1 00:00:00,080 --> 00:00:02,430 The following content is provided under a Creative 2 00:00:02,430 --> 00:00:03,810 Commons license. 3 00:00:03,810 --> 00:00:06,060 Your support will help MIT OpenCourseWare 4 00:00:06,060 --> 00:00:10,150 continue to offer high quality educational resources for free. 5 00:00:10,150 --> 00:00:12,690 To make a donation or to view additional materials 6 00:00:12,690 --> 00:00:16,600 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,600 --> 00:00:17,266 at ocw.mit.edu. 8 00:00:21,130 --> 00:00:24,270 HONG LIU: OK, let us start. 9 00:00:24,270 --> 00:00:29,460 So last time, we finished discussing 10 00:00:29,460 --> 00:00:36,940 quantization of closed strings and open strings. 11 00:00:36,940 --> 00:00:47,190 And we see-- we saw that if you want to quantize strings 12 00:00:47,190 --> 00:00:51,970 consistently in Minkowski spacetime, 13 00:00:51,970 --> 00:00:53,460 then you need 26 dimensions. 14 00:00:56,380 --> 00:01:00,110 We also saw that when you quantize the photon-- when 15 00:01:00,110 --> 00:01:02,350 you quantize open strings, then you 16 00:01:02,350 --> 00:01:07,410 see a vector field, a massless vector field in the spectrum 17 00:01:07,410 --> 00:01:10,600 plus an infinite of [INAUDIBLE]. 18 00:01:10,600 --> 00:01:12,880 And when you quantize closed strings, 19 00:01:12,880 --> 00:01:16,920 then you see massless spin-2 particles, 20 00:01:16,920 --> 00:01:20,440 which is a graviton in your massless spectrum 21 00:01:20,440 --> 00:01:21,940 plus infinite number of [INAUDIBLE]. 22 00:01:26,010 --> 00:01:29,460 So that essentially is the essence of string theory, 23 00:01:29,460 --> 00:01:32,235 is that the strings contain graviton 24 00:01:32,235 --> 00:01:33,740 in a quantum mechanical way. 25 00:01:36,390 --> 00:01:38,970 So it should be consistent-- should be considered 26 00:01:38,970 --> 00:01:42,580 as a serial quantum gravity. 27 00:01:42,580 --> 00:01:44,730 So today, we start talking about another object 28 00:01:44,730 --> 00:01:46,510 in string theory. 29 00:01:46,510 --> 00:01:48,000 It's called the D-branes. 30 00:01:58,970 --> 00:02:00,190 OK? 31 00:02:00,190 --> 00:02:03,350 So we discussed open strings, so let's 32 00:02:03,350 --> 00:02:05,136 go back to the open string. 33 00:02:05,136 --> 00:02:10,500 For the open string, say after we fixed the [INAUDIBLE] 34 00:02:10,500 --> 00:02:14,537 to be flat, the [INAUDIBLE] action 35 00:02:14,537 --> 00:02:15,870 can be written as the following. 36 00:02:25,190 --> 00:02:27,920 OK? 37 00:02:27,920 --> 00:02:30,020 So you can-- when you get the-- when 38 00:02:30,020 --> 00:02:43,600 you obtain the equation of motion for x, 39 00:02:43,600 --> 00:02:46,830 you will have something like this. 40 00:02:49,350 --> 00:02:57,780 So you need to do an integration by parts 41 00:02:57,780 --> 00:03:02,130 plus a total derivative term, or plus a total derivative term 42 00:03:02,130 --> 00:03:07,160 until the derivative term we induce a term on the boundary 43 00:03:07,160 --> 00:03:08,285 if you have an open string. 44 00:03:21,931 --> 00:03:26,170 Say at sigma equal to 0 and pi. 45 00:03:26,170 --> 00:03:28,670 OK? 46 00:03:28,670 --> 00:03:35,280 So this is still d tau. 47 00:03:35,280 --> 00:03:44,230 And so from this term, you get equation of motion. 48 00:03:46,770 --> 00:03:50,406 And then you induce a boundary term, evaluate it. 49 00:03:50,406 --> 00:03:55,110 If it's open string, there's two boundaries, at 0 and pi. 50 00:03:55,110 --> 00:03:57,350 To do such a boundary term at the boundary, 51 00:03:57,350 --> 00:04:01,840 we need to also get rid of-- otherwise, 52 00:04:01,840 --> 00:04:05,240 the equation of motion is not satisfied. 53 00:04:05,240 --> 00:04:07,450 And what we discussed is [INAUDIBLE] 54 00:04:07,450 --> 00:04:09,850 Neumann boundary conditions. 55 00:04:09,850 --> 00:04:13,410 It's just taking the pi of sigma equal to 0, pi of sigma x 56 00:04:13,410 --> 00:04:15,190 mu to be 0. 57 00:04:15,190 --> 00:04:21,046 But in general, for any mu, say for any mu, 58 00:04:21,046 --> 00:04:24,150 you have a choice of four boundary conditions. 59 00:04:24,150 --> 00:04:28,120 You can choose delta x mu equal to 0, 60 00:04:28,120 --> 00:04:33,550 or partial sigma x mu equal to 0. 61 00:04:33,550 --> 00:04:38,790 Say and sigma equal to 0. 62 00:04:38,790 --> 00:04:42,030 And you have the same choices at sigma equal to pi. 63 00:04:44,600 --> 00:04:49,020 So altogether, you have actually four possible boundary 64 00:04:49,020 --> 00:04:51,535 conditions for each direction. 65 00:04:51,535 --> 00:04:54,430 So for each direction, you have four possible boundary 66 00:04:54,430 --> 00:04:55,070 conditions. 67 00:04:55,070 --> 00:04:56,861 If you can see the all possible directions, 68 00:04:56,861 --> 00:04:59,510 there are many, many different boundary conditions. 69 00:04:59,510 --> 00:05:00,431 OK? 70 00:05:00,431 --> 00:05:02,680 There are, in principle, many, many different boundary 71 00:05:02,680 --> 00:05:05,190 conditions. 72 00:05:05,190 --> 00:05:08,420 So let's now look at what the other boundary conditions mean. 73 00:05:08,420 --> 00:05:09,810 OK? 74 00:05:09,810 --> 00:05:12,600 We looked at the case which the-- you just imposed this 75 00:05:12,600 --> 00:05:17,080 everywhere, at both sigma equal to 0 and sigma equal to pi, 76 00:05:17,080 --> 00:05:18,490 and the four directions. 77 00:05:18,490 --> 00:05:21,750 But now let us explore these more general boundary 78 00:05:21,750 --> 00:05:24,430 conditions. 79 00:05:24,430 --> 00:05:30,441 First let's just quickly look at what this boundary-- so this is 80 00:05:30,441 --> 00:05:31,940 called Dirichlet boundary condition, 81 00:05:31,940 --> 00:05:33,537 this is called the Neumann. 82 00:05:33,537 --> 00:05:36,120 So first let's look at what does this boundary condition mean. 83 00:05:36,120 --> 00:05:36,619 OK? 84 00:05:39,950 --> 00:05:42,420 So if we have-- say let's consider some direction, 85 00:05:42,420 --> 00:05:45,740 i, set delta xi equal to 0. 86 00:05:49,270 --> 00:05:55,200 Say at sigma equal to 0, say tau equal to 0. 87 00:05:55,200 --> 00:05:58,310 So this is the example of this Dirichlet boundary 88 00:05:58,310 --> 00:06:00,600 condition in the i direction. 89 00:06:00,600 --> 00:06:07,040 Say suppose that sigma equal to 0, you have-- you impose this. 90 00:06:07,040 --> 00:06:13,170 So what this means is that the vibration of xi at sigma 91 00:06:13,170 --> 00:06:16,030 equal to 0 at all time does not change. 92 00:06:16,030 --> 00:06:20,080 So tau is like a parameter I should watch it-- time, 93 00:06:20,080 --> 00:06:22,290 the watch it motion. 94 00:06:22,290 --> 00:06:26,900 So this tells you that, at all time, the value of the xi 95 00:06:26,900 --> 00:06:29,730 does not change at the endpoint. 96 00:06:29,730 --> 00:06:33,500 So what this means, this directly 97 00:06:33,500 --> 00:06:38,730 translates into xi sigma equal to 0, 98 00:06:38,730 --> 00:06:42,040 and tau should be a constant. 99 00:06:42,040 --> 00:06:43,036 OK? 100 00:06:43,036 --> 00:06:45,690 Should be a constant. 101 00:06:45,690 --> 00:06:47,630 So this is the Dirichlet condition. 102 00:06:52,460 --> 00:06:54,140 And so let's look at some examples 103 00:06:54,140 --> 00:06:58,960 to understand a little bit more explicitly 104 00:06:58,960 --> 00:07:01,900 what this kind of condition-- want 105 00:07:01,900 --> 00:07:04,080 this Dirichlet condition means. 106 00:07:04,080 --> 00:07:05,360 OK? 107 00:07:05,360 --> 00:07:08,010 So let's first consider the example. 108 00:07:08,010 --> 00:07:18,250 Say suppose in all directions, say from 0 to d minus 2, 109 00:07:18,250 --> 00:07:29,630 OK, we use-- suppose use Neumann boundary condition 110 00:07:29,630 --> 00:07:37,680 as what we did before for both sigma equal to 0 and pi. 111 00:07:37,680 --> 00:07:40,865 But for the last coordinate, [INAUDIBLE] 112 00:07:40,865 --> 00:07:48,890 xt minus 1, so let's impose the Dirichlet boundary condition. 113 00:07:48,890 --> 00:07:54,970 So let's take it to be A at sigma-- both sigma 114 00:07:54,970 --> 00:07:55,750 and 0 equal to pi. 115 00:07:55,750 --> 00:07:56,250 OK? 116 00:07:56,250 --> 00:07:58,470 So for the moment, let's take the boundary condition 117 00:07:58,470 --> 00:08:03,830 to be same at both sigma equal to 0 and pi. 118 00:08:03,830 --> 00:08:07,020 So let's look at what did this condition means. 119 00:08:07,020 --> 00:08:08,465 Let's look at this example. 120 00:08:12,980 --> 00:08:16,710 So what this condition means-- so let's draw a figure. 121 00:08:22,490 --> 00:08:24,250 So, I cannot draw many directions. 122 00:08:24,250 --> 00:08:28,250 In particular, I cannot draw 6x26 dimensions. 123 00:08:28,250 --> 00:08:31,895 So let me call the horizontal axis as all the-- 124 00:08:31,895 --> 00:08:37,240 the connection of all the dimensions up to d minus 2. 125 00:08:37,240 --> 00:08:41,419 And under this axis is the d minus 1. 126 00:08:41,419 --> 00:08:41,918 OK? 127 00:08:46,130 --> 00:08:52,750 Then the endpoint, the x d minus 1 equal to a at the endpoint-- 128 00:08:52,750 --> 00:08:56,870 so suppose that this is the line of x equal to a. 129 00:09:02,890 --> 00:09:07,955 So suppose this is the hypersurface for x equals-- x d 130 00:09:07,955 --> 00:09:12,820 minus 1 equal to a, then that boundary condition tells you 131 00:09:12,820 --> 00:09:16,890 that whatever open string the motion, 132 00:09:16,890 --> 00:09:21,580 the endpoint must be restricted to this hypersurface. 133 00:09:21,580 --> 00:09:22,182 OK? 134 00:09:22,182 --> 00:09:23,890 So that's what this-- geometrically, what 135 00:09:23,890 --> 00:09:26,130 this condition means. 136 00:09:26,130 --> 00:09:28,810 It just says, let's consider xt minus 1 137 00:09:28,810 --> 00:09:32,140 equal to a, which is a hypersurface in spacetime, 138 00:09:32,140 --> 00:09:39,760 and then this condition means that the endpoint of the string 139 00:09:39,760 --> 00:09:42,660 must be restricted to this surface. 140 00:09:42,660 --> 00:09:44,180 The interior of the string can be 141 00:09:44,180 --> 00:09:47,680 any-- can be out-- can have no restriction, 142 00:09:47,680 --> 00:09:50,491 but the endpoint have to be on this surface. 143 00:09:50,491 --> 00:09:50,990 OK? 144 00:09:50,990 --> 00:09:51,573 Is this clear? 145 00:09:55,020 --> 00:09:55,670 OK. 146 00:09:55,670 --> 00:09:57,990 So you can generalize. 147 00:09:57,990 --> 00:09:59,410 So this is the first example. 148 00:09:59,410 --> 00:10:00,300 You can generalize. 149 00:10:00,300 --> 00:10:06,262 Let's consider x from 0 to some value, p. 150 00:10:06,262 --> 00:10:08,190 Of course, p is smaller than the total number 151 00:10:08,190 --> 00:10:15,120 of dimensions, which is n for sigma at 0 and the pi. 152 00:10:15,120 --> 00:10:28,170 And then the rest-- say xp plus 1, xp plus 2, to xp minus 1, 153 00:10:28,170 --> 00:10:34,600 these directions we impose the Dirichlet boundary conditions. 154 00:10:34,600 --> 00:10:39,580 And then let me call this one back to a. 155 00:10:39,580 --> 00:10:40,790 So I'm back to a. 156 00:10:40,790 --> 00:10:44,410 For both sigma equal to 0 and pi. 157 00:10:44,410 --> 00:10:44,940 OK? 158 00:10:44,940 --> 00:10:47,460 So a is just some constant vector of the same dimension 159 00:10:47,460 --> 00:10:50,770 as the number of x here. 160 00:10:50,770 --> 00:10:53,270 The same as the number of x here. 161 00:10:53,270 --> 00:10:55,660 And this is just a more general way 162 00:10:55,660 --> 00:10:57,980 to impose this to more directions. 