Discrete Anomaly Matching for the Pouliot Type Dualities

We compute the 't Hooft anomalies of discrete symmetries in the Pouliot type dual theories and check their anomaly matching conditions. The Pouliot type dual theories we will consider in this paper are two dual pairs; the dual pair of ${\cal N}=1$ supersymmetric theories of a $Spin(7)$ gauge theory with spinors and a $SU(N_f-4)$ gauge theory with a symmetric tensor, fundamentals and singlets, and the other dual pair of ${\cal N}=1$ supersymmetric theories of a $Spin(10)$ gauge theory with a spinor and vectors and a $SU(N_f-5)$ gauge theory with a symmetric tensor, fundamentals and singlets. We will show that the both pairs satisfy the discrete anomaly matching conditions.


Introduction
In this paper, we will study the 't Hooft anomalies of independent discrete symmetries of two dual pairs, an N = 1 supersymmetric Spin (7) gauge theory with N f spinors and its dual [1], and an N = 1 supersymmetric Spin(10) gauge theory with a spinor and N f vectors and its dual [2,3]. We will show that the two dual pairs satisfy the 't Hooft anomaly matching conditions of the independent discrete symmetries.
The concern that quantum gravitational effects might break global symmetries 1 and the facts that discrete symmetries play important roles in phenomenological models have led to the idea of discrete gauge symmetries [5], and further to the anomaly cancellation conditions of them [6,7,8] 2 . They have been used to derive the anomaly matching conditions of discrete symmetries [10,11] by extending the discssions given by 't Hooft [12].
The continuous 't Hooft anomaly matching conditions are necessary but severe conditions for possible low-energy theories of a strong coupling high-energy theory to satisfy. In particular, they yield very strong evidences for the Seiberg dualities [13] 3 . The anomaly matching conditions for discrete symmetries may give more evidences for the conjectures, if a dual pair has independent discrete symmetries. In fact, in the papers [10,11], for many dual pairs, the discrete anomaly matching have been checked, and some of them do not pass the tests. Therefore, it is significant to check the discrete anomaly matching for unchecked dual pairs. Among those, we will study the above two pairs, the Spin(7) theory and its dual, and the Spin(10) theory and its dual.
In section 2, we will discuss an independent discrete symmetry of the Spin(7) theory and embed the discrete symmetry into an anomaly free U(1) symmetry by introducing additional fields so as to define the discrete anomalies. We will repeat the same discussions for the dual SU(N f −4) theory. In the magnetic theory, the transformation laws of the fields for the discrete symmetry is not uniquely determined. However, we will show that they all saturate the discrete 't Hooft anomaly matching conditions given by the electric Spin(7) theory. The dual pair of the Spin(7) theory is obtained by the parent dual pair of an N = 1 supersymmetric Spin (8) theory with a spinor and N f vectors and its dual [15] by higgsing the Spin(8) gauge group by 1 See [4] for recent discussions. 2 See [9] for recent discussions on discrete (non-)Abelian symmetries. 3 See [14] for a recent excellent review. the non-zero vacuum expectation value of the spinor. In the Spin(8) theory and the dual theory, there is an anomaly free U(1) global symmetry, and after the higgsing, it is broken into the discrete symmetry of the Spin(7) theory and its dual. The continuous 't Hooft anomalies which the U(1) symmetry takes part in become the discrete anomalies, and therefore, the fact that the parent dual pair satisfies the continuous 't Hooft anomaly matching conditions immediately implies that the Spin(7) and its dual satisfy the discrete anomaly matching. However, there is a subtlety upon defining the discrete anomalies from the continuous anomalies because we need to perform a gauge transformation as well to gain the discrete symmetry. We will elaborate on this issue and show that the dual theories surely satisfy the discrete anomaly matching conditions.
In section 3, we will proceed to the dual pair of the Spin(10) theory, and repeat the same procedure as done for the Spin(7) dual pair. We will embed the discrete symmetry into an anomaly free U(1) symmetry by adding fields to the theories and compute the anomalies.
