0 00:00:02,210 --> 00:00:06,719 Jorge Pullin: All right. So, Speaker, today is Hano Salman, who will speak about Karl's Super. Lqg. 1 00:00:08,310 --> 00:00:16,070 Hanno Sahlmann: Thank you very much. So yeah, Thank you very much for the invitation and opportunity to speak here. That's great. 2 00:00:16,160 --> 00:00:27,690 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 3 00:00:27,710 --> 00:00:38,600 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 4 00:00:38,980 --> 00:00:56,199 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 5 00:00:56,210 --> 00:01:04,220 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. 6 00:01:04,239 --> 00:01:08,939 Hanno Sahlmann: Some of it is in collaboration with me Some of it is I 7 00:01:08,960 --> 00:01:12,349 Hanno Sahlmann: him alone, and the 8 00:01:12,570 --> 00:01:17,099 Hanno Sahlmann: potential mistakes on the slides are mine altogether. 9 00:01:17,980 --> 00:01:19,920 Hanno Sahlmann: So 10 00:01:20,000 --> 00:01:23,240 Hanno Sahlmann: Supergravity, super symmetry. What 11 00:01:23,440 --> 00:01:42,260 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. 12 00:01:42,270 --> 00:01:59,079 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 13 00:01:59,090 --> 00:02:03,370 Hanno Sahlmann: gravitational fields and the rest of the physical fields. 14 00:02:03,540 --> 00:02:06,440 Hanno Sahlmann: So space-time, geometry and the rest. 15 00:02:06,530 --> 00:02:07,670 Hanno Sahlmann: and that's 16 00:02:08,050 --> 00:02:13,780 Hanno Sahlmann: one dichotomy that is to be breach it to to be bridged. Yeah. 17 00:02:14,100 --> 00:02:27,819 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 18 00:02:27,830 --> 00:02:38,239 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 19 00:02:38,250 --> 00:02:48,860 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. 20 00:02:50,040 --> 00:02:51,490 Hanno Sahlmann: and 21 00:02:51,650 --> 00:02:52,660 Hanno Sahlmann: now 22 00:02:53,080 --> 00:03:07,449 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 23 00:03:07,460 --> 00:03:15,230 Hanno Sahlmann: theory, and then you and reinterpret in your dimension some of the fields, as sort of matter feels. 24 00:03:15,380 --> 00:03:22,899 Hanno Sahlmann: and then there's also supersymmetry. And today we're going to talk about super symmetry, of course. 25 00:03:24,140 --> 00:03:26,019 Hanno Sahlmann: Now. 26 00:03:26,820 --> 00:03:30,120 Hanno Sahlmann: what is what is the 27 00:03:30,300 --> 00:03:36,989 Hanno Sahlmann: what is super symmetry. Well, one answer was again 28 00:03:37,050 --> 00:03:47,960 Hanno Sahlmann: given in the context of quantum field theory, Hack and Kuprushanski. Notice that there is a loophole to the Coleman Mandela theorem. 29 00:03:48,060 --> 00:03:52,259 Hanno Sahlmann: and that is, if you don't, just let your symmetries be 30 00:03:52,290 --> 00:03:56,899 Hanno Sahlmann: generated by the algebra. But you allow more general 31 00:03:57,020 --> 00:04:07,420 Hanno Sahlmann: algebraic objects that also have anti commutators. Then, indeed, a mix between space-time symmetries and the other symmetries is possible. 32 00:04:07,910 --> 00:04:20,280 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. 33 00:04:20,529 --> 00:04:31,779 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. 34 00:04:31,870 --> 00:04:51,340 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. 35 00:04:51,400 --> 00:05:08,449 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 36 00:05:10,460 --> 00:05:19,719 Hanno Sahlmann: for 2 odd elements it is, it is symmetric, so it is a graded product. 37 00:05:19,740 --> 00:05:29,150 Hanno Sahlmann: and corresponding to this. There's also a graded Jacobi identity as a sort of replacement of. 38 00:05:31,260 --> 00:05:34,499 Hanno Sahlmann: And so this sounds pretty 39 00:05:34,600 --> 00:05:54,560 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. 40 00:05:54,590 --> 00:06:03,279 Hanno Sahlmann: and we know some operations on these forms. Now we know that we can take the derivatives. We can take the 41 00:06:03,400 --> 00:06:10,610 Hanno Sahlmann: exterior, derivative and we can take these interior products just plugging a tangent vector into a form. 42 00:06:10,710 --> 00:06:16,369 Hanno Sahlmann: and there are these identities, for example, a. D squared sival. 