SOLUTIONS OF FOUR-DIMENSIONAL FIELD THEORIES VIA M THEORY

SOLUTIONS OF FOUR-DIMENSIONAL FIELD THEORIES VIA M THEORY

March 1997 | Edward Witten
This paper presents a solution for the Coulomb branch of a large class of four-dimensional N=2 supersymmetric gauge theories using M theory. The approach involves interpreting these theories as quantum field theories derived from configurations of fourbranes, fivebranes, and sixbranes in Type IIA superstrings, and then reinterpreting these configurations in M theory. This leads to explicit solutions for the Coulomb branch of these theories with zero or negative beta function. The paper begins by discussing the use of brane configurations in Type IIB superstring theory to study quantum field theories on the world-volume of branes. This approach is then extended to Type IIA superstring theory, where fourbranes, fivebranes, and sixbranes are considered. The key idea is to use the duality of Type IIA superstrings to M theory to obtain solutions for four-dimensional field theories. The paper then discusses the construction of models using fourbranes and fivebranes in Type IIA superstring theory on R^10. It introduces the concept of a complex variable v = x^4 + ix^5 and shows how the positions of fourbranes on fivebranes can be described in terms of this variable. The paper also discusses the interpretation of these models in terms of M theory, where the fivebrane world-volume is described as a complex Riemann surface. The paper then analyzes the four-dimensional interpretation of these models, showing how the gauge group and hypermultiplet spectrum can be determined from the brane configurations. It discusses the coupling constant of the gauge group and how it relates to the beta function of the theory. The paper also discusses the interpretation of the models in terms of M theory, showing how the fivebrane world-volume can be described as a complex Riemann surface. The paper concludes by discussing the generalization of these models to a chain of fivebranes connected by fourbranes. It shows how the gauge group and beta function can be determined from the brane configurations and how the models can be described in terms of M theory. The paper also discusses the implications of these results for the understanding of four-dimensional N=2 supersymmetric gauge theories.This paper presents a solution for the Coulomb branch of a large class of four-dimensional N=2 supersymmetric gauge theories using M theory. The approach involves interpreting these theories as quantum field theories derived from configurations of fourbranes, fivebranes, and sixbranes in Type IIA superstrings, and then reinterpreting these configurations in M theory. This leads to explicit solutions for the Coulomb branch of these theories with zero or negative beta function. The paper begins by discussing the use of brane configurations in Type IIB superstring theory to study quantum field theories on the world-volume of branes. This approach is then extended to Type IIA superstring theory, where fourbranes, fivebranes, and sixbranes are considered. The key idea is to use the duality of Type IIA superstrings to M theory to obtain solutions for four-dimensional field theories. The paper then discusses the construction of models using fourbranes and fivebranes in Type IIA superstring theory on R^10. It introduces the concept of a complex variable v = x^4 + ix^5 and shows how the positions of fourbranes on fivebranes can be described in terms of this variable. The paper also discusses the interpretation of these models in terms of M theory, where the fivebrane world-volume is described as a complex Riemann surface. The paper then analyzes the four-dimensional interpretation of these models, showing how the gauge group and hypermultiplet spectrum can be determined from the brane configurations. It discusses the coupling constant of the gauge group and how it relates to the beta function of the theory. The paper also discusses the interpretation of the models in terms of M theory, showing how the fivebrane world-volume can be described as a complex Riemann surface. The paper concludes by discussing the generalization of these models to a chain of fivebranes connected by fourbranes. It shows how the gauge group and beta function can be determined from the brane configurations and how the models can be described in terms of M theory. The paper also discusses the implications of these results for the understanding of four-dimensional N=2 supersymmetric gauge theories.
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