Jorge Pullin: All right. So, Speaker, today is Hano Salman, who will speak about Karl's Super. Lqg.
Hanno Sahlmann: Thank you very much. So yeah, Thank you very much for the invitation and opportunity to speak here. That's great.
Hanno Sahlmann: I'm going to talk about Chiral Super loop quantum gravity. So I will have to tell you a bit about supersymmetry bit about super gravity, and then about loop
Hanno Sahlmann: quantum super gravity, and I hope by the end of the talk I can convince you that we can do interesting things, even in the quantum theory, although we don't
Hanno Sahlmann: by no means have all the answers to to all the things that there's no complete theory, but that will become more clear in a little bit. So I should say that this work is actually mostly due to Constantine, who
Hanno Sahlmann: it's not here right now, but I believe he will join us later. He has some some other urgent appointment right now.
Hanno Sahlmann: Some of it is in collaboration with me Some of it is I
Hanno Sahlmann: him alone, and the
Hanno Sahlmann: potential mistakes on the slides are mine altogether.
Hanno Sahlmann: So
Hanno Sahlmann: Supergravity, super symmetry. What
Hanno Sahlmann: is behind this? So already, Einstein. When he looked at his marvelous field equations he started thinking. And I think, he said at some point, that so, if the geometric side of the equations the left hand side is like a beautiful marble building.
Hanno Sahlmann: and the right hand side seems not as beautiful. It seems more like a wood building, and No, they are joined together in this in this way, and, in fact, he spent quite some time in energy searching for some more unified description of the
Hanno Sahlmann: gravitational fields and the rest of the physical fields.
Hanno Sahlmann: So space-time, geometry and the rest.
Hanno Sahlmann: and that's
Hanno Sahlmann: one dichotomy that is to be breach it to to be bridged. Yeah.
Hanno Sahlmann: And one can see a similar phenomenon in quantum field theory, where, under some reasonable assumptions on the kind of quantum field theory, Coleman and mandola showed that
Hanno Sahlmann: if you look at the global symmetries of this theory, then they also have to be split. So they are the spacetime symmetries that are certainly
Hanno Sahlmann: can be symmetries of the theory, and then there can be some more, but they don't mix as algebra or groups they are commuting with each other.
Hanno Sahlmann: and
Hanno Sahlmann: now
Hanno Sahlmann: that is intriguing, and one maybe wants to go beyond it, and several ideas have been put forward for some sort of unification, and one is certainly Colludes, a client type of ideas where you go to a high dimensional geometric
Hanno Sahlmann: theory, and then you and reinterpret in your dimension some of the fields, as sort of matter feels.
Hanno Sahlmann: and then there's also supersymmetry. And today we're going to talk about super symmetry, of course.
Hanno Sahlmann: Now.
Hanno Sahlmann: what is what is the
Hanno Sahlmann: what is super symmetry. Well, one answer was again
Hanno Sahlmann: given in the context of quantum field theory, Hack and Kuprushanski. Notice that there is a loophole to the Coleman Mandela theorem.
Hanno Sahlmann: and that is, if you don't, just let your symmetries be
Hanno Sahlmann: generated by the algebra. But you allow more general
Hanno Sahlmann: algebraic objects that also have anti commutators. Then, indeed, a mix between space-time symmetries and the other symmetries is possible.
Hanno Sahlmann: And so they went on to classify extensions of the the concrete algebra. In this way, and that's known as the Hag Luboshanski theorem.
Hanno Sahlmann: and at the heart of it what they find is algebraic objects that they find that are compatible with the other rules of quantum theory are super lea algebra.
Hanno Sahlmann: So what is this? This is the algebra is a vector space, and for a super league algebra. This has 2 parts or 2 grades odd and even, and the commutator has to play well with this grading of the vector space.
Hanno Sahlmann: And, in fact, the commutator is not always the product in this algebra. It's not always antisymmetric. It's antisymmetric, for between even elements and between an even and an odd element but
Hanno Sahlmann: for 2 odd elements it is, it is symmetric, so it is a graded product.
Hanno Sahlmann: and corresponding to this. There's also a graded Jacobi identity as a sort of replacement of.
Hanno Sahlmann: And so this sounds pretty
Hanno Sahlmann: complicated, but I wanted to give at least one example that probably everybody has already seen, because we are all working with differential geometry sometimes. And so I want to look at as some the the forms over the tangent space of some manifold.
Hanno Sahlmann: and we know some operations on these forms. Now we know that we can take the derivatives. We can take the
Hanno Sahlmann: exterior, derivative and we can take these interior products just plugging a tangent vector into a form.
Hanno Sahlmann: and there are these identities, for example, a. D squared sival.
Hanno Sahlmann: and also plugging the same vector twice into a form. Also give 0. And so there you already see these things. One would call them super
Hanno Sahlmann: commutative because they Anti.
Hanno Sahlmann: Yeah. Yeah.
Hanno Sahlmann: yeah, they they enter a commute with with themselves, so to speak, and the vector field certainly act as a Ne algebra on on the forms in particular. So that's where the name
Hanno Sahlmann: the algebra comes from, and so
Hanno Sahlmann: you can check the other properties that I've given you. And
Hanno Sahlmann: this object, indeed is a super. These operators formula from a big super algebra. And so Cartons
Hanno Sahlmann: magic equation. Yes, also this context just a a anti commutator.
Sure.
Hanno Sahlmann: So maybe these things are not so strange after all. So let's go on and look at
Hanno Sahlmann: super extensions of the spacetime symmetry. So a minimal extension of the Poncari algebra.
Hanno Sahlmann: and what you will turns out to do. Boshanski, tell us you have to add
Hanno Sahlmann: for
generators
Hanno Sahlmann: to the algebra, and these 4 generators are odd.
Hanno Sahlmann: and
Hanno Sahlmann: these generators have to fulfill certain very specific relations with the rest of the generators.
Hanno Sahlmann: And
Hanno Sahlmann: well, first of all, I show you that
Hanno Sahlmann: it was non-zero
Hanno Sahlmann: commutators, anti commutators show that indeed, we have a mixing of
Hanno Sahlmann: space, time, symmetries, and these new generators they don't commute with each other; and, secondly, from this thing here.
Hanno Sahlmann: this gamma ij. It's just an anti symmetric product anti symmetry, a product of 2 gamma matrices. So you recognize this is a a a a generator of a 4 dimensional spinorial representation of the lowering school.
Hanno Sahlmann: So this equation just says that these queue transform as a as a for Spino under Lawrence transformations.
Hanno Sahlmann: And now
Hanno Sahlmann: this might be a little abstract at first, but it will be very important later in a moment. In the talk you can play the same game for the isometries of anti to sitter. Space
Hanno Sahlmann: is this.
Hanno Sahlmann: This is this ads group, and also here you can add for odd generators, and the commutation relations are then fixed by the rules of the game.
Hanno Sahlmann: and you'll see that they have
Hanno Sahlmann: very similar. There are some additional terms, and they go with the link scale that comes into the picture from the
Hanno Sahlmann: curvature or the radius of this ads space.
ashtekar: So is it so? Is the reason why you do not have
ashtekar: the sitter is because of this.
Hanno Sahlmann: Yeah, exactly. The the reason here the science matter, and for this greater Jacobi identity in particular, and you have to have the negative sign. It does not work with a positive, cosmological, constant.
Yes.
ashtekar: thank you.
Hanno Sahlmann: Thank you. Yeah, please interrupt always and ask
things.
Hanno Sahlmann: So I already said that these things mix space, time and the spot
Hanno Sahlmann: kind of things, so they cannot just be thought of as symmetries of space-time. They
Hanno Sahlmann: are properly thought of symmetries of some kind of super space, or a super manifold. And I say a tiny bit about this in a moment. So that's the kind of classical interpretation of them as some symmetries of some kind of strange space
Hanno Sahlmann: and in quantum field theory, because the cues are odd.
and because they carry a spin where
Hanno Sahlmann: they they espn orio, they transform bosons to fermions, and vice versa. They have to it doesn't.
