[Paper Review] Gravity Induced C-Deformation

Hirosi Ooguri, Cumrun Vafa|arXiv (Cornell University)|Mar 7, 2003

Black Holes and Theoretical Physics18 references20 citations

TL;DR

This paper proposes that gravitational corrections to F-terms in N=1 supersymmetric gauge theories arise from a gravity-induced C-deformation of the gluino chiral ring, generalizing the previously known C-deformation from graviphoton fields. By modifying the anticommutation relation of gluino fields to include gravitino contributions, the authors show that non-planar diagrams in the large N matrix model compute mixed gravitational-glueball superpotentials, confirming a conjecture via large N duality and supergravity consistency.

ABSTRACT

We study F-terms describing coupling of the supergravity to N=1 supersymmetric gauge theories which admit large N expansions. We show that these F-terms are given by summing over genus one non-planar diagrams of the large N expansion of the associated matrix model (or more generally bosonic gauge theory). The key ingredient in this derivation is the observation that the chiral ring of the gluino fields is deformed by the supergravity fields, generalizing the C-deformation which was recently introduced. The gravity induced part of the C-deformation can be derived from the Bianchi identities of the supergravity, but understanding gravitational corrections to the F-terms requires a non-traditional interpretation of these identities.

Motivation & Objective

• To derive gravitational corrections to F-terms in N=1 supersymmetric gauge theories beyond the planar limit.
• To generalize the C-deformation mechanism to include gravitino backgrounds, extending the known graviphoton-induced C-deformation.
• To establish that non-planar diagrams in the large N matrix model compute mixed gravitational-glueball superpotentials.
• To resolve the tension between standard supergravity tensor calculus and the need for non-trivial non-planar contributions in large N duality.
• To show that a non-traditional interpretation of the gluino algebra—relaxing Grassmannian properties—enables consistent supersymmetry and duality in curved backgrounds.

Proposed method

• Introduce a modified chiral ring relation: {Wα, Wβ} = 2WαβγWγ + Fαβ, where Fαβ is the graviphoton field strength and Wαβγ is the gravitino superfield.
• Use this deformed algebra to compute F-term effective superpotentials via Feynman diagram expansions in the large N limit.
• Derive the full gravitational correction series by replacing Fαβ with Fαβ + 2WαβγWγ in the known C-deformation formula.
• Show that the resulting superpotential matches the genus g sum over non-planar diagrams of the matrix model partition function.
• Reconcile the non-traditional algebraic structure (non-Grassmann Wα) with supergravity via a reinterpretation of Bianchi identities.
• Confirm consistency with large N duality by demonstrating that the non-planar contributions match predictions from topological string theory and supergravity.

Experimental results

Research questions

• RQ1 How do gravitational corrections to F-terms in N=1 gauge theories arise from non-planar diagrams in the large N expansion?
• RQ2 What is the generalized C-deformation that includes both graviphoton and gravitino backgrounds?
• RQ3 Why does the standard supergravity tensor calculus fail to capture non-planar corrections, and how can it be reinterpreted to include them?
• RQ4 How does relaxing the Grassmannian property of gluino fields allow for non-trivial F-term contributions in the presence of a constant gravitino background?
• RQ5 What is the precise field-theoretic origin of the mixed gravitational-glueball superpotential terms predicted by large N duality?

Key findings

• The effective superpotential for N=1 gauge theories with gravity is given by a sum over genus g diagrams of the matrix model, with F-terms corrected by (FαβFαβ)g and (FαβFαβ)g−1 WαβγWαβγ terms.
• The gravity-induced C-deformation {Wα, Wβ} = 2WαβγWγ + Fαβ generates both types of gravitational corrections simultaneously when substituted into the C-deformation formula.
• Non-planar contributions survive even when Fαβ = 0, due to the non-traditional algebraic structure of the gluino fields, which no longer require (Wα)² = 0.
• The non-traditional interpretation of the chiral ring relation is necessary for the validity of large N duality in superstring theory, as the standard Grassmannian assumption would kill all non-planar contributions.
• The inclusion of gravitino fields leads to Lorentz-violating photon-graviton interactions in the F-terms, such as terms proportional to F^{U(1)} R, which could serve as signatures of the C-deformation.
• The full correction structure is captured by a generating functional involving both the glueball superfield S and the gravitino superfield Wα, with the full F-term given by an integral over d²θ of a deformed field strength squared raised to genus g.