Conformal field theory in two dimensions/Exactly solvable theories

Spectrums

See Section 2.1 of Ref.

Exercises

• ES1. In the Ising minimal model, what are the OPE spectrums? Write the channel spectrums of all inequivalent 4-point functions.
• ES2. In a generalized minimal model, write the 3 channel spectrums of $\left\langle V_{\langle 3,3\rangle }^{d}V_{\langle 2,4\rangle }^{d}V_{\langle 1,4\rangle }^{d}V_{\langle 2,5\rangle }^{d}\right\rangle$.
• ES3. What are the largest possible non-diagonal spectrums for a CFT with 2 degenerate fields, such that all spins are integer?

ESDI: Conformal dimensions of non-diagonal fields

We consider the non-diagonal primary fields with integer conformal spins, in a CFT that has a degenerate field $V_{\langle 1,3\rangle }^{d}$. We assume $\Re \beta ^{2}>0$.

1. Which non-diagonal fields $V_{(r,s)}$ can exist?
2. Show that the set of their total conformal dimensions $\Delta +{\bar {\Delta }}$ is discrete and bounded from below.
3. Show that there are finitely many fields whose total dimension is below any given bound.
4. Assuming $\beta ^{2}\in \mathbb {R}$, sketch the set of non-diagonal fields on a plot whose axes are $\Delta -{\bar {\Delta }}$ and $\Delta +{\bar {\Delta }}$.

ESDS: D-series minimal models

1. Find the spectrums of D-series minimal models in Wikipedia.
2. What is the intersection of the spectrums of the A-series and D-series minimal models at the same central charge? Is this the spectrum of a consistent CFT?
3. Show that the spectrum of the D-series minimal model DMM$_{6,5}$ is a subset of the spectrum of the 3-state Potts CFT. Explain why the diagonal primary field $V_{P_{(0,{\frac {1}{2}})}}$ appears in the non-diagonal sector of DMM$_{6,5}$. Admitting that $V_{P_{(0,{\frac {1}{2}})}}$ transforms in the standard representation of the symmetric group $S_{3}$, explain why it appears twice in DMM$_{6,5}$.

Loop CFTs

See Section 2.4 of Ref.

EOES: E-series minimal models

1. Find the spectrums of E-series minimal models in Wikipedia.
2. Which diagonal degenerate fields exist in E-series minimal models?
3. Compare the spectrums of the A-series, D-series and E-series minimal models with the same central charge.
4. In the case $q=12$, rewrite the spectrum in terms of non-diagonal fields $V_{(r,s)}$, and compare with loop CFTs.

Liouville theory

See Section 2.2.2 of Ref.

Exercises

• EL1. From OPEs in generalized minimal models, deduce the OPEs in Liouville theory with $c\leq 1$. In particular, show that the measure of integration over momentums is the uniform measure $dP$. (You may use a Fourier transform.)

Path integral approach to w:Liouville theory

Action, parameters, primary fields

The action of Liouville theory, which describes the Weyl anomaly in 2d quantum gravity for the rescaling $g_{\mu \nu }\to e^{\varphi }g_{\mu \nu }$, is

$S[\varphi ]={\frac {1}{4\pi }}\int d^{2}x\,{\sqrt {g}}\left(g^{\mu \nu }\partial _{\mu }\varphi \partial _{\nu }\varphi +QR(g)\varphi +\lambda e^{2b\varphi }\right)$

Here $R(g)$ is the scalar curvature of the metric, while $Q,b,\lambda$ are parameters called the background charge, coupling and cosmological constant. This action justifies the name of Liouville theory, because the associated equation of motion (in the case of a flat metric) is w:Liouville's equation. The derivation of this action is subtle, see Ref. for some ideas and references. Certainly this action should describe a CFT: this will now lead to constraints on the parameters.

