# 3d Self-Dualities

###### Abstract

We investigate self-dualities in three-dimensional supersymmetric gauge theories. The electric and magnetic theories share the same gauge group. The examples include , and with various matter contents. The duality exchanges the role of the baryon and Coulomb branch operators in some examples. In other examples, the Coulomb branch operator becomes an elementary field on the dual side. These self-dualities in turn teach us a correct quantum structure of the Coulomb moduli space of vacua. Some dualities show symmetry enhancement.

Albert Einstein Center for Fundamental Physics

Institute for Theoretical Physics

University of Bern

Sidlerstrasse 5, CH-3012 Bern, Switzerland

###### Contents

## 1 Introduction

Duality is a very powerful tool of studying the low-energy dynamics of the strongly-coupled gauge theory. In supersymmetric gauge theories with four supercharges, this type of duality is called Seiberg duality [1, 2]. In general, the electric and magnetic (dual) theories have the different gauge groups and flow to the same infrared physics. In four dimensional spacetime, “self-dualities” where the electric and magnetic theories share the same gauge group were also found [3, 4] in addition to the usual Seiberg dualities. It is important to study the self-dualities because these examples sometimes exhibit symmetry enhancement in the far-infrared limit and because the symmetry enhancement is in turn related to the existence of many sets of dualities [5, 6, 7, 8, 9].

In this paper, we investigate the possibility of the self-dualities in the 3d supersymmetric gauge theories with various gauge groups and various matter contents. In four dimensions, this was extensively studied in [3, 4]. In three dimensions, this was recently investigated in[10, 11] for the gauge theories with an anti-symmetric matter. The self-duality was studied in [12]. We will mimic their approach and find similar self-dualities. In 4d, there are chiral anomalies and a particular global symmetry is anomalous. The 4d duality does not respect this broken symmetry. Therefore, the naive dimensional reduction of the 4d self-dualities does not hold in 3d. However, we will find that the similar self-duality is indeed applicable. In 3d, there is an additional branch of the moduli space of vacua, which is called a Coulomb branch. The Coulomb brach is a flat direction of the scalar fields in the vector superfields. In [13], we studied the low-energy dynamics of the 3d “chiral” theories with fundamental and anti-fundamental matters. We there found that the “chiral” theory can have the rich structure of the quantum Coulomb branch, compared to the “vector-like” theory whose Coulomb branch is one-dimensional. The richness of the Coulomb branch can remedy seemingly-invalid dualities. In order to obtain the rich Coulomb branch, we have to slightly change the matter contents of the 4d self-duality. The resulting pair of the 3d theories has gauge anomalies if we assume that those self-dualities live in 4d. We will also discuss the connection between the 3d and 4d self-dualities. We will propose self-dualities for , and cases and these self-dualities in turn teach us the correct quantum structure of the Coulomb moduli space.

The rest of this paper is organized as follows. In Section 2, we will review the self-duality in the 3d gauge theory with six fundamental matters, which would be the simplest case of the self-duality. In Section 3, we consider the self-duality in the 3d gauge theory. In Section 4, we will consider the self-duality with a third-order anti-symmetric tensor. In Section 5, we will consider the self-duality with two anti-symmetric tensors. In Section 6, we will consider the self-duality which generalizes the self-duality. In Section 7, we will move on to the self-dualiies of the 3d gauge theories with vector and spinor matters. In Section 8, we will summarize our findings and give possible future directions.

## 2 self-duality

For illustrating how the 3d self-dualities work, let us consider the 3d gauge theory with six fundamental matters (six doublets). The low-energy dynamics of this theory was studied in [14, 15] and the self-duality was discussed in [12] (see also [10, 11]). The Higgs branch is described by a meson composite while the Coulomb branch is parametrized by a single coordinate . When obtains a non-zero vacuum expectation value, the gauge group is broken as . The matter contents and their quantum numbers are summarized in Table 1. The theory exhibits a manifest global symmetry.

1 | ||||
---|---|---|---|---|

1 | 2 | |||

1 | 1 |

We can regard this theory as the 3d gauge theory with four fundamentals and two anti-fundamentals although the explicit flavor symmetry is invisible. The Coulomb branch is completely the same as the previous one while the Higgs branch operator is decomposed into the meson , baryon and anti-baryon operators in Table 2. In the following subsections, we will review the self-dualities of Table 1 or Table 2.

