Tree-level Superpotential Research Articles

A composite axion from a supersymmetric product group

A global U(1)PQ symmetry is protected from gravitational effects in the s-confining SU(N)k product group theory with A + 4Q+Noverline{Q} matter. If the SU(4) family symmetry is gauged and an appropriate tree-level superpotential is added, then the dynamically generated superpotential spontaneously breaks SU(4) × U(1)PQ → SU(3)c and produces a QCD axion. Small values of the CP -violating θ parameter are then possible without any fine-tuning, as long as the product group is suitably large. By introducing a second copy of the s-confining SU(N) product group also coupled to the gauged SU(4), we find that values as small as N = 7 are consistent with overline{theta}<1{0}^{-10} , even under the pessimistic assumption that the dominant contribution to the axion quality is at tree level.

D-Brane superpotentials in Calabi-Yau orientifolds

We develop computational tools for the tree-level superpotential of B-branes in Calabi-Yau orientifolds. Our method is based on a systematic implementation of the orientifold projection in the geometric approach of Aspinwall and Katz. In the process, we lay down some ground rules for orientifold projections in the derived category.

Moduli spaces of gauge theories from Dimer models: proof of the correspondence

Recently, a new way of deriving the moduli space of quiver gauge theories that arise on the world-volume of D3-branes probing singular toric Calabi-Yau cones was conjectured. According to the proposal, the gauge group, matter content and tree-level superpotential of the gauge theory is encoded in a periodic tiling, the dimer graph. The conjecture provides a simple procedure for determining the moduli space of the gauge theory in terms of perfect matchings. For gauge theories described by periodic quivers that can be embedded on a two-dimensional torus, we prove the equivalence between the determination of the toric moduli space with a gauged linear sigma model and the computation of the Newton polygon of the characteristic polynomial of the dimer model. We show that perfect matchings are in one-to-one correspondence with fields in the linear sigma model. Furthermore, we prove that the position in the toric diagram of every sigma model field is given by the slope of the height function of the corresponding perfect matching.

Relating prepotentials and quantum vacua of Script N = 1 gauge theories with different tree-level superpotentials

We consider N=1 supersymmetric U(N) gauge theories with Z_k symmetric tree-level superpotentials W for an adjoint chiral multiplet. We show that (for integer 2N/k) this Z_k symmetry survives in the quantum effective theory as a corresponding symmetry of the effective superpotential W_eff(S_i) under permutations of the S_i. For W(x)=^W(h(x)) with h(x)=x^k, this allows us to express the prepotential F_0 and effective superpotential W_eff on certain submanifolds of the moduli space in terms of an ^F_0 and ^W_eff of a different theory with tree-level superpotential ^W. In particular, if the Z_k symmetric polynomial W(x) is of degree 2k, then ^W is gaussian and we obtain very explicit formulae for F_0 and W_eff. Moreover, in this case, every vacuum of the effective Veneziano-Yankielowicz superpotential ^W_eff is shown to give rise to a vacuum of W_eff. Somewhat surprisingly, at the level of the prepotential F_0(S_i) the permutation symmetry only holds for k=2, while it is anomalous for k>2 due to subtleties related to the non-compact period integrals. Some of these results are also extended to general polynomial relations h(x) between the tree-level superpotentials.

On supersymmetry breaking and the Dijkgraaf-Vafa conjecture

We investigate the Dijkgraaf-Vafa proposal when supersymmetry is broken. We consider U(N) SYM with chiral adjoint matter where the coupling constants in the tree-level superpotential are promoted to chiral spurions. The holomorphic part of the low-energy glueball superpotential can still be analyzed. We compute the holomorphic supersymmetry breaking contributions using methods of the geometry underlying the N=1 effective gauge theory viewed as a Whitham system. We also study the change in the effective glueball superpotential using perturbative supergraph techniques in the presence of spurions.

Computation of Superpotentials for D-Branes

We present a general method for the computation of tree-level superpotentials for the world-volume theory of B-type D-branes. This includes quiver gauge theories in the case that the D-brane is marginally stable. The technique involves analyzing the A-infinity structure inherent in the derived category of coherent sheaves. This effectively gives a practical method of computing correlation functions in holomorphic Chern-Simons theory. As an example, we give a more rigorous proof of previous results concerning 3-branes on certain singularities including conifolds. We also provide a new example.

Quantum Geometries ofA2

We solve = 1 supersymmetric A2 type U(N) × U(N) matrix models obtained by deforming = 2 with symmetric tree level superpotentials of any degree exactly in the planar limit. These theories can be geometrically engineered from string theories by wrapping D-branes over Calabi-Yau threefolds and we construct the corresponding exact quantum geometries.

