1 00:00:00,080 --> 00:00:02,430 The following content is provided under a Creative 2 00:00:02,430 --> 00:00:03,820 Commons license. 3 00:00:03,820 --> 00:00:06,060 Your support will help MIT OpenCourseWare 4 00:00:06,060 --> 00:00:10,150 continue to offer high-quality educational resources for free. 5 00:00:10,150 --> 00:00:12,690 To make a donation or to view additional materials 6 00:00:12,690 --> 00:00:16,600 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,600 --> 00:00:17,255 at ocw.mit.edu. 8 00:00:22,220 --> 00:00:24,130 HONG LIU: OK, let's start. 9 00:00:24,130 --> 00:00:27,490 So first let me just remind you what we 10 00:00:27,490 --> 00:00:34,380 did at the end of last lecture. 11 00:00:34,380 --> 00:00:39,700 So we see that the large N expansion of gauge theory 12 00:00:39,700 --> 00:00:45,460 have essentially exactly the same mathematical structure 13 00:00:45,460 --> 00:00:50,980 with, say, the mathematics of the [? N string ?] scattering. 14 00:00:50,980 --> 00:00:54,250 And so here the observable is a correlation function 15 00:00:54,250 --> 00:00:57,020 of gauging [? invariant ?] operators. 16 00:00:57,020 --> 00:01:01,520 And then these have a large N expansion as follows. 17 00:01:01,520 --> 00:01:04,290 And on this side you have just an N string scattering 18 00:01:04,290 --> 00:01:04,930 amplitude. 19 00:01:04,930 --> 00:01:09,800 Just imagine you have some kind of scattering of strings, 20 00:01:09,800 --> 00:01:12,036 with total number of N strings. 21 00:01:12,036 --> 00:01:13,410 And then this also have expansion 22 00:01:13,410 --> 00:01:18,460 in terms of the string counting in this form. 23 00:01:18,460 --> 00:01:27,330 So now, if we identify-- so if we can identify the g 24 00:01:27,330 --> 00:01:32,810 string as 1/N. So if we identify g string with 1/N, 25 00:01:32,810 --> 00:01:37,040 then these two are essentially the same kind of expansion, OK? 26 00:01:37,040 --> 00:01:45,050 And you also can identify these external strings, 27 00:01:45,050 --> 00:01:51,230 string states, within the large N 28 00:01:51,230 --> 00:01:56,285 theory which we called the glueball states 29 00:01:56,285 --> 00:01:57,410 for single-trace operators. 30 00:02:05,150 --> 00:02:07,715 And then each case is corresponding to [? sum ?] 31 00:02:07,715 --> 00:02:08,465 over the topology. 32 00:02:11,830 --> 00:02:14,120 It's an expansion [? in ?] terms of the topology. 33 00:02:14,120 --> 00:02:29,516 So here is the topology of the worldsheet string. 34 00:02:29,516 --> 00:02:31,390 And here is the topology of Feynman diagrams. 35 00:02:43,010 --> 00:02:45,160 Here is the topology of the Feynman diagrams. 36 00:02:49,800 --> 00:02:53,690 So still at this stage, it's just 37 00:02:53,690 --> 00:02:56,359 like a mathematical correspondence. 38 00:02:56,359 --> 00:02:58,400 We're looking at two completely different things. 39 00:03:01,370 --> 00:03:06,940 But probably there's no-- yeah, no obvious connection 40 00:03:06,940 --> 00:03:11,060 between these two objects we are discussing. 41 00:03:14,446 --> 00:03:18,740 Yeah, we just have a precise mathematical structure. 42 00:03:18,740 --> 00:03:26,760 But one can actually argue that, actually, they also describe 43 00:03:26,760 --> 00:03:32,025 the same physical structure once you realize that when 44 00:03:32,025 --> 00:03:35,740 you sum over all possible Feynman diagrams. 45 00:03:35,740 --> 00:03:46,830 So once you realize that each Feynman diagram, say, 46 00:03:46,830 --> 00:04:02,880 of genus-h can be considered as a partition, 47 00:04:02,880 --> 00:04:10,340 or in other words, triangulization 48 00:04:10,340 --> 00:04:19,055 over genus-h surfaces, [? 2D ?] surfaces. 49 00:04:19,055 --> 00:04:19,555 OK. 50 00:04:26,920 --> 00:04:35,810 So if you write more explicitly this fh, 51 00:04:35,810 --> 00:04:45,787 so if we write explicitly this fh, then this fh, this fnh, 52 00:04:45,787 --> 00:04:47,370 then will be corresponding to your sum 53 00:04:47,370 --> 00:04:58,290 of all Feynman diagrams of genus-h. 54 00:05:06,170 --> 00:05:12,425 Suppose G is the expression for each Feynman diagram. 55 00:05:16,596 --> 00:05:17,470 Say for each diagram. 56 00:05:26,780 --> 00:05:31,220 And then I can just rewrite this. 57 00:05:31,220 --> 00:05:46,170 In some sense, I [? accept ?] all possible triangulation 58 00:05:46,170 --> 00:05:50,140 of [? a genus-g ?] surface. 59 00:05:56,620 --> 00:06:06,355 Say there will be some weight G. And summing over 60 00:06:06,355 --> 00:06:12,030 all possible triangulations of a surface 61 00:06:12,030 --> 00:06:16,720 is essentially-- so this is essentially 62 00:06:16,720 --> 00:06:29,180 the same as this sum over all possible surfaces. 63 00:06:34,980 --> 00:06:38,290 So this is a discrete version. 64 00:06:46,880 --> 00:06:50,150 So sum all possible triangulations 65 00:06:50,150 --> 00:06:54,365 of some genus-g surfaces, or translations 66 00:06:54,365 --> 00:06:58,107 of genus-g surfaces. 67 00:06:58,107 --> 00:07:00,440 Then they can be considered as a discrete version of sum 68 00:07:00,440 --> 00:07:04,112 over all possible surfaces, OK? 69 00:07:04,112 --> 00:07:06,070 AUDIENCE: So you're saying it's like a sum over 70 00:07:06,070 --> 00:07:08,000 [? syntheses, ?] like a simple [? x? ?] 71 00:07:08,000 --> 00:07:09,120 HONG LIU: Exactly. 72 00:07:09,120 --> 00:07:11,350 Exactly. 73 00:07:11,350 --> 00:07:21,080 Yeah, because, say, imagine when you sum over surfaces, 74 00:07:21,080 --> 00:07:23,590 so you sum over all possible metric. 75 00:07:23,590 --> 00:07:25,330 You can put [INAUDIBLE]. 76 00:07:25,330 --> 00:07:27,440 And that's the same way as you sum 77 00:07:27,440 --> 00:07:32,270 over different discretizations of that surface 78 00:07:32,270 --> 00:07:37,260 once you have defined the unit for that discretization. 79 00:07:37,260 --> 00:07:43,140 So if we can identify-- so for now record this Fh. 80 00:07:46,440 --> 00:07:58,060 So this Fh, this Fnh is the path integral 81 00:07:58,060 --> 00:08:06,584 over all genus-h surfaces with some string action, 82 00:08:06,584 --> 00:08:07,875 weighted by some string action. 83 00:08:18,050 --> 00:08:35,545 So if we can, say, identify this G 84 00:08:35,545 --> 00:08:47,710 with some string action-- the exponential of some string 85 00:08:47,710 --> 00:08:48,210 action. 86 00:08:53,050 --> 00:08:55,980 Then we would have-- then one can 87 00:08:55,980 --> 00:09:08,580 conclude that large N gauge theory is just a string 88 00:09:08,580 --> 00:09:09,750 theory, OK? 89 00:09:15,772 --> 00:09:17,855 That large N gauge theory is just a string theory, 90 00:09:17,855 --> 00:09:19,015 if you can do that. 91 00:09:22,510 --> 00:09:30,130 In particular, the large N limits-- 92 00:09:30,130 --> 00:09:34,690 so large N limit here, as we discussed before, 93 00:09:34,690 --> 00:09:38,590 can considered as a classical theory of glueballs. 94 00:09:41,300 --> 00:09:44,610 Or a classical theory of the single-trace operators. 95 00:09:44,610 --> 00:09:47,670 So this would be matched to the classical string theory. 96 00:09:55,590 --> 00:09:58,610 So as we mentioned last time, so I was mentioning before, 97 00:09:58,610 --> 00:10:07,090 this expression-- so just as in the as we discussed 98 00:10:07,090 --> 00:10:09,037 [INAUDIBLE], the [? expansion ?] in g 99 00:10:09,037 --> 00:10:14,850 string the same as expansion in the topology. 100 00:10:14,850 --> 00:10:16,690 And the expansion in the topology 101 00:10:16,690 --> 00:10:18,380 can also be considered as the expansion 102 00:10:18,380 --> 00:10:20,300 of the groups of a string. 103 00:10:20,300 --> 00:10:24,020 Because whenever you add a hole to the genus-- 104 00:10:24,020 --> 00:10:26,650 when you add the genus, and you actually add the string hole, 105 00:10:26,650 --> 00:10:28,655 you add the string loop diagram. 106 00:10:33,050 --> 00:10:35,550 So in this sense, you can [? integrate ?] all these 107 00:10:35,550 --> 00:10:39,300 higher order corrections, as the quantum 108 00:10:39,300 --> 00:10:43,337 correction to this classical string behavior. 109 00:10:43,337 --> 00:10:45,420 So this is just a tree-level amplitude for string. 110 00:10:45,420 --> 00:10:48,340 And this [? goes ?] one into the loops. 111 00:10:48,340 --> 00:10:50,290 Whenever you add this thing, you add the loop. 112 00:10:50,290 --> 00:10:50,790 OK. 113 00:10:50,790 --> 00:10:52,002 Is this clear? 114 00:10:52,002 --> 00:10:53,975 Now, remember what we discussed for the torus. 115 00:10:57,120 --> 00:10:59,940 If you've got a torus, then correspondingly 116 00:10:59,940 --> 00:11:02,960 you have a string split and joined together. 117 00:11:02,960 --> 00:11:07,280 And this split and join process you can also 118 00:11:07,280 --> 00:11:12,250 consider as a string loop, a single string going 119 00:11:12,250 --> 00:11:17,075 around a loop, [? just like ?] in the particle case, OK? 120 00:11:17,075 --> 00:11:19,790 In the standard field theory case. 121 00:11:19,790 --> 00:11:28,450 And so the large N limit, which is the leading order term here, 122 00:11:28,450 --> 00:11:31,034 would map to a leading order in the string scattering. 123 00:11:31,034 --> 00:11:33,450 And the leading order in the string scattering-- they only 124 00:11:33,450 --> 00:11:35,860 consider tree-level [? skin ?] scatterings, 125 00:11:35,860 --> 00:11:40,710 and then corresponding to classical string theory. 126 00:11:40,710 --> 00:11:46,830 And also the single-trace operator 127 00:11:46,830 --> 00:11:53,685 here can be mapped to the string states. 128 00:11:53,685 --> 00:11:55,577 Yeah, can be mapped to the string states. 129 00:12:08,010 --> 00:12:14,780 But this is only-- this is a very nice picture. 130 00:12:14,780 --> 00:12:18,410 But for many years, this was just a dream. 131 00:12:18,410 --> 00:12:25,840 And because this guy looks very different from this guy, 132 00:12:25,840 --> 00:12:29,310 but this is difficult. So this has 133 00:12:29,310 --> 00:12:44,670 some [? identification is ?] difficult for the following 134 00:12:44,670 --> 00:12:45,170 reasons. 135 00:12:52,020 --> 00:12:58,680 So first, so this G just-- say your Feynman diagrams, 136 00:12:58,680 --> 00:13:01,380 amplitude for particular Feynman diagram. 137 00:13:01,380 --> 00:13:14,220 So G is typically expressed as product 138 00:13:14,220 --> 00:13:19,300 of field theory propagators. 139 00:13:24,280 --> 00:13:27,645 So imagine how you evaluate the Feynman diagram. 140 00:13:27,645 --> 00:13:29,020 The Feynman diagram, essentially, 141 00:13:29,020 --> 00:13:32,540 is just a product of the [? propagators. ?] 142 00:13:32,540 --> 00:13:38,657 And then you integrate it [INAUDIBLE] 143 00:13:38,657 --> 00:13:39,740 integrated over spacetime. 144 00:13:46,430 --> 00:13:49,200 So they just take the Yang-Mills theory. 145 00:13:49,200 --> 00:13:55,540 And if you look at the expression for this diagram, 146 00:13:55,540 --> 00:13:58,920 of course, it looks nothing. 147 00:13:58,920 --> 00:14:15,820 So they look nothing like-- OK. 148 00:14:23,000 --> 00:14:26,840 So let me make a few comments about this thing. 149 00:14:26,840 --> 00:14:30,270 Because if you want to match, say if I gave you a Yang-Mills 150 00:14:30,270 --> 00:14:33,712 theory, so I gave you a QCD, then you can write down-- then 151 00:14:33,712 --> 00:14:35,920 you can go to large N. You can write down expressions 152 00:14:35,920 --> 00:14:38,304 for the common diagrams. 153 00:14:38,304 --> 00:14:39,720 But if you say, I want to write it 154 00:14:39,720 --> 00:14:44,720 as a string theory, the first thing you have to say, 155 00:14:44,720 --> 00:14:46,710 what string theory do you want to compare? 156 00:14:51,850 --> 00:14:54,070 So first you have to ask yourself 157 00:14:54,070 --> 00:14:57,350 what string action do you want to compare. 158 00:14:57,350 --> 00:15:06,480 So the string action, as we discussed last time, 159 00:15:06,480 --> 00:15:11,000 this describes the embedding of the worldsheet 160 00:15:11,000 --> 00:15:13,430 into some spacetime. 161 00:15:13,430 --> 00:15:21,600 OK, so this is worldsheet into a spacetime. 162 00:15:27,510 --> 00:15:30,780 So this is also sometimes called the target space. 163 00:15:34,730 --> 00:15:36,290 So this is a spacetime. 164 00:15:36,290 --> 00:15:38,590 This string moves. 165 00:15:38,590 --> 00:15:41,240 And the mathematical of this is just the-- 166 00:15:41,240 --> 00:15:47,640 this is encoded in this mapping X mu sigma tau. 167 00:15:47,640 --> 00:15:50,350 OK, X mu is the coordinate for M. 168 00:15:50,350 --> 00:15:51,925 And then sigma tau is the coordinate 169 00:15:51,925 --> 00:15:55,470 as you parameterize your worldsheet. 170 00:15:55,470 --> 00:15:59,080 So in order to write down action, of course, 171 00:15:59,080 --> 00:16:06,354 you have to choice of space manifold. 172 00:16:06,354 --> 00:16:07,770 You have to choose your spacetime. 173 00:16:11,250 --> 00:16:14,890 And also you have to-- when you fix the spacetime, 174 00:16:14,890 --> 00:16:18,710 you don't have a choice. 175 00:16:18,710 --> 00:16:23,435 And sometimes the way to write down such kind of embedding 176 00:16:23,435 --> 00:16:25,210 is not unique. 177 00:16:25,210 --> 00:16:28,010 The action for such [? finding ?] is unique, 178 00:16:28,010 --> 00:16:35,220 so you only need to choose what action you include. 179 00:16:35,220 --> 00:16:41,700 And also often, in addition to this embedding, 180 00:16:41,700 --> 00:16:51,560 sometimes you can have additional internal degrees 181 00:16:51,560 --> 00:16:52,060 freedom. 182 00:16:58,505 --> 00:16:59,380 living on worldsheet. 183 00:17:04,160 --> 00:17:08,790 For example, you can have some fermions. 184 00:17:08,790 --> 00:17:10,449 Say if you have a superstring, then 185 00:17:10,449 --> 00:17:12,630 you can have some additional fermions 186 00:17:12,630 --> 00:17:17,389 are living on the worldsheet, in addition to this embedding. 187 00:17:22,680 --> 00:17:30,100 So in other words, the choice of this guy in some sense 188 00:17:30,100 --> 00:17:32,410 is infinite. 189 00:17:32,410 --> 00:17:35,845 And without any clue-- so you need 190 00:17:35,845 --> 00:17:39,230 some clue to know what to compare the gauge theory to. 191 00:17:39,230 --> 00:17:43,740 And otherwise, even if this works, 192 00:17:43,740 --> 00:17:46,590 you're searching for some needle in the big ocean. 193 00:17:53,040 --> 00:17:57,250 And then there's another very important reason 194 00:17:57,250 --> 00:18:09,605 why this is difficult, is that this string theory 195 00:18:09,605 --> 00:18:11,140 is formulated in a continuum. 196 00:18:19,804 --> 00:18:21,095 It's formulated in a continuum. 197 00:18:24,080 --> 00:18:32,800 And these Feynman diagrams, even if they're 198 00:18:32,800 --> 00:18:36,160 corresponding to some kind of string theory, 199 00:18:36,160 --> 00:18:38,260 they correspond into a discrete version of that. 200 00:18:43,820 --> 00:18:45,875 So at best, it's a discrete version. 201 00:18:54,170 --> 00:19:11,810 So we expect such a geometric picture for G, 202 00:19:11,810 --> 00:19:20,840 for these Feynman diagrams, to emerge only 203 00:19:20,840 --> 00:19:21,990 at strong couplings. 204 00:19:29,320 --> 00:19:30,545 OK? 205 00:19:30,545 --> 00:19:32,920 Emerge only at strong couplings for the following reason. 206 00:19:36,710 --> 00:19:40,520 So if you look at the Feynman diagram-- 207 00:19:40,520 --> 00:19:43,505 so the simplest Feynman diagram we draw before, 208 00:19:43,505 --> 00:19:46,585 say for example just this diagram. 