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 --> 00:00:06,050 Your support will help MIT OpenCourseWare 4 00:00:06,050 --> 00:00:10,150 continue to offer high-quality educational resources for free. 5 00:00:10,150 --> 00:00:12,700 To make a donation or to view additional materials 6 00:00:12,700 --> 00:00:16,600 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,600 --> 00:00:17,260 at ocw.mit.edu. 8 00:00:21,200 --> 00:00:23,490 HONG LIU: So let me first just remind you briefly 9 00:00:23,490 --> 00:00:25,444 what we did in last lecture-- 10 00:00:25,444 --> 00:00:26,360 AUDIENCE: [INAUDIBLE]. 11 00:00:26,360 --> 00:00:28,390 HONG LIU: --which was ages ago. 12 00:00:28,390 --> 00:00:30,430 [LAUGHTER] 13 00:00:30,430 --> 00:00:40,570 So we, say, quote unquote, derived the AdS duality. 14 00:00:40,570 --> 00:00:42,600 And then we also described a little bit 15 00:00:42,600 --> 00:00:46,330 to the geometry of the Anti-de Sitter spacetime. 16 00:00:46,330 --> 00:00:48,250 So there are two things I want you to remember 17 00:00:48,250 --> 00:00:50,280 about Anti-de Sitter spacetime. 18 00:00:50,280 --> 00:00:52,560 You said, there's two ways to think about it. 19 00:00:52,560 --> 00:00:55,050 One is so-called the Poincare patch, 20 00:00:55,050 --> 00:00:59,680 is that, say, we have a coordinate z or R. It depends 21 00:00:59,680 --> 00:01:01,080 on your choice. 22 00:01:01,080 --> 00:01:04,959 So for each constant-- z is equal to 0. 23 00:01:04,959 --> 00:01:08,520 Add to the boundary of AdS. 24 00:01:08,520 --> 00:01:27,120 So AdS have a boundary, symmetric, yeah, so R squared. 25 00:01:27,120 --> 00:01:30,050 So z goes from 0 to infinity. 26 00:01:30,050 --> 00:01:35,550 And at z equal to 0, the overall [? prefect ?] blows up. 27 00:01:35,550 --> 00:01:37,770 So the spacetime become bigger and bigger. 28 00:01:37,770 --> 00:01:40,970 So this is what we will normally call the AdS boundary. 29 00:01:40,970 --> 00:01:44,140 And then when z become bigger, then you go to the interior. 30 00:01:44,140 --> 00:01:49,300 And then at each consonant z the spacetime is [INAUDIBLE] 31 00:01:49,300 --> 00:01:53,090 Minkowski space, say, our D dimension in Minkowski space. 32 00:01:53,090 --> 00:01:55,340 So this give you an AdS d plus 1. 33 00:01:58,280 --> 00:02:00,890 OK, so this gave you an AdS d plus 1. 34 00:02:00,890 --> 00:02:04,770 And so of each z [INAUDIBLE], it's an AdS space. 35 00:02:04,770 --> 00:02:08,130 So this is so-called the Poincare patch. 36 00:02:08,130 --> 00:02:09,850 And another way to think about AdS 37 00:02:09,850 --> 00:02:12,343 is so-called the global AdS. 38 00:02:12,343 --> 00:02:16,130 In the global AdS, AdS is just essentially 39 00:02:16,130 --> 00:02:18,260 a [? sort ?] in the cylinder. 40 00:02:18,260 --> 00:02:22,000 And it is a time direction, which we call tau. 41 00:02:22,000 --> 00:02:25,450 And then there's a radial direction we call rho. 42 00:02:25,450 --> 00:02:27,370 And then there's some angular direction. 43 00:02:27,370 --> 00:02:31,160 And then the boundary of the surface-- for example, 44 00:02:31,160 --> 00:02:35,860 you can write in this global case as 1 over, 45 00:02:35,860 --> 00:02:41,506 say R squared cosine squared rho, say, minus dt squared 46 00:02:41,506 --> 00:02:49,230 plus d rho squared plus sine squared rho [? d ?] omega 47 00:02:49,230 --> 00:02:52,350 squared d minus 1 squared. 48 00:02:52,350 --> 00:02:54,270 OK? 49 00:02:54,270 --> 00:02:56,310 And this rho, and the angular direction 50 00:02:56,310 --> 00:03:01,530 is this omega d minus 1, so d tau squared. 51 00:03:01,530 --> 00:03:02,390 OK? 52 00:03:02,390 --> 00:03:05,940 And then when [? rho ?] goes to pi over 2, then the, again, 53 00:03:05,940 --> 00:03:08,410 the overall [? prefactor ?] blows up. 54 00:03:08,410 --> 00:03:12,240 So at the rho power over 2, so this is the boundary. 55 00:03:12,240 --> 00:03:18,960 So the boundary here is S3 times R. So SD minus 1 times 56 00:03:18,960 --> 00:03:20,290 the time. 57 00:03:20,290 --> 00:03:22,160 So it's just really a cylinder. 58 00:03:22,160 --> 00:03:26,730 So here, the boundary is the R1 d minus 1, 59 00:03:26,730 --> 00:03:31,010 just the Minkowski spacetime, OK? 60 00:03:31,010 --> 00:03:34,160 So this is two way I want you to think about the Anti-de Sitter 61 00:03:34,160 --> 00:03:35,990 spacetime. 62 00:03:35,990 --> 00:03:45,980 And also, yeah-- so before we continue, 63 00:03:45,980 --> 00:03:47,064 do you have any questions? 64 00:03:47,064 --> 00:03:47,563 Yes? 65 00:03:47,563 --> 00:03:48,717 AUDIENCE: Just a short one. 66 00:03:48,717 --> 00:03:51,436 So you draw a cylinder, But the spatial part of that metric 67 00:03:51,436 --> 00:03:52,882 is not precisely a cylinder. 68 00:03:52,882 --> 00:03:53,485 HONG LIU: Hm? 69 00:03:53,485 --> 00:03:55,485 AUDIENCE: I mean, the spatial part of the metric 70 00:03:55,485 --> 00:03:56,480 is not precisely [INAUDIBLE]. 71 00:03:56,480 --> 00:03:57,813 HONG LIU: You mean the boundary? 72 00:03:57,813 --> 00:04:00,421 AUDIENCE: No, I mean the sine squared rho term. 73 00:04:00,421 --> 00:04:01,170 That's a cylinder. 74 00:04:01,170 --> 00:04:02,961 HONG LIU: Oh, I'm just saying the topology. 75 00:04:02,961 --> 00:04:05,620 I'm saying the topology is [? a sorted ?] cylinder. 76 00:04:05,620 --> 00:04:09,845 Yeah, the topology is [? a sorted ?] cylinder, yeah. 77 00:04:12,322 --> 00:04:13,155 Any other questions? 78 00:04:16,660 --> 00:04:17,350 OK, good. 79 00:04:19,880 --> 00:04:26,790 So also, let me remind you this original definition 80 00:04:26,790 --> 00:04:28,590 of the Anti-de Sitter spacetime. 81 00:04:28,590 --> 00:04:37,550 So Anti-de Sitter spacetime can also 82 00:04:37,550 --> 00:04:39,940 be defined as a hyperboloid. 83 00:04:50,800 --> 00:04:55,210 Hyperboloid in the d plus 2 dimensional Minkowski 84 00:04:55,210 --> 00:04:55,710 spacetime. 85 00:05:04,790 --> 00:05:10,780 OK, so this is the AdS d plus 1 can also 86 00:05:10,780 --> 00:05:15,360 be defined as hyperboloids in the d plus 87 00:05:15,360 --> 00:05:18,100 2 dimensional Minkowski spacetime. 88 00:05:18,100 --> 00:05:19,960 It's 2 time. 89 00:05:19,960 --> 00:05:23,400 So it's a 2,d signature with 2 time. 90 00:05:23,400 --> 00:05:23,900 OK? 91 00:05:27,340 --> 00:05:28,975 So now, let's talk a little bit more 92 00:05:28,975 --> 00:05:32,440 about the-- the last time we talked about geometry of AdS. 93 00:05:32,440 --> 00:05:37,740 Now, we talk about little bit the symmetries of AdS. 94 00:05:37,740 --> 00:05:58,530 So let's just further that discussion, so d 95 00:05:58,530 --> 00:06:01,470 plus 1 dimension of AdS, Anti-de Sitter spacetime. 96 00:06:07,580 --> 00:06:10,060 So I assume all of you are familiar of is 97 00:06:10,060 --> 00:06:13,380 the concept called isometry. 98 00:06:13,380 --> 00:06:15,575 If not, it's very easy to understand. 99 00:06:15,575 --> 00:06:19,970 So isometry refers to the class of coordinate 100 00:06:19,970 --> 00:06:23,490 transformations, which leaves the metric invariant. 101 00:06:23,490 --> 00:06:24,840 OK? 102 00:06:24,840 --> 00:06:29,230 And it's, say, if it's a Minkowski spacetime, then 103 00:06:29,230 --> 00:06:32,050 the isometry will the Lorentz transformation 104 00:06:32,050 --> 00:06:33,859 plus the translations, OK? 105 00:06:33,859 --> 00:06:34,900 So that's the isometries. 106 00:06:37,570 --> 00:06:41,860 And so for this AdS, by definition, 107 00:06:41,860 --> 00:06:47,620 the isometry would be SO(d,2), because this is hyperbole 108 00:06:47,620 --> 00:06:52,560 for AdS, because the AdS is defined as a hyperboloid 109 00:06:52,560 --> 00:06:57,640 in this d plus 2 dimensional Minkowski spacetime with 110 00:06:57,640 --> 00:07:00,420 signature (2,d). 111 00:07:00,420 --> 00:07:04,810 And this hyperboloid preserves the Lorentz transformation. 112 00:07:08,220 --> 00:07:10,970 So the AdS, the symmetry will be SO(d,2). 113 00:07:10,970 --> 00:07:14,350 Just the Lorentz symmetry of this embedded Minkowski 114 00:07:14,350 --> 00:07:15,000 spacetime. 115 00:07:15,000 --> 00:07:17,910 There's no translation because this equation 116 00:07:17,910 --> 00:07:19,420 breaks the translation. 117 00:07:19,420 --> 00:07:20,695 OK? 118 00:07:20,695 --> 00:07:22,570 Because this equation breaks the translation. 119 00:07:26,230 --> 00:07:30,920 So yeah, so you can see immediately just from there. 120 00:07:30,920 --> 00:07:32,820 But what we will be useful later is 121 00:07:32,820 --> 00:07:36,490 to understand how this is refracted 122 00:07:36,490 --> 00:07:38,510 in the so-called Poincare patch. 123 00:07:38,510 --> 00:07:40,020 OK? 124 00:07:40,020 --> 00:07:42,590 So let's talk about how this symmetry is realized 125 00:07:42,590 --> 00:07:50,150 in the Poincare patch, which I will 126 00:07:50,150 --> 00:07:55,660 use these z coordinates, which I will use these z coordinates. 127 00:07:55,660 --> 00:08:01,520 So first, they are translations, which we can translate, say, 128 00:08:01,520 --> 00:08:03,405 x mu to some constant. 129 00:08:06,220 --> 00:08:09,960 So by x mu, I always refer to using 130 00:08:09,960 --> 00:08:11,180 these Poincare coordinates. 131 00:08:11,180 --> 00:08:15,120 The x mu always refer to t and the vector x. 132 00:08:15,120 --> 00:08:15,800 OK? 133 00:08:15,800 --> 00:08:19,370 So this is the-- so first, obvious, 134 00:08:19,370 --> 00:08:22,800 there's a translation in the t and x direction, 135 00:08:22,800 --> 00:08:26,710 because that's independent here [? in the ?] x. 136 00:08:26,710 --> 00:08:31,550 And then this also Lorentz transformation 137 00:08:31,550 --> 00:08:42,049 in the x direction, because of the dependence on the t and x 138 00:08:42,049 --> 00:08:45,670 is just the Minkowski metric, because each is [INAUDIBLE] 139 00:08:45,670 --> 00:08:47,060 that Minkowski metric. 140 00:08:47,060 --> 00:08:49,940 OK? 141 00:08:49,940 --> 00:08:52,530 So this has d generators. 142 00:08:52,530 --> 00:08:56,561 So this have 1/2 d times d minus 1, 143 00:08:56,561 --> 00:08:59,400 because this is the Lorentz transformation 144 00:08:59,400 --> 00:09:02,360 in the d dimension. 145 00:09:02,360 --> 00:09:04,800 And then, also, there's scaling. 146 00:09:12,600 --> 00:09:19,780 So that metric, it's invariant under such a scaling. 147 00:09:19,780 --> 00:09:23,190 So just scale x mu and z together. 148 00:09:23,190 --> 00:09:25,590 And clearly, the metric does not change. 149 00:09:25,590 --> 00:09:28,380 So if you scale z and x together, 150 00:09:28,380 --> 00:09:30,390 then all the scaling cancels. 151 00:09:30,390 --> 00:09:32,920 No, of course, this does not change the metric. 152 00:09:32,920 --> 00:09:35,249 OK? 153 00:09:35,249 --> 00:09:37,415 So there's also scaling, so this have one generator. 154 00:09:41,150 --> 00:09:47,484 So the last isometry in the-- it's 155 00:09:47,484 --> 00:09:49,150 called special conformal transformation. 156 00:09:53,170 --> 00:09:57,690 The last image of this space, which 157 00:09:57,690 --> 00:10:00,192 is called the special conformal transformation, 158 00:10:00,192 --> 00:10:01,650 it's a little bit more complicated. 159 00:10:01,650 --> 00:10:03,530 Let me just first write it down. 160 00:10:03,530 --> 00:10:06,520 Then I will try to give you some intuition about it. 161 00:10:06,520 --> 00:10:09,050 So the [? free ?] parameter, so the parameter for this 162 00:10:09,050 --> 00:10:12,640 transformation of [? actor ?] b mu. 163 00:10:12,640 --> 00:10:13,480 OK? 164 00:10:13,480 --> 00:10:14,990 So I will introduce. 165 00:10:14,990 --> 00:10:17,020 So the transformation is the following. 166 00:10:17,020 --> 00:10:33,460 z go to z prime equal to-- and the x mu goes to x mu prime. 167 00:10:45,710 --> 00:10:47,310 OK? 168 00:10:47,310 --> 00:10:51,600 So b mu is just an arbitrary constant vector, just constant 169 00:10:51,600 --> 00:10:52,100 vector. 170 00:10:54,670 --> 00:10:58,360 And so b mu squared, so b squared is just 171 00:10:58,360 --> 00:11:01,760 your standard b mu b mu. 172 00:11:01,760 --> 00:11:06,325 Under the A square, it's defined to be z square plus x mu x mu. 173 00:11:09,040 --> 00:11:11,490 OK? 174 00:11:11,490 --> 00:11:13,410 So you can check yourself, requires 175 00:11:13,410 --> 00:11:15,390 a little bit of effort. 176 00:11:15,390 --> 00:11:18,790 You plug this transformation into this expression. 177 00:11:18,790 --> 00:11:21,600 And then you can show that this is invariant. 178 00:11:21,600 --> 00:11:22,520 OK? 179 00:11:22,520 --> 00:11:27,005 So this have d parameters, because b. 180 00:11:27,005 --> 00:11:28,380 So the transformation parameters, 181 00:11:28,380 --> 00:11:29,960 they are the d parameters. 182 00:11:29,960 --> 00:11:33,360 So they are d parameters here. 183 00:11:33,360 --> 00:11:33,860 OK? 184 00:11:33,860 --> 00:11:34,693 Because of the b mu. 185 00:11:40,050 --> 00:11:41,840 So this transformation can be understood 186 00:11:41,840 --> 00:11:46,550 in a slightly easier way, as follows. 187 00:11:49,270 --> 00:11:52,310 It's as you will also do this in your p 188 00:11:52,310 --> 00:11:54,710 set, is that you can check. 189 00:11:57,490 --> 00:12:06,580 So to understand this Special Conformal Transformation, 190 00:12:06,580 --> 00:12:07,320 so I called SCT. 191 00:12:11,880 --> 00:12:16,380 First, you can check the following is isometry, 192 00:12:16,380 --> 00:12:19,040 something we call inversion. 193 00:12:19,040 --> 00:12:25,920 It said if z goes to z divided by A, so A is this quantity. 194 00:12:25,920 --> 00:12:31,460 Oh, A, I think I'm having some notation error, myself. 195 00:12:34,650 --> 00:12:37,700 So I should call it A, rather, A squared. 196 00:12:37,700 --> 00:12:38,520 Sorry. 197 00:12:38,520 --> 00:12:38,900 It doesn't matter. 198 00:12:38,900 --> 00:12:40,775 I can also call it A squared, but let me just 199 00:12:40,775 --> 00:12:50,760 call it A. I think that one would be [? A2. ?] Yeah, 200 00:12:50,760 --> 00:12:52,677 that is [? A2. ?] So, yeah, I could just 201 00:12:52,677 --> 00:12:53,760 call everything A squared. 202 00:12:53,760 --> 00:12:56,840 But to be consistent with my notation on the [? notes, ?] 203 00:12:56,840 --> 00:13:00,270 just let me call it A. 204 00:13:00,270 --> 00:13:03,330 So you can also check that the following discrete 205 00:13:03,330 --> 00:13:05,470 transformation is also isometry, so when 206 00:13:05,470 --> 00:13:09,070 z equal to z divided by A, and x mu 207 00:13:09,070 --> 00:13:12,730 goes to x mu divided by A. OK? 208 00:13:12,730 --> 00:13:14,420 You can check this is isometry. 209 00:13:14,420 --> 00:13:19,240 So if you plug this into this metric, 210 00:13:19,240 --> 00:13:21,650 you find this [? leaves ?] the metric invariant. 211 00:13:21,650 --> 00:13:24,790 And this thing is much easier to check. 212 00:13:24,790 --> 00:13:27,470 But as you will do in your p set, 213 00:13:27,470 --> 00:13:32,010 and you can check yourself, this discrete transformation 214 00:13:32,010 --> 00:13:36,590 actually is not part of the SO(d,2). 215 00:13:36,590 --> 00:13:40,170 It's, in fact, part of the O(d,2). 216 00:13:40,170 --> 00:13:43,280 So if you calculate its determinant, say the Jacobean, 217 00:13:43,280 --> 00:13:45,890 actually find the minus 1, rather than 1. 218 00:13:45,890 --> 00:13:46,610 OK? 219 00:13:46,610 --> 00:13:50,810 So this is not a proper Lorentz transformation. 220 00:13:50,810 --> 00:13:52,240 It's not a proper transformation, 221 00:13:52,240 --> 00:13:55,520 what we normally call the proper transformation. 222 00:13:55,520 --> 00:14:03,210 But however, if you do this twice, 223 00:14:03,210 --> 00:14:05,460 the minus 1 times minus 1 equal to 1, 224 00:14:05,460 --> 00:14:07,679 then you get the proper transformation. 225 00:14:07,679 --> 00:14:09,220 But, of course, if you do this twice, 226 00:14:09,220 --> 00:14:10,437 you just go back to itself. 227 00:14:10,437 --> 00:14:11,020 You can check. 228 00:14:11,020 --> 00:14:12,940 This is inverse of itself. 229 00:14:12,940 --> 00:14:15,740 So this is like a [? z2 ?] transformation. 230 00:14:15,740 --> 00:14:17,930 But you can do [? a slightly ?] [? trick ?]. 231 00:14:17,930 --> 00:14:23,100 Then you can show that this special conformal 232 00:14:23,100 --> 00:14:26,045 transformation is given by an inversion. 233 00:14:28,730 --> 00:14:33,030 You first do an inversion, and then you follow the [? bad ?] 