I am taking a string theory module in NUS this semester. For my midterms, I have prepared a presentation explains the why bosonic string theory needs 26 dimensions for “the math to make sense”. Enjoy!

Abstract: In summary, to quantise the Polyakov action for a bosonic string, we had to insert the Faddeev-Popov determinant into the path integral, which ended up being calculated by the bc ghost CFT. bc CFT alone had central charge of −26, but we need the total central charge of Polyakov + bc to be 0 due to the Weyl anomaly. So the Polyakov action needed to have a ”critical central charge” of 26, which corresponded to the coordinates of the string being 26-dimensional.

There are other ways to derive the critical dimension, such as using Lorentz invariance.

Update: Here is a recording