Hanno Sahlmann: and they cannot but do this, so they are. It's a symmetry that mixes. Go on. And
Hanno Sahlmann: now I have to
Hanno Sahlmann: he even a bit more crazy. So
Hanno Sahlmann: what we have seen is the minimal version of extending the spacetime symmetries. You can do more.
Hanno Sahlmann: and the sort of the reli that the one thing you can do is you Don't, take 4 odd generators, but several sets of 4 odd generators.
Hanno Sahlmann: and that is called extended supersymmetry, and depending that you measure the extension by how many of these sets there are, and you call this this curly. And so what we saw before is n equal to one.
Hanno Sahlmann: And now one can think of higher end.
Hanno Sahlmann: And, by the way.
Hanno Sahlmann: many of the statements are very dimension dependent, and so on and so forth. So everything will be in 4 dimensions only.
Hanno Sahlmann: And what about signature? And supposing also? Yes, absolutely so. So, for example, to look forward to to this stuff that that we are going to do would be great to be able to do this in Euclidean signature. But
Hanno Sahlmann: but I I don't think the the right supersymmetry exists there. So so you, in fact, space I could not extend to to in it's not by Point Kari group. I cannot
ashtekar: use just the
ashtekar: Euclidean group, and
ashtekar: I'll have to.
Hanno Sahlmann: So
Hanno Sahlmann: So I I would have to think. But, for example, this statement is that in Euclidean signature you don't have Myorana fermions.
Hanno Sahlmann: and somehow these queues, at least, and then in the normal way, they are Mayorana, the the mayor, Runa fermions, and so you have you? You might have a problem already at that. At that level there are certainly other there.
Hanno Sahlmann: X. Some, some extended supersymmetries that are possible in Euclidean signature, and one can
Hanno Sahlmann: one can go look at the the books with. This has been. This has been worked out.
Hanno Sahlmann: but but it's not not straightforward, so it's not like one to one that you can always do the same in in both signatures.
ashtekar: Thank you.
Hanno Sahlmann: Yeah, I think. Thanks for the
Hanno Sahlmann: of the question.
Hanno Sahlmann: So.
Hanno Sahlmann: speaking of extending so we can extend the number of our generators, and then we can. Once we have those additional generators, we can add
Hanno Sahlmann: more
Hanno Sahlmann: bosonic generators, more even generators. I call them tea here.
Hanno Sahlmann: That however.
Hanno Sahlmann: act non-trivial, or commute non-tribly with the with the queues, so they have some commutator with the queues, and the 2 cues also give
Hanno Sahlmann: keys, and this is called our symmetry. For some reason I I don't know either, but in our case they could be thought of as so n generators. So there's an additional Bosonic
Hanno Sahlmann: League group in play now, and in terms of
Hanno Sahlmann: these allergy process.
Hanno Sahlmann: what we get. We go from one algebra to another. Algebra that is, is, is bigger. And by the way, these names come from
Hanno Sahlmann: the fact that these algebra can be realized as matrices that
Hanno Sahlmann: leave invariant some some inner product, that is, that is symmetric in some dimensions, and
Hanno Sahlmann: antisymmetric in some other dimensions, and that's why they are called ortho-symmetric.
Hanno Sahlmann: algebra, and orthos andplectic groups. But it's by no means obvious to see that that whatever this or the symptic group is equivalent to this
Hanno Sahlmann: supersymmetric extension of the ads. Who are that that somehow I I, at least for me this is something one can go through step by step. But there's not a not some
Hanno Sahlmann: easy way to see this
Hanno Sahlmann: good. So there is one last observation before we go on, and that is that we know in 4 dimensions we can we have the this phenomenon of
Hanno Sahlmann: chirality and self, Dual and anti-self Dual decomposition of the
Hanno Sahlmann: of the Lawrence Jen and Lauren School
Hanno Sahlmann: and on yeah, for the Lawrence school. We can form these complex combinations of goose and boost generators and rotation generators.
Hanno Sahlmann: and that splits the complexification of the of the Lawrence algebra into 2 commuting parts, the self fuel anti-soft your part.
Hanno Sahlmann: and a beautiful thing is that this sort of split
continuous
Hanno Sahlmann: for the rest of the of the algebra, because the these
Hanno Sahlmann: odd generators, they also decompose into vial
parts.
Hanno Sahlmann: some to to chiral parts, and they close as an algebra, together with the other generators. And so one has this
Hanno Sahlmann: chiral decomposition of the complexification of this algebra.
Jerzy Lewandowski: Okay, so, and no, no, no expert.
Jerzy Lewandowski: The question so very open. They say that
Jerzy Lewandowski: there there is a necessary condition which has to be satisfied by the background geometry that it has to admit some
Jerzy Lewandowski: preserved structure depending on
Jerzy Lewandowski: on this, and
Hanno Sahlmann: and you encounter some something like this here or or not, or I'm. Sure this this must be. This must be in the background somewhere. So so so, for example, it is not true that
Hanno Sahlmann: always this end can be arbitrary, for example, or extensions always exist; but I am also, unfortunately not an expert enough to to say this. Maybe we can ask, consenting later when when he comes. But but definitely, this depends on the
Hanno Sahlmann: yeah. So in in this case this is the isometries of something, and or
Hanno Sahlmann: at least part of it, is the isometry of something and
Hanno Sahlmann: and what symmetries? Yeah, what what structure it's on there that that has a bearing on the
Hanno Sahlmann: kind of extensions that you can have. Yes.
Hanno Sahlmann: yeah, Generally speaking, these things are all very sort of rigid, rigid structures and dimension dependent structures.
ashtekar: I thought that in 40 mentions there was originally a problem with this book, dial conditions without supersymmetry.
ashtekar: and that, I thought was resolved precisely by supersymmetry.
ashtekar: But you're saying that Still.
ashtekar: the number and of generate, you know, generation, if you like, maybe constrained by background curvature. Is that what you're saying.
Jerzy Lewandowski: I I know that in some case they require, for instance, a covariantly constant spinner to to exist.
Jerzy Lewandowski: or in some other case currently constant local twist. Or
ashtekar: is that true? We are talking about about global.
Hanno Sahlmann: I mean, we are extending other symmetry. So now we are. We are super extending the isometry group of some of some space time, and that's space time, that that isometry cool or
Hanno Sahlmann: or that space time must probably have very specific properties, so that this is possible.
Hanno Sahlmann: So right now we're only this is only a statement about anti-sitter.
Hanno Sahlmann: and and not about anything else. But yeah, I would imagine that
Hanno Sahlmann: that the properties of that space time that whose isometry group you want to extend, play a big role. Yeah.
Hanno Sahlmann: Okay.
Hanno Sahlmann: So
Hanno Sahlmann: very briefly, super manifold. So there are various ways to define these things, and it's it's important.
Hanno Sahlmann: because
Hanno Sahlmann: because, yeah, that the supersymmetry might be a symmetry of
Hanno Sahlmann: these kind of objects. Also, you want to have eventually fields that take values in a super entrepreneur super group or something like this. And those, then, are also super manifold. So one needs to know what this is, and
Hanno Sahlmann: the the most clear
Hanno Sahlmann: way to define this is like in non commutative geometry. So who has thought about non commutative geometry can understand this immediately. It's about
Hanno Sahlmann: the functions. Now you don't describe the space directly, but you talk about the functions over this space. You can do this classically with all sorts of spaces. They are completely encoded in the structure of the functions continuous functions on that space.
Hanno Sahlmann: and then you can take geometric or other notions
Hanno Sahlmann: on that space and formulate them in terms of the functions. And then, if you have formulated them abstractly enough and and hopefully.
Hanno Sahlmann: reasonably enough.