Although we can not uniquely determine the transformation laws of the magnetic fields, we will show that they all satisfy the discrete anomaly matching conditions, as for the Spin (7) dual theories. Contrary to the Spin(7) dual theories, there are no known parent dual pair, from which the Spin(10) dual pair can be derived. Therefore, we haven't found the extended theories with the embedding U(1) symmetry of the dual pair, where all the continuous 't Hooft anomaly matching conditions including those with the embedding U(1) symmetry are satisfied.
However, in Appendix A, we will construct extended theories of the Spin(10) dual pair, where the continuous 't Hooft anomaly matching conditions which the embedding U(1) symmetry take part in are satisfied exactly. It means that the discrete anomaly matching is automatic after the higgsing of the embedding U(1) symmetries on the both sides. The rest of the continuous 't Hooft anomaly matching are recovered after the decoupling of the additional fields.
Upon computations of anomalies, we will need to use the Dynkin index T G (R) of a representation R of a group G defined by where T a R (a = 1, · · · , dim G) are the generators of the group G in the representation R. We often omit the subscript G of the Dynkin index, when it is obvious. As for the normalization of T (R), we take T ( ) = 1 for the fundamental representation of SU(N), and T (N ) = 2 for the vector representation of Spin(N). They count the number of zero modes of a single fermion in the representation of the group, when the one-instanton background is turned on. For the adjoint representation of Spin(N), we have T (adj) = 2(N − 2). The properties of the spin representations frequently depend on the parity of N of Spin(N). For the spin reprensentataion 2 n−1 of both chiralities of Spin(2n), we have T (2 n−1 ) = 2 n−3 . On the other hand, for the spin representation 2 n of Spin(2n + 1), T (2 n ) = 2 n−2 .
2 The dual pair of the Spin(7) theory with spinors We will consider an N = 1 supersymmetric Spin (7) gauge theory with N f spinors Q i (i = 1, · · · , N f ) with no superpotentials [1]. Besides the continuous global symmetries SU(N f ) × U(1) R , there is an discrete Z 2N f symmetry in the theory. Under the discrete symmetry transformation, the spinors Q i transform as The charge assignments of the spinors Q i are listed in Table 1. Performing the discrete trans- Table 1: The charge assignments of the spinors Q i in the electric Spin(7) theory formation (1) twice, it gives a transformation given by an element of the center Z N f of the flavor SU(N f ) symmetry. It implies that the subgroup Z N f ⊂ Z 2N f can be identified with the center of the flavor group SU(N f ). Therefore, we need to take the quotient of Z 2N f by the subgroup Z N f to find an independent discrete symmetry. where we may identify the cyclic group Z N f with the center of the flavor group SU(N f ). The matter field Q i transforms into −Q i under the remaining subgroup Z 2 . We may identify it with a gauge transformation. In fact, we may take the gamma matrices for the gauge group Spin (7), so that γ 12 = iσ 3 ⊗ 1 2 ⊗ 1 2 , which is one of the generators of the Spin (7). Under the gauge transformation generated by γ 12 , the matter fields Q i transform as When φ = 2π, we see that the gauge transformation yields Q i → −Q i . Therefore, we have found that the discrete symmetry Z 2N f is a subgroup of Spin (7) × SU (N f ), when N f is an odd integer. It implies that there are no independent discrete symmetries for odd N f .
