Exact results in 3d $\mathcal{N}=2$ $Spin(7)$ gauge theories with vector and spinor matters

We study three-dimensional $\mathcal{N}=2$ $Spin(7)$ gauge theories with $N_S$ spinorial matters and with $N_f$ vectorial matters. The quantum Coulomb branch on the moduli space of vacua is one- or two-dimensional depending on the matter contents. For particular values of $(N_f,N_S)$, we find s-confinement phases and derive exact superpotentials. The 3d dynamics of $Spin(7)$ is connected to the 4d dynamics via KK-monopoles. Along the Higgs branch of the $Spin(7)$ theories, we obtain 3d $\mathcal{N}=2$ $G_2$ or $SU(4)$ theories and some of them lead to new s-confinement phases. As a check of our analysis we compute superconformal indices for these theories.


Introduction
Asymptotically-free gauge theories show various phases depending on the matter contents, the (global) structure of the gauge groups, spacetime dimensions, temperature and so on. It is usually difficult to exactly analyze the low-energy dynamics since it is strongly-coupled. In order to extract some analytic results, supersymmetry is a very useful tool. Non-renormalization theorems and holomorphy strongly constrain the SUSY dynamics and enable us to derive exact results [1][2][3]. For theories with four supercharges, supersymmetry can determine an exact form of the superpotential and we can find a quantum moduli space of vacua. In this paper, we are interested in low-energy dynamics of the supersymmetric Spin(7) gauge theories.
In four spacetime dimensions, N = 1 Spin(N) gauge theories with vector matters and with various spinor matters were extensively studied (see [4,5] [6][7][8][9][10][11][12][13][14][15]). For particular matter contents, the theories confine and the low-energy effective description has no gaugeinteraction. For more general matter contents, we sometimes find the Seiberg dual descriptions which are the "chiral" theories and phenomenologically interesting. In three spacetime dimensions, the corresponding Spin(N) gauge theories are not well-studied. In [16] (see also [17]), the 3d N = 2 Spin(N) guage theory with N f vector matters was investigated and its Seiberg duality was proposed by dimensionally reducing the 4d Seiberg duality. However the Spin(N) gauge theories with spinor matters are not studied at all.
In this paper, we study the quantum aspects of the 3d N = 2 Spin(7) gauge theories with spinorial and vectorial matters. Especially we will find new s-confinement phases for these theories and derive exact superpotentials which govern the confined phases. In order to verify the consistency of our analysis, we compute superconformal indices for these theories and for the dual (confined) descriptions. We will observe a complete agreement of the indices. As another check of our findings, we also test the various Higgs branch. Along the Higgs branch we find the s-confinement description of the 3d N = 2 G 2 or SU(4) gauge theories with various matters. For the G 2 Higgs branch, we will reproduce the same superpotential discussed in [18]. Along the SU(4) Higgs branch, we reproduce the known s-confinement phases and also find new s-confinement phases for the 3d N = 2 SU(4) gauge theories with anti-symmetric matters. We also discuss the connection to the 4d N = 1 Spin(7) gauge theories by incorporating the KK-monopoles. This paper is organized as follows. In Section 2, we will briefly review the 4d N = 1 Spin(7) gauge theories with spinorial matters. In Section 3, the Coulomb branch of the Spin(7) vector multiplet is (semi-)classically studied. In Section 4, a 3d N = 2 Spin(7) gauge theory with matters in a spinorial representation is investigated. We also compute the superconformal indices. In Section 5, we study the Spin(7) theory with spinor and vector matters with special attention to the s-confinement phases. In Section 6, we summarize our results and comment on possible future directions.

