# How to expand Lie characters?

The following involves characters of affine Lie algebras, and I will be using as reference the book on CFT by Francesco et at (here are some screenshots if useful). But hopefully the post will be self-contained.

Let $$\Theta^k_\lambda=\sum_{n\in\mathbb Z}\exp\big[-2\pi i(knz+\frac12\lambda z-kn^2\tau-n\lambda\tau-\lambda^2\tau/4k)\big]\tag{14.176}$$ and $$\chi^k_\lambda=\frac{\Theta^{k+2}_{\lambda+1}-\Theta^{k+2}_{-\lambda-1}}{\Theta^2_1-\Theta^2_{-1}}\tag{14.174}$$

The idea is to expand $$\chi$$ for $$\lambda=k=1$$, $$z=0$$, in powers of $$q=e^{2\pi i \tau}$$. The expected result is $$\chi^1_1=q^{5/24}(2+2q+6q^2+8q^3+\cdots)\tag{14.179}$$

How can I use Mathematica to recover eq.$$14.179$$ from the other two equations? The naive approach does not quite work, because $$\chi$$ yields an indeterminate form if we take $$\lambda=1,z=0$$ directly. And the sum does not converge for $$|\mathrm{re}(q)|<1$$ (and it cannot be analytically continued), so the expansion around $$q=0$$ is only asymptotic (not a proper power series in the strict sense).

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Here's the code, if it helps:

Θ[k_, λ_, z_, τ_] := Sum[Exp[-2 π I (k n z + 1/2 λ z - k n^2 τ - n λ τ - λ^2 τ/(4 k))], {n, -∞, ∞}]
χ[k_, λ_, z_, τ_] := (Θ[k + 2, λ + 1, z, τ] - Θ[k + 2, -λ - 1, z, τ])/(Θ[2, 1, z, τ] - Θ[2, -1, z, τ])

• No code for those complex expressions? Jan 24, 2021 at 22:49
• Exp[-2 π I (k n z + 1/2 λ z - k n^2 τ - n λ τ - λ^2 τ/(4 k))] Jan 24, 2021 at 22:59
• I wonder how much of this is expressible in terms of the built-in theta functions or $q$-functions... Jan 30, 2021 at 8:52

Let $$p=e^{2\pi iz}$$ and $$q=e^{2\pi i\tau}$$.

thetaTerm[k_, l_] := p^(-k*n - l/2) q^(k*n^2 + l*n + l^2/(4*k))
numTerm = thetaTerm[k+2, l+1] - thetaTerm[k+2, -l-1] /. {l->1, k->1}
denTerm = thetaTerm[2, 1] - thetaTerm[2, -1]
max = 3
numSum = Sum[numTerm/q^(1/3), {n, -max, max}] // Expand
denSum = Sum[denTerm/q^(1/8), {n, -max, max}] // Expand
lim = Limit[numSum/denSum, p->1]
ser = Series[lim, {q, 0, max}]


$$2+2 q+6 q^2+8 q^3+O\left(q^4\right)$$

The prefactor $$q^{5/24}$$ comes from the $$q^{1/3}$$ in the numerator terms and the $$q^{1/8}$$ in the denominator terms, which I removed.

• great, thanks! :-) Feb 4, 2021 at 21:51