We will compute the discrete 't Hooft anomalies below, whichever the Z 2N f symmetry is independent or not. As is explained in [10], in order to compute Type II discrete anomalies we need to promote the discrete Z 2N f symmetry to a anomalyfree continuous U(1) symmetry. To this end, we will extend the theory by introducing a singlet Φ and a spinor P of the gauge group Spin (7). Under the promoted U(1) symmetry transformation, the fields transform as and Φ carries the U(1) R charge 0, and P the U(1) R charge one. Then, the U(1) R charge assignment for the spinors Q i may be kept intact and the U(1) R symmetry are still anomaly free. By inspection, we can verify that the promoted U(1) symmetry is also anomaly free. We will turn on the superpotential ΦP P . When we will promote the U(1) symmetry to a gauge symmetry by introducing a U(1) gauge superfield and turn on the Fayet-Iliopoulos term in the D-term potential of the gauged U(1) symmetry, we will find the vacuum Φ = 0, where the U(1) symmetry is broken into the discrete Z 2N f symmetry. The spinor P will gain a mass through the superpotential Φ P P and decouple from the rest of the theory in the infrared.
We can utilize the U(1) gauge superfield to compute the Type II discrete anomalies For the computations, we will also multiply the U(1) R charges by N f to make them integers.
Before proceeding to the computations, we notice that the field with the discrete charge q in the representation R of the flavor SU(N f ) group and in the representation R g of the gauge Spin(7) group contributes to the discrete anomalies except for the Z 2N f SU(N f ) 2 anomaly by a mutiple of q dim R dim R g . We see from Table 2 that the combination q dim R dim R g is a multiple of 2N f . Since we count the discrete anomalies modulo 2N f , all the discrete anomalies Note that we can introduce a vector P ′ of the gauge group Spin(7) instead of the spinor P to cancel the U(1) gauge anomaly and add the term ΦP ′ P ′ to the superpotential, which becomes the Majorana mass term after the U(1) symmetry breaking Φ = 0. Since the vector P ′ carries the U(1) R charge 1, it contributes only to the discrete 't Hooft anomalies Z 2N f (gravity) 2 and Z 3 2N f . Its contributions increase the anomalies Z 2N f (gravity) 2 and Z 3 2N f computed for the spinor P by N f and N 3 f , respectively. When N f = 6, the Spin(7) theory is in the confining phase without chiral symmetry breaking [1], and the low-energy physics is described by the mesons M ij ∼ Q i Q j and the baryons B ∼ Q 4 with the superpotential In order to embed the discrete symmetry into an anomaly free U(1) symmetry, we will introduce a singlet X, as listed in Table 3, and replace the superpotential (3) by Upon the U(1) symmetry breaking to the discrete symmetry by X = 0, rescaling the meson M ij and the baryons B ij , it is reduced into the original low energy theory of M ij and B ij . Using the fields M ij , B ij annd X, we find that the discrete 't Hooft anomalies are given by • Z 2N f (gravity) 2 : 2 × 6 × 7 2 + 4 × 6 × 5 2 − 12 = 6 + 7 × 12, They saturate the discrete 't Hooft anomalies of the Spin(7) theory extended by the vector P ′ instead of the spinor P . Thus, the low-energy theory of the meson M ij and the baryons B passes the discrete 't Hooft anomaly matching tests.