Review of 4d N = 1 Spin(7) gauge theories
In this section, we will briefly review the dynamics of the 4d N = 1 Spin(7) gauge theories with spinorial matters. Table 1 shows the matter contents and their quantum numbers. Due to the chiral anomalies in 4d, U(1) and U(1) R global symmetries are anomalous and then we have to combine them into a new U(1) R symmetry In this paper, we are interested in the 3d theories and these U(1) symmetries are not anomalous. Hence we will use spurious charge assignment also in 4d.  (7) theories Table 1, η is a dynamical scale of the Spin(7) gauge group and b is a coefficient of the one-loop beta function, which is given by Quantum dynamics depends on the number of spinor multiplets. We simply enumerate the results and give some comments. For N S ≤ 3, we only need the gauge invariant M SS in order to describe the Higgs branch. The superpotential to govern the low-energy dynamics is From the superpotential, there is no stable SUSY vacuum. At generic points of the moduli space, the gauge group is maximally broken to SU (2). The gaugino condensation of the remaining SU(2) generates this superpotential. For N S = 4, we need also the baryonic operator B S . At generic points of the moduli space, the gauge group is now completely broken and thus we can reliably use the instanton calculation. One-instanton configurations generate (2.4) For N S = 5, the Higgs branch coordinates M SS and B S need one constraint between them. The classical constraint is modified quantum-mechanically and realized by using the Lagrange multiplier X as For N S = 6, the quantum moduli space is the same as the classical one. The classical constraints between the Higgs branch coordinates are depicted as For the 4d N = 1 Spin (7) gauge theory with spinors and vectors, we will not review it here and see [5,15].

Coulomb branch and Monopole operators
In this section, we will define the (semi-)classical Coulomb branch coordinates which correspond to the monopoles with a magnetic charge g i = α * i · H, where α i is a simple root and α * i denotes a dual root 2 α α 2 . H is a Cartan subalgebra. The Coulomb branch operators parametrize the flat directions of the scalar fields from the vector superfields. The adjoint scalar field in a vector superfield is defined as where we used the gauge transformation and diagonalized the adjoint scalar into the Cartan part. In this notation, the weyl chamber is given by where A ij := α i · α * j is a Cartan matrix. The Coulomb brach coordinate for each simple root is equivalent to the on-shell action of each monopole, which is given by where we omitted the normalization of the action. Rigorously speaking, the Coulomb branch operator includes the dual photon which is a dualized scalar from the U(1) photon. Here we omitted it for simplicity since the dual photon dependence is easily restored. Since the Coulomb branch coordinates are originally the member of the vector superfield, it is neutral under the flavor symmetries. However, the zero-modes around the monopole background spontaneously break the flavor symmetries. As a result, the Coulomb branch operators have non-trivial charges under the non-linearly realized flavor symmetries [19] which is the mixing between the original flavor symmetries and the topological U(1) symmetry. The magnitude of the mixing is related to the number of the fermion zero-modes. Hence we need to calculate the zero-modes around the monopole background by employing the Callias index theorem [20][21][22].
The Callias index theorem claims that the number of fermion zero-modes is obtained by the following formula where the summation is taken over all the weight of the matters and g is a magnetic charge of the monopole which we consider. φ is an adjoint scalar field in the vector multiplet. Let us consider the classical Coulomb branch of a 3d N = 2 Spin(7) gauge theory. In our notation, the Weyl chamber which we chose is defined by In order to simplify the Weyl chamber, we sometimes change the variables as In this redefinition, the Weyl chamber is simplified to The Coulomb branch operators are defined as where Z corresponds to a lowest co-root and plays an important role when we study the connection between 3d and 4d theories. Y and Y spin were defined in [16,17], which are the globally defined Coulomb branch coordinates for the 3d N = 2 O(N) or Spin(N) gauge theories with vectorial matters. By using the Callias index theorem, one can compute the fermion zero-modes around each magnetic monopole. Table 2 summarizes the fermion zeromodes for each operator. Notice that we have to divide the Weyl chamber further into two regions depending on the sign of φ 1 − φ 3 for the spinor zero-modes.
For the 3d N = 2 pure Spin(7) theory without matters, all the Coulomb branch operators Y i get two gaugino zero-modes. Thus we have the non-perturbative superpotential like 1 Y i and there is no stable SUSY vacuum.
