Communications in Mathematical Physics

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

Multi-Trace Superpotentials vs. Matrix Models

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

Abstract

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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aharony, O., Antebi, Y.E., Berkooz, M., Fishman, R.: ‘‘Holey sheets’: Pfaffians and subdeterminants as D-brane operators in large N gauge theories.’ arXiv:hep-th/0211152Google Scholar
  2. 2.
    Aharony, O., Berkooz, M., Silverstein, E.: Multiple-trace operators and non-local string theories. hep-th/0105309Google Scholar
  3. 3.
    Argurio, R., Campos, V.L., Ferretti, G., Heise, R.: Exact superpotentials for theories with flavors via a matrix integral. arXiv:hep-th/0210291Google Scholar
  4. 4.
    Argurio, R., Campos, V.L., Ferretti G., Heise, R.: Baryonic corrections to superpotentials from perturbation theory. arXiv:hep-th/0211249Google Scholar
  5. 5.
    Ashok, S.K., Corrado, R., Halmagyi, N., Kennaway, K.D., Romelsberger, C.: Unoriented strings, loop equations, and N=1 superpotentials from matrix models. arXiv:hep-th/0211291Google Scholar
  6. 6.
    Balasubramanian, V., Huang, M.X., Levi, T.S., Naqvi, A.: Open strings from N = 4 super Yang-Mills. JHEP 0208, 037 (2002) [arXiv:hep-th/0204196]CrossRefGoogle Scholar
  7. 7.
    Bena, I., Roiban, R., Tatar, R.: Baryons, boundaries and matrix models. arXiv:hep-th/0211271Google Scholar
  8. 8.
    Berenstein, D.: Reverse geometric engineering of singularities. JHEP 04 052, (2002) http://arXiv.org/abs/hep-th/0201093
  9. 9.
    Berenstein, D.: Quantum moduli spaces from matrix models. arXiv:hep-th/0210183Google Scholar
  10. 10.
    Bitsadze, A. V.: Integral equations of first kind. Carleman, T.: Uber die Abelsche Integralgleichung mit Konstanten Integrationsgrenzen. Mathematische Zeitschrift 15 111–120 (1922)Google Scholar
  11. 11.
    Brezin, E., Itzykson, C., Parisi, G., Zuber, J.B.: Planar diagrams. Commun. Math. Phys. 59, 35 (1978)MATHGoogle Scholar
  12. 12.
    Cachazo, F., Douglas, M.R., Seiberg, N., Witten, E.: Chiral rings and anomalies in supersymmetric gauge theory. arXiv:hep-th/0211170Google Scholar
  13. 13.
    Cachazo, F., Intriligator, K.A., Vafa, C.: A large N duality via a geometric transition. Nucl. Phys. B603, 3–41 (2001) http://arXiv.org/abs/hep-th/0103067
  14. 14.
    Cachazo, F., Vafa, C.: N = 1 and N = 2 geometry from fluxes. http://arXiv.org/abs/hep-th/0206017
  15. 15.
    Das, S.R., Dhar, A., Sengupta, A.M., Wadia, S.R.: New critical behavior in d = 0 large N matrix models. Mod. Phys. Lett. A5, 1041–1056 (1990)Google Scholar
  16. 16.
    Dijkgraaf, R., Vafa, C.: Matrix models, topological strings, and supersymmetric gauge theories. Nucl. Phys. B644, 3–20 (2002) http://arXiv.org/abs/hep-th/0206255
  17. 17.
    Dijkgraaf, R., Vafa, C.: On geometry and matrix models. Nucl. Phys. B644, 21–39 (2002) http://arXiv.org/abs/hep-th/0207106
  18. 18.
    Dijkgraaf, R., Vafa, C.: A perturbative window into non-perturbative physics. http://arXiv.org/abs/hep-th/0208048
  19. 19.
    Dijkgraaf, R., Grisaru, M.T., Lam, C.S., Vafa, C., Zanon, D.: Perturbative computation of glueball superpotentials. http://arXiv.org/abs/hep-th/0211017
  20. 20.
    Dijkgraaf, R., Neitzke, A., Vafa, C.: Large N strong coupling dynamics in non-supersymmetric orbifold field theories. arXiv:hep-th/0211194Google Scholar
  21. 21.
    Dijkgraaf, R., Sinkovics, A., Temurhan, M.: Matrix models and gravitational corrections. arXiv:hep-th/0211241Google Scholar
  22. 22.
    Dijkgraaf, R., Gukov, S., Kazakov, V.A., Vafa, C.: Perturbative analysis of gauged matrix models. arXiv:hep-th/0210238Google Scholar
  23. 23.
    Dorey, N., Hollowood, T.J., Kumar, S.P., Sinkovics, A.: Massive vacua of N = 1* theory and S-duality from matrix models. arXiv:hep-th/0209099Google Scholar
  24. 24.
    Dorey, N., Hollowood, T.J., Prem Kumar, S., Sinkovics, A.: Exact superpotentials from matrix models. arXiv:hep-th/0209089Google Scholar
  25. 25.
    Douglas, M., Shenker, S.: Dynamics of SU(N) supersymmetric gauge theory. hep-th/9503163Google Scholar
  26. 26.
    Feng, B.: Geometric dual and matrix theory for SO/Sp gauge theories. hep-th/0212010Google Scholar
  27. 27.
    Feng, B.: Seiberg duality in matrix model. arXiv:hep-th/0211202Google Scholar
  28. 28.
    Feng, B., He, Y.H.: Seiberg duality in matrix models II. arXiv:hep-th/0211234Google Scholar
  29. 29.
    Ferrari, F.: On exact superpotentials in confining vacua. arXiv:hep-th/0210135Google Scholar
  30. 30.
    Gubser, S.S., Mitra, I.: Double-trace operators and one-loop vacuum energy in AdS/CFT. arXiv:hep-th/0210093Google Scholar
  31. 31.
    Intriligator, K.: Integrating in and exact superpotentials in 4d. hep-th/9407106Google Scholar
  32. 32.
    Ita, H., Nieder, H., Oz, Y.: Perturbative computation of glueball superpotentials for SO(N) and USp(N). arXiv:hep-th/0211261Google Scholar
  33. 33.
    Janik, R.A., Obers, N.A.: SO(N) S uperpotential, Seiberg-Witten Curves and Loop Equations. arXiv:hep-th/0212069Google Scholar
  34. 34.
    Klebanov, I. R., Hashimoto, A.: Nonperturbative solution of matrix models modified by trace squared terms. Nucl. Phys. B434 264–282, (1995) http://arXiv.org/abs/hep-th/9409064
  35. 35.
    Klemm, A., Marino, M., Theisen, S.: Gravitational corrections in supersymmetric gauge theory and matrix models. arXiv:hep-th/0211216Google Scholar
  36. 36.
    McGreevy, J.: Adding flavor to Dijkgraaf-Vafa. hep-th/0211009Google Scholar
  37. 37.
    Seiberg, N., Witten, E.: Monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory. hep-th/9407087; Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD. hep-th/9408099Google Scholar
  38. 38.
    Veneziano, G., Yankielowicz, S.: An effective lagrangian for the pure N=1 supersymmetric Yang-Mills theory. Phys. Lett. B 113, 231 (1982)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Vijay Balasubramanian
    • 1
  • 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

Personalised recommendations