209 00:19:49,220 --> 00:19:52,420 And if you draw it on the sphere, 210 00:19:52,420 --> 00:19:58,380 it separated the sphere into three parts, OK? 211 00:19:58,380 --> 00:20:02,140 So this [? discretizes ?] a sphere into three parts. 212 00:20:02,140 --> 00:20:04,210 And essentially, just as the sphere just becomes 213 00:20:04,210 --> 00:20:06,126 three points, because each particle is wanting 214 00:20:06,126 --> 00:20:09,290 to-- when you're trying to [INAUDIBLE] each part, 215 00:20:09,290 --> 00:20:11,130 you approximate it by one point. 216 00:20:11,130 --> 00:20:14,410 So essentially, in this diagram, you 217 00:20:14,410 --> 00:20:17,220 approximate the whole sphere essentially by three points. 218 00:20:17,220 --> 00:20:17,720 OK. 219 00:20:20,177 --> 00:20:22,510 And of course, it's hard to see your [? magic ?] picture 220 00:20:22,510 --> 00:20:23,980 from here. 221 00:20:23,980 --> 00:20:27,212 And your [? magic picture ?] you expect to emerge, 222 00:20:27,212 --> 00:20:30,350 but your Feynman diagrams become very complicated. 223 00:20:30,350 --> 00:20:34,096 For example, if you have this kind of diagram, 224 00:20:34,096 --> 00:20:35,470 because of the four-point vertex. 225 00:20:35,470 --> 00:20:37,445 In principle, you can have all these diagrams. 226 00:20:37,445 --> 00:20:39,070 And then this [INAUDIBLE] [? wanting ?] 227 00:20:39,070 --> 00:20:41,790 to discretize-- yes, I suppose this is on the torus. 228 00:20:44,380 --> 00:20:46,580 Suppose you have a-- for example, 229 00:20:46,580 --> 00:20:50,074 this could be a Feynman diagram on the torus, OK? 230 00:20:50,074 --> 00:20:51,240 For the vacuum [? energy. ?] 231 00:20:54,860 --> 00:20:59,150 And now this is next some kind of proper discretization. 232 00:20:59,150 --> 00:21:01,610 And this will go to a continuum limit, 233 00:21:01,610 --> 00:21:04,587 say when the number of these box go to infinity. 234 00:21:04,587 --> 00:21:06,170 When the number of box go to infinity, 235 00:21:06,170 --> 00:21:08,390 then you need a number of propagators, 236 00:21:08,390 --> 00:21:11,770 and the number of vertices goes to infinity, OK? 237 00:21:11,770 --> 00:21:14,560 So in order for continuum, a picture 238 00:21:14,560 --> 00:21:17,920 to emerge, so you want those complicated 239 00:21:17,920 --> 00:21:20,960 diagrams-- it's not your number of vertices or large number 240 00:21:20,960 --> 00:21:24,012 of propagators that dominate. 241 00:21:24,012 --> 00:21:25,470 And for those things that dominate, 242 00:21:25,470 --> 00:21:27,240 then you need the strong coupling. 243 00:21:27,240 --> 00:21:31,030 Because with this coupling, this is the leading order diagram. 244 00:21:31,030 --> 00:21:33,750 And there's no geometry from here, OK? 245 00:21:33,750 --> 00:21:36,150 So in order to have the geometry, 246 00:21:36,150 --> 00:21:39,130 you want the diagram are very, very complicated, 247 00:21:39,130 --> 00:21:42,470 so that they really-- [INAUDIBLE] 248 00:21:42,470 --> 00:21:45,510 a triangulation of a surface. 249 00:21:45,510 --> 00:21:48,670 A weak coupled diagram with small number of lines will 250 00:21:48,670 --> 00:21:54,940 cause [? one ?] [? and two ?] are very close triangulization 251 00:21:54,940 --> 00:21:56,970 of a surface. 252 00:21:56,970 --> 00:22:01,050 So we expect this only appears in strong couplings, OK? 253 00:22:08,010 --> 00:22:08,690 Yeah. 254 00:22:08,690 --> 00:22:12,340 AUDIENCE: By the cases like we have to sum over all the 255 00:22:12,340 --> 00:22:13,020 [INAUDIBLE]. 256 00:22:13,020 --> 00:22:15,200 HONG LIU: Yeah, sum over the [INAUDIBLE] diagram. 257 00:22:15,200 --> 00:22:16,220 AUDIENCE: Including those simple ones. 258 00:22:16,220 --> 00:22:17,803 HONG LIU: Including those simple ones. 259 00:22:17,803 --> 00:22:20,620 So that's why you want to-- so if you're 260 00:22:20,620 --> 00:22:28,390 in a weak coupling, then the simple ones-- 261 00:22:28,390 --> 00:22:29,615 so we sum all those diagrams. 262 00:22:29,615 --> 00:22:31,698 And each diagram you can associate with a coupling 263 00:22:31,698 --> 00:22:32,970 power. 264 00:22:32,970 --> 00:22:36,120 So at weak coupling, then the lowest order term 265 00:22:36,120 --> 00:22:37,690 would just dominate. 266 00:22:37,690 --> 00:22:40,670 And the lowest order term have a very simple diagrams. 267 00:22:40,670 --> 00:22:42,794 And then that's because [? one ?] and [? two ?] are 268 00:22:42,794 --> 00:22:46,050 very crude triangulization over the surface. 269 00:22:46,050 --> 00:22:50,150 But if you have a strong coupling-- in particular, 270 00:22:50,150 --> 00:22:53,650 if you have an infinite coupling-- the diagrams, 271 00:22:53,650 --> 00:22:55,914 the infinite number of vertices will dominate. 272 00:22:55,914 --> 00:22:58,080 And then that's because [? one ?] and [? two ?] have 273 00:22:58,080 --> 00:23:00,110 very fine triangulization over the surface. 274 00:23:00,110 --> 00:23:02,334 And then that can go to the [INAUDIBLE]. 275 00:23:02,334 --> 00:23:05,738 AUDIENCE: [INAUDIBLE] interaction a coupling constant 276 00:23:05,738 --> 00:23:08,529 has been [? dragging ?] out from-- 277 00:23:08,529 --> 00:23:09,070 HONG LIU: No. 278 00:23:09,070 --> 00:23:10,397 That's just N dragged out. 279 00:23:10,397 --> 00:23:11,230 AUDIENCE: Oh, I see. 280 00:23:11,230 --> 00:23:13,355 HONG LIU: No, there's what we call this [INAUDIBLE] 281 00:23:13,355 --> 00:23:14,950 still remaining. 282 00:23:14,950 --> 00:23:16,860 By coupling, it's only [? N. ?] 283 00:23:16,860 --> 00:23:19,800 AUDIENCE: [INAUDIBLE] 284 00:23:19,800 --> 00:23:22,639 HONG LIU: No, no, this isn't to [? hold ?] coupling. 285 00:23:22,639 --> 00:23:24,180 In coupling we mean that [INAUDIBLE]. 286 00:23:24,180 --> 00:23:26,555 So example we talk about, [? because one ?] [? and two ?] 287 00:23:26,555 --> 00:23:27,200 [INAUDIBLE]. 288 00:23:27,200 --> 00:23:30,870 Yeah, and then we make more precise. 289 00:23:30,870 --> 00:23:33,820 So in the [? toy ?] example we talked about before. 290 00:23:33,820 --> 00:23:36,870 So previously we talked about this example, 291 00:23:36,870 --> 00:23:44,410 N divided by lambda, trace, say 1/2 partial phi squared, 292 00:23:44,410 --> 00:23:49,210 plus 1/4 phi to the power 4. 293 00:23:49,210 --> 00:23:51,300 And strong coupling means the lambda large. 294 00:23:54,724 --> 00:23:58,360 Because of the N I've already factored out, 295 00:23:58,360 --> 00:23:59,890 so you're coupling just lambda. 296 00:23:59,890 --> 00:24:01,255 AUDIENCE: Oh, I see. 297 00:24:04,900 --> 00:24:05,740 HONG LIU: Yes. 298 00:24:05,740 --> 00:24:11,108 AUDIENCE: So in these [INAUDIBLE] 299 00:24:11,108 --> 00:24:14,524 the propagator in that version would become 300 00:24:14,524 --> 00:24:16,964 the spacetime integration? 301 00:24:16,964 --> 00:24:17,940 HONG LIU: Hm? 302 00:24:17,940 --> 00:24:19,892 AUDIENCE: I was just wondering how 303 00:24:19,892 --> 00:24:23,308 the propagator can [? agree, ?] can match 304 00:24:23,308 --> 00:24:24,780 to the spacetime [INAUDIBLE]. 305 00:24:24,780 --> 00:24:25,654 HONG LIU: Yeah, yeah. 306 00:24:31,290 --> 00:24:34,660 So the propogator-- yeah, propagator you 307 00:24:34,660 --> 00:24:35,760 do in the standard way. 308 00:24:35,760 --> 00:24:38,230 You just write down your propagator, 309 00:24:38,230 --> 00:24:40,350 and then you try to repackage that. 310 00:24:40,350 --> 00:24:44,210 As the question, you said, whatever your rule, 311 00:24:44,210 --> 00:24:46,592 Feynman rule is we just do that Feynman rule. 312 00:24:46,592 --> 00:24:48,050 And you write down this expression. 313 00:24:48,050 --> 00:24:49,970 It's something very complicated. 314 00:24:49,970 --> 00:24:52,390 And then you say, can I find some geometric interpretation 315 00:24:52,390 --> 00:24:54,120 of that? 316 00:24:54,120 --> 00:24:58,230 Yeah, what I'm saying is that doing from this perspective 317 00:24:58,230 --> 00:25:05,140 is very hard because you don't know what thing to compare. 318 00:25:05,140 --> 00:25:09,620 And further, in the second, you expect 319 00:25:09,620 --> 00:25:11,540 that your [INAUDIBLE] would emerge only 320 00:25:11,540 --> 00:25:13,810 in those very complicated diagrams. 321 00:25:13,810 --> 00:25:18,040 And those complicated diagrams we don't know how to deal with. 322 00:25:18,040 --> 00:25:20,315 Because they only emerge in the strong coupling limit, 323 00:25:20,315 --> 00:25:21,690 but in the strong coupling limit, 324 00:25:21,690 --> 00:25:24,430 we don't know how to deal with that. 325 00:25:24,430 --> 00:25:27,720 And so that's why it's also difficult. 326 00:25:27,720 --> 00:25:32,605 But [? nevertheless, ?] for some very simple theories, say, 327 00:25:32,605 --> 00:25:34,720 if you don't consider the Yang-Mills theory, 328 00:25:34,720 --> 00:25:36,220 you don't consider the gauge theory. 329 00:25:36,220 --> 00:25:40,180 But suppose you do consider some matrix integrals. 330 00:25:40,180 --> 00:25:55,705 Say, for very simple systems, like a matrix integral. 331 00:26:02,170 --> 00:26:06,170 So this structure emphasizes-- this structure only 332 00:26:06,170 --> 00:26:10,130 have to do with you have a matrices, OK? 333 00:26:10,130 --> 00:26:13,730 And then you can have matrix-valued fields [? or ?] 334 00:26:13,730 --> 00:26:15,960 this structure will emerge. 335 00:26:15,960 --> 00:26:18,191 Or you only have a matrix integral. 336 00:26:18,191 --> 00:26:20,440 So there no field at all, just have a matrix integral. 337 00:26:20,440 --> 00:26:23,070 That same structure will also emerge. 338 00:26:23,070 --> 00:26:23,970 For example. 339 00:26:23,970 --> 00:26:29,950 I can consider theory-- have a theory like this. 340 00:26:39,270 --> 00:26:40,210 Something like this. 341 00:26:43,880 --> 00:26:46,620 And have a theory like this, OK? 342 00:26:46,620 --> 00:26:50,210 And M is just some [INAUDIBLE] matrices. 343 00:26:50,210 --> 00:26:52,012 So this is just integral. 344 00:26:52,012 --> 00:26:53,720 And the same structure will emerge, also, 345 00:26:53,720 --> 00:26:56,270 in this series when we do large N expansion. 346 00:26:59,510 --> 00:27:02,740 So that structure have nothing to do-- yeah, you can do it. 347 00:27:05,356 --> 00:27:07,730 So matrix integral is much simpler than [INAUDIBLE] field 348 00:27:07,730 --> 00:27:10,980 theory because you have much less degrees freedom. 349 00:27:10,980 --> 00:27:14,960 So for simple systems like, say, your matrix integral or matrix 350 00:27:14,960 --> 00:27:24,540 quantum mechanics, actually, you can 351 00:27:24,540 --> 00:27:27,630 guess the corresponding string theory. 352 00:27:36,999 --> 00:27:38,790 Because also the string theory in that case 353 00:27:38,790 --> 00:27:40,710 is also very simple. 354 00:27:40,710 --> 00:27:44,160 You can guess where is simple string theory. 355 00:27:44,160 --> 00:27:47,460 But it's not possible for field theory. 356 00:27:47,460 --> 00:27:48,961 It's not possible for field theory. 357 00:27:48,961 --> 00:27:49,460 Yes. 358 00:27:49,460 --> 00:27:53,005 AUDIENCE: So what do you mean by matrix quantum mechanics? 359 00:27:53,005 --> 00:27:54,400 Like that, OK. 360 00:27:54,400 --> 00:27:57,670 HONG LIU: So this is a matrix integral. 361 00:27:57,670 --> 00:27:59,880 And I can make it a little bit more complicated. 362 00:27:59,880 --> 00:28:03,839 So I make this M to depend on t, and then this 363 00:28:03,839 --> 00:28:05,255 become a matrix quantum mechanics. 364 00:28:08,560 --> 00:28:15,767 Say trace M dot squared plus M squared plus M4. 365 00:28:15,767 --> 00:28:17,600 Then this become a matrix quantum mechanics, 366 00:28:17,600 --> 00:28:20,510 because it only have time. 367 00:28:20,510 --> 00:28:22,740 And then I can make it more complicated. 368 00:28:22,740 --> 00:28:26,310 I can make M be t, x. 369 00:28:26,310 --> 00:28:29,130 Then this becomes one plus one dimension of field theory. 370 00:28:29,130 --> 00:28:32,020 AUDIENCE: So in what context is this matrix quantum mechanics 371 00:28:32,020 --> 00:28:33,540 [? conflicted? ?] 372 00:28:33,540 --> 00:28:35,550 HONG LIU: Just at some [? toy ?] model. 373 00:28:35,550 --> 00:28:41,570 I just say, and this is a very difficult question. 374 00:28:41,570 --> 00:28:44,690 You said, I don't know how to deal with field theories. 375 00:28:44,690 --> 00:28:46,780 Then this [? part of it's ?] a simple system. 376 00:28:46,780 --> 00:28:48,970 And then just try to use this philosophy, 377 00:28:48,970 --> 00:28:51,450 can see whether it can do it for simple system. 378 00:28:51,450 --> 00:28:54,510 And then you can show that this philosophy actually 379 00:28:54,510 --> 00:28:58,350 works if you do a matrix integral or matrix quantum 380 00:28:58,350 --> 00:28:59,980 mechanics. 381 00:28:59,980 --> 00:29:03,840 Simple enough, matrix integral and matrix quantum mechanics. 382 00:29:03,840 --> 00:29:06,110 OK. 383 00:29:06,110 --> 00:29:07,860 And if you want references, I can give you 384 00:29:07,860 --> 00:29:10,180 references regarding these. 385 00:29:10,180 --> 00:29:13,230 There's a huge, huge amount of works, thousands 386 00:29:13,230 --> 00:29:19,400 of papers, written on this subject in the late '80s 387 00:29:19,400 --> 00:29:20,110 and early '90s. 388 00:29:26,330 --> 00:29:29,050 So those [? toy ?] examples just to show actually 389 00:29:29,050 --> 00:29:31,100 this philosophy works. 390 00:29:31,100 --> 00:29:33,110 I just showed this philosophy works, OK? 391 00:29:37,940 --> 00:29:42,050 But it's not possible if we want to go to higher dimensions. 392 00:29:42,050 --> 00:29:45,320 Actually, there's one paper-- let me just write it here. 393 00:29:45,320 --> 00:29:49,200 So this one paper explains the philosophy. 394 00:29:49,200 --> 00:29:51,890 So here I did not gave you many details, 395 00:29:51,890 --> 00:29:55,000 say, how you write this G down, how you in principle 396 00:29:55,000 --> 00:29:58,310 can match with this thing. 397 00:29:58,310 --> 00:30:00,045 With [? another ?] maybe [INAUDIBLE] 398 00:30:00,045 --> 00:30:04,590 you can make this discussion a little bit more explicit, 399 00:30:04,590 --> 00:30:06,040 but I don't have time. 400 00:30:06,040 --> 00:30:10,610 But if you want, you can take a look at this paper. 401 00:30:10,610 --> 00:30:15,145 So this paper discusses the story for the matrix quantum 402 00:30:15,145 --> 00:30:15,645 mechanics. 403 00:30:18,740 --> 00:30:21,252 But in the section 2 of this paper-- so this 404 00:30:21,252 --> 00:30:22,210 is a paper by Klebanov. 405 00:30:29,260 --> 00:30:32,360 So in the section 2 of this paper, 406 00:30:32,360 --> 00:30:37,240 it explains this mapping of Feynman diagrams 407 00:30:37,240 --> 00:30:39,420 to the string action. 408 00:30:39,420 --> 00:30:43,020 And this discretization picture give you 409 00:30:43,020 --> 00:30:48,080 a nice summary of that philosophy with more details 410 00:30:48,080 --> 00:30:49,800 than I have given to you. 411 00:30:49,800 --> 00:30:51,760 So you can take a look at that. 412 00:30:51,760 --> 00:30:54,430 And this paper also has some other references 413 00:30:54,430 --> 00:30:57,200 if you want to take a look at it. 414 00:30:57,200 --> 00:30:57,700 OK. 415 00:31:02,067 --> 00:31:02,650 Any questions? 416 00:31:07,320 --> 00:31:08,561 Yes. 417 00:31:08,561 --> 00:31:10,602 AUDIENCE: Sorry, but who was the first to realize 418 00:31:10,602 --> 00:31:12,102 this connection between the surfaces 419 00:31:12,102 --> 00:31:13,494 in topology of Feynman diagrams? 420 00:31:13,494 --> 00:31:14,458 HONG LIU: Sorry? 