Conformal field theory in two dimensions/Analytic bootstrap

Belavin-Polyakov-Zamolodchikov differential equations

See Section 3.1 of Ref.

Exercises

• AB1. For $V_{i}$ primary fields, we would like to compute $\left\langle L_{-3}V_{1}(z)V_{2}(0)V_{3}(\infty )V_{4}(1)\right\rangle$ as a differential operator acting on $\left\langle V_{1}(z)V_{2}(0)V_{3}(\infty )V_{4}(1)\right\rangle$. To do this, we may use integrals of the type $\oint _{z_{0}}dy{\frac {y(y-1)(y-a)}{(y-z)^{2}}}\left\langle T(y)V_{1}(z)V_{2}(0)V_{3}(\infty )V_{4}(1)\right\rangle$ with $z_{0}\in \{0,1,\infty ,z\}$. Which value of $a$ allows us to avoid having a contribution of $L_{-2}$? Deduce the desired result.
• AB2. How do hypergeometric blocks behave under field permutations $V_{i}(z_{i})\leftrightarrow V_{j}(z_{j})$? Under reflections of momentums $P_{i}\to -P_{i}$?
• AB3. Compute the inverse of the degenerate fusing matrix $F$, and compare it to $F$.
• AB4. 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 $F^{s\to t}F^{t\to u}=F^{s\to u}$.

More exercises

• AB11. Third-order BPZ equation: see Exercise 2.20 in Ref.
• AB12. 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 $\left\langle V_{\langle 2,1\rangle }^{d}V_{P}V_{\langle r,s\rangle }^{d}V_{P'}\right\rangle$.

1. For which values of $P'$ 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.

Exercises

• AS1. 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 $i\in \{1,2,3,4\}$.
• AS2. How do shift equations behave under parity? i.e. under the exchange of left and right momentums $\forall i,P_{i}\leftrightarrow {\bar {P}}_{i}$.
• AS3. 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 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.

Exercises

• AG1. Using the behaviour of the double Gamma function under $x\to x+\beta ^{\pm 1}$, compute the ratio ${\frac {\Gamma _{\beta }(x+\beta +\beta ^{-1})}{\Gamma _{\beta }(x)}}$ in two different ways, and check that the results agree.
• AG2. Compute the residues of the double Gamma function at its poles in terms of $\Gamma _{\beta }(\beta )$ and the Gamma function.
• AG3. Explicitly compute the nontrivial 3-point structure constants in the minimal models AMM$_{4,3}$ and AMM$_{5,2}$. Check the answers in Ref.

More exercises

• AG11. Behaviour of Liouville 4-point functions at coinciding points: see Exercise 3.4 in Ref.
• AG12. Fusion rules of unitary representations if $c>1$: see Exercise 3.6 in Ref.
• AG13. In Liouville theory with DOZZ-normalized structure constants, compute the limit $\lim _{P_{2}\to P_{(r,s)}}V_{P_{1}}V_{P_{2}}$ with $r,s\in \mathbb {N} ^{*}$. Check the answer in Ref.