely by the spin t of an SU ( 2 ) subgroup of the little group SO ( 4 ) of lightlike vectors in six dimensions. We construct the deformed minimal Unitary Representations and show that they correspond to massless 6D conformal supermultiplets. The minimal Unitary supermultiplet of OSp ( 8 ⁎ | 4 ) is the massless supermultiplet of ( 2 , 0 ) conformal field theory that is believed to be dual to M-theory on AdS 7 × S 4 . We study its deformations in further detail and show that they are isomorphic to the doubleton supermultiplets constructed by using twistorial oscillators.
Y Takaya - One of the best experts on this subject based on the ideXlab platform.
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A new noncommutative product on the fuzzy two-sphere corresponding to the Unitary Representation of SU(2) and the Seiberg–Witten map
Physics Letters B, 2020Co-Authors: Kiyoshi Hayasaka, Ryuichi Nakayama, Y TakayaAbstract:AbstractWe obtain a new explicit expression for the noncommutative (star) product on the fuzzy two-sphere which yields a Unitary Representation. This is done by constructing a star product, ★λ, for an arbitrary Representation of SU(2) which depends on a continuous parameter λ and searching for the values of λ which give Unitary Representations. We will find two series of values: λ=λ(A)j=1/(2j) and λ=λ(B)j=−1/(2j+2), where j is the spin of the Representation of SU(2). At λ=λ(A)j the new star product ★λ has poles. To avoid the singularity the functions on the sphere must be spherical harmonics of order ℓ⩽2j and then ★λ reduces to the star product ★ obtained by Pres̆najder [hep-th/9912050]. The star product at λ=λ(B)j, to be denoted by •, is new. In this case the functions on the fuzzy sphere do not need to be spherical harmonics of order ℓ⩽2j. The star product ★λ has no singularity for negative values of λ and we can move from one Representation λ=λ(B)j to another λ=λ(B)j′ smoothly on the negative λ line. Because in this case there is no cutoff on the order of spherical harmonics, the degrees of freedom of the gauge fields on the fuzzy sphere coincide with those on the commutative sphere. Therefore, although the field theory on the fuzzy sphere is a system with finite degrees of freedom, we can expect the existence of the Seiberg–Witten map between the noncommutative and commutative descriptions of the gauge theory on the sphere. We will derive the first few terms of the Seiberg–Witten map for the U(1) gauge theory on the fuzzy sphere by using power expansion around the commutative point λ=0
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a new noncommutative product on the fuzzy two sphere corresponding to the Unitary Representation of su 2 and the seiberg witten map
Physics Letters B, 2003Co-Authors: Kiyoshi Hayasaka, Ryuichi Nakayama, Y TakayaAbstract:Abstract We obtain a new explicit expression for the noncommutative (star) product on the fuzzy two-sphere which yields a Unitary Representation. This is done by constructing a star product, ★λ, for an arbitrary Representation of SU(2) which depends on a continuous parameter λ and searching for the values of λ which give Unitary Representations. We will find two series of values: λ=λ(A)j=1/(2j) and λ=λ(B)j=−1/(2j+2), where j is the spin of the Representation of SU(2). At λ=λ(A)j the new star product ★λ has poles. To avoid the singularity the functions on the sphere must be spherical harmonics of order l⩽2j and then ★λ reduces to the star product ★ obtained by Presnajder [hep-th/9912050]. The star product at λ=λ(B)j, to be denoted by •, is new. In this case the functions on the fuzzy sphere do not need to be spherical harmonics of order l⩽2j. The star product ★λ has no singularity for negative values of λ and we can move from one Representation λ=λ(B)j to another λ=λ(B)j′ smoothly on the negative λ line. Because in this case there is no cutoff on the order of spherical harmonics, the degrees of freedom of the gauge fields on the fuzzy sphere coincide with those on the commutative sphere. Therefore, although the field theory on the fuzzy sphere is a system with finite degrees of freedom, we can expect the existence of the Seiberg–Witten map between the noncommutative and commutative descriptions of the gauge theory on the sphere. We will derive the first few terms of the Seiberg–Witten map for the U(1) gauge theory on the fuzzy sphere by using power expansion around the commutative point λ=0.
Kiyoshi Hayasaka - One of the best experts on this subject based on the ideXlab platform.