When we turn on the matters in a vectorial representation, Y 3 gets additional zeromodes from the vectorial fermions and W = 1 Y 3 is not allowed. As a result, one dimensional Coulomb branch would remain as the (quantum) moduli space. For an (S)O(7) case with vector matters [17], Y is a globally defined one-dimensional Coulomb branch operator. For a Spin(7) theory with vectorial matters the correct coordinate is Y spin [16]. In these theories, Z appears when we put the corresponding 4d theories on a circle.
Let us next consider the Spin(7) theory with spinorial matters. For φ 1 > φ 3 , Y 1 has zero-modes from the spinor in addition to the gaugino zero-modes. Thus, it is expected that Y 1 is not lifted and that there is a one-dimensional Coulomb branch for φ 1 > φ 3 . The same argument would be available also for φ 1 < φ 3 and Y 3 is un-lifted. In this theory, we need one globally defined coordinate and we will use Z for parametrizing it.
When both the vectors and the spinors are added into the Spin(7) theory, the Coulomb branch becomes more complicated. For φ 1 > φ 3 , Y 1 and Y 3 have more than two fermion zeromodes. Hence they are not lifted while Y 2 is still lifted via the monopole superpotential. For φ 1 < φ 3 , only Y 3 has more than two zero-modes and Y 1,2 are lifted. We therefore need to introduce two coordinates for the description of the (semi-)classical Coulomb moduli. We expect that one of them would be the operator Z. This is because the zero-mode of Z does not depend on the sign of φ 1 − φ 3 so that Z would be globally defined on the whole Coulomb branch. The other one would be described by Y or Y spin . Notice that this analysis is completely (semi-)classical. Therefore the quantum effects might modify these pictures. In fact we will see that the 3d N = 2 Spin(7) gauge theories with N f vectors and N S spinors sometimes show the one-dimensional Coulomb branch.

3d N = 2 Spin(7) theories with spinorial matters
In a previous section, we studied the (semi-)classical Coulomb branch of the Spin(7) theory. Here we examine the quantum aspects of the Spin(7) Coulomb branch. Let us start with the 3d N = 2 Spin(7) gauge theory with spinorial matters. The Higgs branch is parametrized by a meson M SS := SS for N S ≤ 3. The baryonic operator B S := S 4 is also necessary for N S ≥ 4. The matter contents and their quantum numbers are summarized in Table 3. The table also includes the dynamical scale η of the 4d gauge coupling. Since the U(1) symmetries are anomalous in 4d due to the chiral anomalies, the dynamical scale is charged under the U(1) symmetries. For the Coulomb branch, we predict that Z is a correct monopole operator.
For any N S , the superpotential W = ηZ is available, which is dynamically generated from the KK-monopole and necessary when connecting the 3d theory to the 4d theory. From Table  3, we find that the following superpotentials are consistent with all the symmetries.
Consequently, there is no stable SUSY vacuum for N S ≤ 3. The Higgs and Coulomb branches are quantum-mechanically merged for N S = 4. The large values of the Higgs branch is connected to the small value of the Coulomb branch. Importantly the origin of the moduli space is not a vacuum. For N S = 5, the theory is s-confining, where the origin belongs to the vacua. For N S ≥ 6 we have no simple superpotential. In what follows, we will verify our superpotentials above in various ways. It is easy to check the parity anomaly matching for N S = 5. The UV and IR descriptions produce the same anomalies. By adding the term ηZ from the KK-monopole, the 4d superpotentials are correctly reproduced. Next let us introduce a complex mass deformation. We restrict ourself to the case with N S = 5 and introduce a complex mass to the last flavor. By integrating out the massive modes, we arrive at the quantum constraint for N S = 4 as follows.
We can also test the Higgs branch. When a spinor gets a vev M SS,N S N S = v 2 , the theory flows to the 3d N = 2 G 2 gauge theory with N S − 1 fundamentals [18]. The superpotential above correctly explains this flow. For N S = 5, we need the following identification between the Spin(7) and G 2 moduli coordinates.