The dual of the Spin(7) theory for 7 ≤ N f ≤ 14 [1] is an N = 1 supersymmetric SU(N f −4) gauge theory with a s in , N fqi in , and singlets M ij with the superpotential The charge assignments of the fields are listed in Table 4. Table 4: The field content of the dual of the Spin (7) theory The correspondence of the gauge invariant operators M ij ∼ Q i · Q j , Q 4 ∼q N f −4 and the invariance of the superpotential determine the transformation laws of the discrete symmetry up to an integer p ∈ Z. The anomaly free condition of the discrete symmetry by the gauge interaction is The anomaly free condition for the discrete symmetry is obviously satisfied for any integer p, and it is sufficient for computations of Type I discrete 't Hooft anomalies, but, in order to compute Type II discrete anomalies, we need to embed the discrete group into an anomaly free U(1) group, as is done in the electric theory. To this end, we set p = −1 so that the gauge anomaly for the discrete symmetry is strictly vanishing 4 . We thus find the transformation laws of the discrete symmetrỹ We see that the discrete symmetry group is a subgroup of the cyclic group Looking at the exponent of the transformedq i , when we perform the same transformation twice, we see that the resulting transformation can be given by an element of the center of the flavor group SU(N f ) and an element of the center of the gauge group SU(N f − 4). Therefore, it is not an independent discrete symmetry. We have seen that this is also the case for the discrete Z 2N f symmetry in the electric theory. In this sence, the independent discrete symmetries on the both side of the duality are given by the quotient group Z 2 . When N f is an odd integer, we have seen that the quotient group Z 2 is given by the center of the gauge group Spin (7) in the electric theory, and therefore, no independent discrete symmetry in the electric theory are found. In the magnetic theory, when N f is an odd integer, i.e., N f = 2k + 1 (k ∈ Z), the exponent of the transformedq i may be rewritten into and also we may rewrite the exponents of the other transformed fields into We find that the transformation laws of the discrete symmetry can be given by an element Furthermore, as N f and N f − 4 are relatively prime to each other, the group Although there are no independent discrete symmetries for odd N f , we will treat the cases for both odd and even N f on the same footing. We will now promote the discrete symmetry to an anomaly free U(1) symmetry by introducing a singlet chiral superfield X and replacing the term det s in the superpotential by X det s. We suppose that the singlet X carries no U(1) R charge to leave the U(1) R charge of the symmetric tensor s unchanged.
The anomaly free condition of the U(1) symmetry by the SU(N f − 4) gauge interaction and the invariance of the superpotential suggest that the fields under the U(1) transformation should transform as with a transformation parameter ω. Note that the U(1) transformation is distinct from the U(1) transformation in the electric theory, as we can verify from the gauge invariant operators 5 When N f is odd, it is obivous that N f is prime to 2 m , for a positive integer m. It means that there are two However, the reason that we embed the discrete symmetry into an anomaly free U(1) symmetry is just to define Type II discrete 't Hooft anomalies, but not to find the dual theory of the extended electric theory with the extra U(1) symmetry.
Let us introduce a U(1) gauge superfield to promote the U(1) symmetry to a gauge symmetry and introduce the Fayet-Iliopoulos term in the D-term potential of the U(1) gauge symmetry so that there exists a vacuum X = 0, where the U(1) symmetry is broken into the original discrete Z N f (N f −4) symmetry. As was done for the electric theory, we can make use of the back- We will multiply the U(1) R charges by N f . However, the U(1) R charges are not all integers even after multiplying them by N f , contrary to the electric theory. We compute the 't Hooft anomalies for the magnetic theory, In order to examine whether the dual theories satisfy the discrete 't Hooft anomaly matching conditions, we will embed both of the discrete Z 2N f group in the electric theory and the discrete group by multiplying the discrete charges in the electric theory by N f − 4 and those in the magnetic theory by 2, respectively. Then, we find that the . We have seen the ambiguity of the magnetic discrete symmetry by an integer p, and we have taken p = −1 as the simplest choice for an anomaly free U(1) symmetry. We will here consider the case p = −1 and add additional matter fields for an anomaly free U(1) symmetry.
We read a U(1) symmetry transformation for p = −1, with a transformation parameter ω. Since the U(1) symmetry is anomalous, we will add additional fieldsX, F andF , listed in Table 6, to make it anomaly free. We will add the term XFF to the superpotential so as to make them massive after the U(1) symmetry breaking.
Then, the discrete symmetry group is a subgroup of a cyclic group Z 2N f (N f −4) , due to the presence of the extra fundamental pair F ,F . The discrete charges of the magnetic fields are listed in Table 6.
The introduction of a U(1) gauge superfield to promote the U(1) symmetry to a U(1) gauge group is the same as for p = −1, and we will find the vacuum where X = 0, X = 0, breaking we compute the discrete 't Hooft anomalies. We take the U(1) R charges of F andF to be the values in Table 6 in order for the Z 2N f (N f −4) U(1) 2 R anomaly to be a multiple of 2N f (N f − 4), therefore, saturating the matching condition of the anomaly in the electric theory, modulo 2N f (N f − 4), as we will see soon.
