It isn’t widely appreciated, but in the standard model of particle physics coupled to gravity there is actually only one global symmetry: the one described by the conservation of B-L (baryon number minus lepton number). So this is the only known symmetry we are actually saying must be violated!
What Harlow doesn’t mention is that this is a result about AdS [ed. anti-deSitter] gravity, and we live in dS [ed. deSitter], not AdS space, so it doesn’t apply to our world at all. Even if it did apply, and thus would have the single application of telling us B-L is violated, it says nothing about how B-L is violated or what the scale of B-L violation is, so would be pretty much meaningless.
Via Not Even Wrong (this is really a double quotation, the portion in italics is from physicist Daniel Harlow, the remainder is from Peter Woit, the author of the blog).
The terms anti-deSitter and deSitter refer to the topography of the curved spacetime of the universe. The best available experimental evidence suggests that our universe's spacetime has a deSitter topology, although it is very close to flat at a global level. What does this mean? It means that the universe has a positive, sphere-like deSitter curvature, rather than negative, hyperbolic surface-like anti-deSitter curvature. As Wikipedia explains:
In mathematics and physics, n-dimensional anti-de Sitter space (AdSn) is a maximally symmetric Lorentzian manifold with constant negative scalar curvature. Anti-de Sitter space and de Sitter space are named after Willem de Sitter (1872–1934), professor of astronomy at Leiden University and director of the Leiden Observatory. Willem de Sitter and Albert Einstein worked together closely in Leiden in the 1920s on the spacetime structure of the universe.
Manifolds of constant curvature are most familiar in the case of two dimensions, where the surface of a sphere is a surface of constant positive curvature, a flat (Euclidean) plane is a surface of constant zero curvature, and a hyperbolic plane is a surface of constant negative curvature.
Einstein's general theory of relativity places space and time on equal footing, so that one considers the geometry of a unified spacetime instead of considering space and time separately. The cases of spacetime of constant curvature are de Sitter space (positive), Minkowski space (zero), and anti-de Sitter space (negative). As such, they are exact solutions of Einstein's field equations for an empty universe with a positive, zero, or negative cosmological constant, respectively.
A spacetime with an anti-deSitter topography is interesting theoretically because of a correspondence between AdS spaces and conformal field theories that makes it possible to do all sorts of theoretically interesting tricks. As Wikipedia explains:
[T]he anti-de Sitter/conformal field theory correspondence, sometimes called Maldacena duality or gauge/gravity duality, is a conjectured relationship between two kinds of physical theories. On one side are anti-de Sitter spaces (AdS) which are used in theories of quantum gravity, formulated in terms of string theory or M-theory. On the other side of the correspondence are conformal field theories (CFT) which are quantum field theories, including theories similar to the Yang–Mills theories that describe elementary particles.
The duality represents a major advance in our understanding of string theory and quantum gravity. This is because it provides a non-perturbative formulation of string theory with certain boundary conditions and because it is the most successful realization of the holographic principle, an idea in quantum gravity originally proposed by Gerard 't Hooft and promoted by Leonard Susskind.
Empirically there is no experimental evidence of B-L violation, which is preserved in the Standard Model of Particle Physics, and B-L violation is not obviously violated in any viable candidate for a real world gravitational theory standing alone, that I have seen.
B and L are separately conserved in the Standard Model except in very high energy sphaleron interactions (which can't be depicted in Feynman diagrams because they are space-like without a time-like element) that have never been observed in reality, but even those interactions conserve B-L.
The recognition that most or all of string theory vacua exist in anti-deSitter space that does not correspond to reality, that are collectively called the "swampland" is one of the main global and generic challenges to string theory as a viable description of reality in physics today.
Lots of cosmologists deeply want B-L violation to exist, even though it is known that it doesn't happen in any detectable amounts at energies up to the 14 TeV of the Large Hadron Collider (LHC), because this is the easiest way to produce our existing universe, in which matter predominates over anti-matter, with a starting point that consists of pure energy rather than a finite and fixed aggregate baryon number and aggregate lepton number for the universe. (Baryon number is the quarks minus anti-quarks divided by three. Lepton number is leptons minus anti-leptons.)
It is also worth sharing one of my conjectures from my linked swampland post here to provide some out of the box context:
For what it is worth, it is also possible that gravity and the cosmological constant observed are not actually topological effects as in General Relativity, but instead closely approximate a mechanism that involves the behavior of gravitons in a Minkowski spacetime that is itself fundamentally flat, rather than being deSitter or anti-deSitter, even though that graviton behavior is similar to and in most circumstances almost exactly equivalent to, a topologically curved spacetime.