The superpotential reduces to where we absorbed the vev into the monopole operator. This superpotential was first obtained in [18]. The similar argument can be applied also for N S ≤ 4 and the G 2 superpotentials are reproduced. Finally we briefly discuss the theory with N S ≥ 6. In this case one cannot write down the superpotential. From the analysis of the semi-classical Coulomb branch, it is expected that the Coulomb branch is still one-dimensional (it is labeled by Z) and that the quantum moduli space would be identical to the (semi-)classical one. If the fractional power in a superpotential is allowed, one can still write down the "effective" superpotential. For N S = 6, the superpotential is consistent with all the symmetries. By adding a term ηZ, the 4d result (2.6) is reproduced. However, the fractional power leads to the branch-cut singularities on the origin of the moduli space and we have to introduce new massless degrees of freedom along the singularities. Presumably, some Seiberg dual descriptions would explain these massless modes and a certain superconformal fixed point is realized on the origin of the moduli space. We don't discuss it further in this paper and will tackle with this problem elsewhere.

Superconformal Indices
Since the Spin(7) theory with five spinors exhibits the s-confinement phase, the superconformal index is simple enough and it is computed from the dual side. This would be another check of our analysis. For the definitions of the superconformal indices, see for example [23][24][25][26][27][28][29][30]. The index on the dual side has the contributions from the meson M SS,ij , the baryon B i S and the Coulomb branch operator Z. We set R S = 1 8 for simplicity and use a fugacity u for the global U(1) S symmetry which rotates the spinor. The full index (or the index of the dual description) becomes We will briefly explain the low-lying operators below.
• The first term is an identity operator.
• The second term 15u 2 x 1/4 is identified with a meson contribution M SS,ij which has 15 independent components.
• The third term 125u 4 √ x consists of two operators. One is a baryonic operator B i S which contributes to the index as 5u 4 x 1/2 and the other is a square of the mesons M SS ⊗M SS , whose flavor indices are symmetrized. Thus we have 15 × 15| symmetric part = 120 = 50 + 70 ′ in an SU(5) notation.
• The fourth term 1 u 10 + 755u 6 x 3/4 contains the monopole operator which is denoted by Z. The remaining parts are the symmetric products of the Higgs branch operators, Let us move on to the electric side. The index on the electric side is decomposed into the index for each GNO charge (m 1 , m 2 , m 3 ), m i ∈ Z/2. Since we now discuss the Spin (7) gauge group, we have to sum up only the sectors with m 1 + m 2 + m 3 ∈ Z [16]. We need to consider the GNO charges (0, 0, 0), 1 2 , 1 2 , 0 , (1, 1, 0) , 3 2 , 3 2 , 0 and (2, 2, 0) up to O(x 3 ). The index with zero GNO charge becomes The first term is an identity operator and regarded as the state |0, 0, 0 . Since the gauge group is not broken in this sector, we can freely act the Higgs branch operators on the state |0, 0, 0 . For example, 15u 2 x 1/4 is identified with M SS,ij |0, 0, 0 . Next let us study the sectors with non-zero GNO charges.
The sector with a GNO charge 1 2 , 1 2 , 0 contains the monopole operator. The first term x 3/4 u 10 is Z (see Table 3) and the corresponding state is expressed as | 1 2 , 1 2 , 0 . The proceeding two terms 15x needs some explanation. The GNO charge assignment 1 2 , 1 2 , 0 breaks the gauge group to Spin(3) × SU(2) × U(1). The spinor reduces to (2, 2) 0 where we omitted the charged fields since we cannot act the charged fields on | 1 2 , 1 2 , 0 a la [29]. Therefore we cannot totally antisymmetrize the SU(5) flavor indices of the reduced spinors (fourth-order anti-symmetrization is still allowed). Therefore, in the product M SS × B S = 15 ⊗5 = 5 + 70, we have to discard the 5 representation. As a result, The similar argument is available for higher order terms. By summing up these indices, we observe the complete agreement between the electric and magnetic sides. and (4, 1). Since we have already discussed the (N f , N S ) = (0, 5) case, we start with (N f , N S ) = (1, 4). The dynamics for the theories with fewer matters can be obtained from the s-confinement description by integrating out massive fields.