43 00:06:16,710 --> 00:06:25,809 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 44 00:06:25,870 --> 00:06:28,919 Hanno Sahlmann: commutative because they Anti. 45 00:06:29,080 --> 00:06:30,999 Hanno Sahlmann: Yeah. Yeah. 46 00:06:31,180 --> 00:06:43,720 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 47 00:06:43,840 --> 00:06:48,210 Hanno Sahlmann: the algebra comes from, and so 48 00:06:48,610 --> 00:06:53,769 Hanno Sahlmann: you can check the other properties that I've given you. And 49 00:06:53,850 --> 00:07:02,380 Hanno Sahlmann: this object, indeed is a super. These operators formula from a big super algebra. And so Cartons 50 00:07:02,430 --> 00:07:10,200 Hanno Sahlmann: magic equation. Yes, also this context just a a anti commutator. 51 00:07:10,430 --> 00:07:11,180 Sure. 52 00:07:12,670 --> 00:07:20,700 Hanno Sahlmann: So maybe these things are not so strange after all. So let's go on and look at 53 00:07:21,950 --> 00:07:30,550 Hanno Sahlmann: super extensions of the spacetime symmetry. So a minimal extension of the Poncari algebra. 54 00:07:30,590 --> 00:07:37,260 Hanno Sahlmann: and what you will turns out to do. Boshanski, tell us you have to add 55 00:07:37,280 --> 00:07:38,510 Hanno Sahlmann: for 56 00:07:38,570 --> 00:07:40,410 generators 57 00:07:40,600 --> 00:07:45,850 Hanno Sahlmann: to the algebra, and these 4 generators are odd. 58 00:07:46,140 --> 00:07:48,100 Hanno Sahlmann: and 59 00:07:48,580 --> 00:07:55,070 Hanno Sahlmann: these generators have to fulfill certain very specific relations with the rest of the generators. 60 00:07:55,150 --> 00:07:56,640 Hanno Sahlmann: And 61 00:07:56,800 --> 00:08:00,190 Hanno Sahlmann: well, first of all, I show you that 62 00:08:00,490 --> 00:08:02,809 Hanno Sahlmann: it was non-zero 63 00:08:03,360 --> 00:08:07,940 Hanno Sahlmann: commutators, anti commutators show that indeed, we have a mixing of 64 00:08:07,990 --> 00:08:18,939 Hanno Sahlmann: space, time, symmetries, and these new generators they don't commute with each other; and, secondly, from this thing here. 65 00:08:19,340 --> 00:08:38,159 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. 66 00:08:38,169 --> 00:08:47,219 Hanno Sahlmann: So this equation just says that these queue transform as a as a for Spino under Lawrence transformations. 67 00:08:48,780 --> 00:08:51,079 Hanno Sahlmann: And now 68 00:08:51,180 --> 00:09:04,149 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 69 00:09:04,320 --> 00:09:07,439 Hanno Sahlmann: is this. 70 00:09:07,570 --> 00:09:22,669 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. 71 00:09:22,730 --> 00:09:25,389 Hanno Sahlmann: and you'll see that they have 72 00:09:25,430 --> 00:09:35,010 Hanno Sahlmann: very similar. There are some additional terms, and they go with the link scale that comes into the picture from the 73 00:09:35,320 --> 00:09:39,250 Hanno Sahlmann: curvature or the radius of this ads space. 74 00:09:41,610 --> 00:09:45,179 ashtekar: So is it so? Is the reason why you do not have 75 00:09:45,630 --> 00:09:47,720 ashtekar: the sitter is because of this. 76 00:09:47,840 --> 00:10:05,589 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. 77 00:10:05,920 --> 00:10:06,620 Yes. 78 00:10:08,490 --> 00:10:09,159 ashtekar: thank you. 79 00:10:10,230 --> 00:10:12,900 Hanno Sahlmann: Thank you. Yeah, please interrupt always and ask 80 00:10:13,470 --> 00:10:14,370 things. 81 00:10:15,380 --> 00:10:21,850 Hanno Sahlmann: So I already said that these things mix space, time and the spot 82 00:10:22,080 --> 00:10:28,269 Hanno Sahlmann: kind of things, so they cannot just be thought of as symmetries of space-time. They 83 00:10:28,410 --> 00:10:44,829 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 84 00:10:44,870 --> 00:10:50,649 Hanno Sahlmann: and in quantum field theory, because the cues are odd. 85 00:10:51,000 --> 00:10:54,090 and because they carry a spin where 86 00:10:54,140 --> 00:11:01,079 Hanno Sahlmann: they they espn orio, they transform bosons to fermions, and vice versa. They have to it doesn't. 87 00:11:01,280 --> 00:11:08,609 Hanno Sahlmann: and they cannot but do this, so they are. It's a symmetry that mixes. Go on. And 88 00:11:11,150 --> 00:11:12,759 Hanno Sahlmann: now I have to 89 00:11:12,840 --> 00:11:17,949 Hanno Sahlmann: he even a bit more crazy. So 90 00:11:18,050 --> 00:11:25,259 Hanno Sahlmann: what we have seen is the minimal version of extending the spacetime symmetries. You can do more. 91 00:11:25,450 --> 00:11:38,150 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. 