Conformal field theory in two dimensions/Fundamental structures

The Virasoro algebra and its representations

See Sections 1.1.1 to 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 to 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

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

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.