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Jorge Pullin: All right. So, Speaker, today is Hano Salman, who will speak about Karl's Super. Lqg.
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Hanno Sahlmann: Thank you very much. So yeah, Thank you very much for the invitation and opportunity to speak here. That's great.
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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
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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
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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
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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.
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Hanno Sahlmann: Some of it is in collaboration with me Some of it is I
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Hanno Sahlmann: him alone, and the
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Hanno Sahlmann: potential mistakes on the slides are mine altogether.
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Hanno Sahlmann: So
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Hanno Sahlmann: Supergravity, super symmetry. What
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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.
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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
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Hanno Sahlmann: gravitational fields and the rest of the physical fields.
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Hanno Sahlmann: So space-time, geometry and the rest.
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Hanno Sahlmann: and that's
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Hanno Sahlmann: one dichotomy that is to be breach it to to be bridged. Yeah.
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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
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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
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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.
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Hanno Sahlmann: and
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Hanno Sahlmann: now
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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
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Hanno Sahlmann: theory, and then you and reinterpret in your dimension some of the fields, as sort of matter feels.
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Hanno Sahlmann: and then there's also supersymmetry. And today we're going to talk about super symmetry, of course.
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Hanno Sahlmann: Now.
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Hanno Sahlmann: what is what is the
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Hanno Sahlmann: what is super symmetry. Well, one answer was again
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Hanno Sahlmann: given in the context of quantum field theory, Hack and Kuprushanski. Notice that there is a loophole to the Coleman Mandela theorem.
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Hanno Sahlmann: and that is, if you don't, just let your symmetries be
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Hanno Sahlmann: generated by the algebra. But you allow more general
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Hanno Sahlmann: algebraic objects that also have anti commutators. Then, indeed, a mix between space-time symmetries and the other symmetries is possible.
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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.
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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.
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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.
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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
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Hanno Sahlmann: for 2 odd elements it is, it is symmetric, so it is a graded product.
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Hanno Sahlmann: and corresponding to this. There's also a graded Jacobi identity as a sort of replacement of.
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Hanno Sahlmann: And so this sounds pretty
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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.
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Hanno Sahlmann: and we know some operations on these forms. Now we know that we can take the derivatives. We can take the
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Hanno Sahlmann: exterior, derivative and we can take these interior products just plugging a tangent vector into a form.
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Hanno Sahlmann: and there are these identities, for example, a. D squared sival.
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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
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Hanno Sahlmann: commutative because they Anti.
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Hanno Sahlmann: Yeah. Yeah.
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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
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Hanno Sahlmann: the algebra comes from, and so
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Hanno Sahlmann: you can check the other properties that I've given you. And
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Hanno Sahlmann: this object, indeed is a super. These operators formula from a big super algebra. And so Cartons
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Hanno Sahlmann: magic equation. Yes, also this context just a a anti commutator.
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Sure.
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Hanno Sahlmann: So maybe these things are not so strange after all. So let's go on and look at
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Hanno Sahlmann: super extensions of the spacetime symmetry. So a minimal extension of the Poncari algebra.
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Hanno Sahlmann: and what you will turns out to do. Boshanski, tell us you have to add
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Hanno Sahlmann: for
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generators
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Hanno Sahlmann: to the algebra, and these 4 generators are odd.
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Hanno Sahlmann: and
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Hanno Sahlmann: these generators have to fulfill certain very specific relations with the rest of the generators.
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Hanno Sahlmann: And
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Hanno Sahlmann: well, first of all, I show you that
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Hanno Sahlmann: it was non-zero
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Hanno Sahlmann: commutators, anti commutators show that indeed, we have a mixing of
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Hanno Sahlmann: space, time, symmetries, and these new generators they don't commute with each other; and, secondly, from this thing here.
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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.
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Hanno Sahlmann: So this equation just says that these queue transform as a as a for Spino under Lawrence transformations.
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Hanno Sahlmann: And now
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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
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Hanno Sahlmann: is this.
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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.
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Hanno Sahlmann: and you'll see that they have
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Hanno Sahlmann: very similar. There are some additional terms, and they go with the link scale that comes into the picture from the
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Hanno Sahlmann: curvature or the radius of this ads space.
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ashtekar: So is it so? Is the reason why you do not have
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ashtekar: the sitter is because of this.
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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.
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Yes.
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ashtekar: thank you.
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Hanno Sahlmann: Thank you. Yeah, please interrupt always and ask
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things.
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Hanno Sahlmann: So I already said that these things mix space, time and the spot
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Hanno Sahlmann: kind of things, so they cannot just be thought of as symmetries of space-time. They
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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
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Hanno Sahlmann: and in quantum field theory, because the cues are odd.
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and because they carry a spin where
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Hanno Sahlmann: they they espn orio, they transform bosons to fermions, and vice versa. They have to it doesn't.
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Hanno Sahlmann: and they cannot but do this, so they are. It's a symmetry that mixes. Go on. And
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Hanno Sahlmann: now I have to
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Hanno Sahlmann: he even a bit more crazy. So
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Hanno Sahlmann: what we have seen is the minimal version of extending the spacetime symmetries. You can do more.
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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.
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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.
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Hanno Sahlmann: And now one can think of higher end.
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Hanno Sahlmann: And, by the way.
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Hanno Sahlmann: many of the statements are very dimension dependent, and so on and so forth. So everything will be in 4 dimensions only.
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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
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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
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ashtekar: use just the
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ashtekar: Euclidean group, and
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ashtekar: I'll have to.
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Hanno Sahlmann: So
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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.
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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.
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Hanno Sahlmann: X. Some, some extended supersymmetries that are possible in Euclidean signature, and one can
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Hanno Sahlmann: one can go look at the the books with. This has been. This has been worked out.
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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.
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ashtekar: Thank you.
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Hanno Sahlmann: Yeah, I think. Thanks for the
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Hanno Sahlmann: of the question.
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Hanno Sahlmann: So.
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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
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Hanno Sahlmann: more
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Hanno Sahlmann: bosonic generators, more even generators. I call them tea here.
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Hanno Sahlmann: That however.
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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
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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
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Hanno Sahlmann: League group in play now, and in terms of
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Hanno Sahlmann: these allergy process.
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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
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Hanno Sahlmann: the fact that these algebra can be realized as matrices that
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Hanno Sahlmann: leave invariant some some inner product, that is, that is symmetric in some dimensions, and
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Hanno Sahlmann: antisymmetric in some other dimensions, and that's why they are called ortho-symmetric.
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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
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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
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Hanno Sahlmann: easy way to see this
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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
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Hanno Sahlmann: chirality and self, Dual and anti-self Dual decomposition of the
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Hanno Sahlmann: of the Lawrence Jen and Lauren School
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Hanno Sahlmann: and on yeah, for the Lawrence school. We can form these complex combinations of goose and boost generators and rotation generators.