163 00:10:57,980 --> 00:10:58,480 OK? 164 00:11:03,150 --> 00:11:06,280 Is this notation clear? 165 00:11:06,280 --> 00:11:06,780 Yes? 166 00:11:06,780 --> 00:11:09,750 AUDIENCE: Why is there a repeated superscript? 167 00:11:09,750 --> 00:11:11,370 HONG LIU: Oh, sorry. 168 00:11:11,370 --> 00:11:12,680 It's p plus 2. 169 00:11:17,300 --> 00:11:17,806 Yes? 170 00:11:17,806 --> 00:11:19,180 AUDIENCE: Maybe do we talk about, 171 00:11:19,180 --> 00:11:21,431 like, a different 0 pi behaviors, 172 00:11:21,431 --> 00:11:22,555 or is that not interesting? 173 00:11:22,555 --> 00:11:24,138 HONG LIU: No, we will talk about that. 174 00:11:24,138 --> 00:11:26,200 Let's look at the simple case first. 175 00:11:30,310 --> 00:11:32,680 So is this notation clear? 176 00:11:32,680 --> 00:11:38,660 So again, I can do this on the similar plots like this. 177 00:11:38,660 --> 00:11:45,140 So now horizontal axis [INAUDIBLE] x 0 to p, 178 00:11:45,140 --> 00:11:49,750 and the vertical axis I do p plus 1 179 00:11:49,750 --> 00:11:51,850 all the way to d minus 1. 180 00:11:51,850 --> 00:11:55,270 So now those axes are multiple dimensional. 181 00:11:55,270 --> 00:12:03,550 And then this specifies-- so this a 182 00:12:03,550 --> 00:12:14,480 specifies a p plus 1 dimensional hypersurface-- surface. 183 00:12:21,000 --> 00:12:26,010 So the value of those coordinates are fixed, 184 00:12:26,010 --> 00:12:30,550 but the value of those coordinates can be arbitrary. 185 00:12:30,550 --> 00:12:33,816 So this p plus 1 free parameter here. 186 00:12:33,816 --> 00:12:35,040 OK? 187 00:12:35,040 --> 00:12:38,330 And so this is a p plus 1 dimensional surface 188 00:12:38,330 --> 00:12:40,130 in the spacetime. 189 00:12:40,130 --> 00:12:44,870 And this condition means that the endpoint 190 00:12:44,870 --> 00:12:49,180 of whatever open string must be restricted to this surface. 191 00:12:55,130 --> 00:12:55,630 OK? 192 00:13:01,150 --> 00:13:02,260 Is the picture clear? 193 00:13:13,360 --> 00:13:14,190 Yes? 194 00:13:14,190 --> 00:13:17,340 AUDIENCE: [INAUDIBLE] 195 00:13:17,340 --> 00:13:18,105 HONG LIU: Hmm? 196 00:13:18,105 --> 00:13:18,680 AUDIENCE: [INAUDIBLE] 197 00:13:18,680 --> 00:13:19,390 HONG LIU: Yeah. 198 00:13:19,390 --> 00:13:21,540 We are talking about how to impose 199 00:13:21,540 --> 00:13:24,450 the-- what's the meaning of imposing the Dirichlet boundary 200 00:13:24,450 --> 00:13:26,650 condition. 201 00:13:26,650 --> 00:13:29,340 Instead of Neumann boundary condition. 202 00:13:29,340 --> 00:13:31,530 So we now impose the Dirichlet boundary condition, 203 00:13:31,530 --> 00:13:34,300 which goes one into the [INAUDIBLE] of the endpoint 204 00:13:34,300 --> 00:13:36,440 should be 0. 205 00:13:36,440 --> 00:13:39,280 And then this can be specified by the endpoint 206 00:13:39,280 --> 00:13:41,490 if you take some constant value. 207 00:13:41,490 --> 00:13:45,120 And so that's the geometric way to think 208 00:13:45,120 --> 00:13:48,015 about these different Dirichlet boundary conditions. 209 00:13:53,720 --> 00:13:59,230 So if you look at this naively, so previously, when 210 00:13:59,230 --> 00:14:04,630 we can see the Neumann boundary condition for all directions, 211 00:14:04,630 --> 00:14:09,720 then the string-- then there's no restriction to the string. 212 00:14:09,720 --> 00:14:12,360 Then the string can move anywhere. 213 00:14:12,360 --> 00:14:14,760 So now when you-- when in some directions 214 00:14:14,760 --> 00:14:17,220 we impose the Dirichlet condition, 215 00:14:17,220 --> 00:14:20,590 then we see that-- then the motion of the string 216 00:14:20,590 --> 00:14:23,330 get restricted, and that this endpoint now 217 00:14:23,330 --> 00:14:27,410 have to [INAUDIBLE] some soft dimension of the spacetime. 218 00:14:27,410 --> 00:14:29,500 OK? 219 00:14:29,500 --> 00:14:32,390 So at first sight, this may seem to be a nonsensical thing 220 00:14:32,390 --> 00:14:34,080 to do. 221 00:14:34,080 --> 00:14:37,640 Just say-- you ask why somehow I have 222 00:14:37,640 --> 00:14:42,050 to restrict some open strings to some sub-manifold 223 00:14:42,050 --> 00:14:44,430 in the spacetime. 224 00:14:44,430 --> 00:14:46,310 So why should I do that? 225 00:14:46,310 --> 00:14:49,167 In particular, that breaks the Lorentz symmetry. 226 00:14:49,167 --> 00:14:50,750 In particular, that breaks Lorentzian. 227 00:14:50,750 --> 00:14:55,080 So the nice thing about the Neumann boundary condition 228 00:14:55,080 --> 00:14:58,160 we did before, in more directions, so that 229 00:14:58,160 --> 00:15:00,520 respects Lorentz symmetry. 230 00:15:00,520 --> 00:15:03,660 And once we introduce some directions to the Dirichlet 231 00:15:03,660 --> 00:15:05,520 boundary condition, you necessarily 232 00:15:05,520 --> 00:15:12,080 break Lorentz symmetry, because we select out some directions 233 00:15:12,080 --> 00:15:13,320 with some others. 234 00:15:13,320 --> 00:15:14,770 OK? 235 00:15:14,770 --> 00:15:17,620 And so you necessarily break Lorentz symmetry. 236 00:15:17,620 --> 00:15:22,520 You also break translation symmetry because of the-- you 237 00:15:22,520 --> 00:15:24,390 have to specify some sub-manifold. 238 00:15:24,390 --> 00:15:28,500 You also break the translation symmetry. 239 00:15:28,500 --> 00:15:32,420 So at first sight, this seems to be a nonsensical thing to do. 240 00:15:32,420 --> 00:15:35,080 So that's why, for many years, actually people did not 241 00:15:35,080 --> 00:15:42,170 look at this object, even though for a person 242 00:15:42,170 --> 00:15:46,990 to work out string theory on the first day, 243 00:15:46,990 --> 00:15:50,354 you should already realize those are the boundary conditions. 244 00:15:50,354 --> 00:15:51,770 But for many years, people did not 245 00:15:51,770 --> 00:15:54,582 pay attention to this kind of boundary conditions. 246 00:15:54,582 --> 00:15:59,680 It turns out that standard of conditions 247 00:15:59,680 --> 00:16:03,410 plays a key role in string theory. 248 00:16:03,410 --> 00:16:09,640 And the key is to take a slight different perspective 249 00:16:09,640 --> 00:16:11,650 at [INAUDIBLE] sub-manifold. 250 00:16:11,650 --> 00:16:13,910 OK? 251 00:16:13,910 --> 00:16:16,070 So now what you should think about 252 00:16:16,070 --> 00:16:20,490 is-- and now what you think about is the following. 253 00:16:23,170 --> 00:16:28,350 So you think about maybe for the whole spacetime, 254 00:16:28,350 --> 00:16:32,200 the whole spacetime, only closed string 255 00:16:32,200 --> 00:16:35,525 is the most natural object, because it can move anywhere. 256 00:16:38,810 --> 00:16:42,930 But then there's some special places 257 00:16:42,930 --> 00:16:47,860 in the spacetime, some kind of defect. 258 00:16:47,860 --> 00:16:50,690 And at those defect, the closed string 259 00:16:50,690 --> 00:16:53,490 can break to become open string. 260 00:16:53,490 --> 00:16:54,640 OK? 261 00:16:54,640 --> 00:16:56,840 Can become open string. 262 00:16:56,840 --> 00:16:59,017 And if you take that perspective, 263 00:16:59,017 --> 00:17:00,850 then that gives special meaning to such kind 264 00:17:00,850 --> 00:17:03,950 of special sub-manifold which open string can end, 265 00:17:03,950 --> 00:17:07,000 then there's some kind of object there. 266 00:17:07,000 --> 00:17:10,119 Because of the existence of certain object there, 267 00:17:10,119 --> 00:17:15,450 then the open string can end there. 268 00:17:15,450 --> 00:17:16,290 OK? 269 00:17:16,290 --> 00:17:16,834 Yes. 270 00:17:16,834 --> 00:17:19,459 AUDIENCE: So indeed, if you have that perspective, then instead 271 00:17:19,459 --> 00:17:22,530 of having it lie along hyperplanes, 272 00:17:22,530 --> 00:17:24,200 is it more natural to have it just lie 273 00:17:24,200 --> 00:17:25,241 upon some other manifold? 274 00:17:25,241 --> 00:17:27,047 HONG LIU: Oh, you can-- yeah, I'm 275 00:17:27,047 --> 00:17:28,630 just taking you-- I'm just sending you 276 00:17:28,630 --> 00:17:29,420 the simplest example. 277 00:17:29,420 --> 00:17:30,294 AUDIENCE: Sure, sure. 278 00:17:30,294 --> 00:17:32,960 HONG LIU: And once you realize in this simple example, 279 00:17:32,960 --> 00:17:35,160 then you can generalize the [INAUDIBLE] 280 00:17:35,160 --> 00:17:37,890 complicated scenario you can imagine. 281 00:17:37,890 --> 00:17:43,320 Now let's just think about, in the simplest case, imposing 282 00:17:43,320 --> 00:17:47,220 those boundary conditions become natural 283 00:17:47,220 --> 00:17:52,640 if you think those places in spacetime 284 00:17:52,640 --> 00:17:57,380 are special places which something can happen. 285 00:17:57,380 --> 00:18:00,710 It's not the same as other places. 286 00:18:00,710 --> 00:18:05,390 And to say those [INAUDIBLE] give a name. 287 00:18:05,390 --> 00:18:11,250 So you imagine-- so for example, you 288 00:18:11,250 --> 00:18:36,600 can imagine some spacetime defect sitting at, say, 289 00:18:36,600 --> 00:18:46,425 this xp plus 1, x equal to a, sitting at that location. 290 00:18:50,690 --> 00:18:52,170 So where open string can end. 291 00:19:03,920 --> 00:19:06,590 So such a defect, we'll call it D-brane. 292 00:19:11,895 --> 00:19:12,520 We give a name. 293 00:19:23,360 --> 00:19:25,730 So D is clear because it's Dirichlet boundary 294 00:19:25,730 --> 00:19:28,820 condition, and the brane because it's some extended surface. 295 00:19:28,820 --> 00:19:29,320 OK? 296 00:19:35,110 --> 00:19:39,080 And so we often use a notation to say Dp-brane. 297 00:19:42,790 --> 00:19:46,550 So p denotes the [INAUDIBLE] spatial dimensions 298 00:19:46,550 --> 00:19:47,750 along the brane. 299 00:19:47,750 --> 00:19:48,580 OK? 300 00:19:48,580 --> 00:19:51,490 So in this case, there's p spatial dimensions 301 00:19:51,490 --> 00:20:00,360 along the plane-- along this brane, so we call it Dp-brane. 302 00:20:00,360 --> 00:20:00,860 OK? 303 00:20:09,110 --> 00:20:17,330 So right now, Dp-brane is merely just a place 304 00:20:17,330 --> 00:20:19,620 which impose a boundary conditions which 305 00:20:19,620 --> 00:20:24,750 open strings can end. 306 00:20:24,750 --> 00:20:28,610 But we will soon see that actually a Dp-brane 307 00:20:28,610 --> 00:20:32,900 is a dynamic object, and it's actually a dynamic object. 308 00:20:32,900 --> 00:20:36,770 And this actually can move, can fluctuate, can jump around, 309 00:20:36,770 --> 00:20:37,730 et cetera. 310 00:20:37,730 --> 00:20:39,790 OK? 311 00:20:39,790 --> 00:20:44,070 And so also from this perspective, 312 00:20:44,070 --> 00:20:46,295 our previous Neumann boundary condition, 313 00:20:46,295 --> 00:20:50,950 if we have Neumann in all direction, 314 00:20:50,950 --> 00:20:59,060 so if we have Neumann boundary condition in all directions, 315 00:20:59,060 --> 00:21:03,925 then in this case the open string can end everywhere. 316 00:21:10,229 --> 00:21:10,770 End anywhere. 317 00:21:18,250 --> 00:21:27,690 From this new perspective, we call this space-filling brane. 318 00:21:27,690 --> 00:21:30,620 We call this actually a space-filling brane. 319 00:21:34,241 --> 00:21:34,740 OK? 320 00:21:37,950 --> 00:21:40,610 So the Dirichlet boundary-- the Neumann boundary condition 321 00:21:40,610 --> 00:21:43,530 we considered before should be considered as actually you 322 00:21:43,530 --> 00:21:46,040 fill the whole spacetime with such a brane, 323 00:21:46,040 --> 00:21:50,219 and then the string can, in principle, move anywhere. 