4.5 $SU\left(2\right)$ vs $SO\left(3\right)$

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 $SU\left(2\right)$ or $SO\left(3\right)$. This subsection can be skipped on a first reading.

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

represents an electric doublet in $SU\left(2\right)$,

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

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

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

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

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

and still the monodromy are still integer valued:

The BPS mass formula is now

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

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

The standard interpretation is that the Seiberg-Witten curve with cycles $A$ and $B$ as the curve for the pure $SU\left(2\right)$ theory, and that the Seiberg-Witten curve with cycles $A^{\prime }=2A$ and $B^{\prime }=B∕2$ as the curve for the pure $SO\left(3\right)$ theory. The difference manifests in a rather subtle manner.

At the monopole point, with $SO\left(3\right)$ gauge group, the monodromy is $M_{+}^{\prime }$ given above. This means that the charge of the light particle there is 2 with respect to the low-energy $U\left(1\right)$ 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 $U\left(1\right)$ is reduced by a factor of two, as it is embedded in $SO\left(3\right)$ rather than $SU\left(2\right)$. The monopole has $\left(n^{\prime },m^{\prime }\right)=\left(0,2\right)$ and the g.c.d. of $n^{\prime }$ and $m^{\prime }$ is two.

At the dyon point, the monodromy $M_{-}^{\prime }$ is still conjugate to $\left(\begin{array}{cc}\hfill 1\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right)$, that is, the light dyon has charge $1$ with respect to the $U\left(1\right)$. The dyon has $\left(n^{\prime },m^{\prime }\right)=\left(1,2\right)$ and the g.c.d. of $n^{\prime }$ and $m^{\prime }$ 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 ${ℝ}^3\times S^1$ or more complicated manifolds.

Another interesting fact is that this combination $M_{\infty }^{\prime }$, $M_{+}^{\prime }$, and $M_{-}^{\prime }$ is exactly the same as the monodromy matrices of that of the $SU\left(2\right)$ theory with 3 massless flavors, which we discuss in Sec. 8.5. Still, the physics of the pure $SO\left(3\right)$ theory and the $SU\left(2\right)$ 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 $SU\left(2\right)$ 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 $SU\left(2\right)$ curve or the pure $SO\left(3\right)$ curve, given as the first two entries in Fig. 4.11.