Hanno Sahlmann: in terms of the functions. Then you can deform that function algebra to something non commutative. And still these geometric notions
Hanno Sahlmann: makes sense, and
Hanno Sahlmann: it is exactly that for super manifolds. We.
Hanno Sahlmann: you look at
Hanno Sahlmann: you, you you say you you specify an algebra and say that is the algebra of functions over a super manifold and in the simplest example. One has one of these supers super vector spaces here that has key.
Hanno Sahlmann: real directions, and Q. A. P. Even directions and queue up directions. And so you have P. Even coordinates. And Q. Odd
Hanno Sahlmann: coordinates that make up this algebra of functions. Yeah. And so, then, consequently each function expands in a certain way in terms of these coordinates in terms of the the odd coordinates. This terminates this, this expansion
Hanno Sahlmann: and
Hanno Sahlmann: and particular
Hanno Sahlmann: sort of odd super many falls are the ones that have no even directions, and people also call them super points, and the algebra of functions over them are just the grasmanian. So the the totally anti-symmetric
Hanno Sahlmann: functions over some, some vector space. The the underlying space is not so clear what the picture is but we talk about the functions.
Hanno Sahlmann: and there is one technical thing that I will try to be very short about, you know. Do this, you dualize, and then you think about maps between super manifolds.
Hanno Sahlmann: Those are then algebra
in the other direction
Hanno Sahlmann: of these function algebra, because the pull back you want to simulate the pull back, and this this homomorphism property imposes severe constraints. In particular. When you consider
Hanno Sahlmann: cooling back
Hanno Sahlmann: functions on a super manifold to an ordinary manifold.
Hanno Sahlmann: you lose all the odds components of these functions, and that is somewhat unacceptable, because still ultimately, one wants to say that space time is some
Hanno Sahlmann: normal manifold as it's embedded somehow in superspace, and then using sort of the odd
Hanno Sahlmann: fields. For example, the fermion fields would not be acceptable, and one gets around this by doing something that is kind of hard to understand, because it doesn't happen in the ordinary category. But but it apparently needs to be done. In this
Hanno Sahlmann: super category one introduces a parametrization, or one considers relative super manifold, so one throughout the formalism for all of the many fools that one looks at, one
Hanno Sahlmann: ads, or
Hanno Sahlmann: one takes the product with some fixed, simple, super, manifold. But one can turn this parameter and use a different
Hanno Sahlmann: super manifolds, and the maps and the fields and everything they are required to depend on this additional parameter in a very simple, the most simple in a funeral way, and in this way one avoids
Hanno Sahlmann: having this phenomenon that one loses all the
Hanno Sahlmann: all the information about the odd things when one pulls back on to like some some ordinary manifold at the same time, it's an interesting kind of switch. So if one makes this Z actually
Hanno Sahlmann: commutative, an ordinary many fault, then you
Hanno Sahlmann: you have turned off all the super stuff, and you go back down to the normal manifold and normal fields, and so on. But if you turn it up high like a big superpoint, a big press manian, then you get all the non commutative information in this superman.
Hanno Sahlmann: Okay? Good. So that was super symmetry in a in a
Hanno Sahlmann: nutshell.
Hanno Sahlmann: Now, before we go to super gravity, Are there more questions.
ashtekar: Your. Why is there only single parameter in this last map in this embedding map?
ashtekar: And not, I mean
ashtekar: so? There must be some, I mean in other words
ashtekar: you want. F. If not, you want to define f one f 2 up to
the
ashtekar: number of
ashtekar: anti-commuting dimensions. Right? So why is it that one
ashtekar: one parameter suffices to define this and functions?
ashtekar: F. One. F. 2 of
Hanno Sahlmann: oh, so sorry, so
Hanno Sahlmann: so they are not defined by this. The idea is that if you, if you take a a function on this super manifold here
Hanno Sahlmann: and you want to pull it back
Hanno Sahlmann: via a map
Hanno Sahlmann: from an ordinary manifold. So you want to pull this this function back to the ordinary manifold, then you you only keep the the the
Hanno Sahlmann: but you did not want that right? I thought that that one doesn't install them. Yeah. Therefore you are doing something else, and that is your You want to send it so so very roughly speaking, like the this, this generators of the zip. Now they
Hanno Sahlmann: they go into these coefficients, and they also go here on the other side, and they
Hanno Sahlmann: make it in such a way that that that this phenomenon just doesn't happen, but that that also these these coefficients of the of the Thetas is n non-trivial. Odd things can be pulled back.
Hanno Sahlmann: So you you, You introduce additional
ashtekar: odd directions that I some of the odd things even.
Hanno Sahlmann: and then you can preserve that information.
ashtekar: I I I understand it's true. I thought that you're only one parameter. But you're you're parameterizing by super manifold.
ashtekar: Yes.
Hanno Sahlmann: good. So
Hanno Sahlmann: well Where can these super symmetries appear, and how can they appear? They can appear either as global symmetries or they can appear as local symmetries, global symmetries that happens when you are, for example, on Minkowski space, and you add the right number of
Hanno Sahlmann: gauge fields and fermion fields. Then you can have a chance of having a global supersymmetry. And in
Hanno Sahlmann: if you consider gravity plus fermions, then you have a chance to even have
Hanno Sahlmann: local supersymmetry.
Hanno Sahlmann: And that is certainly the case that interests us, and the simplest situation in which this can happen is gravity coupled to a fermion.
Hanno Sahlmann: And I have written here the action e is the
that we know and love
Hanno Sahlmann: say it's this premium, and for generality. I have done this with the cosmological constant, but
Hanno Sahlmann: for the super symmetry here, this is not even so important. What is important is that this this fermion is not just the fermi on, let's say, but it is this fermi on. So it has a form index. It has several space time components.
Hanno Sahlmann: and that must be so so that it's degrees of freedom, a match to those in the frame, so that they can be rotated into each other, which is actually in quoted in these transformation properties, where the the
Hanno Sahlmann: at least here you can see that the frame, when it gets rotated under this super symmetry, picks up some from your own part, and the
Hanno Sahlmann: and and this epsilon, this sort of
Hanno Sahlmann: parameter of this super symmetry transformation is actually a function. It's an odd function on the on the manifold. So that is a point dependent
Hanno Sahlmann: quantity.
Hanno Sahlmann: and
Hanno Sahlmann: certainly also the usual definition of this action.
Hanno Sahlmann: And
Hanno Sahlmann: then there are 2 questions. The one is, Can we quantize this thing with the 2 quantum gravity methods and the other one is, Where does this local symmetry come from? Because you
Hanno Sahlmann: don't see from this action that it would have these these symmetries.
Hanno Sahlmann: and let us answer the first question first, and in principle the answer is a resounding Yes, we can apply the quantum gravity methods to this 150,
Hanno Sahlmann: and this actually started in the eighties soon after a bias discovery of the the new variables, the Jakeobson added the fermions, and so of generalized the the formalism.
Hanno Sahlmann: and
Hanno Sahlmann: then, relatively soon after
Hanno Sahlmann: people. Philip, and also for her and collaborators, discovered that in the constraints of this theory there is some super symmetry. Yeah. And I will show you in a moment.
Hanno Sahlmann: And then they proceeded to quantize this theory, using this super symmetry that that they saw, and
Hanno Sahlmann: And
Hanno Sahlmann: while this is very nice that there's some supersymmetry manifest. The quantization was also somewhat formal.
Hanno Sahlmann: Then from another, there's also some work on on spin forms.
Hanno Sahlmann: and from another sort of perspective is the more recent work of Thomas and collaborators, and they used their methods to to work with real variables even for higher dimensions, and so on
Hanno Sahlmann: to quantize super symmetric theories, and the great thing about this is that everything is vigorously defined as rigorous as as for ordinary, you quantum gravity. The supersymmetry is not many fast.
Hanno Sahlmann: but that's not necessarily a bad thing, as we know from 4 dimensional to your morphism, invariance in the condom gravity. We hope that we can restore this in in our
Hanno Sahlmann: in our theory, although it is not not manifest.