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Hanno Sahlmann: and that splits the complexification of the of the Lawrence algebra into 2 commuting parts, the self fuel anti-soft your part.
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Hanno Sahlmann: and a beautiful thing is that this sort of split
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continuous
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Hanno Sahlmann: for the rest of the of the algebra, because the these
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Hanno Sahlmann: odd generators, they also decompose into vial
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parts.
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Hanno Sahlmann: some to to chiral parts, and they close as an algebra, together with the other generators. And so one has this
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Hanno Sahlmann: chiral decomposition of the complexification of this algebra.
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Jerzy Lewandowski: Okay, so, and no, no, no expert.
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Jerzy Lewandowski: The question so very open. They say that
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Jerzy Lewandowski: there there is a necessary condition which has to be satisfied by the background geometry that it has to admit some
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Jerzy Lewandowski: preserved structure depending on
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Jerzy Lewandowski: on this, and
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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
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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
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Hanno Sahlmann: yeah. So in in this case this is the isometries of something, and or
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Hanno Sahlmann: at least part of it, is the isometry of something and
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Hanno Sahlmann: and what symmetries? Yeah, what what structure it's on there that that has a bearing on the
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Hanno Sahlmann: kind of extensions that you can have. Yes.
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Hanno Sahlmann: yeah, Generally speaking, these things are all very sort of rigid, rigid structures and dimension dependent structures.
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ashtekar: I thought that in 40 mentions there was originally a problem with this book, dial conditions without supersymmetry.
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ashtekar: and that, I thought was resolved precisely by supersymmetry.
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ashtekar: But you're saying that Still.
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ashtekar: the number and of generate, you know, generation, if you like, maybe constrained by background curvature. Is that what you're saying.
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Jerzy Lewandowski: I I know that in some case they require, for instance, a covariantly constant spinner to to exist.
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Jerzy Lewandowski: or in some other case currently constant local twist. Or
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ashtekar: is that true? We are talking about about global.
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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
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Hanno Sahlmann: or that space time must probably have very specific properties, so that this is possible.
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Hanno Sahlmann: So right now we're only this is only a statement about anti-sitter.
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Hanno Sahlmann: and and not about anything else. But yeah, I would imagine that
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Hanno Sahlmann: that the properties of that space time that whose isometry group you want to extend, play a big role. Yeah.
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Hanno Sahlmann: Okay.
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Hanno Sahlmann: So
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Hanno Sahlmann: very briefly, super manifold. So there are various ways to define these things, and it's it's important.
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Hanno Sahlmann: because
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Hanno Sahlmann: because, yeah, that the supersymmetry might be a symmetry of
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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
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Hanno Sahlmann: the the most clear
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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
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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.
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Hanno Sahlmann: and then you can take geometric or other notions
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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.
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Hanno Sahlmann: reasonably enough.
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Hanno Sahlmann: in terms of the functions. Then you can deform that function algebra to something non commutative. And still these geometric notions
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Hanno Sahlmann: makes sense, and
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Hanno Sahlmann: it is exactly that for super manifolds. We.
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Hanno Sahlmann: you look at
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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.
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Hanno Sahlmann: real directions, and Q. A. P. Even directions and queue up directions. And so you have P. Even coordinates. And Q. Odd
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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
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Hanno Sahlmann: and
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Hanno Sahlmann: and particular
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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
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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.
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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.
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Hanno Sahlmann: Those are then algebra
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in the other direction
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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
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Hanno Sahlmann: cooling back
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Hanno Sahlmann: functions on a super manifold to an ordinary manifold.
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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
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Hanno Sahlmann: normal manifold as it's embedded somehow in superspace, and then using sort of the odd
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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
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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
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Hanno Sahlmann: ads, or
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Hanno Sahlmann: one takes the product with some fixed, simple, super, manifold. But one can turn this parameter and use a different
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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
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Hanno Sahlmann: having this phenomenon that one loses all the
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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
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Hanno Sahlmann: commutative, an ordinary many fault, then you
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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.
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Hanno Sahlmann: Okay? Good. So that was super symmetry in a in a
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Hanno Sahlmann: nutshell.
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Hanno Sahlmann: Now, before we go to super gravity, Are there more questions.
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ashtekar: Your. Why is there only single parameter in this last map in this embedding map?
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ashtekar: And not, I mean
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ashtekar: so? There must be some, I mean in other words
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ashtekar: you want. F. If not, you want to define f one f 2 up to
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the
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ashtekar: number of
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ashtekar: anti-commuting dimensions. Right? So why is it that one
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ashtekar: one parameter suffices to define this and functions?
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ashtekar: F. One. F. 2 of
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Hanno Sahlmann: oh, so sorry, so
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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
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Hanno Sahlmann: and you want to pull it back
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Hanno Sahlmann: via a map
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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
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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
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Hanno Sahlmann: they go into these coefficients, and they also go here on the other side, and they
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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.
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Hanno Sahlmann: So you you, You introduce additional
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ashtekar: odd directions that I some of the odd things even.
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Hanno Sahlmann: and then you can preserve that information.
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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.
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ashtekar: Yes.
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Hanno Sahlmann: good. So
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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
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Hanno Sahlmann: gauge fields and fermion fields. Then you can have a chance of having a global supersymmetry. And in
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Hanno Sahlmann: if you consider gravity plus fermions, then you have a chance to even have
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Hanno Sahlmann: local supersymmetry.
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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.
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Hanno Sahlmann: And I have written here the action e is the
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that we know and love
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Hanno Sahlmann: say it's this premium, and for generality. I have done this with the cosmological constant, but
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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.
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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
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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
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Hanno Sahlmann: and and this epsilon, this sort of
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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
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Hanno Sahlmann: quantity.
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Hanno Sahlmann: and
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Hanno Sahlmann: certainly also the usual definition of this action.
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Hanno Sahlmann: And
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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
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Hanno Sahlmann: don't see from this action that it would have these these symmetries.
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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,
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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.
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Hanno Sahlmann: and
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Hanno Sahlmann: then, relatively soon after
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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.
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Hanno Sahlmann: And then they proceeded to quantize this theory, using this super symmetry that that they saw, and
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Hanno Sahlmann: And
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Hanno Sahlmann: while this is very nice that there's some supersymmetry manifest. The quantization was also somewhat formal.
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Hanno Sahlmann: Then from another, there's also some work on on spin forms.
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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
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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.
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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
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Hanno Sahlmann: in our theory, although it is not not manifest.
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Hanno Sahlmann: and I will, in the following, follow somewhat that order path, and try to keep supersymmetry manifest.
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Hanno Sahlmann: and to do that, let's look at the old results
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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
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ashtekar: global supersymmetry. Yes, when we come to local since supersymmetry
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ashtekar: will you?