1 | 1 | 0 | ||||

1 | 0 | 1 | ||||

1 | 1 | 1 | ||||

1 | 1 | 2 | 0 | |||

1 | 1 | 1 | 0 | 2 | ||

1 | 1 | 1 |

### 2.1 self-dual

First, we consider the dual of the theory in Table 1. Since the group is a member of the symplectic groups, we can use the Aharony duality [16] and obtain the dual description. The dual gauge group is again in this case. The dual theory includes the meson and the Coulomb branch operator as elementary fields. Therefore, all the moduli coordinates are introduced as elementary fields on the dual side.

1 | 2 | |||
---|---|---|---|---|

1 | 1 | |||

1 | 1 |

Table 3 shows the matter contents and their quantum numbers of the dual theory. The dual theory has a tree-level superpotential

(2.1) |

which is consistent with all the symmetries in Table 3. One can rewrite the dual in such a way that the global symmetry is manifest (Table 4). The superpotential is decomposed into

(2.2) |

In the following subsections, we will construct other self-dual descriptions where is only introduce as an elementary field or where and are introduced as elementary fields.

1 | 0 | |||||

1 | ||||||

1 | 1 | 1 | ||||

1 | 1 | 2 | 0 | |||

1 | 1 | 1 | ||||

1 | 1 | 1 | ||||

1 | 1 | 1 |

### 2.2 “chiral” self-dual

Next, we consider the “chiral” self-dual description. By regarding the theory with as an theory with four fundamentals and two anti-fundamentals (Table 2), we can apply the “chiral” Seiberg duality [13]. The dual gauge group again becomes . The dual theory contains four fundamentals, two anti-fundamentals and a meson singlet . The dual theory has a tree-level superpotential

(2.3) |

In this self-dual description, only the eight components of , which are denoted by , are introduced as elementary fields. The quantum numbers of the dual fields are summarized in Table 5. The matching of the moduli operators is as follows.

(2.4) |

The role of the Coulomb branch and the anti-baryonic operator is exchanged.

1 | 0 | |||||

1 | ||||||

1 | 1 | 1 | ||||

1 | 1 | 2 | 0 | |||

1 | 1 | 1 | ||||

1 | 1 | 1 |

### 2.3 third self-dual

We can construct the third self-dual description where the (anti-)baryonic operators and the Coulomb branch are introduced as elementary fields. The third dual description includes a tree-level superpotential

(2.5) |

which is consistent with all the symmetries in Table 6. The global symmetry is invisible and only the subgroup is manifest. The global symmetry will be enhanced only in the infrared limit. The meson is decomposed into and in this self-dual description. The Coulomb branch of the third self-dual is lifted by the superpotential.

1 | 0 | |||||

1 | ||||||

1 | 1 | 2 | 0 | |||

1 | 1 | 1 | ||||

1 | 1 | 1 | ||||

1 | 1 | 1 | ||||

1 | 1 | 1 |

## 3 self-duality

In this section, we consider the self-duality in the 3d gauge theory with two anti-symmetric tensors and some (anti-)fundamental matters. We will find the two self-dual examples. The first one is the self-dual of the vector-like theory. The second one is chiral.

### 3.1 with

The first example is the 3d gauge theory with two anti-symmetric tensors and three flavors in the (anti-)fundamental representation. Since the anti-symmetric representation of is real, the theory can be regarded as a “vector-like” theory in a four-dimensional sense. The theory is also regarded as the gauge theory and its Coulomb branch can be studied in the same manner as [17, 18]. The Coulomb branch is now two-dimensional. The first coordinate corresponds to the breaking and is denoted as . The second one corresponds to the breaking and is denoted as . Since the theory is vector-like, these two operators are gauge-invariant. The Higgs branch is described by the following composite operators.

(3.1) |

Table 7 summarizes the quantum numbers of the matter fields and the moduli coordinates.