From the (semi-)classical analysis of the Coulomb branch operators for the simple roots, the Coulomb moduli should be divided into two parts depending on the sign of φ 1 − φ 3 . Thus we expected that two (quantum) Coulomb moduli Z and Y (or Z and Y spin ) are necessary. However, in this phase with (N f , N S ) = (1, 4), we can relate these two coordinates by acting the Higgs branch coordinates on the monopole operator. For example, Y has the same quantum numbers as with some numerical coefficients α and β. The deep reason behind this identification is unclear, but it is allowed at least from a symmetry argument. We predict that the (quantum) Coulomb branch is described by a single Z coordinate. The validity of this prediction can be checked via various deformations and the superconformal indices below.
For the description of the Higgs branch, we define following operators. We listed the quantum numbers of the matter contents and of the moduli coordinates in Table 4. From the table, one can write down the superpotential where the last term is generated by a KK-monopole and absent in a 3d limit. By integrating out the Coulomb branch, we obtain a quantum constraint in 4d. This IR description gives the same parity anomalies as the UV theory. In addition, we cannot satisfy the parity anomaly matching if we introduce two Coulomb branch operators. This is a first non-trivial check of our prediction.
In order to test the superpotential above, let us consider various directions of the Higgs branch, which would justify our analysis. First, we consider introducing the vectorial vev M QQ = v 2 which breaks the Spin(7) group to Spin(6) ∼ = SU(4). The low-energy theory is a 3d N = 2 SU(4) gauge theory with four flavors in a (anti-)fundamental representation, which is s-confining [31]. Since the global symmetries are enhanced to SU(4) × SU(4) in the low-energy limit, we have to rename and decompose the fields as where M j i is regarded as a meson and B,B are (anti-)baryonic operators in the SU(4) theory. By absorbing the vev into the redifinition of the monopole operator, the superpotential reduces to which is precisely the superpotential of the 3d N = 2 SU(4) theory with four flavors [31].
Next, let us focus on the G 2 direction of the Higgs branch, which is achieved by introducing a vev for a single spinorial field as M SS,44 = v 2 . The low-energy theory becomes a 3d N = 2 G 2 gauge theory with four fundamental matters, which is again s-confining [18]. Although the vev breaks the global SU(4) symmetry to SU(3), we again have the enhanced SU(4) symmetry at the low-energy limit since the vector and the spinors become the same representation in G 2 . We need the following identification between the Spin(7) and G 2 moduli coordinates.
The superpotential reduces to which is the superpotential observed in [18]. We can also consider a complex mass deformation for a vectorial matter. By introducing the mass term mM QQ , we find that B ′ S and P A,1 are integrated out. The equation of motion for M QQ leads to a quantum constraint which was observed in a previous section with N S = 4.

(N f , N S ) = (2, 3)
Let us next consider the 3d N = 2 Spin(7) gauge theory with two vectors and three spinors (see Table 5). In this case, we also have a similar relation between Z 3 and Y spin . Therefore, we expect that the quantum Coulomb branch is one-dimensional although the (semi-)classical analysis suggested the two-dimensional coordinates. We use the coordinate Z to parametrize the Coulomb branch. The Higgs branch is described by the following operators Notice that the spinors and the vectors now can be anti-symmetrized and we omitted the gamma matrices above for simplicity. The superpotential consistent with all the symmetries is where the last term exists only when we put the theory on S 1 × R 3 . By integrating out the Coulomb branch operator, we obtain a 4d quantum constraint.
Let us confirm the validity of the superpotential (5.19). The UV and IR descriptions yield the same parity anomalies. As in the previous case, we can test the SU(4) Higgs branch with M QQ,22 = v 2 , where the theory reduces to a 3d N = 2 SU(4) gauge theory with one antisymmetric matter and with three (anti-)fundamental flavors. It is not known in the literature whether this low-energy theory is s-confining or not. However we can show that this theory indeed exhibits an s-confinement phase. Table 6 shows the matter contents and their quantum numbers of the SU(4) theory.
