Dimensional Reduction of Seiberg-Witten Monopole Equations, N = 2 Noncommutative Supersymmetric Field Theories and Young Diagrams

We investigate the Seiberg-Witten monopole equations on noncommutative (N.C.) R 4 at the large N.C. parameter limit, in terms of the equivariant cohomology. In other words, N = 2 supersymmetric U(1) gauge theories with hypermultiplet on N.C.R4 are studied. It is known that after topological twisting partition functions of N > 1 supersymmetric theories on N.C. R2D are invariant under N.C.parameter shift, then the partition functions can be calculated by its dimensional reduction. At the large N.C. parameter limit, the Seiberg-Witten monopole equations are reduced to ADHM equations with the Dirac equation reduced to 0 dimension. The equations are equivalent to the dimensional reduction of non-Abelian U(N) Seiberg-Witten monopole equations in N → ∞. The solutions of the equations are also interpreted as a configuration of brane anti-brane system. The theory has global symmetries under torus actions originated in space rotations and gauge symmetries. We investigate the Seiberg-Witten monopole equations reduced to 0 dimension and the fixed point equations of the torus actions. We show that the Dirac equation reduced to 0 dimension is trivial when the fixed point equations and the ADHM equations are satisfied. It is known that the fixed points of the ADHM data are isolated and are classified by the Young diagrams. We give a new proof of this statement by solving the ADHM equations and the fixed point equations concretely and by giving graphical interpretations of the field components and these equations.