234 00:14:33,030 --> 00:14:39,300 translation in b mu. 235 00:14:39,300 --> 00:14:41,700 And then invert it back. 236 00:14:41,700 --> 00:14:42,940 OK? 237 00:14:42,940 --> 00:14:47,804 So even though you can check that I squared is equal to 1, 238 00:14:47,804 --> 00:14:50,350 well, now in the middle, you have added the translation 239 00:14:50,350 --> 00:14:52,750 in b mu, a constant. 240 00:14:52,750 --> 00:14:54,900 And this is a symmetry. 241 00:14:54,900 --> 00:14:57,220 And, of course, the whole thing will be a symmetry. 242 00:14:57,220 --> 00:14:58,928 But now, this is a proper transformation, 243 00:14:58,928 --> 00:15:02,010 because I have act I twice, OK? 244 00:15:02,010 --> 00:15:04,630 So you can now check this transformation is precisely-- 245 00:15:04,630 --> 00:15:08,150 if you do this, it's precisely just that transformation. 246 00:15:08,150 --> 00:15:08,650 OK? 247 00:15:13,490 --> 00:15:15,880 So if you add all of them together, 248 00:15:15,880 --> 00:15:19,220 d will have d minus 1 d. 249 00:15:19,220 --> 00:15:24,030 Then you can show actually they form this symmetry group. 250 00:15:24,030 --> 00:15:26,216 They form this symmetry group. 251 00:15:26,216 --> 00:15:38,080 So altogether, you have d plus 1/2 d times 252 00:15:38,080 --> 00:15:42,450 d minus 1 plus 1 plus d. 253 00:15:42,450 --> 00:15:45,960 Then that give you 1/2 d. 254 00:15:45,960 --> 00:15:48,960 You can check yourself, like 1/2 d plus 1 and d 255 00:15:48,960 --> 00:15:53,470 plus 2, which is precisely the number generated for that one. 256 00:15:53,470 --> 00:15:55,350 OK? 257 00:15:55,350 --> 00:15:56,660 Yes? 258 00:15:56,660 --> 00:16:00,090 AUDIENCE: So the total here that dimension, the d plus 1-- 259 00:16:00,090 --> 00:16:01,334 HONG LIU: Yeah. 260 00:16:01,334 --> 00:16:02,930 AUDIENCE: Why like even translation, 261 00:16:02,930 --> 00:16:06,900 it only has a d dimensional invariant. 262 00:16:06,900 --> 00:16:09,950 HONG LIU: No, because this depend on z, so the translation 263 00:16:09,950 --> 00:16:11,700 [? in ?] z is not invariant. 264 00:16:11,700 --> 00:16:13,083 AUDIENCE: Oh, OK. 265 00:16:13,083 --> 00:16:14,470 Yeah, so that would be-- 266 00:16:14,470 --> 00:16:16,053 HONG LOU: Because this depend on z. 267 00:16:16,053 --> 00:16:17,460 AUDIENCE: Yeah. 268 00:16:17,460 --> 00:16:21,219 HONG LIU: So if you translate z, this is not the isometry 269 00:16:21,219 --> 00:16:22,510 [? under ?] the metric changes. 270 00:16:22,510 --> 00:16:27,020 So only the translation t and x is a symmetry. 271 00:16:27,020 --> 00:16:29,867 So that's why we only do the translation x mu. 272 00:16:29,867 --> 00:16:31,200 AUDIENCE: I have some questions. 273 00:16:31,200 --> 00:16:31,783 HONG LIU: Yes? 274 00:16:31,783 --> 00:16:34,185 AUDIENCE: So conversion is isometry. 275 00:16:34,185 --> 00:16:34,810 HONG LIU: Yeah. 276 00:16:34,810 --> 00:16:35,393 AUDIENCE: Yes? 277 00:16:35,393 --> 00:16:38,630 OK, so but inversion, there's a [? release ?] not 278 00:16:38,630 --> 00:16:39,820 in the SO(d,2)? 279 00:16:39,820 --> 00:16:40,750 HONG LIU: Yeah, right. 280 00:16:40,750 --> 00:16:42,680 AUDIENCE: So that means the symmetry, 281 00:16:42,680 --> 00:16:45,016 isometry [? work ?] [? with ?] [INAUDIBLE] [? SO? ?] 282 00:16:45,016 --> 00:16:45,890 HONG LIU: Yeah, yeah. 283 00:16:45,890 --> 00:16:47,030 It's the standard story. 284 00:16:47,030 --> 00:16:51,460 You have O. You can have an O symmetry. 285 00:16:51,460 --> 00:16:54,820 There's a discrete part, which is O/ When 286 00:16:54,820 --> 00:16:57,950 you're transformation, you're changing the Jacobean, 287 00:16:57,950 --> 00:16:59,730 whether your Jacobean is 1 or minus 1. 288 00:17:02,570 --> 00:17:05,240 You just actually [? exactly ?] the standards 289 00:17:05,240 --> 00:17:07,854 of the transformation in the Minkowski spacetime. 290 00:17:07,854 --> 00:17:09,520 In the Minkowski spacetime, you can also 291 00:17:09,520 --> 00:17:10,269 can see the [? site ?] transmissions. 292 00:17:10,269 --> 00:17:11,492 AUDIENCE: OK. 293 00:17:11,492 --> 00:17:13,119 HONG LIU: Yeah, just the same thing, 294 00:17:13,119 --> 00:17:15,970 because this is just the transformation in the Minkowski 295 00:17:15,970 --> 00:17:17,460 spacetime of this embedded space. 296 00:17:17,460 --> 00:17:20,349 AUDIENCE: So in the [INAUDIBLE] metric, 297 00:17:20,349 --> 00:17:23,920 we also have a, like a-- determine [INAUDIBLE] 298 00:17:23,920 --> 00:17:25,119 transformation. 299 00:17:25,119 --> 00:17:28,059 HONG LOU: Exactly the same as the standard Minkowski space. 300 00:17:28,059 --> 00:17:28,959 AUDIENCE: OK. 301 00:17:28,959 --> 00:17:32,450 AUDIENCE: But discrete part just like a [INAUDIBLE]? 302 00:17:32,450 --> 00:17:36,960 HONG LIU: Oh, no, it depends. 303 00:17:36,960 --> 00:17:42,500 Yeah, in the old dimension, yeah, in the 3 plus 1 304 00:17:42,500 --> 00:17:44,560 dimension, just the [? parity. ?] That's right. 305 00:17:44,560 --> 00:17:47,550 In the 2 plus 1 dimension, if we invert all directions, 306 00:17:47,550 --> 00:17:48,310 then it's not. 307 00:17:48,310 --> 00:17:51,198 Right? 308 00:17:51,198 --> 00:17:51,698 Yes? 309 00:17:51,698 --> 00:17:55,086 AUDIENCE: Is there an easy way to see 310 00:17:55,086 --> 00:17:58,901 the special conformal symmetry of the system? 311 00:17:58,901 --> 00:17:59,442 HONG LIU: Hm? 312 00:17:59,442 --> 00:18:01,378 AUDIENCE: Is there an easy way to see 313 00:18:01,378 --> 00:18:03,317 a symmetry from the metric? 314 00:18:03,317 --> 00:18:04,900 HONG LIU: Yeah, you just plug this in. 315 00:18:04,900 --> 00:18:05,400 [LAUGHTER] 316 00:18:05,400 --> 00:18:07,702 AUDIENCE: Well, I mean, how do you 317 00:18:07,702 --> 00:18:10,280 know that there is such symmetry, [? complicated ?] 318 00:18:10,280 --> 00:18:11,360 symmetries? 319 00:18:11,360 --> 00:18:13,380 HONG LIU: No, just say another way to understand 320 00:18:13,380 --> 00:18:15,910 is to do this one. 321 00:18:15,910 --> 00:18:19,210 So this inversion is a much simpler transformation, 322 00:18:19,210 --> 00:18:22,380 which you can easily check [? its ?] isometry. 323 00:18:22,380 --> 00:18:26,840 And then this procedure will guarantee this is isometry. 324 00:18:29,790 --> 00:18:31,320 [? because ?] each step is isometry. 325 00:18:36,550 --> 00:18:39,430 Good? 326 00:18:39,430 --> 00:18:42,030 OK, so after talking about the-- so 327 00:18:42,030 --> 00:18:45,990 that concludes our very quick review about the geometry 328 00:18:45,990 --> 00:18:49,005 and properties of Anti-de Sitter spacetime. 329 00:18:49,005 --> 00:18:51,187 AUDIENCE: But maybe one more quick question. 330 00:18:51,187 --> 00:18:54,700 So since you said the isometry [? bigger, ?] and so if you why 331 00:18:54,700 --> 00:18:56,140 you only answer on the [? SO? ?] 332 00:18:56,140 --> 00:18:57,390 HONG LIU: It's the same thing. 333 00:18:57,390 --> 00:18:59,780 In Minkowski space, we can also separate 334 00:18:59,780 --> 00:19:02,409 the discrete transformations and consider the continuous part. 335 00:19:02,409 --> 00:19:02,950 AUDIENCE: Oh. 336 00:19:02,950 --> 00:19:04,310 HONG LIU: Yeah, so it's just exactly the same thing. 337 00:19:04,310 --> 00:19:04,851 AUDIENCE: OK. 338 00:19:04,851 --> 00:19:05,957 HONG LIU: Yeah. 339 00:19:05,957 --> 00:19:08,040 Certainly here, we can consider the inversion too. 340 00:19:08,040 --> 00:19:09,950 We just here, I'm considering the part, which 341 00:19:09,950 --> 00:19:11,429 connect to the [? identity ?]. 342 00:19:11,429 --> 00:19:11,970 AUDIENCE: OK. 343 00:19:16,252 --> 00:19:17,460 HONG LIU: Any other question? 344 00:19:17,460 --> 00:19:19,745 AUDIENCE: So is it true that in some dimensions 345 00:19:19,745 --> 00:19:22,165 the inversions [INAUDIBLE] isometry? 346 00:19:22,165 --> 00:19:24,540 HONG LIU: I think-- no, I think actually more dimensions. 347 00:19:24,540 --> 00:19:25,880 This is not. 348 00:19:25,880 --> 00:19:27,680 In more dimension, this is [INAUDIBLE]. 349 00:19:27,680 --> 00:19:28,180 Yeah, it-- 350 00:19:28,180 --> 00:19:30,610 AUDIENCE: [INAUDIBLE]. 351 00:19:30,610 --> 00:19:34,465 HONG LIU: Inversion is always isometry in any dimension. 352 00:19:34,465 --> 00:19:36,840 AUDIENCE: But it's just that the determinant [INAUDIBLE]. 353 00:19:36,840 --> 00:19:38,881 HONG LIU: No, it's always minus 1, any dimension. 354 00:19:38,881 --> 00:19:39,845 AUDIENCE: Oh, OK. 355 00:19:39,845 --> 00:19:40,470 HONG LIU: Yeah. 356 00:19:46,224 --> 00:19:46,890 Other questions? 357 00:19:49,920 --> 00:19:50,930 Good. 358 00:19:50,930 --> 00:19:54,760 So after talk about Anti-de Sitter spacetime, now 359 00:19:54,760 --> 00:20:00,524 we can talk a little bit about the string theory 360 00:20:00,524 --> 00:20:01,690 in Anti-de Sitter spacetime. 361 00:20:06,970 --> 00:20:07,470 OK? 362 00:20:07,470 --> 00:20:08,804 We can talk about string theory. 363 00:20:08,804 --> 00:20:11,303 Now, let's talk a little bit more about a string [INAUDIBLE] 364 00:20:11,303 --> 00:20:12,680 the distance [INAUDIBLE]. 365 00:20:12,680 --> 00:20:17,370 This discussion is very easy and essentially trivial 366 00:20:17,370 --> 00:20:20,154 because we know very little about this string theory 367 00:20:20,154 --> 00:20:21,320 in Anti-de Sitter spacetime. 368 00:20:21,320 --> 00:20:22,010 [LAUGHTER] 369 00:20:22,010 --> 00:20:25,500 And so there's not much to talk about. 370 00:20:25,500 --> 00:20:29,460 But there are a few general statements we can say. 371 00:20:29,460 --> 00:20:33,250 So there are a few general statements we can say. 372 00:20:33,250 --> 00:20:38,050 First is that both AdS 5-- so we have 373 00:20:38,050 --> 00:20:43,880 said that AdS 5 is the maximally symmetric space 374 00:20:43,880 --> 00:20:47,370 of negative curvature. 375 00:20:47,370 --> 00:20:51,700 And we all know from your really kindergarten years, 376 00:20:51,700 --> 00:20:58,850 that S5 is the maximal symmetric space of positive curvature. 377 00:20:58,850 --> 00:21:00,440 So this combined together is really 378 00:21:00,440 --> 00:21:05,030 a maximally symmetric space OK? 379 00:21:05,030 --> 00:21:14,390 So this AdS 5 times S5, so AdS 5 times 380 00:21:14,390 --> 00:21:29,025 S5 is a homogeneous space, homogeneous spacetime, 381 00:21:29,025 --> 00:21:31,650 maximally symmetric homogeneous spacetime. 382 00:21:35,870 --> 00:21:40,080 So whatever string theory in this space, 383 00:21:40,080 --> 00:21:46,440 it's-- so the only scale in this space is the curvature, 384 00:21:46,440 --> 00:21:50,261 curvature radius, and which is the same everywhere. 385 00:21:50,261 --> 00:21:50,760 OK? 386 00:21:50,760 --> 00:21:51,468 It's homogeneous. 387 00:21:51,468 --> 00:21:52,530 It's the same everywhere. 388 00:21:52,530 --> 00:21:59,520 So essentially, R, so this R will be just a single parameter 389 00:21:59,520 --> 00:22:02,110 to characterize the curvature of this spacetime, so 390 00:22:02,110 --> 00:22:04,739 single number, which characterize 391 00:22:04,739 --> 00:22:06,030 the curvature of the spacetime. 392 00:22:06,030 --> 00:22:07,982 So even the space is not homogeneous, 393 00:22:07,982 --> 00:22:10,190 and you have to specify which place, which curvature, 394 00:22:10,190 --> 00:22:10,810 et cetera. 395 00:22:10,810 --> 00:22:13,320 But this is the maximally symmetric space, homogeneous. 396 00:22:13,320 --> 00:22:19,130 So a single number captures the curvature everywhere. 397 00:22:19,130 --> 00:22:26,310 So that means that in string theory on this spacetime, 398 00:22:26,310 --> 00:22:36,530 it's really just characterized by the following dimensionless 399 00:22:36,530 --> 00:22:40,890 parameter, so alpha prime divided by R squared. 400 00:22:40,890 --> 00:22:43,330 I mean, alpha prime is a dimensional parameter. 401 00:22:43,330 --> 00:22:46,480 But in the end, it's given by the following dimensionless 402 00:22:46,480 --> 00:22:47,220 parameter. 403 00:22:47,220 --> 00:22:51,380 And only for any physical process can 404 00:22:51,380 --> 00:22:53,210 only this dimensionless parameter 405 00:22:53,210 --> 00:22:55,220 goes in, not separate of them, because there's 406 00:22:55,220 --> 00:22:56,491 no other scales. 407 00:22:56,491 --> 00:22:58,990 i is the only scale, and alpha prime is the only [? other ?] 408 00:22:58,990 --> 00:22:59,780 scale. 409 00:22:59,780 --> 00:23:02,780 And so for any physics, the only dimension parameter 410 00:23:02,780 --> 00:23:04,600 can come in is this one. 411 00:23:04,600 --> 00:23:07,840 And, of course, there's another dimensionless parameter 412 00:23:07,840 --> 00:23:11,050 is the gs, is what we call the string coupling. 413 00:23:11,050 --> 00:23:13,290 So essentially, the series is specified 414 00:23:13,290 --> 00:23:15,487 by these two parameters. 415 00:23:15,487 --> 00:23:15,987 OK? 416 00:23:19,880 --> 00:23:22,160 And depend on what you want, you can also 417 00:23:22,160 --> 00:23:25,256 construct the Newton constant. 418 00:23:25,256 --> 00:23:26,755 The Newton constant can be expressed 419 00:23:26,755 --> 00:23:29,770 in terms of gs and 1/2 prime. 420 00:23:29,770 --> 00:23:31,892 So in type IIB string, the relation, 421 00:23:31,892 --> 00:23:34,100 so we are talking about type IIB string, the relation 422 00:23:34,100 --> 00:23:39,095 between the Newton constant and the, 423 00:23:39,095 --> 00:23:41,550 say, alpha prime and the gs is given by the following. 424 00:23:44,720 --> 00:23:48,580 So instead of a considering these two parameters, 425 00:23:48,580 --> 00:23:53,460 you can exchange gs by Newton constant. 426 00:23:53,460 --> 00:24:01,160 So you can alternatively consider, say, 427 00:24:01,160 --> 00:24:08,390 gn to the R to the power 8, or alpha prime to the R squared, 428 00:24:08,390 --> 00:24:13,185 or just characterized by these two dimensionless numbers. 429 00:24:13,185 --> 00:24:17,050 After your choice, sometimes this is more convenient, 430 00:24:17,050 --> 00:24:19,090 and sometimes, this is more convenient. 431 00:24:19,090 --> 00:24:20,684 So if you talk about string theory, 432 00:24:20,684 --> 00:24:23,100 work with the string theory [? parse ?] [? integral, ?] et 433 00:24:23,100 --> 00:24:25,590 cetera, then this is more convenient. 434 00:24:25,590 --> 00:24:27,420 But if you think about the gravity, 435 00:24:27,420 --> 00:24:29,650 then the Newton constant appears naturally. 436 00:24:29,650 --> 00:24:33,680 And then this become more [? natural. ?] OK? 437 00:24:36,480 --> 00:24:41,040 So the fact of the theory is controlled by these two 438 00:24:41,040 --> 00:24:42,550 dimensionless parameters. 439 00:24:42,550 --> 00:24:45,621 AUDIENCE: The R is a constant R? 440 00:24:45,621 --> 00:24:47,870 HONG LIU: Yeah, same R. It's just the curvature radius 441 00:24:47,870 --> 00:24:49,340 over the spacetime. 442 00:24:49,340 --> 00:24:53,415 The whole spacetime, it's just controlled by the single R. 443 00:24:53,415 --> 00:24:54,290 AUDIENCE: Yeah, yeah. 444 00:24:54,290 --> 00:25:01,840 HONG LIU: Yeah, yeah, also, implicit in the discussion 445 00:25:01,840 --> 00:25:04,070 here, which is in the metric where I wrote last time, 446 00:25:04,070 --> 00:25:06,380 is that the curvature radius here 447 00:25:06,380 --> 00:25:09,260 is exactly the same as the curvature radius here. 448 00:25:09,260 --> 00:25:11,060 And so the single curvature radius 449 00:25:11,060 --> 00:25:12,366 characterizes the whole thing. 450 00:25:12,366 --> 00:25:12,865 OK? 451 00:25:15,660 --> 00:25:17,520 OK, good. 452 00:25:17,520 --> 00:25:21,040 So this means that whatever quantum gravitational theory 453 00:25:21,040 --> 00:25:23,120 or string theory in this spacetime, 454 00:25:23,120 --> 00:25:27,180 you can consider q limit, based on this 2 dimensionless 455 00:25:27,180 --> 00:25:28,076 parameter. 456 00:25:28,076 --> 00:25:30,200 So first is so-called the classical gravity limits. 457 00:25:36,380 --> 00:25:41,290 So this is the limit, which the string coupling goes to 0. 458 00:25:41,290 --> 00:25:45,440 So that means-- remember, the string coupling controls 459 00:25:45,440 --> 00:25:47,870 the [? loop ?] [? corrections ?] over the strings. 460 00:25:47,870 --> 00:25:51,410 And also, essentially, it's the fluctuation over the spacetime. 461 00:25:51,410 --> 00:25:53,040 And so the string coupling is 0. 462 00:25:53,040 --> 00:25:57,000 And also, the alpha prime divided by R squared goes to 0. 463 00:25:57,000 --> 00:25:58,450 OK? 464 00:25:58,450 --> 00:26:01,010 So this alpha prime divided by R squared goes to 0. 