Equivalence of effective superpotentials

We show that the low-energy effective superpotential of an N=1 U(N) gauge theory with matter in the adjoint and arbitrary even tree-level superpotential has, in the classically unbroken case, the same functional form as the effective superpotential of a U(N) gauge theory with matter in the fundamental and the same tree-level interactions, up to some rescalings of the couplings. We also argue that the same kind of reasoning can be applied to other cases as well.

Chiral field theories, Konishi anomalies and matrix models

We study a chiral N=1, U(N) field theory in the context of the Dijkgraaf-Vafa correspondence. Our model contains one adjoint, one conjugate symmetric and one antisymmetric chiral multiplet, as well as eight fundamentals. We compute the generalized Konishi anomalies and compare the chiral ring relations they induce with the loop equations of the (intrinsically holomorphic) matrix model defined by the tree-level superpotential of the field theory. Surprisingly, we find that the matrix model is well-defined only if the number of flavors equals two! Despite this mismatch, we show that the 1/N expansion of the loop equations agrees with the generalized Konishi constraints. This indicates that the matrix model - gauge theory correspondence should generally be modified when applied to theories with net chirality. We also show that this chiral theory produces the same gaugino superpotential as a nonchiral SO(N) model with a single symmetric multiplet and a polynomial superpotential.

Effective Superpotentials, Geometry and Integrable Systems

We consider the effective superpotentials of N; = 1 SU(Nc) and U(Nc) supersymmetric gauge theories that are obtained from the N = 2 theory by adding a tree-level superpotential. We show that several of the techniques for computing the effective superpotential are implicitly regularized by 2Nc massive chiral multiplets in the fundamental representation, i.e. the gauge theory is embedded in the finite theory with nontrivial UV fixed point. In the maximally confining phase we obtain explicit general formulae for the effective superpotential, which reduce to previously known results in particular cases. In order to study N = 1 and N = 2 theories with fundamentals, we explicitly factorize the Seiberg-Witten curve for 0 ≤ Nf < 2N c and use the results to rederive the N = 1 superpotential. N = 2 gauge theories have an underlying integrable structure, and we obtain results on a new Lax matrix for Nf = Nc .

Baryons, boundaries and matrix models

A natural extension of the Dijkgraaf–Vafa proposal is to include fields in the fundamental representation of the gauge group. In this paper we use field theory techniques to analyze gauge theories whose tree level superpotential is a generic polynomial in multi-trace operators constructed out of such fields. We show that the effective superpotential is generated by planar diagrams with at most one (generalized) boundary. This justifies the proposal put forward in [I. Bena, R. Roiban, hep-th/0211075]. We then proceed to extend the gauge theory-matrix model duality to include baryonic operators. We obtain the full moduli space of vacua for an U( N) theory with N flavors. We also outline a program leading to a string theory justification of the gauge theory-matrix model correspondence with fundamental matter.

Effective superpotentials via Konishi anomaly

We use Ward identities derived from the generalized Konishi anomaly in order to compute effective superpotentials for SU(N), SO(N) and $Sp(N)$ supersymmetric gauge theories coupled to matter in various representations. In particular we focus on cubic and quartic tree level superpotentials. With this technique higher order corrections to the perturbative part of the effective superpotential can be easily evaluated.

Seiberg duality in matrix models II

In this Letter we continue the investigation, within the context of the Dijkgraaf–Vafa programme, of Seiberg duality in matrix models as initiated in [B. Feng, hep-th/0211202], by allowing degenerate mass deformations. In this case, there are some massless fields which remain and the theory has a moduli space. With this illustrative example, we propose a general methodology for performing the relevant matrix model integrations and addressing the corresponding field theories which have non-trivial IR behaviour, and which may or may not have tree-level superpotentials.

Unoriented strings, loop equations, andN=1superpotentials from matrix models

We apply the proposal of Dijkgraaf and Vafa to analyze N=1 gauge theory with SO(N) and Sp(N) gauge groups with arbitrary tree-level superpotentials using matrix model techniques. We derive the planar and leading non-planar contributions to the large M SO(M) and Sp(M) matrix model free energy by applying the technology of higher-genus loop equations and by straightforward diagrammatics. The loop equations suggest that the RP^2 free energy is given as a derivative of the sphere contribution, a relation which we verify diagrammatically. With a refinement of the proposal of Dijkgraaf and Vafa for the effective superpotential, we find agreement with field theory expectations.

Supersymmetric SO(N)/Sp(N) gauge theory from matrix model: exact mesonic vacua

By performing the matrix integral over the tree-level superpotential of N=1 supersymmetric SO(N)/Sp(N) gauge theories obtained from N=2 SQCD by adding the mass term for the adjoint scalar field, the exact effective superpotential in terms of meson field contains the nonperturbative ADS superpotential as well as the classical tree level superpotential. By completing the meson matrix integral with the help of saddle point equation, we find the free energy contributions from matter part in terms of glueball field, the adjoint field mass and quark mass. By extremizing the effective superpotential with respect to the glueball field, we analyze the vacuum structure and describe the behavior of two limiting cases: zero limit of quark mass and infinity limit of adjoint field mass. We also study the magnetic theory.

