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 GSD 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 GSD 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 GSD?
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.

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 GSD 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 GSD 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 GSD?
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.

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.

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 GSD 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 GSD 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 GSD?
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.