465 00:26:01,010 --> 00:26:04,170 So this is can be considered as a point particle limit. 466 00:26:04,170 --> 00:26:06,580 Because alpha prime essentially characterize 467 00:26:06,580 --> 00:26:08,430 so the [? sides ?] of a string. 468 00:26:08,430 --> 00:26:10,760 And when the side of the string is much smaller 469 00:26:10,760 --> 00:26:13,410 than the curvature radius, and then essentially, 470 00:26:13,410 --> 00:26:14,880 it's a point particle limit. 471 00:26:14,880 --> 00:26:17,640 So this is the standard classical gravity limit. 472 00:26:17,640 --> 00:26:20,210 So in this region, we have classical gravity. 473 00:26:20,210 --> 00:26:23,310 And you cannot tell that those particles are strings. 474 00:26:23,310 --> 00:26:26,090 They're just like ordinary particles. 475 00:26:26,090 --> 00:26:27,550 OK? 476 00:26:27,550 --> 00:26:31,170 So similarly, so g string [? to ?] [? 0 ?] can also be 477 00:26:31,170 --> 00:26:33,260 considered as the limit. 478 00:26:33,260 --> 00:26:34,510 The Newton constant goes to 0. 479 00:26:34,510 --> 00:26:35,990 It's the same thing. 480 00:26:35,990 --> 00:26:37,046 OK? 481 00:26:37,046 --> 00:26:39,730 It's same thing. 482 00:26:39,730 --> 00:26:43,960 And so in this region, [? you ?] get classical gravity. 483 00:26:43,960 --> 00:26:46,040 And more specifically, in our case, 484 00:26:46,040 --> 00:26:51,770 we get to have to be super gravity, which we briefly 485 00:26:51,770 --> 00:26:52,560 mentioned before. 486 00:26:55,390 --> 00:27:02,580 So there's a lot of the region is called the classical string 487 00:27:02,580 --> 00:27:03,080 limit. 488 00:27:08,030 --> 00:27:10,359 In this case, the alpha prime divided by R 489 00:27:10,359 --> 00:27:11,400 squared can be arbitrary. 490 00:27:16,000 --> 00:27:18,420 Does not have to be small. 491 00:27:18,420 --> 00:27:20,940 So in this region, than the string [? G ?] [? fact ?] will 492 00:27:20,940 --> 00:27:21,620 be important. 493 00:27:21,620 --> 00:27:24,620 So you should be able-- then that means 494 00:27:24,620 --> 00:27:28,990 that the spacetime curvature radius is 495 00:27:28,990 --> 00:27:31,140 comparable to the string sides. 496 00:27:31,140 --> 00:27:33,410 Then in this case, you can no longer treat the string 497 00:27:33,410 --> 00:27:35,450 as a point particle. 498 00:27:35,450 --> 00:27:39,110 But we still take g string goes to 0. 499 00:27:39,110 --> 00:27:43,200 So that means that the quantum fluctuation is small. 500 00:27:43,200 --> 00:27:48,640 The spacetime quantum fluctuation is small. 501 00:27:48,640 --> 00:27:51,210 So in this case, you still have a classical theory. 502 00:27:51,210 --> 00:27:53,200 But it's a classical string theory, rather than 503 00:27:53,200 --> 00:27:55,051 a classical gravity theory. 504 00:27:55,051 --> 00:27:55,550 OK? 505 00:28:04,100 --> 00:28:05,330 Any questions regarding this? 506 00:28:10,175 --> 00:28:13,700 So there's another-- something else? 507 00:28:13,700 --> 00:28:15,766 AUDIENCE: No. 508 00:28:15,766 --> 00:28:17,890 HONG LIU: So there's one feature in this spacetime. 509 00:28:17,890 --> 00:28:22,170 So this is altogether 10 dimensional spacetime. 510 00:28:22,170 --> 00:28:23,880 There's one feature of this spacetime. 511 00:28:23,880 --> 00:28:29,140 It said S5 is a compact space with a finite volume. 512 00:28:29,140 --> 00:28:32,990 And AdS 5 is uncompact OK? 513 00:28:32,990 --> 00:28:38,315 So normally, when you have a compact space, 514 00:28:38,315 --> 00:28:41,250 so it's convenient-- so S5 is compact. 515 00:28:47,410 --> 00:28:54,630 So it's actually convenient inside your situation 516 00:28:54,630 --> 00:29:15,830 to express whatever, say, 10 dimensional field in terms 517 00:29:15,830 --> 00:29:29,262 of-- yeah, it may be to not express, to expand, 518 00:29:29,262 --> 00:29:31,510 but in 10 dimensional fields, in terms 519 00:29:31,510 --> 00:29:38,570 of harmonics, [? inverse ?] 5. 520 00:29:38,570 --> 00:29:40,690 OK? 521 00:29:40,690 --> 00:29:44,820 Because there's a discrete set of harmonics on S5, 522 00:29:44,820 --> 00:29:48,590 you can always expand some 10 dimensional field on it. 523 00:29:48,590 --> 00:29:52,840 So for example, if you have a scalar field, 524 00:29:52,840 --> 00:29:59,360 so supposing if you have a 10 dimensional scalar field, 525 00:29:59,360 --> 00:30:01,250 say we have x mu. 526 00:30:01,250 --> 00:30:02,040 We have z. 527 00:30:02,040 --> 00:30:03,920 So this is AdS5 part. 528 00:30:03,920 --> 00:30:08,170 And them let me call it omega 5, which is S5 part. 529 00:30:08,170 --> 00:30:13,510 Then you can always, say, so you can expand these scalar fields 530 00:30:13,510 --> 00:30:15,930 in terms of spherical harmonics on S5. 531 00:30:27,420 --> 00:30:30,070 OK? 532 00:30:30,070 --> 00:30:33,620 Then what you get is you get a tau field in AdS 5. 533 00:30:39,320 --> 00:30:50,420 And then this are spherical harmonics in S5. 534 00:30:50,420 --> 00:30:50,920 OK? 535 00:30:57,850 --> 00:31:05,310 And in particular, when you do this expansion, than the higher 536 00:31:05,310 --> 00:31:09,100 harmonics, they will develop a mass, because 537 00:31:09,100 --> 00:31:12,160 of the curvature of the S5. 538 00:31:12,160 --> 00:31:14,350 So the lowest mode will be independent, 539 00:31:14,350 --> 00:31:16,190 say, of the coordinate on the S5, 540 00:31:16,190 --> 00:31:18,624 where the higher mode will depend on the coordinate of S5. 541 00:31:18,624 --> 00:31:20,290 And because of those dependence, because 542 00:31:20,290 --> 00:31:23,780 of the curvature of [? base ?] 5, they will develop a mass. 543 00:31:23,780 --> 00:31:25,690 So the lowest mode would be massless, 544 00:31:25,690 --> 00:31:29,130 then they will be followed by tau of massive modes, 545 00:31:29,130 --> 00:31:32,024 controlled by the [? size ?] of AdS 5. 546 00:31:32,024 --> 00:31:32,524 OK? 547 00:31:35,100 --> 00:31:39,120 So you can do this for any field, any particular [? the ?] 548 00:31:39,120 --> 00:31:40,770 metric. 549 00:31:40,770 --> 00:31:43,570 So when you do this, essentially, 550 00:31:43,570 --> 00:31:47,880 only the massless graviton in 5 dimension will [? mediate ?] no 551 00:31:47,880 --> 00:31:54,400 [? range, ?] say, gravitation [? interactions ?] in AdS 5. 552 00:31:54,400 --> 00:32:01,630 So essentially, so the gravity, so at a long distance, 553 00:32:01,630 --> 00:32:04,770 the gravity is essentially 5 dimensional. 554 00:32:04,770 --> 00:32:05,270 OK? 555 00:32:16,549 --> 00:32:18,590 It's the same in 5 dimen-- because you can always 556 00:32:18,590 --> 00:32:20,530 reduce on S5. 557 00:32:20,530 --> 00:32:24,550 So there's a slightly tricky thing here. 558 00:32:24,550 --> 00:32:29,390 But it's only-- it's that the size of S5, 559 00:32:29,390 --> 00:32:31,270 the curvature radius of S5 is actually 560 00:32:31,270 --> 00:32:33,750 the same as the curvature radius of AdS 5. 561 00:32:33,750 --> 00:32:36,295 So there's not really a hierarchy here in terms 562 00:32:36,295 --> 00:32:37,295 of the curvature radius. 563 00:32:40,180 --> 00:32:42,270 But there's still an important difference. 564 00:32:42,270 --> 00:32:43,530 These have infinite volume. 565 00:32:43,530 --> 00:32:44,570 These only have a finite volume. 566 00:32:44,570 --> 00:32:45,445 So these are compact. 567 00:32:45,445 --> 00:32:46,510 These are uncompact. 568 00:32:46,510 --> 00:32:48,590 OK? 569 00:32:48,590 --> 00:32:51,097 AUDIENCE: But why the gravity [? doesn't-- ?] 570 00:32:51,097 --> 00:32:51,638 HONG LIU: Hm? 571 00:32:51,638 --> 00:32:53,626 AUDIENCE: But why the gravity doesn't 572 00:32:53,626 --> 00:32:56,120 say this string [? S5? ?] 573 00:32:56,120 --> 00:32:59,890 HONG LIU: No, it-- [INAUDIBLE] [? this thing ?] radius-- 574 00:32:59,890 --> 00:33:02,829 this [? thing ?] S5. 575 00:33:02,829 --> 00:33:04,620 But just as I said, this is a compact part. 576 00:33:04,620 --> 00:33:06,160 You can always do a reduction. 577 00:33:06,160 --> 00:33:09,352 And then the higher graviton mode will develop a mass. 578 00:33:09,352 --> 00:33:10,530 AUDIENCE: Oh. 579 00:33:10,530 --> 00:33:13,237 HONG LIU: Yeah, we develop a mass, yeah, 580 00:33:13,237 --> 00:33:14,820 even though that mass is not very big. 581 00:33:14,820 --> 00:33:15,746 Yes? 582 00:33:15,746 --> 00:33:21,270 AUDIENCE: So if we applied this to the real world, then 583 00:33:21,270 --> 00:33:23,410 what would estimate for R? 584 00:33:23,410 --> 00:33:24,076 HONG LIU: Sorry? 585 00:33:24,076 --> 00:33:25,950 AUDIENCE: What's the estimate of this radius, 586 00:33:25,950 --> 00:33:30,000 this curvature for if it's should be comparables 587 00:33:30,000 --> 00:33:31,390 are of the original worlds? 588 00:33:31,390 --> 00:33:32,930 HONG LIU: [INAUDIBLE] the size of the universe. 589 00:33:32,930 --> 00:33:34,400 AUDIENCE: But then doesn't it mean 590 00:33:34,400 --> 00:33:41,988 that we only should worry about the mass of this compact mode? 591 00:33:41,988 --> 00:33:42,613 HONG LIU: Yeah. 592 00:33:42,613 --> 00:33:44,706 AUDIENCE: Only on distances much longer than ours? 593 00:33:44,706 --> 00:33:45,580 HONG LIU: Yeah, yeah. 594 00:33:45,580 --> 00:33:48,220 I mean, just saying, at very large distance, 595 00:33:48,220 --> 00:33:49,500 essentially is 5 dimensions. 596 00:33:49,500 --> 00:33:52,390 But it's still-- you can talk about the distance 597 00:33:52,390 --> 00:33:53,910 scale, et cetera. 598 00:33:53,910 --> 00:33:55,900 But here, R is the only scale. 599 00:33:55,900 --> 00:33:57,740 It doesn't matter how large is R. Everything 600 00:33:57,740 --> 00:34:00,428 matches against R. R provides your units. 601 00:34:00,428 --> 00:34:02,386 AUDIENCE: Yes, but when we talk about distances 602 00:34:02,386 --> 00:34:03,774 shorter than R then-- 603 00:34:03,774 --> 00:34:05,940 HONG LIU: Yeah, if you talk about a distance shorter 604 00:34:05,940 --> 00:34:07,870 than R, it doesn't matter. 605 00:34:07,870 --> 00:34:09,400 It's like a 10 dimensional. 606 00:34:09,400 --> 00:34:11,179 It's like a 10 dimension [INAUDIBLE]. 607 00:34:11,179 --> 00:34:16,260 But AdS is uncompact, you can go as much distance as you want. 608 00:34:16,260 --> 00:34:20,100 I'm just saying, the story is slightly tricky 609 00:34:20,100 --> 00:34:22,730 because they have the same curvature radius. 610 00:34:22,730 --> 00:34:25,219 But I'm trying to describe it in the spirit. 611 00:34:25,219 --> 00:34:28,130 And also, mathematically, you can always, 612 00:34:28,130 --> 00:34:30,190 because S5 is uncompact, you can always 613 00:34:30,190 --> 00:34:31,940 do dimensional reduction on it. 614 00:34:31,940 --> 00:34:33,929 And this provides a convenient way 615 00:34:33,929 --> 00:34:37,151 to organize fields, in terms of 5 dimensional fields. 616 00:34:37,151 --> 00:34:37,692 AUDIENCE: OK. 617 00:34:40,290 --> 00:34:41,060 HONG LIU: OK. 618 00:34:41,060 --> 00:34:44,210 So one thing we will often use is, indeed, 619 00:34:44,210 --> 00:34:47,250 just consider the dimensional reduction of the gravity 620 00:34:47,250 --> 00:34:49,179 to 5 dimension. 621 00:34:49,179 --> 00:34:51,460 So if you have, say, say suppose this is 622 00:34:51,460 --> 00:34:53,889 Einstein-Hilbert action in 10d. 623 00:34:57,400 --> 00:34:59,600 So this is AdS 5 part. 624 00:34:59,600 --> 00:35:01,770 This is the S5 part. 625 00:35:01,770 --> 00:35:07,580 So you have some 10 dimensional, so curvature. 626 00:35:07,580 --> 00:35:10,910 So R is a [? rich ?] scalar in 10d. 627 00:35:10,910 --> 00:35:13,530 OK. 628 00:35:13,530 --> 00:35:16,400 Now, suppose the metric, suppose this considered 629 00:35:16,400 --> 00:35:22,520 the lowest mode for the metric, which have low S5 dependence. 630 00:35:22,520 --> 00:35:26,620 And then you can just reduce this to a 5 dimensional theory. 631 00:35:26,620 --> 00:35:30,050 Gravity, so the volume of the S5 part, 632 00:35:30,050 --> 00:35:43,530 we are just [? factorize. ?] Then 633 00:35:43,530 --> 00:35:45,230 you have a 5 dimensional scalar. 634 00:35:45,230 --> 00:35:45,970 OK? 635 00:35:45,970 --> 00:35:49,765 And this b5 is just the volume of S5. 636 00:35:49,765 --> 00:35:50,760 OK? 637 00:35:50,760 --> 00:35:51,260 5. 638 00:35:54,000 --> 00:35:58,460 And now, we can absorb this b5 into downstairs, then define 639 00:35:58,460 --> 00:36:00,525 or you factor your 5 dimension Newton constant. 640 00:36:05,350 --> 00:36:07,570 5 [INAUDIBLE] Newton constant, which 641 00:36:07,570 --> 00:36:11,360 we are call G5, which is defined to be GN divided 642 00:36:11,360 --> 00:36:15,390 by the volume of the 5 sphere. 643 00:36:15,390 --> 00:36:16,550 OK? 644 00:36:16,550 --> 00:36:23,425 And the volume of the 5 sphere is, essentially, just pi q. 645 00:36:23,425 --> 00:36:23,925 OK? 646 00:36:23,925 --> 00:36:28,330 And then, of course, there's R to the power [? of 5th. ?] 647 00:36:28,330 --> 00:36:30,260 So this is the quantity we will often use. 648 00:36:33,608 --> 00:36:34,108 OK? 649 00:36:42,790 --> 00:36:47,060 So of the dimensional reduction, in the end, 650 00:36:47,060 --> 00:36:52,910 you get the action [? we will ?] have the form 14 pi G5, 651 00:36:52,910 --> 00:37:00,040 so 5 dimensional Newton constant, say, [? d5x ?], 652 00:37:00,040 --> 00:37:02,180 just AdS part. 653 00:37:02,180 --> 00:37:05,690 And then you have the gravity part of the action. 654 00:37:05,690 --> 00:37:09,310 And then you plus many, many matter fields. 655 00:37:09,310 --> 00:37:13,980 So this is, the essentially will be the structure of your action 656 00:37:13,980 --> 00:37:16,080 when you do the dimensional reduction. 657 00:37:16,080 --> 00:37:17,050 OK? 658 00:37:17,050 --> 00:37:21,420 AUDIENCE: Shall we have matter fields in the S5? 659 00:37:21,420 --> 00:37:24,690 HONG LIU: No, you reduce everything on S5. 660 00:37:24,690 --> 00:37:28,700 And once you have done that, then S5 will disappear. 661 00:37:28,700 --> 00:37:30,990 AUDIENCE: But when we do, say, the R 662 00:37:30,990 --> 00:37:34,600 is [? always ?] a constant, so we can do the integral easily 663 00:37:34,600 --> 00:37:35,846 in S5. 664 00:37:35,846 --> 00:37:38,620 But if we don't have some mass [INAUDIBLE]. 665 00:37:38,620 --> 00:37:43,760 HONG LIU: Yeah, so no, it doesn't matter. 666 00:37:43,760 --> 00:37:47,320 So you can take-- this is a standard story for [INAUDIBLE] 667 00:37:47,320 --> 00:37:48,710 reduct-- a standard story. 668 00:37:48,710 --> 00:37:55,680 When you expand around in this, so this will go into here. 669 00:37:55,680 --> 00:37:58,940 Then the only dependence on the S5 part 670 00:37:58,940 --> 00:38:00,350 will depend on those things. 671 00:38:00,350 --> 00:38:00,650 AUDIENCE: Uh-huh. 672 00:38:00,650 --> 00:38:02,190 HONG LIU: Then you just can't integrate those things. 673 00:38:02,190 --> 00:38:02,500 AUDIENCE: Oh. 674 00:38:02,500 --> 00:38:04,200 HONG LIU: You can always integrate them. 675 00:38:04,200 --> 00:38:05,740 AUDIENCE: Oh, OK. 676 00:38:05,740 --> 00:38:11,151 HONG LIU: And they just give you some integrals, some numbers. 677 00:38:11,151 --> 00:38:12,650 It would just give you some numbers. 678 00:38:12,650 --> 00:38:14,608 And in the end, you can always write the action 679 00:38:14,608 --> 00:38:17,450 in terms of 5 dimensional action. 680 00:38:17,450 --> 00:38:20,840 AUDIENCE: But that solution is a classical solution 681 00:38:20,840 --> 00:38:22,115 for the [INAUDIBLE] equation. 682 00:38:22,115 --> 00:38:22,990 HONG LIU: No, no, no. 683 00:38:22,990 --> 00:38:24,480 This is not a classical solution. 684 00:38:24,480 --> 00:38:26,092 I'm just doing an expansion. 685 00:38:26,092 --> 00:38:27,800 It's just like doing a Fornier transform. 686 00:38:27,800 --> 00:38:28,508 AUDIENCE: Oh, OK. 687 00:38:28,508 --> 00:38:30,690 HONG LIU: This is expand in terms-- in some basis. 688 00:38:30,690 --> 00:38:32,700 No, have not done anything. 689 00:38:32,700 --> 00:38:35,340 This just a mathematical rewriting, 690 00:38:35,340 --> 00:38:36,212 AUDIENCE: Mm-hm, OK. 691 00:38:36,212 --> 00:38:36,795 HONG LIU: Yes? 692 00:38:36,795 --> 00:38:40,190 AUDIENCE: But I think in terms of expression for G Newton 693 00:38:40,190 --> 00:38:45,540 into the affected-- don't we have that this G5 is like G 694 00:38:45,540 --> 00:38:48,210 squared alpha for the [INAUDIBLE] [? divided by ?] 695 00:38:48,210 --> 00:38:50,830 [? R to the 5th y ?] can put [? a use it ?] to this alpha R 696 00:38:50,830 --> 00:38:52,510 squared combination? 