324 00:21:50,219 --> 00:21:51,760 So for example, in the bosonic string 325 00:21:51,760 --> 00:21:57,150 we talked about before, this group because one and two 326 00:21:57,150 --> 00:22:02,440 are D 25 brane, so D 25 brane because there's 327 00:22:02,440 --> 00:22:04,980 25 spatial dimensions. 328 00:22:04,980 --> 00:22:06,750 OK? 329 00:22:06,750 --> 00:22:08,920 Such a brane when you impose the Neumann boundary 330 00:22:08,920 --> 00:22:10,800 condition in all directions. 331 00:22:10,800 --> 00:22:14,050 Or if you are in the super string, which 332 00:22:14,050 --> 00:22:18,480 the total number of spacetime is 10, 333 00:22:18,480 --> 00:22:21,740 and then the spatial number-- spatial dimension is nine, 334 00:22:21,740 --> 00:22:24,560 then you have D9-- then these other D9 branes. 335 00:22:36,810 --> 00:22:37,893 Any questions about this? 336 00:22:48,320 --> 00:22:48,820 Yes? 337 00:22:48,820 --> 00:22:51,634 AUDIENCE: [INAUDIBLE] boundary conditions, 338 00:22:51,634 --> 00:22:54,448 they're imposed at the quantum level, right? 339 00:22:54,448 --> 00:22:57,116 It's not just that the classical solution has 340 00:22:57,116 --> 00:23:00,044 to have its endpoints on this D-brane, 341 00:23:00,044 --> 00:23:02,484 but it's the fluctuations around it [INAUDIBLE] 342 00:23:02,484 --> 00:23:03,894 solution that we consider? 343 00:23:03,894 --> 00:23:05,310 HONG LIU: We will talk about that. 344 00:23:05,310 --> 00:23:06,375 We will talk about that. 345 00:23:14,137 --> 00:23:14,970 Any other questions? 346 00:23:19,160 --> 00:23:21,390 Good. 347 00:23:21,390 --> 00:23:23,080 So let me make a note. 348 00:23:25,880 --> 00:23:28,620 You can, in principle, impose the Dirichlet boundary 349 00:23:28,620 --> 00:23:33,600 condition in any direction you like, except 350 00:23:33,600 --> 00:23:34,650 in the x0 direction. 351 00:23:47,660 --> 00:23:49,010 OK? 352 00:23:49,010 --> 00:23:51,290 It's because x0 cannot be a constant. 353 00:23:51,290 --> 00:23:56,620 x0 have-- so x0 is a time, is our real time 354 00:23:56,620 --> 00:23:58,690 in the Minkowski space. 355 00:23:58,690 --> 00:24:03,080 So the string moves in this Minkowski spacetime, 356 00:24:03,080 --> 00:24:06,240 and this x0 is that time. 357 00:24:06,240 --> 00:24:08,420 And the time cannot stand still. 358 00:24:08,420 --> 00:24:09,310 OK? 359 00:24:09,310 --> 00:24:13,610 And in order to have this, you needed that location 360 00:24:13,610 --> 00:24:15,330 to stand still. 361 00:24:15,330 --> 00:24:18,300 And the time cannot stand still. 362 00:24:18,300 --> 00:24:20,620 So this cannot be a constant. 363 00:24:24,480 --> 00:24:31,130 So you cannot impose the Dirichlet boundary condition 364 00:24:31,130 --> 00:24:33,310 in the time direction. 365 00:24:33,310 --> 00:24:37,690 So the smallest in the Minkowski spacetime, the smallest 366 00:24:37,690 --> 00:24:43,170 would be at D0 brane, which we impose the Dirichlet boundary 367 00:24:43,170 --> 00:24:48,080 condition in all spatial directions except in time. 368 00:24:48,080 --> 00:24:51,120 In time, you always impose the Neumann. 369 00:24:51,120 --> 00:24:54,060 Then the smallest situation will be called 370 00:24:54,060 --> 00:24:58,770 D0 brane, which is a point. 371 00:24:58,770 --> 00:25:01,275 And then the open string can end on this point. 372 00:25:04,448 --> 00:25:04,948 OK? 373 00:25:16,390 --> 00:25:16,890 Yes? 374 00:25:16,890 --> 00:25:18,100 AUDIENCE: So I have question. 375 00:25:18,100 --> 00:25:23,255 So this idea about calling the Neumann boundary conditions 376 00:25:23,255 --> 00:25:26,060 to D25 branes or D9 branes. 377 00:25:26,060 --> 00:25:28,030 So this is a little bit perplexing, 378 00:25:28,030 --> 00:25:31,786 because it seems like not imposing Dirichlet boundary 379 00:25:31,786 --> 00:25:34,841 conditions, this implies that not imposing Dirichlet boundary 380 00:25:34,841 --> 00:25:37,708 conditions is equivalent to applying Neumann boundary 381 00:25:37,708 --> 00:25:40,230 conditions, which is not-- they're not, like-- I mean, 382 00:25:40,230 --> 00:25:41,840 they're like mutually exclusive. 383 00:25:41,840 --> 00:25:43,520 It's not like-- so I mean, I don't 384 00:25:43,520 --> 00:25:47,090 understand why it is that-- so does this thing behave 385 00:25:47,090 --> 00:25:48,350 like a 24 brane? 386 00:25:48,350 --> 00:25:50,934 Like, does it have the same properties as, like, a 24 brane 387 00:25:50,934 --> 00:25:51,892 or something like that? 388 00:25:51,892 --> 00:25:53,550 HONG LIU: Sorry, which? 389 00:25:53,550 --> 00:25:54,300 You mean 25 brane. 390 00:25:54,300 --> 00:25:57,254 AUDIENCE: So what I'm saying is that it seems that the Neumann 391 00:25:57,254 --> 00:25:59,170 boundary conditions have a different character 392 00:25:59,170 --> 00:26:00,770 than-- like a different flavor of how 393 00:26:00,770 --> 00:26:02,935 they work than the Dirichlet boundary conditions. 394 00:26:02,935 --> 00:26:05,060 So it just seems strange to me that you would still 395 00:26:05,060 --> 00:26:07,855 call it a brane, like a D 25 brane when you impose them 396 00:26:07,855 --> 00:26:08,592 or not. 397 00:26:08,592 --> 00:26:10,730 HONG LIU: No. 398 00:26:10,730 --> 00:26:16,340 So in those cases, in those cases-- for example, 399 00:26:16,340 --> 00:26:19,710 in this case, which we Neumann boundary conditions in all 400 00:26:19,710 --> 00:26:22,420 the other directions except one, then 401 00:26:22,420 --> 00:26:24,790 this is a 24 dimensional-- then this 402 00:26:24,790 --> 00:26:30,110 is a 25 dimensional manifold which the string can end on. 403 00:26:30,110 --> 00:26:31,889 And I just-- 404 00:26:31,889 --> 00:26:33,180 AUDIENCE: Oh, I see, of course. 405 00:26:33,180 --> 00:26:33,450 So, right. 406 00:26:33,450 --> 00:26:35,560 You have to apply the Neumann boundary conditions to all 407 00:26:35,560 --> 00:26:36,643 of the other [INAUDIBLE]. 408 00:26:36,643 --> 00:26:37,590 OK, I understand. 409 00:26:37,590 --> 00:26:37,820 Yeah. 410 00:26:37,820 --> 00:26:38,945 HONG LIU: Yeah, yeah, yeah. 411 00:26:38,945 --> 00:26:40,427 Yeah, so it's-- yeah. 412 00:26:40,427 --> 00:26:41,260 Any other questions? 413 00:26:47,850 --> 00:26:50,890 OK, good. 414 00:26:50,890 --> 00:26:53,000 So now let's consider more complicated-- 415 00:26:53,000 --> 00:26:56,150 slightly more complicated boundary conditions. 416 00:26:56,150 --> 00:27:00,180 So by introducing this D-brane, now you 417 00:27:00,180 --> 00:27:06,240 can actually have all different possible boundary conditions. 418 00:27:06,240 --> 00:27:08,980 As we said here, there-- for each direction, 419 00:27:08,980 --> 00:27:13,150 you can have in principle four possible choices. 420 00:27:13,150 --> 00:27:15,997 And then if you have [INAUDIBLE], 421 00:27:15,997 --> 00:27:17,330 so for time direction you don't. 422 00:27:17,330 --> 00:27:20,840 But for the 25 dimensions, 25 spatial dimensions. 423 00:27:20,840 --> 00:27:23,100 And each dimension, you have four choices, 424 00:27:23,100 --> 00:27:26,450 and you have four to the power 25 possible choices 425 00:27:26,450 --> 00:27:28,690 of different boundary conditions. 426 00:27:28,690 --> 00:27:31,810 Now let's talk about the other boundary conditions. 427 00:27:31,810 --> 00:27:35,680 So by using the picture of D-brane, 428 00:27:35,680 --> 00:27:38,610 we can talk about them all. 429 00:27:38,610 --> 00:27:39,835 We can talk about them all. 430 00:27:44,350 --> 00:27:51,630 So then let's consider model 1 branes. 431 00:28:02,350 --> 00:28:03,730 OK? 432 00:28:03,730 --> 00:28:07,960 So let's consider the situation we have two such Dp-brane. 433 00:28:14,080 --> 00:28:19,730 So this direction is still x1-- x01 to p. 434 00:28:19,730 --> 00:28:26,040 And then this is all the other directions. 435 00:28:26,040 --> 00:28:34,900 So suppose we have a brane at a and another brane at b, 436 00:28:34,900 --> 00:28:37,645 at the vector specified by b. 437 00:28:40,590 --> 00:28:42,230 OK? 438 00:28:42,230 --> 00:28:46,880 So now in this case, we can have four types of open strings. 439 00:28:59,360 --> 00:29:01,620 We can have four types of open strings. 440 00:29:01,620 --> 00:29:06,080 For example, I can have an open string which ends of b. 441 00:29:06,080 --> 00:29:09,860 I'm going to have an open string end on a. 442 00:29:09,860 --> 00:29:13,110 But then I can also have an open string 443 00:29:13,110 --> 00:29:17,350 which goes from b to a, and for open string from a to b. 444 00:29:24,710 --> 00:29:26,280 OK? 445 00:29:26,280 --> 00:29:28,430 So in this case, the boundary condition 446 00:29:28,430 --> 00:29:34,400 would be the x01 to p is still a Neumann boundary 447 00:29:34,400 --> 00:29:40,610 condition for both sigma equal to 0 and sigma equal to pi. 448 00:29:43,520 --> 00:29:53,040 But for the other directions, you 449 00:29:53,040 --> 00:29:58,120 can have either-- your sigma 0 can either be in a or b. 450 00:30:01,560 --> 00:30:04,525 Or sigma, you could do pi can either be a or b. 451 00:30:11,280 --> 00:30:13,550 So that gives you the four types of strings. 452 00:30:17,190 --> 00:30:18,297 OK? 453 00:30:18,297 --> 00:30:20,255 That gives you four types of different strings. 454 00:30:25,700 --> 00:30:30,786 Now you also can see the branes of different dimensions. 455 00:30:30,786 --> 00:30:32,660 So now you can see the branes of [INAUDIBLE]. 456 00:30:52,930 --> 00:30:57,440 So for example, say suppose let's think 457 00:30:57,440 --> 00:31:02,261 of the two directions, D minus 2 and D minus 1. 458 00:31:02,261 --> 00:31:04,010 And let's suppose all the other directions 459 00:31:04,010 --> 00:31:06,600 are in the Neumann direction. 460 00:31:06,600 --> 00:31:08,890 So this can see the two kinds of branes. 461 00:31:08,890 --> 00:31:13,960 One is extended in the D minus 2 direction [INAUDIBLE] 462 00:31:13,960 --> 00:31:17,261 some value of D minus 1. 463 00:31:17,261 --> 00:31:18,870 And then you can see another brane, 464 00:31:18,870 --> 00:31:23,440 which is localized in both D minus 1 and D minus 2. 465 00:31:23,440 --> 00:31:23,940 OK? 466 00:31:33,100 --> 00:31:37,390 Then you can again have four types of strings. 467 00:31:37,390 --> 00:31:43,800 [INAUDIBLE] and the ending between these two. 468 00:31:47,697 --> 00:31:49,530 So again, you can see the [INAUDIBLE] string 469 00:31:49,530 --> 00:31:51,279 because we can see the [INAUDIBLE] string. 470 00:31:51,279 --> 00:31:54,630 So you can again see the string like that. 471 00:31:57,430 --> 00:32:03,520 So let me call this brane 1, this brane 2. 472 00:32:03,520 --> 00:32:07,210 So let's consider the 1-2 string. 473 00:32:07,210 --> 00:32:09,150 It reaches from 1 to 2. 474 00:32:09,150 --> 00:32:13,040 Then the 1-2 string, with the corresponding to the boundary 475 00:32:13,040 --> 00:32:22,670 condition, say at D minus 2, say at sigma equal to 0, 476 00:32:22,670 --> 00:32:30,450 you impose the Neumann boundary condition 477 00:32:30,450 --> 00:32:36,660 because it can move freely in the D minus 2 direction. 478 00:32:36,660 --> 00:32:44,140 And but in the-- at sigma equal to pi, then 479 00:32:44,140 --> 00:32:46,280 this should be fixed at some value. 480 00:32:46,280 --> 00:32:50,115 So let's suppose, say, this is a 1, b1. 481 00:32:52,790 --> 00:32:55,040 Then add to that, we are fixed at a1. 482 00:32:57,321 --> 00:32:57,820 OK? 483 00:32:57,820 --> 00:32:59,030 So this is 1-2 string. 