I am reading the paper Fundamental monopoles and multimonopole solutions for arbitrary simple gauge groups - Weinberg, Erick J .

In appendix C of this paper the author states, that the solution obtained (eqn 2.20) by embedding a SU(2) bps monopole, in a gauge theory of higher rank, can be categorized by a natural isospin $t$, and a hypercharge $y$. What is ispospin and hypercharge in this context?

I explain more about the solution. For each simple root $\beta^{(a)}$ of an arbitrary gauge group, you can define a SU(2) subgroup with generators, $t_1, t_2, t_3$ given in eqn 2.20, and a set of scalar and gauge fields, that satisfy the bps monopole equations of unit charge.

HE says the generators belonging to the cartan sub algebra H are isospin singlets with $y=0$. What does this mean? And also for other roots, $t_3$ and $y$ are given by (eqn C.1)

$$t_3 E_{\alpha}=[t_3, E_{\alpha}]=[\frac{\beta \cdot H}{\beta^2},E_{\alpha}]=\frac{\beta \cdot \alpha}{\beta^2}E_{\alpha}$$ $$yE_{\alpha}=(\frac{h \cdot \alpha}{h \cdot \beta}-t_3)E_{\alpha}$$

Here, $\beta$ is any root of the gauge group, $E_{\alpha}$ is the raising operator, and $h$ is the vector formed by the components of the scalar field $\phi$ along generators of $H$.


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