12.3 S-dual of SU(N) with Nf = 2N flavors, part II

12.3.1 For general N

Figure 12.13: S-dual of SU(N) with 2N flavors, explained.

Now we have learned enough techniques to understand the S-dual of SU(N) theory with 2N flavors, see the first row of Fig. 12.13. Originally, we have a sphere with four punctures: two at z = 0, are full punctures, and two at z = q, 1 are simple punctures. We would like to understand the limit q 1. We end up decoupling two simple punctures from the other two. We already learned what happens in this decoupling process.

The simple puncture is a puncture of type (N 1, 1). Decoupling two of them, we generate a puncture of type (N 2, 1, 1). This puncture has a flavor symmetry SU(2) ×U(1) when N > 3, and SU(3) when N = 3. The behavior of the duality when N = 3 is somewhat more peculiar than the other cases. In any case, there is an SU(2) symmetry exchanging the last two columns of height 2, and a weakly-coupled dynamical SU(2) group gauges this SU(2) symmetry. There is in addition one flavor in the doublet representation for this SU(2) gauge group coming from the almost decoupled sphere on the right, see the last row of Fig. 12.13.

The question is the nature of the sphere on the left hand side. It has three punctures: two are full punctures, and one is of type (N 2, 1, 1). Assuming all the mass parameters are zero, we can determine the behavior of fields ϕk(z) easily, as the pole structure at z = is (p2,p3,,pN) = (1, 2,, 2). We see that

ϕ2(z) = 0,ϕk(z) = uk (z 1)k 1zk 1dzk. (12.3.1)

This theory has one dimension-k operator for each k = 3, 4,,N. The flavor symmetry is at least SU(N) ×SU(N) associated to the full punctures, and SU(2) ×U(1) associated to the puncture of type (N 2, 1, 1). Call this funny conformal field theory RN, for which we introduce a graphical notation as in Fig. 12.14. In the original theory, the symmetry SU(N) ×SU(N) ×U(1) was part of the flavor symmetry SU(2N) rotating the whole 2N hypermultiplets in the fundamental representation. We then need to demand that this theory RN has a larger flavor symmetry

SU(2N) ×SU(2) [SU(N) ×SU(N) ×U(1)] ×SU(2). (12.3.2)

Figure 12.14: Strange theory of Chacaltana-Distler, RN.

We finally have the S-duality statement:

SU(N) theory with 2N flavors at the strong coupling q 1 weakly-coupled SU(2) gauge multiplet coupled to one doublet and to the RN theory. (12.3.3)

This general statement was found by Chacaltana and Distler in [72]. We know that the dual SU(2) gauge coupling has zero beta function. Applying the analysis as in Sec. 10.5, we find that the SU(2) flavor symmetry of the RN theory contributes to the running of the SU(2) coupling as if it has effectively three hypermultiplets in the doublet. Equivalently, we have

jμjνRN = 3jμjνfree hyper in a doublet of SU(2) (12.3.4)

where jμ is the SU(2) flavor symmetry current. See Fig. 12.15.

Figure 12.15: The SU(2) flavor symmetry current of the RN theory.

12.3.2 N = 3: Argyres-Seiberg duality

Figure 12.16: S-duality of SU(3) with 6 flavors involves the theory MN(E6).

When N = 3 we can say a little more about this duality. This was originally found by Argyres and Seiberg in [74]; the presentation here follows that given by Gaiotto in [8].

Now the puncture of type (N 2, 1, 1) = (1, 1, 1) is a full puncture. Therefore the theory R3 is given by a sphere with three full punctures, see Fig. 12.16. The structure of ϕk(z) is already given in (12.3.1). Therefore, this theory has just one Coulomb branch operator, of dimension 3.

We know that there is an enhancement of the flavor symmetry SU(3) ×SU(3) associated to two full punctures to SU(6), as in (12.3.2). We have three full punctures. Therefore, it should be that the flavor symmetry F of this theory should be such that we have the following diagram

F SU(6) ×SU(2) SU(3) ×SU(3) ×SU(3) SU(3) ×SU(3) ×U(1) ×SU(2) (12.3.5)

for any choice of two out of three SU(3)s. Fortunately, there is unique such F, that is E6, see Fig. 12.17. There, on the left, we introduce a diagrammatic notation for this theory. On the center and on the right, we have the extended Dynkin diagram of E6 with one node removed.16 We clearly see subgroups SU(3)3 and SU(6) ×SU(2). We already saw above that this theory has only one Coulomb branch operator, and its dimension is three. This nicely fits the feature of a rank-1 superconformal theory announced to exist in Sec. 10.4. This is equivalent to Minahan-Nemeschansky’s theory MN(E6).

Figure 12.17: The theory MN(E6) = R3 = T3

We conclude that we have the following duality:

SU(3) theory with 6 flavors at the strong coupling q 1 weakly-coupled SU(2) gauge multiplet coupled to one doublet and to the theory MN(E6) of Minahan-Nemeschansky. (12.3.6)

We can give a few more checks to this duality. The first one concerns the current two-point functions. Firstly, we computed the current two-point function for the SU(2) flavor symmetry in (12.3.4). Then the whole E6 flavor currents, which include the SU(2) ones, should have the same coefficient in front of the two-point function. Note that SU(6) flavor symmetry of the SU(3) gauge theory with six flavors is also a subgroup of this E6 flavor symmetry. Therefore, we should have

jμSU(6)jνSU(6)SU(3),Nf=6 = 3jμjνfree hyper in the fundamental of SU(6). (12.3.7)

This is indeed the case, since the left hand side can be computed in the extreme weakly-coupled regime, where they just come from three hypermultiplets in the fundamental representation of SU(6).

The second check is about the Higgs branch. The SU(3) theory with six flavors has a Higgs branch of quaternionic dimension

3 6 dim SU(3) = 10. (12.3.8)

Let us perform the computation in the dual side. The theory MN(E6) has a Higgs branch of quaternionic dimension 11, as we tabulated in Table 10.1. We have a doublet of SU(2) in addition, and we perform the hyperkähler quotient with respect to SU(2) gauge group. Therefore the quaternionic dimension is

11 + 2 dim SU(2) = 10, (12.3.9)

which agrees with what we found above in the original gauge theory side. Here we only compared the dimensions, but they can be shown to be equivalent as hyperkähler manifolds, see [75].