To compare these anomalies with those in the electric theory, we will embed the electric Z 2N f group into the Z 2N f (N f −4) group by multiplying the discrete charges by N f −4, and we find that all the discrete 't Hooft anomalies of both electric and magnetic theories match modulo In the previous sections, we have promoted the discrete symmetries into anomaly free U(1) symmetries by extending both of the dual theories to larger theories. However, those extended theories are not dual to each other 6 . But, as we will see below, there is an ideal dual pair of extended theories, an N = 1 supersymmetric Spin(8) gauge theory with a spinor S and N f vectors Q i (i = 1, · · · , N f ) with no superpotential, and its dual theory [15]. See Table 7 for the field content of the Spin(8) theory. 6 For p = 0, we have different U (1) symmetries between the dual theories. For p = 0, although we have the  (8) to Spin(7) by S = 0, (7). The dual of the Spin (8) theory is reduced to the dual of the Spin (7) theory. Then, the non-zero vacuum expectation value S = 0 also breaks the anomaly free U(1) symmetry to the discrete symmetry of the Spin (7) theory. Before the higgsing of the Spin(8) gauge group, all the continuous 't Hooft anomaly matching conditions are satisfied by the Spin(8) theory and its dual theory. Therefore, all the discrete 't Hooft anomaly matching conditions should be satisfied by the Spin(7) theory and its dual SU(N f − 4) theory. However, it is somewhat less trivial to confirm this, as we will see below.
The magnetic theory of the Spin(8) theory is an N = 1 supersymmetry SU(N f − 4) theory with a symmetric tensor s, N f antifundamentalsq i (i = 1, · · · , N f ), and singlets M ij , X with the superpotential [15] M ijq i · s ·q j + X det s.
The singlets M ij , X correspond to the gauge invariant operators of the electric theory, M ij ∼ Q i · Q j , X ∼ S · S. There is also another gauge invariant operator B ∼ S 2 Q 4 ∼q N f −4 . See Table 8 for the field content of the magnetic theory.
The magnetic theory reminds us of the extended dual theory of the Spin(7) theory with p = −1. In order to go down to the dual theory of the Spin(7) theory, we take the vacuum with X = 0, which breaks the U(1) symmetry to the discrete symmetry, under which the fields transform as same U (1) symmetry, the continuous 't Hooft anomaly matching conditions are not satisfied. which is identical to the discrete symmetry for p = −1. Rescaling s,q i and M ij , the superpotential is reduced into M ijq i · s ·q j + det s. Thus, we obtain the dual of the Spin(7) theory.
On the other hand, in the electric Spin(8) theory, the vacuum X = 0 corresponds to the vacuum S = 0 via the correspondence X ∼ S 2 . Then, on the vacuum S = 0, the gauge group Spin(8) is broken to Spin (7). Since X and S transform under the global U(1) symmetry transformation as with a transformation parameter ω, the discrete symmetry (4) corresponds to In order to leave the vacuum expectation value S = 0 invariant, we need to perform the gauge at the same time, which is in a Z 2 subgroup of the center of the Spin(8) group. Let us take a closer look at the gauge transformation (6). To this end, we take the gamma matrices for the gauge group Spin(8) and the chirality matrix Γ 9 = Γ 1 Γ 2 · · · Γ 8 = 1 8 ⊗ σ 3 , where γ m (m = 1, · · · , 7) are the gamma matrices of the Spin(7) group in (2). Let us take the spinor S to be of positive chirality Γ 9 S = S. Then, the gauge transformation generated by Γ 12 transforms the spinor S and the and the Spin (8) We may define the gauge transformation (6) as the limit φ → 2π, and then it is convenient to decompose the representations of the Spin(8) group under Spin(2) × Spin (6), where the Spin(2) group is generated by the Γ 12 . The spinor S is decomposed into 4 1/2 ⊕4 −1/2 , and the vectors Q i into 1 1 ⊕ 1 −1 ⊕ 6 0 . The gaugino is decomposed into 1 0 ⊕ 6 1 ⊕ 6 −1 ⊕ 15 0 . We may regard the charges of the Spin(2) as additional discrete charges to accompany with the discrete transformation (5) in order to leave S = 0 invariant, a more refined definition of (6). The discrete transformation (5) is anomaly free, Although the naive definition (6) of the accompanying