To summarize the complete argument here. Let's pretend Baryon number is an exact symmetry. A neutron star has 10^40-something baryons. It collapses into a black hole which then evaporates into photons and gravitons. During the semi-classical phase, when the black hole is enormous, the emissions cannot be of massive particles, only massless particles can come out, because the mass-scale for emission is the inverse black-hole radius in natural units, which is tiny. At the end of evaporation, when this mass scale is large enough for massive particles to come out, the black hole is very small, and doesn't have enough mass to emit 10^48 baryons. Therefore baryon number is violated in black hole emissions.
This argument shows that baryon number is not exact for sure, and similarly for any global symmetry where all charged particles are massive. It's a complete physical argument, and it can't be sidestepped.
But it leaves a loophole when there is a massless globally charged particle. In this case, you need to think a little more about the details of black hole decay. The decay of black holes is thermal, into photons and gravitons, and any massless mode equally. You can consider forming a large black hole of large mass and global charge from the collision of a large number of such massless particles converging from opposite directions, and this black hole will decay into an equal proportion of these particles and gravitons, which you should always be able to arrange to violate the global symmetry.
The reason it seems this way is that the black hole emissions are thermal, and are produced at thermal equilibrium at the Hawking temperature, regardless of global charge, just by counting field mode degrees of freedom. Choosing a bunch of particles of some global charge, these are not in global equilibrium, because the state is specially chosen to have large value of the charge, and the process of infalling into the black hole removes the charge without a trace. This argument is not airtight, unlike the case with massive globally charged particles.
Black holes preserve gauge symmetries, because these produce a field outside the black hole which keeps track of the total charge inside. But in order to do so, this places a bound on the mass of the lightest charged particle, so that the black hole can properly decay. The bound is that the lightest charged particle must be lighter than it's mass in natural units, in other words, that gravitational attraction is necessarily weaker than the gauge-repulsion.
This means that black holes also can conserve discrete global symmetries, like $Z_n$ symmetries, as these can be the residuals gauge symmetries at low-energies after breaking. When $n$ is not enormously large, these are symmetries that only restrict the very end of the black hole evaporation, and are not inconsistent with the arguments above. When $n$ becomes large, this becomes more and more like a global symmetry, but the same argument gives a similar inequality on the mass of the $Z_n$ charged particles as $n$ gets larger. This suggests that perhaps you can find a situation where there is a global symmetry or near-global with massless or near-massless charged particles, but then it is not about the real world, where all possibly globally charged particles in the standard model are massive (like the baryon).
The inequalities from complete black-hole decay are the most restrictive swampland constraints we know about, and are the most predictive statements string theory can make in general today.
The string theory arguments, in AdS/CFT and worldsheet string theory, seem superficially different from the Hawking semiclassical black-hole argument, but once you understand the holographic principle, they are simply the exact same thing stated mathematically precisely. The string itself is a kind of quantum-sized black hole, and the vertex operators for absorption/emission are representing the amplitude of a particular history of infalling stuff and then Hawking emission, the fact that these are described by simple motion modes of the string is required by the holographic principle.
So the statement that this reproduces the swampland constraints from semiclassical black hole decay is nonperturbative completely persuasive evidence that string theory is consistent as a theory of gravity. The consistency of the two pictures is the content of the 1990s string revolution, which arrived at the string dualities by considerations of BPS states, which are, in string theory, classical extremal black holes, with gauge-charge equal to mass. This is the only thing that was strong enough to persuaded me, personally, that string theory is a consistent theory of quantum gravity, as the constraint of being consistent with semiclassical black-hole decay pretty much uniquely fixes the properties of the mathematical description which string theory has.
Q&A (4869)
