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From wikipedia:

"In order for a string theory to be consistent, the worldsheet theory must be conformally invariant. The obstruction to conformal symmetry is known as the Weyl anomaly and is proportional to the central charge of the worldsheet theory. In order to preserve conformal symmetry the Weyl anomaly, and thus the central charge, must vanish." ... "In string theory, conformal symmetry on the worldsheet is a local Weyl symmetry and the anomaly must therefore cancel if the theory is to be consistent." ... "The required cancellation implies that the spacetime dimensionality must be equal to the critical dimension which is either 26 in the case of bosonic string theory or 10 in the case of superstring theory."

Individually, I have a rudimentary understanding the highlighted concepts. I've truly studied every Leonard Susskind theoretical physics lecture on youtube. Yet, I can not seem to find out anywhere just exactly how and why a Weyl anomaly links up to the concept of central charge, or how central charge is analogous to critical dimensionality, for that matter. Even Susskind never discusses it.

I think if someone could give me even a non-rigorous handle on this, I might finally feel satisfied why ten dimensions in superstring theory is so important. Note: Susskind did explain why 26 dimensions in Bosonic String Theory was critical, btw. https://youtu.be/-I7PjKyCnI0

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  • $\begingroup$ Try reading a string theory textbook like the book by Blumenhagen, Lüst and Theisen. The relevant concepts are explained there. The critical dimension and central charge are linked essentially because (i) the central charge is a sum of central charges due to the fields on the world-sheet $X,\psi$ (ii) the number of bosonic fields $X$ is interpreted as the spacetime dimension. This means that the spacetime dimension and central charge are intimately related. $\endgroup$ – Danu Jan 11 '18 at 23:18
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Let take an closed superstring propagating freely in a flat space-time as an example. There is a gauge, called conformal gauge, where the action of the string is a two-dimensional conformal field theory (CFT) in a cylinder:

$$ S=\frac{1}{4\pi}\int d^2z\left(\frac{2}{\alpha'}\partial X^{m}\bar{\partial}X_m+\psi^{m}\bar{\partial}\psi_m+\tilde{\psi}^{m}\partial\tilde{\psi}_m \right) $$ where the energy-momentum tensor: $$ T(z)=-\frac{1}{\alpha'}\partial X_{m} \partial X^{m}-\frac{1}{2}\psi^{m}\partial\psi_{m} $$ should be imposed to satisfy $T(z)=0$ as a constraint. An additional constraint should be imposed as well: $$ F(z)=i\left(\frac{2}{\alpha'}\right)^{1/2}\psi^{m}\partial X_m =0 $$ but is irrelevant here.

Those are, classically, first-class constraint and can be dealt by the BRST-quantization in the absence of anomalies.

A central charge is a property of CFTs, a number $c$ that measure how the energy-momentum tensor $T(z)$ of the theory deviates from a tensor law under conformal transformations $\delta z=v(z)$:

$$ \delta T(z)=\left[-\frac{c}{12}\partial^3v(z)\right]_{non-tensor\,law}+\left[-2\partial v(z) T(z)-v(z)\partial T(z)\right]_{tensor\,law} $$

After quantization, we can only recover the closed string action from this CFT if $c=15$. If $c\neq 15$ the quantization of this CFT will produce an extra degree of freedom if we try to recover the closed string action, and this degree of freedom breaks Lorentz symmetry. You can see more about here. This happens because in quantum mechanics, symmetries may be incompatible with one another, a phenomena called anomaly. The Lorentz symmetry of the space-time is incompatible with gauge symmetries of the string. All this anomalies cancel for $c=15$. (You need to do the math to know why $15$).

Ok, so each $X_m$ contribute with $+1$ for $c$ and $\psi_m$ contributes $+1/2$. For a space-time with $D$ dimensions, $m=1,2,...,D$, then $c=3D/2$. Imposing $c=15$ implies $D=10$. This means that the gauge symmetries of the string is compatible with Lorentz symmetry just in $10$-dimensional space-time. You can give up about the gauge symmetries or the Lorentz symmetry obtaining a different theory. For giving up the Lorentz symmetry you end up with a non-critical string theory. For giving up the gauge symmetry you end up with a theory that we don't know how to construct consistent interactions (plus, is a theory without massless particles in the low-energy regime).

We have a summary:

  1. to quantize string theory, preserving all the symmetries, the central charge must be $c=15$ for superstrings and $c=26$ for bosonic strings.
  2. Each dimensions contribute $+3/2$ for the superstring and $+1$ for the bosonic string.
  3. Then, superstring must live in $D=10$ space-time and bosonic strings must live in $D=26$ space-time.
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