Analytic bootstrap in 2d CFT

Belavin-Polyakov-Zamolodchikov differential equations

See Section 3.1 of Ref.

Course questions

1. For $V_i$ primary fields, we would like to compute $⟨L_{-3}{V}_1(z)V_2(0)V_3(\mathrm{\infty })V_4(1)⟩$ as a differential operator acting on $⟨V_1(z)V_2(0)V_3(\mathrm{\infty })V_4(1)⟩$. To do this, we may use integrals of the type ${\oint }_{z_0}dy\frac{y(y-1)(y-a)}{(y-z{)}^2}⟨T(y)V_1(z)V_2(0)V_3(\mathrm{\infty })V_4(1)⟩$ with $z_0\in \{0,1,\mathrm{\infty },z\}$. Which value of $a$ allows us to avoid having a contribution of $L_{-2}$? Deduce the desired result.
2. How do hypergeometric blocks behave under field permutations $V_i(z_i)\leftrightarrow V_j(z_j)$? Under reflections of momentums $P_i\to -P_i$?
3. Compute the inverse of the degenerate fusing matrix $F$, and compare it to $F$.
4. Calling $F^{s\to t}$ the degenerate fusing matrix that relates s-channel to t-channel blocks, compute $F^{s\to u}$ and $F^{t\to u}$, and check $F^{s\to t}{F}^{t\to u}=F^{s\to u}$.
5. In a 4-point function with a level 2 degenerate field, compare the ratio of the two t-channel structure constants, with the ratio of the two s-channel structure constants.

Exercises

1. Third-order BPZ equation: see Exercise 2.20 in Ref.
2. BPZ equations from fusion rules: see Exercise 2.22 in Ref.

ABOF: Degenerate 4-point functions with one propagating field

We consider a 4-point function of the type $⟨V_{⟨2,1⟩}^d{V}_P{V}_{⟨r,s⟩}^d{V}_{P^{\prime }}⟩$.

1. For which values of $P^{\prime }$ is the s-channel spectrum made of only one primary field? Choose one such value.
2. Compute the t-channel and u-channel spectrums.
3. Compute the corresponding degenerate fusing matrix.
4. Write the relations between all 4-point structure constants.

Shift equations for structure constants

See Section 3.2 of Ref.

Course questions

1. Check that any 4-point structure constant is invariant under renormalizations of the channel field $V_k\to {\lambda }_k{V}_k$. Furthermore, check that any ratio of 4-point structure constants is invariant under any field renormalization $V_i\to {\lambda }_i{V}_i$ with $i\in \{1,2,3,4\}$.
2. How do shift equations behave under parity? i.e. under the exchange of left and right momentums $\mathrm{\forall }i,P_i\leftrightarrow {\overline{P}}_i$.
3. In how many ways can the ratio $\frac{C_{1^{-}23^{+}}}{C_{1^{+}23^{-}}}$ be computed using 2 shift equations? Check that the different computations yield the same result.

ABUD: Counting independent structure constants

Given 3 numbers $r_i\in \frac{1}{2}{\mathbb{N}}^{*}$ such that $r_1+r_2+r_3\in \mathbb{N}$, we are interested in structure constants of the type $C_{(r_1,s_1)(r_2,s_2)(r_3,s_3)}$ with $s_i\in \frac{1}{r_i}\mathbb{Z}$, modulo shift equations.

1. Write all independent structure constants in the cases $(r_1,r_2,r_3)=(\frac{1}{2},1,\frac{3}{2})$ and $(r_1,r_2,r_3)=(1,2,3)$.
2. What is the number of independent structure constants, as a function of $r_1,r_2,r_3$?

ABDE: Degenerate structure constants

Let $V_1$ be a degenerate field, $V_2$ a diagonal or non-diagonal field, and let $V_3$ be a primary field that appears in the OPE $V_1{V}_2$.

1. Compute the shifts $\frac{C_{123}}{C_{123^{++}}}$ and $\frac{C_{123}}{C_{123^{--}}}$.
2. In which cases are these shifts zero or infinite? Interpret the results in terms of fusion rules.

Double Gamma function and solutions of shift equations

See Section 3.3 of Ref.

Course questions

1. Using the behaviour of the double Gamma function under $x\to x+{\beta }^{\pm 1}$, compute the ratio $\frac{{\mathrm{\Gamma }}_{\beta }(x+\beta +{\beta }^{-1})}{{\mathrm{\Gamma }}_{\beta }(x)}$ in two different ways, and check that the results agree.
2. Compute the residues of the double Gamma function at its poles in terms of ${\mathrm{\Gamma }}_{\beta }(\beta )$ and the Gamma function.
3. Explicitly compute the nontrivial 3-point structure constants in the minimal models AMM${}_{4,3}$ and AMM${}_{5,2}$. Check the answers in Ref.

Exercises

1. Convergence of the conformal block decomposition in Liouville theory: see Exercise 3.3 in Ref.
2. Behaviour of Liouville 4-point functions at coinciding points: see Exercise 3.4 in Ref.
3. Fusion rules of unitary representations if $c>1$: see Exercise 3.6 in Ref.
4. In Liouville theory with DOZZ-normalized structure constants, compute the limit $\underset{P_2\to P_{(r,s)}}{\lim }{V}_{P_1}{V}_{P_2}$ with $r,s\in {\mathbb{N}}^{*}$. Check the answer in Ref.