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ashtekar: There are no such constraints. You can just consider any coverage.
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ashtekar: This is my this is an important transition, and I just want to make sure that I'm.
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Hanno Sahlmann: Yeah. Yeah.
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Okay.
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Hanno Sahlmann: Exactly. Exactly.
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Hanno Sahlmann: Exactly.
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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?
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ashtekar: Well, I I don't know whether I agree, but I I appreciate this clarification.
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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
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Erlangen: L, or the sign of the cosmologically constant? Or do we still have that
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Hanno Sahlmann: we still have that I we, I believe we still have.
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Hanno Sahlmann: Okay. So so, at least in what I'm going to tell you this.
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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.
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Hanno Sahlmann: namely, either this Ponk or a extension, or this ads extension, and
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Hanno Sahlmann: and the ads extension doesn't exist for for Ds: so so yeah, we we have that restriction.
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Hanno Sahlmann: Good. So
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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.
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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.
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Hanno Sahlmann: And what people notice is that the post on brackets
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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.
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Hanno Sahlmann: and when you look at this, then you see that this is exactly a again. This this Chiral chiral component
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Hanno Sahlmann: of the ads symmetry extension that I showed you before.
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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.
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Hanno Sahlmann: And so the answer
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Hanno Sahlmann: to these questions comes from a certain formulation of gravity
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Hanno Sahlmann: with a cosmological constant. So here the cosmological constant is very important
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Hanno Sahlmann: gravity as a gauge theory. So for ads. It is a gauge theory with a gauge group, so 3 comma 2,
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Hanno Sahlmann: however, that gauge and variance is broken in the action down to so 3, one.
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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.
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Hanno Sahlmann: and you can take your gauge field
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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.
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Hanno Sahlmann: and
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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.
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Hanno Sahlmann: Hi
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Hanno Sahlmann: action.
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But for this
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Hanno Sahlmann: large gauge group, and you have
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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.
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Hanno Sahlmann: And you can work out that this action with this decomposition gives back
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Hanno Sahlmann: a gravity on with a negative cosmological, constant. And there is some beautiful underlying geometry that is Carton geometry. And
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Hanno Sahlmann: you can describe this very nicely.
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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
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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
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Hanno Sahlmann: I can certainly take the L going to infinity limit, and I think I still also get
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Hanno Sahlmann: a chiral
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Hanno Sahlmann: decomposition of this concrete, this pon carry extension. So that still works. Yeah.
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Hanno Sahlmann: And the and the theory, then also still has the supersymmetry, but it's it's the geometric explanation, is
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Hanno Sahlmann: It's not so clear. I mean it's clear as a limit of of some of something.
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Deepak Vaid: Can I ask a question? Yes, sure.
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Deepak Vaid: So in in the in, the in the Mcdowell Mansuri mechanism
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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
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Deepak Vaid: on the sign of the
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Deepak Vaid: so. So in this case, why are you restricted only to
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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
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Hanno Sahlmann: local super symmetry.
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Hanno Sahlmann: That's the only reason the Mcdonald'souri works for both signs of the cosmological constant.
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Okay.
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Okay, Great.
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Hanno Sahlmann: Okay. You're welcome. So I will try to speed up a little bit.
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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
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Hanno Sahlmann: an additional component in the connection that corresponds to the fermions.
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Hanno Sahlmann: and
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Hanno Sahlmann: we have to
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Hanno Sahlmann: at
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Hanno Sahlmann: another. Oh, and I Haven't talked about it. So this thing is really an internal hot kind of thing. Here
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Hanno Sahlmann: we have to add another one.
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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.
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Hanno Sahlmann: and
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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.
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Hanno Sahlmann: And again, here the gauge invariance is broken down to
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Hanno Sahlmann: the
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Hanno Sahlmann: just through this projection, just to so 3. One
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Hanno Sahlmann: the many fit the the remaining
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Hanno Sahlmann: cage symmetry. You can now come something new. You can make a host version of this
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Hanno Sahlmann: by replacing this this internal watch, but by some linear combination of internal hodge and just the identity.
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Hanno Sahlmann: and you plug that in, and you
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Hanno Sahlmann: see that still everything works out. The equations of motion are not changed.
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Hanno Sahlmann: You get this super gravity.
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Hanno Sahlmann: but now you have it a parameter Beta. You have the immediately parameter in here.
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Hanno Sahlmann: and then you can Even that's the biggest thing you can do. You can do. This calls extension even with extended
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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
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Hanno Sahlmann: something in this projector, and
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Hanno Sahlmann: you take
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Hanno Sahlmann: the most simple thing that you can come up with, namely, the
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Hanno Sahlmann: it's literally the Hodge. And you take this linear communication.
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Hanno Sahlmann: And
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Hanno Sahlmann: again, this works
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Hanno Sahlmann: and some remarks are that what is nice about this, that all these things include boundary terms.
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Hanno Sahlmann: and that, in fact, these terms are unique when you
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Hanno Sahlmann: demand supersymmetry also on the boundary. So, writing it in this beautiful way
355
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Hanno Sahlmann: automatically gives you sort of preferred boundary terms.
356
00:45:28,850 --> 00:45:30,529
Hanno Sahlmann: And
357
00:45:31,260 --> 00:45:33,750
Hanno Sahlmann: this this
358
00:45:33,870 --> 00:45:48,070
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
359
00:45:48,550 --> 00:45:49,200
yeah
360
00:45:49,350 --> 00:45:52,520
Hanno Sahlmann: amusing permit is something like
361
00:45:52,540 --> 00:45:55,540
Hanno Sahlmann: the the
362
00:45:55,560 --> 00:45:57,949
Hanno Sahlmann: permit for
363
00:45:58,310 --> 00:46:00,459
Hanno Sahlmann: Qcd. Is
364
00:46:01,150 --> 00:46:05,460
Hanno Sahlmann: at least in this supersymmetric context, it is enforced.
365
00:46:05,670 --> 00:46:08,520
Hanno Sahlmann: and however
366
00:46:09,290 --> 00:46:25,850
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
367
00:46:25,860 --> 00:46:30,579
Hanno Sahlmann: that is not the Chiral supersymmetry, except for at the special places.
368
00:46:30,720 --> 00:46:37,720
Hanno Sahlmann: And it those the P. Is really like like in
369
00:46:37,810 --> 00:46:43,259
Hanno Sahlmann: in our usual case projection, and
370
00:46:43,710 --> 00:46:45,609
Hanno Sahlmann: it
371
00:46:45,740 --> 00:47:04,289
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.
372
00:47:04,350 --> 00:47:21,590
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
373
00:47:21,600 --> 00:47:25,959
Hanno Sahlmann: electromagnetic field. If you are for n equal to 2,
374
00:47:26,310 --> 00:47:28,179
Hanno Sahlmann: and you can
375
00:47:29,380 --> 00:47:47,400
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
376
00:47:47,540 --> 00:47:50,919
Hanno Sahlmann: E an electric field.
