9.4 Strong-coupling limit

Figure 9.8: A strongly-coupled limit of SU(2), Nf = 4

So far we mainly considered the weak coupling limit q 0. Instead, consider sending q , as shown in Fig. 9.8. We immediately find that the strong coupling limit q is the weak coupling limit q = 1q 0 of a similarly-looking SU(2) gauge theory with four flavors. Note however that the role of the singularities B and C are exchanged.

Originally, we had four flavors with masses

±μ1, ± μ2, ± μ3, ± μ4. (9.4.1)

The residues of λ at the punctures were then

±μA, ± μB, ± μC, ± μD (9.4.2)


μA = μ1 μ2 2 ,μB = μ1 + μ2 2 ,μC = μ3 + μ4 2 ,μD = μ3 μ4 2 . (9.4.3)

The original masses μi are

μ1 = μA + μB,μ2 = μA + μB,μ3 = μC + μD,μ4 = μC μD. (9.4.4)

Now the singularities B and C are exchanged. Then, the masses μi of the four hypermultiplets of the theory with the coupling q = 1q are instead given by

μ1 = μA + μC = μ1 μ2 2 + μ3 + μ4 2 , (9.4.5) μ2 = μA + μC = μ1 μ2 2 + μ3 + μ4 2 , (9.4.6) μ3 = μB + μD = μ1 + μ2 2 + μ3 μ4 2 , (9.4.7) μ4 = μB μD = μ1 + μ2 2 μ3 μ4 2 . (9.4.8)

The original masses (9.4.1) can be thought of as the weights of the vector representation of SO(8). The dual masses ± μi are then the weights of the spinor representation of SO(8).

Figure 9.9: Monopoles and quarks are exchanged

The dual quarks, therefore, transform in the spinor representation of the flavor SO(8) symmetry. We can identify these dual quarks as the monopoles in the original description. This can be seen by slowly changing the value of q, following how various paths on the sphere change, see Fig. 9.9. Originally, the path connecting branch points close to the singularity B and D was a monopole. Recall also that the semiclassical quantization of the monopole gave us a multiplet in the spinor representation of the flavor symmetry SO(2Nf) as we saw in Sec. 1.3. In the limit q , these monopoles become excitations whose paths are totally contained in the sphere on the right. They are now the quark hypermultiplets in the trifundamental, as shown in Fig. 9.7.

Figure 9.10: W bosons also come from monopoles.

The same manipulation also shows that the SU(2) W-bosons in the dual description came from monopoles in the original description, see Fig. 9.10. Therefore it is important to keep in mind that the dual SU(2) gauge multiplet is not the same physical excitation as the original SU(2) gauge multiplet. Note also that this monopole has twice the magnetic charge of the monopole which became the dual quarks.

Figure 9.11: Triality

There is also a limit where the singularity B approaches the singularity C, q 1. This is again equivalent to a weakly-coupled SU(2) gauge theory with four flavors, but with the role of the singularities are permuted, see Fig. 9.11. The four mass parameters of the hypermultiplets are now given by

μ1 = μA + μD = μ1 μ2 2 + μ3 μ4 2 , (9.4.9) μ2 = μA + μD = μ1 μ2 2 + μ3 μ4 2 , (9.4.10) μ3 = μC + μB = μ3 + μ4 2 + μ1 + μ2 2 , (9.4.11) μ4 = μC μB = μ3 + μ4 2 μ1 μ2 2 . (9.4.12)

These are the weights of the conjugate spinor representation of SO(8).

Therefore, we learned that the strong-weak duality of the SU(2) gauge theory with four flavors,

q q = 1q q = 1 q (9.4.13)

are accompanied by the exchange of the representations of the SO(8) flavor symmetry,

VSC (9.4.14)

where V , S, C are eight dimensional irreducible representations (vector, spinor, conjugate spinor) of SO(8). These exchanges of three irreducible eight dimensional representations are induced by the outer automorphism, and are known as the triality of the group SO(8), whose Dynkin diagram is also shown in Fig. 9.11. This triality was originally found in [3]. The exposition in this subsection followed the one given in [8].

The Higgs branch of this system can be studied in any of these descriptions. Originally we have hypermultiplets qIa, where a = 1, 2 and I = 1,, 8. The gauge invariant combination is

M[IJ] = qIaqJb𝜖ab. (9.4.15)

In the dual, we have hypermultiplets q̃Ĩã, where ã = 1, 2 are for dual SU(2) and Ĩ = 1,, 8 are for the spinor representation of the SO(8) flavor symmetry. The basic gauge invariant is then

M̃[ĨJ̃] = q̃Ĩãq̃J̃b̃𝜖ãb̃. (9.4.16)

Both M[IJ] and M̃[ĨJ̃] are in the adjoint representation of SO(8), and can be naturally identified using the outer automorphism of SO(8). We can check that the constraints satisfied by M[IJ] and M̃[ĨJ̃] are invariant under the outer automorphism. This shows that the Higgs branch are the same as complex spaces.