Imagine three different worlds describe by three theories (I), (II), (III).

**Theory (I) - compact U(1) Chern-Simons**:

A compact U(1) Chern-Simons theory with magnetic monopole charges $m_1$.

$$Z=\exp[i\int\big( \frac{k_{1}}{4\pi} a_1 \wedge d a_1 \big)]$$

**Theory (II) - SU(2) to U(1) Chern-Simons**:

A SU(2) (or SU(N) in general) Chern-Simons theory with magnetic monopole charges $m_2$.

$$Z=\exp[i\int \frac{k_{2}}{4\pi} \big( a_2 \wedge d a_2 +\frac{2}{3}a_2 \wedge a_2 \wedge a_2 \big)]$$

and this SU(2) theory broken down to U(1) symmetry by Higgs mechanism.

**Theory (III) - SO(3) to U(1) Chern-Simons**:

A SO(3) (or SO(N) in general) Chern-Simons theory with magnetic monopole charges $m_3$.

$$Z=\exp[i\int\frac{k_{3}}{4\pi} \big( a_3 \wedge d a_3 +\frac{2}{3}a_3 \wedge a_3 \wedge a_3 \big)]$$

and this SO(3) theory broken down to U(1) symmetry by Higgs mechanism.

Now imagine the Theory (I), Theory (II) and Theory (III) actually live in the same universe but far apart from each other; let us bring the Theory (I), Theory (II) and Theory (III) together and they talk to each other.

**Question**: can we compare their **quantizations on electric charge $e_1$,$e_2$,$e_3$, magnetic monopole charge $m_1$, $m_2$, $m_3$ and their levels $k_1$,$k_2$ and $k_3$?** What are their explicit relations?

ps. the useful fact is that: singular Dirac monopole has magnetic charge $m=2\pi N/e$ (for a compact U(1) theory), and 't Hooft Polyakov monopole has magnetic charge $m=4\pi N/e$ (for a SU(N) theory).

Giving Ref is okay. But **explicit results must be stated and summarized.** Thank you.

This post imported from StackExchange Physics at 2014-06-04 11:39 (UCT), posted by SE-user Idear