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Grosse, Harald; Steinacker, Harold

doi:10.1016/j.nuclphysb.2004.11.058

Cited by 79 publications

(104 citation statements)

References 36 publications

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“…By the dimensional reduction, we obtain YM-higgs on S 2n+1 and CP n and a matrix model. We find the commutative (continuum) limit of gauge theory on fuzzy CP n [28,[31][32][33][34][35] realized in the matrix model coincides with YM-higgs on CP n . Namely, we show that the theory around each monopole vacuum of YM-higgs on CP n is equivalent to the theory around a certain vacuum of the matrix model.…”

Section: Introduction and Conclusionmentioning

confidence: 80%

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Fiber bundles and matrix models

Ishii¹,

Ishiki²,

Shimasaki³

et al. 2008

Phys. Rev. D

103

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We investigate relationship between a gauge theory on a principal bundle and that on its base space. In the case where the principal bundle is itself a group manifold, we also study relations of those gauge theories with a matrix model obtained by dimensionally reducing them to zero dimensions. First, we develop the dimensional reduction of YangMills (YM) on the total space to YM-higgs on the base space for a general principal bundle. Second, we show a relationship that YM on an SU (2) bundle is equivalent to the theory around a certain background of YM-higgs on its base space. This is an extension of our previous work [29], in which the same relationship concerning a U (1) bundle is shown. We apply these results to the case of SU (n + 1) as the total space. By dimensionally reducing YM on SU (n + 1), we obtain YM-higgs on SU (n + 1)/SU (n) ≃ S 2n+1 and on SU (n + 1)/(SU (n) × U (1)) ≃ CP n and a matrix model. We show that the theory around each monopole vacuum of YM-higgs on CP n is equivalent to the theory around a certain vacuum of the matrix model in the commutative limit. By combining this with the relationship concerning a U (1) bundle, we realize YM-higgs on SU (n + 1)/SU (n) ≃ S 2n+1 in the matrix model. We see that the relationship concerning a U (1) bundle can be interpreted as Buscher's

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Section: Introduction and Conclusionmentioning

confidence: 80%

“…The vacuum of YM-higgs on S 2 we take is given by (2.22) with s running from −∞ to ∞, n s = s and N s = N. 4πg 2 S 2 /µ is identified with the coupling constant on S 3 , g 2 S 3 . We decompose the fields on S 2 into the background and the fluctuation, 34) and impose the periodicity (orbifolding) condition on the fluctuation,…”

Section: [J]mentioning

confidence: 99%

Fiber bundles and matrix models

Ishii¹,

Ishiki²,

Shimasaki³

et al. 2008

Phys. Rev. D

103

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show abstract

“…On the other hand, a fuzzy space description for gauge theory was studied in some papers, including [65,66,67,68,69,70,71,72,73,74,75,76,58].…”

Section: General Formalism and Theoretical Predictionsmentioning

confidence: 99%

Quantum Field Theory in a Non-Commutative Space: Theoretical Predictions and Numerical Results on the Fuzzy Sphere

Panero¹

2006

SIGMA

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Abstract. We review some recent progress in quantum field theory in non-commutative space, focusing onto the fuzzy sphere as a non-perturbative regularisation scheme. We first introduce the basic formalism, and discuss the limits corresponding to different commutative or non-commutative spaces. We present some of the theories which have been investigated in this framework, with a particular attention to the scalar model. Then we comment on the results recently obtained from Monte Carlo simulations, and show a preview of new numerical data, which are consistent with the expected transition between two phases characterised by the topology of the support of a matrix eigenvalue distribution.

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“…A similar construction was given in [39] for the case of CP 2 , and applied to S 2 N in a different way in [11]. To justify this claim, we must check that the orbit O captures the correct number of degrees of freedom at least in the commutative limit N → ∞, i.e.…”

Section: Configuration Space Of Gauge Fieldsmentioning

confidence: 99%

“…This means that the hamiltonian function (2.35) defines a periodic flow generated by the action of a one-parameter subgroup C → e i t φ C e − i t φ , t ∈ R. The corresponding equivariant moment map µ : O(Ξ) → u(N ) ∨ is the inclusion map which has the pairings 39) and it defines a representation of the Lie algebra u(N ) through the Poisson algebra corresponding to ω.…”

Section: Symplectic Geometry Of the Configuration Spacementioning

confidence: 99%

Localization for Yang-Mills Theory on the Fuzzy Sphere

Steinacker

Szabo

2007

Commun. Math. Phys.

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We present a new model for Yang-Mills theory on the fuzzy sphere in which the configuration space of gauge fields is given by a coadjoint orbit. In the classical limit it reduces to ordinary Yang-Mills theory on the sphere. We find all classical solutions of the gauge theory and use nonabelian localization techniques to write the partition function entirely as a sum over local contributions from critical points of the action, which are evaluated explicitly. The partition function of ordinary Yang-Mills theory on the sphere is recovered in the classical limit as a sum over instantons. We also apply abelian localization techniques and the geometry of symmetric spaces to derive an explicit combinatorial expression for the partition function, and compare the two approaches. These extend the standard techniques for solving gauge theory on the sphere to the fuzzy case in a rigorous framework.

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Finite gauge theory on fuzzy CP2

Cited by 79 publications

References 36 publications

Fiber bundles and matrix models

Fiber bundles and matrix models

Quantum Field Theory in a Non-Commutative Space: Theoretical Predictions and Numerical Results on the Fuzzy Sphere

Localization for Yang-Mills Theory on the Fuzzy Sphere

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