Mathematical prerequisites for 2d CFT

The prerequisites are in two areas of mathematics:

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

Exercises

MICA: Integrating a complex analytic function

For $a,b\in ℂ$ let us define

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

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 $𝔤$, with a basis $t^a$ obeying commutation relations $[t^a,t^b]=f_c^{ab}{t}^c$. For $\rho$ a representation of $𝔤$, we define

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

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

1. Show that $K$ belongs to the center of the universal enveloping algebra of $𝔤$.
2. Compute $K$ for $𝔤={𝔰𝔩}_2$ 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 $j\in ℂ$ does ${𝔰𝔩}_2$ have an irreducible representation $V_j$ where $J^0$ has the eigenvalues ${\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}}_{V_j}(J^0)=j-\mathbb{N}$?
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 ${\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}}_{R_k}(J^0)$, compute $R_{k_1}\otimes R_{k_2}$ and compute $R_k\otimes V_j$.