6.4 Intermediate 5d Yang-Mills theory and its boundary conditions

6.4.1 Five-dimensional maximally-supersymmetric Yang-Mills

Figure 6.9: 5d maximally supersymmetric Yang-Mills from 6d. A W-boson and a monopole-string are depicted there.

We have so far considered the situations where we put the six-dimensional theory on a two-dimensional space, with coordinates x5 and x6, say. We can take a limit where the x5 direction is much larger than the x6 direction. Then we can first compactify along the x6 direction and consider an intermediate five-dimensional theory, see Fig. 6.9. This is believed to give the maximally supersymmetric 5d Yang-Mills theory with gauge group SU(2).

A string wrapped around the x6 direction gives rise to a massive electric particle, and a string not wrapped around the x6 direction becomes a massive magnetic string. This agrees with a basic feature of the 5d Yang-Mills theory where SU(2) is broken to U(1): First, we have massive W-bosons which are electric. Second, the standard monopole solutions of 4d gauge theory can be regarded as a solution in 5d gauge theory, by declaring that there is no dependence of the fields on the additional fifth direction. Then the solutions should be now regarded as representing a magnetic-monopole-string.

6.4.2 𝒩=4 super Yang-Mills

By imposing periodic boundary condition in the x5 direction, we have the situation of Fig. 6.10. We are compactifying the maximally supersymmetric Yang-Mills in five dimensions on S1. We therefore should obtain 4d 𝒩=4 super Yang-Mills theory.

Figure 6.10: 𝒩=4 SYM from 6d. A W-boson and a monopole are depicted there.

The ultraviolet curve C itself is now a torus T2. Let the complex structure of this T2 be τ. The Seiberg-Witten curve Σ consists of two parallel copies of this torus embedded in X, separated by 2λ in the x direction, where λ is now a constant.

We can consider a cycle Ln,m in C, wrapping n times in the x6 direction and m times in the x5 directions. Then we can consider a ring-shaped membrane over this cycle, which gives rise to particles of masses

Mn,m = 2|na + maD| (6.4.1)


a =Aλ,aD =Bλ = τa. (6.4.2)

The particles with (n,m) = (1, 0) are W-bosons, and the particles with (n,m) = (0, 1) are monopoles. The peculiar feature of this theory is that the monopoles and the W-bosons both come from ring-shaped membranes. In fact, from the 6d point of view, the distinction of the two directions of the torus is completely arbitrary. Then this theory with a given value of τ = τ0, and the theory with another value of τ = 1τ0 are the same after the exchange of the W-bosons and the monopoles.

Indeed they match the property of the 𝒩=4 supersymmetric SU(2) Yang-Mills. This theory is conformal and has an exactly marginal coupling τ. In the semi-classical region, the ratio of the mass of the monopole to that of the W-boson is |τ|. The 𝒩=4 supersymmetric SU(2) Yang-Mills has four Weyl fermions in the adjoint representation. The semiclassical quantization of the monopole solution in this situation, as was recalled briefly in Sec. 1.3, makes the monopole states into a massive 𝒩=4 vector multiplet. This makes it possible to exchange it with the W-boson, which is also in a massive 𝒩=4 vector multiplet. In general we expect that there is a massive 𝒩=4 vector multiplet with mass |na + maD|, for any coprime pair of integers (n,m). This should arise from a semi-classical quantization of charge-m monopole background. This is the celebrated conjecture of Sen [53].

6.4.3 𝒩=2 pure SU(2) theory and the Nf = 1 theory

Figure 6.11: Pure and Nf = 1 SU(2) theories via 5d construction.

The curve of the pure 𝒩=2 SU(2) theory

Λ2 z + Λ2z = x2 u (6.4.3)

and the curve of the 𝒩=2 SU(2) theory with one flavor

2Λ(x μ) z + Λ2z = x2 u (6.4.4)

can be given a similar interpretation. The point is to take x5 = log |z| and x6 = Argz, and compactify along the x6 direction first, see Fig. 6.11.

Let us first consider the pure theory. The term on the left hand side, Λ2z, should be regarded as a boundary condition ‘terminating’ the fifth direction x5, although x5 = log |z| formally extends to . The bulk of the five dimensional theory is maximally supersymmetric. The resulting four-dimensional theory is 𝒩=2, and therefore the boundary breaks half of the supersymmetry, without doing much other than that. A boundary condition which preserves half of the original supersymmetry is called a half-BPS boundary condition. Then we see that the term Λ2z represents a half-BPS boundary condition of the 5d theory.