Hanno Sahlmann: and I will, in the following, follow somewhat that order path, and try to keep supersymmetry manifest.
Hanno Sahlmann: and to do that, let's look at the old results
ashtekar: on this. On this. Can I use it for a second, please? I mean, You're like, raises important question about cause conditions in the background curvature. That was in the context of
ashtekar: global supersymmetry. Yes, when we come to local since supersymmetry
ashtekar: will you?
ashtekar: There are no such constraints. You can just consider any coverage.
ashtekar: This is my this is an important transition, and I just want to make sure that I'm.
Hanno Sahlmann: Yeah. Yeah.
Okay.
Hanno Sahlmann: Exactly. Exactly.
Hanno Sahlmann: Exactly.
ashtekar: I mean, that was one of the things that I was mentioning before about book that and such that every constraints in the Go global case. But in the local case I I don't think that there any constraints do you agree with that, youurek?
ashtekar: Well, I I don't know whether I agree, but I I appreciate this clarification.
Erlangen: Maybe I can ask just a follow up question, so there is also no restriction in local. So much we local super symmetry on
Erlangen: L, or the sign of the cosmologically constant? Or do we still have that
Hanno Sahlmann: we still have that I we, I believe we still have.
Hanno Sahlmann: Okay. So so, at least in what I'm going to tell you this.
Hanno Sahlmann: Yeah, I I think we still do. I think we still do, because otherwise this does not fit together. So we will see that that, in fact, the local local super symmetry that we will have is precisely of the type that I've shown you before.
Hanno Sahlmann: namely, either this Ponk or a extension, or this ads extension, and
Hanno Sahlmann: and the ads extension doesn't exist for for Ds: so so yeah, we we have that restriction.
Hanno Sahlmann: Good. So
Hanno Sahlmann: yeah, let me make it. Let me try to make it short. You do the normal thing that you do you to your canonical analysis, and you have the 3 usual constraints.
Hanno Sahlmann: But you also have 2 additional constraints. They come from some components of the Fermion being non dynamical, being Lagrange multipliers, the left and right super symmetry constraints.
Hanno Sahlmann: And what people notice is that the post on brackets
Hanno Sahlmann: for these constraints have a very suggestive structure, namely, we have the Gauss constraint, but then they go, for example, with the left super symmetry, constraint. It's the super symmetry, constraint, and 2 super symmetry constraints with each other. Give back a Gauss constraint.
Hanno Sahlmann: and when you look at this, then you see that this is exactly a again. This this Chiral chiral component
Hanno Sahlmann: of the ads symmetry extension that I showed you before.
Hanno Sahlmann: So where does this local symmetry come from now? How can we see that it is there? Is it, in particular, a gauge symmetry, because somehow, since it contains the like, I was constrained, it's suggestive that this is part of a larger age, symmetry.
Hanno Sahlmann: And so the answer
Hanno Sahlmann: to these questions comes from a certain formulation of gravity
Hanno Sahlmann: with a cosmological constant. So here the cosmological constant is very important
Hanno Sahlmann: gravity as a gauge theory. So for ads. It is a gauge theory with a gauge group, so 3 comma 2,
Hanno Sahlmann: however, that gauge and variance is broken in the action down to so 3, one.
Hanno Sahlmann: And this to show you how this all works. Yeah, you can not as a lee algebra, but as a vector space. You can divide this so 3, 2 up into something like the translations and the Lawrence generators.
Hanno Sahlmann: and you can take your gauge field
Hanno Sahlmann: and split it up in this way, and then the translational part is supposed to correspond to the fearbine, and the rotational part is supposed to correspond to the spin connection.
Hanno Sahlmann: and
Hanno Sahlmann: this can be worked out. So you the action that you need to write is almost like like young Mills. Yeah, it's F. Which Star F.
Hanno Sahlmann: Hi
Hanno Sahlmann: action.
But for this
Hanno Sahlmann: large gauge group, and you have
Hanno Sahlmann: a projector in this inner product that breaks that that that hits the here, the the the translational part to 0, so breaks the full gauge and variance down to this Subc.
Hanno Sahlmann: And you can work out that this action with this decomposition gives back
Hanno Sahlmann: a gravity on with a negative cosmological, constant. And there is some beautiful underlying geometry that is Carton geometry. And
Hanno Sahlmann: you can describe this very nicely.
ashtekar: So if I were to do it for 0 correspondence for Constant, as you said, that we do have the you know, the supersymmetry there, so we could extend it. But are you saying that then we don't have this manifest
Hanno Sahlmann: understanding of the origin? Yes, so I think I think that would be the the statement. So. So this formulation breaks down. So then I don't know again why this is so, but
Hanno Sahlmann: I can certainly take the L going to infinity limit, and I think I still also get
Hanno Sahlmann: a chiral
Hanno Sahlmann: decomposition of this concrete, this pon carry extension. So that still works. Yeah.
Hanno Sahlmann: And the and the theory, then also still has the supersymmetry, but it's it's the geometric explanation, is
Hanno Sahlmann: It's not so clear. I mean it's clear as a limit of of some of something.
Deepak Vaid: Can I ask a question? Yes, sure.
Deepak Vaid: So in in the in, the in the Mcdowell Mansuri mechanism
Deepak Vaid: you can have either anticipator as so, so 3 to, or you can have a desitter right? I mean, it depends on the
Deepak Vaid: on the sign of the
Deepak Vaid: so. So in this case, why are you restricted only to
Hanno Sahlmann: the super symmet is the super symmetric, the super symmetry requires that I, If I do consider gravity, then I will not get this local
Hanno Sahlmann: local super symmetry.
Hanno Sahlmann: That's the only reason the Mcdonald'souri works for both signs of the cosmological constant.
Okay.
Okay, Great.
Hanno Sahlmann: Okay. You're welcome. So I will try to speed up a little bit.
Hanno Sahlmann: Just tell you the idea. So we just extend. Yeah, we extend this. We now use Osp. One for this extension of the the Ads group as the gauge group. So now we have
Hanno Sahlmann: an additional component in the connection that corresponds to the fermions.
Hanno Sahlmann: and
Hanno Sahlmann: we have to
Hanno Sahlmann: at
Hanno Sahlmann: another. Oh, and I Haven't talked about it. So this thing is really an internal hot kind of thing. Here
Hanno Sahlmann: we have to add another one.
Hanno Sahlmann: and then this gives the right action for super gravity, and this is actually already known. Since the eighties it's called the Doria free formulation.
Hanno Sahlmann: and
Hanno Sahlmann: what maybe it's not so well known is that there is an underlying geometry that is super carton geometry. That is something that Constantine worked out in in detail.
Hanno Sahlmann: And again, here the gauge invariance is broken down to
Hanno Sahlmann: the
Hanno Sahlmann: just through this projection, just to so 3. One
Hanno Sahlmann: the many fit the the remaining
Hanno Sahlmann: cage symmetry. You can now come something new. You can make a host version of this
Hanno Sahlmann: by replacing this this internal watch, but by some linear combination of internal hodge and just the identity.
Hanno Sahlmann: and you plug that in, and you
Hanno Sahlmann: see that still everything works out. The equations of motion are not changed.
Hanno Sahlmann: You get this super gravity.
Hanno Sahlmann: but now you have it a parameter Beta. You have the immediately parameter in here.
Hanno Sahlmann: and then you can Even that's the biggest thing you can do. You can do. This calls extension even with extended
Hanno Sahlmann: 80 as super gravity. Then, in your connection, you have, besides everything else, also a you, one field that's this, our symmetry field. So now you're talking even about electoral magnetism added on top of this, and you have to add
Hanno Sahlmann: something in this projector, and
Hanno Sahlmann: you take
Hanno Sahlmann: the most simple thing that you can come up with, namely, the
Hanno Sahlmann: it's literally the Hodge. And you take this linear communication.