The Coulomb branch Y SU (4) corresponds to the breaking SU(4) → SU(2) × U(1) × U(1). The non-perturbative superpotential becomes (5.20) In deriving the above, we assumed that the Coulomb branch is one-dimensional. This is plausible because the theory flows to a theory with one-dimensional Coulomb branch along the Higgs branch. For instance, when M gets a vev with rank 1, the low-energy theory becomes a 3d N = 2 SU(3) gauge theory with three (anti-)fundamental flavors. This theory has one Coulomb branch coordinate and is also s-confining [31]. When B A orB A gets an expectation value, the theory flows to a 3d N = 2 SU(2) with four fundamental matters, which is again s-confining and has a one-dimensional Coulomb branch. Finally, when T gets a vev, the theory flows to a 3d N = 2 USp(4) theory with six fundamentals, which is s-confining and has a one-dimensional Coulomb branch. We can derive the superpotential (5.20) from (5.19). Since the global symmetry is enhanced to SU(3) × SU(3), we decompose the Higgs branch operators as By properly rescaling the Coulomb branch operator Z, we arrive at the SU(4) superpotential (5.20). We can also test the G 2 Higgs branch M SS,33 = v 2 , where the theory reduces to a 3d N = 2 G 2 gauge theory with four fundamental matters, which is s-confining. We can derive the matter contents and the superpotential of the G 2 theory from our superpotential. We have to decompose the fields as follows.
By substituting these expressions into the superpotential (5.19), the G 2 superpotential is reproduced although unnecessary terms like ) are also generated. Presumably, this is because our description only respects SU(2) × SU(2) × U(1) ⊂ SU(4) of the G 2 theory. In the RG flow, these terms are supposed to be suppressed.

Superconformal Indices of SU(4) with and 3 ( + )
We start with the index of the 3d N = 2 SU(4) gauge theory with one anti-symmetric matter and with three (anti-)fundamental flavors. Since the theory is s-confining, the confined description also yields the same index. The dual index has the contributions from T, M, B A ,B A and Y SU (4) . The dual index becomes I SU (4) dual where we set R A = R Q = 1 6 for simplicity. t is a fugacity for the U(1) axial symmetry and u counts the number of the anti-symmetric tensor. We did not include the fugacity for U(1) baryon symmetry.
For the electric side, we have to sum up the following sectors up to O(x 5/3 ).
The sector with zero GNO charge contains the Higgs branch operators. The second term x 1/3 (9t 2 + u 2 ) corresponds to M and T . The third term 6t 2 u √ x is the baryonic operators B A andB A . The sector with a GNO charge 1 2 , 0, 0 contains the Coulomb branch operator Y SU (4) . We observe exact matching of the indices between the electric and magnetic sides.

(N f , N S ) = (3, 2)
Let us move on to the 3d N = 2 Spin(7) gauge theory with three vectors and two spinors. This case will require two Coulomb branch coordinates even at a quantum level. First, we enumerate the Higgs branch coordinates.  Table 7 below shows the matter contents and their quantum numbers. We also listed the 4d dynamical scale and the moduli coordinates.
From the zero-mode counting of the Coulomb branch operators, we expected that there are two Coulomb branch directions un-lifted. One coordinate would be globally defined on the whole Weyl chamber, which was denoted by Z, and the other is defined on the region of φ 1 > φ 3 , which is Y or Y spin . From various consistency checks, we assume that these two directions are Z and Y in this case. Being different from the other examples, we cannot find any simple relation between them (we need to include at least a fractional power of the Higgs branch operators). Consequently, the two-dimensional coordinates are necessary for the quantum Coulomb branch of (N f , N S ) = (3,2). One can write down the superpotential consistent with all the symmetries listed in Table 7.