421 00:31:14,458 --> 00:31:16,208 AUDIENCE: Who first realized this relation 422 00:31:16,208 --> 00:31:17,350 between topology and-- 423 00:31:17,350 --> 00:31:20,840 HONG LIU: So of course, already when 424 00:31:20,840 --> 00:31:25,040 't Hooft invented this large N expansion, 425 00:31:25,040 --> 00:31:29,920 he already noticed that this is similar to string theory. 426 00:31:29,920 --> 00:31:32,640 So he already commented on that. 427 00:31:32,640 --> 00:31:35,000 And he already commented on that. 428 00:31:35,000 --> 00:31:40,960 And for many years people did not make progress. 429 00:31:40,960 --> 00:31:44,650 For many years, people did not make progress. 430 00:31:44,650 --> 00:31:48,770 But in the late '80s-- in the mid to late '80s, 431 00:31:48,770 --> 00:31:52,740 people started thinking about the question 432 00:31:52,740 --> 00:31:56,290 from this perspective, not from that perspective. 433 00:31:56,290 --> 00:31:59,970 So they started to think about the order 434 00:31:59,970 --> 00:32:00,890 from this perspective. 435 00:32:00,890 --> 00:32:06,050 Because just typical string theory are hard to solve, 436 00:32:06,050 --> 00:32:06,800 et cetera. 437 00:32:06,800 --> 00:32:08,630 So people think, maybe we can actually 438 00:32:08,630 --> 00:32:11,980 understand or generalize our understanding of string theory 439 00:32:11,980 --> 00:32:13,625 by discretize the worldsheets. 440 00:32:17,080 --> 00:32:20,515 And then they just integrate over 441 00:32:20,515 --> 00:32:22,960 all possible triangulization, et cetera. 442 00:32:22,960 --> 00:32:26,030 And then they realized that that thing actually 443 00:32:26,030 --> 00:32:28,180 is like something over Feynman diagrams. 444 00:32:28,180 --> 00:32:30,770 And then for the very simple situations, 445 00:32:30,770 --> 00:32:34,810 say like if you have only a matrix integral, actually 446 00:32:34,810 --> 00:32:38,330 you can make the connection explicit. 447 00:32:38,330 --> 00:32:40,230 So that was in the late '80s. 448 00:32:40,230 --> 00:32:44,210 So people like [? McDowell ?] or [? Kazakov ?] et cetera that 449 00:32:44,210 --> 00:32:46,150 were trying to explore that. 450 00:32:53,790 --> 00:32:55,374 Other questions? 451 00:32:55,374 --> 00:32:57,162 AUDIENCE: I'm having trouble seeing 452 00:32:57,162 --> 00:33:01,918 how the sum over all triangulations [INAUDIBLE] each 453 00:33:01,918 --> 00:33:02,844 surfaces. 454 00:33:02,844 --> 00:33:05,962 How does that correspond to the discrete version of summing 455 00:33:05,962 --> 00:33:06,940 over all [INAUDIBLE]? 456 00:33:06,940 --> 00:33:08,310 HONG LIU: Right. 457 00:33:08,310 --> 00:33:11,250 AUDIENCE: That's the discrete sum over all possible 458 00:33:11,250 --> 00:33:12,580 [? genus-h, ?] right? 459 00:33:12,580 --> 00:33:14,850 HONG LIU: Yeah. 460 00:33:14,850 --> 00:33:17,100 I think this is the example. 461 00:33:17,100 --> 00:33:19,780 Yeah, let's consider torus. 462 00:33:19,780 --> 00:33:24,200 So a torus is a box with this identified with this, 463 00:33:24,200 --> 00:33:25,890 and this identified with that. 464 00:33:25,890 --> 00:33:26,930 OK. 465 00:33:26,930 --> 00:33:30,420 And let me first just draw the simplest partition here. 466 00:33:30,420 --> 00:33:31,390 Just draw like that. 467 00:33:36,150 --> 00:33:36,650 Yeah. 468 00:33:39,620 --> 00:33:43,300 Let me just look at these two things. 469 00:33:43,300 --> 00:33:54,700 So suppose I give each box-- so if I specify each box, 470 00:33:54,700 --> 00:33:57,620 say, give a unit area. 471 00:33:57,620 --> 00:33:58,870 OK? 472 00:33:58,870 --> 00:34:02,190 And I do this one, I do that one, 473 00:34:02,190 --> 00:34:07,190 or I do some other ways to triangulize it. 474 00:34:07,190 --> 00:34:10,429 Then because [? one and two ?] give a different symmetric 475 00:34:10,429 --> 00:34:11,844 to the surface. 476 00:34:11,844 --> 00:34:13,260 And then because [? one and two ?] 477 00:34:13,260 --> 00:34:16,029 integrate over all possible metric on this surface. 478 00:34:16,029 --> 00:34:17,820 And they integrate over all possible metric 479 00:34:17,820 --> 00:34:19,489 on this surface, you can integrate [INAUDIBLE] 480 00:34:19,489 --> 00:34:20,460 all possible surfaces. 481 00:34:22,519 --> 00:34:24,560 AUDIENCE: In the case of the strings for example, 482 00:34:24,560 --> 00:34:26,920 [? we put some ?] over the torus here 483 00:34:26,920 --> 00:34:28,982 and the torus and the torus there. 484 00:34:28,982 --> 00:34:29,690 HONG LIU: No, no. 485 00:34:29,690 --> 00:34:31,909 You only sum over a single torus. 486 00:34:31,909 --> 00:34:34,616 Now, what do you mean by summing over torus here, torus there? 487 00:34:34,616 --> 00:34:36,532 AUDIENCE: I thought like in the path integral, 488 00:34:36,532 --> 00:34:39,376 in the case of the string theory-- 489 00:34:39,376 --> 00:34:42,087 HONG LIU: No, you're only summing over a single torus. 490 00:34:42,087 --> 00:34:44,060 You're only summing over a single surface, 491 00:34:44,060 --> 00:34:46,570 but all possible ways to write-- all possible ways 492 00:34:46,570 --> 00:34:47,550 to draw that surface. 493 00:34:54,860 --> 00:34:57,030 So what you said about summing torus here, 494 00:34:57,030 --> 00:34:59,510 summing torus there, because [INAUDIBLE] what we call 495 00:34:59,510 --> 00:35:01,840 the disconnected amplitudes. 496 00:35:01,840 --> 00:35:05,100 And then you don't need to consider them in physically 497 00:35:05,100 --> 00:35:06,100 disconnected amplitude. 498 00:35:06,100 --> 00:35:07,760 You can just [? exponentiate ?] what 499 00:35:07,760 --> 00:35:09,770 we call by connected amplitude. 500 00:35:09,770 --> 00:35:11,580 And you don't need to do that separately. 501 00:35:11,580 --> 00:35:13,640 So once you know how to do a single one, 502 00:35:13,640 --> 00:35:16,051 and the disconnected one just automatically obtained 503 00:35:16,051 --> 00:35:17,050 by [? exponentiation. ?] 504 00:35:20,006 --> 00:35:20,881 AUDIENCE: [INAUDIBLE] 505 00:35:31,074 --> 00:35:31,740 HONG LIU: Sorry? 506 00:35:31,740 --> 00:35:32,790 No, no. 507 00:35:32,790 --> 00:35:38,575 Here the metric matters, the geometry matters. 508 00:35:38,575 --> 00:35:41,465 It's not just the topology. 509 00:35:41,465 --> 00:35:49,445 AUDIENCE: [INAUDIBLE] Feynman diagram [INAUDIBLE]? 510 00:35:49,445 --> 00:35:50,070 HONG LIU: Yeah. 511 00:35:50,070 --> 00:35:52,403 Yeah, just the key is that the propagator of the Feynman 512 00:35:52,403 --> 00:35:54,775 diagram essentially [? encodes ?] the geometries. 513 00:35:54,775 --> 00:36:00,710 And in encoding a very indirect way. 514 00:36:00,710 --> 00:36:01,210 Yeah. 515 00:36:01,210 --> 00:36:02,770 Just read this part. 516 00:36:02,770 --> 00:36:05,240 This section only have a few pages, 517 00:36:05,240 --> 00:36:08,350 but contain a little bit more details on what I have here. 518 00:36:08,350 --> 00:36:13,800 It requires maybe one more hour to explain this in more detail. 519 00:36:13,800 --> 00:36:15,070 Yeah, this is just that. 520 00:36:15,070 --> 00:36:16,695 I just want to explain this philosophy. 521 00:36:16,695 --> 00:36:19,380 I don't want to go through the details of how you do this. 522 00:36:22,360 --> 00:36:23,525 OK, good. 523 00:36:23,525 --> 00:36:25,816 So now let me just mention a couple of generalizations. 524 00:36:40,540 --> 00:36:42,220 So the first thing you already asked 525 00:36:42,220 --> 00:36:45,870 before, I think maybe both you have asked. 526 00:36:48,692 --> 00:36:50,066 Let me just mention them quickly. 527 00:36:53,650 --> 00:36:57,830 And if you are interested, I can certainly 528 00:36:57,830 --> 00:37:02,400 give you a reference for you to read about them, 529 00:37:02,400 --> 00:37:05,070 or I can put it in [? your P ?] sets. 530 00:37:05,070 --> 00:37:17,100 And so, so far, it's all matrix-valued fields, OK? 531 00:37:17,100 --> 00:37:20,350 But if you can see the theory-- or in other words, 532 00:37:20,350 --> 00:37:22,314 in the mathematical language, say, 533 00:37:22,314 --> 00:37:23,605 it's an adjoint representation. 534 00:37:27,330 --> 00:37:31,030 It's an adjoint representation of the-- 535 00:37:31,030 --> 00:37:34,261 because our symmetries are UN, it's a UN gauge group. 536 00:37:38,500 --> 00:37:40,360 OK? 537 00:37:40,360 --> 00:37:43,490 UN gauge group. 538 00:37:43,490 --> 00:37:47,745 But you can also, for example, in QCD, you also have quarks. 539 00:37:47,745 --> 00:37:51,850 So you also have field in the fundamental representations. 540 00:37:51,850 --> 00:37:59,220 So it can also include field in the fundamental representation. 541 00:37:59,220 --> 00:38:03,762 So rather than matrix-valued, they're N vector. 542 00:38:03,762 --> 00:38:07,160 OK, they're N [? vectors. ?] 543 00:38:07,160 --> 00:38:11,950 So for quarks, of course, for the standard QCD N will be 3, 544 00:38:11,950 --> 00:38:13,380 so you have three quarks. 545 00:38:13,380 --> 00:38:15,265 You have three different colored quarks. 546 00:38:18,850 --> 00:38:22,570 And so then your Feynman diagrams, 547 00:38:22,570 --> 00:38:25,850 in addition to have those matrix [? lines, ?] 548 00:38:25,850 --> 00:38:29,080 which you have a double line. 549 00:38:29,080 --> 00:38:32,790 And now here you only have a single index, OK? 550 00:38:32,790 --> 00:38:34,606 And then you only have a single line. 551 00:38:34,606 --> 00:38:35,980 So the propagator of those quarks 552 00:38:35,980 --> 00:38:37,440 will just have a single line. 553 00:38:37,440 --> 00:38:38,981 And then also in your Feynman diagram 554 00:38:38,981 --> 00:38:41,790 you can have loops over the quarks, et cetera. 555 00:38:41,790 --> 00:38:43,266 So you can again work this out. 556 00:38:43,266 --> 00:38:45,515 And then you find it is a very nice large N expansion. 557 00:38:53,540 --> 00:39:02,260 And then you find the diagrams, the Feynman diagrams. 558 00:39:02,260 --> 00:39:04,310 Now you find in this case the Feynman 559 00:39:04,310 --> 00:39:25,345 diagrams can be classified by 2D surfaces with boundaries. 560 00:39:32,990 --> 00:39:38,400 So essentially, you have-- and let me just say, for example, 561 00:39:38,400 --> 00:39:43,690 this is the vacuum diagrams, for all the vacuum process. 562 00:39:43,690 --> 00:39:48,660 Then you can [INAUDIBLE] or the vacuum diagrams. 563 00:39:52,620 --> 00:39:55,460 And then they can all be [? collectified. ?] 564 00:39:55,460 --> 00:40:00,150 So previously, we have a matrix-valued field. 565 00:40:00,150 --> 00:40:04,445 Then all your vacuum diagrams, they 566 00:40:04,445 --> 00:40:07,550 are corresponding closed surfaces-- so sphere, 567 00:40:07,550 --> 00:40:09,130 torus, et cetera. 568 00:40:09,130 --> 00:40:10,900 But now if you include the quarks, 569 00:40:10,900 --> 00:40:13,310 then those surfaces can have boundaries. 570 00:40:13,310 --> 00:40:16,170 And then [INAUDIBLE] into the quark groups, et cetera. 571 00:40:16,170 --> 00:40:21,270 And then they [? cannot ?] be classified. 572 00:40:21,270 --> 00:40:26,250 And so these also have a counterpart 573 00:40:26,250 --> 00:40:28,970 if you try to map to the string theory. 574 00:40:28,970 --> 00:40:31,345 So this [INAUDIBLE] [? one and ?] [? two, ?] string 575 00:40:31,345 --> 00:40:31,845 theory. 576 00:40:35,230 --> 00:40:47,280 There's string theory with both closed and open strings. 577 00:40:55,150 --> 00:40:57,610 And so essentially those boundaries 578 00:40:57,610 --> 00:40:59,360 give rise to the open strings. 579 00:40:59,360 --> 00:41:01,820 So here, it's all closed strings. 580 00:41:01,820 --> 00:41:02,820 It's all closed surface. 581 00:41:02,820 --> 00:41:05,670 Well, now you can, by adding the open strings, 582 00:41:05,670 --> 00:41:10,840 and then you can, again, have the correspondence 583 00:41:10,840 --> 00:41:12,900 between the two. 584 00:41:12,900 --> 00:41:15,250 OK. 585 00:41:15,250 --> 00:41:17,420 So all the discussion is very similar to what 586 00:41:17,420 --> 00:41:18,330 we discussed before. 587 00:41:18,330 --> 00:41:22,720 We just apply all this the same philosophy to the quarks. 588 00:41:22,720 --> 00:41:23,690 Yes. 589 00:41:23,690 --> 00:41:27,570 AUDIENCE: [INAUDIBLE] do the same trick on string theory 590 00:41:27,570 --> 00:41:30,480 and find some sort of expression which 591 00:41:30,480 --> 00:41:34,360 then will map to some higher order surfaces, [INAUDIBLE]? 592 00:41:37,220 --> 00:41:38,553 HONG LIU: Sorry, say that again? 593 00:41:38,553 --> 00:41:41,511 AUDIENCE: [INAUDIBLE] Feynman diagrams 594 00:41:41,511 --> 00:41:44,469 we move to string theory for surfaces. 595 00:41:44,469 --> 00:41:48,070 Is there some [INAUDIBLE] from surfaces just they go 596 00:41:48,070 --> 00:41:49,670 one more [? step up? ?] 597 00:41:49,670 --> 00:41:53,460 HONG LIU: You mean higher dimensions, not strings. 598 00:41:53,460 --> 00:41:58,260 Yeah, that will become-- of course, 599 00:41:58,260 --> 00:42:00,885 that's a [? lateral ?] idea. 600 00:42:00,885 --> 00:42:03,510 So that will [INAUDIBLE] you can consider [? rather ?] strings, 601 00:42:03,510 --> 00:42:06,320 you can consider two-dimensional surface, 602 00:42:06,320 --> 00:42:09,610 a two-dimensional surface moving in spacetime. 603 00:42:09,610 --> 00:42:12,650 And then [INAUDIBLE] into [? so-called ?] the membrane 604 00:42:12,650 --> 00:42:14,000 theory. 605 00:42:14,000 --> 00:42:17,850 But let's say where it turns out to be-- turns out 606 00:42:17,850 --> 00:42:20,380 string is a nice balance. 607 00:42:20,380 --> 00:42:22,880 It's not too complicated or not too simple. 608 00:42:22,880 --> 00:42:24,340 And it give you lots of structure. 609 00:42:24,340 --> 00:42:26,131 But when you go to membrane, then the story 610 00:42:26,131 --> 00:42:28,060 become too complicated, and nobody knows 611 00:42:28,060 --> 00:42:30,621 how to quantize that theory. 612 00:42:50,615 --> 00:42:58,590 So the second remark is that here we consider UN. 613 00:42:58,590 --> 00:43:00,640 So here our symmetry group is UN. 614 00:43:04,950 --> 00:43:11,450 Because our phi-- phi there is [? commission. ?] So when you 615 00:43:11,450 --> 00:43:14,070 have a [? commission ?] matrix, then there's a difference 616 00:43:14,070 --> 00:43:20,820 between the two indices, so we put one up and one down. 617 00:43:20,820 --> 00:43:24,370 So they are propagators that lead 618 00:43:24,370 --> 00:43:38,580 to-- so it leads to the lines with arrows, 619 00:43:38,580 --> 00:43:42,550 because we need to distinguish upper and lower indices. 620 00:43:42,550 --> 00:43:43,180 OK? 621 00:43:43,180 --> 00:43:45,810 Between the two indices. 622 00:43:45,810 --> 00:43:52,040 But you can also consider, for example, 623 00:43:52,040 --> 00:43:55,870 phi is a symmetric matrix. 624 00:43:55,870 --> 00:43:58,780 Say it's a real symmetric matrix. 625 00:43:58,780 --> 00:44:04,490 It's a real symmetric, or real anti-symmetric. 626 00:44:11,490 --> 00:44:15,260 In those cases, then there's no difference 627 00:44:15,260 --> 00:44:19,050 between the two indices. 628 00:44:19,050 --> 00:44:25,820 And then when you draw a propagator-- so in this case 629 00:44:25,820 --> 00:44:34,320 the symmetry group would be, say, 630 00:44:34,320 --> 00:44:37,500 SON, say, or SPN, et cetera. 631 00:44:41,840 --> 00:44:48,980 And then the propagators, they will no longer 632 00:44:48,980 --> 00:44:51,680 have orientations. 633 00:44:51,680 --> 00:44:52,180 OK? 634 00:44:52,180 --> 00:44:54,900 They will no longer have orientations. 635 00:44:54,900 --> 00:44:57,990 Because you can no longer-- yeah. 636 00:44:57,990 --> 00:45:02,675 So this will give rise-- so let me write it closer. 637 00:45:06,300 --> 00:45:13,810 So this will give rise to unorientable surfaces. 