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A new noncommutative product on the fuzzy two-sphere corresponding to the Unitary Representation of SU(2) and the Seiberg–Witten map
Physics Letters B, 2020Co-Authors: Kiyoshi Hayasaka, Ryuichi Nakayama, Y TakayaAbstract:AbstractWe obtain a new explicit expression for the noncommutative (star) product on the fuzzy two-sphere which yields a Unitary Representation. This is done by constructing a star product, ★λ, for an arbitrary Representation of SU(2) which depends on a continuous parameter λ and searching for the values of λ which give Unitary Representations. We will find two series of values: λ=λ(A)j=1/(2j) and λ=λ(B)j=−1/(2j+2), where j is the spin of the Representation of SU(2). At λ=λ(A)j the new star product ★λ has poles. To avoid the singularity the functions on the sphere must be spherical harmonics of order ℓ⩽2j and then ★λ reduces to the star product ★ obtained by Pres̆najder [hep-th/9912050]. The star product at λ=λ(B)j, to be denoted by •, is new. In this case the functions on the fuzzy sphere do not need to be spherical harmonics of order ℓ⩽2j. The star product ★λ has no singularity for negative values of λ and we can move from one Representation λ=λ(B)j to another λ=λ(B)j′ smoothly on the negative λ line. Because in this case there is no cutoff on the order of spherical harmonics, the degrees of freedom of the gauge fields on the fuzzy sphere coincide with those on the commutative sphere. Therefore, although the field theory on the fuzzy sphere is a system with finite degrees of freedom, we can expect the existence of the Seiberg–Witten map between the noncommutative and commutative descriptions of the gauge theory on the sphere. We will derive the first few terms of the Seiberg–Witten map for the U(1) gauge theory on the fuzzy sphere by using power expansion around the commutative point λ=0
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a new noncommutative product on the fuzzy two sphere corresponding to the Unitary Representation of su 2 and the seiberg witten map
Physics Letters B, 2003Co-Authors: Kiyoshi Hayasaka, Ryuichi Nakayama, Y TakayaAbstract:Abstract We obtain a new explicit expression for the noncommutative (star) product on the fuzzy two-sphere which yields a Unitary Representation. This is done by constructing a star product, ★λ, for an arbitrary Representation of SU(2) which depends on a continuous parameter λ and searching for the values of λ which give Unitary Representations. We will find two series of values: λ=λ(A)j=1/(2j) and λ=λ(B)j=−1/(2j+2), where j is the spin of the Representation of SU(2). At λ=λ(A)j the new star product ★λ has poles. To avoid the singularity the functions on the sphere must be spherical harmonics of order l⩽2j and then ★λ reduces to the star product ★ obtained by Presnajder [hep-th/9912050]. The star product at λ=λ(B)j, to be denoted by •, is new. In this case the functions on the fuzzy sphere do not need to be spherical harmonics of order l⩽2j. The star product ★λ has no singularity for negative values of λ and we can move from one Representation λ=λ(B)j to another λ=λ(B)j′ smoothly on the negative λ line. Because in this case there is no cutoff on the order of spherical harmonics, the degrees of freedom of the gauge fields on the fuzzy sphere coincide with those on the commutative sphere. Therefore, although the field theory on the fuzzy sphere is a system with finite degrees of freedom, we can expect the existence of the Seiberg–Witten map between the noncommutative and commutative descriptions of the gauge theory on the sphere. We will derive the first few terms of the Seiberg–Witten map for the U(1) gauge theory on the fuzzy sphere by using power expansion around the commutative point λ=0.
Ryuichi Nakayama - One of the best experts on this subject based on the ideXlab platform.
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A new noncommutative product on the fuzzy two-sphere corresponding to the Unitary Representation of SU(2) and the Seiberg–Witten map
Physics Letters B, 2020Co-Authors: Kiyoshi Hayasaka, Ryuichi Nakayama, Y TakayaAbstract:AbstractWe obtain a new explicit expression for the noncommutative (star) product on the fuzzy two-sphere which yields a Unitary Representation. This is done by constructing a star product, ★λ, for an arbitrary Representation of SU(2) which depends on a continuous parameter λ and searching for the values of λ which give Unitary Representations. We will find two series of values: λ=λ(A)j=1/(2j) and λ=λ(B)j=−1/(2j+2), where j is the spin of the Representation of SU(2). At λ=λ(A)j the new star product ★λ has poles. To avoid the singularity the functions on the sphere must be spherical harmonics of order ℓ⩽2j and then ★λ reduces to the star product ★ obtained by Pres̆najder [hep-th/9912050]. The star product at λ=λ(B)j, to be denoted by •, is new. In this case the functions on the fuzzy sphere do not need to be spherical harmonics of order ℓ⩽2j. The star product ★λ has no singularity for negative values of λ and we can move from one Representation λ=λ(B)j to another λ=λ(B)j′ smoothly on the negative λ line. Because in this case there is no cutoff on the order of spherical harmonics, the degrees of freedom of the gauge fields on the fuzzy sphere coincide with those on the commutative sphere. Therefore, although the field theory on the fuzzy sphere is a system with finite degrees of freedom, we can expect the existence of the Seiberg–Witten map between the noncommutative and commutative descriptions of the gauge theory on the sphere. We will derive the first few terms of the Seiberg–Witten map for the U(1) gauge theory on the fuzzy sphere by using power expansion around the commutative point λ=0
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a new noncommutative product on the fuzzy two sphere corresponding to the Unitary Representation of su 2 and the seiberg witten map
Physics Letters B, 2003Co-Authors: Kiyoshi Hayasaka, Ryuichi Nakayama, Y TakayaAbstract:Abstract We obtain a new explicit expression for the noncommutative (star) product on the fuzzy two-sphere which yields a Unitary Representation. This is done by constructing a star product, ★λ, for an arbitrary Representation of SU(2) which depends on a continuous parameter λ and searching for the values of λ which give Unitary Representations. We will find two series of values: λ=λ(A)j=1/(2j) and λ=λ(B)j=−1/(2j+2), where j is the spin of the Representation of SU(2). At λ=λ(A)j the new star product ★λ has poles. To avoid the singularity the functions on the sphere must be spherical harmonics of order l⩽2j and then ★λ reduces to the star product ★ obtained by Presnajder [hep-th/9912050]. The star product at λ=λ(B)j, to be denoted by •, is new. In this case the functions on the fuzzy sphere do not need to be spherical harmonics of order l⩽2j. The star product ★λ has no singularity for negative values of λ and we can move from one Representation λ=λ(B)j to another λ=λ(B)j′ smoothly on the negative λ line. Because in this case there is no cutoff on the order of spherical harmonics, the degrees of freedom of the gauge fields on the fuzzy sphere coincide with those on the commutative sphere. Therefore, although the field theory on the fuzzy sphere is a system with finite degrees of freedom, we can expect the existence of the Seiberg–Witten map between the noncommutative and commutative descriptions of the gauge theory on the sphere. We will derive the first few terms of the Seiberg–Witten map for the U(1) gauge theory on the fuzzy sphere by using power expansion around the commutative point λ=0.