Hanno Sahlmann: And
Hanno Sahlmann: again, this works
Hanno Sahlmann: and some remarks are that what is nice about this, that all these things include boundary terms.
Hanno Sahlmann: and that, in fact, these terms are unique when you
Hanno Sahlmann: demand supersymmetry also on the boundary. So, writing it in this beautiful way
Hanno Sahlmann: automatically gives you sort of preferred boundary terms.
Hanno Sahlmann: And
Hanno Sahlmann: this this
Hanno Sahlmann: thing in this P operator in the one sector is literally the Theta ambiguity. So that shows that this intuition that that somehow the the
yeah
Hanno Sahlmann: amusing permit is something like
Hanno Sahlmann: the the
Hanno Sahlmann: permit for
Hanno Sahlmann: Qcd. Is
Hanno Sahlmann: at least in this supersymmetric context, it is enforced.
Hanno Sahlmann: and however
Hanno Sahlmann: this these actions still break manifest super symmetry down to the at least the holes versions break down the the this super symmetry except for, and
Hanno Sahlmann: that is not the Chiral supersymmetry, except for at the special places.
Hanno Sahlmann: And it those the P. Is really like like in
Hanno Sahlmann: in our usual case projection, and
Hanno Sahlmann: it
Hanno Sahlmann: it leaves some of the super symmetry intact, you know, and in particular you can keep sort of the the part of the connection that is left invariant under this, under this projector P.
Hanno Sahlmann: And that is the generalization of the connection. And now it contains, besides, literally, they actually call connection. It contains the spin or and like the chiral component of the spinner, and maybe also a
Hanno Sahlmann: electromagnetic field. If you are for n equal to 2,
Hanno Sahlmann: and you can
Hanno Sahlmann: work out what the action breaks down to in terms of this connection, and then it has the form that we know and recognize. Only, you know, everything is a little super, but here we have curvature, which, and
Hanno Sahlmann: E an electric field.
Hanno Sahlmann: and here we have a cosmological constant term, and then there's a boundary term.
Hanno Sahlmann: And
Hanno Sahlmann: so this
Hanno Sahlmann: shows that I mean coming from this formulation. You see that this, that this supersymmetry in the canonical theory in in these Poisson brackets is really
Hanno Sahlmann: from a gauge symmetry. From this osp 1, 2, or
Hanno Sahlmann: and
Hanno Sahlmann: the that explains these observations that explains the need for the cosmological constant. So in the early literature there is some some sort of ambiguity about
Hanno Sahlmann: whether this also works in other other signatures or other, or without cosmological, constant, and so on. And now we can see very clearly. It does not, I mean. From from this it seems clear that it doesn't work.
Hanno Sahlmann: and the hard reality is then, also that we need reality conditions, and we need complex variables with all the technical problems that are
Hanno Sahlmann: that they have.
Hanno Sahlmann: Now, there is some further nice thing. This
Hanno Sahlmann: chosen boundary term in this formulation is just a John Simon's theory, with a for some reason, an imaginary level
and
Hanno Sahlmann: proportional to the cosmological, constant.
Hanno Sahlmann: and additionally because of the boundary terms
Hanno Sahlmann: for us to have the
Hanno Sahlmann: the equations of motion without any sort of boundary contributions.
Hanno Sahlmann: one has to have an additional boundary
Hanno Sahlmann: condition.
Hanno Sahlmann: and that boundary condition also looks very familiar and natural in the kind of variables that we are used to.
Hanno Sahlmann: And
Hanno Sahlmann: okay. So at this point. We have understood sort of for this to geometry behind these things, so maybe we can now go and do the quantum theory. But before I jump into this? Are there further questions?
Jerzy Lewandowski: So there was a young Muse lagrangian at some point which in port that I mentioned is also conformally invariant. So our conformal
Jerzy Lewandowski: transformations part of the symmetries.
Hanno Sahlmann: I would
Hanno Sahlmann: think that they are probably broken by the somewhere, by the cosmological constant, because you have a link to scale in the game.
Hanno Sahlmann: But I don't. But you also are right to say that
Jerzy Lewandowski: Yeah, maybe there are some invisible constraints, but if which star F usually is conformally in current.
Hanno Sahlmann: Yeah, yeah.
Hanno Sahlmann: Okay, good good point. I have to.
Hanno Sahlmann: I have to think about this.
Erlangen: perhaps perhaps another question on the operating conditions. So of course they are very familiar from isolated
Erlangen: horizons.
Hanno Sahlmann: Yeah.
now.
Erlangen: now there is not. And now we are, I think, at the
Hanno Sahlmann: so. No, I i'm sorry I should have said this more more clearly. So okay, there is a question about an asymptotic boundary, but I don't even we haven't even really thought about those
Hanno Sahlmann: at this point. So what we have in mind here is literally a an inner boundary.
Hanno Sahlmann: I see. Okay, but but but indeed, I mean, there's also a discussion that one should probably have about the
Hanno Sahlmann: asymptotic situation. Yeah.
Erlangen: thank you.
ashtekar: Yeah. I have a question also about the boundary term, which is, I also thought it was in a boundary. But you you mentioned that somehow some supersymmetry requirement fixes the boundary term completely.
ashtekar: Yes; but then, later you said that
ashtekar: this was chosen to be John Simmons.
You?
ashtekar: No, it is not chosen. This is literally you. You
Hanno Sahlmann: You write down this this type of action.
Hanno Sahlmann: and this has a boundary term, and in the Chiral case beta equal plus minus I. It is a transignments term, but then you can independently show that you can't at other
Hanno Sahlmann: terms, and preserve super symmetry. So so in that sense I mean it's it's twice chosen it's chosen just by being included in this Mcdonald's story formulation automatically, and it's chosen because there is no other one that has the same level of symmetry, also on the boundary.
ashtekar: Okay, so the choice is forced on you. It's not chosen, I mean chosen. It seems like you have a choice something else. But this choice is is unique. I mean. So then, from this perspective. There's something interesting, right? I mean, because you're saying that if I started just with without supersymmetry.
ashtekar: then it looks like I can add to the action, a whole bunch of boundary terms. And and but here and
ashtekar: here you are saying that. No, I mean here. If I said that well.
ashtekar: I mean others. You can do this uniquely in the supersymmetric sector, and then project
ashtekar: to the and then say that I get a unique answer, because there's the only one which
Hanno Sahlmann: which can be lifted up to supersymmetric sector. Is that what you're saying?
ashtekar: Okay, so that that's quite exciting. So you're saying that therefore.
ashtekar: also earlier on when you talk about the barbarian, easy, parameter, coming like theta ambiguity.
ashtekar: that all that that also is true, right, namely, that
ashtekar: a priori I could. I met many canonical transformations, for example, but supersymmetry is telling me that. No, there is only one paramet of freedom.
Hanno Sahlmann: Is that correct? I mean, that's very powerful. I think it should be much more emphasized if it is, I see. So so yes, this this term here is this term here
Hanno Sahlmann: is fixed. Yeah, because because we want this, we want this action to describe that unique.
Hanno Sahlmann: that unique, consistent, supersymmetric theory. And then this this thing must be must be here.
ashtekar: Okay, No, because that's very interesting, because originally, when I done the canonical transformation, it was for simplicity, reason to be at the connection variables and searching and self, and motivated myself duality. But now you are saying that there is another way to look at it, which is that
ashtekar: only that
ashtekar: kind of connection is
ashtekar: is the only unique one that will descend from a supersymmetric one.
Hanno Sahlmann: Yes, I think one can say it like this. I I think
Hanno Sahlmann: one could say it.
Hanno Sahlmann: One could say it like this. Yeah, I I will think about it, because certainly so. There is one trivial way to descent, and that is to turn off this
Hanno Sahlmann: to turn off the this premises to something actually bosonic. But then I
Hanno Sahlmann: i'm not sure if I I I have to talk to consenting. If we keep them the electric field. But this, this, this you one field. But I think I I think it is. It is correct what you say. Yes.
ashtekar: second, Thank you.