638 00:45:19,220 --> 00:45:22,166 Say, for example, to classify the diagrams, 639 00:45:22,166 --> 00:45:24,290 you can no longer just use the orientable surfaces. 640 00:45:24,290 --> 00:45:26,620 You also have to include the non-orientable surfaces 641 00:45:26,620 --> 00:45:29,470 to classify the diagrams. 642 00:45:29,470 --> 00:45:35,520 And the [INAUDIBLE] this also have a precise counterpart 643 00:45:35,520 --> 00:45:40,100 into unorientable strings. 644 00:45:40,100 --> 00:45:41,660 No, non-orientable strings. 645 00:45:47,070 --> 00:45:56,220 Yeah, I think non-orientable, non-orientable surfaces. 646 00:45:56,220 --> 00:45:57,410 Also non-orientable strings. 647 00:46:17,300 --> 00:46:18,620 Good. 648 00:46:18,620 --> 00:46:20,780 So I'm emphasizing how difficult it 649 00:46:20,780 --> 00:46:29,430 is if, say, we want to start with QCD 650 00:46:29,430 --> 00:46:35,640 and then try to find the string theory description. 651 00:46:35,640 --> 00:46:41,470 But this still, [? none of ?] this tries-- I just try. 652 00:46:51,930 --> 00:47:14,660 OK, so let's just consider, just take large N generalization 653 00:47:14,660 --> 00:47:16,450 of QCD. 654 00:47:16,450 --> 00:47:18,340 So this, again, will be some UN gauge theory, 655 00:47:18,340 --> 00:47:26,611 UN Yang-Mills theory, say, in 3 plus 1 dimensional Minkowski 656 00:47:26,611 --> 00:47:27,110 spacetime. 657 00:47:31,340 --> 00:47:41,770 And can we say anything about its string theory description? 658 00:47:55,800 --> 00:47:57,380 So [INAUDIBLE]. 659 00:47:57,380 --> 00:48:00,430 So maybe it's difficult, but let's try to guess it. 660 00:48:00,430 --> 00:48:02,340 OK. 661 00:48:02,340 --> 00:48:05,280 So in physics, in many situations, 662 00:48:05,280 --> 00:48:08,990 a seemingly difficult problem, if you know how to guess it, 663 00:48:08,990 --> 00:48:11,800 actually you can get the answer. 664 00:48:11,800 --> 00:48:14,140 On, for example, quantum hole effects, 665 00:48:14,140 --> 00:48:17,220 fractional quantum hole effects, you can just 666 00:48:17,220 --> 00:48:20,490 guess the wave function. 667 00:48:20,490 --> 00:48:23,840 So of course, the simplest guess-- 668 00:48:23,840 --> 00:48:28,140 so this is some gauge theory in 3 plus 1 dimensional Minkowski 669 00:48:28,140 --> 00:48:29,800 spacetime. 670 00:48:29,800 --> 00:48:32,800 So now we say this is a string theory. 671 00:48:32,800 --> 00:48:43,450 So natural guess is that this maybe 672 00:48:43,450 --> 00:48:51,160 is a string theory, again, in the 3 plus 1 673 00:48:51,160 --> 00:48:54,890 dimensional Minkowski spacetime. 674 00:48:54,890 --> 00:48:56,840 OK? 675 00:48:56,840 --> 00:49:00,360 So we just take what-- so these will, of course, 676 00:49:00,360 --> 00:49:09,860 run into a string, propagating in this spacetime, OK? 677 00:49:09,860 --> 00:49:13,590 As I said, when you write down the string theory, 678 00:49:13,590 --> 00:49:16,450 you first have to specify your target space, which, 679 00:49:16,450 --> 00:49:20,680 as the string moves, the larger question would be just, 680 00:49:20,680 --> 00:49:24,240 should it be the gauge theory's Minkowski spacetime. 681 00:49:24,240 --> 00:49:25,930 Maybe this string theory should be. 682 00:49:25,930 --> 00:49:27,720 OK? 683 00:49:27,720 --> 00:49:31,160 And then this. 684 00:49:31,160 --> 00:49:36,110 Then you can just try to-- then you can just 685 00:49:36,110 --> 00:49:38,546 write down the simplest action. 686 00:49:38,546 --> 00:49:48,780 So maybe say Nambu-Goto action, which we wrote last time, OK? 687 00:49:48,780 --> 00:49:55,550 Or the [? old ?] Polyakov action. 688 00:49:55,550 --> 00:50:03,510 So this Nambu-Goto action will result [INAUDIBLE] Polyakov. 689 00:50:03,510 --> 00:50:06,200 And let me not worry about that. 690 00:50:06,200 --> 00:50:08,510 For example, you can just guess, say, maybe 691 00:50:08,510 --> 00:50:12,360 this is a string theory also in the Minkowski spacetime. 692 00:50:12,360 --> 00:50:15,440 Say, consider the simplest action. 693 00:50:21,680 --> 00:50:24,666 Or the equivalent of this, OK? 694 00:50:36,890 --> 00:50:38,720 Then at least what you could try-- now 695 00:50:38,720 --> 00:50:47,554 you actually have an action. 696 00:50:47,554 --> 00:50:49,220 Now you think that you have this object. 697 00:50:49,220 --> 00:50:51,160 Now you think you can compare. 698 00:50:51,160 --> 00:50:52,890 OK, now you can essentially compare. 699 00:50:52,890 --> 00:50:56,110 Say, in QCD you calculated your Feynman diagrams, 700 00:50:56,110 --> 00:50:58,215 and now just compare. 701 00:50:58,215 --> 00:51:00,090 But of course, you still have the difficulty. 702 00:51:00,090 --> 00:51:01,923 Of course, you have to go to strong coupling 703 00:51:01,923 --> 00:51:04,090 to see the geometric limit, et cetera. 704 00:51:04,090 --> 00:51:07,950 But in principle, it's something you can do. 705 00:51:07,950 --> 00:51:10,490 But this actually does not work. 706 00:51:10,490 --> 00:51:13,140 OK? 707 00:51:13,140 --> 00:51:17,280 This does not work, for the following simple reason. 708 00:51:21,280 --> 00:51:30,430 Firstly, that such a string theory-- 709 00:51:30,430 --> 00:51:32,930 so a string theory, actually the remarkable thing 710 00:51:32,930 --> 00:51:37,050 about the string is that if you have a particle, 711 00:51:37,050 --> 00:51:40,800 you can put the particle in any spacetime. 712 00:51:40,800 --> 00:51:43,360 But strings are very picky. 713 00:51:43,360 --> 00:51:46,280 You cannot put them in any spacetime. 714 00:51:46,280 --> 00:51:50,145 And they can only propagate consistently, 715 00:51:50,145 --> 00:51:54,220 quantum mechanically consistently, in some spacetime 716 00:51:54,220 --> 00:51:57,160 but not in others. 717 00:51:57,160 --> 00:52:00,055 So for example, if you want to put the string 718 00:52:00,055 --> 00:52:02,180 to propagate in this 3 plus 1 dimensional Minkowski 719 00:52:02,180 --> 00:52:06,929 spacetime, then you actually find that the theory 720 00:52:06,929 --> 00:52:08,220 is mathematically inconsistent. 721 00:52:10,760 --> 00:52:16,050 So such a string theory is inconsistent. 722 00:52:18,570 --> 00:52:21,200 It's mathematically inconsistent. 723 00:52:21,200 --> 00:52:31,640 Except for the D equal to 26 or 10. 724 00:52:31,640 --> 00:52:32,490 OK? 725 00:52:32,490 --> 00:52:37,410 So 26 if you just purely have the theory, and 10 726 00:52:37,410 --> 00:52:38,690 if you also add some fermion. 727 00:52:43,260 --> 00:52:45,610 So such a string theory does not exist mathematically. 728 00:52:48,640 --> 00:52:50,990 So you say, oh, OK. 729 00:52:50,990 --> 00:52:55,240 You say, I'm a smart fellow. 730 00:52:55,240 --> 00:52:57,904 I can go around this. 731 00:52:57,904 --> 00:52:59,570 Because we want the Minkowski spacetime. 732 00:52:59,570 --> 00:53:01,740 Because those gauge theory propagating the Minkowski 733 00:53:01,740 --> 00:53:06,200 spacetime, so this Minkowski [INAUDIBLE] must be somewhere. 734 00:53:06,200 --> 00:53:10,860 They cannot go away, because all these glueballs [INAUDIBLE] 735 00:53:10,860 --> 00:53:13,450 in this 3 plus 1 dimensional Minkowski spacetime. 736 00:53:13,450 --> 00:53:15,950 And if we want to identify the strings with those glueballs, 737 00:53:15,950 --> 00:53:17,741 those strings must at least [? know ?] some 738 00:53:17,741 --> 00:53:20,780 of this Minkowski spacetime. 739 00:53:20,780 --> 00:53:22,970 And then you say, oh, suppose you tell me 740 00:53:22,970 --> 00:53:25,530 that this string theory is only consistent in 10 dimension. 741 00:53:28,120 --> 00:53:39,880 But then let me take a string theory in 10 dimensions, which 742 00:53:39,880 --> 00:53:42,180 itself consistent. 743 00:53:42,180 --> 00:53:45,970 But I take this 10-dimensional spacetime to have 744 00:53:45,970 --> 00:53:48,120 the form of a 3 plus 1 dimensional 745 00:53:48,120 --> 00:53:49,550 Minkowski spacetime. 746 00:53:49,550 --> 00:53:52,880 And the [? time, ?] some compact manifold, OK? 747 00:53:58,360 --> 00:53:59,290 Some compact manifold. 748 00:54:02,840 --> 00:54:06,510 And in such case-- so if this is a compact manifold, 749 00:54:06,510 --> 00:54:08,760 then the symmetry of this spacetime, 750 00:54:08,760 --> 00:54:11,430 so the spacetime symmetry still only have 751 00:54:11,430 --> 00:54:15,890 the 3 plus 1 dimensional, [? say, ?] Poincare symmetry. 752 00:54:15,890 --> 00:54:19,010 Because if you want to describe the QCD in 3 plus 1 dimension, 753 00:54:19,010 --> 00:54:21,180 QCD has the Poincare symmetry. 754 00:54:21,180 --> 00:54:22,910 You can do Lorentz transformation, 755 00:54:22,910 --> 00:54:24,450 and then you can do rotation. 756 00:54:24,450 --> 00:54:27,010 Or you can do translation. 757 00:54:27,010 --> 00:54:30,110 The string theory should not have more symmetries or less 758 00:54:30,110 --> 00:54:31,110 symmetries than QCD. 759 00:54:31,110 --> 00:54:32,609 They should have the same symmetries 760 00:54:32,609 --> 00:54:35,260 because they are supposed to be the same description. 761 00:54:35,260 --> 00:54:39,360 But if you take the 10-dimensional Minkowski space, 762 00:54:39,360 --> 00:54:40,530 of course, it's not right. 763 00:54:40,530 --> 00:54:42,030 Because the 10-dimensional Minkowski 764 00:54:42,030 --> 00:54:43,930 space have 10-dimensional translation 765 00:54:43,930 --> 00:54:47,450 and 10-dimensional Lorentz symmetry. 766 00:54:47,450 --> 00:54:52,030 But what you can do is that you take this 10-dimensional space 767 00:54:52,030 --> 00:54:54,660 to be a form of the 3 plus 1 dimensional Minkowski 768 00:54:54,660 --> 00:54:58,590 spacetime and times some additional compact manifold, 769 00:54:58,590 --> 00:55:02,890 and then you have solved the symmetry problem. 770 00:55:02,890 --> 00:55:06,510 But except this still does not work 771 00:55:06,510 --> 00:55:13,950 because the string theory, as we know, always contain gravity. 772 00:55:13,950 --> 00:55:18,820 And if you put a string theory on such a compact space N, 773 00:55:18,820 --> 00:55:20,195 [? there would be ?] always leads 774 00:55:20,195 --> 00:55:26,800 to a massless spin-2 particle in this 3 plus 1 dimensional part. 775 00:55:26,800 --> 00:55:31,090 But from Weinberg-Witten theorem we talked in the first lecture, 776 00:55:31,090 --> 00:55:34,090 in the QCD you are not supposed to have 777 00:55:34,090 --> 00:55:39,450 a 3 plus 1 dimensional massless spin-2 particle, OK? 778 00:55:39,450 --> 00:55:41,570 And so this won't work. 779 00:55:41,570 --> 00:55:44,012 So this won't work. 780 00:55:44,012 --> 00:55:46,610 Because this contains-- 781 00:56:19,160 --> 00:56:21,260 In 3 plus 1 dimensional [? Minkowski space, ?] 782 00:56:21,260 --> 00:56:28,840 which does not have-- OK? 783 00:56:28,840 --> 00:56:31,890 Or in the large N [INAUDIBLE]. 784 00:56:31,890 --> 00:56:35,200 So this does not work. 785 00:56:35,200 --> 00:56:36,220 So what to do? 786 00:56:36,220 --> 00:56:36,820 Yes? 787 00:56:36,820 --> 00:56:39,740 AUDIENCE: So does this just mean that it's mathematically 788 00:56:39,740 --> 00:56:40,820 inconsistent? 789 00:56:40,820 --> 00:56:41,705 HONG LIU: No, no. 790 00:56:41,705 --> 00:56:43,913 This does not mean it is mathematically inconsistent. 791 00:56:43,913 --> 00:56:47,300 It just means this string theory cannot not correspond 792 00:56:47,300 --> 00:56:49,590 to the string theory [? describe ?] QCD. 793 00:56:49,590 --> 00:56:55,530 The string theory description-- the equivalent string theory 794 00:56:55,530 --> 00:56:58,150 for QCD cannot have this feature. 795 00:56:58,150 --> 00:57:00,720 Yeah, just say this cannot be the right answer for that 796 00:57:00,720 --> 00:57:01,760 string theory. 797 00:57:01,760 --> 00:57:03,850 This string theory is consistent. 798 00:57:03,850 --> 00:57:04,391 Yes. 799 00:57:04,391 --> 00:57:05,890 AUDIENCE: So is that you were saying 800 00:57:05,890 --> 00:57:08,882 if there is a massless spin-2 particle in that string theory, 801 00:57:08,882 --> 00:57:10,840 there has to be a [? counterpart in the ?] QCD. 802 00:57:10,840 --> 00:57:11,798 HONG LIU: That's right. 803 00:57:11,798 --> 00:57:15,155 AUDIENCE: If there is not a [INAUDIBLE], that won't work 804 00:57:15,155 --> 00:57:15,780 HONG LIU: Yeah. 805 00:57:15,780 --> 00:57:17,910 This cannot be a description of that. 806 00:57:17,910 --> 00:57:21,250 From Weinberg-Witten theorem, we know in QCD there's 807 00:57:21,250 --> 00:57:22,650 no massless spin-2 particle. 808 00:57:25,280 --> 00:57:25,940 Yes. 809 00:57:25,940 --> 00:57:30,487 AUDIENCE: I thought we have talked about maybe we can do 810 00:57:30,487 --> 00:57:35,260 strings to [? find ?] QCD in a different dimension [? in ?] 811 00:57:35,260 --> 00:57:35,760 space. 812 00:57:35,760 --> 00:57:37,145 HONG LIU: We will go into that. 813 00:57:37,145 --> 00:57:38,770 But now they are in the same dimension, 814 00:57:38,770 --> 00:57:43,070 because this Minkowski 4, this will 815 00:57:43,070 --> 00:57:50,370 have-- because this is a compact [? part, ?] it doesn't matter. 816 00:57:50,370 --> 00:57:57,260 So in this part, [? there are ?] massless spin-2 particles. 817 00:57:57,260 --> 00:57:58,782 This does not [? apply ?] in QCD. 818 00:58:02,880 --> 00:58:03,980 So what can you do? 819 00:58:03,980 --> 00:58:07,650 So most people just give up. 820 00:58:07,650 --> 00:58:09,360 Most people give up. 821 00:58:09,360 --> 00:58:15,900 So other than give up, the option is say maybe 822 00:58:15,900 --> 00:58:18,580 this action is too simple. 823 00:58:18,580 --> 00:58:20,600 Maybe you have to look at more exotic action. 824 00:58:26,980 --> 00:58:28,270 OK. 825 00:58:28,270 --> 00:58:30,790 So this is one possibility. 826 00:58:30,790 --> 00:58:32,900 And the second possibility is that maybe you 827 00:58:32,900 --> 00:58:34,610 need to look for some other target space. 828 00:58:41,031 --> 00:58:41,530 OK. 829 00:58:44,080 --> 00:58:45,950 But now, what if you go away from here? 830 00:58:49,459 --> 00:58:51,250 Once you go away from here, everything else 831 00:58:51,250 --> 00:58:57,649 is now becoming such little in the ocean, 832 00:58:57,649 --> 00:58:59,690 because then you don't have much clue what to do. 833 00:58:59,690 --> 00:59:04,330 We just say, your basic guess just could not work. 834 00:59:04,330 --> 00:59:07,689 So for many years, even though this is a very intriguing idea, 835 00:59:07,689 --> 00:59:08,980 people could not make progress. 836 00:59:14,900 --> 00:59:18,207 But now we have hindsight. 837 00:59:18,207 --> 00:59:19,290 But now we have hindsight. 838 00:59:25,110 --> 00:59:30,460 So we know that even this maybe cannot be described 839 00:59:30,460 --> 00:59:39,250 by a four-dimensional-- so even though this cannot have 840 00:59:39,250 --> 00:59:47,820 a-- so this cannot have a massless spin-2 particle 841 00:59:47,820 --> 00:59:51,650 in this 3 plus 1 dimension of Minkowski spacetime. 842 00:59:51,650 --> 00:59:55,330 Maybe you can still have some kind of graviton 843 00:59:55,330 --> 00:59:57,330 in some kind of a five-dimensional spacetime. 844 01:00:00,040 --> 01:00:03,095 You have some five dimensions, in a different dimension. 845 01:00:06,220 --> 01:00:09,940 So there were some rough hints. 846 01:00:09,940 --> 01:00:16,156 Maybe you can consider there's a five-dimensional string theory. 847 01:00:24,640 --> 01:00:27,195 So let me emphasize when we say five or four, 848 01:00:27,195 --> 01:00:32,710 I always mention the non-compact part. 849 01:00:32,710 --> 01:00:37,300 So the compact part, it doesn't count because compact part just 850 01:00:37,300 --> 01:00:40,640 goes for the ride. 851 01:00:40,640 --> 01:00:42,720 What determines the properties, say, 852 01:00:42,720 --> 01:00:44,300 of a massless particle, et cetera, 853 01:00:44,300 --> 01:00:47,319 is the uncompact of the spacetime. 