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Adjoint Representation
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F. Vian – One of the best experts on this subject based on the ideXlab platform.

wilson loops in the Adjoint Representation and multiple vacua in two dimensional yang mills theory
Annals of Physics, 2000CoAuthors: A. Bassetto, Luca Griguolo, F. VianAbstract:Abstract QCD 2 with fermions in the Adjoint Representation is invariant under SU ( N )/ Z N and thereby is endowed with a nontrivial vacuum structure ( k sectors). The static potential between Adjoint charges, in the limit of infinite mass, can be therefore obtained by computing Wilson loops in the pure Yang–Mills theory with the same nontrivial structure. When the (Euclidean) spacetime is compacted on a sphere S 2 , Wilson loops can be exactly expressed in terms of an infinite series of topological excitations (instantons). The presence of k sectors modifies the energy spectrum of the theory and its instanton content. For the exact solution, in the limit in which the sphere is decompacted, a k sector can be mimicked by the presence of k fundamental charges at ∞, according to Witten’s suggestion. However, this property does not hold before decompaction or for the genuine perturbative solution which corresponds to the zeroinstanton contribution on S 2 .

Wilson loops in the Adjoint Representation and multiple vacua in twodimensional YangMills theory
Annals of Physics, 2000CoAuthors: A. Bassetto, Luca Griguolo, F. VianAbstract:$QCD_2$ with fermions in the Adjoint Representation is invariant under $SU(N)/Z_N$ and thereby is endowed with a nontrivial vacuum structure (ksectors). The static potential between Adjoint charges, in the limit of infinite mass, can be therefore obtained by computing Wilson loops in the pure YangMills theory with the same nontrivial structure. When the (Euclidean) spacetime is compactified on a sphere $S^2$, Wilson loops can be exactly expressed in terms of an infinite series of topological excitations (instantons). The presence of ksectors modifies the energy spectrum of the theory and its instanton content. For the exact solution, in the limit in which the sphere is decompactified, a ksector can be mimicked by the presence of kfundamental charges at $\infty$, according to a Witten’s suggestion. However this property neither holds before decompactification nor for the genuine perturbative solution which corresponds to the zeroinstanton contribution on $S^2$.
Kimmo Tuominen – One of the best experts on this subject based on the ideXlab platform.

Perturbative improvement of SU(2) gauge theory with two Wilson fermions in the Adjoint Representation
arXiv: High Energy Physics – Lattice, 2010CoAuthors: Tuomas Karavirta, Annemari Mykkanen, Jarno Rantaharju, Kari Rummukainen, Kimmo TuominenAbstract:We present a perturbative calculation of the improvement coefficients of SU(2) gauge theory with Adjoint Representation Wilsonclover fermions and using Schrodinger functional bounboundary conditions. The computation of the boundary improvement terms is necessary for the full O(a) improvement. With two flavours of Adjoint Representation fermions this theory is called Minimal Walking Technicolor model.

non perturbatively improved clover action for su 2 gauge fundamental and Adjoint Representation fermions
arXiv: High Energy Physics – Lattice, 2010CoAuthors: Tuomas Karavirta, Kimmo Tuominen, Annemari Mykkanen, Jarno Rantaharju, Kari RummukainenAbstract:The research of strongly coupled beyondthestandardmodel theories has generated significant interest in nonabelian gauge field theories with different number of fermions in different Representations. Motivated by the increased interest to various technicolor scenarios, we study the nonperturbative improvement of the Wilsonclover action with SU(2) gauge fields and 2 flavors of fermions in the fundamental and Adjoint Representations. The SheikholeslamiWohlert coefficients are fixed using Schroedinger functional bounboundary conditions. The Adjoint Representation theory is a candidate for a “minimal technicolor” theory, already studied on the lattice using unimproved Wilson fermions.

Nonperturbatively improved clover action for SU(2) gauge + fundamental and Adjoint Representation fermions
arXiv: High Energy Physics – Lattice, 2010CoAuthors: Annemari Mykkanen, Tuomas Karavirta, Jarno Rantaharju, Kari Rummukainen, Kimmo TuominenAbstract:The research of strongly coupled beyondthestandardmodel theories has generated significant interest in nonabelian gauge field theories with different number of fermions in different Representations. Motivated by the increased interest to various technicolor scenarios, we study the nonperturbative improvement of the Wilsonclover action with SU(2) gauge fields and 2 flavors of fermions in the fundamental and Adjoint Representations. The SheikholeslamiWohlert coefficients are fixed using Schroedinger functional bounboundary conditions. The Adjoint Representation theory is a candidate for a “minimal technicolor” theory, already studied on the lattice using unimproved Wilson fermions.
A. Bassetto – One of the best experts on this subject based on the ideXlab platform.

