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Strong Coupling Dynamics of Four-Dimensional N=1 Gauge Theories from M Theory Fivebrane

Kentaro Hori, Hirosi Ooguri, Yaron Oz

TL;DR

<3-5 sentences high-level summary>This paper extends the M-theory fivebrane approach to four-dimensional N=1 gauge theories by introducing an adjoint mass μ to break N=2 to N=1, and shows the resulting fivebrane curve captures the full non-perturbative dynamics, including the Affleck-Dine-Seiberg superpotential for N_f<N_c and the quantum-modified moduli space for N_f=N_c. The authors provide a detailed map between the M5-brane geometry (rotated curves, CP^1 components, and meson/baryon vevs) and four-dimensional field-theory data, including a geometric proof of the s-rule and new non-renormalization insights. They also analyze the μ→∞ limit to recover N=1 SQCD with various flavors, demonstrate dual descriptions, and discuss the Kähler potential beyond holomorphy from the M-theory viewpoint. The work establishes a precise, holomorphic bridge between brane geometry and non-perturbative N=1 dynamics, offering a powerful geometric framework for understanding strong coupling phenomena in four-dimensional gauge theories.

Abstract

It has been known that the fivebrane of type IIA theory can be used to give an exact low energy description of N=2 supersymmetric gauge theories in four dimensions. We follow the recent M theory description by Witten and show that it can be used to study theories with N=1 supersymmetry. The N=2 supersymmetry can be broken to N=1 by turning on a mass for the adjoint chiral superfield in the N=2 vector multiplet. We construct the configuration of the fivebrane for both finite and infinite values of the adjoint mass. The fivebrane describes strong coupling dynamics of N=1 theory with SU(N_c) gauge group and N_f quarks. For N_c > N_f, we show how the brane configuration encodes the information of the Affleck-Dine-Seiberg superpotential. For N_c = and < N_f, we study the deformation space of the brane configuration and compare it with the moduli space of the N=1 theory. We find agreement with field theory results, including the quantum deformation of the moduli space at N_c = N_f. We also prove the type II s-rule in M theory and find new non-renormalization theorems for N=1 superpotentials.

Strong Coupling Dynamics of Four-Dimensional N=1 Gauge Theories from M Theory Fivebrane

TL;DR

<3-5 sentences high-level summary>This paper extends the M-theory fivebrane approach to four-dimensional N=1 gauge theories by introducing an adjoint mass μ to break N=2 to N=1, and shows the resulting fivebrane curve captures the full non-perturbative dynamics, including the Affleck-Dine-Seiberg superpotential for N_f<N_c and the quantum-modified moduli space for N_f=N_c. The authors provide a detailed map between the M5-brane geometry (rotated curves, CP^1 components, and meson/baryon vevs) and four-dimensional field-theory data, including a geometric proof of the s-rule and new non-renormalization insights. They also analyze the μ→∞ limit to recover N=1 SQCD with various flavors, demonstrate dual descriptions, and discuss the Kähler potential beyond holomorphy from the M-theory viewpoint. The work establishes a precise, holomorphic bridge between brane geometry and non-perturbative N=1 dynamics, offering a powerful geometric framework for understanding strong coupling phenomena in four-dimensional gauge theories.

Abstract

It has been known that the fivebrane of type IIA theory can be used to give an exact low energy description of N=2 supersymmetric gauge theories in four dimensions. We follow the recent M theory description by Witten and show that it can be used to study theories with N=1 supersymmetry. The N=2 supersymmetry can be broken to N=1 by turning on a mass for the adjoint chiral superfield in the N=2 vector multiplet. We construct the configuration of the fivebrane for both finite and infinite values of the adjoint mass. The fivebrane describes strong coupling dynamics of N=1 theory with SU(N_c) gauge group and N_f quarks. For N_c > N_f, we show how the brane configuration encodes the information of the Affleck-Dine-Seiberg superpotential. For N_c = and < N_f, we study the deformation space of the brane configuration and compare it with the moduli space of the N=1 theory. We find agreement with field theory results, including the quantum deformation of the moduli space at N_c = N_f. We also prove the type II s-rule in M theory and find new non-renormalization theorems for N=1 superpotentials.

Paper Structure

This paper contains 19 sections, 118 equations, 19 figures.

Table of Contents

  1. Introduction
  2. Field Theory Analysis
  3. $N=2$ Moduli Space of Vacua
  4. Breaking $N=2$ to $N=1$
  5. $N=2$ Higgs Branch via M-Theory Fivebranes
  6. The D6-Branes
  7. The Higgs Branch
  8. Rotating the Brane Configuration
  9. The $\mu \rightarrow \infty$ Limit: $N=1$ SQCD
  10. $SU(N_c)$ Without Matter
  11. $SU(N_c)$ With $N_f$ Massless Matter
  12. Non-Baryonic Branches
  13. Baryonic Branch
  14. $SU(N_c)$ With Massive Matter
  15. Generation of Rational Curves in the $\mu\rightarrow\infty$ Limit
  16. ...and 4 more sections

Figures (19)

  • Figure 1: The type IIA picture of $N=2$ supersymmetric $SU(N_c)$ gauge theory with $N_f$ flavors (Coulomb branch).
  • Figure 2: The $r$-th non-baryonic branch in the type IIA picture.
  • Figure 3: The baryonic branch in the type IIA picture.
  • Figure 4: Resolution of $A_{n-1}$ singularity.
  • Figure 5: Non-baryonic branch root in M-Theory.
  • ...and 14 more figures