854 01:00:47,319 --> 01:00:49,360 Yeah, because this is a 10-dimensional spacetime. 855 01:00:49,360 --> 01:00:50,693 This is already not [INAUDIBLE]. 856 01:00:50,693 --> 01:01:01,890 So maybe we [? change ?] for string theory 857 01:01:01,890 --> 01:01:05,300 in five-dimensional uncompact. 858 01:01:05,300 --> 01:01:07,180 AUDIENCE: Five, so in 4 plus 1? 859 01:01:07,180 --> 01:01:08,470 HONG LIU: Yeah. 860 01:01:08,470 --> 01:01:13,010 In 4 plus 1 uncompact spacetime. 861 01:01:28,450 --> 01:01:29,376 Yes. 862 01:01:29,376 --> 01:01:32,657 AUDIENCE: [INAUDIBLE] compactors. 863 01:01:32,657 --> 01:01:34,031 When you say compact, do you mean 864 01:01:34,031 --> 01:01:35,490 the mathematical definition of compactness? 865 01:01:35,490 --> 01:01:36,698 HONG LIU: Yeah, that's right. 866 01:01:36,698 --> 01:01:40,050 Yeah, I just say there is a finite volume. 867 01:01:40,050 --> 01:01:42,610 Just for our purpose here, we can do it simply. 868 01:01:42,610 --> 01:01:48,500 Just let's imagine-- yeah, compact always 869 01:01:48,500 --> 01:01:51,000 has a finite volume, for example. 870 01:01:51,000 --> 01:01:51,767 Yes? 871 01:01:51,767 --> 01:01:54,509 AUDIENCE: Why can we just ignore the compact dimensions? 872 01:01:54,509 --> 01:01:57,073 Is there any condition on how big they're allowed 873 01:01:57,073 --> 01:02:00,620 to be or something, like limit? 874 01:02:00,620 --> 01:02:03,270 HONG LIU: Yeah, just when you have-- so 875 01:02:03,270 --> 01:02:05,140 if you know a little bit about this thing 876 01:02:05,140 --> 01:02:07,050 called the Kaluza-Klein theory. 877 01:02:07,050 --> 01:02:16,820 And you know that the compact part-- the thing 878 01:02:16,820 --> 01:02:18,640 is that if you have a theory [? based ?] 879 01:02:18,640 --> 01:02:20,880 on uncompact and the compact part, 880 01:02:20,880 --> 01:02:22,630 and then most of the physical properties 881 01:02:22,630 --> 01:02:25,800 is controlled by the physics of uncompact parts. 882 01:02:25,800 --> 01:02:27,680 And this will determine some details 883 01:02:27,680 --> 01:02:30,280 like the detailed spectrum, et cetera. 884 01:02:30,280 --> 01:02:31,980 But the kind of thing we worry about, 885 01:02:31,980 --> 01:02:34,460 whether you have this massless spin-2 particle, et cetera, 886 01:02:34,460 --> 01:02:36,387 will not be determined by this kind of thing. 887 01:02:36,387 --> 01:02:38,004 AUDIENCE: Is there any volume limit 888 01:02:38,004 --> 01:02:39,535 on the compact part, like maximum? 889 01:02:39,535 --> 01:02:41,535 HONG LIU: No, it's fine to have a finite volume. 890 01:02:41,535 --> 01:02:44,897 AUDIENCE: Just finite, but can it be large? 891 01:02:44,897 --> 01:02:46,980 HONG LIU: No matter how large, this have infinite. 892 01:02:46,980 --> 01:02:48,604 It's always much smaller than this one. 893 01:02:51,450 --> 01:02:54,390 Yeah, but now it's just always relative. 894 01:02:54,390 --> 01:02:55,650 It's always relative. 895 01:02:55,650 --> 01:02:56,160 Yes. 896 01:02:56,160 --> 01:02:57,535 AUDIENCE: Tracking back a little, 897 01:02:57,535 --> 01:03:00,740 is there any quick explanation for 26 and 10 are special, 898 01:03:00,740 --> 01:03:02,010 or is it very complicated? 899 01:03:02,010 --> 01:03:03,931 HONG LIU: Um. 900 01:03:03,931 --> 01:03:07,830 [LAUGHTER] 901 01:03:07,830 --> 01:03:09,609 No, it's not complicated. 902 01:03:09,609 --> 01:03:11,650 Actually, we were going to do it in next lecture. 903 01:03:14,980 --> 01:03:18,730 Yeah, next lecture we will see 26, but maybe not 10. 904 01:03:18,730 --> 01:03:21,290 10 is little bit more complicated. 905 01:03:21,290 --> 01:03:23,820 Most people voted for my option one, 906 01:03:23,820 --> 01:03:28,010 so that means you will be able to see the 26. 907 01:03:28,010 --> 01:03:28,910 Right. 908 01:03:28,910 --> 01:03:31,226 AUDIENCE: Who [? discovered ?] 26 and 10? 909 01:03:31,226 --> 01:03:33,892 I mean, they are specific for this [INAUDIBLE] action 910 01:03:33,892 --> 01:03:35,975 rate, so for other action would be something else. 911 01:03:35,975 --> 01:03:42,560 HONG LIU: Specifically for the Nambu-Goto action is 26. 912 01:03:42,560 --> 01:03:45,920 And for the 10, you need to add some additional fermions 913 01:03:45,920 --> 01:03:49,010 and make it into a so-called superstring, then become 10. 914 01:03:49,010 --> 01:03:53,800 And even this 26 one is not completely self-consistent. 915 01:03:53,800 --> 01:03:56,070 And anyway, there's still some little, tiny problems 916 01:03:56,070 --> 01:03:56,990 with this. 917 01:03:56,990 --> 01:03:58,264 Anyway, so normally we use 10. 918 01:04:02,140 --> 01:04:08,420 OK so now, then there's some tantalizing hints 919 01:04:08,420 --> 01:04:13,317 for the-- say, maybe you cannot do it with the 3 plus 1 920 01:04:13,317 --> 01:04:14,650 dimensional uncompact spacetime. 921 01:04:14,650 --> 01:04:18,160 Maybe you can do a 4 plus 1 dimensional uncompact. 922 01:04:18,160 --> 01:04:22,554 So the first is the holographic principle, 923 01:04:22,554 --> 01:04:23,470 where you have length. 924 01:04:28,330 --> 01:04:31,130 Holographic principle we have learned because there we say, 925 01:04:31,130 --> 01:04:34,190 if you want to describe a theory with gravity, 926 01:04:34,190 --> 01:04:36,650 then this gravity should be able to be described 927 01:04:36,650 --> 01:04:39,760 by something on its boundary. 928 01:04:39,760 --> 01:04:42,956 And the string theory is a theory with gravity. 929 01:04:42,956 --> 01:04:44,330 So if the string theory should be 930 01:04:44,330 --> 01:04:49,010 equivalent to some kind of QCD, some kind of gauge theory 931 01:04:49,010 --> 01:04:52,820 without gravity, and then from holographic principle, 932 01:04:52,820 --> 01:04:57,080 this field theory maybe should be one lower dimension, OK? 933 01:04:57,080 --> 01:04:58,588 In one lower dimension. 934 01:05:01,330 --> 01:05:02,480 Is the logic here clear? 935 01:05:02,480 --> 01:05:05,520 AUDIENCE: Wait, can you say that again? 936 01:05:05,520 --> 01:05:08,480 HONG LIU: So here we want to equate large N 937 01:05:08,480 --> 01:05:11,670 QCD with some string theory. 938 01:05:11,670 --> 01:05:14,660 But string theory we know contains gravity. 939 01:05:14,660 --> 01:05:19,550 A list of all our experience contain gravity. 940 01:05:19,550 --> 01:05:22,240 But if you believe that the gravity should satisfy 941 01:05:22,240 --> 01:05:27,580 holographic principle, then the gravity should be equivalent, 942 01:05:27,580 --> 01:05:30,010 according to holographic principle, 943 01:05:30,010 --> 01:05:34,400 gravity in, say, D dimensional spacetime 944 01:05:34,400 --> 01:05:37,490 can be described by something on its boundary, 945 01:05:37,490 --> 01:05:40,879 something one dimension lower. 946 01:05:40,879 --> 01:05:42,920 AUDIENCE: But I thought the holographic principle 947 01:05:42,920 --> 01:05:44,560 was a statement about entropy. 948 01:05:44,560 --> 01:05:46,550 HONG LIU: No, it's a state started 949 01:05:46,550 --> 01:05:48,760 from a statement about entropy. 950 01:05:48,760 --> 01:05:51,330 But then you do a little bit of leap. 951 01:05:51,330 --> 01:05:54,140 So what I call it little bit of a conceptual leap 952 01:05:54,140 --> 01:05:58,410 is that the-- or [? little ?] leap of faith 953 01:05:58,410 --> 01:06:01,490 is that you promote that into the statement that 954 01:06:01,490 --> 01:06:03,200 said the number of degrees freedom 955 01:06:03,200 --> 01:06:07,060 you needed to describe the whole system. 956 01:06:07,060 --> 01:06:09,280 Yeah, so the holographic principle 957 01:06:09,280 --> 01:06:13,260 is that for any region, even the quantum gravity theory, 958 01:06:13,260 --> 01:06:16,960 for any region, you should be able to describe it 959 01:06:16,960 --> 01:06:21,204 by the degrees of freedom living on the boundary of that region. 960 01:06:21,204 --> 01:06:23,620 And degrees freedom living on the boundary of that region, 961 01:06:23,620 --> 01:06:25,130 then it's one dimensional lower. 962 01:06:25,130 --> 01:06:28,920 AUDIENCE: Wait, so can I ask one question about that? 963 01:06:28,920 --> 01:06:31,160 If I have some region, some volume in space, 964 01:06:31,160 --> 01:06:33,450 some closed ball or something. 965 01:06:33,450 --> 01:06:37,334 And I live in a universe which is, for example, a closed-- 966 01:06:37,334 --> 01:06:39,000 like maybe they live on some hypersphere 967 01:06:39,000 --> 01:06:40,160 or something like this. 968 01:06:40,160 --> 01:06:44,190 Then how do I know whether I'm-- how do I know that 969 01:06:44,190 --> 01:06:46,129 the information is encoded? 970 01:06:46,129 --> 01:06:47,920 How do I know whether I'm inside the sphere 971 01:06:47,920 --> 01:06:49,450 or outside of the sphere? 972 01:06:49,450 --> 01:06:51,380 For example, we see that the entropy that 973 01:06:51,380 --> 01:06:53,100 has to do with the sphere basically 974 01:06:53,100 --> 01:06:54,933 tells you about how much information can you 975 01:06:54,933 --> 01:06:56,810 contain inside the sphere. 976 01:06:56,810 --> 01:07:00,054 But if you live in a universe which is closed or something, 977 01:07:00,054 --> 01:07:01,470 then you don't know whether you're 978 01:07:01,470 --> 01:07:03,040 inside or outside the sphere. 979 01:07:03,040 --> 01:07:05,770 HONG LIU: Yeah, but that's a difficult question. 980 01:07:05,770 --> 01:07:08,040 Yeah, if you talk about closed universe here, 981 01:07:08,040 --> 01:07:09,840 we are not talking about closed universe. 982 01:07:09,840 --> 01:07:11,427 AUDIENCE: I see. 983 01:07:11,427 --> 01:07:12,010 HONG LIU: Yes. 984 01:07:12,010 --> 01:07:13,884 AUDIENCE: I thought the holographic principle 985 01:07:13,884 --> 01:07:17,130 is that the number of degrees freedom inside the region 986 01:07:17,130 --> 01:07:18,914 is actually bounded by the area. 987 01:07:18,914 --> 01:07:20,330 HONG LIU: Right, it's bounded by-- 988 01:07:20,330 --> 01:07:21,871 AUDIENCE: Yeah, but why is it that we 989 01:07:21,871 --> 01:07:25,169 use that degree of freedom living on the boundary? 990 01:07:25,169 --> 01:07:27,210 HONG LIU: There are several formulations of that. 991 01:07:27,210 --> 01:07:29,251 First is that the total number of degrees freedom 992 01:07:29,251 --> 01:07:32,150 in this region is bounded by the area. 993 01:07:32,150 --> 01:07:34,720 And then you can go to the next step, which is maybe 994 01:07:34,720 --> 01:07:37,510 the whole region can be just described 995 01:07:37,510 --> 01:07:40,170 by these degrees of freedom living 996 01:07:40,170 --> 01:07:41,915 on the boundary on that region. 997 01:07:41,915 --> 01:07:46,070 AUDIENCE: Is that because, say, the state of density 998 01:07:46,070 --> 01:07:50,840 on the boundary [INAUDIBLE] the state on the boundary 999 01:07:50,840 --> 01:07:53,720 is proportional to the area of the boundary? 1000 01:07:53,720 --> 01:07:54,440 HONG LIU: Yeah. 1001 01:07:54,440 --> 01:07:54,970 Exactly. 1002 01:07:54,970 --> 01:07:56,896 That's right. 1003 01:07:56,896 --> 01:08:01,020 AUDIENCE: So here our goal is to recover the large N theory in 3 1004 01:08:01,020 --> 01:08:02,740 plus 1 dimensions without gravity. 1005 01:08:02,740 --> 01:08:04,090 So we have no gravity. 1006 01:08:04,090 --> 01:08:05,260 You can't 3 plus 1. 1007 01:08:05,260 --> 01:08:06,710 HONG LIU: Right. 1008 01:08:06,710 --> 01:08:09,240 Yeah, so if that is supposed to be equivalent to the gravity 1009 01:08:09,240 --> 01:08:10,740 theory, and the gravity [? theory ?] 1010 01:08:10,740 --> 01:08:13,440 to find the holographic principle, 1011 01:08:13,440 --> 01:08:16,510 and then the natural guess is that 1012 01:08:16,510 --> 01:08:18,740 this non-gravitational field theory should 1013 01:08:18,740 --> 01:08:22,100 live in one dimensional lower. 1014 01:08:22,100 --> 01:08:22,917 OK? 1015 01:08:22,917 --> 01:08:23,750 So this is one hint. 1016 01:08:26,899 --> 01:08:31,220 And the second is actually from the consistency 1017 01:08:31,220 --> 01:08:32,300 of string theory itself. 1018 01:08:45,380 --> 01:08:48,020 So this is a little bit technical. 1019 01:08:48,020 --> 01:08:50,260 Again, we will only be able to explain it 1020 01:08:50,260 --> 01:08:54,460 a little bit later, when we talk about more details about string 1021 01:08:54,460 --> 01:08:55,580 theory. 1022 01:08:55,580 --> 01:08:58,979 You can [? tell, ?] even though the string theory in this space 1023 01:08:58,979 --> 01:09:02,386 is inconsistent. 1024 01:09:05,090 --> 01:09:09,370 But there's a simple way. 1025 01:09:09,370 --> 01:09:13,279 This is-- it's not a simple way. 1026 01:09:13,279 --> 01:09:14,990 So what's happening is the following. 1027 01:09:14,990 --> 01:09:18,290 So if you consider, say, a string 1028 01:09:18,290 --> 01:09:26,450 propagating in this spacetime, and there are some symmetries 1029 01:09:26,450 --> 01:09:28,750 on the worldsheet. 1030 01:09:28,750 --> 01:09:30,990 And only in the 10 and 26 dimension, 1031 01:09:30,990 --> 01:09:35,680 those symmetries are satisfied quantum mechanically. 1032 01:09:35,680 --> 01:09:38,191 And in other dimensions, those symmetries, somehow, 1033 01:09:38,191 --> 01:09:39,649 even though classically it's there, 1034 01:09:39,649 --> 01:09:41,930 but quantum mechanically it's gone. 1035 01:09:41,930 --> 01:09:43,680 And those symmetries become-- because they 1036 01:09:43,680 --> 01:09:45,193 are gone quantum mechanically, then 1037 01:09:45,193 --> 01:09:46,359 it leads to inconsistencies. 1038 01:09:48,880 --> 01:09:54,189 And it turns out that there's some other way 1039 01:09:54,189 --> 01:09:58,410 you can make that consistent, to make that symmetry still 1040 01:09:58,410 --> 01:10:01,941 to be valid, is by adding some new degrees of freedom. 1041 01:10:01,941 --> 01:10:02,440 OK? 1042 01:10:02,440 --> 01:10:03,856 It's just there's some new degrees 1043 01:10:03,856 --> 01:10:05,785 freedom dynamically generated. 1044 01:10:05,785 --> 01:10:07,160 And then that new degrees freedom 1045 01:10:07,160 --> 01:10:10,830 turned out to behave like an additional dimension. 1046 01:10:10,830 --> 01:10:12,200 OK. 1047 01:10:12,200 --> 01:10:13,970 Yeah, this will make no sense to you. 1048 01:10:13,970 --> 01:10:18,970 I'm just saying a consistency of string theory actually 1049 01:10:18,970 --> 01:10:21,060 sometimes can give you one additional dimension. 1050 01:10:21,060 --> 01:10:23,684 AUDIENCE: What is the difference between these inconsistencies, 1051 01:10:23,684 --> 01:10:25,630 talking about anomalies and-- 1052 01:10:25,630 --> 01:10:27,380 HONG LIU: It is anomalies. 1053 01:10:27,380 --> 01:10:30,210 But here it's called gauge anomalies. 1054 01:10:30,210 --> 01:10:34,970 It's gauge anomalies is at the local symmetry anomalies, 1055 01:10:34,970 --> 01:10:35,965 which is inconsistent. 1056 01:10:35,965 --> 01:10:37,506 AUDIENCE: So just-- maybe this is not 1057 01:10:37,506 --> 01:10:39,960 the time to ask this-- but are the degrees of freedom 1058 01:10:39,960 --> 01:10:43,530 that you need to save you from this inconsistency problem. 1059 01:10:43,530 --> 01:10:47,280 So do they have to be extra dimensions of space? 1060 01:10:47,280 --> 01:10:51,080 Or what I'm saying is that if we need to do string theory in 10 1061 01:10:51,080 --> 01:10:53,390 dimensions, is it really four dimensions plus six 1062 01:10:53,390 --> 01:10:54,350 degrees of freedom? 1063 01:10:54,350 --> 01:10:57,590 Or are they actually six bona fide spatial dimensions? 