92 00:11:38,160 --> 00:11:52,190 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. 93 00:11:52,640 --> 00:11:56,329 Hanno Sahlmann: And now one can think of higher end. 94 00:11:56,360 --> 00:11:57,989 Hanno Sahlmann: And, by the way. 95 00:11:58,170 --> 00:12:06,390 Hanno Sahlmann: many of the statements are very dimension dependent, and so on and so forth. So everything will be in 4 dimensions only. 96 00:12:06,750 --> 00:12:23,969 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 97 00:12:24,220 --> 00:12:37,400 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 98 00:12:37,450 --> 00:12:38,970 ashtekar: use just the 99 00:12:39,220 --> 00:12:40,839 ashtekar: Euclidean group, and 100 00:12:41,050 --> 00:12:43,650 ashtekar: I'll have to. 101 00:12:47,770 --> 00:12:48,710 Hanno Sahlmann: So 102 00:12:48,890 --> 00:12:57,949 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. 103 00:12:58,030 --> 00:13:14,869 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. 104 00:13:14,880 --> 00:13:21,820 Hanno Sahlmann: X. Some, some extended supersymmetries that are possible in Euclidean signature, and one can 105 00:13:22,010 --> 00:13:26,939 Hanno Sahlmann: one can go look at the the books with. This has been. This has been worked out. 106 00:13:27,080 --> 00:13:35,009 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. 107 00:13:36,530 --> 00:13:37,200 ashtekar: Thank you. 108 00:13:40,050 --> 00:13:42,080 Hanno Sahlmann: Yeah, I think. Thanks for the 109 00:13:42,220 --> 00:13:43,539 Hanno Sahlmann: of the question. 110 00:13:43,730 --> 00:13:45,440 Hanno Sahlmann: So. 111 00:13:48,500 --> 00:13:59,349 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 112 00:13:59,590 --> 00:14:00,560 Hanno Sahlmann: more 113 00:14:00,660 --> 00:14:05,630 Hanno Sahlmann: bosonic generators, more even generators. I call them tea here. 114 00:14:05,800 --> 00:14:08,509 Hanno Sahlmann: That however. 115 00:14:09,570 --> 00:14:18,640 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 116 00:14:18,750 --> 00:14:33,129 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 117 00:14:33,320 --> 00:14:38,740 Hanno Sahlmann: League group in play now, and in terms of 118 00:14:38,810 --> 00:14:41,499 Hanno Sahlmann: these allergy process. 119 00:14:41,540 --> 00:14:53,519 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 120 00:14:53,790 --> 00:15:01,259 Hanno Sahlmann: the fact that these algebra can be realized as matrices that 121 00:15:01,400 --> 00:15:10,769 Hanno Sahlmann: leave invariant some some inner product, that is, that is symmetric in some dimensions, and 122 00:15:10,890 --> 00:15:17,630 Hanno Sahlmann: antisymmetric in some other dimensions, and that's why they are called ortho-symmetric. 123 00:15:18,760 --> 00:15:30,410 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 124 00:15:30,420 --> 00:15:42,480 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 125 00:15:42,740 --> 00:15:45,119 Hanno Sahlmann: easy way to see this 126 00:15:50,280 --> 00:16:03,409 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 127 00:16:03,420 --> 00:16:11,699 Hanno Sahlmann: chirality and self, Dual and anti-self Dual decomposition of the 128 00:16:11,770 --> 00:16:15,560 Hanno Sahlmann: of the Lawrence Jen and Lauren School 129 00:16:15,770 --> 00:16:26,469 Hanno Sahlmann: and on yeah, for the Lawrence school. We can form these complex combinations of goose and boost generators and rotation generators. 130 00:16:26,540 --> 00:16:37,400 Hanno Sahlmann: and that splits the complexification of the of the Lawrence algebra into 2 commuting parts, the self fuel anti-soft your part. 131 00:16:37,810 --> 00:16:42,219 Hanno Sahlmann: and a beautiful thing is that this sort of split 132 00:16:42,290 --> 00:16:44,240 continuous 133 00:16:44,300 --> 00:16:50,010 Hanno Sahlmann: for the rest of the of the algebra, because the these 134 00:16:50,240 --> 00:16:55,589 Hanno Sahlmann: odd generators, they also decompose into vial 135 00:16:55,630 --> 00:16:57,470 parts. 