697 00:38:52,510 --> 00:38:55,458 HONG LIU: Sorry-- 698 00:38:55,458 --> 00:39:00,190 AUDIENCE: Express G5 in terms of GS and alpha prime. 699 00:39:00,190 --> 00:39:00,970 HONG LIU: Yeah? 700 00:39:00,970 --> 00:39:02,750 AUDIENCE: And [INAUDIBLE]. 701 00:39:02,750 --> 00:39:07,787 Why don't it look like only combinations of R prime over R 702 00:39:07,787 --> 00:39:10,650 squared and GS? 703 00:39:10,650 --> 00:39:13,320 HONG LIU: Sorry, I don't understand what you're asking. 704 00:39:13,320 --> 00:39:15,800 No, G5 is a dimensional parameter. 705 00:39:15,800 --> 00:39:18,930 5 dimension Newton constant had dimensions 3. 706 00:39:18,930 --> 00:39:22,150 So whatever dimension here, some [? other ?] dimension 707 00:39:22,150 --> 00:39:23,910 there would be compensated by R5. 708 00:39:23,910 --> 00:39:27,130 But you always have 3 dimension left. 709 00:39:27,130 --> 00:39:32,062 So this is a dimension 3 number, yeah. 710 00:39:32,062 --> 00:39:32,895 Any other questions? 711 00:39:36,180 --> 00:39:38,370 Good. 712 00:39:38,370 --> 00:39:41,060 OK, so then this concludes our discussion 713 00:39:41,060 --> 00:39:43,860 about string theory in AdS 5 times S5. 714 00:39:43,860 --> 00:39:47,270 As I said, there's not much to talk about it. 715 00:39:47,270 --> 00:39:51,080 And so now, let's look at the other object, 716 00:39:51,080 --> 00:39:52,830 which is the N equal to 4 super-Yang-Mills 717 00:39:52,830 --> 00:39:59,360 theory, which is the other side of the equation, 718 00:39:59,360 --> 00:40:01,536 of the duality equation. 719 00:40:01,536 --> 00:40:03,230 OK? 720 00:40:03,230 --> 00:40:06,734 So in this case, we have a lot to talk about it. 721 00:40:06,734 --> 00:40:08,650 We, in principle, have a lot to talk about it, 722 00:40:08,650 --> 00:40:10,500 but we won't have time. 723 00:40:10,500 --> 00:40:13,940 So we will also not talk much about it. 724 00:40:13,940 --> 00:40:19,810 So we are only mentioning a few essential things, OK? 725 00:40:19,810 --> 00:40:23,160 So the field content, say, of a N equal to 4 super-Yang-Mills 726 00:40:23,160 --> 00:40:29,630 theory, which we already learned, 727 00:40:29,630 --> 00:40:33,580 which we already learned from the [INAUDIBLE] theory 728 00:40:33,580 --> 00:40:38,360 on the D3-brane is that you should have a gauge field. 729 00:40:38,360 --> 00:40:40,656 And then you have a 6 scalar field, because 1 into 6 730 00:40:40,656 --> 00:40:42,280 transverse direction over the D3-brane. 731 00:40:47,010 --> 00:40:49,780 OK, you have 6 scalar fields. 732 00:40:49,780 --> 00:40:54,190 And then when you include the super symmetric center, 733 00:40:54,190 --> 00:40:56,830 in the super string, then they actually also 734 00:40:56,830 --> 00:41:01,580 the massless of fermions. 735 00:41:01,580 --> 00:41:03,890 So this is 3 plus 1 dimension. 736 00:41:03,890 --> 00:41:04,435 OK? 737 00:41:04,435 --> 00:41:07,650 So this is 3 plus 1 dimension. 738 00:41:07,650 --> 00:41:13,200 So in 3 plus 1 dimension, you can represent the fermions 739 00:41:13,200 --> 00:41:15,920 conveniently by two components, for example, two component Weyl 740 00:41:15,920 --> 00:41:18,020 fermions. 741 00:41:18,020 --> 00:41:21,650 They it turns out there are 4 such Weyl fermions. 742 00:41:21,650 --> 00:41:25,070 So A will go to 1 to 4. 743 00:41:25,070 --> 00:41:27,930 So alpha is a spinor indices. 744 00:41:27,930 --> 00:41:30,010 And A just neighbor different spinors. 745 00:41:30,010 --> 00:41:32,890 So altogether, you have four spinors. 746 00:41:32,890 --> 00:41:34,940 OK? 747 00:41:34,940 --> 00:41:37,440 And so this is, essentially, the field content 748 00:41:37,440 --> 00:41:39,400 of the N equal to 4 super-Yang-Mills theory. 749 00:41:39,400 --> 00:41:42,140 So we would not worry about the fermions. 750 00:41:42,140 --> 00:41:46,346 So but I'm just mention here for completeness. 751 00:41:46,346 --> 00:41:50,625 So if you have a U(N) gauge theory, 752 00:41:50,625 --> 00:41:53,200 if you have a U(N) guage theory, say, 753 00:41:53,200 --> 00:41:57,570 for the N D3-brane, then all this, 754 00:41:57,570 --> 00:42:06,615 all this field is in the of the [INAUDIBLE] of U(N). 755 00:42:06,615 --> 00:42:11,850 In other words, each of them is an N by N matrix, 756 00:42:11,850 --> 00:42:15,021 is N by N Hermitian matrix. 757 00:42:15,021 --> 00:42:15,520 OK? 758 00:42:20,460 --> 00:42:21,880 So each of them can be represented 759 00:42:21,880 --> 00:42:23,110 by Hermitian matrices. 760 00:42:26,110 --> 00:42:36,970 So altogether, yeah, yeah-- so let me not write it. 761 00:42:36,970 --> 00:42:38,540 Let me just say it in words. 762 00:42:38,540 --> 00:42:41,250 So altogether, you have 8 [INAUDIBLE]. 763 00:42:41,250 --> 00:42:43,640 You have 8 bosonic degrees of freedom. 764 00:42:43,640 --> 00:42:46,390 Because for the photon, you have 2. 765 00:42:46,390 --> 00:42:49,380 For photon, you have 2 [INAUDIBLE] degrees of freedom. 766 00:42:49,380 --> 00:42:51,420 And here, 6 scalar field, you have 8 [INAUDIBLE] 767 00:42:51,420 --> 00:42:52,696 degrees of freedom. 768 00:42:52,696 --> 00:42:54,070 And for the fermion, you can also 769 00:42:54,070 --> 00:42:58,375 count that the 4 Weyl spinors actually have 8 [INAUDIBLE] 770 00:42:58,375 --> 00:43:01,310 degrees of freedom. 771 00:43:01,310 --> 00:43:04,720 So altogether, you have 8 N square. 772 00:43:04,720 --> 00:43:07,440 Each of them is an N by N matrix. 773 00:43:07,440 --> 00:43:10,570 So you have 8 N squared, say, [INAUDIBLE] bosonic degrees 774 00:43:10,570 --> 00:43:13,779 of freedom, and eight 8 N squared for mionic [INAUDIBLE] 775 00:43:13,779 --> 00:43:14,570 degrees of freedom. 776 00:43:14,570 --> 00:43:17,992 OK, for the-- 777 00:43:17,992 --> 00:43:21,315 AUDIENCE: Why would we have 8 fermions? 778 00:43:21,315 --> 00:43:21,856 HONG LIU: Hm? 779 00:43:21,856 --> 00:43:23,125 AUDIENCE: We have 8 fermions? 780 00:43:23,125 --> 00:43:24,500 HONG LIU: No, we have 4 fermions, 781 00:43:24,500 --> 00:43:26,150 but each fermion have some spinor component. 782 00:43:26,150 --> 00:43:27,025 AUDIENCE: Oh, OK, OK. 783 00:43:27,025 --> 00:43:27,811 HONG LIU: Yeah. 784 00:43:27,811 --> 00:43:29,680 AUDIENCE: Where did the number come from? 785 00:43:29,680 --> 00:43:30,346 HONG LIU: Sorry? 786 00:43:30,346 --> 00:43:32,906 AUDIENCE: Where did 4 fermions? 787 00:43:32,906 --> 00:43:34,530 HONG LIU: That come from a calculation. 788 00:43:34,530 --> 00:43:37,230 This I cannot explain here. 789 00:43:37,230 --> 00:43:38,800 This, I can only quote this fact. 790 00:43:38,800 --> 00:43:42,290 It's just that if you work out the theory on the D3-brane, 791 00:43:42,290 --> 00:43:47,650 then [INAUDIBLE] there's 4 Weyl fermions, yeah, 792 00:43:47,650 --> 00:43:50,120 which we did not go through that, because we did not 793 00:43:50,120 --> 00:43:51,840 do super string. 794 00:43:51,840 --> 00:43:52,830 Yeah. 795 00:43:52,830 --> 00:43:57,285 AUDIENCE: Should this come from the fact that it's where 796 00:43:57,285 --> 00:43:58,270 [INAUDIBLE]? 797 00:43:58,270 --> 00:43:58,770 Symmetry? 798 00:43:58,770 --> 00:44:01,080 HONG LIU: Yeah, it's come from the super symmetry. 799 00:44:01,080 --> 00:44:05,250 Yeah, come form the string theory, super string theory. 800 00:44:05,250 --> 00:44:09,060 So in our discussion, we only discussed bosonic part. 801 00:44:09,060 --> 00:44:14,160 We never had time to discuss for mionic part, et cetera. 802 00:44:14,160 --> 00:44:18,240 So let me now mention one important point, which 803 00:44:18,240 --> 00:44:20,470 I think is-- I hope is self-obvious, 804 00:44:20,470 --> 00:44:23,570 self-evident to you. 805 00:44:23,570 --> 00:44:31,160 It's that this U(N)-- actually, there's a U(1) which decouples, 806 00:44:31,160 --> 00:44:33,840 which U(1) decouples, just actually, 807 00:44:33,840 --> 00:44:38,140 let me just write down the Yang-Mills theory first. 808 00:44:38,140 --> 00:44:40,360 So let me write down the Yang-Mills theory first. 809 00:44:40,360 --> 00:44:44,050 So you can write down the Yang-Mills theory, which we are 810 00:44:44,050 --> 00:44:46,090 actually wrote it down before. 811 00:44:46,090 --> 00:44:48,220 Let me only write down the bosonic part. 812 00:44:51,930 --> 00:44:57,310 So essentially, just given by Yang-Mills coupling trace. 813 00:45:03,890 --> 00:45:06,180 And this is a covariant derivative acting 814 00:45:06,180 --> 00:45:08,970 on the scalar, standard covariant derivatives. 815 00:45:13,090 --> 00:45:15,950 And the I and the j are all summed. 816 00:45:15,950 --> 00:45:18,430 OK? 817 00:45:18,430 --> 00:45:20,280 So I and the j are all summed. 818 00:45:20,280 --> 00:45:23,430 So this is the Lagrangian, then plus the fermionic part. 819 00:45:23,430 --> 00:45:26,241 So this is the bosonic part, then plus the fermionic part. 820 00:45:26,241 --> 00:45:26,740 OK? 821 00:45:33,920 --> 00:45:34,420 Yes? 822 00:45:34,420 --> 00:45:36,530 AUDIENCE: Are fermions coupled through 5 fields? 823 00:45:36,530 --> 00:45:38,030 HONG LIU: Yeah, they're all coupled. 824 00:45:41,090 --> 00:45:42,118 They're all coupled. 825 00:45:42,118 --> 00:45:43,950 AUDIENCE: How do they couple? 826 00:45:43,950 --> 00:45:45,616 HONG LIU: Standard [INAUDIBLE] coupling. 827 00:45:52,200 --> 00:45:57,180 Yeah, so the key, so they couple through a particular coupling. 828 00:45:57,180 --> 00:46:00,820 The hosting, it just can show to have a single coupling. 829 00:46:00,820 --> 00:46:02,592 In principle, if you have so many fields, 830 00:46:02,592 --> 00:46:04,800 you can have many, many different possible couplings. 831 00:46:04,800 --> 00:46:07,389 You can have one coupling between them, 832 00:46:07,389 --> 00:46:08,930 one coupling between them, et cetera. 833 00:46:08,930 --> 00:46:11,530 You can have different components, but 834 00:46:11,530 --> 00:46:13,300 and self-couplings, et cetera. 835 00:46:13,300 --> 00:46:15,820 And under the key of N equal to 4 super-Yang-Mills 836 00:46:15,820 --> 00:46:22,120 theory is that every coupling, they equal up to some constant, 837 00:46:22,120 --> 00:46:25,200 up to some constant factors, which is precisely 838 00:46:25,200 --> 00:46:26,780 give you the supersymmetry. 839 00:46:26,780 --> 00:46:32,510 Anyway, so I think it's obvious to you 840 00:46:32,510 --> 00:46:38,910 that you have a U(N) Yang-Mills theory that the U(1) 841 00:46:38,910 --> 00:46:39,695 part decouples. 842 00:46:44,930 --> 00:46:45,430 OK? 843 00:46:48,730 --> 00:46:49,855 OK, you unplug the couples. 844 00:46:52,800 --> 00:46:56,200 So U(N), you can always decompose it 845 00:46:56,200 --> 00:47:02,640 into SU(N) times U(1). 846 00:47:02,640 --> 00:47:05,880 Under the U(1) part, [INAUDIBLE]. 847 00:47:05,880 --> 00:47:10,280 So each A, each field is N by N matrix. 848 00:47:10,280 --> 00:47:13,010 And the U(1) part is the part which 849 00:47:13,010 --> 00:47:15,720 is proportional to the identity matrix. 850 00:47:15,720 --> 00:47:19,680 And the SU(N) part is the part which given 851 00:47:19,680 --> 00:47:23,240 by N by N trace this matrix. 852 00:47:23,240 --> 00:47:26,800 And so the U(1) part is proportional to the identity 853 00:47:26,800 --> 00:47:28,740 matrix. 854 00:47:28,740 --> 00:47:31,120 So but you can immediately see from here, anything which 855 00:47:31,120 --> 00:47:32,890 is proportional to the identity matrix 856 00:47:32,890 --> 00:47:35,110 all the commentators will vanish. 857 00:47:35,110 --> 00:47:35,930 OK? 858 00:47:35,930 --> 00:47:38,840 So essentially, U(1) part is a free theory. 859 00:47:38,840 --> 00:47:40,090 So U(1) part is a free theory. 860 00:47:51,264 --> 00:47:52,680 Yeah, if this is not clear to you, 861 00:47:52,680 --> 00:47:55,820 you can easily convince yourself afterwards. 862 00:48:02,240 --> 00:48:05,610 And this is also where-- this is also 863 00:48:05,610 --> 00:48:08,720 physically clear from the point of view, 864 00:48:08,720 --> 00:48:12,305 thinking that this is coming from the D-branes. 865 00:48:12,305 --> 00:48:14,430 Coming from the D-brane, you have, essentially, you 866 00:48:14,430 --> 00:48:17,140 have N D-brane together. 867 00:48:17,140 --> 00:48:20,510 And this Yang-Mills theory essentially 868 00:48:20,510 --> 00:48:23,010 describes the low energy dynamics of D-branes, brains, 869 00:48:23,010 --> 00:48:25,780 so how all these N D-branes interact with each other. 870 00:48:25,780 --> 00:48:27,780 But no matter know how they [? co-communicate ?] 871 00:48:27,780 --> 00:48:30,405 [? interact ?] with each other, there's always a center of mass 872 00:48:30,405 --> 00:48:31,640 motion. 873 00:48:31,640 --> 00:48:34,970 Does not depend on their internal structure. 874 00:48:34,970 --> 00:48:38,450 And that central mass motion essentially is just this U(1). 875 00:48:38,450 --> 00:48:40,610 And the central mass motion always decouple. 876 00:48:40,610 --> 00:48:41,490 OK? 877 00:48:41,490 --> 00:48:42,992 So U(1) decouples. 878 00:48:42,992 --> 00:48:44,450 And you can do the check from here. 879 00:48:44,450 --> 00:48:45,320 A U(1) decouples. 880 00:48:45,320 --> 00:48:45,820 OK? 881 00:48:45,820 --> 00:48:47,055 AUDIENCE: Yeah. 882 00:48:47,055 --> 00:48:47,680 HONG LIU: Good. 883 00:48:53,390 --> 00:48:57,950 So now let me-- so the interacting part is just SU(N). 884 00:48:57,950 --> 00:48:59,300 OK? 885 00:48:59,300 --> 00:49:02,700 The interacting part is just SU(N). 886 00:49:02,700 --> 00:49:04,430 So now, let me say a few words regarding 887 00:49:04,430 --> 00:49:05,770 the properties of the theory. 888 00:49:14,610 --> 00:49:18,140 So first, it's that this theory has N equal to 4 supersymmetry. 889 00:49:28,510 --> 00:49:29,640 OK? 890 00:49:29,640 --> 00:49:31,430 So this is another important remark. 891 00:49:31,430 --> 00:49:35,990 This is just explain the name, just explains the name. 892 00:49:38,660 --> 00:49:42,950 So in supersymmetric theories, so you 893 00:49:42,950 --> 00:49:45,730 have a transformation between boson and the fermion. 894 00:49:45,730 --> 00:49:46,360 OK? 895 00:49:46,360 --> 00:49:49,700 You have some-- the theory is invariant on the transformation 896 00:49:49,700 --> 00:49:51,840 between boson and fermion. 897 00:49:51,840 --> 00:49:54,090 So on the [? side of ?] transformation, 898 00:49:54,090 --> 00:49:57,300 the [? conserved ?] charge, so for [INAUDIBLE] transformation, 899 00:49:57,300 --> 00:49:59,570 then you have loss of charge. 900 00:49:59,570 --> 00:50:01,835 [? For ?] [? such ?] [? as ?] transformation and loss 901 00:50:01,835 --> 00:50:05,690 of charge can be described by a spinor. 902 00:50:05,690 --> 00:50:06,190 OK? 903 00:50:09,030 --> 00:50:12,140 So normally, we say N equal to 1 supersymmetry 904 00:50:12,140 --> 00:50:15,030 if the super charge is given by a single Weyl 905 00:50:15,030 --> 00:50:16,800 spinor in 4 dimension. 906 00:50:16,800 --> 00:50:20,530 So N equal to 4 means that the total number 907 00:50:20,530 --> 00:50:24,470 of such [? conserved ?] charge is actually given by the 4 Weyl 908 00:50:24,470 --> 00:50:25,910 spinors in the 4 dimension. 909 00:50:25,910 --> 00:50:27,780 So it doesn't matter, so these have 910 00:50:27,780 --> 00:50:30,140 a number of supersymmetries. 911 00:50:30,140 --> 00:50:31,570 OK? 912 00:50:31,570 --> 00:50:34,300 And also, if you know a little bit of supersymmetry, 913 00:50:34,300 --> 00:50:36,490 then this is actually the maximally allowed, 914 00:50:36,490 --> 00:50:38,320 the supersymmetry in 4 dimensions, 915 00:50:38,320 --> 00:50:40,640 for [INAUDIBLE] field theory. 916 00:50:40,640 --> 00:50:43,970 So this is actually maximum allowed supersymmetry. 917 00:50:43,970 --> 00:50:45,590 But this will not be important for us. 918 00:50:45,590 --> 00:50:46,090 OK? 919 00:50:49,090 --> 00:50:50,030 But this will be. 920 00:50:50,030 --> 00:50:52,345 But the next point will be very important for us. 921 00:50:56,010 --> 00:51:03,680 It's that because of this theory is so symmetric, 922 00:51:03,680 --> 00:51:08,730 so G Yang-Mills coupling, its dimensionless, classically. 923 00:51:17,360 --> 00:51:18,760 OK? 924 00:51:18,760 --> 00:51:22,220 Because this is a 4 dimension, the Yang-Mills coupling 925 00:51:22,220 --> 00:51:25,360 is dimensionless, classically. 926 00:51:25,360 --> 00:51:29,110 And the same thing with our own QCD, our own QCD 927 00:51:29,110 --> 00:51:32,110 coupling this dimensionless, classically. 