484 00:32:59,030 --> 00:33:01,907 Then you can have one end which is a Neumann boundary 485 00:33:01,907 --> 00:33:04,240 condition, but the other end which is Dirichlet boundary 486 00:33:04,240 --> 00:33:04,740 condition. 487 00:33:07,400 --> 00:33:07,900 OK? 488 00:33:07,900 --> 00:33:09,280 Is this clear to you? 489 00:33:09,280 --> 00:33:13,870 And then for this 2-1 string that will be opposite, 490 00:33:13,870 --> 00:33:16,870 for the 2-1 string for the D minus 2 direction at sigma 491 00:33:16,870 --> 00:33:19,640 equal to 0 will be the Dirichlet condition, 492 00:33:19,640 --> 00:33:22,070 but at sigma equal to pi would be the Neumann condition. 493 00:33:36,300 --> 00:33:38,940 Any questions about this? 494 00:33:38,940 --> 00:33:41,630 So these are pretty simple. 495 00:33:41,630 --> 00:33:46,770 You just need to get used to it and spend a couple minutes 496 00:33:46,770 --> 00:33:48,200 drawing a couple of diagrams. 497 00:33:48,200 --> 00:33:50,450 Then you will be an expert. 498 00:33:50,450 --> 00:33:51,877 You will be an expert. 499 00:33:51,877 --> 00:33:52,960 This is pretty elementary. 500 00:33:56,647 --> 00:33:57,480 So important remark. 501 00:34:04,580 --> 00:34:12,760 So a Dp-brane, as we said before, 502 00:34:12,760 --> 00:34:14,190 it breaks translation symmetry. 503 00:34:27,010 --> 00:34:29,239 Translation and the Lorentz symmetries. 504 00:34:40,032 --> 00:34:40,532 OK? 505 00:34:45,750 --> 00:34:54,570 Of the original Minkowski spacetime to your sub-group. 506 00:34:54,570 --> 00:34:58,630 So the subgroup is that you have-- still 507 00:34:58,630 --> 00:35:01,700 have translational symmetry and the Lorentz symmetry 508 00:35:01,700 --> 00:35:03,980 along the brane direction. 509 00:35:03,980 --> 00:35:06,080 So along the brane direction, you still 510 00:35:06,080 --> 00:35:08,330 can translate-- your string can still 511 00:35:08,330 --> 00:35:09,990 translate in that direction. 512 00:35:09,990 --> 00:35:13,350 And there's still Lorentz symmetry in that direction. 513 00:35:13,350 --> 00:35:18,000 So there's still Poincare symmetry in 1,p-- 1 comma p-- 514 00:35:18,000 --> 00:35:22,620 so in the time and the p spatial dimensions along the brane. 515 00:35:22,620 --> 00:35:25,890 And then you have rotational symmetries in the direction 516 00:35:25,890 --> 00:35:27,570 transverse to the brane. 517 00:35:27,570 --> 00:35:31,120 So that's D minus 1 minus p. 518 00:35:31,120 --> 00:35:36,550 So that's the rotation among those coordinates. 519 00:35:36,550 --> 00:35:38,050 So you can have the Lorentz symmetry 520 00:35:38,050 --> 00:35:40,660 along those directions, then the rotational symmetry 521 00:35:40,660 --> 00:35:41,751 is along those directions. 522 00:35:41,751 --> 00:35:42,250 OK? 523 00:35:42,250 --> 00:35:43,920 So this is for Dp-brane. 524 00:35:43,920 --> 00:35:47,495 So this is the remaining symmetry. 525 00:35:50,390 --> 00:35:54,590 So this is remaining symmetry say if you have 526 00:35:54,590 --> 00:35:58,275 a Dp-brane in the spacetime. 527 00:35:58,275 --> 00:36:01,850 Of course, if you have multiple Dp-brane, et cetera, 528 00:36:01,850 --> 00:36:05,480 then it's broken even more. 529 00:36:05,480 --> 00:36:05,980 Yes? 530 00:36:05,980 --> 00:36:11,980 AUDIENCE: [INAUDIBLE] branes are [INAUDIBLE] open not 531 00:36:11,980 --> 00:36:12,480 [INAUDIBLE]. 532 00:36:12,480 --> 00:36:13,440 HONG LIU: Oh. 533 00:36:13,440 --> 00:36:15,259 Yeah, we can talk-- here, I'm just 534 00:36:15,259 --> 00:36:16,800 talking about the simplest situation. 535 00:36:16,800 --> 00:36:18,890 I'm imposing the simplest boundary condition, just 536 00:36:18,890 --> 00:36:21,060 like these kind of boundary conditions. 537 00:36:21,060 --> 00:36:23,170 And then these kind of boundary conditions 538 00:36:23,170 --> 00:36:26,380 is-- applies to the infinite-- yeah, 539 00:36:26,380 --> 00:36:29,620 you can-- once later we talk about the D-brane can 540 00:36:29,620 --> 00:36:32,010 be dynamically excited, then you can see the complicated 541 00:36:32,010 --> 00:36:34,830 configurations. 542 00:36:34,830 --> 00:36:35,810 Yes? 543 00:36:35,810 --> 00:36:43,650 AUDIENCE: So [INAUDIBLE] boundary condition [INAUDIBLE]. 544 00:36:43,650 --> 00:36:44,630 HONG LIU: Yeah. 545 00:36:44,630 --> 00:36:59,392 AUDIENCE: [INAUDIBLE] x plus [INAUDIBLE]. 546 00:36:59,392 --> 00:37:00,615 HONG LIU: Ah. 547 00:37:00,615 --> 00:37:02,320 You are saying something like this-- 548 00:37:02,320 --> 00:37:05,060 you're saying some combination of this? 549 00:37:05,060 --> 00:37:07,360 AUDIENCE: [INAUDIBLE] x plus [INAUDIBLE]. 550 00:37:07,360 --> 00:37:08,920 HONG LIU: Yeah, yeah, yeah. 551 00:37:08,920 --> 00:37:11,540 That can happen. 552 00:37:11,540 --> 00:37:14,930 That does not happen in the straight Minkowski space, 553 00:37:14,930 --> 00:37:16,840 because in the straight Minkowski space 554 00:37:16,840 --> 00:37:18,360 this is the action. 555 00:37:18,360 --> 00:37:21,720 So you only have one or the other. 556 00:37:21,720 --> 00:37:25,510 But if you, say, have Minkowski but you 557 00:37:25,510 --> 00:37:30,070 turn on some electric field, then 558 00:37:30,070 --> 00:37:33,000 you can actually have such kind of mixed boundary conditions. 559 00:37:33,000 --> 00:37:33,500 Yeah. 560 00:37:33,500 --> 00:37:37,260 Just they exist-- it depends on the specific dynamical 561 00:37:37,260 --> 00:37:38,370 situation. 562 00:37:38,370 --> 00:37:42,179 So this is empty, flat Minkowski space, 563 00:37:42,179 --> 00:37:44,220 then you can only have those boundary conditions. 564 00:37:44,220 --> 00:37:45,648 AUDIENCE: And when [INAUDIBLE]? 565 00:37:50,884 --> 00:37:51,700 HONG LIU: Sorry? 566 00:37:51,700 --> 00:37:53,053 AUDIENCE: When [INAUDIBLE]. 567 00:37:55,759 --> 00:37:58,252 HONG LIU: That I'm not sure. 568 00:37:58,252 --> 00:38:00,640 That is different. 569 00:38:00,640 --> 00:38:04,720 Yeah, because that does not satisfy that boundary 570 00:38:04,720 --> 00:38:05,270 condition. 571 00:38:05,270 --> 00:38:08,912 And so we have a look at the specific situation. 572 00:38:08,912 --> 00:38:12,100 We have to look at the specific situation. 573 00:38:12,100 --> 00:38:13,830 Just for the flat Minkowski space, 574 00:38:13,830 --> 00:38:16,490 these are the only boundary conditions you can have. 575 00:38:16,490 --> 00:38:19,800 And we will not have-- we will not 576 00:38:19,800 --> 00:38:22,820 be able to do it in this course, but you 577 00:38:22,820 --> 00:38:26,942 can see the situation because of the-- in open string, 578 00:38:26,942 --> 00:38:28,400 you can have electromagnetic field. 579 00:38:28,400 --> 00:38:29,316 You can have a photon. 580 00:38:29,316 --> 00:38:32,410 So you can, in principle, turn on electromagnetic fields. 581 00:38:32,410 --> 00:38:34,199 And then we modify the boundary condition. 582 00:38:34,199 --> 00:38:36,407 Then you can have actually mixed boundary conditions. 583 00:38:39,536 --> 00:38:41,710 AUDIENCE: Do people ever consider 584 00:38:41,710 --> 00:38:45,960 this is a [INAUDIBLE] between two D-branes as a [INAUDIBLE] 585 00:38:45,960 --> 00:38:46,850 defect? 586 00:38:46,850 --> 00:38:48,700 Because [INAUDIBLE] between-- 587 00:38:48,700 --> 00:38:51,270 HONG LIU: You mean string between different-- string 588 00:38:51,270 --> 00:38:53,020 before-- between different D-branes. 589 00:38:53,020 --> 00:38:55,645 AUDIENCE: Yes. 590 00:38:55,645 --> 00:38:57,800 HONG LIU: You can see that the D-branes themselves 591 00:38:57,800 --> 00:39:01,400 have topological defect, rather than the string between them 592 00:39:01,400 --> 00:39:03,480 has the topological defect. 593 00:39:03,480 --> 00:39:06,300 The D-branes themselves are defects. 594 00:39:06,300 --> 00:39:10,340 Strings are just-- as we will talk about-- we will talk about 595 00:39:10,340 --> 00:39:13,785 it now, the strings on the D-brane, 596 00:39:13,785 --> 00:39:15,660 they're just the excitations of the D-branes. 597 00:39:22,390 --> 00:39:25,610 So now after talking about those boundary conditions, 598 00:39:25,610 --> 00:39:27,580 then we can again go through the procedure 599 00:39:27,580 --> 00:39:32,580 of quantizing open strings with those boundary conditions. 600 00:39:32,580 --> 00:39:33,785 Then we see what do we get. 601 00:39:54,110 --> 00:39:55,610 Now let's talk about the open string 602 00:39:55,610 --> 00:39:57,810 spectrum on the Dp-brane. 603 00:39:57,810 --> 00:40:04,220 So as you can see already from those pictures, 604 00:40:04,220 --> 00:40:09,410 because of the restriction on the endpoints, so essentially 605 00:40:09,410 --> 00:40:12,090 those strings, they can only move along 606 00:40:12,090 --> 00:40:14,520 the direction of the brane. 607 00:40:14,520 --> 00:40:16,920 They can only move in the direction along the brane. 608 00:40:16,920 --> 00:40:21,320 So essentially, they can be considered 609 00:40:21,320 --> 00:40:22,710 as living on the brane. 610 00:40:22,710 --> 00:40:23,530 OK? 611 00:40:23,530 --> 00:40:26,450 So those open strings are living on those Dp-branes. 612 00:40:26,450 --> 00:40:27,340 OK? 613 00:40:27,340 --> 00:40:29,660 So when we quantize those open strings, 614 00:40:29,660 --> 00:40:32,450 then we will get the excitations of those open strings, 615 00:40:32,450 --> 00:40:35,170 the state of those open strings, on the D-branes. 616 00:40:35,170 --> 00:40:35,670 OK? 617 00:40:38,450 --> 00:40:40,280 So let me first introduce some notations. 618 00:40:40,280 --> 00:40:44,490 Previously, we have the notation mu refers to all directions. 619 00:40:44,490 --> 00:40:47,960 So now I separate into [INAUDIBLE] and a. 620 00:40:47,960 --> 00:40:52,520 And [INAUDIBLE] is the Neumann boundary-- Neumann directions 621 00:40:52,520 --> 00:40:55,700 from 0 to p. 622 00:40:55,700 --> 00:41:00,420 And the a are the Dirichlet directions from p plus 1 to D 623 00:41:00,420 --> 00:41:03,070 minus 1. 624 00:41:03,070 --> 00:41:06,290 So there's D minus 1 minus p of them. 625 00:41:06,290 --> 00:41:08,670 So D minus 1 minus p of them. 626 00:41:12,527 --> 00:41:14,110 So for the Neumann boundary condition, 627 00:41:14,110 --> 00:41:17,980 it's just identical to what we had before. 628 00:41:17,980 --> 00:41:20,780 So let me just write it down to remind you. 629 00:41:47,190 --> 00:41:47,690 OK. 630 00:41:47,690 --> 00:41:51,600 So these are the, say, the most general solution. 631 00:41:51,600 --> 00:41:55,410 So in this case in particular, if you remember you have xa 632 00:41:55,410 --> 00:41:57,482 you could do xr. 633 00:41:57,482 --> 00:41:58,940 So the left moving and right moving 634 00:41:58,940 --> 00:42:03,180 are the same, controlled by the same function. 635 00:42:03,180 --> 00:42:06,410 And so that's why you get this cosine sig cosine. 636 00:42:06,410 --> 00:42:09,180 And when you take the derivative of a sigma becomes sine, 637 00:42:09,180 --> 00:42:10,730 then vanishes at the endpoints. 638 00:42:10,730 --> 00:42:11,230 OK? 639 00:42:15,560 --> 00:42:19,460 So for this story, it's just exactly identical as before. 640 00:42:19,460 --> 00:42:24,380 So now let's look at the Dirichlet conditions. 641 00:42:24,380 --> 00:42:28,300 So for the Dirichlet conditions, let's again 642 00:42:28,300 --> 00:42:37,990 write down the most general possible boundary conditions 643 00:42:37,990 --> 00:42:41,610 for Dirichlet boundary-- yeah, first write down 644 00:42:41,610 --> 00:42:46,710 the most general solution for such a field. 