Supersymmetric gauge theories with flavors and matrix models

doi:10.1016/j.nuclphysb.2004.07.015

Nuclear Physics B

Volume 698, Issues 1–2, 18 October 2004, Pages 3-52

https://doi.org/10.1016/j.nuclphysb.2004.07.015 Get rights and content

Abstract

We present two results concerning the relation between poles and cuts by using the example of $N = 1$ $U (N_{c})$ gauge theories with matter fields in the adjoint, fundamental and antifundamental representations. The first result is the on-shell possibility of poles, which are associated with flavors and on the second sheet of the Riemann surface, passing through the branch cut and getting to the first sheet. The second result is the generalization of hep-th/0311181 (Intriligator, Kraus, Ryzhov, Shigemori, and Vafa) to include flavors. We clarify when there are closed cuts and how to reproduce the results of the strong coupling analysis by matrix model, by setting the glueball field to zero from the beginning. We also make remarks on the possible stringy explanations of the results and on generalization to $SO (N_{c})$ and $USp (2 N_{c})$ gauge groups.

Introduction

String theory can be a powerful tool to understand four-dimensional supersymmetric gauge theory which exhibits rich dynamics and allows an exact analysis. In [1], using the generalized Konishi anomaly and matrix model [2], $N = 1$ supersymmetric $U (N_{c})$ gauge theory with matter fields in the adjoint, fundamental and antifundamental representations was studied. The resolvents in the quantum theory live on the two-sheeted Riemann surface defined by the matrix model curve. Their quantum behavior is characterized by the structure around the branch cuts and poles, which are related to the RR flux contributions in the Calabi–Yau geometry and flavor fields, respectively. A pole associated with flavor on the first sheet is related to the Higgs vacua (corresponding to classical nonzero vacuum expectation value of the fundamental) while a pole on the second sheet is related to the pseudo-confining vacua where the classically vanishing vacuum expectation value of the fundamental gets nonzero values due to quantum correction.

It is known [1] that Higgs vacua and pseudo-confining vacua, which are distinct in the classical theory, are smoothly transformed into each other in the quantum theory. This transition is realized on the Riemann surface by moving poles located on the second sheet to pass the branch cuts and enter the first sheet. This process was analyzed in [1] at the off-shell level by fixing the value of glueball fields during the whole process. However, in an on-shell process, the position of poles and the width and position of branch cuts are correlated (when the flavor poles are moved, the glueball field is also changed). It was conjectured in [1] that for a given branch cut, there is an upper bound for the number of poles (the number of flavors) which can pass through the cut from the second sheet to the first sheet.

Our first aim of this paper is to confirm this conjecture and give the corresponding upper bound for various gauge groups (in particular, we will concentrate on the $U (N_{c})$ gauge group). The main result is that if $N_{f} ⩾ N_{c}$ , the poles will not be able to pass through the cut to the first sheet where $N_{c}$ is the effective fluxes associated with the cut (and can be generalized to other gauge groups).

Another important development was made in [3], which was inspired by [4]. In [3], which we will refer to as IKRSV, it was shown that, to correctly compute the prediction of string theory (matrix model), it is crucial to determine whether the glueball is really a good variable or not. A prescription was given, regarding when a glueball field corresponding to a given branch cut should be set to zero before extremizing the off-shell glueball superpotential. The discussion of IKRSV was restricted to $N = 1$ gauge theories with an adjoint and no flavors, so the generalization to the case with fundamental flavors is obviously the next task.

Our second aim of this paper is to carry out this task. The main result is the following. Assuming $N_{f}$ poles around a cut associated with gauge group $U (N_{c, i})$ , when $N_{f} ⩾ N_{c, i}$ there are situations in which we should set $S_{i} = 0$ in matrix model computations. More concretely, situations with $S_{i} = 0$ belong to either of the following two branches: the baryonic branch for $N_{c, i} ⩽ N_{f} < 2 N_{c, i}$ , or the $r = N_{c, i}$ nonbaryonic branch for $N_{f} ⩾ 2 N_{c, i}$ . Moreover, when $S_{i} = 0$ , the gauge group is completely broken and there should exist some extra, charged massless field which is not incorporated in matrix model.

In Section 2, as background, we review basic materials for $N = 1$ supersymmetric $U (N_{c})$ gauge theory with an adjoint chiral superfield, and $N_{f}$ flavors of quarks and anti-quarks. The chiral operators and the exact effective glueball superpotential are given. We study the vacuum structure of the gauge theory at classical and quantum levels. We review also the main results of IKRSV. In addition to all these reviews, we present our main motivations of this paper.

