Conformal field theory in two dimensions/Fundamental structures

The Virasoro algebra and its representations

See Sections 1.1.1, 1.1.2, 1.1.3 and 1.1.4 of Ref.

Exercises

• FV1. Does the Virasoro algebra's central term affect global conformal transformations?
• FV2. Write a basis of a Verma module's level 5.
• FV3. From its explicit expression, check that a level-3 null vector is annihilated by $L_{1}$ and $L_{2}$. Is it necessary to check it for $L_{3}$ as well?
• FV4. In the case $c=28$ i.e. $\beta ^{2}=-{\frac {1}{2}}$, find the levels of all the null vectors in the Verma module with dimension $\Delta _{(3,3)}=\Delta _{(1,7)}=-9$. How many degenerate quotients does this Verma module have?

More exercises

• FV11. Eigenvalues of $L_{0}$ differ by integers in indecomposable representations of the Virasoro algebra: see Exercise 1.7 of Ref.
• FV12. Decomposing Virasoro representations into ${\mathfrak {sl}}_{2}$ representations: see Exercise 1.8 of Ref.
• FV13. The existence of a Hermitian form on the space of states is a necessary condition for unitarity, and occurs also in many non-unitary CFTs. Show that if there is a Hermitian form that is compatible with the Virasoro algebra and is such that $L_{0}$ is self-conjugate, then the central charge is real. For a more precise statement of the problem and a few hints, see Exercise 1.9 of Ref.
• FV14. Alternative spanning set of a Verma module: see Exercise 2.2 of Ref.

Fields and operator product expansions

See Sections 1.2.1, 1.2.2 and 1.2.3 of Ref.

Exercises

• FF1. Given two CFTs, each one with its own Virasoro algebra and spectrum, let the product CFT be a CFT whose spectrum is the tensor product of the two spectrums. Which Virasoro algebra describes conformal symmetry in the product CFT? What is its central charge?
• FF2. For $V_{\Delta }$ a primary field, write the OPE of the energy-momentum tensor $T$ with $L_{-2}V_{\Delta }$, and compare with the OPE of $T$ with itself.
• FF3. Let $C_{12}^{k}(z_{1},z_{2})$ be a coefficient in the OPE of 2 primary fields $V_{1}(z_{1})V_{2}(z_{2})$ as $z_{1}\to z_{2}$. Is $C_{12}^{k}(z_{1},z_{2})$ related to $C_{21}^{k}(z_{2},z_{1})$? To $C_{12}^{k}(z_{2},z_{1})$?
• FF4. Assuming the coefficients $f^{L}$ are known, compute the first few orders of the OPEs $L_{-1}V_{\Delta _{1}}(z_{1})V_{\Delta _{2}}(z_{2})$ and $V_{\Delta _{1}}(z_{1})L_{-1}V_{\Delta _{2}}(z_{2})$.
• FF5. Virasoro algebra and $TT$ OPE: see Exercise 2.11 of Ref.

FOGO: Global vs local conformal symmetry in OPEs

In a CFT with local conformal symmetry, we recall that the contribution of $V_{\Delta }$ and its descendants in an OPE of 2 primary fields reads

$V_{\Delta _{1}}(z_{1})V_{\Delta _{2}}(z_{2})\supset C_{\Delta _{1},\Delta _{2}}^{\Delta }z_{12}^{\Delta -\Delta _{1}-\Delta _{2}}{\Bigg (}V_{\Delta }(z_{2})+\sum _{L\in {\mathcal {L}}\backslash \{1\}}z_{12}^{|L|}f_{\Delta _{1},\Delta _{2}}^{\Delta ,L}LV_{\Delta }(z_{2}){\Bigg )}$

where ${\mathcal {L}}$ is a basis of Virasoro creation operators.

1. What would be the analogous formula in a CFT with only global conformal symmetry? Show that its universal coefficients are parametrized by integers $k\in \mathbb {N} ^{*}$, and write these coefficients as ${\tilde {f}}^{k}={\tilde {f}}_{\Delta _{1},\Delta _{2}}^{\Delta ,k}={\tilde {f}}_{\Delta _{1},\Delta _{2}}^{\Delta ,L_{-1}^{k}}$.
2. Compute the coefficients ${\tilde {f}}^{1}$ and ${\tilde {f}}^{2}$, and compare them with $f^{L_{-1}},f^{L_{-1}^{2}}$.
3. In order to explain why ${\tilde {f}}^{2}\neq f^{L_{-1}^{2}}$, find how the coefficients $f^{L}$ behave under a change of basis $L_{-2}\to L_{-2}+\alpha L_{-1}^{2}$. Which value $\alpha _{0}$ leads to ${\tilde {f}}^{2}=f^{L_{-1}^{2}}$?
4. Show that $\left(L_{-2}+\alpha _{0}L_{-1}^{2}\right)V_{\Delta }$ is a global primary field.

Fusion rules and minimal models

Fusion rules

See Sections 1.2.4 and 1.2.5 of Ref.

(Generalized) minimal models

See Sections 2.2.1 and 2.2.3 of Ref.

Case of the critical Ising model

On a lattice, the Ising model may be defined by the Hamiltonian

$H=-\sum _{<i,j>}\sigma _{i}\sigma _{j}$

where $\sigma _{i}\in \{-1,1\}$ is the value of a spin on a lattice site, and the sum is over pairs of neighbouring sites. The overall minus sign makes the interaction ferromagnetic: two neighbouring spins minimize their energy by being aligned i.e. $\sigma _{i}=\sigma _{j}$. The partition function is

$Z=\sum _{\{\sigma _{i}\}_{i}}e^{-\beta H(\{\sigma _{i}\}_{i})}$

where $0<\beta <\infty$ is the inverse temperature.