1064 01:10:57,590 --> 01:11:00,180 HONG LIU: Oh, this is a very good question. 1065 01:11:00,180 --> 01:11:04,920 So if you have-- yeah, this something 1066 01:11:04,920 --> 01:11:08,772 we would be a little bit more clear just even in-- oh, it's 1067 01:11:08,772 --> 01:11:10,180 very late. 1068 01:11:10,180 --> 01:11:12,550 Even the second part of this lecture 1069 01:11:12,550 --> 01:11:15,190 is that here you have four degrees of freedom, 1070 01:11:15,190 --> 01:11:17,440 you have six degrees of freedom. 1071 01:11:17,440 --> 01:11:20,632 But turns out, if you only consider this guy, 1072 01:11:20,632 --> 01:11:23,090 then this four degrees freedom by itself is not consistent. 1073 01:11:23,090 --> 01:11:24,990 It's [? its own ?] violation of the symmetry at the quantum 1074 01:11:24,990 --> 01:11:25,900 level. 1075 01:11:25,900 --> 01:11:28,320 And then you need to add more, and then one more, 1076 01:11:28,320 --> 01:11:31,050 because of course one and two have extra dimension. 1077 01:11:31,050 --> 01:11:35,710 Anyway, we can make it more explicit in next lecture. 1078 01:11:35,710 --> 01:11:39,920 Here I just throw a remark here. 1079 01:11:39,920 --> 01:11:46,610 Anyway, this guy-- this is purely hindsight. 1080 01:11:46,610 --> 01:11:50,580 Nobody have realized this point, this first point, 1081 01:11:50,580 --> 01:11:53,610 nobody have realized it before this holographic duality 1082 01:11:53,610 --> 01:11:55,410 was discovered. 1083 01:11:55,410 --> 01:11:56,900 Nobody really made this connection. 1084 01:11:59,950 --> 01:12:03,200 And at this point, saying there should 1085 01:12:03,200 --> 01:12:04,790 be a five-dimensional string theory 1086 01:12:04,790 --> 01:12:11,990 describing gauge theory, that was made just 1087 01:12:11,990 --> 01:12:13,860 before the discovery. 1088 01:12:13,860 --> 01:12:15,485 I will mention that a little bit later. 1089 01:12:18,940 --> 01:12:23,560 Anyway, so now let's-- let me just maybe finish this, 1090 01:12:23,560 --> 01:12:24,550 and we have a break. 1091 01:12:30,890 --> 01:12:32,690 So now let's consider-- suppose there is 1092 01:12:32,690 --> 01:12:36,340 a five-dimensional spacetime, string theory 1093 01:12:36,340 --> 01:12:39,170 in some five-dimensional spacetime, 1094 01:12:39,170 --> 01:12:47,685 say 4 plus 1 dimensional spacetime that describes QCD. 1095 01:12:50,360 --> 01:12:53,410 Then what should be the property of this Y? 1096 01:12:53,410 --> 01:12:58,550 So this Y denotes some manifold Y. OK? 1097 01:12:58,550 --> 01:13:03,850 So as I mentioned, it must have at least all the symmetries 1098 01:13:03,850 --> 01:13:06,010 of the QCD, but not more. 1099 01:13:06,010 --> 01:13:10,180 Should have exactly the same amount of symmetries. 1100 01:13:10,180 --> 01:13:12,430 So that means it must have the translation and Lorentz 1101 01:13:12,430 --> 01:13:14,440 symmetries of QCD. 1102 01:13:14,440 --> 01:13:16,420 OK? 1103 01:13:16,420 --> 01:13:22,730 So that means the only metric I can write down 1104 01:13:22,730 --> 01:13:23,972 must be of this form. 1105 01:13:35,660 --> 01:13:38,570 The only metric I can write down, 1106 01:13:38,570 --> 01:13:40,190 the metric must be have this form. 1107 01:13:43,130 --> 01:13:45,480 So this az just some function. 1108 01:13:45,480 --> 01:13:49,060 And z is the extra dimension to a Minkowski spacetime. 1109 01:13:49,060 --> 01:13:52,170 And this is some Minkowski metric for 3 plus 1 dimension. 1110 01:13:54,810 --> 01:13:58,170 AUDIENCE: You mean it's like a prototype to four dimension, 1111 01:13:58,170 --> 01:14:01,525 we have to get the Minkowski space. 1112 01:14:01,525 --> 01:14:02,150 HONG LIU: Yeah. 1113 01:14:02,150 --> 01:14:05,040 Just say whatever this space, whatever 1114 01:14:05,040 --> 01:14:09,400 is the symmetry of this-- so the symmetry of this spacetime 1115 01:14:09,400 --> 01:14:11,770 must have the Poincare-- must have 1116 01:14:11,770 --> 01:14:14,190 all the symmetry of the 3 plus 1 dimensional 1117 01:14:14,190 --> 01:14:15,770 Minkowski spacetime. 1118 01:14:15,770 --> 01:14:18,550 Then the simplest way, you're saying that the only way 1119 01:14:18,550 --> 01:14:21,040 to do it is just you put the Minkowski spacetime there 1120 01:14:21,040 --> 01:14:23,140 as a subspace. 1121 01:14:23,140 --> 01:14:26,130 And then you have one additional space, 1122 01:14:26,130 --> 01:14:30,240 and then you can have one additional dimension. 1123 01:14:30,240 --> 01:14:33,530 And then, because you have to maintain 1124 01:14:33,530 --> 01:14:37,170 the symmetries and [INAUDIBLE] to be thinking then 1125 01:14:37,170 --> 01:14:40,796 you can convince yourself that the only additional degrees 1126 01:14:40,796 --> 01:14:42,170 freedom in the metric [INAUDIBLE] 1127 01:14:42,170 --> 01:14:44,060 is the overall function. 1128 01:14:44,060 --> 01:14:48,720 So the function of this z, and nothing else. 1129 01:14:48,720 --> 01:14:50,510 OK. 1130 01:14:50,510 --> 01:14:53,500 AUDIENCE: Can that be part of kind 1131 01:14:53,500 --> 01:14:55,440 of a scalar in Minkowski space? 1132 01:14:55,440 --> 01:14:57,110 HONG LIU: Yeah. 1133 01:14:57,110 --> 01:15:00,820 Let me just say, this is most general 1134 01:15:00,820 --> 01:15:13,160 metric, consistent with four dimensional, 3 plus 1 1135 01:15:13,160 --> 01:15:18,951 dimensional, Poincare symmetries. 1136 01:15:30,510 --> 01:15:34,590 AUDIENCE: Why this additional dimension always in a space 1137 01:15:34,590 --> 01:15:35,290 part? 1138 01:15:35,290 --> 01:15:37,685 Can it be in a time-like part? 1139 01:15:37,685 --> 01:15:39,665 Like a 3 plus 2? 1140 01:15:46,820 --> 01:15:49,406 HONG LIU: Both arguments suggest it's a space part. 1141 01:15:49,406 --> 01:15:51,530 So because this is just the boundary of some region 1142 01:15:51,530 --> 01:15:53,970 there's a spatial dimension [? reduction ?], not time. 1143 01:16:01,130 --> 01:16:04,280 So is this clear to you? 1144 01:16:04,280 --> 01:16:07,140 Because you won't have a Minkowski spacetime, 1145 01:16:07,140 --> 01:16:09,850 so you must have a Minkowski here. 1146 01:16:09,850 --> 01:16:11,970 And then in the prefactor of the Minkowski, 1147 01:16:11,970 --> 01:16:14,390 you can multiply by anything, any function, 1148 01:16:14,390 --> 01:16:17,880 but this function cannot depend on the X. 1149 01:16:17,880 --> 01:16:20,130 It can only depend on this extra dimension. 1150 01:16:20,130 --> 01:16:23,400 Because if you have anything which depend on capital X, 1151 01:16:23,400 --> 01:16:26,671 then you have violated the Poincare symmetry. 1152 01:16:26,671 --> 01:16:28,420 You have violated the translation [? X. ?] 1153 01:16:31,250 --> 01:16:35,290 So the only function you can put before this Minkowski spacetime 1154 01:16:35,290 --> 01:16:38,949 is a function of this additional dimension. 1155 01:16:38,949 --> 01:16:40,990 And then by redefining this additional dimension, 1156 01:16:40,990 --> 01:16:44,360 I can always put this overall factor in the front. 1157 01:16:44,360 --> 01:16:46,280 Yeah, so this tells you that this 1158 01:16:46,280 --> 01:16:48,070 is the most general metric. 1159 01:16:48,070 --> 01:16:49,697 OK? 1160 01:16:49,697 --> 01:16:52,030 So if it's not clear to you, think about it a little bit 1161 01:16:52,030 --> 01:16:52,530 afterwards. 1162 01:16:55,420 --> 01:16:58,250 So these are the most as you can do. 1163 01:17:01,100 --> 01:17:01,850 So that's the end. 1164 01:17:01,850 --> 01:17:06,520 So you say, you cannot determine az, et cetera. 1165 01:17:06,520 --> 01:17:12,840 So this is as most you can say for the QCD. 1166 01:17:12,840 --> 01:17:33,950 But if the theory, if the field theory is scale invariant, 1167 01:17:33,950 --> 01:17:39,850 say, conformal field theory, that normally we call CFT, OK? 1168 01:17:39,850 --> 01:17:41,675 So conformal field theory. 1169 01:17:44,590 --> 01:17:47,800 Then we can show this metric. 1170 01:17:47,800 --> 01:17:49,387 So let me call this equation 1. 1171 01:17:52,970 --> 01:17:59,605 Then 1 must be [INAUDIBLE] spacetime. 1172 01:18:03,770 --> 01:18:05,893 AUDIENCE: [INAUDIBLE] symmetry on the boundary 1173 01:18:05,893 --> 01:18:08,190 as well, [INAUDIBLE]? 1174 01:18:08,190 --> 01:18:10,720 HONG LIU: Yeah, I'm going to show that. 1175 01:18:10,720 --> 01:18:15,860 So if the field theory is scale invariant, that 1176 01:18:15,860 --> 01:18:18,600 means that the fields theory have some additional symmetry, 1177 01:18:18,600 --> 01:18:21,470 should be satisfied by this metric. 1178 01:18:21,470 --> 01:18:24,590 And then I will show that this additional scaling symmetry 1179 01:18:24,590 --> 01:18:27,780 will make this to precisely a so-called anti-de Sitter 1180 01:18:27,780 --> 01:18:29,660 spacetime. 1181 01:18:29,660 --> 01:18:31,760 AUDIENCE: Field theory, and then the 3 plus 1. 1182 01:18:31,760 --> 01:18:32,610 HONG LIU: Yeah. 1183 01:18:32,610 --> 01:18:33,110 Right. 1184 01:18:33,110 --> 01:18:39,400 If the field theory, say the-- QCD does not have a scale. 1185 01:18:39,400 --> 01:18:41,944 It's not scaling right, so I do not say a QCD anymore. 1186 01:18:41,944 --> 01:18:43,360 Just say, suppose some other field 1187 01:18:43,360 --> 01:18:46,020 theory, which have large N expansion, which 1188 01:18:46,020 --> 01:18:48,270 is also scale invariant. 1189 01:18:48,270 --> 01:18:51,290 And then the corresponding string theory 1190 01:18:51,290 --> 01:18:53,864 must be in anti-de Sitter spacetime. 1191 01:18:53,864 --> 01:18:55,864 AUDIENCE: Are we ever going to come back to QCD, 1192 01:18:55,864 --> 01:18:56,810 or is that a-- 1193 01:18:56,810 --> 01:18:59,480 HONG LIU: No, that's it. 1194 01:18:59,480 --> 01:19:04,310 Maybe we'll come back to QCD, but in a somewhat indirect way. 1195 01:19:04,310 --> 01:19:09,571 Yeah, not to your real-life, beloved QCD. 1196 01:19:09,571 --> 01:19:11,570 AUDIENCE: So no one's solved that problem still? 1197 01:19:11,570 --> 01:19:14,300 HONG LIU: Yeah, no one's solved that problem yet. 1198 01:19:14,300 --> 01:19:15,425 So you still have a chance. 1199 01:19:19,590 --> 01:19:21,040 So that remains very simple. 1200 01:19:21,040 --> 01:19:24,925 So let me just say, then we will have a break. 1201 01:19:24,925 --> 01:19:25,800 Then we will be done. 1202 01:19:29,510 --> 01:19:32,830 I think I'm going very slowly today. 1203 01:19:32,830 --> 01:19:57,550 So scale invariant theory-- is invariant under the scaling 1204 01:19:57,550 --> 01:20:02,940 for any constant, constant lambda. 1205 01:20:09,780 --> 01:20:12,200 So scale invariant theory should be 1206 01:20:12,200 --> 01:20:14,390 invariant under such a scaling. 1207 01:20:14,390 --> 01:20:17,270 And then now we want to require this metric also 1208 01:20:17,270 --> 01:20:19,100 have this scaling. 1209 01:20:19,100 --> 01:20:20,550 OK? 1210 01:20:20,550 --> 01:20:28,068 So now, we require 1 also have such scaling. 1211 01:20:32,040 --> 01:20:33,152 That's scaling symmetry. 1212 01:20:41,520 --> 01:20:46,940 OK, so we just do a scaling X mu go to lambda X mu. 1213 01:20:50,600 --> 01:20:55,320 And then this term will give me additional lambda squared. 1214 01:20:55,320 --> 01:20:57,920 So we see, in order for this to be the same as before, the z 1215 01:20:57,920 --> 01:21:01,530 should scale the same, OK? 1216 01:21:01,530 --> 01:21:07,026 So in order for this to be-- so we need z to scale as the same, 1217 01:21:07,026 --> 01:21:10,780 in order I can scale this lambda out. 1218 01:21:10,780 --> 01:21:13,220 After I scale this lambda out, I also 1219 01:21:13,220 --> 01:21:25,080 need that a lambda z should be equal to 1 over lambda az, OK? 1220 01:21:25,080 --> 01:21:27,320 So the scaling symmetry of that equation 1221 01:21:27,320 --> 01:21:28,810 requires these two conditions. 1222 01:21:28,810 --> 01:21:32,980 So on the scaling of z, this a lambda z 1223 01:21:32,980 --> 01:21:34,470 should satisfy this condition. 1224 01:21:34,470 --> 01:21:36,400 Then the lambda will cancel. 1225 01:21:36,400 --> 01:21:38,750 So this condition is important because we 1226 01:21:38,750 --> 01:21:40,160 did scale them homogeneously. 1227 01:21:40,160 --> 01:21:42,930 Otherwise, of course, lambda will not drop out. 1228 01:21:42,930 --> 01:21:47,120 And the second condition just makes sure lambda is canceled. 1229 01:21:47,120 --> 01:21:49,240 OK, is it clear? 1230 01:21:49,240 --> 01:21:58,470 So now this condition just determined 1231 01:21:58,470 --> 01:22:06,130 that az must be a simple power, must 1232 01:22:06,130 --> 01:22:09,525 be written as R divided by z. 1233 01:22:09,525 --> 01:22:10,952 See, R is some constant. 1234 01:22:21,790 --> 01:22:24,765 And now we can write down the full metric. 1235 01:22:27,540 --> 01:22:29,535 So now I've determined this function up 1236 01:22:29,535 --> 01:22:30,535 to our overall constant. 1237 01:22:33,040 --> 01:22:38,650 So the full metric is dS square equal to R squared divided 1238 01:22:38,650 --> 01:22:48,390 by z squared dz squared plus eta mu, mu, dX mu, dX mu. 1239 01:22:48,390 --> 01:22:52,090 And this is precisely AdS metric, 1240 01:22:52,090 --> 01:22:55,500 written in certain coordinates. 1241 01:22:55,500 --> 01:22:59,064 And then this R, then you adjust the curvature radius of AdS. 1242 01:23:02,940 --> 01:23:05,120 So if you don't know about anti-de Sitter spacetime, 1243 01:23:05,120 --> 01:23:07,320 it doesn't matter. 1244 01:23:07,320 --> 01:23:11,320 So this is the metric, and the name of this metric 1245 01:23:11,320 --> 01:23:14,180 is anti-de Sitter. 1246 01:23:14,180 --> 01:23:19,400 And later we will explain the properties 1247 01:23:19,400 --> 01:23:23,080 of the anti-de Sitter spacetime. 1248 01:23:23,080 --> 01:23:28,660 So now we find, so now we reach a conclusion, 1249 01:23:28,660 --> 01:23:32,850 is that if I have a large N conformal field 1250 01:23:32,850 --> 01:23:37,600 theory in Minkowski D-dimensional space, time. 1251 01:23:37,600 --> 01:23:40,040 So this can be applied to any dimensional. 1252 01:23:40,040 --> 01:23:44,610 It's not necessary [? to be ?] 3 plus 1. 1253 01:23:44,610 --> 01:23:53,220 In D-- so this, if it can be described by a string theory, 1254 01:23:53,220 --> 01:23:58,710 should be string theory in AdS d plus 1. 1255 01:24:01,270 --> 01:24:05,380 And in particular, the 1/N here is related to the g strings 1256 01:24:05,380 --> 01:24:08,300 here, the string coupling here. 1257 01:24:08,300 --> 01:24:10,140 So this is what we concluded. 1258 01:24:20,220 --> 01:24:21,020 Yes? 1259 01:24:21,020 --> 01:24:22,920 AUDIENCE: So all we've shown is that there 1260 01:24:22,920 --> 01:24:25,834 is no obvious inconsistency with that correspondence. 1261 01:24:25,834 --> 01:24:27,750 HONG LIU: What do you mean there's no obvious? 1262 01:24:27,750 --> 01:24:29,960 AUDIENCE: As in, we didn't illustrate any way that they-- 1263 01:24:29,960 --> 01:24:32,340 HONG LIU: Sure, I'm just saying this is a necessary condition. 1264 01:24:32,340 --> 01:24:34,370 AUDIENCE: Right, so at least that is necessary. 1265 01:24:34,370 --> 01:24:36,410 HONG LIU: Yeah, this is a necessary condition. 1266 01:24:36,410 --> 01:24:42,860 So if you can describe a large N CFT by our string theory-- 1267 01:24:42,860 --> 01:24:47,660 and it should be a string theory-- yeah, 1268 01:24:47,660 --> 01:24:49,450 this proposal works. 1269 01:24:49,450 --> 01:24:51,650 This proposal passed the minimal test. 1270 01:24:53,644 --> 01:24:54,810 AUDIENCE: I have a question. 