Fiber Bundles and Matrix Models

Abstract

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 Yang-Mills (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 (hep-th/0703021), 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 $SU(n+1)/SU(n)\simeq S^{2n+1}$ and on $SU(n+1)/(SU(n)\times U(1))\simeq C{P}^n$ and a matrix model. We show that the theory around each monopole vacuum of YM-higgs on $C{P}^n$ is equivalent to the theory around a certain vacuum of the matrix model in the commutative limit. By combing this with the relationship concerning a U(1) bundle, we realize YM-higgs on $SU(n+1)/SU(n)\simeq S^{2n+1}$ in the matrix model. We see that the relationship concerning a U(1) bundle can be interpreted as Buscher's T-duality.

53 pages, references added, typos corrected

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• N=4 Super Yang-Mills from the Plane Wave Matrix Model
• The fuzzy S^2 structure of M2-M5 systems in ABJM membrane theories
• Deconfinement phase transition in N=4 super Yang-Mills theory on RxS^3 from supersymmetric matrix quantum mechanics
• Kac-Moody Extensions of 3-Algebras and M2-branes
• Testing a novel large-N reduction for N=4 super Yang-Mills theory on RxS^3
• Hopf Maps, Lowest Landau Level, and Fuzzy Spheres
• Exact results for perturbative partition functions of theories with SU(2|4) symmetry
• Large N reduction on group manifolds
• Direct test of the AdS/CFT correspondence by Monte Carlo studies of N=4 super Yang-Mills theory
• Black resonators and geons in AdS${}_5$
• Large N reduction for Chern-Simons theory on S^3
• A Novel Large-N Reduction on S^3: Demonstration in Chern-Simons Theory
• Four-dimensional N=1 super Yang-Mills from matrix model
• Large N reduction on coset spaces
• Formulation of Supersymmetry on a Lattice as a Representation of a Deformed Superalgebra
• Large-N reduction for N=2 quiver Chern-Simons theories on S^3 and localization in matrix models
• Localization and Large N reduction on S^3 for the Planar and M-theory limit
• Bounces/Dyons in the Plane Wave Matrix Model and SU(N) Yang-Mills Theory
• Two-Dimensional Gauge Theory and Matrix Model
• Finite-matrix formulation of gauge theories on a non-commutative torus with twisted boundary conditions
• Index theorem in spontaneously symmetry-broken gauge theories on a fuzzy 2-sphere
• The origin of space-time as seen from matrix model simulations
• Interior analysis, stretched technique and bubbling geometries
• Ginsparg-Wilson relation on a fuzzy 2-sphere for adjoint matter
• Quantum corrections of (fuzzy) spacetimes from a supersymmetric reduced model with Filippov 3-algebra
• Category of Quantizations and Inverse Problem
• Equivalence of large-$N$ gauge theories on a group manifold and its coset space