377
00:47:51,810 --> 00:47:57,149
Hanno Sahlmann: and here we have a cosmological constant term, and then there's a boundary term.
378
00:47:58,750 --> 00:48:00,389
Hanno Sahlmann: And
379
00:48:00,980 --> 00:48:02,069
Hanno Sahlmann: so this
380
00:48:02,460 --> 00:48:16,630
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
381
00:48:16,660 --> 00:48:22,759
Hanno Sahlmann: from a gauge symmetry. From this osp 1, 2, or
382
00:48:22,910 --> 00:48:24,660
Hanno Sahlmann: and
383
00:48:24,830 --> 00:48:35,220
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
384
00:48:35,690 --> 00:48:51,940
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.
385
00:48:52,490 --> 00:49:04,179
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
386
00:49:05,520 --> 00:49:06,599
Hanno Sahlmann: that they have.
387
00:49:08,680 --> 00:49:12,809
Hanno Sahlmann: Now, there is some further nice thing. This
388
00:49:12,890 --> 00:49:23,160
Hanno Sahlmann: chosen boundary term in this formulation is just a John Simon's theory, with a for some reason, an imaginary level
389
00:49:23,400 --> 00:49:25,009
and
390
00:49:25,820 --> 00:49:28,690
Hanno Sahlmann: proportional to the cosmological, constant.
391
00:49:29,170 --> 00:49:34,029
Hanno Sahlmann: and additionally because of the boundary terms
392
00:49:34,230 --> 00:49:37,139
Hanno Sahlmann: for us to have the
393
00:49:37,950 --> 00:49:44,410
Hanno Sahlmann: the equations of motion without any sort of boundary contributions.
394
00:49:44,490 --> 00:49:47,500
Hanno Sahlmann: one has to have an additional boundary
395
00:49:47,580 --> 00:49:48,750
Hanno Sahlmann: condition.
396
00:49:48,950 --> 00:49:57,380
Hanno Sahlmann: and that boundary condition also looks very familiar and natural in the kind of variables that we are used to.
397
00:49:59,870 --> 00:50:01,339
Hanno Sahlmann: And
398
00:50:02,120 --> 00:50:17,850
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?
399
00:50:19,940 --> 00:50:31,019
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
400
00:50:31,370 --> 00:50:34,430
Jerzy Lewandowski: transformations part of the symmetries.
401
00:50:40,210 --> 00:50:41,269
Hanno Sahlmann: I would
402
00:50:41,410 --> 00:50:49,259
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.
403
00:50:50,010 --> 00:50:54,520
Hanno Sahlmann: But I don't. But you also are right to say that
404
00:51:00,150 --> 00:51:08,759
Jerzy Lewandowski: Yeah, maybe there are some invisible constraints, but if which star F usually is conformally in current.
405
00:51:09,240 --> 00:51:10,649
Hanno Sahlmann: Yeah, yeah.
406
00:51:13,540 --> 00:51:16,289
Hanno Sahlmann: Okay, good good point. I have to.
407
00:51:16,770 --> 00:51:18,389
Hanno Sahlmann: I have to think about this.
408
00:51:19,170 --> 00:51:28,140
Erlangen: perhaps perhaps another question on the operating conditions. So of course they are very familiar from isolated
409
00:51:28,220 --> 00:51:29,470
Erlangen: horizons.
410
00:51:29,650 --> 00:51:30,399
Hanno Sahlmann: Yeah.
411
00:51:30,620 --> 00:51:32,089
now.
412
00:51:32,240 --> 00:51:37,039
Erlangen: now there is not. And now we are, I think, at the
413
00:51:37,540 --> 00:51:53,470
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
414
00:51:53,540 --> 00:51:58,610
Hanno Sahlmann: at this point. So what we have in mind here is literally a an inner boundary.
415
00:51:58,710 --> 00:52:06,570
Hanno Sahlmann: I see. Okay, but but but indeed, I mean, there's also a discussion that one should probably have about the
416
00:52:07,440 --> 00:52:10,600
Hanno Sahlmann: asymptotic situation. Yeah.
417
00:52:11,220 --> 00:52:12,129
Erlangen: thank you.
418
00:52:12,450 --> 00:52:24,539
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.
419
00:52:24,610 --> 00:52:27,290
ashtekar: Yes; but then, later you said that
420
00:52:27,730 --> 00:52:30,259
ashtekar: this was chosen to be John Simmons.
421
00:52:30,400 --> 00:52:31,500
You?
422
00:52:31,560 --> 00:52:37,020
ashtekar: No, it is not chosen. This is literally you. You
423
00:52:37,400 --> 00:52:41,340
Hanno Sahlmann: You write down this this type of action.
424
00:52:41,390 --> 00:52:53,639
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
425
00:52:53,840 --> 00:53:11,760
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.
426
00:53:12,250 --> 00:53:29,050
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.
427
00:53:29,060 --> 00:53:36,959
ashtekar: then it looks like I can add to the action, a whole bunch of boundary terms. And and but here and
428
00:53:38,170 --> 00:53:41,830
ashtekar: here you are saying that. No, I mean here. If I said that well.
429
00:53:42,310 --> 00:53:46,870
ashtekar: I mean others. You can do this uniquely in the supersymmetric sector, and then project
430
00:53:46,960 --> 00:53:53,469
ashtekar: to the and then say that I get a unique answer, because there's the only one which
431
00:53:53,570 --> 00:54:00,959
Hanno Sahlmann: which can be lifted up to supersymmetric sector. Is that what you're saying?
432
00:54:00,980 --> 00:54:04,370
ashtekar: Okay, so that that's quite exciting. So you're saying that therefore.
433
00:54:04,420 --> 00:54:09,659
ashtekar: also earlier on when you talk about the barbarian, easy, parameter, coming like theta ambiguity.
434
00:54:10,780 --> 00:54:13,909
ashtekar: that all that that also is true, right, namely, that
435
00:54:14,270 --> 00:54:23,490
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.
436
00:54:23,650 --> 00:54:37,699
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
437
00:54:37,710 --> 00:54:45,509
Hanno Sahlmann: is fixed. Yeah, because because we want this, we want this action to describe that unique.
438
00:54:46,960 --> 00:54:54,039
Hanno Sahlmann: that unique, consistent, supersymmetric theory. And then this this thing must be must be here.
439
00:54:54,470 --> 00:55:08,660
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
440
00:55:08,820 --> 00:55:10,129
ashtekar: only that
441
00:55:10,590 --> 00:55:12,639
ashtekar: kind of connection is
442
00:55:12,860 --> 00:55:16,749
ashtekar: is the only unique one that will descend from a supersymmetric one.