1271 01:24:54,810 --> 01:24:57,482 So when Maldacena presumably actually did figure this out, 1272 01:24:57,482 --> 01:24:59,940 you said that this resulted from the holographic principle, 1273 01:24:59,940 --> 01:25:02,420 like it was just figured out right before he did it. 1274 01:25:02,420 --> 01:25:04,330 Was he aware of the holographic-- 1275 01:25:04,330 --> 01:25:06,610 HONG LIU: No, here is what I'm going to talk. 1276 01:25:06,610 --> 01:25:21,240 So Maldacena, in 1997, Maldacena found precisely-- in 1997, 1277 01:25:21,240 --> 01:25:25,205 Maldacena found a few examples of this, precisely realized 1278 01:25:25,205 --> 01:25:26,220 this. 1279 01:25:26,220 --> 01:25:31,310 And not using this mass or using some completely indirect way, 1280 01:25:31,310 --> 01:25:32,455 which we will explain next. 1281 01:25:35,450 --> 01:25:39,490 So he found this through some very indirect way. 1282 01:25:39,490 --> 01:25:43,440 But in principle, one could have realized this 1283 01:25:43,440 --> 01:25:47,010 if one kept those things in mind. 1284 01:25:47,010 --> 01:25:49,470 So now let me tell you a little bit of the history, 1285 01:25:49,470 --> 01:25:52,500 and then we will have a break. 1286 01:25:52,500 --> 01:25:53,490 Then we can go home. 1287 01:25:57,764 --> 01:25:59,680 It depends on whether you want a break or not. 1288 01:25:59,680 --> 01:26:01,989 Maybe you don't want a break. 1289 01:26:01,989 --> 01:26:03,905 Yeah, let me tell you a little bit of history. 1290 01:26:07,280 --> 01:26:12,740 So yeah, just to save time, let me not write it down, just 1291 01:26:12,740 --> 01:26:14,650 say it. 1292 01:26:14,650 --> 01:26:18,830 So in the late '60s to early '70s, so 1293 01:26:18,830 --> 01:26:24,740 string theory was developed to understand strong interactions. 1294 01:26:24,740 --> 01:26:30,650 So understanding strong interactions was the problem. 1295 01:26:30,650 --> 01:26:34,190 At the time, people were developing string theory 1296 01:26:34,190 --> 01:26:37,350 to try to understand strong interactions. 1297 01:26:37,350 --> 01:26:44,190 So in 1971, our friend Frank, Frank Wilczek, 1298 01:26:44,190 --> 01:26:47,070 and other people, they discover the asymptotic freedom. 1299 01:26:47,070 --> 01:26:50,727 And they established the Yang-Mills theory 1300 01:26:50,727 --> 01:26:52,310 as a description of strong interaction 1301 01:26:52,310 --> 01:26:55,180 which now have our QCD. 1302 01:26:55,180 --> 01:26:58,845 And so that's essentially eliminated the hope of string 1303 01:26:58,845 --> 01:27:02,690 theory to describe QCD. 1304 01:27:02,690 --> 01:27:05,160 Because the QCD seems to be very different. 1305 01:27:05,160 --> 01:27:07,030 You [? need ?] the help of string theory 1306 01:27:07,030 --> 01:27:10,320 to describe strong interaction because the QCD 1307 01:27:10,320 --> 01:27:13,650 [INAUDIBLE] gauge theory, it's very different from the string 1308 01:27:13,650 --> 01:27:15,480 theory. 1309 01:27:15,480 --> 01:27:19,330 So people soon abandoned the string theory. 1310 01:27:19,330 --> 01:27:23,050 So now we go to 1974. 1311 01:27:23,050 --> 01:27:30,310 So 1974, a big number of things were discovered in 1974. 1312 01:27:30,310 --> 01:27:32,900 So 1974 was a golden year. 1313 01:27:32,900 --> 01:27:36,580 So first is 't Hooft realized his large N expansion 1314 01:27:36,580 --> 01:27:38,080 and then realized that this actually 1315 01:27:38,080 --> 01:27:40,490 looks like string theory. 1316 01:27:40,490 --> 01:27:44,690 And then completely independently, 1317 01:27:44,690 --> 01:27:46,710 Scherk, Schwarz, and [? Yoneya, ?] 1318 01:27:46,710 --> 01:27:48,630 they realized that string theory should 1319 01:27:48,630 --> 01:27:51,200 considered a theory of gravity, rather than 1320 01:27:51,200 --> 01:27:54,660 a theory of strong interaction. 1321 01:27:54,660 --> 01:27:58,890 So they realized actually-- it's ironic, 1322 01:27:58,890 --> 01:28:01,920 people started doing string theory in the '60s and '70s, et 1323 01:28:01,920 --> 01:28:02,420 cetera. 1324 01:28:02,420 --> 01:28:04,610 But only in 1974 people realized, 1325 01:28:04,610 --> 01:28:06,616 ah, string theory always have a gravity 1326 01:28:06,616 --> 01:28:08,490 and should be considered a theory of gravity. 1327 01:28:08,490 --> 01:28:10,480 Anyway, so in 1974, they realized 1328 01:28:10,480 --> 01:28:14,210 the string theory should be considered as a gravity. 1329 01:28:14,210 --> 01:28:18,270 So that was a very, very exciting realization, 1330 01:28:18,270 --> 01:28:21,020 because then you can have [? quantum ?] gravity. 1331 01:28:21,020 --> 01:28:24,620 But by that time, people had given up on string theory. 1332 01:28:24,620 --> 01:28:28,490 So nobody cared about this important observation. 1333 01:28:28,490 --> 01:28:31,290 Nobody cared about this important observation. 1334 01:28:31,290 --> 01:28:33,680 So, also in the same year, in 1974, 1335 01:28:33,680 --> 01:28:36,420 Hawking discovered his Hawking radiation. 1336 01:28:36,420 --> 01:28:40,245 And they established that black hole mechanics is really 1337 01:28:40,245 --> 01:28:41,580 a thermodynamics. 1338 01:28:41,580 --> 01:28:43,440 Then really established that the black hole 1339 01:28:43,440 --> 01:28:46,260 is a thermodynamic object, 1340 01:28:46,260 --> 01:28:51,070 And in 1974 there's also a lot of important discovery-- 1341 01:28:51,070 --> 01:28:52,730 which is related to MIT, so that's 1342 01:28:52,730 --> 01:28:57,890 why I'm mentioning it-- is that people first really saw quarks 1343 01:28:57,890 --> 01:29:02,770 experimentally, is that, again, our friend, colleague Samuel 1344 01:29:02,770 --> 01:29:06,720 Ting at Brookhaven, which they discovered 1345 01:29:06,720 --> 01:29:10,820 a so-called charmonium, which is a bounce state of the charm 1346 01:29:10,820 --> 01:29:12,830 quark and the anti-charm quark. 1347 01:29:12,830 --> 01:29:14,970 And because the charm quark is very heavy, 1348 01:29:14,970 --> 01:29:16,760 so they form a hydrogen-like structure. 1349 01:29:16,760 --> 01:29:21,070 So in some sense, the charmonium is the first-- you first 1350 01:29:21,070 --> 01:29:23,590 directly see the quarks. 1351 01:29:23,590 --> 01:29:27,280 And actually, even after the 1971, after asymptotic freedom, 1352 01:29:27,280 --> 01:29:29,590 many people do not believe QCD. 1353 01:29:29,590 --> 01:29:31,010 They did not believe in quarks. 1354 01:29:31,010 --> 01:29:33,930 They say, if there's quarks, why don't we see them? 1355 01:29:33,930 --> 01:29:43,360 And then in 1974, Samuel Ting discovered this charmonium 1356 01:29:43,360 --> 01:29:45,690 in October. 1357 01:29:45,690 --> 01:29:47,750 And so people call it the October Revolution. 1358 01:29:47,750 --> 01:29:51,110 [LAUGHTER] 1359 01:29:54,120 --> 01:29:56,818 Do you know why they laugh? 1360 01:29:56,818 --> 01:29:57,318 OK. 1361 01:30:01,190 --> 01:30:02,260 Anyway. 1362 01:30:02,260 --> 01:30:02,760 Yeah. 1363 01:30:09,330 --> 01:30:11,400 Yeah, because I saw your emotions, 1364 01:30:11,400 --> 01:30:15,080 I think you have very good composure. 1365 01:30:15,080 --> 01:30:20,900 Anyway, in the same year, in 1974, Wilson 1366 01:30:20,900 --> 01:30:23,790 proposed what we now call the lattice QCD, 1367 01:30:23,790 --> 01:30:25,760 so he put the QCD on the lattice. 1368 01:30:25,760 --> 01:30:30,070 And then he invented, and then he 1369 01:30:30,070 --> 01:30:31,570 developed a very beautiful technique 1370 01:30:31,570 --> 01:30:36,520 to show from this putting QCD on the lattice 1371 01:30:36,520 --> 01:30:41,900 that, actually, the quark can be confined through the strings. 1372 01:30:41,900 --> 01:30:45,750 So the quarks in QCD can be confined through the strings. 1373 01:30:45,750 --> 01:30:48,850 And that essentially revived the idea maybe 1374 01:30:48,850 --> 01:30:52,480 the QCD can be a string theory, because the quarks are 1375 01:30:52,480 --> 01:30:54,550 confined through the strings. 1376 01:30:54,550 --> 01:30:58,460 And this all happened in 1974. 1377 01:30:58,460 --> 01:31:03,206 So then I mentioned the same, in the late '80s 1378 01:31:03,206 --> 01:31:04,830 and the early '90s, people were looking 1379 01:31:04,830 --> 01:31:07,290 at these so-called matrix models, the matrix 1380 01:31:07,290 --> 01:31:08,500 integrals, et cetera. 1381 01:31:08,500 --> 01:31:10,910 Then they showed they related to lower dimensional string 1382 01:31:10,910 --> 01:31:12,280 theory. 1383 01:31:12,280 --> 01:31:16,360 But nobody-- yeah, they showed this related 1384 01:31:16,360 --> 01:31:19,060 to some kind of lower dimensional string theory. 1385 01:31:19,060 --> 01:31:24,740 And then in 1993 and 1994, then 't Hooft 1386 01:31:24,740 --> 01:31:28,060 had this crazy idea of this holographic principle. 1387 01:31:28,060 --> 01:31:32,080 And he said maybe, things about the quantum gravity 1388 01:31:32,080 --> 01:31:35,220 can be described by things living on the boundary. 1389 01:31:35,220 --> 01:31:36,770 And again, it's a crazy idea. 1390 01:31:36,770 --> 01:31:39,970 Very few people paid attention to it. 1391 01:31:39,970 --> 01:31:45,880 But the only person who picked it up is Leonard Susskind. 1392 01:31:45,880 --> 01:31:51,520 And then he tried to come up with some sort of experiments 1393 01:31:51,520 --> 01:31:53,800 to show that that idea is not so crazy. 1394 01:31:57,220 --> 01:32:00,080 Actually, Susskind wrote a very sexy name for his paper. 1395 01:32:00,080 --> 01:32:02,660 It's called "The World As a Hologram." 1396 01:32:02,660 --> 01:32:08,890 And so that paper received some attention, 1397 01:32:08,890 --> 01:32:13,990 but still, still, people did not know what to make of it. 1398 01:32:13,990 --> 01:32:16,920 And then in 1995, Polchinski discovers so-called D-branes. 1399 01:32:23,990 --> 01:32:27,280 And then we go to 1997. 1400 01:32:27,280 --> 01:32:32,910 So in 1997, first in June, so as I said, 1401 01:32:32,910 --> 01:32:35,560 that QCD may be some kind of string theory. 1402 01:32:35,560 --> 01:32:38,000 This idea is a long idea, starting 1403 01:32:38,000 --> 01:32:42,160 from the 't Hooft and large N expansion, and also 1404 01:32:42,160 --> 01:32:44,630 from the Wilson's picture of confining strings 1405 01:32:44,630 --> 01:32:47,320 from the lattice QCD, etc. 1406 01:32:47,320 --> 01:32:48,870 But it's just a very hard problem. 1407 01:32:48,870 --> 01:32:51,927 If from QCD, how can you come up with a string theory? 1408 01:32:51,927 --> 01:32:52,760 It's just very hard. 1409 01:32:52,760 --> 01:32:54,680 Very few people are working on it. 1410 01:32:54,680 --> 01:32:58,640 So in 1997, in June, Polyakov finally, he 1411 01:32:58,640 --> 01:33:01,389 said, had a breakthrough. 1412 01:33:01,389 --> 01:33:02,930 He said that this consistent [? of ?] 1413 01:33:02,930 --> 01:33:06,700 string theory give you one extra dimension, 1414 01:33:06,700 --> 01:33:08,910 you should consider a five-dimensional string 1415 01:33:08,910 --> 01:33:12,220 theory rather than a four-dimensional string theory. 1416 01:33:12,220 --> 01:33:17,160 And then he gave up some arguments, anyway. 1417 01:33:17,160 --> 01:33:23,837 And he almost always actually write down this metric 1418 01:33:23,837 --> 01:33:26,420 And maybe he already wrote down this metric, I don't remember. 1419 01:33:26,420 --> 01:33:28,510 Anyway, he was very close to that. 1420 01:33:28,510 --> 01:33:30,710 But then in November, then Maldacena 1421 01:33:30,710 --> 01:33:33,660 came up with this idea of CFT. 1422 01:33:33,660 --> 01:33:35,700 And then he provided [? explicit ?] 1423 01:33:35,700 --> 01:33:40,020 examples of certain large N gauge theories, which 1424 01:33:40,020 --> 01:33:42,600 is scale invariant and some string 1425 01:33:42,600 --> 01:33:45,640 theory in certain anti-de Sitter spacetime. 1426 01:33:45,640 --> 01:33:49,691 And as I said, through the understanding 1427 01:33:49,691 --> 01:33:50,440 of these D-branes. 1428 01:33:53,200 --> 01:33:54,952 But even Maldacena's paper, he did 1429 01:33:54,952 --> 01:33:59,840 not-- he was still thinking from the picture of large N gauge 1430 01:33:59,840 --> 01:34:02,010 theory corresponding to some string theory. 1431 01:34:02,010 --> 01:34:06,680 He did not make the connection to the holographic principle. 1432 01:34:06,680 --> 01:34:10,430 He did not make a connection to the holographic principle. 1433 01:34:10,430 --> 01:34:14,640 But very soon, in February 1998, Witten wrote the paper, 1434 01:34:14,640 --> 01:34:16,020 and he made the connection. 1435 01:34:16,020 --> 01:34:19,730 He said, ah, this is precisely the holographic principle. 1436 01:34:19,730 --> 01:34:22,910 And this example, he said, ah, this example is precisely 1437 01:34:22,910 --> 01:34:25,860 the holographic principle Susskind 1438 01:34:25,860 --> 01:34:27,410 and 't Hooft was talking about. 1439 01:34:31,520 --> 01:34:34,620 So that's a brief history of how people actually 1440 01:34:34,620 --> 01:34:36,970 reached this point. 1441 01:34:36,970 --> 01:34:40,870 So the next stage, what we are going to do 1442 01:34:40,870 --> 01:34:47,430 is to try to derive [INAUDIBLE]. 1443 01:34:47,430 --> 01:34:51,530 So now we can-- as I said, we have two options. 1444 01:34:51,530 --> 01:34:54,330 We can just start from here, assuming 1445 01:34:54,330 --> 01:34:59,180 there is CFT [? that's ?] equivalent to some string 1446 01:34:59,180 --> 01:35:00,150 theory. 1447 01:35:00,150 --> 01:35:04,080 And then we can see how we can develop this further. 1448 01:35:04,080 --> 01:35:06,570 And this is one option we can take. 1449 01:35:06,570 --> 01:35:09,620 And our other option is to really see 1450 01:35:09,620 --> 01:35:12,550 how this relation actually arises from string theory. 1451 01:35:12,550 --> 01:35:17,190 And many people voted for the second option, 1452 01:35:17,190 --> 01:35:18,980 which in my [? email ?] is option one. 1453 01:35:18,980 --> 01:35:23,860 So you want to see how this is actually 1454 01:35:23,860 --> 01:35:26,140 deduced from string theory. 1455 01:35:26,140 --> 01:35:31,750 So now we will do that, OK? 1456 01:35:31,750 --> 01:35:35,910 But I should warn you, there will be some technicality 1457 01:35:35,910 --> 01:35:38,440 you have to tolerate. 1458 01:35:38,440 --> 01:35:41,445 You wanted to see how this is derived, OK? 1459 01:35:44,000 --> 01:35:48,000 So we do a lot of [? 20 ?] minutes today? 1460 01:35:48,000 --> 01:35:49,881 Without break? 1461 01:35:49,881 --> 01:35:50,380 Good. 1462 01:35:50,380 --> 01:35:50,980 OK. 1463 01:35:50,980 --> 01:35:52,730 Yeah, next time, I will remember to break. 1464 01:36:11,910 --> 01:36:12,410 OK. 1465 01:36:15,870 --> 01:36:17,970 So now we are going to derive this. 1466 01:36:17,970 --> 01:36:23,410 So first just as a preparation, I 1467 01:36:23,410 --> 01:36:26,580 need to tell you a little bit more about string theory. 1468 01:36:26,580 --> 01:36:31,070 In particular, the spectrum of closed strings, 1469 01:36:31,070 --> 01:36:32,285 closed and open strings. 1470 01:36:39,580 --> 01:36:43,180 And so this is where the gravity-- 1471 01:36:43,180 --> 01:36:45,440 and from a closed string you will see the gravity, 1472 01:36:45,440 --> 01:36:47,773 and from the open string, you will see the gauge theory. 1473 01:36:52,420 --> 01:36:52,920 OK. 1474 01:36:56,240 --> 01:36:59,560 We will see gravity and gauge theory. 1475 01:36:59,560 --> 01:37:03,130 So these are the first things we will do. 1476 01:37:03,130 --> 01:37:12,600 So the second thing we will do-- so the second thing we will do 1477 01:37:12,600 --> 01:37:14,410 is to understand the physics of D-branes. 1478 01:37:20,670 --> 01:37:24,300 So D-brane is some object in string theory. 