This conjecture is just me on physics and doesn't come from any authoritative source, so take it for what it is worth, which is not very much. But, it it were correct, it would eliminate one of the biggest conceptual and mathematical barriers to integrating the Standard Model, which is formulated in Minkowski space (which observes special relativity but not general relativity), but not in a spacetime with a curved topology.
Also, even if spacetime is actually slightly deSitter in real life as General Relativity supposes, a quantum gravity theory formulated in Minkowski space might be close enough to reality to be a good approximation of reality, especially in the weak gravitational field domain of applicability for distances less than or equal to the order magnitude of the immediate vicinity of galaxy clusters (which is all that matters except in the immediate vicinity of black holes, neutron stars and similarly massive object, at scales smaller than cosmological ones, unless one is extremely precise). And, in these strong fields and for universal scale distances, we know experimentally that classical General Relativity is a very good approximation of reality, so resort to quantum gravity in lieu of classical General Relativity isn't usually necessary except in the immediate first moments after the Big Bang.
In the same vein, even though we have very good reason to think that a graviton, if it exists, is a spin-2 particle known as a "tensor" particle, there is also good reason to think that a scalar (i.e. spin-0) graviton approximation is a very good approximation of a spin-2 graviton over a wide domain of applicability, particularly in weak fields. This is because in most contexts, the contribution of the mass-energy of particles to the stress-energy tensor of General Relativity far exceeds that of the other aspects of the stress-energy tensor such as linear and angular momentum, electromagnetic flux, sheer stress, torsion and pressure. A scalar graviton theory is basically a static equilibrium approximation of a full spin-2 graviton theory, which is fine when a system is very far from moving at relativistic speeds (i.e. speeds close to the speed of light) and is in a state reasonably close to equilibrium (such as a fairly stable solar system or galaxy).
One reason to doubt the purely geometric interpretation of gravity from General Relativity is that MOND's external field effect (EFE) violates the "strong equivalence principle" of General Relativity (i.e. that the outcome of any local experiment (gravitational or not) in a freely falling laboratory is independent of the velocity of the laboratory and its location in spacetime) and there are several recent astronomy observations which suggest that the EFE is a real phenomena.
This matters because there are lot of very difficult technical mathematical challenges to formulating gravity as a theory of a spin-2 graviton in a curved spacetime that are drastically simplified if the approximation of a spin-0 graviton in Minkowski space, that would make it much more feasible to do calculations with the theory. So, if this approximation is a very good one in a wide domain of applicability involving weak gravitational fields, we could have a very workable quantum gravity theory for a very wide range of applications, even though there would be some extreme circumstances in which it would be flawed and the model's analytical and qualitative predictions could not be trusted.
Indeed, in practice, it is common place in many astronomy applications involving scales of galaxy cluster distances of less that involve weak gravitational fields to approximate General Relativity with Newtonian gravity in any case. But, using a spin-0 graviton in Minkowski space approximation is still much more precise, and captures far more non-Newtonian quantum gravity effects, than using a Newtonian gravity approximation does. Since a quantum gravity theory with a spin-0 graviton in Minkowski space would still involve a self-interacting carrier boson, just like QCD does, even this greatly oversimplified approximation of quantum gravity still involves very challenging mathematics that only specialist physics and mathematics PhDs would be capable of mastering and applying confidently and accurately.
A good analogy would be the simplifications that can arise when considering electromagnetic phenomena without considering polarization. It isn't perfect and ignores some important experimentally testable phenomena, but it is also much easier to understand, makes the mathematics of applying it a lot simpler, and can be safely neglected without significant loss of accuracy in many applications.
UPDATE July 19, 2019: Sabine Hossenfelder has some cogent discussion of why theories related to anti-deSitter space may provide little insight into the real world. An excerpt:
[D]iscontinuous limits should make you skeptical about any supposed insights gained into quantum gravity by using calculations in Anti de Sitter space.Anti De Sitter (AdS) space, to remind you, is a space with a negative cosmological constant. It is popular among string theorists because they know how to make calculations in this space. Trouble is, the cosmological constant in our universe is positive. And there is no reason to think the limit of taking the cosmological constant from negative values to positive values is continuous. Indeed, it almost certainly is not because the very reason that string theorists prefer calculations in AdS is that this space provides additional structure that exists for any negative value of the cosmological constant, and suddenly vanishes if the value is zero.
String theorists usually justify working with a negative cosmological constant by arguing it can teach us something about quantum gravity in general. That may be so or it may not be so.