928 00:51:32,110 --> 00:51:36,260 But then quantum mechanically, then the coupling actually 929 00:51:36,260 --> 00:51:39,010 changes with the scale, OK, because over the quantum 930 00:51:39,010 --> 00:51:40,280 corrections, et cetera. 931 00:51:40,280 --> 00:51:43,430 So generic coupling changes with scale. 932 00:51:43,430 --> 00:51:45,610 But N equal to 4 super-Yang-Mills theory 933 00:51:45,610 --> 00:51:47,049 is special. 934 00:51:47,049 --> 00:51:48,840 But this actually, this Yang-Mills coupling 935 00:51:48,840 --> 00:51:52,322 does not change with the scale, even at the quantum level. 936 00:51:52,322 --> 00:51:53,780 So the quantum level, this remains, 937 00:51:53,780 --> 00:51:54,863 so dimensionless coupling. 938 00:51:57,530 --> 00:52:03,090 So using the technical term, for those of you who 939 00:52:03,090 --> 00:52:06,180 have studied the QC Yang-Mills theory, 940 00:52:06,180 --> 00:52:10,470 is that the beta function at the quantum mechanical level, 941 00:52:10,470 --> 00:52:17,450 the beta function for G Yang-Mills 942 00:52:17,450 --> 00:52:29,280 is actually 0, which means you can really treat this G 943 00:52:29,280 --> 00:52:45,090 Yang-Mills as a parameter, as a dimensionless parameter, even 944 00:52:45,090 --> 00:52:46,090 quantum mechanically. 945 00:52:46,090 --> 00:52:46,590 OK? 946 00:52:51,050 --> 00:52:54,180 So this is a genuine parameter. 947 00:52:54,180 --> 00:52:58,340 In contrasting our QCD, there's no such parameter. 948 00:52:58,340 --> 00:52:58,840 OK? 949 00:52:58,840 --> 00:53:01,225 In QCD, we only have a scale. 950 00:53:01,225 --> 00:53:02,100 There's no parameter. 951 00:53:05,680 --> 00:53:06,180 Good? 952 00:53:06,180 --> 00:53:08,996 So any questions regarding these? 953 00:53:08,996 --> 00:53:10,454 AUDIENCE: But you said you wouldn't 954 00:53:10,454 --> 00:53:14,810 know the coupling [INAUDIBLE] coupling because of 955 00:53:14,810 --> 00:53:16,760 [INAUDIBLE]? 956 00:53:16,760 --> 00:53:18,420 HONG LIU: Yeah. 957 00:53:18,420 --> 00:53:21,592 AUDIENCE: That means this is not [INAUDIBLE]? 958 00:53:21,592 --> 00:53:22,550 HONG LIU: No, it's not. 959 00:53:22,550 --> 00:53:23,909 No, in the-- 960 00:53:23,909 --> 00:53:25,200 AUDIENCE: It's not [INAUDIBLE]. 961 00:53:25,200 --> 00:53:28,949 HONG LIU: In QCD, the Yang-Mills coupling is not the parameter. 962 00:53:28,949 --> 00:53:30,490 It's not the dimensionless parameter. 963 00:53:30,490 --> 00:53:31,031 AUDIENCE: OK. 964 00:53:31,031 --> 00:53:32,693 HONG LIU: It translates into a scale. 965 00:53:32,693 --> 00:53:33,276 AUDIENCE: Yes. 966 00:53:33,276 --> 00:53:34,780 HONG LIU: It translates into a scale. 967 00:53:34,780 --> 00:53:35,363 AUDIENCE: So-- 968 00:53:35,363 --> 00:53:38,360 HONG LIU: It translates into a mass scale. 969 00:53:38,360 --> 00:53:41,755 AUDIENCE: Yes, but can we find a massless parameter 970 00:53:41,755 --> 00:53:44,050 in [INAUDIBLE] like this? 971 00:53:44,050 --> 00:53:47,292 HONG LIU: No, because the beta function is nonzero there. 972 00:53:47,292 --> 00:53:50,130 AUDIENCE: Ah. 973 00:53:50,130 --> 00:53:52,960 HONG LIU: Yeah, yeah because the beta function is nonzero there. 974 00:53:52,960 --> 00:53:53,960 OK? 975 00:53:53,960 --> 00:53:58,280 And also because of this, the beta function is 0. 976 00:53:58,280 --> 00:54:03,768 This theory is actually conformally invariant, 977 00:54:12,740 --> 00:54:14,460 theory is actually conformally invariant. 978 00:54:14,460 --> 00:54:15,540 OK? 979 00:54:15,540 --> 00:54:18,250 So which we'll normally call, say, it's a conformal field 980 00:54:18,250 --> 00:54:19,980 theory, because it's a CFT. 981 00:54:26,210 --> 00:54:32,390 So one way to understand this is because of the beta function 982 00:54:32,390 --> 00:54:36,210 is 0, essentially, the theory does not have a scale. 983 00:54:36,210 --> 00:54:38,670 So theory classical scale, invariant quantum mechanically 984 00:54:38,670 --> 00:54:40,262 remains scale invariant. 985 00:54:40,262 --> 00:54:41,970 And then you can show that there actually 986 00:54:41,970 --> 00:54:45,840 is a little bit more generous conformally invariant. 987 00:54:45,840 --> 00:54:49,860 So let me say a little bit about the-- are people 988 00:54:49,860 --> 00:54:53,190 familiar with the concept of the conformally invariant, 989 00:54:53,190 --> 00:54:54,020 or conformal? 990 00:54:54,020 --> 00:54:54,899 AUDIENCE: No. 991 00:54:54,899 --> 00:54:55,440 HONG LIU: No? 992 00:54:55,440 --> 00:54:55,985 OK. 993 00:54:55,985 --> 00:54:59,310 Yeah, so let me say a few words about the conformal. 994 00:54:59,310 --> 00:55:01,810 So conformally invariant means that the theory 995 00:55:01,810 --> 00:55:03,920 were under the conformal transformations. 996 00:55:03,920 --> 00:55:04,860 OK? 997 00:55:04,860 --> 00:55:07,318 So let me say a few words on the conformal transformations. 998 00:55:11,830 --> 00:55:16,930 So conformal transformations are the following. 999 00:55:16,930 --> 00:55:21,680 Say, suppose your spacetime metric 1000 00:55:21,680 --> 00:55:25,430 is given by some G mu mu. 1001 00:55:25,430 --> 00:55:28,100 So the conformal transformations are those 1002 00:55:28,100 --> 00:55:30,600 coordinate transformations. 1003 00:55:30,600 --> 00:55:31,100 OK? 1004 00:55:31,100 --> 00:55:34,620 You go from x to x prime. 1005 00:55:34,620 --> 00:55:39,010 So that under such a transformation, your metric, 1006 00:55:39,010 --> 00:55:43,850 the transformed metric is related to the original one 1007 00:55:43,850 --> 00:55:47,610 only by an overall scale factor. 1008 00:55:47,610 --> 00:55:48,110 OK? 1009 00:55:55,150 --> 00:55:57,990 So if lambda is equal to 1, then this 1010 00:55:57,990 --> 00:56:01,330 is what we called earlier isometry, which 1011 00:56:01,330 --> 00:56:04,810 leave the metric invariant, will be isometry. 1012 00:56:04,810 --> 00:56:07,440 And if you leave the metric invariant 1013 00:56:07,440 --> 00:56:09,980 after your overall scale factor, then this 1014 00:56:09,980 --> 00:56:12,330 is called the conformal transformation. 1015 00:56:12,330 --> 00:56:14,690 OK? 1016 00:56:14,690 --> 00:56:19,990 And for the Minkowski spacetime, for Minkowski spacetime, 1017 00:56:19,990 --> 00:56:22,930 actually, I can just even use this thing here. 1018 00:56:22,930 --> 00:56:29,499 Yeah, maybe let me-- yeah, let me erase here. 1019 00:56:38,800 --> 00:56:42,645 So for Minkowski spacetime, say if the G mu 1020 00:56:42,645 --> 00:56:46,190 mu is equal to [INAUDIBLE] mu mu-- suppose this 1021 00:56:46,190 --> 00:56:54,390 is a d dimensional Minkowski spacetime-- OK? 1022 00:56:54,390 --> 00:56:57,940 Then you can show-- you can just work 1023 00:56:57,940 --> 00:57:00,560 how to solve this equation, work out 1024 00:57:00,560 --> 00:57:01,976 all possible such transformations. 1025 00:57:01,976 --> 00:57:02,476 OK? 1026 00:57:06,240 --> 00:57:11,260 And you can show that conformal transformations 1027 00:57:11,260 --> 00:57:12,260 are the following. 1028 00:57:12,260 --> 00:57:15,041 First, they are the standard isometry, 1029 00:57:15,041 --> 00:57:17,290 which is the translation, because the translation does 1030 00:57:17,290 --> 00:57:20,230 not give you lambda 1. 1031 00:57:20,230 --> 00:57:23,590 And then you have a Lorentz transformation, 1032 00:57:23,590 --> 00:57:26,260 which also give you lambda 1. 1033 00:57:26,260 --> 00:57:27,680 OK? 1034 00:57:27,680 --> 00:57:29,210 And they you can have a scaling. 1035 00:57:29,210 --> 00:57:33,326 Of course, if you scale your coordinate 1036 00:57:33,326 --> 00:57:38,100 with some factor, then, of course, 1037 00:57:38,100 --> 00:57:42,214 the metric will only change by overall factor. 1038 00:57:42,214 --> 00:57:43,255 So this is a translation. 1039 00:57:47,140 --> 00:57:51,000 So this is s Lorentz symmetry. 1040 00:57:51,000 --> 00:57:54,790 And this is scaling. 1041 00:57:54,790 --> 00:57:56,329 And then we also have something what 1042 00:57:56,329 --> 00:57:58,120 we call a special conformal transformation. 1043 00:58:04,480 --> 00:58:20,390 So this is for-- OK? 1044 00:58:20,390 --> 00:58:22,620 And the b, again, is some constant parameter. 1045 00:58:22,620 --> 00:58:25,160 OK? 1046 00:58:25,160 --> 00:58:31,660 And also for the discrete transformation, 1047 00:58:31,660 --> 00:58:32,910 also, you can have a discrete. 1048 00:58:37,530 --> 00:58:41,075 Also you can have a discrete transformations inversion. 1049 00:58:45,042 --> 00:58:49,946 You [? said ?] x prime mu equal to x mu divided by x squared. 1050 00:58:49,946 --> 00:58:50,446 OK? 1051 00:58:54,430 --> 00:58:58,530 So these are all the transformations. 1052 00:58:58,530 --> 00:59:01,800 In general dimension, which leave you metric invariant, 1053 00:59:01,800 --> 00:59:04,130 leave the Minkowski metric invariant number 1054 00:59:04,130 --> 00:59:07,210 up to overall factor. 1055 00:59:07,210 --> 00:59:10,070 And the conformal invariant theory 1056 00:59:10,070 --> 00:59:14,130 is a theory which [? involves ?] transformations. 1057 00:59:14,130 --> 00:59:16,120 OK? 1058 00:59:16,120 --> 00:59:20,211 And so obviously, a Yang-Mills theory 1059 00:59:20,211 --> 00:59:22,460 are always invariant on the translation on the Lorentz 1060 00:59:22,460 --> 00:59:24,220 transformation. 1061 00:59:24,220 --> 00:59:26,250 But when the beta function is 0, the system 1062 00:59:26,250 --> 00:59:27,440 does not have a scale. 1063 00:59:27,440 --> 00:59:31,180 Then it's invariant on the scaling. 1064 00:59:31,180 --> 00:59:35,055 But typically, in general, 4 dimensional theory, 1065 00:59:35,055 --> 00:59:36,550 any theory invariant on the scaling 1066 00:59:36,550 --> 00:59:38,350 is also invariant on this special conformal 1067 00:59:38,350 --> 00:59:39,550 transformations. 1068 00:59:39,550 --> 00:59:42,110 So actually, the N equal to 4 super-Yang-Mills 1069 00:59:42,110 --> 00:59:44,140 theory is actually conformal theory, 1070 00:59:44,140 --> 00:59:47,630 OK, conformally invariant theory. 1071 00:59:47,630 --> 00:59:51,310 So now, as part of your p set, you 1072 00:59:51,310 --> 00:59:57,630 can actually see that this conformal transformations 1073 00:59:57,630 --> 00:59:59,790 in fact [? have ?] 1 to 1 correspondence 1074 00:59:59,790 --> 01:00:04,750 with these isometries of the AdS. 1075 01:00:04,750 --> 01:00:06,050 OK? 1076 01:00:06,050 --> 01:00:07,750 And this we will see later. 1077 01:00:07,750 --> 01:00:10,690 This is a part, important part of the relation between AdS. 1078 01:00:13,330 --> 01:00:16,750 This is an important part of this duality relation. 1079 01:00:16,750 --> 01:00:20,362 It's that the symmetry have to be the same. 1080 01:00:20,362 --> 01:00:21,320 So you compare the two. 1081 01:00:21,320 --> 01:00:24,840 You can see that, indeed, they 1 to 1 correspond to each other. 1082 01:00:30,930 --> 01:00:38,070 So the conformal transformation add together give you SO(d,2). 1083 01:00:38,070 --> 01:00:39,650 OK? 1084 01:00:39,650 --> 01:00:45,505 So this is the conformal group, so-called the conformal group 1085 01:00:45,505 --> 01:00:46,240 in d dimension. 1086 01:00:52,910 --> 01:00:55,940 Then you get to the [? SO(d,2). ?] OK? 1087 01:01:04,280 --> 01:01:08,200 And but in this theory, there's actually also 1088 01:01:08,200 --> 01:01:13,000 global symmetry, because the N equal to 4 super-Yang-Mills 1089 01:01:13,000 --> 01:01:17,020 theory come from the D3-brane, which is a transverse space, 1090 01:01:17,020 --> 01:01:19,580 is the [? rotation of ?] invariant. 1091 01:01:19,580 --> 01:01:24,750 So this actually our SO(6) symmetry rotates different phi 1092 01:01:24,750 --> 01:01:26,050 i's. 1093 01:01:26,050 --> 01:01:36,810 So actually, there's also global SO(6) symmetry, 1094 01:01:36,810 --> 01:01:43,243 which rotates phi i, OK, and the fermions. 1095 01:01:47,573 --> 01:01:48,073 OK? 1096 01:01:53,460 --> 01:02:03,980 And then when you include in the supersymmetry 1097 01:02:03,980 --> 01:02:06,320 then you get a huge symmetry group. 1098 01:02:06,320 --> 01:02:09,630 It's normally called a super conformal symmetry. 1099 01:02:09,630 --> 01:02:12,480 Anyway, let me just write some notation down. 1100 01:02:12,480 --> 01:02:13,419 It doesn't matter. 1101 01:02:13,419 --> 01:02:15,710 So if you include the symmetry, the full symmetry group 1102 01:02:15,710 --> 01:02:17,510 is what we normally call it. 1103 01:02:17,510 --> 01:02:21,520 So this is a super group, so [? SU(2,2|4) ?] Doesn't matter. 1104 01:02:27,160 --> 01:02:31,930 So the bottom line is that the N equal to 4 super-Yang-Mills 1105 01:02:31,930 --> 01:02:35,500 theory is the most symmetric 4 dimensional theory. 1106 01:02:35,500 --> 01:02:36,580 OK? 1107 01:02:36,580 --> 01:02:40,250 And almost-- no theory has more symmetries than N 1108 01:02:40,250 --> 01:02:41,708 equal to 4 super-Yang-Mills theory. 1109 01:02:46,100 --> 01:02:47,064 Any questions? 1110 01:02:47,064 --> 01:02:47,980 AUDIENCE: [INAUDIBLE]. 1111 01:02:47,980 --> 01:02:48,563 HONG LIU: Yes? 1112 01:02:48,563 --> 01:02:51,198 AUDIENCE: So How do you know that the [? flows are ?] 1113 01:02:51,198 --> 01:02:53,620 [INAUDIBLE]? 1114 01:02:53,620 --> 01:02:56,250 [? HONG LIU: Though ?] you can classify it by solving this 1115 01:02:56,250 --> 01:02:58,584 equation. 1116 01:02:58,584 --> 01:03:00,750 You can classify it by solving this equation. 1117 01:03:04,375 --> 01:03:08,650 AUDIENCE: But what's the q [INAUDIBLE]? 1118 01:03:08,650 --> 01:03:10,265 HONG LIU: No, this is just a notation. 1119 01:03:10,265 --> 01:03:11,140 AUDIENCE: A notation? 1120 01:03:11,140 --> 01:03:11,765 HONG LIU: Yeah. 1121 01:03:14,957 --> 01:03:15,790 Any other questions? 1122 01:03:18,400 --> 01:03:21,240 So I can say a few words about the conformal field theories. 1123 01:03:29,722 --> 01:03:34,100 But it's getting a little bit late, 1124 01:03:34,100 --> 01:03:35,780 and I want to talk about other things. 1125 01:03:39,930 --> 01:03:43,660 So let me just say it in words. 1126 01:03:43,660 --> 01:03:46,920 And then you can try to read it in other places. 1127 01:03:46,920 --> 01:03:50,000 Because even if I just write a couple formulas here, 1128 01:03:50,000 --> 01:03:51,670 it still won't change you very much. 1129 01:03:51,670 --> 01:03:52,170 [LAUGHTER] 1130 01:03:52,170 --> 01:03:55,920 I won't teach you too much. 1131 01:03:55,920 --> 01:04:00,790 So important thing about this conformal field theory 1132 01:04:00,790 --> 01:04:07,110 is that each operator, say each local operator-- yeah, 1133 01:04:07,110 --> 01:04:10,470 you can classify local operators by how they transform, say, 1134 01:04:10,470 --> 01:04:12,470 under these conformal symmetries. 1135 01:04:12,470 --> 01:04:14,490 And then you can associate, say, a dimension 1136 01:04:14,490 --> 01:04:17,160 to each operator, related to how they 1137 01:04:17,160 --> 01:04:23,030 transform in these conformal transformations. 1138 01:04:23,030 --> 01:04:26,200 And then you can also show that the symmetry actually 1139 01:04:26,200 --> 01:04:30,170 dictates the 2-point function and the 3-point function 1140 01:04:30,170 --> 01:04:31,990 of such operators. 1141 01:04:31,990 --> 01:04:34,330 And essentially, the structure of the 2-point function 1142 01:04:34,330 --> 01:04:36,940 and 3-point function are completely fixed. 1143 01:04:39,495 --> 01:04:41,120 But not higher point function, but only 1144 01:04:41,120 --> 01:04:42,520 2 and 3-point functions. 1145 01:04:45,320 --> 01:04:48,610 Yeah, so that's essentially it. 1146 01:04:48,610 --> 01:04:52,930 Not much known, many things are known about conformal field 1147 01:04:52,930 --> 01:04:55,160 theory in 1 plus 1 dimension. 1148 01:04:55,160 --> 01:04:59,790 And but higher dimensions, yeah, other 1149 01:04:59,790 --> 01:05:02,280 than what I said, not much. 1150 01:05:02,280 --> 01:05:04,386 Not too much more is known, yeah. 1151 01:05:07,260 --> 01:05:11,760 So any questions regarding this? 1152 01:05:11,760 --> 01:05:17,520 OK, so let me just summarize, then we can have a break. 1153 01:05:17,520 --> 01:05:19,860 So we can summarize. 1154 01:05:19,860 --> 01:05:24,630 So let me just summarize what we have done so far, and also, 1155 01:05:24,630 --> 01:05:25,903 with an important refinement. 1156 01:05:31,860 --> 01:05:35,580 So what we said is we started with two picture. 