645 00:42:46,710 --> 00:42:53,280 So you have xa then 2ipa tau. 646 00:42:53,280 --> 00:43:00,800 Then you have [INAUDIBLE] tau [INAUDIBLE] sigma 647 00:43:00,800 --> 00:43:08,000 and [INAUDIBLE] sigma. 648 00:43:08,000 --> 00:43:10,130 OK? 649 00:43:10,130 --> 00:43:15,510 But now the boundary condition we need to impose 650 00:43:15,510 --> 00:43:21,320 is xa at sigma equal to 0 and pi. 651 00:43:21,320 --> 00:43:31,080 Sigma 0 tau and sigma pi tau equal to ba. 652 00:43:31,080 --> 00:43:33,660 So here, I call them a, but let me call them 653 00:43:33,660 --> 00:43:38,882 b because [INAUDIBLE] write a to the index a. 654 00:43:38,882 --> 00:43:41,720 So let me just them b, the vector b. 655 00:43:41,720 --> 00:43:43,629 OK? 656 00:43:43,629 --> 00:43:45,670 So now we have to impose this boundary condition. 657 00:43:48,250 --> 00:43:51,480 So you can immediately see that because of this boundary 658 00:43:51,480 --> 00:43:54,270 condition xa have to be ba. 659 00:43:56,990 --> 00:43:59,660 OK, xa have to be ba. 660 00:43:59,660 --> 00:44:01,430 And this pa have to be 0. 661 00:44:04,170 --> 00:44:06,670 pa have to be 0, otherwise you won't satisfy this condition, 662 00:44:06,670 --> 00:44:09,780 because this changes with time. 663 00:44:09,780 --> 00:44:11,810 And then you want the sum of them 664 00:44:11,810 --> 00:44:16,130 to be 0 at x equal to-- sigma equal to 0. 665 00:44:16,130 --> 00:44:22,510 So that gives you xao equal to [INAUDIBLE] xr. 666 00:44:22,510 --> 00:44:26,520 So this is from the boundary condition at sigma equal to 0. 667 00:44:26,520 --> 00:44:27,740 OK? 668 00:44:27,740 --> 00:44:33,170 And then the boundary condition is sigma equal to pi, 669 00:44:33,170 --> 00:44:36,370 you deduce both of them are periodic boundary conditions. 670 00:44:36,370 --> 00:44:41,960 They are periodic functions in 2pi. 671 00:44:41,960 --> 00:44:44,130 OK? 672 00:44:44,130 --> 00:44:47,510 They're periodic in 2pi. 673 00:44:47,510 --> 00:44:49,260 You can see it easily. 674 00:44:49,260 --> 00:44:51,810 So at the tau-- sigma equal to 0, 675 00:44:51,810 --> 00:44:55,760 we just have xr tau plus xao tau equal to-- yeah, 676 00:44:55,760 --> 00:44:58,460 it should be equal to 0, so you xao equal to [INAUDIBLE] xr. 677 00:45:04,610 --> 00:45:07,070 So now, you can just write down explicitly 678 00:45:07,070 --> 00:45:10,730 just-- still, this is a periodic function in 2pi, 679 00:45:10,730 --> 00:45:13,250 so the expansion is essentially the same. 680 00:45:13,250 --> 00:45:19,690 So the only difference for those directions 681 00:45:19,690 --> 00:45:23,220 is that now you have b alpha, ba. 682 00:45:23,220 --> 00:45:26,460 Now you don't have this term, because this term is 0. 683 00:45:26,460 --> 00:45:33,020 And then a not equal to 0 alpha [INAUDIBLE] divided by n 684 00:45:33,020 --> 00:45:37,670 minus [INAUDIBLE] tau sine in sigma. 685 00:45:37,670 --> 00:45:41,020 OK, there's some pre-factors here I forgot to write. 686 00:45:41,020 --> 00:45:43,270 Some alpha prime [INAUDIBLE] by convention, 687 00:45:43,270 --> 00:45:44,740 let's not worry about those things. 688 00:45:56,350 --> 00:45:56,850 OK? 689 00:46:01,420 --> 00:46:03,380 So now you do quantization. 690 00:46:06,970 --> 00:46:15,110 So the quantization in Neumann boundary condition, 691 00:46:15,110 --> 00:46:18,950 of course identical to before-- oh yeah, 692 00:46:18,950 --> 00:46:21,610 first let me just explain the pa equal 0. 693 00:46:21,610 --> 00:46:24,340 I just want to make one remark. 694 00:46:24,340 --> 00:46:27,590 So this is, of course, expected. 695 00:46:27,590 --> 00:46:30,770 Because if the string is limited in moving on this plane, 696 00:46:30,770 --> 00:46:33,590 then of course it cannot move in the perpendicular direction, 697 00:46:33,590 --> 00:46:36,270 so cannot have central mass, momentum in the-- so cannot 698 00:46:36,270 --> 00:46:39,320 have central mass, momentum in the perpendicular direction. 699 00:46:39,320 --> 00:46:42,600 So this is indeed what we intuitively expect. 700 00:46:42,600 --> 00:46:44,170 OK? 701 00:46:44,170 --> 00:46:48,370 The central mass momentum can only be in the one direction, 702 00:46:48,370 --> 00:46:50,330 or can only be in the Neumann directions. 703 00:46:53,257 --> 00:46:55,465 So the quantization of the Neumann boundary condition 704 00:46:55,465 --> 00:46:57,820 is exactly the same as before. 705 00:46:57,820 --> 00:47:07,740 And actually, quantization in Dirichlet directions 706 00:47:07,740 --> 00:47:14,535 which this is set up also the same, also exactly the same. 707 00:47:18,350 --> 00:47:20,900 Because essentially the only change is cosine 708 00:47:20,900 --> 00:47:22,040 sigma goes to sine sigma. 709 00:47:22,040 --> 00:47:24,280 You can walk through that procedure, 710 00:47:24,280 --> 00:47:26,740 and you find that nothing really changes. 711 00:47:26,740 --> 00:47:31,860 So it's exactly the same as that of n. 712 00:47:37,231 --> 00:47:37,730 OK? 713 00:47:42,080 --> 00:47:47,490 In particular, the commutation relations 714 00:47:47,490 --> 00:47:52,000 between all those modes does not change at all, 715 00:47:52,000 --> 00:47:53,600 just [INAUDIBLE] exactly the same 716 00:47:53,600 --> 00:48:02,350 as what you would have in the Neumann case. 717 00:48:02,350 --> 00:48:04,400 So this is also [INAUDIBLE]. 718 00:48:07,360 --> 00:48:10,000 This is [INAUDIBLE]. 719 00:48:10,000 --> 00:48:13,510 So imposing such boundary conditions, 720 00:48:13,510 --> 00:48:16,570 essentially you're only restricted to the central mass 721 00:48:16,570 --> 00:48:18,820 motion, so you can no longer move in the perpendicular 722 00:48:18,820 --> 00:48:20,380 directions. 723 00:48:20,380 --> 00:48:22,220 But the string can oscillate. 724 00:48:22,220 --> 00:48:24,610 But string can still oscillate in all 10 dimensions. 725 00:48:24,610 --> 00:48:27,990 There's no difference in the way the string oscillates. 726 00:48:27,990 --> 00:48:31,560 And so strings still oscillate in 10 dimensions, 727 00:48:31,560 --> 00:48:33,450 so everything is the same. 728 00:48:36,910 --> 00:48:37,730 Yeah. 729 00:48:37,730 --> 00:48:41,500 A string is still oscillating in all directions. 730 00:48:41,500 --> 00:48:45,395 In particular, the zero point energy is the same. 731 00:48:49,050 --> 00:48:51,110 Zero point energy on the worksheet is the same. 732 00:48:51,110 --> 00:48:51,610 OK? 733 00:49:00,270 --> 00:49:04,330 So this means that the mass formula we derived before 734 00:49:04,330 --> 00:49:07,410 for the Neumann boundary condition 735 00:49:07,410 --> 00:49:10,560 will exactly apply without any change. 736 00:49:10,560 --> 00:49:11,060 OK? 737 00:49:11,060 --> 00:49:13,396 Without any change. 738 00:49:13,396 --> 00:49:15,020 In particular, the mass of the spectrum 739 00:49:15,020 --> 00:49:19,600 will also be essentially-- the mass of the spectrum 740 00:49:19,600 --> 00:49:23,992 will also be essentially exactly the same as before. 741 00:49:23,992 --> 00:49:24,660 One second. 742 00:49:28,980 --> 00:49:38,910 So let me just mention one point, that this is identical. 743 00:49:38,910 --> 00:49:40,840 This is identical. 744 00:49:40,840 --> 00:49:45,440 This is actually related to that for the Dirichlet boundary 745 00:49:45,440 --> 00:49:47,550 condition on both sides. 746 00:49:47,550 --> 00:49:49,620 This is still a periodic boundary-- 747 00:49:49,620 --> 00:49:51,740 a periodic function in 2pi. 748 00:49:51,740 --> 00:49:54,810 And so you get this. 749 00:49:54,810 --> 00:49:58,710 In such a case, say if one endpoint is the Neumann 750 00:49:58,710 --> 00:50:01,330 boundary condition but the other endpoint 751 00:50:01,330 --> 00:50:03,090 is a Dirichlet boundary condition, 752 00:50:03,090 --> 00:50:05,450 then the story is a little bit more subtle. 753 00:50:05,450 --> 00:50:06,280 OK? 754 00:50:06,280 --> 00:50:12,420 And then you actually get to the-- then the frequency 755 00:50:12,420 --> 00:50:16,040 of each modes become half integer rather than integer. 756 00:50:16,040 --> 00:50:18,930 And then you will get the difference [INAUDIBLE], 757 00:50:18,930 --> 00:50:21,370 and there the spectrum will change, et cetera. 758 00:50:21,370 --> 00:50:23,400 OK? 759 00:50:23,400 --> 00:50:26,360 So I will not do it now. 760 00:50:26,360 --> 00:50:28,840 Maybe I will put it in your [INAUDIBLE] p set, 761 00:50:28,840 --> 00:50:31,880 so that you can do it yourself. 762 00:50:31,880 --> 00:50:32,380 Yeah? 763 00:50:32,380 --> 00:50:33,213 You have a question? 764 00:50:33,213 --> 00:50:35,590 AUDIENCE: [INAUDIBLE] question was what happens 765 00:50:35,590 --> 00:50:37,970 when p equals 0 [INAUDIBLE]. 766 00:50:41,302 --> 00:50:42,260 HONG LIU: Oh, yeah. 767 00:50:42,260 --> 00:50:43,840 Yeah, for the p equal to 0, yeah, 768 00:50:43,840 --> 00:50:46,180 indeed it's a little bit tricky to do the [INAUDIBLE]. 769 00:50:46,180 --> 00:50:46,680 Yeah. 770 00:50:49,304 --> 00:50:51,720 Yeah, for p equal to 0, it's tricky to do the [INAUDIBLE]. 771 00:50:51,720 --> 00:50:55,640 But you can do-- using other quantization procedures, 772 00:50:55,640 --> 00:50:57,284 then you can do the same thing. 773 00:50:57,284 --> 00:50:59,700 Yeah, but the conclusion does not change between different 774 00:50:59,700 --> 00:51:01,170 p's. 775 00:51:01,170 --> 00:51:02,150 AUDIENCE: [INAUDIBLE]. 776 00:51:02,150 --> 00:51:03,130 HONG LIU: Yes? 777 00:51:03,130 --> 00:51:05,907 AUDIENCE: Maybe I missed this, but for the two different brane 778 00:51:05,907 --> 00:51:08,240 that you have a string going from one brane to another-- 779 00:51:08,240 --> 00:51:09,120 HONG LIU: Yeah. 780 00:51:09,120 --> 00:51:11,073 AUDIENCE: You said from a point to a brane, the story changes. 781 00:51:11,073 --> 00:51:13,037 What about, like, from one brane to another? 782 00:51:13,037 --> 00:51:15,640 Is it exactly the same as this, or is it-- 783 00:51:15,640 --> 00:51:19,700 HONG LIU: Oh yeah, that's also a good question, 784 00:51:19,700 --> 00:51:21,580 which I might already have something 785 00:51:21,580 --> 00:51:23,910 like this in the current p set. 786 00:51:23,910 --> 00:51:27,070 If not, it's something I'll put in the next p set 787 00:51:27,070 --> 00:51:28,255 because you asked. 788 00:51:31,710 --> 00:51:35,380 Yeah, you see there's a distance between them. 789 00:51:35,380 --> 00:51:38,581 So now you actually need to include the sigma term. 790 00:51:38,581 --> 00:51:42,110 Now you actually need to include the sigma term here 791 00:51:42,110 --> 00:51:44,180 to account this difference. 792 00:51:44,180 --> 00:51:47,710 Because now the 0 and the pi is no longer the same. 793 00:51:47,710 --> 00:51:49,810 Yeah, here I don't need to put in the sigma term 794 00:51:49,810 --> 00:51:51,540 because the 0 and the pi are the same. 795 00:51:54,222 --> 00:51:56,074 AUDIENCE: So the zero point would change-- 796 00:51:56,074 --> 00:51:57,990 HONG LIU: Yeah, zero point energy will change. 797 00:51:57,990 --> 00:52:00,810 And essentially, it's given by the mass of the strings 798 00:52:00,810 --> 00:52:02,970 stretched between them. 799 00:52:02,970 --> 00:52:06,507 Yeah, so that's-- yeah, that's a very instructive exercise. 800 00:52:06,507 --> 00:52:08,590 Actually maybe I already put in the current p set. 801 00:52:08,590 --> 00:52:10,280 I don't remember. 802 00:52:10,280 --> 00:52:11,674 Anyway. 803 00:52:11,674 --> 00:52:12,340 Other questions? 804 00:52:17,060 --> 00:52:17,990 Good. 805 00:52:17,990 --> 00:52:20,626 So now let's look at the-- more explicitly at the open string 806 00:52:20,626 --> 00:52:21,125 spectrum. 