9.3 Weak-coupling limit and trifundamentals

Figure 9.6: Weakly-coupled limit of SU(2) Nf = 4

Let us now take the limit q 0 to decouple the gauge SU(2), see Fig. 9.6. On the left hand side, we have a sphere parameterized by z, with four points at z = 0,q, 1 and . On the right hand side, we have two spheres, parameterized by z and z. We put the points A, B, G on the first sphere, at z = 0, 1 and , and then the points G, C, D on the second sphere, at z = 0, 1 and . Then we glue the neighborhoods of G and G by declaring

zz = q. (9.3.1)

Defining z = z, we see that four points A,B,C,D are exactly as in the first description. In this limit, around the tube connecting G and G, λ ±adzz. Then in the sphere containing A, B and G, we have three singularities, with residues of λ given by

±μ1 + μ2 2 , ±μ1 μ2 2 , ± a, (9.3.2)

each corresponding to the symmetry SU(2)A, SU(2)B and SU(2)G, respectively. Here SU(2)G was originally the gauge symmetry.

Figure 9.7: The ultraviolet curve of the trifundamental, together with the BPS paths representing hypermultiplets

We were talking about the Nf = 4 theory. Then each of the sphere with three punctures should be associated to the Nf = 2 hypermultiplet system, see Fig. 9.7; note that this is not coupled to any gauge group. Let us recall the structure of the hypermultiplets again. We start from two hypermultiplets (Qia,Q̃ai) in the doublet of SU(2), i = 1, 2 and a = 1, 2. We combine them to qIa, a = 1, 2 and I = 1,, 4, making SU(2) ×SO(4) symmetry manifest. We then decompose the SO(4) index I into the pair (α,u) where α = 1, 2 and u = 1, 2: we have the trifundamental qaαu. The mass term for this hypermultiplet is

μabqaαuqbβv𝜖αβ𝜖uv + μ̃αβqaαuqbβv𝜖ab𝜖uv + μ̂uvqaαuqbβv𝜖ab𝜖αβ, (9.3.3)


μba = adiag(1,1),μ̃βα = μ1 μ2 2 diag(1,1),μ̂βα = μ1 + μ2 2 diag(1,1). (9.3.4)

Then (a,b) are the indices for SU(2)G, (α,β) for SU(2)A, and (u,v) for SU(2)B. The physical masses of these fields are given by

±a ±μ1 μ2 2 ±μ1 + μ2 2 = {±a ± μ1,±a ± μ2}. (9.3.5)

which are the masses for the two doublets of SU(2) with bare masses μ1,2.

The curve of the system, shown in Fig. 9.7 is given by

λ2 ϕ(z) = 0, (9.3.6)

where ϕ(z) has the asymptotic behavior

ϕ(z) μ̃2 z2 dz2, μ̂2 (z 1)2dz2, μ2 w2dw2 (9.3.7)

at z = 0, z = 1, z = respectively. Here w = 1z as always, and we set μ = a, μ̃ = (μ1 μ2)2 and μ̂ = (μ1 + μ2)2. Note that these asymptotic conditions uniquely fix the quadratic differential ϕ(z) to be

ϕ(z) = μ2z2 + (μ̂2 μ̃2 μ2)z + μ̃2 z2(z 1)2 dz2 (9.3.8)

As was discussed before, the BPS particles of this system can be found by solving the BPS equation (6.1.6)

Arg λ ds = ei𝜃 (9.3.9)

for a given 𝜃. As ϕ(z) given above has two branch points only, the solution to the BPS equation should start from one and end on the other. A computer simulation shows that there are always four and only four such solutions, corresponding to the hypermultiplets with masses given in (9.3.5).