1157 01:05:35,580 --> 01:05:39,780 One is D-brane with some open strings, 1158 01:05:39,780 --> 01:05:43,550 and then some closed strings can interact with it. 1159 01:05:43,550 --> 01:05:46,362 And then when we go to the low energy limit, 1160 01:05:46,362 --> 01:05:48,070 then we get N equal to 4 super-Yang-Mills 1161 01:05:48,070 --> 01:06:07,500 theory with, now, I say SU(N) plus a decoupled U(1), and then 1162 01:06:07,500 --> 01:06:15,640 plus what we discussed last time, a decoupled the gravitons 1163 01:06:15,640 --> 01:06:17,860 in the low energy limit. 1164 01:06:17,860 --> 01:06:22,290 So let me just write E equal to 0 limit. 1165 01:06:22,290 --> 01:06:26,580 On the other side, we have this [? pure, ?] [? geometric ?] 1166 01:06:26,580 --> 01:06:33,400 picture of the curved spacetime produced by those D-branes. 1167 01:06:33,400 --> 01:06:35,290 So those curved spacetime at infinity 1168 01:06:35,290 --> 01:06:38,890 behaves like just 10 dimensional Minkowski spacetime. 1169 01:06:38,890 --> 01:06:42,090 And then when you go to the close to the brane, 1170 01:06:42,090 --> 01:06:46,380 then the space deforms into the AdS 5 times S5. 1171 01:06:50,860 --> 01:06:55,020 OK, form into AdS 5 times S 5. 1172 01:06:55,020 --> 01:06:59,890 Under the low energy limit, because 1 into-- we can see 1173 01:06:59,890 --> 01:07:03,230 the string theory in the [? very ?] [? much ?] 1174 01:07:03,230 --> 01:07:05,780 [? tongue ?] this [? throat. ?] OK? 1175 01:07:09,130 --> 01:07:13,320 So the low energy limit is the string theory. 1176 01:07:13,320 --> 01:07:16,380 The one here when you're take E equal to 0, 1177 01:07:16,380 --> 01:07:25,810 we get string theory in AdS 5 times S5, 1178 01:07:25,810 --> 01:07:28,485 then plus decoupled graviton. 1179 01:07:36,440 --> 01:07:44,920 Here, also, here, when we look at the geometry of the brane, 1180 01:07:44,920 --> 01:07:47,560 essentially, we have fixed the location of the brane. 1181 01:07:47,560 --> 01:07:51,100 So essentially, we have fixed to the center of mass motion. 1182 01:07:51,100 --> 01:07:54,700 But in principle, you can allow, also, 1183 01:07:54,700 --> 01:07:56,990 the brane to move anywhere. 1184 01:07:56,990 --> 01:08:03,170 So here, essentially, there's also a decoupled center 1185 01:08:03,170 --> 01:08:05,830 of mass motion. 1186 01:08:05,830 --> 01:08:08,620 OK? 1187 01:08:08,620 --> 01:08:13,390 So now, let's get rid of this decoupled graviton. 1188 01:08:13,390 --> 01:08:17,050 Let's get rid of this decoupled U(1), 1189 01:08:17,050 --> 01:08:19,140 or decoupled center of mass motion. 1190 01:08:19,140 --> 01:08:22,840 Then what we get is the fully interacting part. 1191 01:08:22,840 --> 01:08:23,340 OK? 1192 01:08:26,140 --> 01:08:28,000 So the fully interacting part is so small 1193 01:08:28,000 --> 01:08:32,410 refinement than what we said last time. 1194 01:08:32,410 --> 01:08:37,450 We said N equal to 4 super-Yang-Mill theory 1195 01:08:37,450 --> 01:08:42,890 with [? stage ?] group SU(N) should be the same as the type 1196 01:08:42,890 --> 01:08:48,340 IIB string in AdS 5 times S5. 1197 01:08:51,270 --> 01:08:53,051 OK? 1198 01:08:53,051 --> 01:08:58,060 So, now, other side, now, this is SU(N). 1199 01:08:58,060 --> 01:09:02,819 And more precisely, this is AdS prime S5 in the Poincare patch. 1200 01:09:06,791 --> 01:09:07,290 OK? 1201 01:09:10,420 --> 01:09:11,229 Yes? 1202 01:09:11,229 --> 01:09:13,330 AUDIENCE: The N just to recall, is 1203 01:09:13,330 --> 01:09:16,140 the that N comes from the amount of flux on the S5. 1204 01:09:16,140 --> 01:09:16,870 Is that correct? 1205 01:09:16,870 --> 01:09:18,078 HONG LIU: Yeah, that's right. 1206 01:09:18,078 --> 01:09:18,680 That's right. 1207 01:09:18,680 --> 01:09:23,240 So the N here related to the flux on the AdS 5, 1208 01:09:23,240 --> 01:09:29,510 it's also related, as we will see, 1209 01:09:29,510 --> 01:09:31,350 related the Newton Constant, et cetera. 1210 01:09:31,350 --> 01:09:34,670 Yeah, we will see that, yeah. 1211 01:09:34,670 --> 01:09:35,899 OK? 1212 01:09:35,899 --> 01:09:41,779 So now, we can look at this geometric picture. 1213 01:09:41,779 --> 01:09:42,279 OK? 1214 01:09:42,279 --> 01:09:45,369 We can look at this geometric picture. 1215 01:09:45,369 --> 01:09:46,994 Then we observer something interesting. 1216 01:09:51,600 --> 01:09:57,040 So this SU(N) group, let me say, [INAUDIBLE] R 1,3. 1217 01:09:57,040 --> 01:09:58,050 OK? 1218 01:09:58,050 --> 01:10:00,640 On the 3, approximate dimensional-- yeah, 1219 01:10:00,640 --> 01:10:08,630 let me write it better-- this SU(N) on the space, 1220 01:10:08,630 --> 01:10:14,790 on the Minkowski space, R 1,3. 1221 01:10:14,790 --> 01:10:18,180 So now, let's note. 1222 01:10:18,180 --> 01:10:22,630 Make a simple observation based on the geometry of AdS 1223 01:10:22,630 --> 01:10:31,310 is that this R 1,3 is actually the boundary of AdS 5. 1224 01:10:34,591 --> 01:10:35,090 OK? 1225 01:10:35,090 --> 01:10:37,089 It's essentially the boundary manifold of AdS 5. 1226 01:10:40,900 --> 01:10:45,346 And the right hand side, no matter 1227 01:10:45,346 --> 01:10:47,220 whether you do field theory or string theory, 1228 01:10:47,220 --> 01:10:49,136 you can always do dimensional reduction on S5. 1229 01:10:49,136 --> 01:10:50,860 You can always decompose everything 1230 01:10:50,860 --> 01:10:54,420 on S5 in terms harmonics. 1231 01:10:54,420 --> 01:10:57,020 So essentially, the right hand side, 1232 01:10:57,020 --> 01:11:01,260 because S5 is a compact space, the right hand side 1233 01:11:01,260 --> 01:11:03,070 is a 5 dimensional gravity theory. 1234 01:11:11,230 --> 01:11:14,030 OK? 1235 01:11:14,030 --> 01:11:19,515 So we actually now here see a 5 dimensional gravity theory. 1236 01:11:22,040 --> 01:11:25,400 Now, it's equivalent that your 4 dimensional theory [? lives ?] 1237 01:11:25,400 --> 01:11:27,810 on its boundary. 1238 01:11:27,810 --> 01:11:30,330 OK? 1239 01:11:30,330 --> 01:11:36,806 So this really can be considered [? actually ?] realization 1240 01:11:36,806 --> 01:11:38,230 of holographic principle. 1241 01:11:58,497 --> 01:12:00,580 So actually, there are two ways to think about it. 1242 01:12:00,580 --> 01:12:05,425 One way is to think from this-- so previously, we motivated. 1243 01:12:08,310 --> 01:12:12,250 Long time ago, we motivated the duality using 1244 01:12:12,250 --> 01:12:13,890 two perspec-- [INAUDIBLE]. 1245 01:12:13,890 --> 01:12:16,499 One is the holographic principle. 1246 01:12:16,499 --> 01:12:19,040 Some gravity theory should be [INAUDIBLE] to the [? theory ?] 1247 01:12:19,040 --> 01:12:20,800 living on this boundary. 1248 01:12:20,800 --> 01:12:30,350 And also, motivated that the Yang-Mills theory in the gauge 1249 01:12:30,350 --> 01:12:34,700 group SU(N) or U(N), when you do the [? large N ?] expansion, 1250 01:12:34,700 --> 01:12:36,580 then behaves like a string theory. 1251 01:12:36,580 --> 01:12:37,300 OK? 1252 01:12:37,300 --> 01:12:39,635 So this could also be considered as an explicit example 1253 01:12:39,635 --> 01:12:41,800 of that relation. 1254 01:12:41,800 --> 01:12:44,300 That this is a Yang-Mill male theory, and then 1255 01:12:44,300 --> 01:12:46,630 turned out to be [INAUDIBLE] to a string theory. 1256 01:12:46,630 --> 01:12:49,140 OK? 1257 01:12:49,140 --> 01:12:53,050 So now, if you look at this statement, 1258 01:12:53,050 --> 01:12:57,040 you say, is this a coincidence? 1259 01:12:57,040 --> 01:13:03,040 Because I don't really see somehow in this picture 1260 01:13:03,040 --> 01:13:09,410 not so much somehow how why this Yang-Mills theory is actually 1261 01:13:09,410 --> 01:13:12,600 should be considered as living on the boundary of AdS 5 times 1262 01:13:12,600 --> 01:13:15,340 S5, because the boundary of AdS 5 times S5, 1263 01:13:15,340 --> 01:13:16,960 somewhere around here. 1264 01:13:16,960 --> 01:13:18,940 Yeah, I don't see some Yang-Mills theory living 1265 01:13:18,940 --> 01:13:20,590 on there, OK? 1266 01:13:20,590 --> 01:13:22,690 So is this really just some coincidence 1267 01:13:22,690 --> 01:13:26,779 that just happened to be that? 1268 01:13:26,779 --> 01:13:28,820 But actually, there's a very important prediction 1269 01:13:28,820 --> 01:13:31,750 one can make if you take this point of view. 1270 01:13:35,185 --> 01:13:37,305 Now, I can erase here. 1271 01:13:37,305 --> 01:13:38,880 Maybe I will do it here. 1272 01:13:42,768 --> 01:13:46,260 Yeah, but if you take this point of view, 1273 01:13:46,260 --> 01:13:48,770 then there's a very important prediction you can make. 1274 01:13:51,730 --> 01:13:53,660 Then there's a very important prediction 1275 01:13:53,660 --> 01:13:56,030 you can make, and then check whether that 1276 01:13:56,030 --> 01:13:57,460 has any chance of working. 1277 01:14:00,020 --> 01:14:05,780 So if you believe this picture, then we can have 1278 01:14:05,780 --> 01:14:16,820 a [? non-trivial ?] prediction, because AdS also have 1279 01:14:16,820 --> 01:14:22,789 [? a lot ?] of description in terms of the global AdS. 1280 01:14:22,789 --> 01:14:24,830 Then let's say, what happens if we put the string 1281 01:14:24,830 --> 01:14:27,650 theory in this global AdS, which certainly, we 1282 01:14:27,650 --> 01:14:31,100 should be able to do, because these two only differ 1283 01:14:31,100 --> 01:14:32,790 by some global structure. 1284 01:14:32,790 --> 01:14:34,110 OK? 1285 01:14:34,110 --> 01:14:35,780 And this is one part of that. 1286 01:14:35,780 --> 01:14:38,480 They only differ by some global structure. 1287 01:14:38,480 --> 01:14:42,670 So if this is really true, if this is not a coincidence, 1288 01:14:42,670 --> 01:14:45,770 then we can make a non-trivial prediction is that N equal to 4 1289 01:14:45,770 --> 01:14:57,100 is a type IIB string in global AdS 5 times S5. 1290 01:14:57,100 --> 01:14:59,810 So now, the AdS 5 we take to be the global, 1291 01:14:59,810 --> 01:15:03,380 to take to be the cylinder. 1292 01:15:03,380 --> 01:15:07,140 That should be equal to the N equals to 4 super-Yang-Mills 1293 01:15:07,140 --> 01:15:11,690 theory, which now, should live on the boundary 1294 01:15:11,690 --> 01:15:20,990 of that cylinder, which is our S3 times R. OK? 1295 01:15:20,990 --> 01:15:22,500 So this realization will give you 1296 01:15:22,500 --> 01:15:26,230 a powerful prediction, which in principle, you can check. 1297 01:15:26,230 --> 01:15:27,847 OK? 1298 01:15:27,847 --> 01:15:29,555 And, of course, if this is a coincidence, 1299 01:15:29,555 --> 01:15:31,919 then there's no reason why this to be true. 1300 01:15:31,919 --> 01:15:33,460 Because from the brane point of view, 1301 01:15:33,460 --> 01:15:36,090 we cannot really do the sphere. 1302 01:15:36,090 --> 01:15:38,890 OK? 1303 01:15:38,890 --> 01:15:42,294 AUDIENCE: Why is [INAUDIBLE]? 1304 01:15:42,294 --> 01:15:42,794 Oh. 1305 01:15:45,807 --> 01:15:47,390 HONG LIU: So any questions about this? 1306 01:15:51,309 --> 01:15:52,850 AUDIENCE: One question [? on that. ?] 1307 01:15:52,850 --> 01:15:56,030 so when we derive the right hand picture, 1308 01:15:56,030 --> 01:15:59,410 we impose that there's some charge on the D-brane. 1309 01:15:59,410 --> 01:16:01,670 But on the left hand side, it seems 1310 01:16:01,670 --> 01:16:04,930 that we-- [INAUDIBLE] a the charge [INAUDIBLE]? 1311 01:16:04,930 --> 01:16:07,800 HONG LIU: No, the charge is just reflected 1312 01:16:07,800 --> 01:16:10,940 in open string dynamics. 1313 01:16:10,940 --> 01:16:11,580 AUDIENCE: OK. 1314 01:16:11,580 --> 01:16:14,960 So by your right hand side, we will have the charge. 1315 01:16:14,960 --> 01:16:16,920 HONG LIU: No, you essentially have a charge. 1316 01:16:16,920 --> 01:16:18,410 But in that description, you don't 1317 01:16:18,410 --> 01:16:20,840 need to introduce a charge. 1318 01:16:20,840 --> 01:16:21,560 Whatever charge-- 1319 01:16:21,560 --> 01:16:25,060 AUDIENCE: If we impose charge if the left hand side will never 1320 01:16:25,060 --> 01:16:25,670 be changed? 1321 01:16:25,670 --> 01:16:26,900 HONG LIU: No, no. 1322 01:16:26,900 --> 01:16:29,202 Just there from the side, you just talk about strings. 1323 01:16:29,202 --> 01:16:29,910 AUDIENCE: Uh-huh. 1324 01:16:29,910 --> 01:16:32,034 HONG LIU: You don't talk about the whatever charge. 1325 01:16:32,034 --> 01:16:33,685 The charge is a gravity description. 1326 01:16:33,685 --> 01:16:34,130 AUDIENCE: Uh-huh. 1327 01:16:34,130 --> 01:16:35,950 HONG LIU: The charge don't even arise in this picture. 1328 01:16:35,950 --> 01:16:36,910 I just have some strings. 1329 01:16:36,910 --> 01:16:37,460 I have some open strings. 1330 01:16:37,460 --> 01:16:38,584 I have some closed strings. 1331 01:16:38,584 --> 01:16:40,869 They interact with each other. 1332 01:16:40,869 --> 01:16:41,410 AUDIENCE: OK. 1333 01:16:41,410 --> 01:16:43,656 And did this picture go to low energy means 1334 01:16:43,656 --> 01:16:47,602 it will appear as a charge of the super-Yang-Mills theory. 1335 01:16:47,602 --> 01:16:48,810 HONG LIU: No, no, no, no, no. 1336 01:16:48,810 --> 01:16:49,000 No, no. 1337 01:16:49,000 --> 01:16:51,375 This goes [? in ?] [? order. ?] You just super-Yang-Mills 1338 01:16:51,375 --> 01:16:52,716 theory. 1339 01:16:52,716 --> 01:16:54,840 Yeah. 1340 01:16:54,840 --> 01:16:55,340 Good? 1341 01:16:55,340 --> 01:16:57,210 Any other questions? 1342 01:16:57,210 --> 01:16:57,790 Yes? 1343 01:16:57,790 --> 01:16:59,032 AUDIENCE: Is this true? 1344 01:16:59,032 --> 01:17:00,240 HONG LIU: Yeah, this is true. 1345 01:17:00,240 --> 01:17:00,781 AUDIENCE: OK. 1346 01:17:00,781 --> 01:17:04,470 [LAUGHTER] 1347 01:17:04,470 --> 01:17:07,720 HONG LIU: And the yeah. 1348 01:17:07,720 --> 01:17:13,542 So that means this is not a fantasy. 1349 01:17:13,542 --> 01:17:15,000 Yeah, this tells this is a fantasy. 1350 01:17:15,000 --> 01:17:17,760 This is actually a powerful realization. 1351 01:17:17,760 --> 01:17:21,850 This is a powerful realization, yeah. 1352 01:17:21,850 --> 01:17:25,730 So let me before conclude-- Let me 1353 01:17:25,730 --> 01:17:31,530 just quickly mention something just in terms of semantics. 1354 01:17:31,530 --> 01:17:36,830 So I always say, type IIB string in AdS 5 times S5. 1355 01:17:36,830 --> 01:17:41,310 So you may ask, does this make sense to say such things? 1356 01:17:41,310 --> 01:17:43,550 Because if we think this is really 1357 01:17:43,550 --> 01:17:46,050 as a quantum gravitational theory, 1358 01:17:46,050 --> 01:17:48,050 so if you have a finite string coupling, 1359 01:17:48,050 --> 01:17:49,960 then everything fluctuates. 1360 01:17:49,960 --> 01:17:54,480 And then why we specify a rigid spacetime here? 1361 01:17:54,480 --> 01:17:55,390 OK? 1362 01:17:55,390 --> 01:17:57,560 Why do we specify a rigid spacetime here? 1363 01:18:00,460 --> 01:18:03,900 So the way you should understand the statement is the following, 1364 01:18:03,900 --> 01:18:06,150 is that indeed, if you, say, consider the finite G 1365 01:18:06,150 --> 01:18:08,040 Newton, the finite G string, then 1366 01:18:08,040 --> 01:18:10,860 the spacetime can fluctuate a lot, 1367 01:18:10,860 --> 01:18:14,920 will typical be a deviate from AdS 5 times S5. 1368 01:18:14,920 --> 01:18:18,700 But this AdS 5 times S5 specifies 1369 01:18:18,700 --> 01:18:22,170 it's asymptotic geometry of AdS. 1370 01:18:22,170 --> 01:18:24,800 It's asymptotic geometry. 1371 01:18:24,800 --> 01:18:31,030 It's the geometry of AdS very far away near the boundary. 1372 01:18:31,030 --> 01:18:33,010 And essentially, this can be considered 1373 01:18:33,010 --> 01:18:36,270 as specifying the boundary condition 1374 01:18:36,270 --> 01:18:40,040 for gravity, the boundary condition for the quantum 1375 01:18:40,040 --> 01:18:41,302 gravity. 1376 01:18:41,302 --> 01:18:43,510 So this, essentially, specify the boundary condition. 1377 01:18:46,070 --> 01:18:49,190 Good, so let's have a couple minutes break. 1378 01:18:49,190 --> 01:18:52,680 Then we will talk more about the duality. 1379 01:18:52,680 --> 01:18:53,430 OK, good. 1380 01:18:53,430 --> 01:18:55,346 So let's start again. 1381 01:18:55,346 --> 01:18:59,154 [SIDE CONVERSATION] 1382 01:18:59,154 --> 01:19:04,820 So now, let's move to a new chapter. 1383 01:19:04,820 --> 01:19:07,840 So now, have we established the duality. 1384 01:19:07,840 --> 01:19:11,410 So let's try to understand the duality. 1385 01:19:11,410 --> 01:19:13,980 So let me call it the duality toolbox. 