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.

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    By 'living in the same universe' do you mean there is underlying 'grand unified' theory that is broken in three different ways? If so, then the areas of the universe with different theories would be separated by domain walls, so we really could not 'bring them together'. Or do you mean something else? – user23660 Dec 27 '13 at 3:29
  • @user23660: Thanks for the nice comments. Actually I am not sure whether the three theories can really be brought together to compare with each others. (If not, you can still answer the questions that they cannot be compared.) What I wonder is that whether there is a universal smallest quantization scale for e, m, k (electric, magnetic and level quantization), so the three theories can put on this scales to compare? ps. I read some paper that they mentioned how the levels are matched for a compact U(1) C-S and SU(2)->U(1) C-S theory. Thanks! – wonderich Dec 27 '13 at 4:27
  • @user23660: if there is an extra Maxwell |F|^2 term for each theory (say, three theories share the same unbroken U(1) gauge field a), could we still detect the force between e,m charges and use this E-M force to tell their quantized values relatively between Theory (I), Theory (II) and Theory (III)? even if they are separated by (if there is any) domain wall? – wonderich Dec 27 '13 at 4:52

This is apparently too long for a comment, so it's going to be fleshed out into at least (hopefully) a partial answer.

A big problem I see with this is determining which theories can live in the same universe. In this way, I think Liang Kong's Mathematical Theory of Anyon Condensation provides a way forward.

To reconstruct your set up:

Consider three topological phases whose excitations are described by the modular tensor categories $\mathcal C_1$, $\mathcal C_2$, $\mathcal C_3$ associated with the Chern-Simons theories above. Now consider that these TPs are all condensed phases in some larger system with a suitable separation. This seems to be your first (or perhaps zeroeth) question - whether or not such a system exists.

From a categorical perspective I think this can be rephrased as whether or not there exists an MTC - let's call it $\mathcal D$ - which contains appropriate special Frobenius (equivalently connected commutative separable) algebras $A_1$, $A_2$, $A_3$ whose categories of local modules $mod_{l} A_1$, $mod_{l} A_2$, and $mod_{l} A_3$ are equivalent to $\mathcal C_1$, $\mathcal C_2$, and $\mathcal C_3$ as MTCs.

One way I can think of to construct such a category would probably be to take the Deligne tensor product of categories $\mathcal D =(\mathcal C_1\boxtimes \mathcal C_2) \boxtimes \mathcal C_3$. I believe then that $\mathcal C_1$, $\mathcal C_2$ and $\mathcal C_3$ should have the structure of bimodule categories over this - via the appropriate projection functors from $\mathcal D\rightarrow C_i$ - and thus this should guarantee the existence of the appropriate algebras.

Anyway, From Kong's analysis this should give a topological phase - i.e. a universe - which is capable of supporting condensed phases described by the three different types of Chern-Simons TQFT you describe above.


By "bringing them together" what I assume you mean is that the three phases each share a domain wall. This is where restrictions on the level of the theory will arise (this is probably obvious) and will require an analysis of their module categories. By talk to one another what I assume you mean is that excitations in one phase can be passed back and forth between the others. I believe this can also be analyzed using the method from Kong but before going on I want to make sure I'm not going too far afield since the analysis above is not strictly confined to the context of Chern-Simons theories.

  • Dear Matthew, thanks for the nice reply. +1. It is nice to have a category interpretation. (What I had in mind earlier was a TQFT field theory approach.) Do you think you can still tell the e charge/m charge quantization and level k for those theories using category? How? Many thanks! – wonderich Jan 3 '14 at 6:01
  • I believe that the answer to determining $k$ of the theory should be "Yes." The MTC for a CS theory with group $G$ at level k appears to be a quotient of the sub-category of representations of the Lie algebra of $G$ with some highest weight. Determining the three theories and their levels $(G_1,k_1)$, $(G_2,k_2)$, and $(G_3,k_3)$ is then a matter of determining which theories can share a domain wall. From Kong, this should be reducible to a statement about the module categories of the MTCs corresponding to the theories. I'm going to edit the answer to give a better description of this. – Matthew Titsworth Jan 7 '14 at 0:19
  • Assume for the moment though that we have three CS theories $(G_1,k_1)$, $(G_2, k_2)$, and $(G_3,k_3)$ which can share a boundary. Shouldn't the e/m charge quantization be derivable given the theory and it's level? – Matthew Titsworth Jan 7 '14 at 0:21
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    Thanks Matthew, actually I am not sure how. Would you provide the results if you know? (I had some thoughts from C-S but not sure the answer is right. I may post it to have other people judging my thought if there still lacks of response.) – wonderich Jan 10 '14 at 5:10

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