136 00:16:57,650 --> 00:17:09,299 Hanno Sahlmann: some to to chiral parts, and they close as an algebra, together with the other generators. And so one has this 137 00:17:09,839 --> 00:17:14,329 Hanno Sahlmann: chiral decomposition of the complexification of this algebra. 138 00:17:21,119 --> 00:17:24,530 Jerzy Lewandowski: Okay, so, and no, no, no expert. 139 00:17:24,609 --> 00:17:28,589 Jerzy Lewandowski: The question so very open. They say that 140 00:17:28,750 --> 00:17:36,770 Jerzy Lewandowski: there there is a necessary condition which has to be satisfied by the background geometry that it has to admit some 141 00:17:37,350 --> 00:17:40,050 Jerzy Lewandowski: preserved structure depending on 142 00:17:40,330 --> 00:17:42,910 Jerzy Lewandowski: on this, and 143 00:17:43,230 --> 00:17:58,840 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 144 00:17:58,870 --> 00:18:18,440 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 145 00:18:19,300 --> 00:18:24,619 Hanno Sahlmann: yeah. So in in this case this is the isometries of something, and or 146 00:18:24,670 --> 00:18:27,759 Hanno Sahlmann: at least part of it, is the isometry of something and 147 00:18:27,780 --> 00:18:34,510 Hanno Sahlmann: and what symmetries? Yeah, what what structure it's on there that that has a bearing on the 148 00:18:34,780 --> 00:18:37,759 Hanno Sahlmann: kind of extensions that you can have. Yes. 149 00:18:42,230 --> 00:18:51,810 Hanno Sahlmann: yeah, Generally speaking, these things are all very sort of rigid, rigid structures and dimension dependent structures. 150 00:18:52,550 --> 00:18:58,950 ashtekar: I thought that in 40 mentions there was originally a problem with this book, dial conditions without supersymmetry. 151 00:18:59,320 --> 00:19:03,279 ashtekar: and that, I thought was resolved precisely by supersymmetry. 152 00:19:03,320 --> 00:19:05,470 ashtekar: But you're saying that Still. 153 00:19:05,510 --> 00:19:13,539 ashtekar: the number and of generate, you know, generation, if you like, maybe constrained by background curvature. Is that what you're saying. 154 00:19:15,750 --> 00:19:23,240 Jerzy Lewandowski: I I know that in some case they require, for instance, a covariantly constant spinner to to exist. 155 00:19:23,890 --> 00:19:28,200 Jerzy Lewandowski: or in some other case currently constant local twist. Or 156 00:19:28,350 --> 00:19:32,400 ashtekar: is that true? We are talking about about global. 157 00:19:33,390 --> 00:19:46,869 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 158 00:19:47,330 --> 00:19:53,220 Hanno Sahlmann: or that space time must probably have very specific properties, so that this is possible. 159 00:19:53,500 --> 00:19:58,869 Hanno Sahlmann: So right now we're only this is only a statement about anti-sitter. 160 00:19:59,000 --> 00:20:03,930 Hanno Sahlmann: and and not about anything else. But yeah, I would imagine that 161 00:20:03,960 --> 00:20:13,370 Hanno Sahlmann: that the properties of that space time that whose isometry group you want to extend, play a big role. Yeah. 162 00:20:21,540 --> 00:20:22,480 Hanno Sahlmann: Okay. 163 00:20:22,760 --> 00:20:24,680 Hanno Sahlmann: So 164 00:20:24,920 --> 00:20:33,539 Hanno Sahlmann: very briefly, super manifold. So there are various ways to define these things, and it's it's important. 165 00:20:33,580 --> 00:20:34,600 Hanno Sahlmann: because 166 00:20:35,430 --> 00:20:39,979 Hanno Sahlmann: because, yeah, that the supersymmetry might be a symmetry of 167 00:20:40,150 --> 00:20:56,229 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 168 00:20:56,660 --> 00:20:58,620 Hanno Sahlmann: the the most clear 169 00:20:58,670 --> 00:21:11,310 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 170 00:21:11,500 --> 00:21:30,570 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. 171 00:21:30,580 --> 00:21:34,819 Hanno Sahlmann: and then you can take geometric or other notions 172 00:21:34,930 --> 00:21:44,959 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. 