Communications in Mathematical Physics

, Volume 242, Issue 1–2, pp 361–392 | Cite as

Multi-Trace Superpotentials vs. Matrix Models

  • Vijay BalasubramanianEmail author
  • Jan de Boer
  • Bo Feng
  • Yang-Hui He
  • Min-xin Huang
  • Vishnu Jejjalaa
  • Asad Naqvi


We consider ᵊ9=1 supersymmetric U(N) field theories in four dimensions with adjoint chiral matter and a multi-trace tree-level superpotential. We show that the computation of the effective action as a function of the glueball superfield localizes to computing matrix integrals. Unlike the single-trace case, holomorphy and symmetries do not forbid non-planar contributions. Nevertheless, only a special subset of the planar diagrams contributes to the exact result. In addition, the computation of the superpotential localizes to doing matrix integrals. In view of the results of Dijkgraaf and Vafa for single-trace theories, one might have naively expected that these matrix integrals are related to the free energy of a multi-trace matrix model. We explain why this naive identification does not work. Rather, an auxiliary single-trace matrix model with additional singlet fields can be used to exactly compute the field theory superpotential. Along the way we also describe a general technique for computing the large-N limits of multi-trace Matrix models and raise the challenge of finding the field theories whose effective actions they may compute. Since our models can be treated as ᵊ9=1 deformations of pure ᵊ9=2 gauge theory, we show that the effective superpotential that we compute also follows from the ᵊ9=2 Seiberg-Witten solution. Finally, we observe an interesting connection between multi-trace local theories and non-local field theory.


Field Theory Gauge Theory Matrix Model Effective Action Matrix Integral 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Vijay Balasubramanian
    • 1
    Email author
  • Jan de Boer
    • 2
  • Bo Feng
    • 3
  • Yang-Hui He
    • 1
  • Min-xin Huang
    • 1
  • Vishnu Jejjalaa
    • 4
  • Asad Naqvi
    • 1
  1. 1.David Rittenhouse LaboratoriesThe University of PennsylvaniaPhiladelphiaUSA
  2. 2.Institute of Theoretical PhysicsUniversity of AmsterdamAmsterdamThe Netherlands
  3. 3.Institute for Advanced StudyPrincetonUSA
  4. 4.Department of Physics, Institute for Particle Physics and AstrophysicsVirginia TechBlacksburgUSA

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