807 00:52:55,200 --> 00:52:56,860 So again, it's the same thing. 808 00:52:56,860 --> 00:53:04,260 You just [INAUDIBLE] and 1, mu 1. 809 00:53:04,260 --> 00:53:14,930 You can act m1 times n k, mu k, mk. 810 00:53:14,930 --> 00:53:18,740 The only difference is now your vacuum state 811 00:53:18,740 --> 00:53:22,850 is only-- can only have momentum in the p alpha direction. 812 00:53:22,850 --> 00:53:24,970 That's essentially the only difference. 813 00:53:24,970 --> 00:53:25,470 OK? 814 00:53:29,470 --> 00:53:32,640 And again, the mass square now it just matters 815 00:53:32,640 --> 00:53:35,560 p alpha-- p alpha. 816 00:53:35,560 --> 00:53:39,680 OK, only around the Neumann directions. 817 00:53:39,680 --> 00:53:42,990 So this is really-- then each mode 818 00:53:42,990 --> 00:53:46,950 on the string-- each oscillation mode of the string 819 00:53:46,950 --> 00:53:53,200 now should really be corresponding to a particle 820 00:53:53,200 --> 00:54:09,810 or field in p plus 1 dimensional world-volume of the Dp-brane. 821 00:54:13,980 --> 00:54:14,547 OK? 822 00:54:14,547 --> 00:54:16,130 So we call-- this is the world-volume. 823 00:54:16,130 --> 00:54:18,390 We call the directions, all these directions 824 00:54:18,390 --> 00:54:22,480 together, as the world-volume of the Dp-brane. 825 00:54:22,480 --> 00:54:27,180 So since the string can only move along the Dp-brane, 826 00:54:27,180 --> 00:54:29,840 so they can only correspond into the field of particles 827 00:54:29,840 --> 00:54:31,030 within the Dp-brane. 828 00:54:31,030 --> 00:54:32,910 They cannot move in the opposite direction. 829 00:54:32,910 --> 00:54:35,455 They cannot move in the perpendicular direction. 830 00:54:38,510 --> 00:54:39,240 OK? 831 00:54:39,240 --> 00:54:42,910 So [INAUDIBLE] lives on the brane. 832 00:54:42,910 --> 00:54:43,410 OK? 833 00:54:46,150 --> 00:54:49,970 And in this-- then all of those things 834 00:54:49,970 --> 00:54:55,260 may be considered as the excitations of the branes. 835 00:54:55,260 --> 00:54:56,112 OK? 836 00:54:56,112 --> 00:54:57,036 AUDIENCE: [INAUDIBLE] 837 00:54:57,036 --> 00:54:57,812 HONG LIU: Hm? 838 00:54:57,812 --> 00:55:00,270 AUDIENCE: [INAUDIBLE] string can be part of the excitation. 839 00:55:00,270 --> 00:55:01,240 HONG LIU: Yeah, that's right. 840 00:55:01,240 --> 00:55:01,590 Yeah. 841 00:55:01,590 --> 00:55:04,173 Because you can excite certain strings and not excite strings, 842 00:55:04,173 --> 00:55:05,156 et cetera. 843 00:55:05,156 --> 00:55:06,780 And they're all moving along the brane. 844 00:55:12,150 --> 00:55:17,340 And the important thing is that the states 845 00:55:17,340 --> 00:55:25,984 should fall into repetitions now of this symmetry, 846 00:55:25,984 --> 00:55:27,775 because that's the only remaining symmetry. 847 00:55:34,620 --> 00:55:41,590 [INAUDIBLE] when p under the rotation 848 00:55:41,590 --> 00:55:44,650 in the transverse directions. 849 00:55:44,650 --> 00:55:45,150 OK? 850 00:55:48,870 --> 00:55:53,760 And again, just by this, you can show 851 00:55:53,760 --> 00:55:58,290 that it actually makes sense only in the 26th dimension. 852 00:55:58,290 --> 00:55:59,034 OK? 853 00:55:59,034 --> 00:56:00,950 Again, makes sense only in the 26th dimension. 854 00:56:03,670 --> 00:56:07,680 So now let's again look at the massless states. 855 00:56:07,680 --> 00:56:10,235 So as I mentioned before, the mass [INAUDIBLE] 856 00:56:10,235 --> 00:56:12,360 exactly the same as before, because the oscillation 857 00:56:12,360 --> 00:56:14,630 modes behaves exactly the same. 858 00:56:14,630 --> 00:56:17,730 So the mass [INAUDIBLE] exact behaves the same as before. 859 00:56:17,730 --> 00:56:21,995 So the massless states, that is the following. 860 00:56:25,511 --> 00:56:26,010 Yeah? 861 00:56:26,010 --> 00:56:28,860 AUDIENCE: When we say the particles in [INAUDIBLE] 862 00:56:28,860 --> 00:56:29,970 dimensional world-volume-- 863 00:56:29,970 --> 00:56:30,756 HONG LIU: Yeah. 864 00:56:30,756 --> 00:56:36,975 AUDIENCE: But [INAUDIBLE] particles [INAUDIBLE] object, 865 00:56:36,975 --> 00:56:38,800 whereas the open string can-- you 866 00:56:38,800 --> 00:56:43,090 know, one end of the open string located 867 00:56:43,090 --> 00:56:45,710 at one point [INAUDIBLE] the other one [INAUDIBLE] 868 00:56:45,710 --> 00:56:47,110 very far away. 869 00:56:47,110 --> 00:56:49,130 But is that picture [INAUDIBLE]? 870 00:56:49,130 --> 00:56:51,540 HONG LIU: No. 871 00:56:51,540 --> 00:56:53,600 That's a good question. 872 00:56:53,600 --> 00:56:59,480 So when I say particles, I'm talking about the center 873 00:56:59,480 --> 00:57:01,460 of mass motion. 874 00:57:01,460 --> 00:57:02,920 OK? 875 00:57:02,920 --> 00:57:04,855 Therefore, each oscillation modes 876 00:57:04,855 --> 00:57:06,730 that you see, that's corresponding to-- yeah, 877 00:57:06,730 --> 00:57:11,810 for particular oscillation modes, then 878 00:57:11,810 --> 00:57:17,090 the-- other than those quantum number-- yeah, 879 00:57:17,090 --> 00:57:17,970 let me finish this. 880 00:57:17,970 --> 00:57:20,584 I will answer that question. 881 00:57:20,584 --> 00:57:24,110 I will write this down explicitly. 882 00:57:24,110 --> 00:57:26,330 So let's talk about massless states. 883 00:57:26,330 --> 00:57:32,010 So again, using [INAUDIBLE]. 884 00:57:32,010 --> 00:57:34,250 So again, we are using the [INAUDIBLE]. 885 00:57:34,250 --> 00:57:41,810 You can separate the [INAUDIBLE] and i in the Neumann direction. 886 00:57:41,810 --> 00:57:44,640 And in the Dirichlet direction, again it's 887 00:57:44,640 --> 00:57:47,216 just the p plus 1 to D minus 1. 888 00:57:47,216 --> 00:57:48,960 OK? 889 00:57:48,960 --> 00:57:52,120 And then the massless modes would be the same as before. 890 00:57:52,120 --> 00:57:58,760 You have [INAUDIBLE] 1i acting on p alpha. 891 00:57:58,760 --> 00:58:03,660 And the alpha minus 1 a acting on p alpha. 892 00:58:08,160 --> 00:58:12,230 And the plus infinite number. 893 00:58:14,780 --> 00:58:18,570 Plus infinite number of massless modes. 894 00:58:18,570 --> 00:58:20,880 OK? 895 00:58:20,880 --> 00:58:24,450 So let me just illustrate this point. 896 00:58:24,450 --> 00:58:32,630 So this [INAUDIBLE] as before, the [INAUDIBLE] 897 00:58:32,630 --> 00:58:36,466 further breaks this Poincare symmetry to only a sub-group. 898 00:58:36,466 --> 00:58:37,840 But you can already see from here 899 00:58:37,840 --> 00:58:40,970 this must be corresponding to a vector particle. 900 00:58:40,970 --> 00:58:44,337 So this is like a massless vector. 901 00:58:53,400 --> 00:58:54,260 Massless vector. 902 00:58:57,370 --> 00:59:01,770 Yeah, so to answer your question, so example, 903 00:59:01,770 --> 00:59:05,040 for typical modes like this, the only quantum number 904 00:59:05,040 --> 00:59:12,020 you have is p alpha and this vector indices. 905 00:59:12,020 --> 00:59:14,970 This vector index. 906 00:59:14,970 --> 00:59:18,980 So from the perspective of the quantum number-- 907 00:59:18,980 --> 00:59:22,760 and this really is just like a vector particle. 908 00:59:26,140 --> 00:59:27,450 Just like a vector particle. 909 00:59:30,180 --> 00:59:35,710 And then you can describe it by a vector field, et cetera. 910 00:59:35,710 --> 00:59:38,210 It is very good question. 911 00:59:38,210 --> 00:59:43,260 Say suppose in this situation which I have a string, 912 00:59:43,260 --> 00:59:45,590 suppose a string is really very known. 913 00:59:45,590 --> 00:59:47,220 It's very big. 914 00:59:47,220 --> 00:59:51,160 Then I should be able to see this one-dimensional structure. 915 00:59:51,160 --> 00:59:54,549 Indeed, then that means that, even 916 00:59:54,549 --> 00:59:57,090 though this corresponding-- this is just corresponding to one 917 00:59:57,090 --> 00:59:59,070 particular mode of the string. 918 00:59:59,070 --> 01:00:01,670 And to see the string structure you have to be able to see, 919 01:00:01,670 --> 01:00:05,720 you can excite this into other different modes, et cetera. 920 01:00:09,995 --> 01:00:13,125 AUDIENCE: [INAUDIBLE] stretch the string. 921 01:00:13,125 --> 01:00:13,873 HONG LIU: Yeah. 922 01:00:13,873 --> 01:00:15,030 AUDIENCE: I mean, [INAUDIBLE]-- 923 01:00:15,030 --> 01:00:17,155 HONG LIU: Yeah, stretching the string corresponding 924 01:00:17,155 --> 01:00:19,130 to exciting the string in a different way. 925 01:00:19,130 --> 01:00:21,400 It means leaving this particular state. 926 01:00:21,400 --> 01:00:22,640 AUDIENCE: Oh, I see. 927 01:00:22,640 --> 01:00:23,920 HONG LIU: Yeah. 928 01:00:23,920 --> 01:00:25,470 I'm talking about the interpretation 929 01:00:25,470 --> 01:00:27,710 of this particular state, which is 930 01:00:27,710 --> 01:00:29,280 [INAUDIBLE] spacetime particle. 931 01:00:29,280 --> 01:00:33,180 AUDIENCE: Actually this data conversely 932 01:00:33,180 --> 01:00:36,416 decided the other configuration of-- 933 01:00:36,416 --> 01:00:38,040 HONG LIU: Yeah. 934 01:00:38,040 --> 01:00:39,720 It determines all the configuration. 935 01:00:39,720 --> 01:00:41,459 It determines all the oscillation modes 936 01:00:41,459 --> 01:00:42,250 of the open string. 937 01:00:42,250 --> 01:00:43,624 Just the other oscillation modes, 938 01:00:43,624 --> 01:00:45,850 they are all not excited. 939 01:00:45,850 --> 01:00:47,935 But of course, if when [INAUDIBLE] 940 01:00:47,935 --> 01:00:49,560 this one-dimensional object [INAUDIBLE] 941 01:00:49,560 --> 01:00:51,430 find the ways to excite them. 942 01:00:51,430 --> 01:00:53,170 And then you know this is not a particle, 943 01:00:53,170 --> 01:00:54,878 it's a one-dimensional object, et cetera. 944 01:00:56,845 --> 01:00:58,970 But in order to see this one-dimensional structure, 945 01:00:58,970 --> 01:01:01,369 you have to probe the final structure that just loads 946 01:01:01,369 --> 01:01:02,035 quantum numbers. 947 01:01:04,594 --> 01:01:06,760 If those are the only quantum numbers you can probe, 948 01:01:06,760 --> 01:01:07,843 then [INAUDIBLE] particle. 949 01:01:13,630 --> 01:01:15,980 So this is the massless vector. 950 01:01:15,980 --> 01:01:21,730 But now those guys, they have index a, which 951 01:01:21,730 --> 01:01:24,010 is not in the Poincare here. 952 01:01:24,010 --> 01:01:26,080 So those are now scalar fields. 953 01:01:31,910 --> 01:01:32,410 OK? 954 01:01:32,410 --> 01:01:36,585 Those are scalar fields on the world-volume of the D-branes. 955 01:01:39,810 --> 01:01:40,900 OK? 956 01:01:40,900 --> 01:01:55,320 So now we find-- so previously, when 957 01:01:55,320 --> 01:01:57,670 we can see the all Neumann directions, we only 958 01:01:57,670 --> 01:01:59,680 see a vector field. 959 01:01:59,680 --> 01:02:02,030 So moving more 26th dimension. 960 01:02:02,030 --> 01:02:06,820 But now-- so these massless modes, 961 01:02:06,820 --> 01:02:15,750 they give rise to a vector field, 962 01:02:15,750 --> 01:02:21,510 the massless vector field A alpha, 963 01:02:21,510 --> 01:02:31,272 and D minus p minus 1 scalar fields phi a. 964 01:02:31,272 --> 01:02:31,772 OK? 965 01:02:39,360 --> 01:02:41,200 So essentially, when you introduce D-brane 966 01:02:41,200 --> 01:02:43,340 compared to the space-fitting brane cage, which 967 01:02:43,340 --> 01:02:47,360 we can see the before in the Neumann direction, 968 01:02:47,360 --> 01:02:51,220 just some component of those cage fields 969 01:02:51,220 --> 01:02:54,350 now become the oscillation in the transverse direction, 970 01:02:54,350 --> 01:02:57,130 and now they just essentially become scalar fields. 