 Okay, thank you. Many thanks to the organizers for this opportunity. This is based on this paper that we wrote with Rachel at Columbia. Okay, so the subject of my talk is partially mass risk gravity, which I'm assuming not everyone here or most people here are not familiar with. So I'm planning to spend a good fraction of my talk simply introducing the subject, okay? Okay, so my starting point is a massive spin-to-fill on a constant curvature background. So imagine splitting the metric into a constant curvature background, which is either the seater or anti-the seater plus a fluctuation, which is our graviton, okay? Okay, so as we learn on Monday, the free theory is described by the third spowly action. The only new thing here is that instead of partial derivatives, we have covariant derivatives defined with respect to our background geometry. And we also have a curvature term that comes from linearizing the cosmological constant term in the Einstein-Hilbert action, okay? So this R bar is a constant. It's just the richest scalar of our background space, okay? And let me remind you that we also have a mass term now for a massive graviton, which is given by this expression here, with m being the mass of the graviton. Okay, so we also learned about this Stuckelberg method, which is a nice way to study the degrees of freedom of our theory and also their stability. In particular, for gravity, we take our tensor field and replace it with this expression. So we introduce a new vector field and also a new scalar field. And the only new thing here is that we should use covariant derivatives rather than partial derivatives because our background is now curved, okay? So Rachel showed that there exists this relativistic limit which formally amounts to taking the mass of the graviton to zero. And in this limit, these three fields become the independent helicities of our massive graviton. So in particular, the tensor field becomes the helicities plus and minus two. The vector field becomes the helicities plus and minus one. And the scalar is just the longitudinal mode, the helicity zero, okay? So for generic values of the rigid scalar of the background and the graviton mass, these helicities are all physical, okay? They are all propagating degrees of freedom. And therefore, if you do the math, you find that the theory has five degrees of freedom as you would expect for a massive graviton, okay? And the theory is massive gravity. Okay, so this is, let me emphasize, this is for generic values of curvature and mass of the graviton. So what about the stability? So this is like a plot in the theory space of massive gravity, so to speak. So the horizontal axis is the mass of the graviton and the vertical axis is the curvature of the background. So when you are in the upper half plane, this is the seater with positive curvature and in the lower half plane is anti-the seater with negative curvature. So it can be shown that the helicity plus and minus one mode of the graviton is stable provided that the mass of the graviton squared is positive. So that rules out this whole region on the left, okay? So yeah, so in this region, the helicity one is unstable. It's a ghost and therefore you can rule out that region. The helicity zero is stable provided that the mass squared of the graviton satisfies this inequality. This is called the Higuchi bound and therefore on this plot, this region here in blue is ruled out. So in this region, the helicity zero is unstable. It's a ghost, okay? So that gives you this wide region here as the space of theories which are stable, okay? So note that these bounds are strict. So this is strictly greater, strictly greater, but you may ask what happens when we saturate the bounds and it turns out interesting things happen because we get gauge symmetries when we hit the bounds. So in particular, when this is well known, when the mass of the graviton is zero, you know what you get, you get GR. That's the theory of a mass less graviton which has this diffeomorphism symmetry, linearized diffeomorphism symmetry because so far I'm dealing only with free theories. And this theory has two degrees of freedom rather than five. So when you are on this line here, M equals zero line, you're five degrees of freedom. Some of them become unphysical, they become pure gauge and you end up with two physical degrees of freedom corresponding to a mass less graviton. Perhaps more interesting is when you saturate the Higuchi bound when M squared equals one sixth of the curvature radius. In this case, this corresponds to this line here which node is in the sitter half of our theory space and this corresponds to this partially mass less theory of gravity and this has this curious gauge symmetry. It's a scalar gauge symmetry because the gauge parameter in this case is a scalar and it involves two derivatives in this case, okay? So it's kind of curious in the sense that you have a massive field with a gauge symmetry living in the sitter, okay? This is partially mass less gravity. Okay, so in a nutshell, the main properties of this theory, as I just said, the graviton has a mass that is given by this value. It's a fraction, a specific fraction of the curvature of the Ritchie scalar of the background which can also be written in terms of the cosmological constant like so. The theory has a scalar gauge symmetry, as I just said, which has two derivatives and this gauge symmetry removes the Helicity zero mode of the graviton. So this is a spin two field that has four degrees of freedom, has the Helicity plus minus one and plus minus two, but not the longitudinal mode, okay? So those were the main basic aspects of this theory, but there are also some interesting phenomenological consequences. The first one, probably the most interesting one is the cosmological constant problem. So I just told you that the cosmological constant is related to the graviton mass by this equation. So they are