443
00:55:17,630 --> 00:55:21,309
Hanno Sahlmann: Yes, I think one can say it like this. I I think
444
00:55:21,440 --> 00:55:22,909
Hanno Sahlmann: one could say it.
445
00:55:23,150 --> 00:55:33,559
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
446
00:55:33,970 --> 00:55:43,019
Hanno Sahlmann: to turn off the this premises to something actually bosonic. But then I
447
00:55:43,030 --> 00:55:55,920
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.
448
00:55:56,430 --> 00:55:57,500
ashtekar: second, Thank you.
449
00:56:01,230 --> 00:56:02,990
Hanno Sahlmann: Good. So
450
00:56:05,430 --> 00:56:07,689
Hanno Sahlmann: let us.
451
00:56:08,450 --> 00:56:11,099
Hanno Sahlmann: So I'm: basically
452
00:56:11,510 --> 00:56:15,410
Hanno Sahlmann: almost out of time. So let me be quick here.
453
00:56:16,770 --> 00:56:20,559
Hanno Sahlmann: Now we want to quantize this, and there are some things that work.
454
00:56:20,830 --> 00:56:40,109
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
455
00:56:40,560 --> 00:56:48,000
Hanno Sahlmann: the harm measure on super groups fails to be positive.
456
00:56:48,970 --> 00:56:54,849
Hanno Sahlmann: just a fact of life. And so there's another complication that one has to deal with. One has to
457
00:56:55,090 --> 00:57:11,489
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.
458
00:57:11,940 --> 00:57:27,989
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
459
00:57:28,130 --> 00:57:45,879
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.
460
00:57:46,030 --> 00:57:56,030
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.
461
00:57:56,270 --> 00:57:57,750
Hanno Sahlmann: and
462
00:57:58,110 --> 00:58:05,200
Hanno Sahlmann: they are are created also relations of these things, and they look
463
00:58:06,190 --> 00:58:14,060
Hanno Sahlmann: exactly like our normal ones, and you can make some algebra. They are also right and left invariant
464
00:58:14,090 --> 00:58:20,589
Hanno Sahlmann: vector fields on the super groups. You can make a Holonomy flux algebra.
465
00:58:20,720 --> 00:58:26,080
Hanno Sahlmann: and you can even have generalized connections
466
00:58:26,170 --> 00:58:36,449
Hanno Sahlmann: that actually mathematically works out quite beautifully. But, as I said, we have problems with the the Hilbert space structure.
467
00:58:36,650 --> 00:58:52,910
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.
468
00:58:52,940 --> 00:58:57,999
Hanno Sahlmann: and there is a basic question whether these should be finite, dimensional representations.
469
00:58:58,240 --> 00:59:01,969
Hanno Sahlmann: yeah, or infinite, dimensional. And one also wants them to
470
00:59:02,070 --> 00:59:13,239
Hanno Sahlmann: have the a tensor category, so that multiplication with components of a Holonomy doesn't make one leave that that category.
471
00:59:13,440 --> 00:59:17,910
Hanno Sahlmann: But it seems in both cases there are options for this.
472
00:59:18,280 --> 00:59:33,740
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.
473
00:59:33,750 --> 00:59:51,210
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
474
00:59:51,420 --> 01:00:00,999
Hanno Sahlmann: in general. This J. Is now a complex number. So, then, this is not positive or anything like this.
475
01:00:02,300 --> 01:00:06,859
Hanno Sahlmann: And so the super area in general is also not real.
476
01:00:07,140 --> 01:00:08,740
Hanno Sahlmann: but
477
01:00:08,800 --> 01:00:14,110
Hanno Sahlmann: that is to be expected because we have not talked about reality conditions.
478
01:00:14,400 --> 01:00:16,089
Hanno Sahlmann: And
479
01:00:16,500 --> 01:00:19,940
Hanno Sahlmann: interestingly, there are J's
480
01:00:21,410 --> 01:00:24,909
Hanno Sahlmann: of a certain form. Yeah, that is given here.
481
01:00:25,270 --> 01:00:33,740
Hanno Sahlmann: The complex jays that give a real area spectrum of this of this form.
482
01:00:34,470 --> 01:00:36,390
Hanno Sahlmann: And
483
01:00:37,160 --> 01:00:41,319
Hanno Sahlmann: okay and sorry, I always say area. And I have to say, this is not
484
01:00:41,390 --> 01:00:43,390
Hanno Sahlmann: area. This is
485
01:00:43,440 --> 01:00:49,550
Hanno Sahlmann: super area now. So this is. This is the length of this super electric field.
486
01:00:51,370 --> 01:00:57,220
Hanno Sahlmann: Now it in a sense it bites us that we are in the supersymmetric situation because
487
01:00:57,560 --> 01:01:03,270
Hanno Sahlmann: a priori, the area of something is not an invariant. It's not a gauge and variant
488
01:01:03,310 --> 01:01:06,029
Hanno Sahlmann: concept. This super area is
489
01:01:06,150 --> 01:01:07,830
perhaps
490
01:01:08,320 --> 01:01:09,709
Hanno Sahlmann: one can.
491
01:01:10,510 --> 01:01:25,290
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
492
01:01:25,400 --> 01:01:34,940
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.
493
01:01:36,660 --> 01:01:37,580
Hanno Sahlmann: And
494
01:01:38,250 --> 01:01:43,029
Hanno Sahlmann: we went on to use this to talk about entropy of
495
01:01:43,080 --> 01:01:44,549
Hanno Sahlmann: of
496
01:01:44,760 --> 01:01:48,770
Hanno Sahlmann: in our boundaries of space-time, because the
497
01:01:48,950 --> 01:02:01,889
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
498
01:02:02,300 --> 01:02:11,229
Hanno Sahlmann: some kind of area operator, but there are problems, namely, the transignment theory is for a non compact
499
01:02:11,870 --> 01:02:17,860
Hanno Sahlmann: super group, and the level is imaginary. And so what doesn't know anything about this
500
01:02:17,940 --> 01:02:24,249
Hanno Sahlmann: mathematically, and we have the reality conditions that we haven't taken care of.
501
01:02:25,010 --> 01:02:32,959
Hanno Sahlmann: and what we decided is we the wind to this work by
502
01:02:33,020 --> 01:02:47,379
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
503
01:02:47,390 --> 01:02:57,840
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.
504
01:02:57,980 --> 01:03:07,460
Hanno Sahlmann: and then they get they get an entropy proportional to quarter of the area without.
505
01:03:07,490 --> 01:03:19,850
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.
506
01:03:19,960 --> 01:03:26,030
Hanno Sahlmann: and then we just had to find what can play the role of su 2 here
507
01:03:26,600 --> 01:03:29,119
Hanno Sahlmann: and there is, indeed, there is a real
508
01:03:29,160 --> 01:03:32,069
Hanno Sahlmann: compact super group
509
01:03:32,240 --> 01:03:41,279
Hanno Sahlmann: that is a real form of this of this super gauge super group that we are talking about.