gauge transformation may be anomalous, we can verify that the refined one keeps it anomaly free. For example, the spinor S contributes to the gauge anomaly as They imply that the contributions from the Spin(2) charges, i.e., the accompanying gauge transformation, to the discrete 't Hooft anomalies are canceled to give nothing. For example, the Z 3 2N f anomaly is computed by (U(1) + Spin(2)) 3 , where U(1) denotes the global U(1) charges, which are the same as the discrete charges appearing in the discrete transformation (5) and Spin(2) means the Spin(2) charges of the accompanying gauge transformation. Then, (2), and Spin(2) 3 are zero, and it becomes U(1) 3 . Thus, we may ignore the Spin(2) charges to compute the discrete anomalies, and therefore, the discrete 't Hooft anomalies of the Spin (7) (7) theory and its dual theory. In fact, we can verify that the discrete 't Hooft matching conditions are satisfied by the direct computations • Z 2N f (gravity) 2 = U(1)(gravity) 2 the electric theory : the magnetic theory : the magnetic theory : the magnetic theory : the magnetic theory : Under the gauge transformation (7), the gauge invariant operator S 2 Q 4 is left invariant by definition. Therefore, under the discrete transformation (5), it transforms as The contribution from the spinor S to the discrete charge of S 2 Q 4 accounts for the discrepancy of the U(1) symmetries between the Spin(7) theory and its dual, upon the promotion of the discrete symmetries into anomaly free U(1) symmetries, when choosing p = −1, as we have seen in the previous sections.
In summary, we have seen that the discrete 't Hooft anomaly matching between the Spin (7) theory and its dual can beautifully be shown by using the parent dual pair, the Spin(8) theory and its dual.
3 The dual pair of the Spin(10) with a spinor and vectors We will consider an N = 1 supersymmetric Spin(10) gauge theory with a single spinor S and N f vectors Q i (i = 1, · · · , N f ) with no superpotentials and its dual theory [2,3]. See Table  9 for the charge assignments of the matter fields in the electric Spin(10) theory. There is an independent Z 2N f symmetry in the theory. The discrete Z 2N f symmetry transformation act on the vectors Q i and the spinor S as which is anomaly free discrete symmetry. We will introduce a singlet Φ and a vector P with the superpotential ΦP P in order to promote the discrete symmetry Z 2N f to an anomaly free U(1) symmetry for computations of Type II discrete 't Hooft anomalies so that the fields Φ, P cancel the gauge anomaly of the promoted U(1) symmetry by the Spin(10) gauge interactions. Table 9: The electric Spin(10) theory with a spinor and N f vectors In order to make the U(1) R charges integers, we will multiply them by N f . Since the spinor S does not give any contributions to the discrete anomalies, we may divide the U(1) charges by two. Each of the fields gives a multiple of q dim R dim R g as its contribution to all the discrete anomalies except for Z 2N f SU(N f ) 2 , where q is the discrete charge of the field, dim R is the dimension of its representation of the flavor symmetry group SU(N f ) and dim R g is the dimension of its representation of the gauge group Spin(10). As we can see from Table 9, the quantity q dim R dim R g for each of all the fields is always a multiple of 2N f , and therefore, all the discrete anomalies except for Z 2N f SU(N f ) 2 are zero modulo 2N f . In fact, we compute the discrete 't Hooft anomalies; • Z 2N f SU(N f ) 2 : 10, The magnetic theory exists for 7 ≤ N f ≤ 21, and it is an N = 1 supersymmetric SU(N f −5) gauge theory with N f antifundamentalsq i , a single fundamental q, a symmetric tensor s and The charge assignments for the magnetic fields are listed in Table 10. The gauge invariant which, together with the invariance of the superpotential, determines the discrete symmetry transformation of the other magnetic fields up to an integer p. The discrete symmetry transformations with different values of p are related to one another by the gauge transformations given by elements of the center of the gauge group SU(N f − 5), and therefore, there exists only one independent discrete symmetry out of them.