In two dimensions, there is a second-order phase transition at $\beta _{c}={\frac {\log(1+{\sqrt {2}})}{2}}$, the solution of $\sinh(2\beta _{c})=1$. Assuming that the model is described by a CFT at the phase transition, the question is which CFT?

First we would like to find the central charge. This may be inferred from the free energy of the model on a cylinder of circumference $L$ and length $M$ lattice spacings,

$f=-{\frac {\log Z}{ML}}\ {\underset {1\ll L\ll M}{=}}\ {\text{constant}}-{\frac {\pi c}{6L^{2}}}+o\left({\frac {1}{L^{2}}}\right)$

The appearance of the central charge may be traced back to the a conformal transformation from the plane to the cylinder. In the Ising model, the free energy may be computed numerically, or from Onsager's exact solution of the lattice model. The result is $c={\frac {1}{2}}$.

The model has a $\mathbb {Z} _{2}$ symmetry $\sigma _{i}\to -\sigma _{i}$, which should survive in the CFT. Then let us consider some fields. It is natural to define a spin field associated to $\sigma _{i}$, whose correlation functions may be computed in the lattice model. In particular, the 2-point function is found to behave as

$\left\langle \sigma _{i}\sigma _{j}\right\rangle ={\frac {1}{Z}}\sum _{\{\sigma _{i}\}_{i}}e^{-\beta H(\{\sigma _{i}\}_{i})}\sigma _{i}\sigma _{j}\ {\underset {|i-j|\to \infty }{\propto }}\ |i-j|^{-{\frac {1}{4}}}$

Again this can be deduced from numerics, or from an exact solution. This suggests that the CFT has a corresponding primary field of left and right dimensions $\Delta ={\bar {\Delta }}={\frac {1}{16}}$. Similarly, we may define an energy field $\epsilon _{i}=-\sum _{<i,j>}\sigma _{i}\sigma _{j}$, leading to a primary field of dimensions $\Delta ={\bar {\Delta }}={\frac {1}{2}}$. It is however not so easy to show that all local fields, built from spin variables in the neighbourhood of a lattice site, reduce to the 3 primary fields $1,\sigma ,\epsilon$, or Virasoro descendants thereof. We could try to argue for OPEs from the lattice: $\sigma \sigma \sim 1+\epsilon$ is fairly natural, but $\epsilon \epsilon \sim 1$ not so much.

So, there are reasonable arguments that local observables in the critical limit of the Ising model are described by the $c={\frac {1}{2}}$ minimal model. However, we can also define nonlocal observables. If a spin cluster is a maximal connected set of spins with the same value, what is the probability that $N$ given sites $i_{1},i_{2},\dots ,i_{N}$ belong to the same spin cluster? In the critical limit, this tends to an $N$-point function in a CFT that is larger than the minimal model. This CFT may be called a loop CFT or critical loop model. The definition of a CFT that we would call the w:Two-dimensional critical Ising model therefore depends on which observables we consider.

Exercises

• FM1. Write the fusion products of the degenerate representation ${\mathcal {R}}_{\langle 3,2\rangle }^{d}$ with a Verma module or with another degenerate representation. In which cases are there fewer than 6 terms?
• FM2. Assume the central charge is generic, and consider the fusion ring of degenerate representations. Find all its subrings. (For example, the field $V_{\langle 2,1\rangle }^{d}$ generates a subring whose basis is $\{V_{\langle r,1\rangle }^{d}\}_{r\in \mathbb {N} ^{*}}$.) Which subrings are finite-dimensional? finitely generated?
• FM3. What is the smallest Kac table that is not displayed in the article w:minimal model (physics)?

More exercises

• FM11. Structure of fully degenerate representations: see Exercise 2.5 of Ref.
• FM12. Characters of Virasoro representations: see Exercise 2.6 of Ref.
• FM13. In which minimal models do fusion rules have a $\mathbb {Z} _{2}$ symmetry like in the Ising case?
• FM14. Write the fusion rules of the minimal model AMM$_{5,3}$.

Correlation functions and conformal blocks

See Section 1.3 of Ref.

Exercises

• FC1. Assuming we know $Z=\left\langle \prod _{i=1}^{N}V_{\Delta _{i}}(z_{i})\right\rangle$, compute $Z(y_{1},y_{2})=\left\langle T(y_{1})T(y_{2})\prod _{i=1}^{N}V_{\Delta _{i}}(z_{i})\right\rangle$. Check that the answer is invariant under $y_{1}\leftrightarrow y_{2}$.
• FC2. In two dimensions, write the generators of the conformal algebra $D,M_{\mu ,\nu },K_{\mu },P_{\mu }$, in terms of Virasoro generators $L_{n},{\bar {L}}_{n}$.
• FC3. Assuming $\left\langle V_{i}V_{i}V_{k}\right\rangle \neq 0$ for some primary fields $V_{i},V_{k}$, what can we say on the conformal spin of $V_{k}$? (Use permutation symmetry.)

More exercises

• FC11. Behaviour of the energy-momentum tensor at infinity: see Exercise 2.10 of Ref.
• FC12. Creation operators as differential operators: see Exercise 2.13 of Ref.
• FC13. Logarithmic correlation functions: see Exercise 2.15 of Ref.
• FC14. Permutations and 4-point functions: see Exercise 2.16 of Ref.