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N=1 Field Theories and Fluxes in IIB String Theory

Richard Corrado, Nick Halmagyi

TL;DR

This work develops a field-theory framework for N=1 fixed points obtained by mass-deforming N=2 quiver theories, clarifying how IIB complex 3-form flux selects ALE versus generalized conifold moduli spaces. It uses ${\it a}$-maximization to fix exact R-charges and analyzes the marginal operator spectrum that parameterizes fixed-point manifolds, including the conifold subspace ${\mathbb P}^{n-2} \subset {\mathbb P}^{n-1}$. The paper shows that deformations by irrelevant operators do not generate new fixed points (via an ${a}$-theorem argument) and derives quantum-corrected moduli spaces for nonconformal theories, including Seiberg duality cascades that often end on conifold-like geometries with monodromic fibrations. These results bridge the field-theory dynamics with their gravity duals, highlighting how flux, flux-induced potentials, and monodromy shape IR physics and offering a pathway to explicit holographic realizations. The findings underscore the role of flux in obstructing complex structure deformations and identify precise conditions under which the moduli space remains an ALE fiber or deforms to a generalized conifold.

Abstract

Deformation of N=2 quiver gauge theories by adjoint masses leads to fixed manifolds of N=1 superconformal field theories. We elaborate on the role of the complex three-form flux in the IIB duals to these fixed point theories, primarily using field theory techniques. We study the moduli space at a fixed point and find that it is either the two (complex) dimensional ALE space or three-dimensional generalized conifold, depending on the type of three-form flux that is present. We describe the exactly marginal operators that parameterize the fixed manifolds and find the operators which preserve the dimension of the moduli space. We also study deformations by arbitrary superpotentials W(Φ_i) for the adjoints. We invoke the a-theorem to show that there are no dangerously irrelevant operators like TrΦ_i^{k+1}, k>2 in the N=2 quiver gauge theories. The moduli space of the IR fixed point theory generally contains orbifold singularities if W(Φ_i) does not give a mass to the adjoints. Finally we examine some nonconformal N=1 quiver theories. We find evidence that the moduli space at the endpoint of a Seiberg duality cascade is always a three-dimensional generalized conifold. In general, the low-energy theory receives quantum corrections. In several non-cascading theories we find that the moduli space is a generalized conifold realized as a monodromic fibration.

N=1 Field Theories and Fluxes in IIB String Theory

TL;DR

This work develops a field-theory framework for N=1 fixed points obtained by mass-deforming N=2 quiver theories, clarifying how IIB complex 3-form flux selects ALE versus generalized conifold moduli spaces. It uses -maximization to fix exact R-charges and analyzes the marginal operator spectrum that parameterizes fixed-point manifolds, including the conifold subspace . The paper shows that deformations by irrelevant operators do not generate new fixed points (via an -theorem argument) and derives quantum-corrected moduli spaces for nonconformal theories, including Seiberg duality cascades that often end on conifold-like geometries with monodromic fibrations. These results bridge the field-theory dynamics with their gravity duals, highlighting how flux, flux-induced potentials, and monodromy shape IR physics and offering a pathway to explicit holographic realizations. The findings underscore the role of flux in obstructing complex structure deformations and identify precise conditions under which the moduli space remains an ALE fiber or deforms to a generalized conifold.

Abstract

Deformation of N=2 quiver gauge theories by adjoint masses leads to fixed manifolds of N=1 superconformal field theories. We elaborate on the role of the complex three-form flux in the IIB duals to these fixed point theories, primarily using field theory techniques. We study the moduli space at a fixed point and find that it is either the two (complex) dimensional ALE space or three-dimensional generalized conifold, depending on the type of three-form flux that is present. We describe the exactly marginal operators that parameterize the fixed manifolds and find the operators which preserve the dimension of the moduli space. We also study deformations by arbitrary superpotentials W(Φ_i) for the adjoints. We invoke the a-theorem to show that there are no dangerously irrelevant operators like TrΦ_i^{k+1}, k>2 in the N=2 quiver gauge theories. The moduli space of the IR fixed point theory generally contains orbifold singularities if W(Φ_i) does not give a mass to the adjoints. Finally we examine some nonconformal N=1 quiver theories. We find evidence that the moduli space at the endpoint of a Seiberg duality cascade is always a three-dimensional generalized conifold. In general, the low-energy theory receives quantum corrections. In several non-cascading theories we find that the moduli space is a generalized conifold realized as a monodromic fibration.

Paper Structure

This paper contains 16 sections, 114 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Review of ${\cal N}=2$ Quiver Gauge Theories
  3. Flux Deformation of IIB Orbifolds
  4. An illustration for $A_1$
  5. The Mass Deformation and ${\cal N}=1$ Fixed Points
  6. Analysis by $a$--maximization
  7. Moduli space geometry
  8. The Spectrum of Marginal Operators at the Fixed Points
  9. Deformation of ${\cal N}=2$ Conformal Theories by General $W(\Phi_i)$
  10. Restrictions from conformal invariance
  11. The IR fixed point moduli spaces revisited
  12. Nonconformal Theories and Quantum Moduli Spaces
  13. The structure of the effective superpotential
  14. Seiberg Duality
  15. Infrared Moduli Spaces
  16. ...and 1 more sections

Figures (1)

  • Figure 1: The ${\mathbb {P}}^1$ of fixed points for the adjoint mass deformation \ref{['eq:aonegeneral']} of the $A_1$ quiver theory. The PW and KW points are indicated, as well as the generic point $(m_1,m_2)$.