Fundamental structures of 2d CFT

The Virasoro algebra and its representations

See Sections 1.1.1 to 1.1.4 of Ref.

Course questions

1. Does the Virasoro algebra's central term affect global conformal transformations?
2. Given two CFTs, each one with its own Virasoro algebra, what is the Virasoro algebra for the product CFT? What is its central charge?
3. Write a basis of a Verma module's level 5.
4. 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?
5. In the case $c=28$ find the levels of all the null vectors in the Verma module with dimension ${\mathrm{\Delta }}_{(3,3)}$. How many degenerate quotients does this Verma module have?

Exercises

1. Eigenvalues of $L_0$ differ by integers in indecomposable representations of the Virasoro algebra: see Exercise 1.7 of Ref.
2. Decomposing Virasoro representations into ${𝔰𝔩}_2$ representations: see Exercise 1.8 of Ref.
3. 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.
4. Alternative spanning set of a Verma module: see Exercise 2.2 of Ref.

Fields and operator product expansions

See Sections 1.2.1 to 1.2.3 of Ref.

Course questions

1. For $V_{\mathrm{\Delta }}$ a primary field, write the OPE of the energy-momentum tensor $T$ with $L_{-2}{V}_{\mathrm{\Delta }}$, and compare with the OPE of $T$ with itself.
2. Is $C_{12}^k(z_1,z_2)$ related to $C_{21}^k(z_2,z_1)$? To $C_{12}^k(z_2,z_1)$?
3. Assuming the coefficients $f^L$ are known, compute the first few orders of the OPEs $L_{-1}{V}_{{\mathrm{\Delta }}_1}(z_1)V_{{\mathrm{\Delta }}_2}(z_2)$ and $V_{{\mathrm{\Delta }}_1}(z_1)L_{-1}{V}_{{\mathrm{\Delta }}_2}(z_2)$.

FOGO: Global vs local conformal symmetry in OPEs

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

$V_{{\mathrm{\Delta }}_1}(z_1)V_{{\mathrm{\Delta }}_2}(z_2)\supset C_{{\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2}^{\mathrm{\Delta }}{z}_{12}^{\mathrm{\Delta }-{\mathrm{\Delta }}_1-{\mathrm{\Delta }}_2}(V_{\mathrm{\Delta }}(z_2)+\sum _{L\in ℒ\mathrm{\setminus }\{1\}}{z}_{12}^{|L|}{f}_{{\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2}^{\mathrm{\Delta },L}L{V}_{\mathrm{\Delta }}(z_2))$

where $ℒ$ 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 ${\tilde{f}}^k={\tilde{f}}_{{\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2}^{\mathrm{\Delta },k}={\tilde{f}}_{{\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2}^{\mathrm{\Delta },L_{-1}^k}$.
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 ${\tilde{f}}^2\ne f^{L_{-1}^2}$, 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 $(L_{-2}+{\alpha }_0{L}_{-1}^2)V_{\mathrm{\Delta }}$ is a global primary field.

Other exercises

1. Virasoro algebra and $TT$ OPE: see Exercise 2.11 of Ref.

Fusion rules and minimal models

See Sections 1.2.4, 1.2.5, 2.2.1 and 2.2.3 of Ref.

Course questions

1. Write the fusion products of the degenerate representation ${ℛ}_{⟨3,2⟩}^d$ with a Verma module or with another degenerate representation. In which cases are there fewer than 6 terms?
2. At generic central charge, find all subrings of the fusion ring of degenerate representations. Are there any finite-dimensional subrings?
3. What is the smallest Kac table that is not displayed in the article w:minimal model (physics)?

Exercises

1. Structure of fully degenerate representations: see Exercise 2.5 of Ref.
2. Characters of Virasoro representations: see Exercise 2.6 of Ref.
3. Unitarity and negative conformal dimensions in minimal models: see Exercise 3.11 of Ref.
4. In which minimal models do fusion rules have a ${\mathbb{Z}}_2$ symmetry like in the Ising case?
5. Write the fusion rules of the minimal model AMM${}_{5,3}$.

Correlation functions and conformal blocks

See Section 1.3 of Ref.

Course questions

1. Assuming we know $Z=⟨\prod _{i=1}^N{V}_{{\mathrm{\Delta }}_i}(z_i)⟩$, compute $Z(y_1,y_2)=⟨T(y_1)T(y_2)\prod _{i=1}^N{V}_{{\mathrm{\Delta }}_i}(z_i)⟩$.
2. Write the generators of the conformal algebra $D,M_{\mu ,\nu },K_{\mu },P_{\mu }$, in terms of Virasoro generators $L_n,{\overline{L}}_n$.
3. Assuming $⟨V_i{V}_i{V}_k⟩\ne 0$ for some primary fields $V_i,V_k$, what can we say on the conformal spin of $V_k$?

Exercises

1. Behaviour of the energy-momentum tensor at infinity: see Exercise 2.10 of Ref.
2. Creation operators as differential operators: see Exercise 2.13 of Ref.
3. Logarithmic correlation functions: see Exercise 2.15 of Ref.
4. Permutations and 4-point functions: see Exercise 2.16 of Ref.