related by order one numerical coefficient and therefore having a small cosmological constant is equivalent to having a small graviton mass. But this is great because having a small graviton mass is technically natural. The reason being that when the mass of the graviton is exactly zero, you know that you recover the diffeomorphism symmetry of GR. So it's kind of when the mass of the graviton is zero, you have more symmetry so to speak and that implies that a small m square is technically natural. It does not receive large quantum corrections. So if you start with a small mass, then you know that that's going to be under control under quantum corrections. That's what we mean by technically natural, okay? A second interesting consequence is the fifth force problem. Rachel talked about this on Monday. This is the BDBZ discontinuity. And so as you learn, in massive gravity, the helicity is zero mode. It remains coupled at short distances. It remains coupled to matter at short distances. So the Lagrangian looks roughly like this. This is what you would expect to see at short distances. We would expect to have GR at short distances. However, we have also this coupling between the longitudinal mode and the trace of the stress energy tensor. And this gives rise to a fifth force, which we don't observe, of course. As we learn also, this is solving nonlinear massive gravity by the Weinstein mechanism, which consists of the sort of killing of these longitudinal mode at short distances by means of the nonlinearities of this field. So around a source, say a star of mass m, the profile of the field looks roughly like this. It goes like one over r at large distances, but at short distances, it goes to zero as some power. So this renders the field and the corresponding force unobservable. So the Weinstein mechanism is great because it solves this fifth force problem. However, they're surprised to pay, which is generically perturbations in this nonlinear region are superluminal. This is not really a theorem, but it's kind of a very generic property of this Weinstein mechanism. But the good thing of portioning massless gravity is that we never have to worry about this because we just don't have this longitudinal mode. So that's great. We never have to really worry about this issue of having superluminality in the nonlinear regime. Okay. Let me skip the details here. There are also some nice analogies with electromagnetism. So for example, you can write the partially massless action like in Maxwell-like form, F squared. So if you define this tensor F as the anti-symmetric, the derivative of H, then the action can be written as F squared, roughly speaking. And this F is an invariant field strength, just like the F mu nu of electromagnetism. It is invariant under the gauge symmetry of partially massless. And there's also a duality invariance. If you go to Hamiltonian formalism, you can define as sort of EMB fields in terms of which the action has these duality invariance pretty much like in EMM. Okay. So, so far I've been telling you about the free theory. So the action that I wrote down was just the quadratic action. But the question, the obvious question is whether there is a nonlinear theory of partially massless gravity. That is, can we add to our quadratic action that we know? Can we add higher order interactions to this field? Well, this is a gauge symmetry. So we should also expect to have nonlinear corrections to our gauge symmetry. Like it happens in young males, for example. So this delta zero H is the gauge symmetry that we know. The one that corresponds to the quadratic action. But we in principle should expect to have higher order corrections to this symmetry. Okay. So know that this lowest order symmetry does not involve any powers of H on the right hand side. But in principle, we could have a gauge symmetry that involves the field itself, like in young males, for example, or GR. Okay. So unfortunately, so far we only have negative results. So first it was shown that this partially, this nonlinear theory of partially massless gravity cannot belong to the DRGT class of ghost free massive gravity. That's one thing. More generally, even if you don't worry about ghosts, it was shown that you can make a very generic, very general answer. So assume that the kinetic Lagrangian is a Hilbert and add just any potential that you want. And it was shown that a theory of this class cannot have this partially massless symmetry at nonlinear order. And even more generally, it was shown that even if you relax the assumption about the kinetic terms, so the only constraint being now that the action has two derivatives, it was shown that partially massless gravity cannot have this form at the nonlinear order. Okay. So these are very general no-go results if you think about it because the only assumption here is that the action has at most two derivatives. So imagine writing anything you want with up to two derivatives. The claim is that that's not going to have the partially massless symmetry at nonlinear order. Okay. So the method here, the method we used to prove this is based on this closure condition on gauge symmetries, which is kind of a nice method because it doesn't make any assumption about the form of the action. That's why it's so general in a sense. It only relies on the form of the symmetry. So just to explain the method quickly, let me do the spin one example. So the starting point here is the Maxwell action, which you know has this usual gauge symmetry. And we look for nonlinear extensions of this symmetry. So we write generically, we can write a nonlinear gauge symmetry in this way where this beta and alpha are the functions that we want to determine. So beta is some arbitrary function of powers of the field and alpha likewise involves one derivative and powers of the field, like so. And our goal is to determine this symmetry. Okay. And here the method is based on this closure