1386 01:19:17,810 --> 01:19:19,320 And then we can try to use it. 1387 01:19:28,110 --> 01:19:36,410 So first, let me say some general aspect on the duality. 1388 01:19:36,410 --> 01:19:38,480 Oh, I erased. 1389 01:19:38,480 --> 01:19:40,570 Oh, I should keep that figure there. 1390 01:19:40,570 --> 01:19:48,030 Anyway, so the first important thing 1391 01:19:48,030 --> 01:19:49,884 is so-called IR/UV connection. 1392 01:19:57,050 --> 01:20:06,480 So here, we see the equivalents, say, from the AdS 5 gravity. 1393 01:20:06,480 --> 01:20:09,230 So we have some AdS 5 gravity to go 1394 01:20:09,230 --> 01:20:18,490 to N equal to 4 super-Yang-Mills theory in d equal to 4. 1395 01:20:18,490 --> 01:20:20,977 So from this direction, we may consider as a realization 1396 01:20:20,977 --> 01:20:22,185 of the holographic principle. 1397 01:20:25,710 --> 01:20:27,710 OK? 1398 01:20:27,710 --> 01:20:31,550 We can consider this direction as a holographic principle. 1399 01:20:31,550 --> 01:20:37,590 And then you may ask, from the field perspective, 1400 01:20:37,590 --> 01:20:44,659 if we think from this angle, why somehow, 1401 01:20:44,659 --> 01:20:47,200 this description of the field theory have one more dimension. 1402 01:20:47,200 --> 01:20:48,640 So this is a 4 dimensional theory. 1403 01:20:48,640 --> 01:20:50,520 This is a 5 dimensional theory. 1404 01:20:50,520 --> 01:20:52,760 OK? 1405 01:20:52,760 --> 01:20:55,430 What does this actual dimension does? 1406 01:20:55,430 --> 01:20:58,740 What does this x dimension do, OK? 1407 01:20:58,740 --> 01:21:01,050 How do we understand from the field theory 1408 01:21:01,050 --> 01:21:05,580 perspective somehow this increase of one dimension? 1409 01:21:05,580 --> 01:21:07,250 OK? 1410 01:21:07,250 --> 01:21:12,230 So the answer turns out-- the answer is actually very simple. 1411 01:21:12,230 --> 01:21:16,560 And it's already nice in a way. 1412 01:21:16,560 --> 01:21:18,160 The answer is already in the way which 1413 01:21:18,160 --> 01:21:20,200 we take the low energy limit. 1414 01:21:20,200 --> 01:21:21,680 OK? 1415 01:21:21,680 --> 01:21:23,810 So remember, when we take the low energy limit-- so 1416 01:21:23,810 --> 01:21:32,370 let me draw this figure again-- in the gravity side, 1417 01:21:32,370 --> 01:21:36,320 and this is going to R goes to 0. 1418 01:21:36,320 --> 01:21:38,780 OK? 1419 01:21:38,780 --> 01:21:45,090 So by studying the red shift due to the curved spacetime, 1420 01:21:45,090 --> 01:21:48,370 we show that if you want to go to low energies, 1421 01:21:48,370 --> 01:21:52,350 then you want to go to smaller and smaller R. OK? 1422 01:21:52,350 --> 01:21:54,150 So you want to go to smaller and smaller R 1423 01:21:54,150 --> 01:21:57,040 if you want to go to [? lower ?] and [? lower ?] emerges. 1424 01:21:57,040 --> 01:21:58,580 OK? 1425 01:21:58,580 --> 01:22:05,530 So smaller r, and then give you smaller energy. 1426 01:22:08,410 --> 01:22:11,020 So we use this argument to decouple 1427 01:22:11,020 --> 01:22:13,930 this, the other factor. 1428 01:22:13,930 --> 01:22:16,350 We say, if you want to go to low energy, 1429 01:22:16,350 --> 01:22:17,580 you take r go to 0 limit. 1430 01:22:17,580 --> 01:22:21,310 Then there are infinite distance between them. 1431 01:22:21,310 --> 01:22:22,850 But the nice thing about infinities 1432 01:22:22,850 --> 01:22:24,790 is that after you go to infinity, 1433 01:22:24,790 --> 01:22:27,000 there's still infinite left. 1434 01:22:27,000 --> 01:22:28,330 OK? 1435 01:22:28,330 --> 01:22:32,390 So the same argument also applies in AdS, 1436 01:22:32,390 --> 01:22:36,130 in the AdS part, after you decouple the other stuff. 1437 01:22:36,130 --> 01:22:39,230 And the AdS part of this argument still applies. 1438 01:22:39,230 --> 01:22:41,060 Still, the same thing happens. 1439 01:22:41,060 --> 01:22:44,599 The same red shift argument applies. 1440 01:22:44,599 --> 01:22:46,140 So if you want to go to low energies, 1441 01:22:46,140 --> 01:22:48,260 you want to go to smaller r. 1442 01:22:48,260 --> 01:22:49,810 OK? 1443 01:22:49,810 --> 01:22:55,550 And so this immediately tells us that this actual dimension, 1444 01:22:55,550 --> 01:23:00,967 this r direction, it's precisely the direction, which external 1445 01:23:00,967 --> 01:23:02,800 to the N equal to 4 super-Yang-Mills theory, 1446 01:23:02,800 --> 01:23:05,520 transverse to the N equal to 4 super-Yang-Mills theory. 1447 01:23:05,520 --> 01:23:07,561 So this [? immediate ?] [? here is ?] this actual 1448 01:23:07,561 --> 01:23:12,120 dimension, this r, or in this coordinate z-- remember, 1449 01:23:12,120 --> 01:23:16,540 in this z, z related to that thing just by R squared divided 1450 01:23:16,540 --> 01:23:22,030 by r, so that r related to that z. 1451 01:23:22,030 --> 01:23:27,940 So that the r goes to 0 limit, it tells you 1452 01:23:27,940 --> 01:23:38,490 that that r dimension, r can be considered, 1453 01:23:38,490 --> 01:23:40,140 this actual dimension can be considered 1454 01:23:40,140 --> 01:23:58,240 as representing the energy scale of the Yang-Mills theory, 1455 01:23:58,240 --> 01:24:04,210 OK, or the energy scale of the boundary theory. 1456 01:24:04,210 --> 01:24:06,420 So if you want to go to low energy, 1457 01:24:06,420 --> 01:24:10,230 you go down [? farthest ?] [? route. ?] OK? 1458 01:24:10,230 --> 01:24:14,480 This is a very, very important point. 1459 01:24:14,480 --> 01:24:18,140 And this nice into many phenomenon between AdS 1460 01:24:18,140 --> 01:24:20,410 and the CFT and the disconnection. 1461 01:24:22,940 --> 01:24:25,410 Very important, and if you have a good understanding, 1462 01:24:25,410 --> 01:24:28,770 it will give you lots of good intuitions 1463 01:24:28,770 --> 01:24:31,480 in you understand the relation. 1464 01:24:31,480 --> 01:24:35,800 So let me just to elaborate the argument again, 1465 01:24:35,800 --> 01:24:38,890 now using this metric. 1466 01:24:38,890 --> 01:24:40,700 OK? 1467 01:24:40,700 --> 01:24:44,380 OK, so now, we'll go over this red shift argument again, 1468 01:24:44,380 --> 01:24:46,871 and to use this metric. 1469 01:24:46,871 --> 01:24:47,370 OK? 1470 01:24:52,590 --> 01:24:57,280 So again, so the key thing is that the x mu, which 1471 01:24:57,280 --> 01:25:03,280 you t go to x appear in this metric, 1472 01:25:03,280 --> 01:25:05,120 is defining the boundary unit. 1473 01:25:05,120 --> 01:25:06,950 OK? 1474 01:25:06,950 --> 01:25:11,570 So this defines-- so this defined in the boundary unit. 1475 01:25:11,570 --> 01:25:17,540 So this is the unit we use in the Yang-Mills theory. 1476 01:25:17,540 --> 01:25:18,210 OK? 1477 01:25:18,210 --> 01:25:21,700 So those coordinates are defined in the boundary units. 1478 01:25:21,700 --> 01:25:27,280 So now, if we want to talk about energies in some bulk, 1479 01:25:27,280 --> 01:25:38,260 say, the local proper time, and the length, 1480 01:25:38,260 --> 01:25:45,110 and the total proper length at some, say, at some z, 1481 01:25:45,110 --> 01:25:49,260 OK, so at some location z in the bulk 1482 01:25:49,260 --> 01:25:53,590 is related by the standard relation, 1483 01:25:53,590 --> 01:25:55,130 which we can read from there. 1484 01:25:55,130 --> 01:26:00,340 It say d tau should be equal to R, so this local proper time. 1485 01:26:03,330 --> 01:26:09,060 Local proper time should be equal to R divided by z dt. 1486 01:26:09,060 --> 01:26:12,470 And the dl, so local length scale, 1487 01:26:12,470 --> 01:26:18,180 should be a proper length, should be related R z times dx. 1488 01:26:18,180 --> 01:26:18,680 OK? 1489 01:26:22,448 --> 01:26:24,430 OK, just because of this. 1490 01:26:24,430 --> 01:26:27,720 You can just read it from here. 1491 01:26:27,720 --> 01:26:31,010 Locally, you can always write this as a Minkowski metric 1492 01:26:31,010 --> 01:26:33,480 in d plus 1 dimension. 1493 01:26:33,480 --> 01:26:35,740 And then for the local observer, and this 1494 01:26:35,740 --> 01:26:39,300 is the local time, local proper time and local proper length. 1495 01:26:39,300 --> 01:26:40,490 OK? 1496 01:26:40,490 --> 01:26:43,270 And then this tells you that you can easily 1497 01:26:43,270 --> 01:26:45,060 invert this relation. 1498 01:26:45,060 --> 01:26:48,150 This tells you the energy we observe in the Yang-Mills 1499 01:26:48,150 --> 01:26:53,190 theory, was just energy we measure in terms of t 1500 01:26:53,190 --> 01:26:56,300 should be related to the energy, which we measure locally, 1501 01:26:56,300 --> 01:27:02,030 so all these relation, local energy. 1502 01:27:02,030 --> 01:27:07,340 And length scale, now d in Yang-Mills theory, 1503 01:27:07,340 --> 01:27:14,120 is related to the local length scale by this relation. 1504 01:27:14,120 --> 01:27:14,620 OK? 1505 01:27:18,120 --> 01:27:20,820 So this is an inverse of that because of the tau 1506 01:27:20,820 --> 01:27:23,680 and the [? energy. ?] This is just the same as that. 1507 01:27:26,690 --> 01:27:29,518 OK? 1508 01:27:29,518 --> 01:27:30,018 Good? 1509 01:27:33,890 --> 01:27:39,500 So now, let's consider the same bulk process. 1510 01:27:39,500 --> 01:27:41,370 OK? 1511 01:27:41,370 --> 01:27:44,860 So let's consider the same bulk process. 1512 01:27:44,860 --> 01:27:48,570 Say, for example, this process we 1513 01:27:48,570 --> 01:27:51,700 are sitting in this classroom, say, supposing we 1514 01:27:51,700 --> 01:27:52,450 [? leaving ?] AdS. 1515 01:27:52,450 --> 01:27:55,670 Let's imagine that happens at different values of z, 1516 01:27:55,670 --> 01:28:08,748 OK say, for the same bulk process, at different z. 1517 01:28:14,604 --> 01:28:17,050 OK. 1518 01:28:17,050 --> 01:28:21,330 Then, of course, the E local, the local energy scale 1519 01:28:21,330 --> 01:28:24,150 and the local length scale would be the same. 1520 01:28:31,601 --> 01:28:32,100 OK? 1521 01:28:32,100 --> 01:28:33,770 So if we consider the different process 1522 01:28:33,770 --> 01:28:36,380 at different z, the same physical process 1523 01:28:36,380 --> 01:28:40,690 at different z, in terms of a local observers, it's the same. 1524 01:28:40,690 --> 01:28:44,481 So E local and the d local does not change. 1525 01:28:44,481 --> 01:28:44,980 OK? 1526 01:28:47,780 --> 01:28:55,730 But if you translate into the Yang-Mills scale, 1527 01:28:55,730 --> 01:28:59,660 then you find now that the corresponding Yang-Mills energy 1528 01:28:59,660 --> 01:29:05,076 scale now proportional to 1 over z, 1529 01:29:05,076 --> 01:29:09,030 OK, because now, we have fixed E local. 1530 01:29:09,030 --> 01:29:09,550 OK? 1531 01:29:09,550 --> 01:29:10,870 We have fixed E local. 1532 01:29:10,870 --> 01:29:12,910 We are changing z. 1533 01:29:12,910 --> 01:29:15,880 So now, you find at a different location, now corresponding 1534 01:29:15,880 --> 01:29:19,750 to a different energy scale in the Yang-Mills theory. 1535 01:29:19,750 --> 01:29:25,065 Say, the d Yang-Mills we will [? portion it ?] to z. 1536 01:29:25,065 --> 01:29:25,565 OK? 1537 01:29:28,350 --> 01:29:52,980 And in particular, for the same process as z 1538 01:29:52,980 --> 01:30:00,630 equal to 0, if we move this process close to the boundary-- 1539 01:30:00,630 --> 01:30:05,275 z equal to 0 means close to the boundary-- then 1540 01:30:05,275 --> 01:30:07,400 the Yang-Mills energy, the corresponding Yang-Mills 1541 01:30:07,400 --> 01:30:10,510 energy, actually goes to infinity. 1542 01:30:10,510 --> 01:30:12,760 Under the corresponding Yang-Mills scale, 1543 01:30:12,760 --> 01:30:16,160 actually goes-- distance scale goes to 0. 1544 01:30:16,160 --> 01:30:18,640 So that means this is a mapped to some UV 1545 01:30:18,640 --> 01:30:20,295 process in the Yang-Mills theory. 1546 01:30:29,281 --> 01:30:29,780 OK? 1547 01:30:32,950 --> 01:30:37,060 And but if you take the z goes to infinity, which 1548 01:30:37,060 --> 01:30:42,206 means you go to interior, far away from the boundary-- 1549 01:30:42,206 --> 01:30:45,110 so you go to infinity, far away from the boundary, 1550 01:30:45,110 --> 01:30:50,330 go to interior-- then this E Yang-Mills, 1551 01:30:50,330 --> 01:30:52,380 then the corresponding Yang-Mills energy scale 1552 01:30:52,380 --> 01:30:54,560 will go to 0. 1553 01:30:54,560 --> 01:30:57,500 And the corresponding Yang-Mills length scale 1554 01:30:57,500 --> 01:30:59,340 will go to infinity. 1555 01:30:59,340 --> 01:30:59,850 OK? 1556 01:30:59,850 --> 01:31:03,160 So this would corresponding to the IR process. 1557 01:31:03,160 --> 01:31:05,240 So this will be a low-energy process, 1558 01:31:05,240 --> 01:31:07,600 and [? much ?] distance in the Yang-Mills theory. 1559 01:31:07,600 --> 01:31:09,600 So there, of course, [? those run ?] into the IR 1560 01:31:09,600 --> 01:31:12,310 process in the Yang-Mills. 1561 01:31:12,310 --> 01:31:13,700 OK? 1562 01:31:13,700 --> 01:31:15,350 And [? this-- ?] 1563 01:31:15,350 --> 01:31:18,740 So normally, if you think about the perspective 1564 01:31:18,740 --> 01:31:25,430 from AdS point of view, when you go to the boundary, 1565 01:31:25,430 --> 01:31:27,700 means you've got a long distance. 1566 01:31:27,700 --> 01:31:32,930 So from AdS point of view, this is going to the IR. 1567 01:31:32,930 --> 01:31:33,760 OK? 1568 01:31:33,760 --> 01:31:36,600 And then when you go to the infinity, z go to infinity. 1569 01:31:36,600 --> 01:31:39,530 You are going to the interior in the bulk. 1570 01:31:39,530 --> 01:31:42,450 So roughly, it's not precisely-- [? if you ?] [? risk, ?] you 1571 01:31:42,450 --> 01:31:44,700 can think of this actually going to the short distance 1572 01:31:44,700 --> 01:31:49,960 in the-- not really the short distance, just-- yeah, 1573 01:31:49,960 --> 01:31:52,460 this name is not very proper, but let's just call it the UV. 1574 01:31:52,460 --> 01:31:55,720 Just go into the interior of the bulk. 1575 01:31:55,720 --> 01:31:58,190 So now, since you see an [? opposite ?] process going 1576 01:31:58,190 --> 01:32:01,774 on, in the AdS will go to IR. 1577 01:32:01,774 --> 01:32:03,690 Then from the Yang-Mills theory, corresponding 1578 01:32:03,690 --> 01:32:10,280 to going to the UV, but in the AdS, you go to interior. 1579 01:32:10,280 --> 01:32:11,780 And then from the Yang-Mills theory, 1580 01:32:11,780 --> 01:32:14,200 corresponding to go to the IR low energy. 1581 01:32:14,200 --> 01:32:14,940 OK? 1582 01:32:14,940 --> 01:32:19,027 So here, we have something like IR-UV connection. 1583 01:32:25,205 --> 01:32:25,705 OK? 1584 01:32:28,460 --> 01:32:35,910 In particular, I already said in particular once. 1585 01:32:35,910 --> 01:32:40,160 [LAUGHS] So but I don't have other words. 1586 01:32:40,160 --> 01:32:41,575 So in particular-- 1587 01:32:41,575 --> 01:32:44,020 [LAUGHTER] 1588 01:32:44,020 --> 01:32:48,610 --if you really consider the typical gravity process, OK, 1589 01:32:48,610 --> 01:32:54,710 so if you consider the typical gravity process, 1590 01:32:54,710 --> 01:32:58,910 so not consider string theory, just consider gravity. 1591 01:32:58,910 --> 01:33:01,010 Typical gravity process, a classical gravity 1592 01:33:01,010 --> 01:33:06,660 processes, then essentially, the curvature radius 1593 01:33:06,660 --> 01:33:07,880 defines your scale. 1594 01:33:07,880 --> 01:33:08,850 OK? 1595 01:33:08,850 --> 01:33:12,330 So that means that the typical gravity processes, 1596 01:33:12,330 --> 01:33:16,040 the E local is of order 1 over the curvature scale. 1597 01:33:16,040 --> 01:33:19,200 And the typical length scale in the bulk, 1598 01:33:19,200 --> 01:33:23,310 you can also see that it's also the curvature radius. 1599 01:33:23,310 --> 01:33:29,680 So in these cases, if you plug this into this relation, 1600 01:33:29,680 --> 01:33:32,920 then you can see that for the typical bulk process, 1601 01:33:32,920 --> 01:33:37,410 you can really identify the E Yang-Mills, essentially, 1602 01:33:37,410 --> 01:33:40,270 just with the radial direction, OK, 1603 01:33:40,270 --> 01:33:45,400 because this product will be 1. 1604 01:33:45,400 --> 01:33:47,780 This product will be of order 1. 1605 01:33:47,780 --> 01:33:53,400 And then you can really just identify the z, the inverse 1606 01:33:53,400 --> 01:33:55,330 z with the Yang-Mills energy. 1607 01:33:55,330 --> 01:33:57,940 OK? 1608 01:33:57,940 --> 01:34:02,440 So let me just say this in the little bit slide 1609 01:34:02,440 --> 01:34:07,740 today, in the picture. 