173 00:21:45,570 --> 00:21:47,170 Hanno Sahlmann: reasonably enough. 174 00:21:47,220 --> 00:21:56,260 Hanno Sahlmann: in terms of the functions. Then you can deform that function algebra to something non commutative. And still these geometric notions 175 00:21:56,280 --> 00:21:58,280 Hanno Sahlmann: makes sense, and 176 00:21:58,550 --> 00:22:02,029 Hanno Sahlmann: it is exactly that for super manifolds. We. 177 00:22:02,050 --> 00:22:04,460 Hanno Sahlmann: you look at 178 00:22:04,560 --> 00:22:23,280 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. 179 00:22:23,290 --> 00:22:35,750 Hanno Sahlmann: real directions, and Q. A. P. Even directions and queue up directions. And so you have P. Even coordinates. And Q. Odd 180 00:22:35,950 --> 00:22:55,060 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 181 00:22:55,980 --> 00:22:57,249 Hanno Sahlmann: and 182 00:22:57,310 --> 00:23:00,060 Hanno Sahlmann: and particular 183 00:23:00,240 --> 00:23:17,880 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 184 00:23:18,000 --> 00:23:28,090 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. 185 00:23:30,170 --> 00:23:44,969 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. 186 00:23:45,190 --> 00:23:47,499 Hanno Sahlmann: Those are then algebra 187 00:23:48,360 --> 00:23:50,790 in the other direction 188 00:23:50,910 --> 00:24:06,670 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 189 00:24:06,860 --> 00:24:08,090 Hanno Sahlmann: cooling back 190 00:24:08,230 --> 00:24:12,499 Hanno Sahlmann: functions on a super manifold to an ordinary manifold. 191 00:24:13,890 --> 00:24:26,930 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 192 00:24:27,110 --> 00:24:34,750 Hanno Sahlmann: normal manifold as it's embedded somehow in superspace, and then using sort of the odd 193 00:24:34,760 --> 00:24:51,730 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 194 00:24:52,020 --> 00:25:06,590 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 195 00:25:06,880 --> 00:25:08,349 Hanno Sahlmann: ads, or 196 00:25:08,470 --> 00:25:21,109 Hanno Sahlmann: one takes the product with some fixed, simple, super, manifold. But one can turn this parameter and use a different 197 00:25:21,120 --> 00:25:39,040 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 198 00:25:39,050 --> 00:25:42,880 Hanno Sahlmann: having this phenomenon that one loses all the 199 00:25:42,900 --> 00:25:59,920 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 200 00:25:59,930 --> 00:26:03,990 Hanno Sahlmann: commutative, an ordinary many fault, then you 201 00:26:04,020 --> 00:26:20,900 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. 202 00:26:23,560 --> 00:26:29,449 Hanno Sahlmann: Okay? Good. So that was super symmetry in a in a 203 00:26:29,630 --> 00:26:30,940 Hanno Sahlmann: nutshell. 204 00:26:30,960 --> 00:26:35,979 Hanno Sahlmann: Now, before we go to super gravity, Are there more questions. 205 00:26:39,380 --> 00:26:43,279 ashtekar: Your. Why is there only single parameter in this last map in this embedding map? 206 00:26:43,410 --> 00:26:44,759 ashtekar: And not, I mean 207 00:26:44,820 --> 00:26:47,129 ashtekar: so? There must be some, I mean in other words 208 00:26:47,220 --> 00:26:52,640 ashtekar: you want. F. If not, you want to define f one f 2 up to 209 00:26:52,790 --> 00:26:54,099 the 210 00:26:54,120 --> 00:26:55,540 ashtekar: number of 211 00:26:56,060 --> 00:27:00,729 ashtekar: anti-commuting dimensions. Right? So why is it that one 212 00:27:02,270 --> 00:27:06,579 ashtekar: one parameter suffices to define this and functions? 213 00:27:07,370 --> 00:27:08,870 ashtekar: F. One. F. 2 of 214 00:27:10,050 --> 00:27:13,379 Hanno Sahlmann: oh, so sorry, so 215 00:27:13,900 --> 00:27:24,720 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 216 00:27:24,990 --> 00:27:27,699 Hanno Sahlmann: and you want to pull it back 217 00:27:28,130 --> 00:27:29,650 Hanno Sahlmann: via a map 218 00:27:29,780 --> 00:27:44,870 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 219 00:27:44,880 --> 00:28:00,140 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 220 00:28:00,150 --> 00:28:07,430 Hanno Sahlmann: they go into these coefficients, and they also go here on the other side, and they 221 00:28:07,450 --> 00:28:25,649 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. 