510
01:03:41,670 --> 01:03:47,160
Hanno Sahlmann: and a compact, a compact, real form of this complexification here.
511
01:03:47,330 --> 01:03:49,090
Hanno Sahlmann: and
512
01:03:49,420 --> 01:03:56,700
Hanno Sahlmann: and the irreducible representations are indeed labeled by jails. And so we can
513
01:03:56,880 --> 01:03:58,120
Hanno Sahlmann: use that
514
01:03:58,990 --> 01:04:11,920
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.
515
01:04:12,080 --> 01:04:16,819
Hanno Sahlmann: and we do the analytic continuation.
516
01:04:16,940 --> 01:04:36,830
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.
517
01:04:36,840 --> 01:04:44,929
Hanno Sahlmann: After the allergy continuation, the right leading order contribution gets reproduced, and there are also.
518
01:04:44,990 --> 01:04:46,049
Hanno Sahlmann: when you.
519
01:04:46,650 --> 01:04:52,820
Hanno Sahlmann: when you make some further assumptions, there are some lower order terms that you can calculate.
520
01:04:52,930 --> 01:05:01,019
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.
521
01:05:01,150 --> 01:05:04,399
Hanno Sahlmann: but it's never the less such as stiff.
522
01:05:05,200 --> 01:05:06,689
Hanno Sahlmann: And
523
01:05:06,790 --> 01:05:09,909
Hanno Sahlmann: with that I I I want to
524
01:05:09,990 --> 01:05:13,330
Hanno Sahlmann: summarize and and stop.
525
01:05:13,400 --> 01:05:15,009
Hanno Sahlmann: So
526
01:05:15,290 --> 01:05:16,519
Hanno Sahlmann: I guess we have
527
01:05:16,580 --> 01:05:19,569
Hanno Sahlmann: clarified the this geometric
528
01:05:20,050 --> 01:05:23,719
Hanno Sahlmann: nature of the super-ashical connection
529
01:05:23,950 --> 01:05:32,330
Hanno Sahlmann: can keep supersymmetry partially manifest. But there is the problem of non- compactness and non
530
01:05:32,350 --> 01:05:35,449
on in reality conditions
531
01:05:37,520 --> 01:05:45,280
Hanno Sahlmann: there is some nice mathematical physics going on here with the super connections in the super parallel transport.
532
01:05:45,550 --> 01:05:46,500
Hanno Sahlmann: and
533
01:05:47,140 --> 01:06:05,850
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
534
01:06:05,860 --> 01:06:11,679
Hanno Sahlmann: dealing with a super symmetric like holes from the quantum gravity.
535
01:06:12,260 --> 01:06:20,520
Hanno Sahlmann: and they are certainly more to do so. Perhaps one can clarify under which conditions these inner boundaries are really
536
01:06:20,540 --> 01:06:21,819
Hanno Sahlmann: like holes.
537
01:06:21,990 --> 01:06:27,569
Hanno Sahlmann: play the whole horizons. It would be very interesting to
538
01:06:27,760 --> 01:06:34,570
Hanno Sahlmann: do this, counting for n equal 2, because this is where the string theory calculations are done.
539
01:06:34,790 --> 01:06:51,259
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
540
01:06:51,910 --> 01:06:55,089
Hanno Sahlmann: some level on which they they meet.
541
01:06:55,440 --> 01:06:57,080
Hanno Sahlmann: And
542
01:06:58,210 --> 01:07:04,119
Hanno Sahlmann: there's one thing that goes through my mind that somehow we don't live in a supersymmetric world.
543
01:07:04,210 --> 01:07:09,549
Hanno Sahlmann: So somehow one has to break supersymmetry, and maybe
544
01:07:09,560 --> 01:07:25,780
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.
545
01:07:26,120 --> 01:07:37,599
Hanno Sahlmann: some various representations or charges of the individual fields. Yeah. So, for example, if I have a a a super group
546
01:07:37,720 --> 01:07:54,189
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
547
01:07:54,250 --> 01:08:07,189
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.
548
01:08:07,590 --> 01:08:12,980
Hanno Sahlmann: and certainly ultimately one would like to have a different space structure for for this
549
01:08:13,150 --> 01:08:14,789
Hanno Sahlmann: this theory tool.
550
01:08:15,690 --> 01:08:19,960
Hanno Sahlmann: So sorry for rushing through the last part.
551
01:08:20,050 --> 01:08:26,319
Hanno Sahlmann: Thank you very much for for listening and yeah, happy for your questions.
552
01:08:34,250 --> 01:08:40,390
Ivan Agullo: Yeah, thank you, kind of. And I think her he had to leave. So I am in charge of the
553
01:08:40,720 --> 01:08:48,640
Ivan Agullo: the question and session, so that we have several questions, and but but the more questions are surely welcome.
554
01:08:49,910 --> 01:08:51,460
And so okay.
555
01:08:52,830 --> 01:08:55,029
Deepak Vaid: So I have a I have a question.
556
01:08:55,090 --> 01:08:55,979
Ivan Agullo: Go ahead.
557
01:08:57,840 --> 01:08:59,379
Deepak Vaid: So
558
01:08:59,939 --> 01:09:07,879
Deepak Vaid: how do we? How do we think of about these supersymmetric spin networks like I mean Normally, we have
559
01:09:08,370 --> 01:09:13,639
Deepak Vaid: spins labeling our edges, and you know interplayers for the word disease.
560
01:09:13,670 --> 01:09:15,859
Deepak Vaid: So how you know what
561
01:09:16,620 --> 01:09:17,639
Deepak Vaid: What does
562
01:09:17,689 --> 01:09:18,439
you know?
563
01:09:18,470 --> 01:09:21,010
Deepak Vaid: What degree of freedom can does
564
01:09:21,510 --> 01:09:23,160
Deepak Vaid: super symmetry add to the
565
01:09:23,500 --> 01:09:24,540
Deepak Vaid: to the structure.
566
01:09:27,160 --> 01:09:33,439
Hanno Sahlmann: Well, so so now this connection field contains
567
01:09:33,540 --> 01:09:37,689
Hanno Sahlmann: also fermionic degrees of freedom. So this
568
01:09:37,880 --> 01:09:48,950
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
569
01:09:49,000 --> 01:09:51,489
of
570
01:09:52,060 --> 01:10:04,889
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.
571
01:10:05,160 --> 01:10:10,410
Hanno Sahlmann: and the labels are going to be labels of the Super.