1479 01:37:24,300 --> 01:37:29,490 And it turned out to play a very, very special role, 1480 01:37:29,490 --> 01:37:33,480 to connect the gravity and the string theory. 1481 01:37:33,480 --> 01:37:34,120 OK. 1482 01:37:34,120 --> 01:37:37,010 Connect the gravity and the string theory. 1483 01:37:37,010 --> 01:37:39,800 Because this is the connection between the gravity 1484 01:37:39,800 --> 01:37:41,790 and the string theory. 1485 01:37:41,790 --> 01:37:45,380 And in string theory, this [? object will ?] 1486 01:37:45,380 --> 01:37:47,060 deeply and precisely play this role, 1487 01:37:47,060 --> 01:37:49,101 which connects the gravity and the string theory. 1488 01:37:49,101 --> 01:37:51,060 So that's why you can deduce such a relation. 1489 01:37:51,060 --> 01:37:53,086 OK. 1490 01:37:53,086 --> 01:37:54,710 Yeah, so this is the two things we will 1491 01:37:54,710 --> 01:37:56,620 do before we can derive this. 1492 01:38:01,000 --> 01:38:07,810 So this is, say, the rough plan we 1493 01:38:07,810 --> 01:38:11,830 will do before we can derive this gravity. 1494 01:38:11,830 --> 01:38:14,920 So first let's tell you a little bit more about string theory. 1495 01:38:25,420 --> 01:38:28,930 So at beginning, just say some more general setup 1496 01:38:28,930 --> 01:38:30,230 of string theory. 1497 01:38:36,470 --> 01:38:56,320 So let's consider a string moving in a spacetime, which 1498 01:38:56,320 --> 01:39:02,560 I denote by M, say, with the metric ds squared 1499 01:39:02,560 --> 01:39:05,630 equal to g mu mu. 1500 01:39:05,630 --> 01:39:11,670 And this can depend on X, dX mu, dX mu. 1501 01:39:11,670 --> 01:39:13,440 OK? 1502 01:39:13,440 --> 01:39:17,400 So you can imagine some general curved spacetime. 1503 01:39:17,400 --> 01:39:22,075 Say mu and nu will go from 0 to 1, to D minus 1. 1504 01:39:22,075 --> 01:39:31,090 So D is the total number of space dimensions for this M. 1505 01:39:31,090 --> 01:39:49,380 So the motion of the string, as we said quite a few times now, 1506 01:39:49,380 --> 01:39:54,750 is the embedding of the worldsheet to the spacetime. 1507 01:39:54,750 --> 01:39:59,760 So this is in the form of X mu sigma tau. 1508 01:39:59,760 --> 01:40:03,010 OK, you parameterize the worldsheet by two coordinates. 1509 01:40:03,010 --> 01:40:08,170 So I will also write it as X mu sigma a. 1510 01:40:08,170 --> 01:40:13,260 And the sigma a is equal to sigma 0, and the sigma 1511 01:40:13,260 --> 01:40:15,500 1 is equal to tau sigma, OK? 1512 01:40:18,715 --> 01:40:19,965 And we will use this notation. 1513 01:40:24,300 --> 01:40:31,922 So now imagine a surface embedded in some spacetime. 1514 01:40:31,922 --> 01:40:33,380 And this is the embedding equation. 1515 01:40:33,380 --> 01:40:37,770 Because if you know those functions, 1516 01:40:37,770 --> 01:40:40,810 then you know precisely how the surface are embedded, OK? 1517 01:40:43,620 --> 01:40:47,870 And because the original spacetime have a metric, 1518 01:40:47,870 --> 01:40:56,845 then this induced metric on the worldsheet. 1519 01:40:59,780 --> 01:41:04,086 And this induced metric is very easy to write down. 1520 01:41:04,086 --> 01:41:07,335 You just plug in this function into here. 1521 01:41:09,920 --> 01:41:11,330 And when you take the derivative, 1522 01:41:11,330 --> 01:41:15,080 you only worry that sigma and tau, 1523 01:41:15,080 --> 01:41:18,820 because then that means you're restricted on the surface, 1524 01:41:18,820 --> 01:41:20,980 when your only [? value is ?] sigma and tau. 1525 01:41:20,980 --> 01:41:23,900 And then you can plug this into there. 1526 01:41:23,900 --> 01:41:28,130 So you can get the metric, then can be written in this form. 1527 01:41:28,130 --> 01:41:30,930 Here's sigma a and this sigma b. 1528 01:41:30,930 --> 01:41:31,660 OK? 1529 01:41:31,660 --> 01:41:35,440 So remember, sigma a and sigma b just tau and sigma. 1530 01:41:35,440 --> 01:41:47,490 And this hab is just equal to g mu mu, X, partial a, X mu, 1531 01:41:47,490 --> 01:41:50,230 partial b, X nu. 1532 01:41:50,230 --> 01:41:51,937 OK? 1533 01:41:51,937 --> 01:41:53,020 So this is trivial to see. 1534 01:41:53,020 --> 01:41:57,960 Just plug this into there, to the variation with sigma 1535 01:41:57,960 --> 01:42:01,000 and tau, you just get that, and it's that. 1536 01:42:01,000 --> 01:42:02,200 OK? 1537 01:42:02,200 --> 01:42:03,200 Is it clear? 1538 01:42:12,200 --> 01:42:20,420 So this Nambu-Goto action is the tension-- tension 1539 01:42:20,420 --> 01:42:26,765 we always write this 1 over 2 pi alpha prime-- dA. 1540 01:42:26,765 --> 01:42:32,230 So alpha prime is the [INAUDIBLE] dimensions square. 1541 01:42:32,230 --> 01:42:38,270 So we often also write alpha prime as ls square. 1542 01:42:38,270 --> 01:42:40,520 So alpha prime, just a parameter, too. 1543 01:42:40,520 --> 01:42:44,770 Parameterize to [? load ?] the tension of the string. 1544 01:42:44,770 --> 01:42:48,410 So this area, of course, you can just 1545 01:42:48,410 --> 01:42:55,350 write it as d squared sigma. 1546 01:42:55,350 --> 01:42:58,210 So again, you use the notation d squared sigma 1547 01:42:58,210 --> 01:43:01,140 just d sigma d tau. 1548 01:43:01,140 --> 01:43:05,520 d squared sigma minus delta h. 1549 01:43:05,520 --> 01:43:06,020 OK. 1550 01:43:06,020 --> 01:43:07,603 So this is just the area, because this 1551 01:43:07,603 --> 01:43:11,354 is the induced metric on the worldsheet. 1552 01:43:11,354 --> 01:43:13,770 Then you take the determinant, and that give you the area. 1553 01:43:13,770 --> 01:43:15,519 So this is the standard geometric formula. 1554 01:43:19,240 --> 01:43:22,850 So now let me call this equation 1. 1555 01:43:22,850 --> 01:43:25,237 So I have a [? lot ?] equation 1 before, 1556 01:43:25,237 --> 01:43:26,320 but this is a new chapter. 1557 01:43:31,783 --> 01:43:32,470 OK. 1558 01:43:32,470 --> 01:43:36,570 So this is the explicit form of this Nambu-Goto action. 1559 01:43:36,570 --> 01:43:40,160 But this action is a little bit awkward, 1560 01:43:40,160 --> 01:43:43,720 because involving the square root. 1561 01:43:43,720 --> 01:43:46,660 A square root, it's considered to be 1562 01:43:46,660 --> 01:43:48,980 not a good thing in physics. 1563 01:43:48,980 --> 01:43:51,030 Because when you write down action, 1564 01:43:51,030 --> 01:43:52,280 because it's a non-polynomial. 1565 01:43:54,400 --> 01:43:56,602 We typically like polynomial things. 1566 01:43:56,602 --> 01:43:59,060 Because the only integral we can do is a Gaussian integral, 1567 01:43:59,060 --> 01:44:00,351 and the Gaussian is polynomial. 1568 01:44:03,590 --> 01:44:08,810 So this is inconvenient, so one can rewrite it a little bit. 1569 01:44:08,810 --> 01:44:10,358 So you write down the answer. 1570 01:44:17,850 --> 01:44:22,500 So we can rewrite it in the polynomial form. 1571 01:44:22,500 --> 01:44:25,120 And this polynomial form is corresponding-- 1572 01:44:25,120 --> 01:44:29,930 it's called the Polyakov action, so I call it SP, 1573 01:44:29,930 --> 01:44:31,900 even though Polyakov had nothing to do with it. 1574 01:44:36,270 --> 01:44:39,810 And this action can be written in the following form. 1575 01:44:39,810 --> 01:44:41,600 And let me write down the answer. 1576 01:44:41,600 --> 01:44:43,076 Then I will show the equivalent. 1577 01:44:47,666 --> 01:44:51,390 AUDIENCE: Wasn't it invented by Leonard Susskind? 1578 01:44:51,390 --> 01:44:53,340 HONG LIU: No, it's not Leonard Susskind. 1579 01:44:53,340 --> 01:44:55,840 [INTERPOSING VOICES] 1580 01:44:55,840 --> 01:44:57,600 AUDIENCE: Why is it called Polyakov-- 1581 01:44:57,600 --> 01:44:59,900 HONG LIU: Polyakov-- yeah, actually Polyakov 1582 01:44:59,900 --> 01:45:01,180 had something to do with it. 1583 01:45:05,030 --> 01:45:08,060 Polyakov used it mostly [INAUDIBLE] first. 1584 01:45:15,120 --> 01:45:19,110 OK, so you can rewrite it as that, in this form. 1585 01:45:19,110 --> 01:45:22,030 And the gamma ab is a new variable introduced. 1586 01:45:22,030 --> 01:45:25,090 It's a Lagrangian multiplier. 1587 01:45:25,090 --> 01:45:25,590 OK. 1588 01:45:28,320 --> 01:45:30,300 So let me point out a few things. 1589 01:45:30,300 --> 01:45:33,980 So this structure is precisely just this hab. 1590 01:45:33,980 --> 01:45:35,700 So that's if you look at this structure, 1591 01:45:35,700 --> 01:45:41,010 so this structure is precisely what I called hab. 1592 01:45:41,010 --> 01:45:44,690 So now the claim is adding [INAUDIBLE] 1593 01:45:44,690 --> 01:45:48,770 to original variable with just X. Now 1594 01:45:48,770 --> 01:45:52,520 I introduce a new variable, gamma. 1595 01:45:52,520 --> 01:45:54,480 And gamma is like a Lagrangian multiplier, 1596 01:45:54,480 --> 01:45:57,730 because there's no connected term for gamma. 1597 01:45:57,730 --> 01:46:02,860 So if I eliminate gamma, then I will recover this. 1598 01:46:02,860 --> 01:46:04,430 OK, so this is the claim. 1599 01:46:04,430 --> 01:46:05,470 So now let me show that. 1600 01:46:08,650 --> 01:46:12,980 This is very easy to see. 1601 01:46:12,980 --> 01:46:19,410 Because if you just do the variation of gamma, 1602 01:46:19,410 --> 01:46:22,400 do the variation of gamma ab. 1603 01:46:22,400 --> 01:46:23,060 OK. 1604 01:46:23,060 --> 01:46:26,730 So whenever I wrote in this is in [? upstairs, ?] 1605 01:46:26,730 --> 01:46:27,940 it always means the inverse. 1606 01:46:27,940 --> 01:46:31,352 OK, this is the standard notation for the metric. 1607 01:46:31,352 --> 01:46:33,560 So if you look at the equation of motion, [INAUDIBLE] 1608 01:46:33,560 --> 01:46:36,930 by variation of this gamma ab, then what you'll find 1609 01:46:36,930 --> 01:46:39,929 is that the gamma ab-- just do the variation of that action. 1610 01:46:39,929 --> 01:46:41,762 You find the equation of motion for gamma ab 1611 01:46:41,762 --> 01:46:42,845 is given by the following. 1612 01:46:45,950 --> 01:46:48,350 So hab, just that guy. 1613 01:46:48,350 --> 01:46:51,374 And the lambda is arbitrary constant, or lambda 1614 01:46:51,374 --> 01:46:52,856 is arbitrary function. 1615 01:47:06,220 --> 01:47:09,480 So this I'm sure you can do. 1616 01:47:09,480 --> 01:47:11,060 You just do the variation. 1617 01:47:11,060 --> 01:47:13,060 You find that equation. 1618 01:47:13,060 --> 01:47:16,940 So now we can just verify this actually works. 1619 01:47:16,940 --> 01:47:22,360 When you substitute this into here, OK, into here. 1620 01:47:22,360 --> 01:47:25,560 So this gamma ab, when you take the inverse, then [? cause ?] 1621 01:47:25,560 --> 01:47:30,020 one into the inverse, hab, inverse hab contracted 1622 01:47:30,020 --> 01:47:32,800 with this hab just give you 2. 1623 01:47:32,800 --> 01:47:38,401 And that 2-- did I put that 2 in the right place? 1624 01:47:41,390 --> 01:47:43,122 That gave you 2. 1625 01:47:43,122 --> 01:47:49,350 And that have a 2 on-- yeah, I'm confused about 2 now. 1626 01:47:49,350 --> 01:47:53,020 Oh, no, no, it's fine. 1627 01:47:53,020 --> 01:47:56,310 Anyway, so this contracted with that, 1628 01:47:56,310 --> 01:48:03,850 so gamma ab contracted with hab give you 2 divided by lambda, 1629 01:48:03,850 --> 01:48:06,490 times 2. 1630 01:48:06,490 --> 01:48:07,720 OK? 1631 01:48:07,720 --> 01:48:11,690 Because you just invert this guy and invert the lambda and 2. 1632 01:48:11,690 --> 01:48:17,190 And then square root of minus gamma give me 1/2 1633 01:48:17,190 --> 01:48:21,460 lambda, square root minus h. 1634 01:48:21,460 --> 01:48:21,960 OK? 1635 01:48:24,650 --> 01:48:28,230 So sometimes I also approximate. 1636 01:48:28,230 --> 01:48:30,275 I will not write this determinant explicitly. 1637 01:48:30,275 --> 01:48:32,650 When I write [? less h, ?] it means the determinant of h. 1638 01:48:32,650 --> 01:48:34,970 And the minus gamma, determinant of gamma. 1639 01:48:34,970 --> 01:48:35,770 OK? 1640 01:48:35,770 --> 01:48:39,350 So you multiply these two together, so these two cancel. 1641 01:48:39,350 --> 01:48:43,240 And this two, multiply this 4 pi alpha prime, 1642 01:48:43,240 --> 01:48:46,194 and then get back that, OK? 1643 01:48:46,194 --> 01:48:47,110 So they're equivalent. 1644 01:48:49,980 --> 01:48:51,930 Clear? 1645 01:48:51,930 --> 01:48:54,264 So this gives you [? SNG. ?] 1646 01:49:00,660 --> 01:49:10,350 So now the key-- so now if you look at this form, 1647 01:49:10,350 --> 01:49:17,410 this really have a polynomial form for X, OK? 1648 01:49:17,410 --> 01:49:21,940 So now let me call this equation 2. 1649 01:49:21,940 --> 01:49:26,280 So equation 2, if you look at that expression, 1650 01:49:26,280 --> 01:49:28,880 just has the form-- so this is just 1651 01:49:28,880 --> 01:49:33,560 like a two-dimensional field theory-- has 1652 01:49:33,560 --> 01:49:38,320 the form of a two-dimensional scalar field 1653 01:49:38,320 --> 01:49:50,917 theory in the curved spacetime. 1654 01:49:57,740 --> 01:50:03,030 Of course, the curved spacetime is just our worldsheet sigma 1655 01:50:03,030 --> 01:50:10,298 with metric gamma ab, OK? 1656 01:50:19,200 --> 01:50:26,400 So this is just like-- but the key here, 1657 01:50:26,400 --> 01:50:29,225 so sometimes 2 is called the nonlinear sigma model, just 1658 01:50:29,225 --> 01:50:32,480 traditionally, a theory of the form 1659 01:50:32,480 --> 01:50:35,120 that equation 2 is called the nonlinear sigma model. 1660 01:50:35,120 --> 01:50:38,640 Nonlinear because typically this metric can depend on X, 1661 01:50:38,640 --> 01:50:40,550 and so dependence on X is nonlinear. 1662 01:50:40,550 --> 01:50:43,802 So it's called nonlinear sigma model. 1663 01:50:43,802 --> 01:50:51,328 But I would say it's both gamma ab and X are dynamical. 1664 01:50:59,200 --> 01:51:00,570 Are dynamical variables. 1665 01:51:08,310 --> 01:51:12,900 So that means when you do the path integral, 1666 01:51:12,900 --> 01:51:19,926 so in the path integral quantization, 1667 01:51:19,926 --> 01:51:21,800 you need to integrate over all possible gamma 1668 01:51:21,800 --> 01:51:24,010 ab and all possible X mu. 1669 01:51:24,010 --> 01:51:26,820 Not only integrate over all possible X mu, 1670 01:51:26,820 --> 01:51:31,270 but also integrate all possible gamma ab with this action. 1671 01:51:39,770 --> 01:51:41,860 OK. 1672 01:51:41,860 --> 01:51:46,060 So this is a two-dimensional [? world ?] 1673 01:51:46,060 --> 01:51:47,810 with some scalar field. 1674 01:51:47,810 --> 01:51:52,060 And you integrate over all possible metric, 1675 01:51:52,060 --> 01:51:56,100 so over all possible intrinsic metric in that [? world. ?] 1676 01:51:56,100 --> 01:52:07,810 So this can also be considered as 2D gravity, 1677 01:52:07,810 --> 01:52:16,538 two-dimensional gravity, coupled to D scalar fields. 1678 01:52:25,570 --> 01:52:29,400 So now we see that when you rewrite anything 1679 01:52:29,400 --> 01:52:36,120 in this polynomial form, in this Polyakov form, 1680 01:52:36,120 --> 01:52:39,730 the problem of quantizing the string 1681 01:52:39,730 --> 01:52:44,010 become the problem of quantizing two-dimensional gravity coupled 1682 01:52:44,010 --> 01:52:47,050 to D scalar fields. 1683 01:52:47,050 --> 01:52:48,020 OK. 1684 01:52:48,020 --> 01:52:52,470 So this may look very scary, but it turns out actually 1685 01:52:52,470 --> 01:52:55,150 two-dimensional gravity is very simple. 1686 01:52:55,150 --> 01:52:56,490 So it's actually not scary. 1687 01:52:56,490 --> 01:52:58,700 So in the end, for many situations, 1688 01:52:58,700 --> 01:53:01,637 this just reduced to, say, a quantizing scalar 1689 01:53:01,637 --> 01:53:03,220 field with a little bit of subtleties. 1690 01:53:06,860 --> 01:53:08,782 So yeah, let's stop here.