Since our intension for the promotion of the discrete symmetry to an anomaly free U(1) symmetry is just to define Type II discrete 't Hooft anomalies in the magnetic theory, we do not have to take the U(1) symmetry to be the promoted U(1) symmetry of the electric theory through the correspondence of the gauge invariant operators. When we choose p = −1 and promote it to the U(1) symmetry transformation of the magnetic theory with a transformation parameter ω, we find that the promoted U(1) symmetry is anomaly free, and we need to introduce no more fields to cancel the gauge anomaly. However, the gauge invariant operator B mag ∼q N f −5 transforms under the magnetic U(1) symmetry into the transform of B ele ∼ S 2 Q 5 under the electric U(1) symmetry multiplied by e −2πiω .
For the promotion to the anomaly free U(1) symmetry, we will also replace the term det s in the superpotential by X det s with a singlet X, transforming under the promoted magnetic U(1) symmetry as In order to compute the Type II discrete 't Hooft anomalies, we will introduce a U(1) gauge superfield for the promoted magnetic U(1) symmetry and will turn on the Fayet-Iliopoulos term in the D-term potential of the U(1) gauge symmetry. Then, we will find a vacuum with X = 0, which breaks the U(1) symmetry down back to the original discrete symmetry in the magnetic theory. In the infrared, the theory is reduced into the original magnetic theory.
The discrete symmetry group is a subgroup of a cyclic group Z 2N f (N f −5) . However, when we perform the above discrete symmetry transformation twice, the resulting transformation can be given by an element of the center of the flavor group and an element of the center of the gauge group, and therefore, it is not an independent discrete symmetry, anymore.
As is done for the electric theory, we will multiply the U(1) R charges by N f and divide the U(1) charges by two. Let Ψ be one of the magnetic fields with the discrete charge q in the representation R of the flavor symmetry group SU(N f ) and in the representation R g of the gauge symmetry group SU(N f − 5). The field Ψ gives its contributions to all the discrete 't Hooft anomaly except for Z 2N f (N f −5) SU(N f ) 2 by a multiple of q dim R dim R g . As we can see from Table 10, the combination q dim R dim R g is a multiple of N f (N f − 5), and therefore, checking the discrete anomaly matching except for Z 2N f (N f −5) SU(N f ) 2 is to examine whether it is an even or odd multiple of N f (N f − 5) for the anomalies including no U(1) R . Although the U(1) R charges of the magnetic fields are not all integers, even after multiplying them by N f , we find that the anomalies including the U(1) R symmetry are also integers by the computations, In order to check the discrete 't Hooft matching conditions between the dual theories, we will embed the Z 2N f symmetry group of the electric theory into the Z 2N f (N f −5) group by multiplying the discrete charges of the electric fields by N f − 5. Then, we can see that the Incidentally, let us consider the discrete 't Hooft anomalies for the other cases with p = −1.
When n ≡ p+1 = 0, the invariance of the term X det s in the superpotential requires the discrete charge of X to be −2N f (N f − 5)(1 − 2n), as in Table 11. Then, the promoted U(1) symmetry is anomalous by the SU(N f − 5) gauge interactions, and therefore, we need to introduce more matter fields listed in Table 11 to cancel the gauge anomaly in order to promote the discrete symmetry to an anomaly U(1) symmetry. We will also add the termXFF to the superpotential so that the extra matter fieldsX, F ,F decouple in the vacuum X = 0 at the low energies. The discrete charges of F andF are determined by cancellation of the gauge anomaly of the promoted U(1) symmetry and by the requirement that performing the discrete symmetry transformation twice gives the gauge transformation of the element of the center of the gauge group SU(N f − 5). We have chosen the U(1) R charges of F andF so as to saturate the 't Hooft anomaly matching condition for The remaining procedure we have to carry out for the computation of the discrete anomalies is almost the same as what was done for p = −1, except that we take the vacuum such that X = 0, X = 0.
