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

TL;DR

This paper explores the dimensional reduction of Seiberg-Witten monopole equations on noncommutative space, linking them to ADHM equations, and classifies solutions using Young diagrams, revealing insights into supersymmetric gauge theories and brane configurations.

Contribution

It provides a new proof that fixed points of ADHM data are classified by Young diagrams, with explicit solutions and graphical interpretations, in the context of noncommutative supersymmetric theories.

Findings

Seiberg-Witten equations reduce to ADHM equations at large noncommutative parameter.

Fixed points are classified by Young diagrams, with a new proof provided.

Solutions are interpreted as brane anti-brane configurations.

Abstract

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. R}^4 are studied. It is known that after topological twisting partition functions of N}>1 supersymmetric theories on N.C. R^2D 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 -> \infty. 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. For finite N, 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.