TITLE: String theory "computation" for math undergrad audience
QUESTION [23 upvotes]: I am giving a talk on String theory to a math undergraduate audience. I am looking for a nice and suprising mathematical computation, maybe just a surprising series expansion, which is motivated by string theory and which can be motivated and explained relatively easily. Examples of what I have in mind are the results in Dijkgraaf's "Mirror symmetry and the elliptic curve", or the "genus expansion" of the MacMahon function (aka DT/GW for affine three-space), but I am not sure I can fit either into the time I have. Any thoughts?

REPLY [2 votes]: Derive the Casimir Energy in Bosonic String Theory. 
You start with the $\hat L_0$ operator and get rid  of the non-vacuum $\displaystyle\frac{\alpha_0^2}{2}+\sum_{n=1}^\infty\alpha_{-n}\cdot\alpha_n$, then you use a Ramanujam sum to do $\zeta$-function renormalisation,  from which you find out that the vacuum energy denoted by $\varepsilon_0$ is 
$$\varepsilon_0=-\frac{d-2}{24}$$       
However, the most interesting part comes when you go around deriving the critical dimension of Bosonic String Theory. 
After which, the expression surprisingly simplifyies to a $-1$.  
For a more detailed derivation of the above stuff, see these lecture notes/. (Section 4) (Equation 4.5-4.10)