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Exactly Marginal Operators and Duality in Four Dimensional N=1 Supersymmetric Gauge Theory

Robert G. Leigh, Matthew J. Strassler

TL;DR

Manifolds of fixed points generated by exactly marginal operators are widespread in four-dimensional N=1 SUSY gauge theories. The authors develop a holomorphy-based framework linking beta-functions to operator anomalous dimensions, enabling systematic identification of marginal directions and exploration of both weakly and strongly coupled fixed lines, including non-renormalizable cases. They present numerous explicit models (vector-like and chiral), show how fixed curves can interpolate between weak and strong coupling, and propose a mechanism connecting N=1 Seiberg duality to N=2 S-duality via fixed-point curves and meson mass deformations. The work suggests a unifying perspective on four-dimensional conformal dynamics and dualities, with implications for finite theories and beyond, while outlining several conjectures and directions for future research.

Abstract

We show that manifolds of fixed points, which are generated by exactly marginal operators, are common in N=1 supersymmetric gauge theory. We present a unified and simple prescription for identifying these operators, using tools similar to those employed in two-dimensional N=2 supersymmetry. In particular we rely on the work of Shifman and Vainshtein relating the $\bt$-function of the gauge coupling to the anomalous dimensions of the matter fields. Finite N=1 models, which have marginal operators at zero coupling, are easily identified using our approach. The method can also be employed to find manifolds of fixed points which do not include the free theory; these are seen in certain models with product gauge groups and in many non-renormalizable effective theories. For a number of our models, S-duality may have interesting implications. Using the fact that relevant perturbations often cause one manifold of fixed points to flow to another, we propose a specific mechanism through which the N=1 duality discovered by Seiberg could be associated with the duality of finite N=2 models.

Exactly Marginal Operators and Duality in Four Dimensional N=1 Supersymmetric Gauge Theory

TL;DR

Manifolds of fixed points generated by exactly marginal operators are widespread in four-dimensional N=1 SUSY gauge theories. The authors develop a holomorphy-based framework linking beta-functions to operator anomalous dimensions, enabling systematic identification of marginal directions and exploration of both weakly and strongly coupled fixed lines, including non-renormalizable cases. They present numerous explicit models (vector-like and chiral), show how fixed curves can interpolate between weak and strong coupling, and propose a mechanism connecting N=1 Seiberg duality to N=2 S-duality via fixed-point curves and meson mass deformations. The work suggests a unifying perspective on four-dimensional conformal dynamics and dualities, with implications for finite theories and beyond, while outlining several conjectures and directions for future research.

Abstract

We show that manifolds of fixed points, which are generated by exactly marginal operators, are common in N=1 supersymmetric gauge theory. We present a unified and simple prescription for identifying these operators, using tools similar to those employed in two-dimensional N=2 supersymmetry. In particular we rely on the work of Shifman and Vainshtein relating the -function of the gauge coupling to the anomalous dimensions of the matter fields. Finite N=1 models, which have marginal operators at zero coupling, are easily identified using our approach. The method can also be employed to find manifolds of fixed points which do not include the free theory; these are seen in certain models with product gauge groups and in many non-renormalizable effective theories. For a number of our models, S-duality may have interesting implications. Using the fact that relevant perturbations often cause one manifold of fixed points to flow to another, we propose a specific mechanism through which the N=1 duality discovered by Seiberg could be associated with the duality of finite N=2 models.

Paper Structure

This paper contains 37 sections, 65 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Vanishing of Scaling Anomalies and Anomalous Dimensions
  3. Conditions for fixed points and marginal operators
  4. Discussion
  5. The Fixed Lines of N=4 and N=2 Theories
  6. Theories with Weakly Coupled Fixed Lines in N=1 Supersymmetry
  7. $SU(3)$ with $N_f=9$
  8. $SU(3)\times SU(3)$ with $3\times[({\bf 3,3})+({\bf\overline3,\overline3})]$
  9. $SU(2)\times SU(2)$ with a $({\bf 3,3})$
  10. N=4 with a d-type coupling
  11. An N=2 model with an additional operator
  12. A chiral example
  13. An $SU(4)$ model with undetermined $R$ charge
  14. Other models with undetermined $R$ charge
  15. From Weakly to Strongly Coupled Fixed Lines
  16. ...and 22 more sections