12.4 Applications

12.4.1 TN

Figure 12.18: Duality producing TN theory

We can now have some fun manipulating punctures. For example, consider a gauge theory with gauge group SU(N)N 2, with bifundamental hypermultiplets between consecutive groups, together with N additional fundamental hypermultiplets for the first and the last one, see the first row of Fig. 12.18. The Seiberg-Witten solution is easily given: it is given by a sphere of type SU(N) theory, with two full punctures and N 1 simple punctures. We go to a duality frame where we decouple all of these N 1 simple punctures. Applying the decoupling procedure we learned in Sec. 12.2, we find that we generate a quiver tail with gauge group

SU(N 1) ×SU(N 2) ×SU(2), (12.4.1)

with bifundamental hypermultiplets between two consecutive groups and one doublet for the last SU(2). The first SU(N 1) gauges an SU(N 1) subgroup of the flavor symmetry SU(N) of the puncture of type (1, 1,, 1), i.e. the full puncture.

Figure 12.19: The TN theory

In this way, we can construct a theory described by a sphere with three full punctures. This is called the TN theory, see Fig. 12.19. Note that R3 = T3. As we have three full punctures, the flavor symmetry is at least SU(N)3. When N = 3, we saw above that this flavor symmetry enhances to E6. When N 4, there are more than one gauge group in the original gauge theory. Therefore, we do not have an enhancement from SU(N) ×SU(N) to any other group. This matches with the fact that there is no group containing SU(N)3 such that SU(N)2 enhances to SU(2N) when N 4. Putting the punctures at z = 0, 1,, we see that ϕk has the form

ϕk = uk(1) + + uk(k 2)zk 3 zk 1(z 1)k 1 dzk. (12.4.2)

Therefore this theory has one Coulomb branch operator of dimension 3, two Coulomb branch operators of dimension 4, …, and N 2 Coulomb branch operators of dimension N.

Figure 12.20: S-duality of coupled copies of TN theory

Now we can take two copies of this TN theory and couple them by an SU(N) gauge multiplet. In the 6d construction, we just have four full punctures on the sphere. Therefore, we have the S-duality structure exactly as in SU(2) theory with four flavors, exchanging all four punctures. In fact, T2 theory is just the trifundamental hypermultiplet Qijk.

12.4.2 MN(E7)

Figure 12.21: Duality producing the MN(E7)

Next, consider the duality shown in Fig. 12.21. We end up with a three-punctured sphere with two full puncture and one puncture of type (2, 2). In the original gauge theory, we have six fundamental flavors coupling to the SU(4) gauge multiplet with SU(6) flavor symmetry. To construct the ultraviolet curve, we split these six flavors into four flavors and two flavors, and applied the rule shown in the third row of Fig. 12.7. Therefore, we see that the theory represented by the three-punctured sphere have a flavor symmetry F of the form

F SU(6) ×SU(3) SU(4) ×SU(2) ×SU(4) SU(4) ×SU(2) ×U(1) ×SU(3) . (12.4.3)

Figure 12.22: The theory MN(E7).

Thankfully, there is a unique such group F, that is E7, see Fig. 12.22. We can of course compute the number of Coulomb branch operators this theory has, by studying ϕk(z). Here, let us try a different procedure. Originally, we had the gauge group SU(4) ×SU(2). Therefore, the numbers of the Coulomb branch operators of dimension 2,3,4 were respectively 2, 1, 1. On the dual side, the quiver tail contains SU(3) ×SU(2), which has two operators of dimension 2 and one operator of dimension 1. The theory represented by the three-punctured sphere should account for the difference. Therefore there is just one Coulomb branch operator, of dimension 4. This again 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(E7). We can also check the agreement of the current two-point functions and the dimensions of the Higgs branch, as we did at the end of Sec. 12.3.2.

12.4.3 MN(E8)

Figure 12.23: Duality producing the MN(E8) theory

Generalizing this to the E8 symmetry is by now rather straightforward. We perform the duality as shown in Fig. 12.23. In the dual side, we have a three-punctured sphere with one full puncture, another of type (2, 2, 2), and of type (3, 3). We see that the flavor symmetry F of the theory should satisfy

F SU(5) ×SU(5) SU(2) ×SU(3) ×SU(6) SU(2) ×SU(3) ×U(1) ×SU(5) . (12.4.4)

This nicely fits the structure of Minahan-Nemeschansky’s theory MN(E8), see Fig. 12.24. Checks of various properties are left as an exercise to the reader.