572
01:10:10,590 --> 01:10:18,999
Hanno Sahlmann: a super gauge group, like like representations of that super gauge group, and
573
01:10:19,600 --> 01:10:24,580
Hanno Sahlmann: as as such, since we don't know the product, we don't
574
01:10:24,630 --> 01:10:37,389
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
575
01:10:37,400 --> 01:10:53,880
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
576
01:10:53,890 --> 01:10:59,019
Hanno Sahlmann: of this, for example, an su 2 subgroup. And I want to ask, how does this
577
01:10:59,030 --> 01:11:14,889
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
578
01:11:14,900 --> 01:11:21,550
Hanno Sahlmann: the so if you're singling out generators in this in this super algebra super coop?
579
01:11:23,530 --> 01:11:24,800
Deepak Vaid: Okay?
580
01:11:25,160 --> 01:11:27,060
Deepak Vaid: And can I ask second question?
581
01:11:28,550 --> 01:11:32,899
Hanno Sahlmann: There's also a question on this on the chat
582
01:11:33,060 --> 01:11:34,290
Hanno Sahlmann: So maybe.
583
01:11:34,450 --> 01:11:37,840
Hanno Sahlmann: or is it related to to your
584
01:11:38,480 --> 01:11:45,059
Deepak Vaid: well, it's about the isolated horizon boundary condition, and
585
01:11:45,470 --> 01:11:59,659
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.
586
01:12:02,430 --> 01:12:03,760
Deepak Vaid: at what sorry
587
01:12:04,440 --> 01:12:08,889
Hanno Sahlmann: at the isolated horizon boundary condition in supergravity.
588
01:12:12,730 --> 01:12:16,170
Deepak Vaid: No, I mean I mean you, you You You showed that relationship earlier right
589
01:12:17,450 --> 01:12:20,450
ashtekar: that that that was for any boundary. So
590
01:12:21,080 --> 01:12:25,669
Deepak Vaid: right, right? So so. My my My question was, that since you are an anti-discer.
591
01:12:26,420 --> 01:12:29,320
Deepak Vaid: would it not also apply to the asymptotic boundary
592
01:12:33,620 --> 01:12:34,929
Deepak Vaid: that that's my question.
593
01:12:39,510 --> 01:12:48,560
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
594
01:12:48,770 --> 01:12:51,699
Hanno Sahlmann: the asymptotic boundary.
595
01:12:52,370 --> 01:12:58,409
Hanno Sahlmann: the not the inner boundary, but the asymptotic one. And then, yeah, there, there's also the
596
01:12:58,570 --> 01:13:00,260
Hanno Sahlmann: Yes, boundary.
597
01:13:00,600 --> 01:13:03,279
Hanno Sahlmann: There is a boundary term, and that
598
01:13:03,360 --> 01:13:19,210
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.
599
01:13:19,220 --> 01:13:37,009
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.
600
01:13:37,610 --> 01:13:42,020
Ivan Agullo: But just the did this.
601
01:13:42,680 --> 01:13:59,629
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
602
01:13:59,640 --> 01:14:03,479
Hanno Sahlmann: so, what about what about those
603
01:14:03,900 --> 01:14:10,559
Hanno Sahlmann: th, those symmetries of what about those constraints? I suppose so. So, indeed!
604
01:14:11,380 --> 01:14:13,700
Hanno Sahlmann: What so
605
01:14:14,280 --> 01:14:15,170
Hanno Sahlmann: so up
606
01:14:16,360 --> 01:14:26,110
Hanno Sahlmann: in terms of constraints. What I said is that somehow one of the 2 supersymmetry constraints gets absorbed into the
607
01:14:26,160 --> 01:14:31,809
Hanno Sahlmann: into the this gauge symmetry. But there's also the other one.
608
01:14:31,840 --> 01:14:45,659
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
609
01:14:45,730 --> 01:14:52,630
Hanno Sahlmann: what is left to do now is to deal with that. If youomorphism, constraint, and the
610
01:14:53,190 --> 01:15:00,630
Hanno Sahlmann: and the the other supersymmetry constraint, so there are more, indeed, there are more.
611
01:15:00,840 --> 01:15:14,719
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.
612
01:15:19,990 --> 01:15:22,949
ashtekar: Okay, Can I just ask a couple of questions, or there are other people?
613
01:15:23,750 --> 01:15:25,280
Ivan Agullo: No, no, please go.
614
01:15:25,320 --> 01:15:45,409
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?
615
01:15:45,470 --> 01:15:52,970
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.
616
01:15:53,120 --> 01:15:54,170
ashtekar: on the other hand.
617
01:15:54,570 --> 01:15:58,249
ashtekar: if I just declare some representations as being sacrificed.
618
01:15:58,390 --> 01:16:04,140
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?
619
01:16:07,120 --> 01:16:15,110
Hanno Sahlmann: Yeah, Not sure because of 2 province. One is that one still has to.
620
01:16:15,190 --> 01:16:23,569
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
621
01:16:23,660 --> 01:16:42,249
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.
622
01:16:42,260 --> 01:16:50,359
Hanno Sahlmann: but i'm not sure that I can. Going from there build up a consistent inner product for many edges.
623
01:16:50,490 --> 01:16:52,280
Hanno Sahlmann: and i'm also.
624
01:16:52,500 --> 01:17:13,619
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
625
01:17:14,190 --> 01:17:16,019
Hanno Sahlmann: and an inner product.
626
01:17:16,300 --> 01:17:28,340
Hanno Sahlmann: But the other question is whether the multiplication operator works so so because in these non compact groups one can
627
01:17:28,970 --> 01:17:31,250
Hanno Sahlmann: sometimes
628
01:17:31,500 --> 01:17:32,340
Hanno Sahlmann: or
629
01:17:32,470 --> 01:17:52,449
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
630
01:17:52,460 --> 01:17:54,469
Hanno Sahlmann: that are
631
01:17:54,530 --> 01:18:01,989
Hanno Sahlmann: that that I want to have in my class. So. So one has to have sort of a a a
632
01:18:02,070 --> 01:18:06,670
Hanno Sahlmann: category of representations that are closed undertaking the tensor product.
633
01:18:08,200 --> 01:18:13,729
ashtekar: Okay, so that. But I mean most of these issues that you're talking about. They are already for the
634
01:18:13,880 --> 01:18:26,849
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
635
01:18:27,570 --> 01:18:41,390
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.
636
01:18:41,430 --> 01:18:56,549
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.
637
01:18:56,620 --> 01:19:06,629
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
638
01:19:06,710 --> 01:19:07,720
ashtekar: if I look at
639
01:19:07,820 --> 01:19:18,059
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.
640
01:19:18,150 --> 01:19:19,510
you know.
641
01:19:19,690 --> 01:19:23,290
ashtekar: real phenomenologists have taken hints from that theory
642
01:19:23,520 --> 01:19:36,520
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
643
01:19:36,530 --> 01:19:43,590
ashtekar: the spin networks and supersymmetric spin networks, etc. I mean, what you're doing is really unifying this penis, and the
644
01:19:43,880 --> 01:19:59,470
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
645
01:20:00,380 --> 01:20:17,609
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?
