Mathematical prerequisites for 2d CFT

The prerequisites are in two areas of mathematics:

• Complex analysis: contour integrals of complex analytic functions on $\mathbb {C}$.
• Lie algebras and their representations.

Exercises

MICA: Integrating a complex analytic function

For $a,b\in \mathbb {C}$ let us define

$f(a,b|x)={\frac {1}{(x^{4}+a^{4})(x^{2}+b^{2})^{2}}}\quad ,\quad g(a,b)=\int _{-\infty }^{\infty }f(a,b|x)dx$

1. What are the poles and residues of $f(a,b|x)$ as a function of $x$?
2. Compute $g(a,b)$ and discuss its analytic properties.

MARE: A Lie algebra and its representations

Consider a finite-dimensional Lie algebra ${\mathfrak {g}}$, with a basis $t^{a}$ obeying commutation relations $[t^{a},t^{b}]=f_{c}^{ab}t^{c}$. For $\rho$ a representation of ${\mathfrak {g}}$, we define

$g^{ab}=\mathrm {Tr} _{\rho }(t^{a}t^{b})\quad ,\quad K=g_{ab}^{-1}t^{a}t^{b}$

assuming the matrix $g^{ab}$ is invertible.

1. Show that $K$ belongs to the center of the universal enveloping algebra of ${\mathfrak {g}}$.
2. Compute $K$ for ${\mathfrak {g}}={\mathfrak {sl}}_{2}$ and $\rho =R_{2}$ the fundamental representation, i.e. the irreducible representation of dimension 2. Use a basis $J^{0},J^{+},J^{-}$ such that $[J^{0},J^{\pm }]=\pm J^{\pm }$ and $[J^{+},J^{-}]=2J^{0}$.
3. For which values of $j\in \mathbb {C}$ does ${\mathfrak {sl}}_{2}$ have an irreducible representation $V_{j}$ where $J^{0}$ has the eigenvalues $\mathrm {Spec} _{V_{j}}(J^{0})=j+\mathbb {N}$?
4. Compute the value of $K$ in $R_{2}$ and $V_{j}$. Diagonalize $K$ and $J^{0}$ in $R_{2}\otimes V_{j}$, and deduce $R_{2}\otimes V_{j}=\oplus _{\pm }V_{j\pm {\frac {1}{2}}}$.
5. By induction on $k\in \mathbb {N}$, decompose $R_{2}^{\otimes k}$ into irreducible representations. This should include an irreducible representation $R_{k+1}$ of dimension $k+1$. Compute $\mathrm {Spec} _{R_{k}}(J^{0})$, compute $R_{k_{1}}\otimes R_{k_{2}}$ and compute $R_{k}\otimes V_{j}$.
6. Let $W$ be the 2-dimensional representation with a basis $(v,J^{0}v)$ such that $(J^{0})^{2}v=J^{\pm }v=0$. Show that $W$ is reducible but indecomposable. Same question for $W\otimes R_{k}$ and $W\otimes V_{j}$. What are the submodules of $W\otimes W$?