[Paper Review] Gaugino Condensates and Fluxes in N = 1 Effective Superpotentials

Jean-Pierre Derendinger, Costas Kounnas+1 more|arXiv (Cornell University)|Dec 31, 2005

Black Holes and Theoretical Physics6 citations

TL;DR

This paper investigates the interplay between gaugino condensates and fluxes in N=1 effective supergravity theories derived from heterotic and type II orientifold compactifications. By applying the N=4 supergravity gauging formalism, it demonstrates that nonperturbative corrections from gaugino condensation can stabilize moduli, control the supersymmetry breaking scale, and avoid common pathologies like runaway moduli or anti-de Sitter vacua—offering three distinct scaling behaviors for the gravitino mass depending on the source of supersymmetry breaking.

ABSTRACT

In the framework of orbifold compactifications of heterotic and type II orientifolds, we study effective N = 1 supergravity potentials arising from fluxes and gaugino condensates. These string solutions display a broad phenomenology which we analyze using the method of N = 4 supergravity gaugings. We give examples in type II and heterotic compactifications of combined fluxes and condensates leading to vacua with naturally small supersymmetry breaking scale controlled by the condensate, cases where the supersymmetry breaking scale is specified by the fluxes even in the presence of a condensate and also examples where fluxes and condensates conspire to preserve supersymmetry.

Motivation & Objective

• To understand how nonperturbative gaugino condensates and fluxes jointly shape the effective superpotential in N=1 compactifications of heterotic and type II strings.
• To resolve longstanding issues in flux-induced vacua—such as runaway moduli, instability, and fine-tuning—by incorporating nonperturbative corrections.
• To classify the scaling behavior of the gravitino mass in relation to fluxes and condensates, identifying three distinct regimes depending on the source of supersymmetry breaking.
• To demonstrate that nonperturbative effects are not always the source of supersymmetry breaking, but can instead stabilize moduli and alter mass hierarchies.
• To provide a systematic framework for analyzing soft terms and vacuum structure by unifying flux and condensate contributions in the superpotential.

Proposed method

• Uses the N=4 supergravity gauging procedure to derive the N=1 effective superpotential from underlying string compactifications with fluxes.
• Applies the universal structure of N=4 supergravity to determine the Kähler potential and superpotential unambiguously in terms of flux quantum numbers.
• Integrates nonperturbative superpotential contributions of the form $ W_{\text{nonpert}} = \mu^3 \exp(-24\pi^2 Z / b_0) $, where $ Z $ is a modulus field (e.g., dilaton $ S $ in heterotic, combinations of $ T, U, S $ in type II).
• Analyzes vacuum stability by examining the Kähler potential, superpotential, and F-term potential, ensuring positivity and moduli stabilization.
• Performs explicit calculations in orbifold compactifications of heterotic and type II orientifolds, including plane-symmetric and generalized Calabi–Yau geometries with torsion.
• Derives the gravitino mass $ m_{3/2} $ as a function of moduli and fluxes, identifying three distinct scaling behaviors: $ m_{3/2} \propto 1/\sqrt{V} $, $ m_{3/2} \propto w(S)/\sqrt{V} $, and $ m_{3/2} \propto w(S)^2/\sqrt{V} $.

Experimental results

Research questions

• RQ1 How do gaugino condensates and fluxes combine in the effective superpotential of N=1 compactifications to influence vacuum stability?
• RQ2 Can nonperturbative corrections from gaugino condensation resolve the runaway behavior and anti-de Sitter instabilities commonly found in flux-induced vacua?
• RQ3 What are the distinct scaling behaviors of the gravitino mass when supersymmetry breaking is driven by fluxes, condensates, or both?
• RQ4 Under what conditions can gaugino condensation preserve supersymmetry or induce it indirectly through flux interactions?
• RQ5 How does the inclusion of nonperturbative terms alter the moduli stabilization picture derived from fluxes alone?

Key findings

• Nonperturbative corrections from gaugino condensation can stabilize moduli and prevent runaway behavior, even in the presence of fluxes.
• The gravitino mass scales as $ m_{3/2} \propto 1/\sqrt{V} $ when supersymmetry breaking is driven by fluxes and nonperturbative effects are secondary.
• In type II models, the gravitino mass scales as $ m_{3/2} \propto w(S)/\sqrt{V} $, where $ w(S) $ is the nonperturbative superpotential, enabling mass hierarchy independent of volume.
• In heterotic compactifications, a novel scaling $ m_{3/2} \propto w(S)^2/\sqrt{V} $ emerges when gaugino condensation directly breaks supersymmetry, leading to stronger mass hierarchies.
• The analysis shows that fluxes and condensates can conspire to preserve supersymmetry, even when individually they would break it.
• The inclusion of nonperturbative terms is essential from the outset: they can drastically alter the vacuum structure and moduli stabilization pattern, even if they do not trigger supersymmetry breaking.