where the last term appears when we put a theory on S 1 × R 3 and it is absent in a 3d discussion. We can check the validity of this s-confinement phase in various ways. First, remember that the 4d superpotential for (N f , N S ) = (3, 2) takes the following form [5], and this is easily reproduced by integrating out the two Coulomb branch operators. Second, we consider the Higgs branch along which the gauge group is broken to G 2 . This can be achieved by higgsing the spinorial matter, let's say M SS,22 = v 2 . In order to properly obtain the G 2 superpotential we have to rename the fields as where Y and P S3,11 become massive and integrated out. By substituting these expressions into the superpotential, we arrive at the G 2 superpotential [18]. Next, let us study another Higgs branch M QQ,33 = v 2 along which the Spin(7) group is broken to SU(4). The low-energy theory becomes a 3d N = 2 SU(4) gauge theory with two antisymmetric matters and with two (anti-)fundamental flavors. This was studied in [32] (see also [33]). Table 8 shows the matter contents, moduli coordinates and their quantum numbers. This theory has a two-dimensional Coulomb branch parametrized by Y andỸ . These two monopole operators correspond to the breaking SU(4) → SU(2) × U(1) × U(1) and SU(4) → SU(2) × SU(2) × U(1) respectively. This theory is known to be s-confining and the effective superpotential becomes where we neglected the relative coefficients for simplicity. Table 8: SU(4) with 2 and 2 ( + ) Since the global symmetries are enhanced to SU ( By substituting these expressions into the superpotential, we reproduce the superpotential (5.40).

Superconformal Indices
As another non-trivial check of our analysis, we study the superconformal indices of the 3d N = 2 Spin(7) gauge theory with (N f , N S ) = (3,2). Since the dual description has no gauge group, the index is simple and expanded as

(N f , N S ) = (4, 1)
In this subsection, we will investigate the 3d N = 2 Spin(7) gauge theory with four vectors and one spinor. In order to describe the Higgs branch of the moduli space, we need to define the following gauge invariant operators Notice that only the symmetric product of the spinor is available. The theory has the SU(4) × U(1) Q × U(1) S × U(1) R global symmetries. Table 9 shows the quantum numbers of the moduli coordinates. Table 9: Quantum numbers for (N f , N S ) = (4, 1) From the analysis of the Coulomb branch corresponding to the semi-classical monopoles, one might expect that two-dimensional subspace of the classical Coulomb moduli remains flat and these are parametrized by Z and Y spin . In this case, however one can identify these two Coulomb branch operators as Z ∼ Y spin M SS . Therefore it is plausible to expect that the quantum Coulomb branch is one-dimensional. The superpotential consistent with all the symmetries takes where the term proportional to η is generated by a KK-monopole and absent in a 3d limit.
Originally the KK-monopole contribution is ηZ but now it is expressed in terms of Y spin . We can easily check the parity anomaly matching between the UV theory and the IR description (5.54). One might consider that the quantum Coulomb branch is described by Y instead of Y spin . However, in this case, we cannot satisfy the parity anomaly matching for k U (1) R U (1) R . By integrating out the Coulomb branch Y spin , we reproduce the 4d result with a single quantum constraint [5] M 2 SS det M QQ + P 2 M QQ − R 2 + ηM SS = 0. (5.55) Therefore, the identification, Z ∼ Y spin M SS , properly reduces the 3d result to the 4d constraint. Let us check the complex mass deformation for the spinorial matter, which leads to the 3d N = 2 Spin(7) gauge theory with four vector matters. The superpotential becomes which lead to P i = R = 0 and M SS is integrated out. The low-energy superpotential results in This is consistent with the observation in [17] with modification of the Coulomb branch operator. This difference is due to the fact that we deal with not an SO(7) group but a Spin (7) group.
Next, we will test the Higgs branch. When the spinor gets a vev M SS = v 2 , the gauge group is broken to G 2 . The low-energy limit becomes a 3d N = 2 G 2 gauge theory with four fundamentals from the vector matters. Under the breaking we have the following identification between the Spin(7) and G 2 theories The superpotential reduces to which is precisely the G 2 superpotential observed in [18]. Let us consider the different direction of the Higgs branch M QQ,44 = v 2 , along which the gauge group is broken as Spin (7) → SU(4). The low-energy theory becomes a 3d N = 2 SU(4) gauge theory with three antisymmetric matters and one (anti-)fundamental flavor. Since the UV theory is s-confining, the low-energy SU(4) theory is also confining. We can directly show that this theory indeed exhibits an s-confinement phase. Table 10 shows the matter contents of the SU(4) theory and their quantum numbers.