1 | 1 | 1 | 0 | 0 | ||||

1 | 1 | 0 | 1 | 0 | ||||

1 | 1 | 0 | 1 | |||||

1 | 1 | 0 | 1 | 1 | ||||

1 | 1 | 2 | 1 | 1 | ||||

1 | 1 | 1 | 2 | 0 | 0 | |||

1 | 1 | 1 | 2 | 0 | ||||

1 | 1 | 1 | 0 | 2 | ||||

1 | 1 | 1 | 1 | |||||

1 | 1 | 1 | 1 |

The dual description is again given by the 3d gauge theory with two anti-symmetric tensors and three flavors in the (anti-)fundamnetal representation. The dual theory includes the gauge-singlet chiral superfields and as elementary fields. These are identified with the moduli coordinates on the electric side. The magnetic description includes a tree-level superpotential

(3.2) |

where are the magnetic Coulomb branches corresponding to the breaking and , respectively. The quantum numbers of the magnetic matter contents are summarized in Table 8. The charge assignment of the magnetic matter is determined by the above superpotential. The matching of the chiral rings between the electric and magnetic theories is manifest from Table 8. The non-trivial identification of the gauge invariant operator is .

Let us check the validity of this self-duality. First, the parity anomaly matching is satisfied. We can test the branch which can be realized by giving the expectation value to one of the anti-symmetric matters. Both the electric and magnetic theories flow to the theory with an anti-symmetric matter and six fundamentals. The self-duality was recently studied in [10, 11] and we can reproduce it. We can also connect this duality to the 4d self-duality of the 4d theory [3] with via dimensional reduction and a real mass deformation [19, 20]. By putting the 4d self-dual pair on a circle, we obtain the self-duality with monopole superpotential. In order to get rid of the monopole effects, we introduce a positive real mass to one fundamental matter and a negative real mass to one anti-fundamental matter. The electric theory flows to Table 7. On the magnetic side, the real masses are introduced also for the gauge singlets. The massless components of the meson fields are decomposed into and . The magnetic theory flows to Table 8. The mechanism of the dynamical generation of the monopole superpotential in (3.2) is unclear in this dimensional reduction process but it is consistent with all the symmetries.

As another consistency check, we can compare the superconformal indices [21, 22, 23, 24] of the electric and magnetic theories. The both sides produce the same index

(3.3) |

where is a fugacity for the first symmetry of the anti-symmetric matters and are the fugacities for the remaining symmetries. The r-charges are fixed to for convenience. We computed the indices up to and found an exact agreement. The low-lying terms are easily identified with the moduli operators defined in Table 7 and Table 8. In the second term, corresponds to the Coulomb branch . In the fourth term, is interpreted as and .

1 | 1 | 1 | 0 | 0 | ||||

1 | 1 | 0 | ||||||

1 | 1 | |||||||

1 | 1 | 0 | 1 | |||||

1 | 1 | 2 | 1 | 1 | ||||

1 | 1 | 1 | 2 | 0 | ||||

1 | 1 | 1 | 0 | 2 | ||||

1 | 1 | 1 | 1 | |||||

1 | 1 | 1 | ||||||

1 | 1 | 1 | 2 | 0 | 0 | |||

1 | 1 | 1 | 1 | |||||

1 | 1 | 1 | 1 | 4 | 3 | 3 |

### 3.2 with

Next, we consider the self-duality in the 3d gauge theory with two anti-symmetric matters, four fundamental matters and two anti-fundamental matters. The theory has no tree-level superpotential. The similar theory was studied in four-dimensions [3], where the theory was vector-like (four flavors) due to the gauge anomaly constraint. Of course, we can regard this theory as the theory with two vectors, four spinors and two complex conjugate spinors. The “chiral-ness” of the theory will allow us to construct various self-dual description.

We first investigate the structure of the Coulomb moduli space of this theory. The bare Coulomb branch leads to the gauge symmetry brealking

(3.4) | ||||

(3.5) | ||||

(3.6) | ||||

(3.7) |

and we denote the corresponding operator as . Along this Coulomb branch, the massive components are integrated out and the mixed Chern-Simons term is induced between the two gauge groups

(3.8) |

The mixed Chern-Simons term makes the bare Coulomb branch operator gauge non-invariant. The charge of is . In order to construct the gauge invariant moduli coordinates, we can use the massless components from the anti-fundamental and anti-symmetric matters, or . The dressed (gauge-invariant) Coulomb branch operators become

(3.9) | |||

(3.10) |

where the flavor indices of the anti-quark chiral superfields are totally anti-symmetrized while has a flavor index of the anti-symmetric matter. The Higgs branch is described by the following composite operators.

