[Paper Review] Geometric Transitions and N=1 Quiver Theories

Freddy Cachazo, Sheldon Katz+1 more|ArXiv.org|Aug 16, 2001

Black Holes and Theoretical Physics30 references115 citations

TL;DR

This paper proposes large N dualities for N=1 quiver gauge theories arising from D5-branes wrapped on 2-cycles of A-D-E fibered Calabi-Yau threefolds, using geometric transitions involving blowdowns of 2-cycles and blowups of 3-cycles. It provides exact predictions for the vacuum structure of U(N) gauge theories with two adjoint fields and superpotentials of the form W = P_{p+2}(X) + P_{q+2}(Y) + P_{r+2}(X+Y), resolving long-standing challenges in analyzing such strongly coupled systems.

ABSTRACT

We construct N=1 supersymmetric theories on worldvolumes of D5 branes wrapped around 2-cycles of threefolds which are A-D-E fibrations over a plane. We propose large N duals as geometric transitions involving blowdowns of two cycles and blowups of three-cycles. This yields exact predictions for a large class of N=1 supersymmetric gauge systems including U(N) gauge theories with two adjoint matter fields deformed by superpotential terms, which arise in A-D-E fibered geometries with non-trivial monodromies.

Motivation & Objective

• To extend large N dualities beyond N=2 and N=4 theories to include N=1 supersymmetric gauge theories with two adjoint fields.
• To construct geometric transitions that serve as exact large N duals for U(N) gauge theories with specific superpotentials involving two adjoint fields.
• To resolve the vacuum structure of N=1 gauge theories with two adjoint fields, which are neither close to N=4 nor N=2 and thus difficult to analyze with standard methods.
• To generalize the geometric engineering of N=1 theories by incorporating non-trivial monodromies in A-D-E fibrations over a plane.

Proposed method

• Utilizes A-D-E singularities in two dimensions, deformed via relevant parameters t_i or α_i corresponding to holomorphic volumes of 2-cycles.
• Constructs 3-fold geometries by fibering A-D-E singularities over a plane, leading to N=1 quiver gauge theories on D5-brane worldvolumes.
• Applies geometric transitions involving blowdowns of 2-cycles and blowups of 3-cycles to construct large N duals.
• Relates the superpotential W to fluxes H via W = ∫ H ∧ Ω, with H determined by RR and B-fluxes through S^3 cycles.
• Identifies the number of normalizable deformations in the geometry with the number of inequivalent branches in the Higgs branch, using the singularity ring with charge constraints.
• Extremizes the superpotential to determine the coefficients of the deformed geometry, yielding exact results for the quantum-corrected superpotential.

Experimental results

Research questions

• RQ1 How can large N dualities be extended to N=1 supersymmetric gauge theories with two adjoint fields?
• RQ2 What geometric transitions correspond to the large N duals of N=1 quiver theories arising from A-D-E fibrations with monodromy?
• RQ3 How can the vacuum structure of U(N) gauge theories with superpotential W = P_{p+2}(X) + P_{q+2}(Y) + P_{r+2}(X+Y) be exactly computed?
• RQ4 What is the role of monodromy in the fibration of A-D-E singularities in constructing N=1 quiver theories?
• RQ5 How do the number of normalizable deformations in the geometry relate to the number of inequivalent Higgs branch representations?

Key findings

• The number of normalizable deformations in the geometry is 3n+3 for the case P(z) = z^n, Q(z) = 0, matching the number of inequivalent one- and two-dimensional representations in the Higgs branch.
• The superpotential W is derived from fluxes via W = ∫ H ∧ Ω, and its extremization determines the coefficients of the deformed geometry, yielding exact results.
• For the affine A-D-E case, both positive and negative roots contribute to fluxes, allowing for positive or negative effective ranks, while the superpotential structure remains identical to the non-affine case.
• The method successfully computes the quantum-corrected superpotential for U(N) N=1 gauge theories with two adjoint fields and superpotential W = P_{p+2}(X) + P_{q+2}(Y) + P_{r+2}(X+Y).
• The geometric transition framework provides exact information on the vacuum structure of theories that are neither close to N=4 nor N=2, resolving a long-standing challenge in the field.
• The correspondence between the holomorphic volumes α_i of 2-cycles and the deformation parameters t_i or α_i is established, with α_i related to the simple roots of the A-D-E Dynkin diagram.