wilson loops in the Adjoint Representation and multiple vacua in two dimensional yang mills theory
Annals of Physics, 2000CoAuthors: A. Bassetto, Luca Griguolo, F. VianAbstract:Abstract QCD 2 with fermions in the Adjoint Representation is invariant under SU ( N )/ Z N and thereby is endowed with a nontrivial vacuum structure ( k sectors). The static potential between Adjoint charges, in the limit of infinite mass, can be therefore obtained by computing Wilson loops in the pure Yang–Mills theory with the same nontrivial structure. When the (Euclidean) spacetime is compacted on a sphere S 2 , Wilson loops can be exactly expressed in terms of an infinite series of topological excitations (instantons). The presence of k sectors modifies the energy spectrum of the theory and its instanton content. For the exact solution, in the limit in which the sphere is decompacted, a k sector can be mimicked by the presence of k fundamental charges at ∞, according to Witten’s suggestion. However, this property does not hold before decompaction or for the genuine perturbative solution which corresponds to the zeroinstanton contribution on S 2 .

Wilson loops in the Adjoint Representation and multiple vacua in twodimensional YangMills theory
Annals of Physics, 2000CoAuthors: A. Bassetto, Luca Griguolo, F. VianAbstract:$QCD_2$ with fermions in the Adjoint Representation is invariant under $SU(N)/Z_N$ and thereby is endowed with a nontrivial vacuum structure (ksectors). The static potential between Adjoint charges, in the limit of infinite mass, can be therefore obtained by computing Wilson loops in the pure YangMills theory with the same nontrivial structure. When the (Euclidean) spacetime is compactified on a sphere $S^2$, Wilson loops can be exactly expressed in terms of an infinite series of topological excitations (instantons). The presence of ksectors modifies the energy spectrum of the theory and its instanton content. For the exact solution, in the limit in which the sphere is decompactified, a ksector can be mimicked by the presence of kfundamental charges at $\infty$, according to a Witten’s suggestion. However this property neither holds before decompactification nor for the genuine perturbative solution which corresponds to the zeroinstanton contribution on $S^2$.
Mojtaba Nouraddini – One of the best experts on this subject based on the ideXlab platform.

right suq 2 and left suq 1 2 invariances of the q hilbert schmidt scalar products for an Adjoint Representation of the quantum algebra uq su2
Journal of Geometry and Physics, 2016CoAuthors: H. Fakhri, Mojtaba NouraddiniAbstract:Abstract The Jordan–Schwinger realization is used to construct tensor operators as the even and odd dimensional irreducible submodules of an Adjoint Representation of the quantum algebra U q ( s u 2 ) . All U q ( s u 2 ) submodules are equipped with the socalled left and right q Hilbert–Schmidt scalar products by using the Wigner–Eckart theorem. The bases of all irreducible submodules of the Adjoint Representation are orthonormal with respect to the left q Hilbert–Schmidt scalar product, and are orthogonal, but not normalized, with respect to the right one. Consequently, only with respect to the left q Hilbert–Schmidt scalar product, the Adjoint Representation of the quantum algebra U q ( s u 2 ) on the tensor operators is a ∗ –Representation. We show that both left and right q Hilbert–Schmidt scalar products are right S U q ( 2 ) invariant and left S U q − 1 ( 2 ) invariant. Moreover, every irreducible submodule of the Adjoint Representation of the quantum algebra U q ( s u 2 ) as an associative algebra with unit, is a left quantum space for O ( S U q − 1 ( 2 ) ) and a right quantum space for O ( S U q ( 2 ) ) . Finally, it is shown that there is a natural compatibility between the coproducts and the Haar measures of the quantum groups O ( S U q − 1 ( 2 ) ) and O ( S U q ( 2 ) ) and the definitions of the left and right q Hilbert–Schmidt scalar products on the tensor operators of the Hopf algebra U q ( s u 2 ) .

From the WignerEckart theorem to the HilbertSchmidt scalar product for an Adjoint Representation of the quantum algebra Up,q(su2)
International Journal of Geometric Methods in Modern Physics, 2016CoAuthors: H. Fakhri, Mojtaba NouraddiniAbstract:Irreducible tensor operators as the irreducible submodules of an Adjoint Representation of the twoparametric quantum ∗algebra Up,q(su2) are constructed by using its Jordan–Schwinger formulation on two independent (p,q)oscillator ∗algebras. All Up,q(su2)submodules are equipped with an appropriate Hilbert–Schmidt scalar product with the help of the Wigner–Eckart theorem. We show that with respect to this scalar product, not only the bases of all irreducible submodules of the Adjoint Representation are orthonormal, but also the Adjoint Representation is a ∗Representation.

