4.5 SU(2) vs SO(3)

At this point, it might be useful to discuss a rather sutble point concerning the Seiberg-Witten curve of the theory which depends on the precise choice of the gauge group to be SU(2) or SO(3). This subsection can be skipped on a first reading.

In this section, our choice of the charges has been that

(n,m) = (1, 0) (4.5.1)

represents an electric doublet in SU(2),

(n,m) = (0, 1) (4.5.2)

represents a ’t Hooft-Polyakov monopole associated to the breaking SU(2) U(1).

That said, the dynamical particles in the theory all has the charge of the form

(n,m) = (2k,m) (4.5.3)

for some integers k and m. Furthermore, as we do not have any dynamical fields in the doublet of SU(2), we can consider an external monopole with charge

(n,m) = (0, 12). (4.5.4)

This still satisfies the Dirac quantization condition with respect to any of the dynamical particles in the theory, whose chages are given by (4.5.3).

Correspondingly, the monodromy matrices

M = 1 4 0 1 ,M+ = 1 01 1 ,M = 14 1 3 (4.5.5)

all had an integral multiple of 4 in the upper right corner.

Therefore, we can do the following. We define rescaled electric and magnetic charges via

(n,m) = (n2, 2m) (4.5.6)

and still the monodromy are still integer valued:

M = 1 1 0 1 ,M+ = 1 04 1 ,M = 11 4 3 . (4.5.7)

The BPS mass formula is now

|na + maD| = |na + maD|where(a,aD) = (2a,aD2) (4.5.8)

and correspondingly, the new A and B cycles are related to the old ones via

A = 2A,B = B2. (4.5.9)

The respective Seiberg-Witten curves, as quotients of the complex plane as in Fig. 4.8, are given in Fig. 4.11.

Figure 4.11: Various choices of the Seiberg-Wittencurves. From the left to right: the pure SU(2) theory, the pure SO(3) theory, and the choice in [2]

The standard interpretation is that the Seiberg-Witten curve with cycles A and B as the curve for the pure SU(2) theory, and that the Seiberg-Witten curve with cycles A = 2A and B = B2 as the curve for the pure SO(3) theory. The difference manifests in a rather subtle manner.

At the monopole point, with SO(3) gauge group, the monodromy is M+ given above. This means that the charge of the light particle there is 2 with respect to the low-energy U(1) field: the entry 4 in the lower left corner is given by the square of the charges. This is due to the fact that the periodicity of low-energy U(1) is reduced by a factor of two, as it is embedded in SO(3) rather than SU(2). The monopole has (n,m) = (0, 2) and the g.c.d. of n and m is two.

At the dyon point, the monodromy M is still conjugate to 11 0 1 , that is, the light dyon has charge 1 with respect to the U(1). The dyon has (n,m) = (1, 2) and the g.c.d. of n and m is one. Therefore one loses the physical equivalence of the monopole point and the dyon point. These subtle differences affect the system more drastically when the system is put on 3 × S1 or more complicated manifolds.

Another interesting fact is that this combination M, M+, and M is exactly the same as the monodromy matrices of that of the SU(2) theory with 3 massless flavors, which we discuss in Sec. 8.5. Still, the physics of the pure SO(3) theory and the SU(2) theory with 3 massless flavors are drastically different, as we will learn later. This shows an obvious point that the structure of the Coulomb branch alone does not fix the entire theory.

Finally, we should mention that in the very orignal paper on the pure SU(2) theory [2] another curve is used, which had 2A and B as two cycles, as shown in Fig. 4.11. This choice is adapted to the spectrum of the dynamical particles (4.5.3), but it is now known to be a not very well motivated when we consider the theory on nontrivial manifolds and the properties of line operators. Therefore, it is advisable to stick to either the pure SU(2) curve or the pure SO(3) curve, given as the first two entries in Fig. 4.11.