View source for Mathematical prerequisites for 2d CFT


The prerequisites are in two areas of mathematics:
* Complex analysis: contour integrals of complex analytic functions on <math>\mathbb{C}</math>.
* Lie algebras and their representations.

== Exercises ==

=== MICA: Integrating a complex analytic function ===

For <math>a,b\in\mathbb{C}</math> let us define 
:<math> 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 
</math>
# What are the poles and residues of <math>f(a,b|x)</math> as a function of <math>x</math>?
# Compute <math>g(a,b)</math> and discuss its analytic properties. 

=== MARE: A Lie algebra and its representations ===

Consider a finite-dimensional Lie algebra <math>\mathfrak{g}</math>, with a basis <math>t^a</math> obeying commutation relations <math>[t^a, t^b] =f^{ab}_c t^c</math>. For <math>\rho</math> a representation of <math>\mathfrak{g}</math>, we define
:<math>g^{ab}=\mathrm{Tr}_\rho (t^a t^b) \quad , \quad K = g^{-1}_{ab}t^at^b
</math>
assuming the matrix <math>g^{ab}</math> is invertible.
# Show that <math>K</math> belongs to the center of the universal enveloping algebra of <math>\mathfrak{g}</math>.
# Compute <math>K</math> for <math>\mathfrak{g}=\mathfrak{sl}_2</math> and <math>\rho=R_2</math> the fundamental representation, i.e. the irreducible representation of dimension 2. Use a basis <math>J^0, J^+,J^-</math> such that <math>[J^0,J^\pm] = \pm J^\pm</math> and <math>[J^+,J^-]=2J^0</math>.
# For which values of <math>j\in\mathbb{C}</math> does <math>\mathfrak{sl}_2</math> have an irreducible representation <math>V_j</math> where <math>J^0</math> has the eigenvalues <math>\mathrm{Spec}_{V_j}(J^0) = j-\mathbb{N}</math>? 
# Compute the value of <math>K</math> in <math>R_2</math> and <math>V_j</math>. Diagonalize <math>K</math> and <math>J^0</math> in <math>R_2\otimes V_j</math>, and deduce <math>R_2\otimes V_j=\oplus_\pm V_{j\pm \frac12}</math>.
# By induction on <math>k\in \mathbb{N}</math>, decompose <math>R_2^{\otimes k}</math> into irreducible representations. This should include an irreducible representation <math>R_{k+1}</math> of dimension <math>k+1</math>. Compute <math>\mathrm{Spec}_{R_k}(J^0)</math>, compute <math>R_{k_1}\otimes R_{k_2}</math> and compute <math>R_k\otimes V_j</math>.

[[Category: CFT course]]