The Phase Structure of Mass-Deformed SU(2) × SU(2) Quiver Theory

The phase structure of the finite SU(2)xSU(2) theory with N=2 supersymmetry, broken to N=1 by mass terms for the adjoint-valued chiral multiplets, is determined exactly by compactifying the theory on a circle of finite radius. The exact low-energy superpotential is constructed by identifying it as a linear combination of the Hamiltonians of a certain symplectic reduction of the spin generalized elliptic Calogero-Moser integrable system. It is shown that the theory has four confining, two Higgs and two massless Coulomb vacua which agrees with a simple analysis of the tree-level superpotential of the four-dimensional theory. In each vacuum, we calculate all the condensates of the adjoint-valued scalars.

Comments on Effective Superpotentials via Matrix Models

Dijkgraaf and Vafa have conjectured that the effective superpotentials for = 1 four-dimensional supersymmetric gauge theories can be given by the planar diagrams of matrix models. We examine some special models with cubic and quartic tree level superpotentials for adjoint chiral superfield Φ. We consider the effective superpotentials for the classical vacuum Φ = 0 for U(N) and SO(N)/Sp(N) gauge theories. We evaluate the effective superpotentials exactly in terms of the matrix model and in terms of closed string theory on Calabi-Yau geometry with fluxes. As a result we find their perfect agreements.

Comments on Effective Superpotentials via Matrix Models

Dijkgraaf and Vafa have conjectured that the effective superpotentials for N=1 four-dimensional supersymmetric gauge theories can be given by the planar diagrams of matrix models. We examine some special models with cubic and quartic tree level superpotentials for adjoint chiral superfield \Phi. We consider the effective superpotentials for the classical vacuum \Phi=0 for U(N) and SO(N)/Sp(N) gauge theories. We evaluate the effective superpotentials exactly in terms of the matrix model and in terms of closed string theory on Calabi-Yau geometry with fluxes. As a result we find their perfect agreements.

Higgs mechanism and Luna strata in gauge theories

The classical moduli space M of a supersymmetric gauge theory with trivial superpotential can be stratified according to the unbroken gauge subgroup at different vacua. We apply known results about this stratification to obtain the W \neq 0 theory classical moduli space M^W \subset M, working entirely with the composite gauge invariant operators X that span M, assuming we do not known their elementary matter chiral field content. In this construction, the patterns of gauge symmetry breaking of the W \neq 0 theory are determined, Higgs flows in these theories show important differences from the W=0 case. The methods here introduced provide an alternative way to construct tree level superpotentials that lift all classical flat directions leaving a candidate theory for dynamical supersymmetry breaking, and are also useful to identify heavy composite fields to integrate out from effective superpotentials when the elementary field content of the composites is unknown. We also show how to recognize the massless singlets after Higgs mechanism at a vacuum X \in M^W among the moduli \delta X using the stratification of M, and establish conditions under which the space of non singlet massless fields after Higgs mechanism (unseen as moduli \delta X) is null. A small set of theories with so called "unstable" representations of the complexified gauge group is shown to exhibit unexpected properties regarding the dimension of their moduli space, and the presence of non singlet massless fields after Higgs mechanism at all of their vacua.

N = 1 supersymmetric SU(4) × SU(2) L × SU (2) R effective theory from the weakly coupled heterotic superstring

In the context of the free-fermionic formulation of the heterotic superstring, we construct a three-generation N = 1 supersymmetric SU(4) × SU(2) L × SU(2) R model supplemented by an SU(8) hidden gauge symmetry and five Abelian factors. The symmetry breaking to the standard model is achieved using vacuum expectation values of a Higgs pair in ( 4,2 R) + ( 4,2 R) at a high scale. One linear combination of the Abelian symmetries is anomalous and is broken by vacuum expectation values of singlet fields along the flat directions of the superpotential. All consistent string vacua of the model are completely classified by solving the corresponding system of F- and D-flatness equations including non-renormalizable terms up to sixth order. The requirement of existence of electroweak massless doublets imposes further restrictions to the phenomenologically viable vacua. The third generation fermions receive masses from the tree-level superpotential. Further, a complete calculation of all non-renormalizable fermion mass terms up to fifth order shows that in certain string vacua the hierarchy of the fermion families is naturally obtained in the model as the second and third generation fermions earn their mass from fourth- and fifth-order terms. Along certain flat directions it is shown that the ratio of the SU(4) breaking scale and the reduced Planck mass is equal to the up quark ratio m c m t at the string scale. An additional prediction of the model, is the existence of a U(1) symmetry carried by the fields of the hidden sector, ensuring thus the stability of the lightest hidden state. It is proposed that the hidden states may account for the invisible matter of the universe.