Figure 12.24: E8 theory

12.4.4 The singular limit of SU(N) with even number of flavors

Finally, let us study a non-conformal example. Consider SU(N) theory with Nf = 2n flavors, with N > n. The curve is

ΛN n i=1n(x + μ + μi) z +ΛN n i=1n(x+μ+μ̃i)nz = xN+u2xN 2++uN (12.4.5)

with the differential λ = xdzz. Here we demanded iμi + μ̃i = 0 and split the U(1) mass term as μ. Clearly something happens when uNn = 2ΛN n around z 1. This point was first considered in [76]. The correct physics was first discussed in [77]. We will see below that the low-energy limit is an infrared-free SU(2) gauge theory coupled to the theories Rn and XNn+4.

To study the infrared behavior, we let

uNn,old = 2ΛN n + uNn,new,z = 1 + δz (12.4.6)

and assume the scaling

μi 𝜖,uN 𝜖n,uN1 𝜖n 1,,uNn+2 𝜖2, (12.4.7)


u2 𝜖2,u3 𝜖3,,uNn+2 𝜖N n + 2. (12.4.8)

We then need to assume

𝜖N n + 2 𝜖2. (12.4.9)

In particular we have

𝜖 𝜖 1. (12.4.10)

In the region x 𝜖, we can approximate the curve (12.4.5) as

ΛN n i=1n(x + μ + μi) z + ΛN n i=1n(x + μ + μ̃i)z = (2ΛN n + uNn)xn + uNn+2xn 2 + + uN(12.4.11)

with the scaling (12.4.7). When this is written as a degree-n equation for x, the coefficient of the xn term is given by

ΛN n z + ΛN nz 2ΛN n uNn (12.4.12)

In the limit 𝜖 0, two zeros of (12.4.12) collide at z = 1. This is exactly the situation we studied in Sec. 12.3 for SU(n) theory with 2n flavors in the q 1 limit. We see that we generate the Rn theory coupled to SU(2) gauge group; the operator uNn+2 is now regarded as the Coulomb branch vev of this SU(2). The parameters μi and μ̃i are now the mass parameters for the SU(2n) symmetry of the Rn theory.

In the region x 𝜖, the curve (12.4.5) can be approximated as

cδz2 = (xN n + u2xN n 2 + + uNn+1 x + uNn+2 x2 ), (12.4.13)

where the differential λ = xdδz and c is an unimportnat constant. We already encountered this in Sec. 11.5; this is the curve describing the Argyres-Douglas point of SU(N n + 1) theory with 2 flavors. Equivalently, we called this theory XNn+4 in Sec. 10.5.

Figure 12.25: The most singular point of SU(N) with Nf = 2n flavors

Summarizing, we see that the limiting theory has the structure given in Fig. 12.25. Namely, there is a weakly-coupled SU(2) gauge group, connecting the region x 𝜖 given by a sphere of 6d theory of type SU(n), representing the Rn theory, to the region x 𝜖, given by a sphere of 6d theory of type SU(2), representing the theory XNn+4.

In the intermediate region 𝜖 x 𝜖, the curve is just

δz2 uNn+2 x2 (12.4.14)

with λ = δzdx uNn+2dxx. We see that there is an SU(2) gauge group, with

a 1 2πi δzdx x uNn+2. (12.4.15)

The dual coordinate aD is then given roughly by

aD 2 2πix𝜖x 𝜖uNn+2dx x 2 2πia log 𝜖 𝜖. (12.4.16)

Using a 𝜖 and the relation (12.4.9), we see

aD = 2 2πi N n N n + 2a log a + . (12.4.17)

Recall that the running is given by

aD 2 2πi(4 Nf)a log a + (12.4.18)

for SU(2) theory with Nf flavors. This system then effectively has

Nf = 5N 5n + 8 N n + 2 > 4. (12.4.19)

The SU(2) is now infrared free. Note that this is correctly the sum of the effective number of flavors of the RN theory and the XNn+4 theory, as computed already. Indeed, it is 3 for the RN theory, and 2(N n + 1)(N n + 2) for the XNn+4 theory, see (12.3.4) and (10.5.8), respectively.