Localization for Yang-Mills Theory on the Fuzzy Sphere

Abstract

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.

55 pages. V2: references added; V3: minor corrections, reference added; Final version to be published in Communications in Mathematical Physics

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Cited by in corpus (14)

• Geometry in transition: A model of emergent geometry
• Matrix Models, Gauge Theory and Emergent Geometry
• On the fermion spectrum of spontaneously generated fuzzy extra dimensions with fluxes
• Exact Partition Functions for Gauge Theories on ${\mathbb{R}}_{\lambda }^3$
• Noncommutative gauge theories on ${\mathbb{R}}_{\lambda }^3$: Perturbatively finite models
• On Matrix Model Formulations of Noncommutative Yang-Mills Theories
• Emergent Geometry from Quantized Spacetime
• Closed star product on noncommutative ${\mathbb{R}}^3$ and scalar field dynamics
• Two-Dimensional Gauge Theory and Matrix Model
• Impact of Supersymmetry on Emergent Geometry in Yang-Mills Matrix Models II
• Emergent fuzzy geometry and fuzzy physics in $4$ dimensions
• Spectral theorem in noncommutative field theories: Jacobi dynamics
• Remarks on the eigenvalues distributions of D\leq 4 Yang-Mills matrix models
• The loop equations for noncommutative geometries on quivers