In Section 3, we apply the formula for the off-shell superpotential obtained in [1] to the case with quadratic tree level superpotential, and solve the equation of motion derived from it. We consider what happens if one moves $N_{f}$ poles associated with flavors on the second sheet through the cut onto the first sheet, on-shell. Also, in Section 3.4, we briefly touch the matter of generalizing IKRSV in the one cut model.

In Section 4, we consider cubic tree level superpotential. On the gauge theory side, the factorization of the Seiberg–Witten curve provides an exact superpotential. We reproduce this superpotential by matrix model, by extremizing the effective glueball superpotential with respect to glueball fields after setting the glueball field to zero when necessary. We present explicit results for $U (3)$ theory with all possible breaking patterns and different number of flavors ( $N_{f} = 1, 2, 3, 4$ , and 5).

In Section 5, after giving concluding remarks, we repeat the procedure we did in previous sections for $SO (N_{c}) / USp (2 N_{c})$ theories, briefly.

In the appendix, we present some proofs and detailed calculations which are necessary for the analysis in Section 4.

Since string theory results in the dual Calabi–Yau geometry are equivalent to the matrix model results, we refer to them synonymously through the paper. There exist many related works to the present paper. For a list of references, we refer the reader to [5].

Section snippets

Background

In this section, we will summarize the relevant background needed for the study of $N = 1$ supersymmetric gauge theory with matter fields.

One cut model—quadratic tree level superpotential

In this section we will study whether a cut closes up if one tries to pass too many poles through it. If the poles are near the cut, the precise form of the tree level superpotential (namely the polynomials $W (z)$ , $m^{\tilde{f}}_{f} (z)$ ) is inessential and we can simplify the problem to the quadratic tree level superpotential given by (2.14). For this superpotential, we will compute the effective glueball superpotential using the formalism reviewed in the previous section. Then, by solving the equation of

Two cut model—cubic tree level superpotential

Now, let us move on to $U (N_{c})$ theory with cubic tree level superpotential, where we have two cuts. We will demonstrate that for each closed cut we can set $S = 0$ by hand to reproduce the correct gauge theory superpotential using matrix model.

Specifically, we take the tree level superpotential to be $W_{tree} = Tr [W (Φ)] - \sum_{I = 1}^{N_{f}} {\tilde{Q}}_{I} (Φ - z_{f}) Q^{I},$ $W (z) = \frac{g}{3} z^{3} + \frac{m}{2} z^{2},$ $W^{'} (z) = g z (z + \frac{m}{g}) \equiv g (z - a_{1}) (z - a_{2}) .$ Here we wrote down $W (z)$ in terms of $g_{2} = g$ , $g_{1} = m$ for definiteness, but mostly we will work with the last expression in terms of

Conclusion and some remarks

In this paper, taking $N = 1$ $U (N_{c})$ gauge theory with an adjoint and flavors, we studied the on-shell process of passing $N_{f}$ flavor poles on top of each other on the second sheet through a cut onto the first sheet. This corresponds to a continuous transition from the pseudo-confining phase with $U (N_{c})$ unbroken to the Higgs phase with $U (N_{c} - N_{f})$ unbroken (we are focusing on one cut). We confirmed the conjecture of [1] that for $N_{f} < N_{c}$ the poles can go all the way to infinity on the first sheet, while for $N$

Acknowledgments

We would like to thank Freddy Cachazo, Eric D'Hoker, Ken Intriligator, Romuald Janik, Per Kraus, and Hirosi Ooguri for enlightening discussions. This research of C.A. was supported by a grant in aid from the Monell Foundation through Institute for Advanced Study, by SBS Foundation, and by Korea Research Foundation Grant (KRF-2002-015-CS0006). The work of B.F. is supported by the Institute for Advanced Study under NSF grant PHY-0070928. The work of Y.O. was supported by JSPS Research Fellowships

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Supersymmetric gauge theories with flavors and matrix models

Abstract

Introduction

Section snippets

Background

One cut model—quadratic tree level superpotential

Two cut model—cubic tree level superpotential

Conclusion and some remarks

Acknowledgments

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Nucl. Phys. B

Nucl. Phys. B

Nucl. Phys. B

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JHEP

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On the matter of the Dijkgraaf–Vafa conjecture

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