12.1 S-dual of SU(N) with Nf = 2N flavors, part I

12.1.1 Rewriting of the curve

We learned in the last section that the curve of SU(N) theory with 2N flavors is given by:

i=1N(x̃ μ̃i) z̃ + f i=1N(x̃ μ̃i)z̃ = x̃N + ũ2x̃N 2 + + uN (12.1.1)

where f is a complex number; the differential is λ̃ = x̃dzz. This theory is superconformal, and f is a function of the UV coupling constant τUV . We would like to understand the strong-coupling limits of this theory.

As we did in Sec. 9, it is convenient to rewrite the curve in terms of the Seiberg-Witten differential λ, to the form

λN + ϕ2(z)λN 2 + + ϕN(z) = 0. (12.1.2)

We start from (12.1.1). First we gather terms with the same power of x̃:

(1 1 z̃ fz̃)x̃N + 1x̃N 1 + 2x̃N 2 + + N = 0 (12.1.3)


1 = μ̃i z̃ + fz̃ μ̃i. (12.1.4)

We divide the whole equation by (1 1z̃ fz̃) and define x = x̃ + 1(1 1z̃ fz̃)N. We now have

xN + 2xN 2 + + N = 0 (12.1.5)

where k has poles of order k at two zeros z̃1,2 of 1 1z̃ fz̃ = 0, due to the shift from x̃ to x. We set z = z̃z̃1 so that one zero is now at 1, and another is at q = z̃2z̃1.

Introducing λ = xdzz, we have an equation of the form (12.1.2); ϕk(z) has poles of order at most k at z = 0, q, 1 and . Consider the case when all μ̃i and μ̃i are generic, and assume q 1. Then it is straightforward to determine how λ behaves close to each of the singularity. As we are solving a degree-N equation, we have N residues at each singularity. They are given by

μ1, μ2, ,μN1, μN,z 0, μ, μ, ,μ, (1 N)μ,z q, μ, μ, ,μ, (1 N)μ,z 1, μ1,μ2,,μN1, μN,z . (12.1.6)


μi = μ̃i 1 N iμ̃i + O(q), μi = 0; (12.1.7) μ = 1 N iμ̃i + O(q) (12.1.8)

and similarly for the μi, μ. Note that μi and μ are the mass parameters which enter the BPS mass formula. We found that they are related to the parameters μ̃i via a finite renormalization.

When N = 2, the structure of the residues at all four punctures were of the same type, as they are all given by (m,m). For N > 2, we see that the structure of the residues at z = 0, and the structure at z = q, 1 are different. The former is of the form (m1,,mN) with mi = 0, and the latter is of the form m(1, 1,, 1 N).

It is also instructive to consider the completely massless case, when we have μ̃i = μ̃i = 0 for all i. The original curve is just

xN z + fxNz = xN + u2xN 2 + + uN. (12.1.9)

After the same manipulation as above, we find

ϕk(z) = uk (z q)(z 1) dzk zk 1. (12.1.10)


ϕk(z) has poles of orderk 1when z = 0,, ϕk(z) has poles of order1 when z = q, 1. (12.1.11)

We observe here again that the behavior of the poles are all the same when N = 2, while the behavior at z = 0, and the behavior at z = 1,q are distinct when N > 2.

Figure 12.1: The ultraviolet curve of SU(N) theory with 2N flavors

We have 2N mass terms in the system. First of all we split them into N mass terms encoded in the region z 0, and another N mass terms in the region z . Correspondingly, we started from the flavor symmetry U(2N) and decomposed it into U(N) ×U(N). We further decompose each of U(N) into SU(N) and U(1). Combined, we use the decomposition of the flavor symmetry of the form

U(2N) U(N) ×U(N) [SU(N)A ×U(1)B] × [U(1)C ×SU(N)D]. (12.1.12)

The residues of λ at the puncture A at z = 0 and those at the puncture D at z = encode the mass terms for SU(N)A,D respectively, whereas those at the puncture B at z = q and those at the puncture C at z = 1 encode the mass terms for U(1)B,C; compare (12.1.6).

We then say that the singularity at z = 0 carry the SU(N) symmetry, the one at z = q carry the U(1) symmetry, and similarly for those at z = 1, = . We can visualize the situation as in Fig. 12.1. We call the punctures at z = 0, the full punctures, and those at z = q, 1 the simple punctures. In the 6d viewpoint, these are four-dimensional defect objects extending along the Minkowski 3, 1, and they carry respective flavor symmetries on them.

When N = 2, the original symmetry is not just U(2) but SO(4). Accordingly, the split U(2) SU(2) ×U(1) is enhanced to the following structure

SO(4) SU(2)×SU(2) = U(2) SU(2)× U(1) (12.1.13)

and therefore the distinction of the types of punctures is gone.

12.1.2 Weak-coupling limit

Figure 12.2: Weakly coupled limit

Clearly f q e2πiτUV in the weak coupling region, see Fig. 12.2. When the coupling is extremely weak, we can think that the four-punctured sphere on the left is composed of two three-punctured spheres. In the tube region connecting the two, the behavior of λ is essentially given just by

ϕk(z) ukdzk zk . (12.1.14)


(x ai) = xN + u2xN 2 + + uN, (12.1.15)

we find that the residues of λ in the tube region is given by a1,,aN. Therefore, we find full punctures after we split off two spheres.

The resulting three-punctured sphere has one simple puncture and two full punctures. Therefore it should carry U(1) ×SU(N) ×SU(N) symmetry. The four-punctured sphere represents the SU(N) theory with 2N flavors. The tube region carries the SU(N) vector multiplet. Then each three-punctured sphere just represents N flavors, i.e. hypermultiplets (Qia,Q̃ai) where a,i = 1,,N. Then two SU(N) symmetries can be identified with those acting on the index a and i respectively, and the U(1) symmetry is such that Q has charge + 1 while Q̃ has charge 1.

The ultraviolet curve of the SU(N) theory with 2N flavors, shown in Fig. 12.1, is composed of two copies of this three-punctured sphere. The 2N hypermultiplets are split into N hypermultiplets (Qia,Q̃ai) charged under SU(N)A and U(1)B, and another N hypermultiplets (Qia,Q̃ai) charged under SU(N)D and U(1)C.

Figure 12.3: S-duality of SU(N) 2N flavors

12.1.3 A strong-coupling limit

Let us consider what happens when q . As shown in Fig. 12.3, it just ends up exchanging the puncture B and C, at the same time redefining the coupling q via q = 1q. This means that this strongly-coupled limit turns out to be another weakly-coupled SU(N) gauge theory with 2N flavors. This time, the 2N hypermultiplets are split into N hypermultiplets (qia,q̃ai) and another N hypermultiplets (qia,q̃ai), but notice that the first N are charged under SU(N)A and U(1)C while the second N are charged under SU(N)D and U(1)B. As we learned for the case of the SU(2) theory with four flavors in Sec. 9.4, the new quarks are magnetic from the point of view of the original theory.

Figure 12.4: Another limit of SU(N) 2N flavors:

We would like to understand the limit q 1 too. We need to split the four-punctured sphere as shown in Fig. 12.4. But the configuration of punctures are not what we already know: we have two full punctures on one side, and two simple punctures on the other side. We need to study more about the 6d construction before answering what happens in the limit.