971 01:02:57,130 --> 01:02:57,630 OK? 972 01:02:57,630 --> 01:02:59,980 Is this clear? 973 01:02:59,980 --> 01:03:03,560 So now instead you have a smaller vector field, 974 01:03:03,560 --> 01:03:05,570 plus a bunch of scalar fields. 975 01:03:10,100 --> 01:03:21,190 As we said before, [INAUDIBLE] defective action, 976 01:03:21,190 --> 01:03:35,530 so the [INAUDIBLE] defective action for such massless modes 977 01:03:35,530 --> 01:03:39,380 must be-- have the structure, say-- 978 01:03:39,380 --> 01:03:42,560 so this is in the p plus 1 dimensional world-volume 979 01:03:42,560 --> 01:03:47,370 of the string, say, of a Maxwell series. 980 01:03:52,820 --> 01:03:57,160 And they're just the same as with scalar fields. 981 01:04:00,210 --> 01:04:04,390 And then of course, they are more complicated contributions. 982 01:04:04,390 --> 01:04:07,419 Just this is a no energy. 983 01:04:07,419 --> 01:04:09,085 Then the structure of those excitations, 984 01:04:09,085 --> 01:04:11,970 they must be described by this kind of low-energy factor 985 01:04:11,970 --> 01:04:14,012 series. 986 01:04:14,012 --> 01:04:15,470 And this is, again, just determined 987 01:04:15,470 --> 01:04:17,220 by the universality of massless modes. 988 01:04:26,390 --> 01:04:32,160 So now there's a very-- so now let me-- 989 01:04:32,160 --> 01:04:38,880 the first node-- let me emphasize this point, 990 01:04:38,880 --> 01:04:45,260 is that the number of massless modes, massless scalar fields. 991 01:04:50,020 --> 01:04:53,960 So we have such a bunch of scalar fields. 992 01:04:53,960 --> 01:05:09,550 It's exactly the number of transverse directions 993 01:05:09,550 --> 01:05:10,580 to the D-brane. 994 01:05:15,970 --> 01:05:17,365 OK? 995 01:05:17,365 --> 01:05:19,740 So essentially, associated with each transverse direction 996 01:05:19,740 --> 01:05:21,550 you have scalar excitations. 997 01:05:21,550 --> 01:05:24,860 You have massless scalar fields. 998 01:05:24,860 --> 01:05:26,860 OK? 999 01:05:26,860 --> 01:05:28,340 So let me emphasize this point. 1000 01:05:36,400 --> 01:05:41,820 So those scalar fields, some of you maybe already have guessed, 1001 01:05:41,820 --> 01:05:46,530 they have a very simple physical explanation-- 1002 01:05:46,530 --> 01:05:52,090 they have a very simple physical interpretation. 1003 01:05:52,090 --> 01:05:53,880 And those scalar fields, they can 1004 01:05:53,880 --> 01:05:58,890 be considered as describing the motions and the fluctuations 1005 01:05:58,890 --> 01:06:02,211 of the D-brane in the transverse directions. 1006 01:06:02,211 --> 01:06:02,710 OK? 1007 01:06:08,600 --> 01:06:10,080 So for each transverse direction, 1008 01:06:10,080 --> 01:06:13,320 you have such massless scalar fields. 1009 01:06:13,320 --> 01:07:13,110 And each of scalar fields-- so more precisely, 1010 01:07:13,110 --> 01:07:18,170 so more explicitly, just as a cartoon picture, 1011 01:07:18,170 --> 01:07:20,130 suppose let's consider, say, the p 1012 01:07:20,130 --> 01:07:24,010 minus 1 direction, p plus 1 direction. 1013 01:07:24,010 --> 01:07:27,120 This direction, p plus 1 direction. 1014 01:07:27,120 --> 01:07:30,280 So suppose you have a D-brane sitting here 1015 01:07:30,280 --> 01:07:32,520 at some value of a. 1016 01:07:32,520 --> 01:07:34,290 OK? 1017 01:07:34,290 --> 01:07:42,050 So when we impose the Dirichlet boundary condition-- yeah, 1018 01:07:42,050 --> 01:07:46,460 so those are the x1 01 p directions. 1019 01:07:46,460 --> 01:07:48,850 OK? 1020 01:07:48,850 --> 01:07:50,670 Yeah. 1021 01:07:50,670 --> 01:07:55,720 Yeah, this is the other x0 and p directions. 1022 01:07:55,720 --> 01:08:01,440 You have a D-brane sitting here at some value of a. 1023 01:08:01,440 --> 01:08:05,775 So when we impose the Dirichlet boundary conditions, 1024 01:08:05,775 --> 01:08:07,400 that boundary condition does not depend 1025 01:08:07,400 --> 01:08:09,160 on the value of x0 and p. 1026 01:08:09,160 --> 01:08:11,010 So they're the same everywhere. 1027 01:08:11,010 --> 01:08:16,439 So that's why I draw it flat, exact flat. 1028 01:08:16,439 --> 01:08:19,170 But now, if you have some excitations in the phi 1029 01:08:19,170 --> 01:08:22,460 p plus 1 direction, so phi p plus 1 1030 01:08:22,460 --> 01:08:29,100 is a field in the world-volume of the half-- 1031 01:08:29,100 --> 01:08:32,370 can have p alpha momentum. 1032 01:08:32,370 --> 01:08:34,470 You can go to coordinate space. 1033 01:08:34,470 --> 01:08:39,689 Then their function's just x1, xp. 1034 01:08:39,689 --> 01:08:41,490 OK? 1035 01:08:41,490 --> 01:08:43,580 xi, xp. 1036 01:08:43,580 --> 01:08:49,920 And then the profile of this phi essentially 1037 01:08:49,920 --> 01:08:58,020 tells you that the D-brane-- say if you have a scalar field 1038 01:08:58,020 --> 01:09:00,910 excitation like this, then the shape of the D-brane 1039 01:09:00,910 --> 01:09:04,290 actually is no longer strictly flat. 1040 01:09:04,290 --> 01:09:07,510 Actually have a profile given by these scalar fields. 1041 01:09:14,859 --> 01:09:18,750 So we are not justify this precise-- 1042 01:09:18,750 --> 01:09:22,500 so this can be justified mathematically to really show 1043 01:09:22,500 --> 01:09:23,939 that this is the case. 1044 01:09:23,939 --> 01:09:24,970 OK? 1045 01:09:24,970 --> 01:09:27,279 And I will not do that here, but let 1046 01:09:27,279 --> 01:09:32,291 me just mention some support for this picture. 1047 01:09:36,219 --> 01:09:44,430 So-- OK. 1048 01:09:44,430 --> 01:09:53,410 So first you can show at [INAUDIBLE] level 1049 01:09:53,410 --> 01:09:59,030 having such excitations-- OK? 1050 01:09:59,030 --> 01:10:02,556 So having such excitations, so we can just call it 1051 01:10:02,556 --> 01:10:03,555 [INAUDIBLE] excitations. 1052 01:10:09,990 --> 01:10:13,020 So this is a picture in momentum space. 1053 01:10:13,020 --> 01:10:21,560 [INAUDIBLE] excitations modifies the Dirichlet boundary 1054 01:10:21,560 --> 01:10:24,450 condition. 1055 01:10:24,450 --> 01:10:25,010 OK? 1056 01:10:25,010 --> 01:10:26,620 So this you can show explicitly. 1057 01:10:32,690 --> 01:10:36,210 So this is first point, which I will not go into there. 1058 01:10:39,670 --> 01:10:43,360 And the second point is that you can actually work out 1059 01:10:43,360 --> 01:10:46,180 the precise pre-factor. 1060 01:10:46,180 --> 01:10:52,110 There's a way, say by doing string theory calculation, 1061 01:10:52,110 --> 01:10:54,020 say by computing the [INAUDIBLE], 1062 01:10:54,020 --> 01:10:56,930 there's a way you can actually work out 1063 01:10:56,930 --> 01:10:58,620 the pre-factor, precise pre-factor. 1064 01:11:01,210 --> 01:11:06,500 So let me call this the equation star. 1065 01:11:06,500 --> 01:11:15,780 So you can actually work out-- so that was just by general-- 1066 01:11:15,780 --> 01:11:17,960 this is just by general principle. 1067 01:11:17,960 --> 01:11:20,390 Say if you have a massless spectrum, then the [INAUDIBLE] 1068 01:11:20,390 --> 01:11:22,750 must behave like this. 1069 01:11:22,750 --> 01:11:25,580 But this pre-factor is not fixed. 1070 01:11:25,580 --> 01:11:27,720 OK? 1071 01:11:27,720 --> 01:11:34,077 But you can actually, with the interpretation phi, 1072 01:11:34,077 --> 01:11:36,660 [INAUDIBLE] fluctuations of this Dirichlet boundary condition, 1073 01:11:36,660 --> 01:11:41,260 you can actually fix the precise pre-factor here. 1074 01:11:41,260 --> 01:11:55,690 So working out the [INAUDIBLE] you 1075 01:11:55,690 --> 01:11:58,665 can show that actually this action can 1076 01:11:58,665 --> 01:12:01,570 be written as the following. 1077 01:12:01,570 --> 01:12:09,410 Tp-- again, it's integrated only over the [INAUDIBLE] 1078 01:12:09,410 --> 01:12:11,250 of the D-branes, p plus 1. 1079 01:12:14,069 --> 01:12:15,860 And actually first we find actually there's 1080 01:12:15,860 --> 01:12:19,020 a constant term. 1081 01:12:19,020 --> 01:12:21,420 We will call that action precisely. 1082 01:12:21,420 --> 01:12:35,120 And then you have this F term under this-- and let me-- yeah, 1083 01:12:35,120 --> 01:12:38,400 actually I should have done it there. 1084 01:12:38,400 --> 01:12:42,780 So there is a summation a over here, right? 1085 01:12:42,780 --> 01:12:54,406 So this Tp, this pre-factor, is actually-- 1086 01:12:54,406 --> 01:12:56,030 we can work out the precise pre-factor. 1087 01:12:59,750 --> 01:13:10,504 And this pre-factor is given by the mass of our Dp-brane 1088 01:13:10,504 --> 01:13:11,170 per unit volume. 1089 01:13:15,090 --> 01:13:18,590 Or in other words, the brane tension. 1090 01:13:18,590 --> 01:13:22,210 So I will explain where does this come from, where-- 1091 01:13:22,210 --> 01:13:24,235 how do we define brane tension. 1092 01:13:24,235 --> 01:13:26,030 But for now, let me just introduce 1093 01:13:26,030 --> 01:13:27,630 this concept of brane tension. 1094 01:13:30,084 --> 01:13:31,500 So if you work out the pre-factor, 1095 01:13:31,500 --> 01:13:36,050 you find the pre-factor is precisely the brane tension, 1096 01:13:36,050 --> 01:13:37,270 OK? 1097 01:13:37,270 --> 01:13:39,070 Brane tension. 1098 01:13:39,070 --> 01:13:44,770 So now if you look at this equation, so then this 1, 1099 01:13:44,770 --> 01:13:47,840 it's just like a static mass term. 1100 01:13:47,840 --> 01:13:50,950 This 1 is like a static mass term. 1101 01:13:50,950 --> 01:13:55,420 So you also find this 1 in this [INAUDIBLE] action. 1102 01:13:55,420 --> 01:14:00,860 But this 1 is very important because of the-- so this 1 1103 01:14:00,860 --> 01:14:02,460 is like static term. 1104 01:14:02,460 --> 01:14:05,790 And now if you look at this term, 1105 01:14:05,790 --> 01:14:11,850 so now if you consider the situation which A alpha is 1106 01:14:11,850 --> 01:14:15,170 equal to 0-- so you don't have an excitation of the vector 1107 01:14:15,170 --> 01:14:19,558 field, but phi a only depend on t. 1108 01:14:19,558 --> 01:14:21,040 Only depend on the time. 1109 01:14:25,000 --> 01:14:27,670 And then this effective action then 1110 01:14:27,670 --> 01:14:34,620 will become Tp, then the volume of the p, 1111 01:14:34,620 --> 01:14:39,303 then dt 1/2 phi dot a square. 1112 01:14:42,201 --> 01:14:45,110 OK? 1113 01:14:45,110 --> 01:14:51,240 And so Vp is the volume of Dp-brane. 1114 01:14:56,710 --> 01:14:57,880 Volume of the Dp-brane. 1115 01:14:57,880 --> 01:15:00,447 And this tension times Dp-brane-- so this 1116 01:15:00,447 --> 01:15:01,155 is just the mass. 1117 01:15:06,760 --> 01:15:08,570 So this is just the mass. 1118 01:15:08,570 --> 01:15:18,330 So you can rewrite this just 1/2 and p-brane phi dot square. 1119 01:15:23,200 --> 01:15:29,330 And then D-brane [INAUDIBLE] Tp times the volume. 1120 01:15:35,760 --> 01:15:39,655 So you see, this is precisely like a particle. 1121 01:15:42,360 --> 01:15:46,600 Like the whole brane moving-- the [INAUDIBLE] motion 1122 01:15:46,600 --> 01:15:50,620 moving of some object. 1123 01:15:50,620 --> 01:15:51,660 OK? 1124 01:15:51,660 --> 01:15:53,960 So again, this give you the interpretation 1125 01:15:53,960 --> 01:16:01,280 that the phi can be interpreted as the [INAUDIBLE] describe 1126 01:16:01,280 --> 01:16:04,240 the motion of the brane. 1127 01:16:04,240 --> 01:16:04,740 OK? 1128 01:16:30,390 --> 01:16:30,890 OK. 1129 01:16:34,030 --> 01:16:35,850 So do you have any questions about this? 1130 01:16:35,850 --> 01:16:36,350 Yes? 1131 01:16:36,350 --> 01:16:38,558 AUDIENCE: [INAUDIBLE] introduce the dynamics of brane 1132 01:16:38,558 --> 01:16:39,308 from [INAUDIBLE]. 