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How to see the ground state degeneracy (GSD) from a $BF$ theory in $2+1$ $d$?

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I have seen many times the $BF$ theory has non-trivial ground state degeneracy (typically on torus), but I can not see how the conclusion come out. Recently I found a paper by Hansson, Oganesyan and Sondhi,[Superconductors are topologically ordered] in which the superconductor is described by a Maxwell$-BF$ theory. They have a section of the GCD in a $BF$ theory in $2+1$ $d$. But actually I still have questions to understand it. 


The $BF$ theory in $2+1$ $d$ is given by the action
$$
S = \frac{1}{\pi} \int d^3 x \epsilon^{\mu \nu \sigma} b_{\mu} \partial_{\nu} a_{\sigma}, \qquad (1)
$$
where $a_{\mu}$ and $b_{\mu}$ are $U(1)$ gauge fields. 
$\mu,\nu,\sigma = 0,x,y$.


Working on $2-$torous, as in the section [IV.A] in Hansson's paper, the $BF$ theory can be written in the form 
$$
S = \frac{1}{\pi}\int d^3x[\epsilon^{ij} \dot{a}_i b_j+
a_0 \epsilon^{ij} \partial_i b_j + b_0 \epsilon^{ij} \partial_i a_j],
$$
where $\dot{a} = \partial_0 a$ and $i,j = x,y$.  They interpret $a_0$ and $b_0$ are multipliers for constraints 
$\epsilon^{ij} \partial_i b_j = 0$ and $\epsilon^{ij} \partial_i a_j = 0$. 
Upon inserting $a_i = \partial_i \Lambda_a + \bar{a}_i/L$
and $b_i = \partial_i \Lambda_b + \bar{b}_i/L$, 
where $\Lambda_{a/b}$ are periodic functions on the torus, $\bar{a_i}$ and $\bar{b_i}$ are spatially constant, $L$ denotes the size of the system, the above $BF$ theory reduces to
$$
S = \frac{1}{\pi}\int d^3 x \epsilon^{ij} \dot{\bar{a}}_i \bar{b}_j. \qquad (2)
$$

Then they say from the Eq.(2) one can obtain the commutation relation ( [Eq. (38)] in  their paper)
$$
[\bar{a}_x, \frac{1}{\pi}\bar{b}_y] = i, \quad
[\bar{a}_y,-\frac{1}{\pi}\bar{b}_x] = i. \qquad (3)
$$


Moreover, from the commutation relations Eq. (3), one can have ( [Eq. (39)] in  their paper)

$$
A_x B_y + B_y A_x = 0, \quad
A_y B_x + B_x A_y = 0, \qquad (4)
$$
where $A_i = e^{i\bar{a}_i}$ and $B_i = e^{i\bar{b}_i}$.
They claim that relations Eq. (4) indicates a $2\times2 = 4-$fold GCD and "$B_i$ can be interpreted either as measuring the $b$-flux or inserting an $a-$flux."


There are several points that I don't understand.

 1. How can I get communication relations Eq. (3) from the action Eq. (2)?
 2. Why relations Eq. (4) indicate a $4-$fold GCD?
 3. How should I understand the statement "$B_i$ can be interpreted either as measuring the $b$-flux or inserting an $a-$flux."?

I would be very appreciate if anyone can give me some hints or suggest me some relevant references.

asked Oct 21, 2014 in Theoretical Physics by hongchan (90 points) [ revision history ]
edited Oct 21, 2014 by hongchan

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