Hanno Sahlmann: Good. So
Hanno Sahlmann: let us.
Hanno Sahlmann: So I'm: basically
Hanno Sahlmann: almost out of time. So let me be quick here.
Hanno Sahlmann: Now we want to quantize this, and there are some things that work.
Hanno Sahlmann: But there are also some things that don't work, and chiefly what doesn't work is the Us. Space and consistent family of measures, and so on, and that fails, partly because of the non compactness of the gauge group, and partially also because the the
Hanno Sahlmann: the harm measure on super groups fails to be positive.
Hanno Sahlmann: just a fact of life. And so there's another complication that one has to deal with. One has to
Hanno Sahlmann: potentially find some crime structure or something like this. But that is a a a problem. Here let me be a little bit more specific. So, for example, we love Holonomy, and one can make colonies of this super connection.
Hanno Sahlmann: and they naturally contain the other field. So, for example, the the fermion field, and you can actually go ultimately and work out and see what they are, and you can see that there is a bosonic
Hanno Sahlmann: holonomy involved. And then there is parallel transport along the edge of this Fermion somehow, and it's such a way that everything goes harmonically together, so that this thing is covariant under the action of the of the super group.
Hanno Sahlmann: So that is quite nice. And there was lots of confusion about how this these super holonomy's work, but consenting, just figured this out.
Hanno Sahlmann: and
Hanno Sahlmann: they are are created also relations of these things, and they look
Hanno Sahlmann: exactly like our normal ones, and you can make some algebra. They are also right and left invariant
Hanno Sahlmann: vector fields on the super groups. You can make a Holonomy flux algebra.
Hanno Sahlmann: and you can even have generalized connections
Hanno Sahlmann: that actually mathematically works out quite beautifully. But, as I said, we have problems with the the Hilbert space structure.
Hanno Sahlmann: Nevertheless, one can consider some representations. Yeah. And and work with representations of these super groups, and already link in this morning started this. Define some super spin networks.
Hanno Sahlmann: and there is a basic question whether these should be finite, dimensional representations.
Hanno Sahlmann: yeah, or infinite, dimensional. And one also wants them to
Hanno Sahlmann: have the a tensor category, so that multiplication with components of a Holonomy doesn't make one leave that that category.
Hanno Sahlmann: But it seems in both cases there are options for this.
Hanno Sahlmann: and we have looked at one the the non extended situation in particular, and have worked out some kind of principal series of of representations.
Hanno Sahlmann: There. There are some special representations that have finite dimension that they were already known, but we have written down a more general one, and what one can do is one can work out what the Casimia gives in these representations
Hanno Sahlmann: in general. This J. Is now a complex number. So, then, this is not positive or anything like this.
Hanno Sahlmann: And so the super area in general is also not real.
Hanno Sahlmann: but
Hanno Sahlmann: that is to be expected because we have not talked about reality conditions.
Hanno Sahlmann: And
Hanno Sahlmann: interestingly, there are J's
Hanno Sahlmann: of a certain form. Yeah, that is given here.
Hanno Sahlmann: The complex jays that give a real area spectrum of this of this form.
Hanno Sahlmann: And
Hanno Sahlmann: okay and sorry, I always say area. And I have to say, this is not
Hanno Sahlmann: area. This is
Hanno Sahlmann: super area now. So this is. This is the length of this super electric field.
Hanno Sahlmann: Now it in a sense it bites us that we are in the supersymmetric situation because
Hanno Sahlmann: a priori, the area of something is not an invariant. It's not a gauge and variant
Hanno Sahlmann: concept. This super area is
perhaps
Hanno Sahlmann: one can.
Hanno Sahlmann: Yeah. So one would have to think whether one can somehow rotate. H rotate always, so that one only has a component in the sort of in the bosonic direction. I'm not sure that that's possible. Then one could somehow identify
Hanno Sahlmann: these 2 things. But it is a graded graded area here that we are talking about, but that's the only thing that is.
Hanno Sahlmann: And
Hanno Sahlmann: we went on to use this to talk about entropy of
Hanno Sahlmann: of
Hanno Sahlmann: in our boundaries of space-time, because the
Hanno Sahlmann: I mean, we have all the ingredients to to do the same thing that we do in the real theory. Yeah, we have a trans assignments boundary theory. We have this Co. Boundary conditions. We have
Hanno Sahlmann: some kind of area operator, but there are problems, namely, the transignment theory is for a non compact
Hanno Sahlmann: super group, and the level is imaginary. And so what doesn't know anything about this
Hanno Sahlmann: mathematically, and we have the reality conditions that we haven't taken care of.
Hanno Sahlmann: and what we decided is we the wind to this work by
Hanno Sahlmann: telling we and collaborators where they, where they observe that they can do a State counting for real embassy parameter, and then analytically continue the
Hanno Sahlmann: the music parameter as well as the representation labels in the state, counting formula in such a way that in some sense area stays real.
Hanno Sahlmann: and then they get they get an entropy proportional to quarter of the area without.
Hanno Sahlmann: Well, the amazing permit has gone because it's analytically continued to I. So they just get get the right Beaconstein hawking formula on the nose.
Hanno Sahlmann: and then we just had to find what can play the role of su 2 here
Hanno Sahlmann: and there is, indeed, there is a real
Hanno Sahlmann: compact super group
Hanno Sahlmann: that is a real form of this of this super gauge super group that we are talking about.
Hanno Sahlmann: and a compact, a compact, real form of this complexification here.
Hanno Sahlmann: and
Hanno Sahlmann: and the irreducible representations are indeed labeled by jails. And so we can
Hanno Sahlmann: use that
Hanno Sahlmann: we write down the number of States. It's the number of transignment states, like the the number of conformal, the dimension of the conformal blocks of this and this and that.
Hanno Sahlmann: and we do the analytic continuation.
Hanno Sahlmann: and there is some real work involved using, I mean writing down the characters of the super group and doing the State counting, and so on, and so forth. But at the end of the day the same phenomenon happens that happen for these, these these other people.
Hanno Sahlmann: After the allergy continuation, the right leading order contribution gets reproduced, and there are also.
Hanno Sahlmann: when you.
Hanno Sahlmann: when you make some further assumptions, there are some lower order terms that you can calculate.
Hanno Sahlmann: Certainly it's not clear that one counts anything as so. This is only an analytic continuation of something that really counted States.
Hanno Sahlmann: but it's never the less such as stiff.
Hanno Sahlmann: And
Hanno Sahlmann: with that I I I want to
Hanno Sahlmann: summarize and and stop.
Hanno Sahlmann: So
Hanno Sahlmann: I guess we have
Hanno Sahlmann: clarified the this geometric
Hanno Sahlmann: nature of the super-ashical connection
Hanno Sahlmann: can keep supersymmetry partially manifest. But there is the problem of non- compactness and non
on in reality conditions
Hanno Sahlmann: there is some nice mathematical physics going on here with the super connections in the super parallel transport.
Hanno Sahlmann: and
Hanno Sahlmann: we have not talked about isolated horizons or anything, so I can't even say that this argument there is counting like whole micro states. It's counting as sort of transignment states, on a on an inner boundary. But at least, these are the steps towards
Hanno Sahlmann: dealing with a super symmetric like holes from the quantum gravity.
Hanno Sahlmann: and they are certainly more to do so. Perhaps one can clarify under which conditions these inner boundaries are really
Hanno Sahlmann: like holes.
Hanno Sahlmann: play the whole horizons. It would be very interesting to
Hanno Sahlmann: do this, counting for n equal 2, because this is where the string theory calculations are done.
Hanno Sahlmann: and then also to understand if on some level, they some some sort of I mean, if the string theory, calculation, and the quantum gravity calculation are just completely orthogonal, and get to the same result, or whether there is some
Hanno Sahlmann: some level on which they they meet.
