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edited Mar 29, 2013 at 21:02

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I want to know how one can write down a CFT such that its conserved currents will satisfy some chosen (affine) Lie algebra $G$.

On the few pages leading up to page 192 in here in here one can see see the analysis of the CFT obtained in the compactified directions of closed bosonic strings.

From what one sees in these notes it seems that a CFT with the above properties will exist if it is put on a torus $\mathbb{R}^n/\Lambda$ where $n=rank (G)$ and $\Lambda =$root lattice of $G$.

Is the above correct? If it is correct then how does one write down the corresponding Lagrangian and the currents?
But in the string context where these attached notes are based one is forced to have simply laced $G$ and hence only the $A,D,E$ series. How would one do this for say the group $G_2$? (..being concerned with just a CFT and not connected to string theory..)
The restriction of being on the $A$, $D$, $E$ series is related to the fact that in string context one has to tune all the compactification radius to the same self-dual point. If $i$ indexes the compact directions , $1\leq i \leq n$ then the current operators are possibly like $:\partial _{z} X_i (z):$ and $:e^{i\vec{\alpha}.\vec{X}}:$ where the first set is one for each Cartan and the send set is one for each root $\alpha$. But I wonder where would the radius of the circles go in these currents.
Finally where in this process can one tune the level of the affine Lie algebra? What choice fixes that?

I want to know how one can write down a CFT such that its conserved currents will satisfy some chosen (affine) Lie algebra $G$.

On the few pages leading up to page 192 in here one can see see the analysis of the CFT obtained in the compactified directions of closed bosonic strings.

From what one sees in these notes it seems that a CFT with the above properties will exist if it is put on a torus $\mathbb{R}^n/\Lambda$ where $n=rank (G)$ and $\Lambda =$root lattice of $G$.

Is the above correct? If it is correct then how does one write down the corresponding Lagrangian and the currents?
But in the string context where these attached notes are based one is forced to have simply laced $G$ and hence only the $A,D,E$ series. How would one do this for say the group $G_2$? (..being concerned with just a CFT and not connected to string theory..)
The restriction of being on the $A$, $D$, $E$ series is related to the fact that in string context one has to tune all the compactification radius to the same self-dual point. If $i$ indexes the compact directions , $1\leq i \leq n$ then the current operators are possibly like $:\partial _{z} X_i (z):$ and $:e^{i\vec{\alpha}.\vec{X}}:$ where the first set is one for each Cartan and the send set is one for each root $\alpha$. But I wonder where would the radius of the circles go in these currents.
Finally where in this process can one tune the level of the affine Lie algebra? What choice fixes that?

I want to know how one can write down a CFT such that its conserved currents will satisfy some chosen (affine) Lie algebra $G$.

On the few pages leading up to page 192 in here one can see see the analysis of the CFT obtained in the compactified directions of closed bosonic strings.

From what one sees in these notes it seems that a CFT with the above properties will exist if it is put on a torus $\mathbb{R}^n/\Lambda$ where $n=rank (G)$ and $\Lambda =$root lattice of $G$.

Is the above correct? If it is correct then how does one write down the corresponding Lagrangian and the currents?
But in the string context where these attached notes are based one is forced to have simply laced $G$ and hence only the $A,D,E$ series. How would one do this for say the group $G_2$? (..being concerned with just a CFT and not connected to string theory..)
The restriction of being on the $A$, $D$, $E$ series is related to the fact that in string context one has to tune all the compactification radius to the same self-dual point. If $i$ indexes the compact directions , $1\leq i \leq n$ then the current operators are possibly like $:\partial _{z} X_i (z):$ and $:e^{i\vec{\alpha}.\vec{X}}:$ where the first set is one for each Cartan and the send set is one for each root $\alpha$. But I wonder where would the radius of the circles go in these currents.
Finally where in this process can one tune the level of the affine Lie algebra? What choice fixes that?

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asked Mar 26, 2013 at 22:22

Anirbit

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CFTs corresponding to affine Lie algebras

I want to know how one can write down a CFT such that its conserved currents will satisfy some chosen (affine) Lie algebra $G$.

On the few pages leading up to page 192 in here one can see see the analysis of the CFT obtained in the compactified directions of closed bosonic strings.

From what one sees in these notes it seems that a CFT with the above properties will exist if it is put on a torus $\mathbb{R}^n/\Lambda$ where $n=rank (G)$ and $\Lambda =$root lattice of $G$.

Is the above correct? If it is correct then how does one write down the corresponding Lagrangian and the currents?
But in the string context where these attached notes are based one is forced to have simply laced $G$ and hence only the $A,D,E$ series. How would one do this for say the group $G_2$? (..being concerned with just a CFT and not connected to string theory..)
The restriction of being on the $A$, $D$, $E$ series is related to the fact that in string context one has to tune all the compactification radius to the same self-dual point. If $i$ indexes the compact directions , $1\leq i \leq n$ then the current operators are possibly like $:\partial _{z} X_i (z):$ and $:e^{i\vec{\alpha}.\vec{X}}:$ where the first set is one for each Cartan and the send set is one for each root $\alpha$. But I wonder where would the radius of the circles go in these currents.
Finally where in this process can one tune the level of the affine Lie algebra? What choice fixes that?