646
01:20:17,620 --> 01:20:21,319
ashtekar: Because if I the super symmetric version, my
647
01:20:21,440 --> 01:20:23,490
ashtekar: then again, what is.
648
01:20:24,050 --> 01:20:29,290
ashtekar: I mean lattice stage 3. For example, what is going to happen along the edges?
649
01:20:29,360 --> 01:20:37,030
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.
650
01:20:37,510 --> 01:20:42,500
ashtekar: in super symmetric yeah, and equal to 4. Young misery is exactly soluble for some reason.
651
01:20:42,570 --> 01:20:55,609
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
652
01:20:56,090 --> 01:20:59,070
ashtekar: drawing lessons from N. Equal to 4 young meals
653
01:20:59,110 --> 01:21:12,629
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.
654
01:21:12,700 --> 01:21:19,939
ashtekar: I mean, that's the reason why you know the first idea, c. Ft. Correspondence was for, and you could the 4 supersymmetric.
655
01:21:20,180 --> 01:21:30,249
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
656
01:21:30,330 --> 01:21:40,549
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.
657
01:21:41,300 --> 01:21:49,929
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
658
01:21:49,970 --> 01:21:55,559
Hanno Sahlmann: the super symmetry. Then without yeah, we have to think about in this in this direction here.
659
01:21:57,040 --> 01:22:10,799
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
660
01:22:10,810 --> 01:22:29,949
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
661
01:22:29,970 --> 01:22:39,689
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.
662
01:22:40,620 --> 01:22:42,509
Hanno Sahlmann: some, some good.
663
01:22:42,590 --> 01:22:48,029
Hanno Sahlmann: I have some some good idea, I don't know, but this is all
664
01:22:48,160 --> 01:22:49,579
not easy.
665
01:22:50,580 --> 01:23:06,950
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.
666
01:23:08,810 --> 01:23:17,220
Ivan Agullo: Yeah, yeah, just just wanted to ask one technical question, and then on up
667
01:23:17,280 --> 01:23:18,580
Deepak Vaid: procedural comment.
668
01:23:18,790 --> 01:23:21,769
Deepak Vaid: So the technical question is that
669
01:23:21,960 --> 01:23:24,399
Deepak Vaid: just to follow up on my earlier
670
01:23:24,710 --> 01:23:28,299
Deepak Vaid: remark about modifying the spin network structure?
671
01:23:28,880 --> 01:23:33,640
Deepak Vaid: So we know that when you introduce a cost magical, constant, right? What happens
672
01:23:33,680 --> 01:23:36,330
Deepak Vaid: is one of the ways to look at it is that
673
01:23:36,610 --> 01:23:41,820
Deepak Vaid: the the gauge group. Su 2 gets becomes defined right, it becomes su 2. Q.
674
01:23:43,160 --> 01:23:45,480
Deepak Vaid: It happens in some
675
01:23:46,100 --> 01:23:55,379
Deepak Vaid: and and and and then, in order to regulate the 30 you need to
676
01:23:57,540 --> 01:23:59,389
Deepak Vaid: frame the edges
677
01:23:59,580 --> 01:24:03,060
Deepak Vaid: it rather than having them be. You know one-dimensional objects.
678
01:24:03,390 --> 01:24:05,120
Deepak Vaid: So
679
01:24:05,480 --> 01:24:07,350
Deepak Vaid: you know it would be
680
01:24:07,370 --> 01:24:13,679
Deepak Vaid: I mean I don't know if that su. 2 queue thing works for negative customers will be constant or not.
681
01:24:14,020 --> 01:24:17,030
Deepak Vaid: but it if it does, then it would be nice to see
682
01:24:17,530 --> 01:24:20,880
Deepak Vaid: whether you know that is related to.
683
01:24:21,840 --> 01:24:25,950
Deepak Vaid: If if if that gives us something
684
01:24:26,690 --> 01:24:32,009
Hanno Sahlmann: for the for the quantum
685
01:24:32,540 --> 01:24:47,360
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
686
01:24:47,450 --> 01:25:00,669
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
687
01:25:00,680 --> 01:25:09,449
Hanno Sahlmann: for K to infinity. So I suppose small.
688
01:25:10,180 --> 01:25:12,320
Hanno Sahlmann: That's like, okay, yeah.
689
01:25:14,370 --> 01:25:19,220
Deepak Vaid: yeah, Kate K: it goes as elsewhere, right? So on my land, I think. Yeah.
690
01:25:19,380 --> 01:25:28,680
Hanno Sahlmann: But then, and it's large con somehow this doesn't seem right, but but in any case so so so perhaps.
691
01:25:28,690 --> 01:25:42,630
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.
692
01:25:44,420 --> 01:25:52,150
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
693
01:25:52,300 --> 01:25:55,790
Deepak Vaid: where I live and work. People are very, very
694
01:25:56,140 --> 01:25:59,809
Deepak Vaid: rigid about profit channels, and I I am not.
695
01:26:00,700 --> 01:26:03,679
Deepak Vaid: I think, that it would really really serve our community
696
01:26:03,700 --> 01:26:08,500
Deepak Vaid: if we took all these amazing talks that we have recorded.
697
01:26:08,520 --> 01:26:10,390
Deepak Vaid: and we posted them on Youtube.
698
01:26:10,680 --> 01:26:15,079
Deepak Vaid: I I think it would. It would give us give our work tremendous visibility.
699
01:26:15,280 --> 01:26:22,220
Deepak Vaid: and and and and you know we could really really use that. So I mean again, this is just a you know
700
01:26:22,780 --> 01:26:23,389
oh.
701
01:26:23,920 --> 01:26:27,939
Deepak Vaid: so a very humble suggestion to the Organizing Committee
702
01:26:27,960 --> 01:26:31,249
Deepak Vaid: that this is this is something they might they might consider
703
01:26:32,700 --> 01:26:35,679
Deepak Vaid: it. Will, it will definitely, you know, increase
704
01:26:36,080 --> 01:26:38,719
Deepak Vaid: the visibility of of flu quantum gravity
705
01:26:39,170 --> 01:26:43,109
Ivan Agullo: I'll bring. I'll bring that to the up to the committee.
706
01:26:43,350 --> 01:26:46,000
Ivan Agullo: That's a good suggestion. Thank you. Appreciate it.
707
01:26:46,400 --> 01:26:54,010
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.
708
01:26:54,560 --> 01:27:03,919
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.
709
01:27:04,050 --> 01:27:05,070
Ivan Agullo: Wonderful?
710
01:27:06,620 --> 01:27:09,509
Deepak Vaid: Okay, bye, bye, bye, bye.