222 00:28:26,740 --> 00:28:30,140 Hanno Sahlmann: So you you, You introduce additional 223 00:28:30,260 --> 00:28:35,489 ashtekar: odd directions that I some of the odd things even. 224 00:28:35,620 --> 00:28:39,370 Hanno Sahlmann: and then you can preserve that information. 225 00:28:39,440 --> 00:28:45,550 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. 226 00:28:45,580 --> 00:28:50,050 ashtekar: Yes. 227 00:28:56,320 --> 00:28:59,080 Hanno Sahlmann: good. So 228 00:28:59,680 --> 00:29:16,999 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 229 00:29:17,870 --> 00:29:27,130 Hanno Sahlmann: gauge fields and fermion fields. Then you can have a chance of having a global supersymmetry. And in 230 00:29:27,240 --> 00:29:33,000 Hanno Sahlmann: if you consider gravity plus fermions, then you have a chance to even have 231 00:29:33,170 --> 00:29:35,309 Hanno Sahlmann: local supersymmetry. 232 00:29:35,630 --> 00:29:46,339 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. 233 00:29:46,430 --> 00:29:52,399 Hanno Sahlmann: And I have written here the action e is the 234 00:29:52,990 --> 00:29:55,480 that we know and love 235 00:29:55,510 --> 00:30:05,269 Hanno Sahlmann: say it's this premium, and for generality. I have done this with the cosmological constant, but 236 00:30:05,630 --> 00:30:24,730 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. 237 00:30:24,740 --> 00:30:44,130 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 238 00:30:44,550 --> 00:30:56,230 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 239 00:30:57,470 --> 00:31:00,839 Hanno Sahlmann: and and this epsilon, this sort of 240 00:31:01,230 --> 00:31:11,530 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 241 00:31:11,560 --> 00:31:12,630 Hanno Sahlmann: quantity. 242 00:31:13,760 --> 00:31:15,410 Hanno Sahlmann: and 243 00:31:15,550 --> 00:31:21,249 Hanno Sahlmann: certainly also the usual definition of this action. 244 00:31:22,730 --> 00:31:23,700 Hanno Sahlmann: And 245 00:31:24,060 --> 00:31:35,309 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 246 00:31:35,330 --> 00:31:40,800 Hanno Sahlmann: don't see from this action that it would have these these symmetries. 247 00:31:41,160 --> 00:31:53,150 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, 248 00:31:53,160 --> 00:32:10,210 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. 249 00:32:10,300 --> 00:32:11,790 Hanno Sahlmann: and 250 00:32:12,090 --> 00:32:15,970 Hanno Sahlmann: then, relatively soon after 251 00:32:16,020 --> 00:32:30,850 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. 252 00:32:32,670 --> 00:32:42,209 Hanno Sahlmann: And then they proceeded to quantize this theory, using this super symmetry that that they saw, and 253 00:32:43,240 --> 00:32:44,700 Hanno Sahlmann: And 254 00:32:44,790 --> 00:32:54,009 Hanno Sahlmann: while this is very nice that there's some supersymmetry manifest. The quantization was also somewhat formal. 255 00:32:54,580 --> 00:32:59,679 Hanno Sahlmann: Then from another, there's also some work on on spin forms. 256 00:32:59,930 --> 00:33:18,259 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 257 00:33:18,270 --> 00:33:34,539 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. 258 00:33:34,550 --> 00:33:46,599 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 259 00:33:46,970 --> 00:33:50,520 Hanno Sahlmann: in our theory, although it is not not manifest. 260 00:33:50,660 --> 00:34:01,110 Hanno Sahlmann: and I will, in the following, follow somewhat that order path, and try to keep supersymmetry manifest. 261 00:34:01,380 --> 00:34:07,210 Hanno Sahlmann: and to do that, let's look at the old results 262 00:34:07,410 --> 00:34:20,429 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 263 00:34:20,480 --> 00:34:25,750 ashtekar: global supersymmetry. Yes, when we come to local since supersymmetry 264 00:34:25,790 --> 00:34:27,060 ashtekar: will you? 265 00:34:27,230 --> 00:34:30,749 ashtekar: There are no such constraints. You can just consider any coverage. 