The term Λ2z is obtained by the flip x5 x5, and therefore should represent the same boundary condition. In the end, we see that the system is a compactification of the maximally supersymmetric SU(2) Yang-Mills on a segment, terminated by two boundary conditions breaking half of the supersymmetry, realizing 4d pure SU(2) Yang-Mills.

Next, let us consider the one-flavor theory. The term Λ2z is the same as the pure case, so it should give the same half-BPS boundary condition. The boundary condition at z 0 is different: now we have a term of the form Λ(x μ)z. This should mean that one hypermultiplet with the mass μ in the doublet of SU(2) lives on this boundary, coupling to the bulk five-dimensional gauge multiplets.

6.4.4 The SU(2) theories with Nf = 2, 3, 4

From this interpretation, it is easy to get the 6d realization of SU(2) theory with Nf = 2, 3, 4 flavors, namely the theory with Nf = 2, 3, 4 hypermultiplets in the doublet representation. In terms of 𝒩=1 chiral multiplets, we have (Qia,Q̃ai) for a = 1, 2 and i = 1,,Nf, with the superpotential

i QiΦQ̃i + μiQiQ̃i (6.4.5)

where μi are mass terms.

Let us start with the Nf = 2 theory. We know how to introduce one hypermultiplet in the doublet at the boundary on the side z = 0. To do the same on the side z = , we just a change of variables z 1z. We end up with the setup shown on the left-hand side of Fig. 6.12, with the curve given by

2Λ(x μ1) z + 2Λ(x μ2)z = x2 u (6.4.6)

with the Seiberg-Witten differential λ = xdzz.

Figure 6.12: Nf = 2 theory.

The same curve can be rewritten using another variable z = (x μ2)z(2Λ):

(x μ1)(x μ2) z + 4Λ2z = x2 u. (6.4.7)

But now we can consider x5 = log |z|, x6 = Argz to reduce first to a theory on C parameterized by z, and then to a five-dimensional theory on a segment parameterized by |z|. In this interpretation, the boundary condition on the z = side is the same one in the pure SU(2) case. Therefore, the boundary condition on the z = 0 side given by the term (x μ1)(x μ2)z should be the half-BPS condition such that two hypermultiplets in the doublet of SU(2) live on the boundary.

The description of the system is not complete until we give the one-form λ describing the variable tension. In (6.4.6) it is 2πiλ = xdzz and in (6.4.7) it is 2πiλ = xdzz. Both are obtained by integrating dx d log z = dx d log z, see (6.2.1). The two differentials are not quite equal, however:

λ λ = 1 2πixd log z z = 1 2πixd log(x μ2). (6.4.8)

The difference is independent of u, and its non-zero residue is at μ2 at x = μ2. This means that, given a cycle L on the Seiberg-Witten curve Σ, we have

Lλ Lλ = kμ2 (6.4.9)

where k is an integer. Recall that the BPS mass formula is governed by the expansion

Lλ = na + maD + f1μ1 + f2μ2 (6.4.10)

where f1,2 are flavor charges, see (2.3.9). Therefore, the choice between the two Seiberg-Witten differentials λ and λ affects the mapping of the flavor charge f2 and the cycle L, but not much else. In general, a change in the Seiberg-Witten differential by a form which is independent of u and whose residues are integral linear combinations of the hypermultiplet masses are safe. We will encounter them repeatedly later.

Figure 6.13: Nf = 3 theory and Nf = 4 theory.

Now that we have a boundary condition representing the existence of two doublet hypermultiplets, it is easy to guess the curve of the Nf = 3 theory and Nf = 4 theory. We just have to combine various boundary conditions which we already found, as in Fig. 6.13. For the Nf = 3 theory we find

(x μ1)(x μ2) z + 2Λ(x μ3)z = x2 u, (6.4.11)

and for the Nf = 4 theory we find

f (x μ1)(x μ2) z + f (x μ3)(x μ4)z = x2 u (6.4.12)

where we put complex numbers f and f. One of them can be eliminated by a rescaling of z.

Our next task is to check that the curves thus obtained via the 6d construction have the correct properties to describe the respective four-dimensional theories. Before proceeding, we need to learn more about the Higgs branch of 𝒩=2 theories in general.