Multi-Trace Superpotentials vs. Matrix Models

Abstract

We consider 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 non-planar contributions. Nevertheless, only a special subset of the planar diagrams contributes to the exact result. Some of the data of this subset can be computed from the large-N limit of an associated multi-trace Matrix model. However, the prescription differs in important respects from that of Dijkgraaf and Vafa for single-trace superpotentials in that the field theory effective action is not the derivative of a multi-trace matrix model free energy. The basic subtlety involves the correct identification of the field theory glueball as a variable in the Matrix model, as we show via an auxiliary construction involving a single-trace matrix model with additional singlet fields which are integrated out to compute the multi-trace results. 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 N = 1 deformations of pure N =2 gauge theory, we show that the effective superpotential that we compute also follows from the N = 2 Seiberg-Witten solution. Finally, we observe an interesting connection between multi-trace local theories and non-local field theory.Comment: 35 pages, LaTeX, 6 EPS figures. v2: typos fixed, v3: typos fixed, references added, Sec. 5 added explaining how multi-trace theories can be linearized in traces by addition of singlet fields and the relation of this approach to matrix model

Note on Seiberg duality in matrix model

“…According to the field theory analysis in [5][6][7], because fields M are gauge singlets, we should leave M untouched at the level of free energy and add them back to the effective action directly by the prescription (1.1). This point has also been emphasized in [18,19]. Using this new understanding, we redo the integration of magnetic matrix model in [16,17] at Appendix to show the consistence.…”

“…The baryonic deformation has been addressed in [21][22][23][24] where it has been shown that although the baryonic deformation makes the boundary condition in matrix models very tricky, there is a way to sum up relative contributions for field theory in matrix model expansions. The multi-trace deformation was investigated in [18,32] where it was pointed out [18] that the direct integration in the multi-trace matrix model does not give back correct results in field theory. But by linearization we can reduce the multi-trace matrix model to the single-trace matrix model, thus the standard result can be applied.…”

On the matter of the Dijkgraaf-Vafa conjecture

“…The example given in Appendix A should help to clarify this. We should emphasize that the above procedure must by done diagram by diagram -there is no obvious way to directly relate the entire glueball superpotential to the matrix model free energy; the situation is similar to [11] in this respect.…”

Explicit factorization of Seiberg–Witten curves with matter from random matrix models

“…There the explicit factorization of curves without matter of [19] was known. Factorization properties were also used in the context of SO(N c ) theories [20] and multi-trace operators [21] to obtain the appropriate effective superpotentials W ef f (S).…”