Hanno Sahlmann: And
Hanno Sahlmann: there's one thing that goes through my mind that somehow we don't live in a supersymmetric world.
Hanno Sahlmann: So somehow one has to break supersymmetry, and maybe
Hanno Sahlmann: this could suggest some particular breaking patterns, because if we have super spin networks, and we somehow say, now we want to consider all this feels separately. Then, somehow, with the one representation of the super group.
Hanno Sahlmann: some various representations or charges of the individual fields. Yeah. So, for example, if I have a a a super group
Hanno Sahlmann: representation labeled by J. Then typically what that means for the su 2 subgroup that they S. J. And the J. Plus one half or something like this that is sitting in there. And so also for the charge of this, our symmetry, and for the
Hanno Sahlmann: it's been, or as well, I suppose, and maybe they have some relations, or they they will have some relations, and that would say something about how this symmetry would be broken.
Hanno Sahlmann: and certainly ultimately one would like to have a different space structure for for this
Hanno Sahlmann: this theory tool.
Hanno Sahlmann: So sorry for rushing through the last part.
Hanno Sahlmann: Thank you very much for for listening and yeah, happy for your questions.
Ivan Agullo: Yeah, thank you, kind of. And I think her he had to leave. So I am in charge of the
Ivan Agullo: the question and session, so that we have several questions, and but but the more questions are surely welcome.
And so okay.
Deepak Vaid: So I have a I have a question.
Ivan Agullo: Go ahead.
Deepak Vaid: So
Deepak Vaid: how do we? How do we think of about these supersymmetric spin networks like I mean Normally, we have
Deepak Vaid: spins labeling our edges, and you know interplayers for the word disease.
Deepak Vaid: So how you know what
Deepak Vaid: What does
you know?
Deepak Vaid: What degree of freedom can does
Deepak Vaid: super symmetry add to the
Deepak Vaid: to the structure.
Hanno Sahlmann: Well, so so now this connection field contains
Hanno Sahlmann: also fermionic degrees of freedom. So this
Hanno Sahlmann: representation label that we choose in our spin network, that now also says something about the fermionic field. So so spin networks are not just flux cubes of
of
Hanno Sahlmann: gravitational area, but they are also flux tubes of the of the spin or field, and maybe the electromagnetic field. If If you would take this, our symmetry into account.
Hanno Sahlmann: and the labels are going to be labels of the Super.
Hanno Sahlmann: a super gauge group, like like representations of that super gauge group, and
Hanno Sahlmann: as as such, since we don't know the product, we don't
Hanno Sahlmann: know exactly which class of representations we should be looking at. But there are candidates, and then one can say more about about this. But you could, for example, If you think about
Hanno Sahlmann: the breaking of the symmetry, then you could ask these questions Here you could say I am in the given representation of the super group. And now I want to find out, how does it decompose in terms of now a single or a subc
Hanno Sahlmann: of this, for example, an su 2 subgroup. And I want to ask, how does this
Hanno Sahlmann: big representation decompose into representations of the subgroup, and so on? And that would then presumably tell you something about how the quantum numbers of these individual fields are related to each other. If you are breaking it
Hanno Sahlmann: the so if you're singling out generators in this in this super algebra super coop?
Deepak Vaid: Okay?
Deepak Vaid: And can I ask second question?
Hanno Sahlmann: There's also a question on this on the chat
Hanno Sahlmann: So maybe.
Hanno Sahlmann: or is it related to to your
Deepak Vaid: well, it's about the isolated horizon boundary condition, and
Hanno Sahlmann: well, I think I think this is very, very quick, because I have not talked about this, and I have nothing to say about this in the super symmetric context. I'm also not aware that anybody has looked at this.
Deepak Vaid: at what sorry
Hanno Sahlmann: at the isolated horizon boundary condition in supergravity.
Deepak Vaid: No, I mean I mean you, you You You showed that relationship earlier right
ashtekar: that that that was for any boundary. So
Deepak Vaid: right, right? So so. My my My question was, that since you are an anti-discer.
Deepak Vaid: would it not also apply to the asymptotic boundary
Deepak Vaid: that that's my question.
Hanno Sahlmann: Oh, so that yeah. So so so I You you guys are right. So one should also think about the think about the
Hanno Sahlmann: the asymptotic boundary.
Hanno Sahlmann: the not the inner boundary, but the asymptotic one. And then, yeah, there, there's also the
Hanno Sahlmann: Yes, boundary.
Hanno Sahlmann: There is a boundary term, and that
ashtekar: also fine. Yes, No, no, but you'll have to make conformal completion. Everything will be different. It's not. It's not okay about my. You don't have to put conditions on the here. You don't have to put any conditions on the in a boundary. It just everything is smooth, or everything is.
ashtekar: whereas they the form of condition, which, if you like, in the confirm with completion, is it? It just gets transferred to the behavior of the conformal rescaling. So now I think what work is more thought is needed. There is not, you cannot just so the same.
Ivan Agullo: But just the did this.
Hanno Sahlmann: that there is a question before by by Thomas steaming the chat. So perhaps we should. Yeah. So Thomas is asking a very good question, and I I should have said this. So Thomas is asking about the hyper surface deformation, algebra. So
Hanno Sahlmann: so, what about what about those
Hanno Sahlmann: th, those symmetries of what about those constraints? I suppose so. So, indeed!
Hanno Sahlmann: What so
Hanno Sahlmann: so up
Hanno Sahlmann: in terms of constraints. What I said is that somehow one of the 2 supersymmetry constraints gets absorbed into the
Hanno Sahlmann: into the this gauge symmetry. But there's also the other one.
Hanno Sahlmann: And so I think if you take the commute, the the anti commutator of those 2 super symmetry constraints. Then you get the Hamiltonian constraint, so you could argue that
Hanno Sahlmann: what is left to do now is to deal with that. If youomorphism, constraint, and the
Hanno Sahlmann: and the the other supersymmetry constraint, so there are more, indeed, there are more.
Hanno Sahlmann: There is one more constraint that you have to deal with. It might not be the Hamiltonian constraint, but the the second supersymmetry constraint, but that it's also. That is also a thing you eventually have to do.
ashtekar: Okay, Can I just ask a couple of questions, or there are other people?
Ivan Agullo: No, no, please go.
ashtekar: Okay. So the first, just to make sure that we understood what what you're saying, you said. But I mean the the problem of actually finding the generally in a product, you know, is is is open because we don't know the measure, but that problem is also open in the says self, dual gravity in in because of non compactness, right?
ashtekar: But then you said that Well, but you can look at some representations, I mean. So you're not solving that problem. But so.
ashtekar: on the other hand.
ashtekar: if I just declare some representations as being sacrificed.
ashtekar: then do I have a little bit of space then, and everything is. I can just do the calculations then, over what? Exactly is it?
Hanno Sahlmann: Yeah, Not sure because of 2 province. One is that one still has to.
Hanno Sahlmann: I think I can do this on one edge. I can. Maybe so this is what I did. Yeah, I just considered one edge, and I said, okay, this is
Hanno Sahlmann: this, for since I identified the this area operator with the second, what the the the quadratic casimia is sort of inside the the area operator then given certain representations. This gives certain eigenvalues. That is a statement I can make.
Hanno Sahlmann: but i'm not sure that I can. Going from there build up a consistent inner product for many edges.
Hanno Sahlmann: and i'm also.
Hanno Sahlmann: I said, so you went on a fixed graph, forgetting about, I think, on the it's it's already better, because there I could just, for example, declare that certain certain representations are orthogonal to each other. I suppose then, I could this way build up
Hanno Sahlmann: and an inner product.
