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Multi-trace superpotentials vs. matrix models. (English) Zbl 1037.81090

Summary: We consider \(\mathcal N=1\) 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 nonplanar 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 \(\mathcal N=1\) deformations of pure \(\mathcal N=2\) gauge theory, we show that the effective superpotential that we compute also follows from the \(\mathcal N=2\) Seiberg-Witten solution. Finally, we observe an interesting connection between multi-trace local theories and nonlocal field theory.

MSC:

81T60 Supersymmetric field theories in quantum mechanics
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
81T13 Yang-Mills and other gauge theories in quantum field theory
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[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/0211152
[2] Aharony, O., Berkooz, M., Silverstein, E.: Multiple-trace operators and non-local string theories. hep-th/0105309 · Zbl 0889.53056
[3] Argurio, R., Campos, V.L., Ferretti, G., Heise, R.: Exact superpotentials for theories with flavors via a matrix integral. arXiv:hep-th/0210291 · Zbl 1222.81263
[4] Argurio, R., Campos, V.L., Ferretti G., Heise, R.: Baryonic corrections to superpotentials from perturbation theory. arXiv:hep-th/0211249 · Zbl 1222.81263
[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/0211291
[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] · Zbl 1226.81258 · doi:10.1088/1126-6708/2002/08/037
[7] Bena, I., Roiban, R., Tatar, R.: Baryons, boundaries and matrix models. arXiv:hep-th/0211271 · Zbl 1045.81566
[8] Berenstein, D.: Reverse geometric engineering of singularities. JHEP 04 052, (2002) http://arXiv.org/abs/hep-th/0201093
[9] Berenstein, D.: Quantum moduli spaces from matrix models. arXiv:hep-th/0210183
[10] Bitsadze, A. V.: Integral equations of first kind. Carleman, T.: Uber die Abelsche Integralgleichung mit Konstanten Integrationsgrenzen. Mathematische Zeitschrift 15 111–120 (1922) · JFM 48.0457.01
[11] Brezin, E., Itzykson, C., Parisi, G., Zuber, J.B.: Planar diagrams. Commun. Math. Phys. 59, 35 (1978) · Zbl 0997.81548 · doi:10.1007/BF01614153
[12] Cachazo, F., Douglas, M.R., Seiberg, N., Witten, E.: Chiral rings and anomalies in supersymmetric gauge theory. arXiv:hep-th/0211170
[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 · Zbl 0983.81050
[14] Cachazo, F., Vafa, C.: N = 1 and N = 2 geometry from fluxes. http://arXiv.org/abs/hep-th/0206017
[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) · Zbl 1020.81740
[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 · Zbl 0999.81068
[17] Dijkgraaf, R., Vafa, C.: On geometry and matrix models. Nucl. Phys. B644, 21–39 (2002) http://arXiv.org/abs/hep-th/0207106 · Zbl 0999.81069
[18] Dijkgraaf, R., Vafa, C.: A perturbative window into non-perturbative physics. http://arXiv.org/abs/hep-th/0208048 · Zbl 1301.81195
[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 · Zbl 1058.81585
[20] Dijkgraaf, R., Neitzke, A., Vafa, C.: Large N strong coupling dynamics in non-supersymmetric orbifold field theories. arXiv:hep-th/0211194
[21] Dijkgraaf, R., Sinkovics, A., Temurhan, M.: Matrix models and gravitational corrections. arXiv:hep-th/0211241 · Zbl 1061.81070
[22] Dijkgraaf, R., Gukov, S., Kazakov, V.A., Vafa, C.: Perturbative analysis of gauged matrix models. arXiv:hep-th/0210238 · Zbl 1244.81050
[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/0209099
[24] Dorey, N., Hollowood, T.J., Prem Kumar, S., Sinkovics, A.: Exact superpotentials from matrix models. arXiv:hep-th/0209089
[25] Douglas, M., Shenker, S.: Dynamics of SU(N) supersymmetric gauge theory. hep-th/9503163 · Zbl 1009.81571
[26] Feng, B.: Geometric dual and matrix theory for SO/Sp gauge theories. hep-th/0212010
[27] Feng, B.: Seiberg duality in matrix model. arXiv:hep-th/0211202 · Zbl 1031.81041
[28] Feng, B., He, Y.H.: Seiberg duality in matrix models II. arXiv:hep-th/0211234 · Zbl 1072.81547
[29] Ferrari, F.: On exact superpotentials in confining vacua. arXiv:hep-th/0210135 · Zbl 1005.81082
[30] Gubser, S.S., Mitra, I.: Double-trace operators and one-loop vacuum energy in AdS/CFT. arXiv:hep-th/0210093
[31] Intriligator, K.: Integrating in and exact superpotentials in 4d. hep-th/9407106
[32] Ita, H., Nieder, H., Oz, Y.: Perturbative computation of glueball superpotentials for SO(N) and USp(N). arXiv:hep-th/0211261 · Zbl 1226.81140
[33] Janik, R.A., Obers, N.A.: SO(N) S uperpotential, Seiberg-Witten Curves and Loop Equations. arXiv:hep-th/0212069 · Zbl 1006.81050
[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 · Zbl 1020.81751
[35] Klemm, A., Marino, M., Theisen, S.: Gravitational corrections in supersymmetric gauge theory and matrix models. arXiv:hep-th/0211216
[36] McGreevy, J.: Adding flavor to Dijkgraaf-Vafa. hep-th/0211009 · Zbl 1226.81214
[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/9408099 · Zbl 0996.81511
[38] Veneziano, G., Yankielowicz, S.: An effective lagrangian for the pure N=1 supersymmetric Yang-Mills theory. Phys. Lett. B 113, 231 (1982) · doi:10.1016/0370-2693(82)90828-0
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