(3.11) |

Table 9 shows the quantum numbers of the matter contents and the moduli coordinates.

1 | 1 | 1 | 0 | 0 | ||||

1 | 1 | 0 | 1 | 0 | ||||

1 | 1 | 0 | 1 | |||||

1 | 1 | 0 | 1 | 1 | ||||

1 | 1 | 2 | 1 | 1 | ||||

1 | 1 | 1 | 2 | 0 | 0 | |||

1 | 1 | 1 | 1 | 0 | 4 | 0 | ||

1 | 1 | 1 | 2 | 0 | ||||

1 | 1 | 1 | 1 | 0 | 2 | |||

charge: 2 | 1 | 1 | 1 | |||||

1 | 1 | 1 | ||||||

1 | 1 | 1 | 1 | 0 |

#### 3.2.1 First self-duality

We start with the first dual description where the meson fields , the baryon fields and the dressed Coulomb branch are introduced as elementary fields. The dual theory includes the tree-level superpotential

(3.12) |

The quantum numbers of the dual matter contents are summarized in Table 10. The charge assignment is completely fixed by requiring that should be identified with and from the above superpotential.

The analysis of the Coulomb branch is the same as the electric theory but needs a different interpretation. The bare Coulomb branch corresponds to the breaking and requires the “dressing” procedure. The dressed operators are defined as

(3.13) | |||

(3.14) |

The Coulomb branch dressed by the anti-symmetric matter is lifted and excluded from the low-energy spectrum by the tree-level superpotential. The other operator is identified with the baryon operator . This is a bit surprising since does not include the dual quark superfield at all. The dual baryon is identified with the dressed Coulomb branch . The matching of the other operators is found in Table 10.

We can test various consistency conditions. The first self-dual description satisfies the parity anomaly matching. This 3d self-duality is connected to the 4d self-duality [3] via dimensional reduction as in the previous subsection. In this case, we have to introduce the real masses to the subgroup of the (anti-quark) flavor symmetry. The electric theory flows to Table 9. On the magnetic side, the anti-baryon operator is decomposed into and and we recover the description in Table 10.

1 | 1 | 1 | 0 | 0 | ||||

1 | 1 | 0 | ||||||

1 | 1 | |||||||

1 | 1 | 0 | 1 | 1 | ||||

1 | 1 | 2 | 1 | 1 | ||||

1 | 1 | 1 | 2 | 0 | ||||

1 | 1 | 1 | 1 | 0 | 2 | |||

1 | 1 | 1 | ||||||

1 | 1 | |||||||

1 | 1 | 0 | ||||||

1 | 1 | 0 | ||||||

1 | 1 | 1 | 0 | |||||

1 | 1 | 1 | 2 | 0 | 0 | |||

1 | 1 | 1 | 1 | 0 | ||||

charge: 2 | 1 | 1 | 1 | |||||

1 | 1 | 1 | ||||||

1 | 1 | 1 | 1 | 0 |

#### 3.2.2 Second self-duality

Let us consider the second self-dual description. The dual description is given by the 3d gauge theory with two anti-symmetric matters, four fundamental matters, two anti-fundamental matters and two meson singlets . The dual theory includes the tree-level superpotential

(3.15) |

Table 11 shows the quantum numbers of the matter contents and the moduli coordinates. The charge assignment is determined by the above superpotential and by requiring the identification

(3.16) |

The Coulomb branch of the second dual description is defined in the same manner.

(3.17) | |||

(3.18) |

The operator matching is manifest from Table 11 and reads

Let us study the validity of the second self-dual. The parity anomaly matching condition is satisfied. This 3d self-duality is also connected to the 4d self-duality [3] as in the case of the first self-dual. By giving the expectation value to one of the anti-symmetric matters, we can reproduce the self-duality [10, 11].

1 | 1 | 1 | 0 | 0 | ||||

1 | 1 | 0 | ||||||

1 | 1 | |||||||

1 | 1 | 0 | 1 | 1 | ||||

1 | 1 | 2 | 1 | 1 | ||||

1 | 1 |