[Paper Review] A Perturbative Window into Non-Perturbative Physics

Robbert Dijkgraaf, Cumrun Vafa|ArXiv.org|Aug 7, 2002

Black Holes and Theoretical PhysicsPhysics and Astronomy50 references365 citations

TL;DR

This paper proposes that the exact effective superpotential in ${\cal N}=1$ supersymmetric gauge theories can be computed perturbatively by summing planar diagrams of the same theory, mapping the computation to a matrix model with the tree-level superpotential as its action. The key result is that perturbative genus expansions yield exact non-perturbative instanton effects, revealing dualities like Seiberg-Witten and Montonen-Olive duality from a purely perturbative, planar perspective without requiring dual theories or conjectures.

ABSTRACT

We argue that for a large class of N=1 supersymmetric gauge theories the effective superpotential as a function of the glueball chiral superfield is exactly given by a summation of planar diagrams of the same gauge theory. This perturbative computation reduces to a matrix model whose action is the tree-level superpotential. For all models that can be embedded in string theory we give a proof of this result, and we sketch an argument how to derive this more generally directly in field theory. These results are obtained without assuming any conjectured dualities and can be used as a systematic method to compute instanton effects: the perturbative corrections up to n-th loop can be used to compute up to n-instanton corrections. These techniques allow us to see many non-perturbative effects, such as the Seiberg-Witten solutions of N=2 theories, the consequences of Montonen-Olive S-duality in N=1* and Seiberg-like dualities for N=1 theories from a completely perturbative planar point of view in the same gauge theory, without invoking a dual description.

Motivation & Objective

• To establish a direct link between perturbative planar diagrams and exact non-perturbative dynamics in ${\cal N}=1$ supersymmetric gauge theories.
• To demonstrate that the effective superpotential in terms of the glueball superfield $S$ is exactly given by summing all planar diagrams of the same gauge theory.
• To show that this perturbative computation reduces to a matrix model whose action is the tree-level superpotential, enabling exact resummation.
• To derive non-perturbative phenomena—such as Seiberg-Witten solutions and Montonen-Olive duality—without invoking dual theories or string theory.
• To provide a systematic method to compute instanton corrections using $n$-loop perturbative results.

Proposed method

• The effective superpotential is computed as a function of the glueball chiral superfield $S = \frac{1}{32\pi^2} \mathrm{Tr} \, W_\alpha W^\alpha$ using planar diagram summation.
• The planar diagrams are shown to reduce to a matrix model with action equal to the tree-level superpotential.
• For theories embeddable in string theory, the result is proven via the chain of dualities involving geometric transition and topological string theory.
• A field-theoretic argument is sketched to derive the result without string theory, relying on holomorphy and localization.
• Genus $g$ perturbative diagrams are shown to compute $g$-instanton corrections, with genus one diagrams yielding $R^2$ couplings to gravity.
• The matrix model's large $N$ solution is interpreted as defining a Calabi-Yau threefold geometry that encodes the non-perturbative dynamics.

Experimental results

Research questions

• RQ1 Can the exact effective superpotential in ${\cal N}=1$ gauge theories be computed purely perturbatively via planar diagrams?
• RQ2 Does summing planar diagrams in the same gauge theory reproduce non-perturbative effects like instantons and dualities?
• RQ3 Can the matrix model derived from planar diagrams capture the full non-perturbative structure, including Seiberg-Witten and Montonen-Olive dualities?
• RQ4 Is the perturbative genus expansion equivalent to a fractional instanton expansion in the absence of dual descriptions?
• RQ5 Can this method systematically compute $n$-instanton corrections from $n$-loop perturbation theory?

Key findings

• The effective superpotential in ${\cal N}=1$ theories is exactly given by summing all planar diagrams of the same theory, without requiring dual descriptions.
• This planar diagram summation reduces to a matrix model with action equal to the tree-level superpotential, enabling exact resummation.
• The genus $g$ contribution in the perturbative expansion computes the $g$-instanton correction, establishing a direct link between perturbation theory and non-perturbative effects.
• For the ${\cal N}=1^*$ theory, the genus one correction reproduces the $f(\tau_0) = -\log \eta(\tau_0/2)$ coupling, matching known results from $\eta$-function determinants.
• The matrix model's large $N$ solution yields a Calabi-Yau threefold geometry that encodes the non-perturbative dynamics, with periods related to $A$- and $B$-cycle integrals of the holomorphic three-form.
• Modular transformations in the $S$-duality group act on the matrix model's periods, showing that the full duality structure emerges from the planar limit.