The Z 2N f (N f −5) SU(N f ) 2 anomaly for p = −1 is computed to give which is equal modulo 2N f (N f − 5) to the one for the case p = −1 (n = 0). For the rest of the discrete anomalies, multiplying the U(1) R charges by N f and dividing the U(1) charges by two, we find that they are all zero modulo 2N f (N f − 5), satisfying all the 't Hooft anomaly matching conditions.

Discussions
We have studied the discrete anomaly matching of the two dual pairs. One of them is the Spin(7) gauge theory with spinors and the SU(N f − 4) gauge theory with a symmetric tensor, fundamentals and singlets [1]. The other is the Spin(10) gauge theory with a spinor and vectors and the SU(N f − 5) gauge theory with a symmetric tensor, fundamentals and singlets [2,3].
We have shown that both of the dual pairs satisfy the discrete anomaly matching conditions.
For the dual pair of the Spin(7) theory, we have done this in two ways. In one way, we have embedded the discrete symmetries into an anomaly free U(1) symmetries by additional fields, which decouple after the U(1) symmetry breaking into the discrete symmetries, on the both sides of the duality. The extended theories are not dual to each other, and we have to compute the continuous 't Hooft anomalies in order to ensure the anomaly matching conditions. In the other way, we take another dual pair [15] of the Spin(8) gauge theory with a spinor and vectors and the SU(N f − 4) gauge theory with a symmetric tensor, anti-fundamentals and singlets, which is reduced to the Spin(7) dual pair by higgsing the Spin(8) gauge group to Spin (7).
The Spin(8) dual pair has an anomaly free U(1) symmetry, which is broken to the discrete symmetries of the Spin (7)  symmetries become the discrete anomalies after the higgsing of the U(1) symmetries, and we easily see that the discrete anomaly matching is achieved.
Another dual pair of an N = 1 supersymmetric G 2 gauge theory with fundamentals in the representation 7 and an N = 1 supersymmetric SU(N f − 3) gauge theory with a symmetric tensor, fundamentals and singlets is reduced from the Spin(7) dual pair [1] 7 by higgsing the Spin(7) gauge group. Therefore, it should be straightforward to check the discrete anomaly matching in a similar way to what we have done for the Spin(8) dual pair. This is also the case for other Pouliot type dualities [17] reduced from the Spin(10) dual pair. 7 See also [16], for the confining phases of the G 2 gauge theory.
There are no known dual pair, from which the Spin(10) dual pair is derived. Although we wish that the extended theories in Appendix A would give an insight into the discovery of such a parent dual pair, we guess that the gauge group of the electric Spin(10) theory should be larger than the Spin(10) group 8 , so that the higgsing in the electric theory should correspond to the decoupling of massive states in the magnetic theory.
Finally, we may extend the studies in this paper to a dual pair of an N = 1 supersymmetric Spin(10) gauge theory with more than one spinor and vectors, and the dual theory [19].
However, we will leave this subject for future investigation.

Acknowledgement
The author thanks Yukawa Institute for Theoretical Physics, Kyoto University for hospitality during the course of this project. 2 • U(1) 2 X U(1): We see that there are discrepancies in the anomalies except for U(1) X SU(N f ), U(1) X (gravity) 2 and U(1) 2 X U(1), between both the sides. We will add the singlet fieldsΦ, Ψ andΨ, listed in Table 12, and the termΦΨΨ to the superpotential in the electric theory. The fields contribute to the 't Hooft anomalies by theory to fill the gaps in the continuous 't Hooft anomalies