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I was reading the lecture notes on Quantum field theory by David Tong. In the section on Casimir force he derived the force of attraction felt by the plates due to the field vacuum energy in $1+1$ dimensions. The force is given by $$F=\frac{\pi}{24d^2} +...$$ He then says that the $24$ in the denominator has something to do with the $26$ dimensions in bosonic string theory. Since I have no knowledge of string theory and it's working, can someone explain this connection between the Casimir force and the number of dimensions?

I know that I will not understand the connection mathematically, so please explain it in a way that a beginner in quantum field theory may understand.

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It is always the same $24$, and it is all over the place. I'm sure this number has some deeper origin, but I usually associate it to the fact that $\mathrm{SL}_2\mathbb Z=\mathbb Z_4*_{\mathbb Z_2}\mathbb Z_6$ (read, the modular group is the amalgamated product of these cyclic groups along $\mathbb Z_2$). We see that $4\times 6=24$, which shows up everywhere when studying 2d stuff (in particular, the torus, whose mapping class group is the modular group above). It probably also has something to do with the E8 and Leech lattices (with rank $8$ and $24$, respectively). Also, recall that String Theory is effectively a 2d CFT (and the torus is the first surface with a non-trivial topology). For other occurrences of the number 24 (or 12) in physics and mathematics see e.g. this talk by Baez (and this PSE post for an explicit computation). Finally, for another example that does not seem to be mentioned in the above references, let me note that anomalies are classified by generalised homology theories in one higher dimension, and we have $\Omega_3^\mathrm{fr}=\mathbb Z_{24}$, again the same integer (this is the cobordism group of framed 3-manifolds; the framing anomaly in 3d is the same thing as the chiral anomaly in 2d).

But anyway, the quick-and-dirty way to find this number is to use zeta-regularisation: $$ \sum_{n=1}^\infty n\sim \zeta(-1)=-\frac{1}{12} $$

This sum appears both when computing the Casimir force (cf. this PSE post), and the string critical dimension. In particular, the Casimir can be shown to be proportional to $F_D\propto \zeta(-D+1)$ (which becomes $\zeta(-1)$ in $D=2$), while the critical dimension of the bosonic string is found by insisting that the Virasoro central charge vanishes $$ c\propto \frac{D-2}{2}\zeta(-1)+1\equiv 0 $$ from where $D=2-2/\zeta(-1)=26$ follows. We thus see that in both cases one is required to evaluate $\zeta(-1)$, which is $-1/12$. This is the "something" in Casimir that has to do with String Theory.

It is important to insist that these two phenomena are not in fact independent. As discussed by Tong (cf. §4.4.1), the central charge of a CFT is a Casimir force!

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  • $\begingroup$ maybe i can add that a direct computation of the 1/12 factor (and independently of any regularisation) in a simple example (a single scalar with $c=1$) is given here: physics.stackexchange.com/a/483863/83405. this also shows that the 1/12 follows from local physics and is insensitive to topology. (which is why strings propagate consistently in D=26 dimensions on all Riemann surfaces.) so I'm not sure i see how this is related to SL(2,Z) (?) $\endgroup$ Dec 1 '19 at 11:51
  • $\begingroup$ @Wakabaloola You see, that's the cute thing about anomalies: they are RG invariant. So you can indeed compute them in a purely local way by looking at the deep UV, or in a purely topological way by looking at the deep IR. You get the same answer, by the standard 't Hooft argument. Therefore, while it is true that $c$ follows from local physics, it is wrong to say that it is insensitive to topology. It is sensitive to both local physics and topology, precisely because it is an anomaly (i.e., scale independent). $\endgroup$ Dec 1 '19 at 14:59
  • $\begingroup$ @Wakabaloola That being said, the fact that ST is consistent on all Riemann surfaces requires it to be consistent on a torus (by surgery, i.e., you build your surface by cutting and pasting tori, etc.). This is the reason we typically find strong consistency conditions by looking at the one-loop calculation, which (presumably) ensure consistency at all orders (even if the details are not 100% understood to the best of my knowledge). $\endgroup$ Dec 1 '19 at 15:01
  • $\begingroup$ @Wakabaloola Around each handle, every Riemann surface looks like a torus. The mapping class group $Sp_{2g}\mathbb Z$ has distinguished subgroups $SL_2\mathbb Z$, generated by the corresponding Dehn twists. So generic modular invariance requires in particular $SL_2\mathbb Z$ invariance. Does this make any sense to you? This is far from my area of expertise so I might be messing things up somewhat. (Hopefully I am not completely off!) $\endgroup$ Dec 1 '19 at 15:04
  • $\begingroup$ as always your comments are interesting and insightful. (And i agree that proving modular invariance of the one-point one-loop amplitude supplemented by OPE associativity arguments is expected to be sufficient to ensure modular invariance at arbitrary loop order.) But how would you compute the conformal anomaly in a purely topological way? $\endgroup$ Dec 1 '19 at 16:34

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