266 00:34:30,770 --> 00:34:36,029 ashtekar: This is my this is an important transition, and I just want to make sure that I'm. 267 00:34:36,120 --> 00:34:37,509 Hanno Sahlmann: Yeah. Yeah. 268 00:34:37,530 --> 00:34:38,169 Okay. 269 00:34:38,260 --> 00:34:40,049 Hanno Sahlmann: Exactly. Exactly. 270 00:34:41,560 --> 00:34:43,469 Hanno Sahlmann: Exactly. 271 00:34:44,960 --> 00:34:57,720 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? 272 00:34:58,100 --> 00:35:04,739 ashtekar: Well, I I don't know whether I agree, but I I appreciate this clarification. 273 00:35:04,970 --> 00:35:17,459 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 274 00:35:17,510 --> 00:35:22,040 Erlangen: L, or the sign of the cosmologically constant? Or do we still have that 275 00:35:22,550 --> 00:35:26,689 Hanno Sahlmann: we still have that I we, I believe we still have. 276 00:35:26,900 --> 00:35:32,370 Hanno Sahlmann: Okay. So so, at least in what I'm going to tell you this. 277 00:35:33,360 --> 00:35:50,439 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. 278 00:35:50,500 --> 00:35:55,580 Hanno Sahlmann: namely, either this Ponk or a extension, or this ads extension, and 279 00:35:55,610 --> 00:36:03,870 Hanno Sahlmann: and the ads extension doesn't exist for for Ds: so so yeah, we we have that restriction. 280 00:36:09,350 --> 00:36:11,740 Hanno Sahlmann: Good. So 281 00:36:13,080 --> 00:36:24,580 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. 282 00:36:24,590 --> 00:36:40,550 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. 283 00:36:40,630 --> 00:36:46,390 Hanno Sahlmann: And what people notice is that the post on brackets 284 00:36:46,490 --> 00:37:06,179 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. 285 00:37:06,400 --> 00:37:20,170 Hanno Sahlmann: and when you look at this, then you see that this is exactly a again. This this Chiral chiral component 286 00:37:20,180 --> 00:37:26,399 Hanno Sahlmann: of the ads symmetry extension that I showed you before. 287 00:37:27,680 --> 00:37:45,250 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. 288 00:37:46,900 --> 00:37:50,530 Hanno Sahlmann: And so the answer 289 00:37:50,660 --> 00:37:56,660 Hanno Sahlmann: to these questions comes from a certain formulation of gravity 290 00:37:56,940 --> 00:38:01,799 Hanno Sahlmann: with a cosmological constant. So here the cosmological constant is very important 291 00:38:01,870 --> 00:38:12,560 Hanno Sahlmann: gravity as a gauge theory. So for ads. It is a gauge theory with a gauge group, so 3 comma 2, 292 00:38:12,960 --> 00:38:20,390 Hanno Sahlmann: however, that gauge and variance is broken in the action down to so 3, one. 293 00:38:20,580 --> 00:38:37,590 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. 294 00:38:37,790 --> 00:38:40,849 Hanno Sahlmann: and you can take your gauge field 295 00:38:40,870 --> 00:38:54,759 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. 296 00:38:55,390 --> 00:38:56,830 Hanno Sahlmann: and 297 00:38:57,130 --> 00:39:06,330 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. 298 00:39:06,390 --> 00:39:07,189 Hanno Sahlmann: Hi 299 00:39:07,250 --> 00:39:08,180 Hanno Sahlmann: action. 300 00:39:08,310 --> 00:39:10,220 But for this 301 00:39:10,280 --> 00:39:14,670 Hanno Sahlmann: large gauge group, and you have 302 00:39:14,730 --> 00:39:29,780 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. 303 00:39:30,270 --> 00:39:38,889 Hanno Sahlmann: And you can work out that this action with this decomposition gives back 304 00:39:38,970 --> 00:39:49,670 Hanno Sahlmann: a gravity on with a negative cosmological, constant. And there is some beautiful underlying geometry that is Carton geometry. And 305 00:39:49,720 --> 00:39:52,549 Hanno Sahlmann: you can describe this very nicely. 306 00:39:54,210 --> 00:40:05,569 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 307 00:40:05,830 --> 00:40:21,579 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 308 00:40:21,590 --> 00:40:29,180 Hanno Sahlmann: I can certainly take the L going to infinity limit, and I think I still also get 309 00:40:29,220 --> 00:40:30,819 Hanno Sahlmann: a chiral 310 00:40:31,010 --> 00:40:41,530 Hanno Sahlmann: decomposition of this concrete, this pon carry extension. So that still works. Yeah. 311 00:40:41,560 --> 00:40:51,450 Hanno Sahlmann: And the and the theory, then also still has the supersymmetry, but it's it's the geometric explanation, is 312 00:40:51,880 --> 00:40:58,020 Hanno Sahlmann: It's not so clear. I mean it's clear as a limit of of some of something. 