Hanno Sahlmann: But the other question is whether the multiplication operator works so so because in these non compact groups one can
Hanno Sahlmann: sometimes
Hanno Sahlmann: or
Hanno Sahlmann: Yeah. So so when when I, when I multiply this representation with the Holonomy in another representation, and I make something like the tensor product, and I have to make sure that this tensor product doesn't map us out of the class of representations that
Hanno Sahlmann: that are
Hanno Sahlmann: that that I want to have in my class. So. So one has to have sort of a a a
Hanno Sahlmann: category of representations that are closed undertaking the tensor product.
ashtekar: Okay, so that. But I mean most of these issues that you're talking about. They are already for the
Hanno Sahlmann: non-supposymmetric case. If the gauge group is not compact, I mean, I I don't see any difference. I'm. In super submit to making any. No, it I mean, there's a slight slight thing that that
Hanno Sahlmann: I mean, yeah, it's also that that the hard measure that one would like to use, and these constructions has more weird properties for the super groups. But I think this can be dealt with, and in this in this all.
Hanno Sahlmann: Ou Sp: 1, 2 case that consenting and I have looked at it. We could work with this harm measure also with our problem, and so on. Yeah. So so I agree. It's basically a similar problem. As as in the normal.
ashtekar: the the last thing is just a suggestion. So yes, which might be very useful actually, which is that you? We know that in
ashtekar: if I look at
ashtekar: the N. Equal to 4 supersymmetric theory, that is somehow exactly soluble. And now that doesn't have much to do with the real world. And so, you know, one might want to take it too seriously. But, on the other hand.
you know.
ashtekar: real phenomenologists have taken hints from that theory
ashtekar: about how to break supersymmetry, and what may be true in the non supersymmetry case, and so on. So here you could do something similar, right? I mean you could. This this question that Deepak was asking before about
ashtekar: the spin networks and supersymmetric spin networks, etc. I mean, what you're doing is really unifying this penis, and the
ashtekar: the bosonic sectors. So normally, what we have is really the gauge bosons are along the links. They are the ones that are using the parent transport and the for me on sit at the word is this. But now everything is mixed up here right? So the question is what
ashtekar: I mean. Take the more pop, aggressive or positive aspects, saying that Well, now, let me just see that this theory is better just mixing everything else. And what is it that I'll learn from here right? I mean, so that there you can go to young this series, because there's exactly the same thing. It's true in this year, right?
ashtekar: Because if I the super symmetric version, my
ashtekar: then again, what is.
ashtekar: I mean lattice stage 3. For example, what is going to happen along the edges?
ashtekar: He's not going to. There's a parallel transport on connection. It is going to be Paradise part of the connection as well as the spin. And so somehow.
ashtekar: in super symmetric yeah, and equal to 4. Young misery is exactly soluble for some reason.
ashtekar: and there's a huge simplification that is occurring if I look at it in the unified way rather than a broken way. If, as soon as I break it, everything becomes much more hard. So so, taking a more aggressive attitude here, and looking at it from the
ashtekar: drawing lessons from N. Equal to 4 young meals
Hanno Sahlmann: might be really helpful up here. I think that that, I think is that we will look into this. This is a good point, then one can have these things for more present. Is this: in in 4 dimensions or 40 mentions.
ashtekar: I mean, that's the reason why you know the first idea, c. Ft. Correspondence was for, and you could the 4 supersymmetric.
ashtekar: and then in 40 mentioned it was for that. But but but you can forget about that, and you can just look at it equal to 4 Supreme. You know that that is really
ashtekar: It's much, much, much better behaved than the one, or actually, it's much simpler than the one without super symmetry. So there may be some.
Hanno Sahlmann: Yeah, that would be nice if, if, in fact, this super symmetry could enable us to solve a problem that is easier to solve with
Hanno Sahlmann: the super symmetry. Then without yeah, we have to think about in this in this direction here.
Ivan Agullo: So there is a a response by Thomas. I don't know if you want to say something about that, and I completely agree with Thomas, so it's, so he says, I think if I understand correctly, that's a very non-trivial. So so even if
Hanno Sahlmann: you will find a measure that, or like you, find consistent measures on the graphs, so that you have a Hilbert Space structure. Then the next problem is, your your flux operators, and can they be made? Can they be made densely defined, and
Hanno Sahlmann: the metric or self a joint or something. And and this is by no means clear. So I I agree with that that I mean one, for it has to have some.
Hanno Sahlmann: some, some good.
Hanno Sahlmann: I have some some good idea, I don't know, but this is all
not easy.
Hanno Sahlmann: and that's why it's. It's very nice that there is this complementary approach where woman says, okay, we we don't keep super symmetry manifest. But at least we have a hybrid space, and we can write on operators and preview and resonations, and so on and so forth.
Ivan Agullo: Yeah, yeah, just just wanted to ask one technical question, and then on up
Deepak Vaid: procedural comment.
Deepak Vaid: So the technical question is that
Deepak Vaid: just to follow up on my earlier
Deepak Vaid: remark about modifying the spin network structure?
Deepak Vaid: So we know that when you introduce a cost magical, constant, right? What happens
Deepak Vaid: is one of the ways to look at it is that
Deepak Vaid: the the gauge group. Su 2 gets becomes defined right, it becomes su 2. Q.
Deepak Vaid: It happens in some
Deepak Vaid: and and and and then, in order to regulate the 30 you need to
Deepak Vaid: frame the edges
Deepak Vaid: it rather than having them be. You know one-dimensional objects.
Deepak Vaid: So
Deepak Vaid: you know it would be
Deepak Vaid: I mean I don't know if that su. 2 queue thing works for negative customers will be constant or not.
Deepak Vaid: but it if it does, then it would be nice to see
Deepak Vaid: whether you know that is related to.
Deepak Vaid: If if if that gives us something
Hanno Sahlmann: for the for the quantum
Hanno Sahlmann: group happens, because that's somehow cuts off the that they are finitely many reducible representations. So so I don't know I mean this will not help us with this compactness, thing or or something. But
Hanno Sahlmann: in principle, I mean even in this Shawn Simon's quantum turn. Simon's theory. Technically, there should be a quantum group. Yeah, I the formula I've shown you is for for
Hanno Sahlmann: for K to infinity. So I suppose small.
Hanno Sahlmann: That's like, okay, yeah.
Deepak Vaid: yeah, Kate K: it goes as elsewhere, right? So on my land, I think. Yeah.
Hanno Sahlmann: But then, and it's large con somehow this doesn't seem right, but but in any case so so so perhaps.
Hanno Sahlmann: as as I said, non-commutative geometry, and the super symmetry for me from my very naive standpoint, seem to be not so far away in terms of the basic idea. And maybe there is some overlap also there.
Deepak Vaid: And and Finally, I just want to make one procedural comment, which is I'm. I apologize for not going through proper channels
Deepak Vaid: where I live and work. People are very, very
Deepak Vaid: rigid about profit channels, and I I am not.
Deepak Vaid: I think, that it would really really serve our community
Deepak Vaid: if we took all these amazing talks that we have recorded.
Deepak Vaid: and we posted them on Youtube.
Deepak Vaid: I I think it would. It would give us give our work tremendous visibility.
Deepak Vaid: and and and and you know we could really really use that. So I mean again, this is just a you know
oh.
Deepak Vaid: so a very humble suggestion to the Organizing Committee
Deepak Vaid: that this is this is something they might they might consider
Deepak Vaid: it. Will, it will definitely, you know, increase
Deepak Vaid: the visibility of of flu quantum gravity
Ivan Agullo: I'll bring. I'll bring that to the up to the committee.
Ivan Agullo: That's a good suggestion. Thank you. Appreciate it.
Ivan Agullo: I think we are 30 min past the time, so it's like with time to stop. So thank you, Hano, and and see you in a sense.
Hanno Sahlmann: Yeah, thank you very much for the for the many questions also thought provoking one. So I have to go back and and think it was very nice. Thank you very much.
Ivan Agullo: Wonderful?
Deepak Vaid: Okay, bye, bye, bye, bye.