condition that the commutator of two gauge transformations has to be itself a gauge transformation. That's kind of an integrability condition for a gauge symmetry. And it follows from Frobeni's theorem. Okay. So the goal is to solve this equation for our unknown functions, alpha and beta. And skipping all the algebra, the result turns out to be just the trivial one. Beta has to be just the Kronecker delta and alpha has to be zero. And as a result, the most general gauge symmetry for a single spin one field turns out to be just the usual Maxwell symmetry, this one. And as a consequence, this tells you that the most general low energy action that you can write down for a spin one field is just a Maxwell action. There's nothing else you can write down. Okay. And this all follows from this closure condition on the gauge symmetry. You can play the same game with multiple spin one fields. Let me skip the details here. It turns out that you can derive young mills using this method. Right, so this is the result. If you apply the closure condition to multiple spin ones, you find the young mills gauge symmetry. And therefore, as a result, you get that at low energies, the action has to be the young mills theory. A third example is the spin two, a mass of the spin two in this case, which has linearized the filmorphisms as a symmetry. And playing the same game, you turn the crank, turns out that you can derive GR. So in this case, you find that the most general gauge symmetry for a mass of spin two field turns out to be non-linear diffeomorphisms. So you can derive GR using this method, starting from the free action of a spin two field. And this is kind of nice because in our starting point, in the linearized action, there's no reference to manifolds and geometry. So it's kind of similar to the desert construction that Rachel described on Monday, in which this geometric picture of GR sort of emerges as a consequence, rather than as an assumption, okay? So this is the result for a mass of spin two case. And again, at low energies, this symmetry implies that the action has to be Einstein-Hilbert. So you can derive Maxwell theory, you can derive young mills, you can derive GR using this method of the closure condition. Okay, so just to finish here, what we did with Rachel was to apply this closure condition to partially massless. So our starting point is the quadratic partially massless action, which has this lowest order symmetry. And in this case, the general form of the non-linear symmetry now involves three unknown functions because it has two derivatives. So it's a bit more complicated. But nevertheless, you turn the crank and you can show that the general result turns out to be this for these unknown functions B, D and C. And let me remind you this F tensor is the Maxwell-like tensor for partially massless, this anti-symmetric derivative. So this is the punchline. The most general gauge symmetry for a partially massless spin two turns out to be this. This is the lowest order one corresponding to the quadratic action. And we found this unique non-linear extension that involves this tensor F. So this is quadratic in the fields. And let me point out this, just to remind you this phi is the gauge function. It's a scalar. And this gamma is just an arbitrary constant. Okay, so this was the symmetry. What about the action? So I've been selling you this method because you don't make any assumptions about the form of the action. So it's kind of super general. You only rely on the symmetries and you don't make any assumptions about the action. But at the end of the day, you obviously want the action, right? So to find the action, what you can use is the net identity. So let me remind you that if you have a gauge symmetry, there exists a corresponding net identity, which in GR, this is just the Bianca identity. So if you have the filmorphisms, you know that you can derive the Bianca identity like this. And in general, net identities are always of the form differential operator hitting the equation of motion equals zero. This is always the case. For partially massless, the net identity gives you this mess here. Again, this F is this anti-symmetric combination and E3 is the cubic part of the equation of motion. This is what we are after. Remember, we are trying to find nonlinear extensions for partially massless gravity. We have the quadratic equation of motion that's known and we're trying to find nonlinear interactions. So this is what we are after. And the result turns out to be there's no solution. If you stick to actions with two derivatives, it turns out that there's no solution for the equation of motion at cubic order. So that's our no-go result. Yeah, bummer. Okay, just to wrap up in one minute. So I told you about partially massless gravity. This is a theory of a massive graviton that lives on the cedar and enjoys a gauge symmetry, a scalar gauge symmetry. I told you about the free theory because that's the only thing that is known so far. The obvious question is whether there is an interacting theory and I hope I convinced you that this is a well-motivated question. But we have these very general no-go results that any nonlinear extensions must necessarily have more than two derivatives. So some future prospects. Well, one obvious one that follows from this is that whether there could be higher order derivatives interactions. Second, you could consider coupling two other fields. So far, I only consider a single partially massless field. But you can consider, you can imagine coupling this field to say a massless graviton or two other partially massless fields and so on. And finally, I didn't talk about this, but it turns out that these partially massless fields exist for any higher spin, for any spin greater or equal than two. So you can imagine playing this same game for higher spin fields and see whether you can get interactions for these fields. That's it, thank you.