What is the relationship between the Liouville CFT and the Zograf-Takhtajan action?

Question

I am bit confused as to if both the things are the same since it seems that people refer to both as being given by the ``Liouville action",

\(\frac{1}{4\pi}\int d^2z ( \vert \partial \phi \vert ^2 + \mu e^{\phi } ) + bounday-terms \)

- Is the above action (which I would have thought is for the Liouville CFT) the same as $S_{ZT}$ which is said to satisfy the identity $S_{E}(M_3) = \frac{c}{3}S_{ZT}(M_2,\Gamma_{i=1,...,g})$?

In the above equality $S_E$ is the Einstein gravity action evaluated on-shell on a 3-manifold which is formed by "filling" in the interiors of a choice $\Gamma_{i=1,..,g}$ of $g$ non-contractible cycles on a genus-g Riemann surface $M_2$.  (...in the context of $AdS/CFT$ one would want to interprete $M_2$ as being the conformal boundary of the space-time $M_3$...)

If $S_{ZT}$ is indeed the same as the Liouville CFT action then how does one understand the need to choose these non-contractible cycles to define that?

- In the same strain I would want to know what exactly is the meaning of the conjecture made in equation 5.1 in this paper,  http://arxiv.org/abs/1303.6955

Comments

Hi curiousgradstudent, welcome to PhysicsOverflow :-).

I hope you do not mind that I tried to help you with the LaTex, as the Liouville action did not compile first. If you click on the TEX button in the editor, this lets you write centered equations and also gives you a preview to see if the LaTex compiles. When using the TEX button, you do not have to use the dollars.

Cheers

---

@dilaton, how do you write centered Latex using the editor?

Answer 1

1. By the uniformization theorem, any Riemann surface, $M_2$, possibly with punctures (excluding some cases such a sphere with less than 3 punctures, torus etc.) is realised as the quotient of Poincare upper half space $\mathbb{H}_2$ by a Fuchsian group $\Gamma$ i.e, $M=\Gamma \backslash \mathbb{H}_2$.
2. $\mathbb{H}_2$ is the boundary of three-dimensional hyperbolic space $\mathbb{H}_3$ (AdS${}_3$) and suppose that the action of $\Gamma$ extends to a nice (properly discontinuous) action on $\mathbb{H}_3$. Let $M_3 = \Gamma\backslash \mathbb{H}_3$.
3. Roughly speaking, the (quasi-Fuchsian holography) theorem 5.1 of this paper (by Takhtajan-Teo) is that the regularized Einstein-Hilbert action on $M_3$, $\mathcal{E}[\phi]$, is equal to the Liouville action, $S[\phi]$, on $M_2$ up to an extra piece.

Off-hand, I don't recall the difference between quasi-Fuchsian and Fuchsian. So I am only attempting to sketch the result in simple terms hiding technicalities. I hope that is fine.

Comments

@suresh1 Thanks for the reply. So what you are calling the "Liouville action" is it the same as what is called the ``Zograf-Takhtajan" action and that is same as the Liouville CFT action as I have written at the top? What is the relation between the two? (in the equation 5.1 in my linked paper isn't the LHS the negative of the free energy of an arbitrary $1+1$ CFT?) But what does it mean when in my linked paper it is said that, ``this does not mean that quantum Liouville theory is equivalent to either gravity or the CFT" I thought that the point was that large-central charge Liouville CFT is indeed $2+1$ quantum gravity?

---

@suresh Thanks for the reply. So what you are calling the "Liouville action" is it the same as what is called the ``Zograf-Takhtajan" action and that is same as the Liouville CFT action as I have written at the top? What is the relation between the two? (in the equation 5.1 in my linked paper isn't the LHS the negative of the free energy of an arbitrary 1+1 CFT?) But what does it mean when in my linked paper it is said that, ``this does not mean that quantum Liouville theory is equivalent to either gravity or the CFT" I thought that the point was that large-central charge Liouville CFT is indeed 2+1 quantum gravity?

---

$S[\phi]$ is more or less what you defined -- some extra terms are added to make it globally defined (see the linked paper).  I consider $\mathcal{E}[\phi]$ is the ZT action and the theorem "equates" the two. I have not looked at the linked paper and so will not comment on it. 

---

@suresh So the paper you linked to is the proof that the on-shell value of the Einstein gravity action on a 3-manifold is the same as the Zograf-Takhtajan action on some notion of boundary of it? And its a separate proof that the Zograf-Takhtajan action on a 2-manifold is the same as the Liouville CFT action? It would be great if you could give references for the different parts of this story...

---

This appears to be a comment to another answer rather than an answer.

---

@Danu I have converted @curiousgradstudent 's answer into a comment.