The Liouville action may be treated as a free term, plus an exponential interaction. The free term is classically Weyl invariant, because $g\mapsto \lambda g\implies R(g)\mapsto \lambda ^{-1}R(g)$. Conformal transformations are generated by the energy-momentum tensor

$T^{\mu \nu }=-{\frac {2}{\sqrt {g}}}{\frac {\delta S}{\delta g_{\mu \nu }}}$

In the flat metric with complex coordinates, this tensor is

$T_{z{\bar {z}}}=T_{{\bar {z}}z}=0\quad ,\quad T=T_{zz}=(\partial _{z}\varphi )^{2}+Q\partial _{z}^{2}\varphi \quad ,\quad {\bar {T}}=T_{{\bar {z}}{\bar {z}}}=(\partial _{\bar {z}}\varphi )^{2}+Q\partial _{\bar {z}}^{2}\varphi$

The central charge $c=1+6Q^{2}$ is found by computing the OPE of $T$ with itself in the quantum theory. Expressions that are quadratic or exponential in $\varphi$ now make sense as normal-ordered products.

Then consider the interaction term. For the theory to remain conformal, this term must be marginal, i.e. it must have conformal dimension one. By computing its OPE with $T$, it can be shown that $e^{2\alpha \varphi }$ is a primary field with the conformal dimension

$\Delta (\alpha )=\alpha (Q-\alpha )$

The condition $\Delta (b)=1$ then leads to a relation between the background charge and coupling constant,

$Q=b+b^{-1}$

This is a hint that Liouville theory is in fact invariant under $b\to b^{-1}$, which is not manifest in the path integral formulation. What about the cosmological constant? In fact it is not an important parameter. If the metric is flat so that $R(g)=0$, the cosmological constant can be absorbed by $\varphi \to \varphi -{\frac {\log \lambda }{2b}}$. So the only parameter of Liouville theory is the central charge.

Spectrum

In the free theory, a primary field is characterized by its momentum $\alpha$, and we have momentum conservation:

$\left\langle \prod _{i=1}^{N}e^{2\alpha \varphi (z_{i})}\right\rangle \neq 0\implies \sum _{i=1}^{N}\alpha _{i}=Q$

In Liouville theory however, two primary fields with the same dimension are proportional, so we have a reflection relation

$e^{2\alpha \varphi (z)}\ {\underset {\text{Liouville}}{=}}\ R_{\alpha }e^{2(Q-\alpha )\varphi (z)}$

From the point of view of the action, this is due to the exponential interaction. We can view a state as a wave propagating in $\varphi$-space. The interaction term creates a Liouville wall that prevents the wave from reaching $\varphi =+\infty$. The reflection coefficient $R_{\alpha }$ characterizes how the wave is reflected back towards $\varphi =-\infty$. And of course, the interaction breaks momentum conservation.

Correlation functions

Since symmetries of quantum Liouville theory are not manifest in the path integral formalism, and the exponential interaction has a rather violent effect on the spectrum, it is hard to compute correlation functions perturbatively from the path integral. Nevertheless, this is how Liouville theory was first solved by Dorn-Otto and Zamolodchikov-Zamolodchikov in the 1990s, using a fair amount of guesswork. A similar approach was previously applied to minimal models by Dotsenko-Fateev in the 1980s under the name of the Coulomb gas method.

The idea is to write a Liouville correlation function as a perturbed free field correlation function:

$\left\langle \prod _{i=1}^{N}e^{2\alpha \varphi (z_{i})}\right\rangle =\sum _{k=0}^{\infty }{\frac {\lambda ^{k}}{(4\pi )^{k}k!}}\left\langle \left(\int d^{2}z\,e^{2b\varphi (z)}\right)^{k}\prod _{i=1}^{N}e^{2\alpha \varphi (z_{i})}\right\rangle _{\text{free}}$

A free correlation function reduces to a Gaussian path integral, and can be computed explicitly:

$\left\langle \prod _{i=1}^{N}e^{2\alpha \varphi (z_{i})}\right\rangle _{\text{free}}=\delta \left(\textstyle {\sum }_{i=1}^{N}\alpha _{i}-Q\right)\prod _{i<j}|z_{i}-z_{j}|^{-4\alpha _{i}\alpha _{j}}$

Then it remains to perform the integrals from the interaction term. To interpret such calculations, the main difficulties come from the delta function. We only get a nonzero result if

$\sum _{i=1}^{N}\alpha _{i}=Q-2kb\quad {\text{with}}\quad k\in \mathbb {N}$

Of course there is no such condition in Liouville theory. The interpretation is that we are actually computing a residue of a Liouville correlation function at a pole. Knowing all such residues is not enough for fully determining the correlation functions, this is why some guesswork is needed.