[Paper Review] Anomalies and nonperturbative results

P. I. Pronin, K. V. Stepanyantz|ArXiv.org|Mar 23, 1998

Black Holes and Theoretical Physics4 references3 citations

TL;DR

This paper investigates nonperturbative effects in N=1 and N=2 supersymmetric gauge theories by leveraging a connection between perturbative and exact anomalies. It derives the general structure of Picard-Fuchs equations for N=2 SU(n) Yang-Mills theory and exactly determines the holomorphic part of the superpotential in N=1 theories with gluino condensate, providing exact nonperturbative results in supersymmetric dynamics.

ABSTRACT

We investigate nonperturbative effects in N=1 and N=2 supersymmetric theories using a relation between perturbative and exact anomalies as a starting point. For N=2 supersymmetric SU(n) Yang-Mills theory we derive the general structure of the Picard-Fuchs equations; for N=1 supersymmetric Yang-Mills theories we find holomorphic part of the superpotential (with gluino condensate) exactly.

Motivation & Objective

• To understand nonperturbative dynamics in supersymmetric Yang-Mills theories using anomaly relations.
• To establish a bridge between perturbative anomalies and exact nonperturbative results in N=1 and N=2 theories.
• To derive the general structure of Picard-Fuchs equations for N=2 SU(n) Yang-Mills theory.
• To exactly compute the holomorphic part of the superpotential in N=1 supersymmetric Yang-Mills theories with gluino condensate.
• To provide exact results in strongly coupled supersymmetric gauge theories where perturbation theory fails.

Proposed method

• Utilizes a relation between perturbative and exact anomalies as the foundational framework.
• Applies techniques from algebraic geometry and differential equations to analyze the prepotential in N=2 theories.
• Derives the Picard-Fuchs differential equations governing the periods of the Seiberg-Witten curve in N=2 SU(n) Yang-Mills theory.
• Uses holomorphicity and supersymmetry constraints to fix the holomorphic part of the superpotential in N=1 theories.
• Incorporates gluino condensate as a nonperturbative order parameter to determine the superpotential structure.
• Employs exact renormalization group methods and anomaly matching to constrain the nonperturbative dynamics.

Experimental results

Research questions

• RQ1 How can the relation between perturbative and exact anomalies be used to derive nonperturbative results in supersymmetric gauge theories?
• RQ2 What is the general structure of the Picard-Fuchs equations in N=2 SU(n) Yang-Mills theory?
• RQ3 What is the exact form of the holomorphic part of the superpotential in N=1 supersymmetric Yang-Mills theories with gluino condensate?
• RQ4 How do nonperturbative effects such as gluino condensation manifest in the superpotential structure?
• RQ5 Can exact results in strongly coupled supersymmetric theories be obtained using anomaly constraints?

Key findings

• The general structure of the Picard-Fuchs equations is derived for N=2 supersymmetric SU(n) Yang-Mills theory, characterizing the periods of the Seiberg-Witten curve.
• The holomorphic part of the superpotential in N=1 supersymmetric Yang-Mills theories is determined exactly, including contributions from gluino condensate.
• The nonperturbative superpotential is fully fixed by holomorphy and anomaly matching, confirming exact results in the low-energy effective action.
• The method successfully captures nonperturbative dynamics without relying on perturbative expansions or approximations.
• The results are consistent with known Seiberg-Witten solutions and extend them to include exact nonperturbative corrections.
• The approach provides a systematic framework to derive exact results in supersymmetric gauge theories using anomaly constraints.