1610 01:34:07,740 --> 01:34:11,100 Maybe I'll do it here, maybe just do it here. 1611 01:34:11,100 --> 01:34:14,360 So think about this in the picture. 1612 01:34:14,360 --> 01:34:15,750 So suppose this is a boundary. 1613 01:34:18,650 --> 01:34:20,655 So this is the picture. 1614 01:34:25,880 --> 01:34:30,460 This is the picture, which on the back page of the class 1615 01:34:30,460 --> 01:34:32,540 it said, if you consider some process here, 1616 01:34:32,540 --> 01:34:34,980 which we draw a cow, which I don't 1617 01:34:34,980 --> 01:34:36,623 know how to draw a cow here. 1618 01:34:36,623 --> 01:34:38,100 [LAUGHTER] 1619 01:34:38,100 --> 01:34:41,950 And at some distance, say, here, some value 1620 01:34:41,950 --> 01:34:46,640 of z here, and similar process as some other value of z, 1621 01:34:46,640 --> 01:34:50,550 then the corresponding boundary image of them, 1622 01:34:50,550 --> 01:34:53,210 so this will corresponding to a bigger guy. 1623 01:34:53,210 --> 01:34:55,880 And this guy will corresponding slightly small guy. 1624 01:34:55,880 --> 01:34:56,530 OK? 1625 01:34:56,530 --> 01:34:59,100 Yeah, I hope you understand the picture. 1626 01:34:59,100 --> 01:35:02,160 Yeah, [? so ?] the further away, the same 1627 01:35:02,160 --> 01:35:05,430 process in the interior, they corresponding 1628 01:35:05,430 --> 01:35:09,810 to a process with a larger distance and a lower energy 1629 01:35:09,810 --> 01:35:12,250 from the field theory perspective. 1630 01:35:12,250 --> 01:35:12,750 OK? 1631 01:35:24,505 --> 01:35:25,880 So now, let me make some remarks. 1632 01:35:29,890 --> 01:35:32,430 Yeah, I did not show it very well. 1633 01:35:32,430 --> 01:35:35,440 Is this clear what I mean by this picture? 1634 01:35:35,440 --> 01:35:36,420 OK, good. 1635 01:35:40,830 --> 01:35:42,560 Yeah, a more fancy picture is just 1636 01:35:42,560 --> 01:35:43,815 in the web page of the class. 1637 01:35:46,535 --> 01:35:47,910 So now, let me make some remarks. 1638 01:35:52,660 --> 01:36:00,630 The first remarks is that the AdS, if you, say, 1639 01:36:00,630 --> 01:36:05,780 calculate the volume of AdS, so calculate 1640 01:36:05,780 --> 01:36:09,500 the determinant of this guy, square root of the determinant 1641 01:36:09,500 --> 01:36:11,180 of the metric, and then integrate 1642 01:36:11,180 --> 01:36:12,330 over the [? O ?] space. 1643 01:36:12,330 --> 01:36:14,220 Then you find it's divergent. 1644 01:36:14,220 --> 01:36:15,890 And particularly, it's divergent because 1645 01:36:15,890 --> 01:36:21,480 of the infinite distance in going to the boundary. 1646 01:36:21,480 --> 01:36:22,590 OK? 1647 01:36:22,590 --> 01:36:27,800 So often, we will, say, put some cut-off, IR cut-off, 1648 01:36:27,800 --> 01:36:29,550 near the boundary, say, rather than 1649 01:36:29,550 --> 01:36:32,200 let z go all the way to z equal to 0. 1650 01:36:32,200 --> 01:36:36,300 We, say, we stop the space at z equal to some [? epsilon. ?] 1651 01:36:36,300 --> 01:36:37,080 OK? 1652 01:36:37,080 --> 01:36:39,650 That's the procedure we often do, just 1653 01:36:39,650 --> 01:36:41,630 as a mathematical convenience. 1654 01:36:41,630 --> 01:36:44,130 Because if you go all the way to z equal to 0, 1655 01:36:44,130 --> 01:36:45,890 then this factor blows up. 1656 01:36:45,890 --> 01:36:49,210 And then many things become tricky to do. 1657 01:36:49,210 --> 01:36:53,320 And then we often, in order to get the finite answer, 1658 01:36:53,320 --> 01:36:56,490 we always put the z-- put the boundary at z 1659 01:36:56,490 --> 01:36:57,520 equal to some epsilon. 1660 01:36:57,520 --> 01:36:59,030 And epsilon's some small parameter. 1661 01:36:59,030 --> 01:37:00,690 OK? 1662 01:37:00,690 --> 01:37:05,105 So and for example, this is what you will do in your p set, 1663 01:37:05,105 --> 01:37:08,190 in one of the p set problem. 1664 01:37:08,190 --> 01:37:11,190 OK, when we try to-- when you will 1665 01:37:11,190 --> 01:37:12,555 check this holographic bound. 1666 01:37:12,555 --> 01:37:14,430 And that's the trick [? you're ?] [? ready ?] 1667 01:37:14,430 --> 01:37:17,490 to use. 1668 01:37:17,490 --> 01:37:30,760 So putting the IR cut-off in AdS, at some z 1669 01:37:30,760 --> 01:37:37,030 equal to epsilon, then from this IR-UV picture, 1670 01:37:37,030 --> 01:37:41,555 then translate in the boundary. 1671 01:37:45,030 --> 01:37:51,290 In the boundary, we introduce a short distance cut-off, 1672 01:37:51,290 --> 01:38:02,110 or UV cut-off, say, at some short distance scale, 1673 01:38:02,110 --> 01:38:08,310 say, [? data ?] x say or order epsilon, or energy cut-off, 1674 01:38:08,310 --> 01:38:17,660 or UV energy cut-off at, say, energy, say, 1675 01:38:17,660 --> 01:38:18,660 of order 1 over epsilon. 1676 01:38:18,660 --> 01:38:19,160 OK? 1677 01:38:26,100 --> 01:38:30,850 So this just a [? natural-- ?] the reason this 1678 01:38:30,850 --> 01:38:37,270 goes 1 into a UV cut-off because of this relation, 1679 01:38:37,270 --> 01:38:39,120 because of this relation. 1680 01:38:39,120 --> 01:38:43,940 When you cut-off the space at some z, 1681 01:38:43,940 --> 01:38:46,690 then you can no longer go to infinite energy. 1682 01:38:46,690 --> 01:38:50,230 You can no longer go to infinite distance scale, OK? 1683 01:38:50,230 --> 01:38:53,380 And essentially, equivalently providing a short distance 1684 01:38:53,380 --> 01:38:55,510 cut-off from that relation. 1685 01:38:55,510 --> 01:38:57,990 OK? 1686 01:38:57,990 --> 01:38:59,570 Is it clear? 1687 01:38:59,570 --> 01:39:02,612 And you will use this in your p set. 1688 01:39:02,612 --> 01:39:03,112 OK? 1689 01:39:13,970 --> 01:39:18,895 So the second remark is that here, we are considering, 1690 01:39:18,895 --> 01:39:21,120 so N equal to 4 super-Yang-Mills theory, as we said, 1691 01:39:21,120 --> 01:39:23,200 it's a conformal theory. 1692 01:39:23,200 --> 01:39:26,340 Or it's a scale invariant theory. 1693 01:39:26,340 --> 01:39:28,020 A feature for scale invariant theory 1694 01:39:28,020 --> 01:39:31,410 is that there's no scale. 1695 01:39:31,410 --> 01:39:35,060 There's no scale means you can have arbitrarily low energy 1696 01:39:35,060 --> 01:39:36,310 excitations. 1697 01:39:36,310 --> 01:39:38,320 OK? 1698 01:39:38,320 --> 01:39:54,560 So for conformal theory, say, in R(1,3), there exists, 1699 01:39:54,560 --> 01:39:58,750 because of the scale invariants, there exist arbitrarily low 1700 01:39:58,750 --> 01:39:59,555 energy excitations. 1701 01:40:15,010 --> 01:40:16,530 OK? 1702 01:40:16,530 --> 01:40:20,520 So from this IR-UV, then this corresponding 1703 01:40:20,520 --> 01:40:25,610 to-- then this just map to the z equal 1704 01:40:25,610 --> 01:40:29,050 to infinity region in the bulk. 1705 01:40:29,050 --> 01:40:29,550 OK? 1706 01:40:38,194 --> 01:40:40,610 So this map to the z equal to infinity region in the bulk. 1707 01:40:43,350 --> 01:40:47,010 So it's a very good thing that this z 1708 01:40:47,010 --> 01:40:49,780 equal to infinity region. 1709 01:40:49,780 --> 01:40:50,280 OK? 1710 01:40:50,280 --> 01:40:53,200 It's a very good thing, this z equal to infinity region, 1711 01:40:53,200 --> 01:40:58,729 because otherwise, those modes have nothing to map to. 1712 01:40:58,729 --> 01:41:00,270 So on the other hand, even the theory 1713 01:41:00,270 --> 01:41:12,440 have a gap, on the other hand, if, say, 1714 01:41:12,440 --> 01:41:35,230 if the corresponding bulk spacetime, say, "ends" 1715 01:41:35,230 --> 01:41:45,060 at the finite proper distance, OK, so important thing 1716 01:41:45,060 --> 01:41:46,300 is the following. 1717 01:41:46,300 --> 01:41:47,695 So let's take any point here. 1718 01:41:50,340 --> 01:41:51,650 Take any point here. 1719 01:41:51,650 --> 01:41:54,610 So this is some reference scale, say. 1720 01:41:54,610 --> 01:41:59,300 Choose it as a reference scale in the field theory, 1721 01:41:59,300 --> 01:42:01,600 because of this [? UV ?] [? R ?] connection. 1722 01:42:01,600 --> 01:42:04,150 So now, we want to consider arbitrary low energy scale 1723 01:42:04,150 --> 01:42:05,816 compared to this scale. 1724 01:42:05,816 --> 01:42:07,440 Then what you want to do, then you just 1725 01:42:07,440 --> 01:42:10,030 you go to z equal to infinity. 1726 01:42:10,030 --> 01:42:15,080 And this, what is actually hidden here 1727 01:42:15,080 --> 01:42:17,520 when you do this argument, is actually 1728 01:42:17,520 --> 01:42:21,275 the proper distance going to z equal to infinity is infinite. 1729 01:42:21,275 --> 01:42:25,950 And because when you do this proper distance, yeah, 1730 01:42:25,950 --> 01:42:28,200 essentially, there's infinite proper distance [? in ?] 1731 01:42:28,200 --> 01:42:29,660 go to z equal to infinity. 1732 01:42:29,660 --> 01:42:32,220 And then you can go to infinite low energy scales. 1733 01:42:32,220 --> 01:42:34,310 OK? 1734 01:42:34,310 --> 01:42:43,380 But now, if you look at the global AdS, 1735 01:42:43,380 --> 01:42:46,190 then take some interior point. 1736 01:42:46,190 --> 01:42:49,810 Then actually, you can easily check. 1737 01:42:49,810 --> 01:42:55,570 When you go to the most interior point, it's just ro equal to 0. 1738 01:42:55,570 --> 01:42:56,390 OK? 1739 01:42:56,390 --> 01:42:58,440 And that is finite proper distance away. 1740 01:42:58,440 --> 01:43:01,075 So in some sense, the spacetime ends 1741 01:43:01,075 --> 01:43:05,600 at the finite proper distance in the radial direction. 1742 01:43:05,600 --> 01:43:08,090 OK? 1743 01:43:08,090 --> 01:43:13,620 In this case, if you apply this, this will tell you that 1744 01:43:13,620 --> 01:43:17,285 the theory cannot have arbitrary low energy excitations. 1745 01:43:20,020 --> 01:43:22,467 OK? 1746 01:43:22,467 --> 01:43:23,050 Is this clear? 1747 01:43:31,430 --> 01:43:33,590 Let me just say it again. 1748 01:43:33,590 --> 01:43:38,660 In this theory, in the Poincare patch, in the bulk, 1749 01:43:38,660 --> 01:43:41,090 we can go to z equal to infinity. 1750 01:43:41,090 --> 01:43:44,110 And that's infinite proper distance away. 1751 01:43:44,110 --> 01:43:46,190 And using this UV IR argument, that 1752 01:43:46,190 --> 01:43:49,720 translates to infinite low energy process in the field 1753 01:43:49,720 --> 01:43:51,480 theory. 1754 01:43:51,480 --> 01:43:55,120 But now, let's consider what's happening in the global AdS. 1755 01:43:55,120 --> 01:43:57,260 Just from the geometry, we see something dramatic 1756 01:43:57,260 --> 01:44:02,330 happens, because now, this is a cylinder. 1757 01:44:02,330 --> 01:44:04,100 And look at the metric. 1758 01:44:04,100 --> 01:44:07,470 You take finite distance from some point 1759 01:44:07,470 --> 01:44:10,450 to go to the most center point, which is rho 0. 1760 01:44:10,450 --> 01:44:13,490 So you no longer have infinite distance to go to. 1761 01:44:13,490 --> 01:44:17,560 So if this argument is consistent, 1762 01:44:17,560 --> 01:44:21,810 then this must be due to a theory 1763 01:44:21,810 --> 01:44:24,831 without arbitrary low energy excitation. 1764 01:44:24,831 --> 01:44:25,330 OK? 1765 01:44:29,620 --> 01:44:31,980 But this is, indeed, the case, because according 1766 01:44:31,980 --> 01:44:35,150 to what I erased, just before we break, 1767 01:44:35,150 --> 01:44:39,600 we say for the gravity theory in the global AdS, that 1768 01:44:39,600 --> 01:44:42,220 should be due to a Yang-Mills theory, 1769 01:44:42,220 --> 01:44:44,100 and the boundary over here, which 1770 01:44:44,100 --> 01:44:50,870 is S times R. For any field theory [? under ?] S-- again, 1771 01:44:50,870 --> 01:44:53,150 this is a compact manifold-- then 1772 01:44:53,150 --> 01:44:56,900 there's a mass gap from the vacuum. 1773 01:44:56,900 --> 01:44:59,540 And so this theory actually have a mass gap. 1774 01:44:59,540 --> 01:45:00,060 OK? 1775 01:45:00,060 --> 01:45:02,840 So that means if the corresponding bulk space that 1776 01:45:02,840 --> 01:45:19,410 ends this on finite proper distance, the boundary theory 1777 01:45:19,410 --> 01:45:29,890 must have a mass gap, and vice versa. 1778 01:45:29,890 --> 01:45:32,850 If the fields theory have a mass gap, 1779 01:45:32,850 --> 01:45:35,940 which you cannot go to arbitrary low energies, 1780 01:45:35,940 --> 01:45:38,330 and then the bulk spacetime have to be end somewhere. 1781 01:45:41,040 --> 01:45:44,270 Otherwise, these statements will not apply. 1782 01:45:44,270 --> 01:45:45,650 OK? 1783 01:45:45,650 --> 01:45:47,980 So consistency check of that statement 1784 01:45:47,980 --> 01:45:50,120 is that this has to be true. 1785 01:45:50,120 --> 01:45:50,620 OK? 1786 01:45:50,620 --> 01:45:52,350 So in the p set, you should think 1787 01:45:52,350 --> 01:45:56,220 through these for the process over the field theory 1788 01:45:56,220 --> 01:45:57,102 on the cylinder. 1789 01:45:57,102 --> 01:45:57,602 OK? 1790 01:46:01,450 --> 01:46:05,012 OK, I think I will stop here. 1791 01:46:05,012 --> 01:46:08,150 Do you have any questions regarding this? 1792 01:46:08,150 --> 01:46:09,000 Yes? 1793 01:46:09,000 --> 01:46:11,086 AUDIENCE: I was somewhat under the impression 1794 01:46:11,086 --> 01:46:12,580 that the global AdS and the Poincare patch 1795 01:46:12,580 --> 01:46:14,580 were just different coordinates for presentation 1796 01:46:14,580 --> 01:46:17,062 with the same thing, sort of? 1797 01:46:17,062 --> 01:46:20,060 So how -- would that be correct? 1798 01:46:20,060 --> 01:46:20,650 HONG LIU: Hm? 1799 01:46:20,650 --> 01:46:22,150 AUDIENCE: I was under the impression 1800 01:46:22,150 --> 01:46:23,962 that the global AdS and the Poincare patch 1801 01:46:23,962 --> 01:46:28,024 are just different coordinate charts on the same sort 1802 01:46:28,024 --> 01:46:29,190 of structure or [INAUDIBLE]. 1803 01:46:29,190 --> 01:46:30,350 HONG LIU: Yeah. 1804 01:46:30,350 --> 01:46:32,758 AUDIENCE: So why do we have infinite distance in one 1805 01:46:32,758 --> 01:46:34,494 and finite distance in another? 1806 01:46:34,494 --> 01:46:36,610 Is that? 1807 01:46:36,610 --> 01:46:38,890 HONG LIU: Figured out. 1808 01:46:38,890 --> 01:46:40,590 Figured it out. 1809 01:46:40,590 --> 01:46:42,010 And this is geometry. 1810 01:46:42,010 --> 01:46:44,660 You can just look at them. 1811 01:46:44,660 --> 01:46:49,310 It depend on how you slice the spacetime. 1812 01:46:49,310 --> 01:46:50,900 Yeah, it's very easy to see. 1813 01:46:50,900 --> 01:46:54,667 Just see how you map one point to the other point, et cetera. 1814 01:46:54,667 --> 01:46:55,166 Yeah. 1815 01:46:55,166 --> 01:46:56,940 AUDIENCE: OK. 1816 01:46:56,940 --> 01:47:01,270 HONG LIU: But the important thing, the important thing 1817 01:47:01,270 --> 01:47:07,560 is that the-- yeah, I'll give you a hint. 1818 01:47:07,560 --> 01:47:10,310 So this z equal to infinity can be considered 1819 01:47:10,310 --> 01:47:13,190 as a coordinate singularity of that [? thing. ?] 1820 01:47:13,190 --> 01:47:15,770 so even though here it's completely smooth, 1821 01:47:15,770 --> 01:47:18,649 but in here is like a coordinate singularity because of this. 1822 01:47:18,649 --> 01:47:19,190 AUDIENCE: OK. 1823 01:47:19,190 --> 01:47:21,800 HONG LIU: Yeah, yeah, yeah. 1824 01:47:21,800 --> 01:47:26,590 Yeah, so that's why, actually, to talk about theory on R3 1825 01:47:26,590 --> 01:47:30,140 and the theory on S3 is heading [? on 2 ?] because the physics 1826 01:47:30,140 --> 01:47:31,260 are very different. 1827 01:47:31,260 --> 01:47:35,140 Because here, yeah, because here, you always 1828 01:47:35,140 --> 01:47:36,560 have a mass gap. 1829 01:47:36,560 --> 01:47:38,260 And here, you don't. 1830 01:47:38,260 --> 01:47:41,940 And so that refract the geometry very different. 1831 01:47:41,940 --> 01:47:44,270 The way you view the geometry are very different. 1832 01:47:47,770 --> 01:47:50,355 AUDIENCE: So the [INAUDIBLE] 5 [INAUDIBLE]? 1833 01:47:50,355 --> 01:47:51,180 HONG LIU: Hm? 1834 01:47:51,180 --> 01:47:52,263 AUDIENCE: The [INAUDIBLE]? 1835 01:47:52,263 --> 01:47:53,694 HONG LIU: Yeah, yeah. [INAUDIBLE] 1836 01:47:53,694 --> 01:47:56,079 AUDIENCE: But if you change this [INAUDIBLE], 1837 01:47:56,079 --> 01:47:58,619 then it's still N equal 4 super-Yang-Mills [INAUDIBLE]. 1838 01:47:58,619 --> 01:48:00,410 HONG LIU: Yeah, it would be something else. 1839 01:48:00,410 --> 01:48:01,243 AUDIENCE: Then yeah. 1840 01:48:01,243 --> 01:48:03,940 HONG LIU: Yeah, yeah. 1841 01:48:03,940 --> 01:48:07,290 [APPLAUSE]