313 00:41:02,980 --> 00:41:05,970 Deepak Vaid: Can I ask a question? Yes, sure. 314 00:41:06,780 --> 00:41:11,970 Deepak Vaid: So in in the in, the in the Mcdowell Mansuri mechanism 315 00:41:12,020 --> 00:41:22,990 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 316 00:41:23,030 --> 00:41:24,959 Deepak Vaid: on the sign of the 317 00:41:25,960 --> 00:41:33,299 Deepak Vaid: so. So in this case, why are you restricted only to 318 00:41:33,420 --> 00:41:46,519 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 319 00:41:46,550 --> 00:41:49,200 Hanno Sahlmann: local super symmetry. 320 00:41:49,750 --> 00:41:55,779 Hanno Sahlmann: That's the only reason the Mcdonald'souri works for both signs of the cosmological constant. 321 00:41:57,820 --> 00:41:58,509 Okay. 322 00:41:59,080 --> 00:42:00,359 Okay, Great. 323 00:42:00,490 --> 00:42:05,940 Hanno Sahlmann: Okay. You're welcome. So I will try to speed up a little bit. 324 00:42:06,330 --> 00:42:25,580 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 325 00:42:25,590 --> 00:42:31,519 Hanno Sahlmann: an additional component in the connection that corresponds to the fermions. 326 00:42:31,730 --> 00:42:33,279 Hanno Sahlmann: and 327 00:42:33,430 --> 00:42:35,030 Hanno Sahlmann: we have to 328 00:42:35,230 --> 00:42:36,069 Hanno Sahlmann: at 329 00:42:36,100 --> 00:42:43,439 Hanno Sahlmann: another. Oh, and I Haven't talked about it. So this thing is really an internal hot kind of thing. Here 330 00:42:43,490 --> 00:42:46,709 Hanno Sahlmann: we have to add another one. 331 00:42:46,810 --> 00:43:03,379 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. 332 00:43:03,980 --> 00:43:05,270 Hanno Sahlmann: and 333 00:43:05,550 --> 00:43:15,530 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. 334 00:43:15,740 --> 00:43:22,250 Hanno Sahlmann: And again, here the gauge invariance is broken down to 335 00:43:22,670 --> 00:43:23,240 Hanno Sahlmann: the 336 00:43:24,810 --> 00:43:29,780 Hanno Sahlmann: just through this projection, just to so 3. One 337 00:43:32,190 --> 00:43:34,740 Hanno Sahlmann: the many fit the the remaining 338 00:43:35,550 --> 00:43:43,059 Hanno Sahlmann: cage symmetry. You can now come something new. You can make a host version of this 339 00:43:43,400 --> 00:43:55,069 Hanno Sahlmann: by replacing this this internal watch, but by some linear combination of internal hodge and just the identity. 340 00:43:55,270 --> 00:44:00,810 Hanno Sahlmann: and you plug that in, and you 341 00:44:01,410 --> 00:44:07,040 Hanno Sahlmann: see that still everything works out. The equations of motion are not changed. 342 00:44:07,070 --> 00:44:10,130 Hanno Sahlmann: You get this super gravity. 343 00:44:10,890 --> 00:44:15,770 Hanno Sahlmann: but now you have it a parameter Beta. You have the immediately parameter in here. 344 00:44:16,900 --> 00:44:23,930 Hanno Sahlmann: and then you can Even that's the biggest thing you can do. You can do. This calls extension even with extended 345 00:44:23,940 --> 00:44:43,390 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 346 00:44:43,530 --> 00:44:47,399 Hanno Sahlmann: something in this projector, and 347 00:44:47,720 --> 00:44:49,750 Hanno Sahlmann: you take 348 00:44:49,840 --> 00:44:54,200 Hanno Sahlmann: the most simple thing that you can come up with, namely, the 349 00:44:54,330 --> 00:44:58,109 Hanno Sahlmann: it's literally the Hodge. And you take this linear communication. 350 00:44:58,460 --> 00:45:00,029 Hanno Sahlmann: And 351 00:45:00,120 --> 00:45:02,310 Hanno Sahlmann: again, this works 352 00:45:02,380 --> 00:45:11,829 Hanno Sahlmann: and some remarks are that what is nice about this, that all these things include boundary terms. 353 00:45:12,340 --> 00:45:16,789 Hanno Sahlmann: and that, in fact, these terms are unique when you 354 00:45:16,830 --> 00:45:22,980 Hanno Sahlmann: demand supersymmetry also on the boundary. So, writing it in this beautiful way 355 00:45:23,080 --> 00:45:28,579 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.