1133 01:16:46,210 --> 01:16:47,196 HONG LIU: Sorry? 1134 01:16:47,196 --> 01:16:51,140 AUDIENCE: If we [INAUDIBLE] brane [INAUDIBLE], 1135 01:16:51,140 --> 01:16:53,605 then with the pre-factors [INAUDIBLE]. 1136 01:17:03,310 --> 01:17:04,120 HONG LIU: No. 1137 01:17:04,120 --> 01:17:07,600 Those pre-factors depend on your normalization. 1138 01:17:07,600 --> 01:17:10,230 Those pre-factors depend on normalization. 1139 01:17:10,230 --> 01:17:14,350 So you can normalize them in some particular way. 1140 01:17:14,350 --> 01:17:18,420 You can normalize them-- here, what 1141 01:17:18,420 --> 01:17:21,640 I'm saying is that there is a specific normalization, which 1142 01:17:21,640 --> 01:17:24,725 then the phi can precisely be interpreted 1143 01:17:24,725 --> 01:17:28,744 as the motion of the D-brane-- yeah, phi always corresponding 1144 01:17:28,744 --> 01:17:31,285 to the motion of the D-brane, but this specific normalization 1145 01:17:31,285 --> 01:17:35,210 in phi can be really interpreted as a dynamical version 1146 01:17:35,210 --> 01:17:37,590 of this coordinate in the transverse direction. 1147 01:17:37,590 --> 01:17:39,600 And with that normalization, then you 1148 01:17:39,600 --> 01:17:42,610 see that this [INAUDIBLE] action is 1149 01:17:42,610 --> 01:17:46,241 like the motion of a massive object. 1150 01:17:46,241 --> 01:17:46,740 OK? 1151 01:17:46,740 --> 01:17:50,670 It's like the motion of a massive object. 1152 01:17:50,670 --> 01:17:52,670 And this gives you the confirmation 1153 01:17:52,670 --> 01:17:55,560 that the phi indeed is degrees freedom-- should 1154 01:17:55,560 --> 01:17:58,600 be considered degrees freedom to describe 1155 01:17:58,600 --> 01:18:00,350 the motion of the D-brane. 1156 01:18:00,350 --> 01:18:01,221 Yes? 1157 01:18:01,221 --> 01:18:05,550 AUDIENCE: That's-- I mean, that is only in the [INAUDIBLE]. 1158 01:18:05,550 --> 01:18:07,474 If that analogy continued, [INAUDIBLE] 1159 01:18:07,474 --> 01:18:10,360 higher excitations of those closed-- open strings? 1160 01:18:10,360 --> 01:18:11,880 HONG LIU: Yeah. 1161 01:18:11,880 --> 01:18:14,370 So that's a good question, which I don't have time 1162 01:18:14,370 --> 01:18:18,320 to discuss now but I can-- but I will 1163 01:18:18,320 --> 01:18:21,540 describe at the beginning of next lecture. 1164 01:18:21,540 --> 01:18:24,600 So this is a no-energy story. 1165 01:18:24,600 --> 01:18:28,520 Then you can actually include a subset of higher-order terms. 1166 01:18:28,520 --> 01:18:30,740 Then you find that there's an infinite number 1167 01:18:30,740 --> 01:18:33,020 of higher-order terms you can sum, 1168 01:18:33,020 --> 01:18:35,760 and then this precisely generalize to the motion 1169 01:18:35,760 --> 01:18:37,497 of a [INAUDIBLE] object. 1170 01:18:37,497 --> 01:18:40,080 Yeah, because if you want to-- yeah, because this is no-energy 1171 01:18:40,080 --> 01:18:42,310 [INAUDIBLE] have to be non-relativistic. 1172 01:18:42,310 --> 01:18:44,160 But then you can generalize, generally 1173 01:18:44,160 --> 01:18:48,050 if you sum over an infinite number of higher terms, then 1174 01:18:48,050 --> 01:18:51,930 this describes [INAUDIBLE] motion of object. 1175 01:18:51,930 --> 01:18:54,490 And again, that's another independent confirmation 1176 01:18:54,490 --> 01:18:58,020 that this is the-- describe the motion of the D-brane. 1177 01:18:58,020 --> 01:18:58,695 Yes? 1178 01:18:58,695 --> 01:19:02,890 AUDIENCE: So I'm kind of fundamentally confused 1179 01:19:02,890 --> 01:19:05,320 about how-- so we started with a D-brane which 1180 01:19:05,320 --> 01:19:08,155 was a topological defect in our background manifold. 1181 01:19:08,155 --> 01:19:08,820 HONG LIU: Yeah. 1182 01:19:08,820 --> 01:19:10,333 AUDIENCE: So like a [INAUDIBLE] or something. 1183 01:19:10,333 --> 01:19:11,060 HONG LIU: Yeah. 1184 01:19:11,060 --> 01:19:14,654 AUDIENCE: And we are looking at motion of strings 1185 01:19:14,654 --> 01:19:15,820 in this background manifold. 1186 01:19:15,820 --> 01:19:16,640 HONG LIU: Yeah. 1187 01:19:16,640 --> 01:19:19,500 AUDIENCE: And this is-- so why do 1188 01:19:19,500 --> 01:19:23,475 we all of a sudden have the string motion affect 1189 01:19:23,475 --> 01:19:24,100 the background? 1190 01:19:24,100 --> 01:19:26,570 Like we never even introduced any coupling to the background. 1191 01:19:26,570 --> 01:19:27,194 HONG LIU: Yeah. 1192 01:19:27,194 --> 01:19:32,030 So that's precisely-- so this is a very good question. 1193 01:19:32,030 --> 01:19:38,150 This is precisely the remarkable thing about this string theory. 1194 01:19:38,150 --> 01:19:42,950 So yeah, so I was going to say at the beginning 1195 01:19:42,950 --> 01:19:46,570 of next lecture, but now let me say it. 1196 01:19:46,570 --> 01:19:50,650 So at the beginning, you introduce the D-brane just 1197 01:19:50,650 --> 01:19:53,150 by some rigid boundary conditions. 1198 01:19:53,150 --> 01:19:56,440 Some rigid boundary conditions. 1199 01:19:56,440 --> 01:19:58,890 And looks like this is a non-dynamical object 1200 01:19:58,890 --> 01:20:02,840 because you are introducing some rigid boundary conditions. 1201 01:20:02,840 --> 01:20:07,120 But now, when you quantize the strings under D-branes, 1202 01:20:07,120 --> 01:20:13,830 and then you find that those strings now give the D-brane 1203 01:20:13,830 --> 01:20:15,435 itself [INAUDIBLE]. 1204 01:20:18,510 --> 01:20:20,857 AUDIENCE: Is it because there's a graviton here? 1205 01:20:20,857 --> 01:20:22,940 HONG LIU: No, it has nothing to do with gravitons. 1206 01:20:22,940 --> 01:20:24,148 [INAUDIBLE] do with graviton. 1207 01:20:24,148 --> 01:20:27,150 Generally, you start with rigid boundary conditions. 1208 01:20:27,150 --> 01:20:30,360 But now you find [INAUDIBLE] boundary conditions. 1209 01:20:30,360 --> 01:20:33,620 Then you can generate-- then you have some excitations. 1210 01:20:33,620 --> 01:20:36,510 Then the coherent motions of these excitations 1211 01:20:36,510 --> 01:20:42,090 now give this object dynamics. 1212 01:20:42,090 --> 01:20:45,015 So that's a precisely remarkable thing about string theory. 1213 01:20:45,015 --> 01:20:47,860 AUDIENCE: What if you have two strings both ending 1214 01:20:47,860 --> 01:20:49,286 on the same brane? 1215 01:20:49,286 --> 01:20:50,207 HONG LIU: No, no, no. 1216 01:20:50,207 --> 01:20:51,790 No, this is including all the strings. 1217 01:20:51,790 --> 01:20:54,205 You can have as many strings as you want. 1218 01:20:54,205 --> 01:20:55,830 AUDIENCE: Well I mean, we're quantizing 1219 01:20:55,830 --> 01:20:57,206 one particular object, and we're looking 1220 01:20:57,206 --> 01:20:59,242 at a spectrum of a single string starting here-- 1221 01:20:59,242 --> 01:21:01,325 HONG LIU: Yeah, each string have such excitations. 1222 01:21:01,325 --> 01:21:03,270 AUDIENCE: But then each different string 1223 01:21:03,270 --> 01:21:05,330 will generate a different dynamics for the-- 1224 01:21:05,330 --> 01:21:06,550 HONG LIU: No, they're the same dynamic. 1225 01:21:06,550 --> 01:21:07,320 We just have two particles. 1226 01:21:07,320 --> 01:21:08,700 It's just like we have one photon here, 1227 01:21:08,700 --> 01:21:09,110 another photon there. 1228 01:21:09,110 --> 01:21:10,022 It's the same thing. 1229 01:21:14,130 --> 01:21:19,480 And this is the same thing about how string theory generates 1230 01:21:19,480 --> 01:21:21,440 gravity. 1231 01:21:21,440 --> 01:21:25,010 So let me just say a few words here. 1232 01:21:25,010 --> 01:21:28,550 So at the beginning, when we quantize the closed string, 1233 01:21:28,550 --> 01:21:31,400 we can see the closed string moving, again, 1234 01:21:31,400 --> 01:21:35,320 in the rigid Minkowski spacetime. 1235 01:21:35,320 --> 01:21:37,280 Yeah, because that [INAUDIBLE] region. 1236 01:21:37,280 --> 01:21:41,010 We just consider in the region in Minkowski spacetime. 1237 01:21:41,010 --> 01:21:43,670 But now, when you quantize that closed string, 1238 01:21:43,670 --> 01:21:47,130 when you see that the closed string actually half spin-2 two 1239 01:21:47,130 --> 01:21:49,440 excitations. 1240 01:21:49,440 --> 01:21:51,700 Half spin-2 excitations. 1241 01:21:51,700 --> 01:21:56,610 That means that string actually has gravitational effect. 1242 01:21:56,610 --> 01:22:02,010 Unless you have spin-2 excitations, then 1243 01:22:02,010 --> 01:22:06,430 Einstein-- already from Einstein, 1244 01:22:06,430 --> 01:22:11,662 gravity tells us that the spacetime is no longer rigid, 1245 01:22:11,662 --> 01:22:15,380 that actually you can deform the Minkowski 1246 01:22:15,380 --> 01:22:18,170 spacetime by putting in those excitations, 1247 01:22:18,170 --> 01:22:20,400 these spin-2 excitations. 1248 01:22:20,400 --> 01:22:23,390 And if you put enough of them, then [INAUDIBLE] 1249 01:22:23,390 --> 01:22:26,280 deform the Minkowski spacetime to some other curve spacetime, 1250 01:22:26,280 --> 01:22:30,440 then you can actually describe generically 1251 01:22:30,440 --> 01:22:32,360 the quantum gravitational dynamics 1252 01:22:32,360 --> 01:22:34,350 of a general curved spacetime. 1253 01:22:34,350 --> 01:22:35,980 So here, it's the same thing. 1254 01:22:35,980 --> 01:22:37,780 We start with a reach of the D-brane. 1255 01:22:37,780 --> 01:22:40,070 Then we quantize the excitations-- then 1256 01:22:40,070 --> 01:22:42,410 we quantize the degrees freedom on the D-brane. 1257 01:22:42,410 --> 01:22:45,000 And then we find that those degrees freedom 1258 01:22:45,000 --> 01:22:47,250 corresponding to the fluctuation of the D-brane. 1259 01:22:47,250 --> 01:22:49,530 And if you have enough of them-- say 1260 01:22:49,530 --> 01:22:54,030 if you have phi only depend on p to be the same everywhere, 1261 01:22:54,030 --> 01:22:58,170 so that's some coherent motion for every point on this brane. 1262 01:22:58,170 --> 01:23:02,530 So then if you build enough [INAUDIBLE] excitations, 1263 01:23:02,530 --> 01:23:05,780 then you can actually make the D-brane move [INAUDIBLE] 1264 01:23:05,780 --> 01:23:08,544 become a fully dynamic object. 1265 01:23:08,544 --> 01:23:10,210 So that's precisely the remarkable thing 1266 01:23:10,210 --> 01:23:11,110 about string theory. 1267 01:23:11,110 --> 01:23:11,385 Yes? 1268 01:23:11,385 --> 01:23:12,260 AUDIENCE: [INAUDIBLE] 1269 01:23:18,969 --> 01:23:21,510 HONG LIU: No, we don't know how to describe a dynamical brane 1270 01:23:21,510 --> 01:23:22,301 from the beginning. 1271 01:23:24,830 --> 01:23:26,370 No, that's how we do it. 1272 01:23:26,370 --> 01:23:30,050 We first start with some-- so this 1273 01:23:30,050 --> 01:23:33,420 is called a background dependent quantization. 1274 01:23:33,420 --> 01:23:36,496 You start with a specific solution. 1275 01:23:36,496 --> 01:23:38,120 You start with a specific configuration 1276 01:23:38,120 --> 01:23:41,910 of the D-brane, which you happen to be able to quantize. 1277 01:23:41,910 --> 01:23:43,650 Then you can find all the excitations 1278 01:23:43,650 --> 01:23:45,910 around that particular configuration, 1279 01:23:45,910 --> 01:23:48,520 and then you can turn on those excitations 1280 01:23:48,520 --> 01:23:49,955 to move away from those points. 1281 01:23:53,100 --> 01:23:55,190 So that's essentially what we're doing here. 1282 01:23:55,190 --> 01:23:55,690 OK. 1283 01:23:55,690 --> 01:23:57,323 Let's stop.