From the classical analysis of the SU(4) Coulomb brach (see [32,34]), one might expect that there are two types of Coulomb branch corresponding to However, Table 10 suggests that these two variables are related as Y ∼Ŷ M 0 . Consequently the quantum Coulomb branch becomes one-dimensional. We obtain the confining superpotential W =Ŷ (T 3 M 2 0 + T M 2 2 + BB). (5.64) One can flow to this superpotential also from the UV description of (5.54). In order to show this, we have to rename the fields as follows By substituting these expressions, we reproduce the superpotential (5.64).

Superconformal Indices of SU(4) with 3 and ( + )
Let us first study the superconformal indices of the 3d N = 2 SU(4) gauge theory with three antisymmetric matters and with one (anti-)fundamental flavor. Since the theory is s-confining, the index must be equivalent to the index of the dual description with matters has the one-dimensional (quantum) Coulomb branch parametrized by Z. For N S ≤ 3, we found no stable SUSY vacuum. For N S = 4, the Higgs branch and the Coulomb branch are merged. For N S = 5, the theory is s-confining. For the theory with both spinors and vectors, the Coulomb branch becomes two-dimensional at least semi-classically and needs two coordinates Z and Y (or Y spin ). However, sometimes we can relate these two coordinates quantum-mechanically by taking the product of the Higgs and Coulomb branch coordinates. If this is possible, the Coulomb branch becomes one-dimensional. Especially we focused on the s-confinement phases which appear for (N f , N S ) = (0, 5), (1,4), (2,3), (3,2) and (4, 1). We found and tested various s-confinement phases for the Spin(7) theories. As a byproduct, we could obtain the s-confinement phases for the 3d N = 2 SU(4) gauge theories with n anti-symmetric matters and with 4 − n (anti-)fundamental flavors. For n = 1, 3, the sconfinement phases were not known in the literature. We also tested the validity of our analysis by computing the superconformal indices. The indices are perfectly consistent with our prediction on the Coulomb branch coordinates and also consistent with the s-confinement phases which we found.
In this paper, we expected that two-dimensional coordinates are semi-classically described by Z and Y (or Z and Y spin ). Since the Z coordinate is globally defined without depending on the sign of φ 1 − φ 3 , it is plausible to expect that Z is necessary in any cases. However we could not find a priori way for choosing Y or Y spin for the description of the remaining Coulomb branch. Just from various consistencies (including the SCI calculation, parity anomaly matching, deformations), we decided which one is more appropriate. For instance, Z and Y are presumably the natural coordinates for (N f , N S ) = (3, 2) while Z and Y spin are chosen for (N f , N S ) = (4, 1) and Z was equivalent to Y spin M SS . However, these decisions and reasoning were not conclusive. It would be nice if we gain a clear understanding of the quantum Coulomb branch.
It is interesting to study 3d N = 2 Spin(N) (N > 7) theories with vector matters and with spinor matters. In the case of Spin(2N) groups, two types of spinor representations are available. Hence the phase diagrams would be more richer than the Spin(2N + 1) cases. We will soon come back to this generalization elsewhere.
It is worth searching for Seiberg dual descriptions for the 3d N = 2 Spin(7) gauge theories with spinorial matters. In 4d, the dual theory has an SU(N S − 4) gauge group with N S anti-fundamental matters and with a matter in a symmetric representation. When we naively put a dual theory on a circle, the resulting Coulomb branch would be more than one-dimensional because a symmetric tensor divides the (classical) Coulomb branch. Furthermore the Coulomb branch operators are dressed by Higgs branch operators [34] because the matter contents are "chiral" in a 4d sense. Deriving the 3d duality from the 4d duality becomes very complicated in this case. We don't have any simple 3d dual to the Spin(7) now but would like to report some progresses along this direction in the near future.