Right SUq(2) – and left SUq−1(2) invariances of the q Hilbert–Schmidt Scalar products for an Adjoint Representation of the quantum algebra Ŭq(su2)
Journal of Geometry and Physics, 2016CoAuthors: H. Fakhri, Mojtaba NouraddiniAbstract:Abstract The Jordan–Schwinger realization is used to construct tensor operators as the even and odd dimensional irreducible submodules of an Adjoint Representation of the quantum algebra U q ( s u 2 ) . All U q ( s u 2 ) submodules are equipped with the socalled left and right q Hilbert–Schmidt scalar products by using the Wigner–Eckart theorem. The bases of all irreducible submodules of the Adjoint Representation are orthonormal with respect to the left q Hilbert–Schmidt scalar product, and are orthogonal, but not normalized, with respect to the right one. Consequently, only with respect to the left q Hilbert–Schmidt scalar product, the Adjoint Representation of the quantum algebra U q ( s u 2 ) on the tensor operators is a ∗ –Representation. We show that both left and right q Hilbert–Schmidt scalar products are right S U q ( 2 ) invariant and left S U q − 1 ( 2 ) invariant. Moreover, every irreducible submodule of the Adjoint Representation of the quantum algebra U q ( s u 2 ) as an associative algebra with unit, is a left quantum space for O ( S U q − 1 ( 2 ) ) and a right quantum space for O ( S U q ( 2 ) ) . Finally, it is shown that there is a natural compatibility between the coproducts and the Haar measures of the quantum groups O ( S U q − 1 ( 2 ) ) and O ( S U q ( 2 ) ) and the definitions of the left and right q Hilbert–Schmidt scalar products on the tensor operators of the Hopf algebra U q ( s u 2 ) .
Kari Rummukainen – One of the best experts on this subject based on the ideXlab platform.

Perturbative improvement of SU(2) gauge theory with two Wilson fermions in the Adjoint Representation
arXiv: High Energy Physics – Lattice, 2010CoAuthors: Tuomas Karavirta, Annemari Mykkanen, Jarno Rantaharju, Kari Rummukainen, Kimmo TuominenAbstract:We present a perturbative calculation of the improvement coefficients of SU(2) gauge theory with Adjoint Representation Wilsonclover fermions and using Schrodinger functional boundary conditions. The computation of the boundary improvement terms is necessary for the full O(a) improvement. With two flavours of Adjoint Representation fermions this theory is called Minimal Walking Technicolor model.

non perturbatively improved clover action for su 2 gauge fundamental and Adjoint Representation fermions
arXiv: High Energy Physics – Lattice, 2010CoAuthors: Tuomas Karavirta, Kimmo Tuominen, Annemari Mykkanen, Jarno Rantaharju, Kari RummukainenAbstract:The research of strongly coupled beyondthestandardmodel theories has generated significant interest in nonabelian gauge field theories with different number of fermions in different Representations. Motivated by the increased interest to various technicolor scenarios, we study the nonperturbative improvement of the Wilsonclover action with SU(2) gauge fields and 2 flavors of fermions in the fundamental and Adjoint Representations. The SheikholeslamiWohlert coefficients are fixed using Schroedinger functional boundary conditions. The Adjoint Representation theory is a candidate for a “minimal technicolor” theory, already studied on the lattice using unimproved Wilson fermions.

Nonperturbatively improved clover action for SU(2) gauge + fundamental and Adjoint Representation fermions
arXiv: High Energy Physics – Lattice, 2010CoAuthors: Annemari Mykkanen, Tuomas Karavirta, Jarno Rantaharju, Kari Rummukainen, Kimmo TuominenAbstract:The research of strongly coupled beyondthestandardmodel theories has generated significant interest in nonabelian gauge field theories with different number of fermions in different Representations. Motivated by the increased interest to various technicolor scenarios, we study the nonperturbative improvement of the Wilsonclover action with SU(2) gauge fields and 2 flavors of fermions in the fundamental and Adjoint Representations. The SheikholeslamiWohlert coefficients are fixed using Schroedinger functional boundary conditions. The Adjoint Representation theory is a candidate for a “minimal technicolor” theory, already studied on the lattice using unimproved Wilson fermions.