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 --> 00:00:06,050 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 --> 00:00:12,690 To make a donation or to view additional materials 6 00:00:12,690 --> 00:00:16,610 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,610 --> 00:00:17,266 at ocw.mit.edu. 8 00:00:21,590 --> 00:00:27,310 HONG LIU: So last time, we talked about-- we introduced 9 00:00:27,310 --> 00:00:29,950 the concept of D-branes. 10 00:00:29,950 --> 00:00:34,720 And then we quantized open strings on the D-branes. 11 00:00:34,720 --> 00:00:40,880 And we see the massive spectrum on the D-branes 12 00:00:40,880 --> 00:00:45,520 includes, say, massless gauge field, 13 00:00:45,520 --> 00:00:50,030 and also some massless scalar fields. 14 00:00:50,030 --> 00:00:58,030 And then I described that one can interpolate 15 00:00:58,030 --> 00:00:59,770 the dynamics of the scalar fields 16 00:00:59,770 --> 00:01:02,500 actually as the motion of the D-branes. 17 00:01:02,500 --> 00:01:04,860 So in other words, at the beginning, 18 00:01:04,860 --> 00:01:08,700 even when we quantize the open string, 19 00:01:08,700 --> 00:01:11,230 we started with a rigid boundary condition, 20 00:01:11,230 --> 00:01:13,550 so we started with a rigid ring. 21 00:01:13,550 --> 00:01:19,470 But now, after you quantize it, then you get the fluctuations. 22 00:01:19,470 --> 00:01:21,291 And because of those fluctuations, 23 00:01:21,291 --> 00:01:22,540 the D-branes become dynamical. 24 00:01:25,200 --> 00:01:27,000 Those fluctuations of the D-branes 25 00:01:27,000 --> 00:01:29,750 make it into a dynamic object in principle 26 00:01:29,750 --> 00:01:33,140 to make it move, or et cetera. 27 00:01:33,140 --> 00:01:35,630 So now let's say a little bit regarding 28 00:01:35,630 --> 00:01:38,120 the math of a D-brane. 29 00:01:38,120 --> 00:01:42,680 In the gravitational theory, anything gravitates. 30 00:01:42,680 --> 00:01:44,080 D-brane will have energy. 31 00:01:44,080 --> 00:01:46,440 It will have a mass, et cetera. 32 00:01:46,440 --> 00:01:50,210 so let's talk about what should be the mass of a D-brane 33 00:01:50,210 --> 00:01:51,435 if it's a dynamic object. 34 00:02:07,760 --> 00:02:12,340 So here, there's a very simple and intuitive answer. 35 00:02:12,340 --> 00:02:18,450 So on the D-brane, there are many open strings. 36 00:02:18,450 --> 00:02:20,600 In principle, there are an infinite number 37 00:02:20,600 --> 00:02:25,970 of open string excitations live on the D-brane. 38 00:02:25,970 --> 00:02:27,750 And each of them can be considered, say, 39 00:02:27,750 --> 00:02:31,070 as a space time field living on the D-brane, et cetera. 40 00:02:31,070 --> 00:02:42,990 So actual definition for the mass of a D-brane 41 00:02:42,990 --> 00:02:48,010 is that this should be the energy of a D-brane, which 42 00:02:48,010 --> 00:02:50,930 essentially should be the ground state, the energy of the ground 43 00:02:50,930 --> 00:02:52,280 state of the D-brane. 44 00:02:52,280 --> 00:02:54,220 The energy of the ground state of a D-brane 45 00:02:54,220 --> 00:02:57,150 could be corresponding to the energy of the D-brane none 46 00:02:57,150 --> 00:03:00,300 of those strings are excited. 47 00:03:00,300 --> 00:03:08,656 That should correspond to the vacuum energy 48 00:03:08,656 --> 00:03:17,006 of open strings living on it. 49 00:03:24,290 --> 00:03:29,440 So this is very intuitive definition 50 00:03:29,440 --> 00:03:31,520 and obviously makes sense. 51 00:03:31,520 --> 00:03:35,800 So we can write the mass of a D-brane, DP-brane, 52 00:03:35,800 --> 00:03:38,890 as the tension, which is the mass per unit 53 00:03:38,890 --> 00:03:42,490 volume times the total volume. 54 00:03:42,490 --> 00:03:45,080 And this should be equal to the vacuum 55 00:03:45,080 --> 00:03:47,590 energy of all the open strings. 56 00:03:50,890 --> 00:03:55,170 So say each open string excitation corresponding 57 00:03:55,170 --> 00:03:55,680 to a field. 58 00:03:55,680 --> 00:03:56,785 You have a tachyon. 59 00:03:56,785 --> 00:03:58,196 You have gauge field. 60 00:03:58,196 --> 00:03:59,820 You have massive scalar field, and also 61 00:03:59,820 --> 00:04:01,420 infinite number of massive fields. 62 00:04:01,420 --> 00:04:04,460 All those fields, they have vacuum energy. 63 00:04:04,460 --> 00:04:07,235 So you need to sum all of them together. 64 00:04:09,910 --> 00:04:12,400 The sum of those vacuum energies would 65 00:04:12,400 --> 00:04:14,174 be the mass of the D-brane. 66 00:04:16,959 --> 00:04:20,560 So this can be achieved just by doing the vacuum 67 00:04:20,560 --> 00:04:23,980 diagram of open strings. 68 00:04:23,980 --> 00:04:26,660 So we describe in the closed stream case, 69 00:04:26,660 --> 00:04:29,340 if you want to find the vacuum energy in the closed string, 70 00:04:29,340 --> 00:04:34,010 you just sum of all possible. 71 00:04:34,010 --> 00:04:35,910 So this will be just the vacuum diagram. 72 00:04:51,036 --> 00:04:54,580 In other words, so the difference 73 00:04:54,580 --> 00:04:57,840 between the open string is that open strings have boundary 74 00:04:57,840 --> 00:05:02,810 and closed strings are closed. 75 00:05:02,810 --> 00:05:14,450 So that means the sum of all two dimensional surfaces 76 00:05:14,450 --> 00:05:32,610 with this one boundaries but no external open strings. 77 00:05:35,440 --> 00:05:40,180 This is the natural definition of the vacuum diagram, 78 00:05:40,180 --> 00:05:42,860 as we would do. 79 00:05:42,860 --> 00:05:44,890 And we will do in the Euclidean path integral. 80 00:05:44,890 --> 00:05:47,222 So you can do this in the Euclidean path integral. 81 00:05:53,980 --> 00:05:56,480 You sum over all surfaces. 82 00:05:56,480 --> 00:06:00,770 In the case when you need the sum of all surfaces, 83 00:06:00,770 --> 00:06:03,460 in some sense, the only way we know how to define such a sum 84 00:06:03,460 --> 00:06:05,270 is to do the Euclidean path integral. 85 00:06:13,930 --> 00:06:21,820 And this sum-- so previously, we talked about the vacuum 86 00:06:21,820 --> 00:06:24,280 energy of the closed stream, the sum 87 00:06:24,280 --> 00:06:30,150 of all possible closed surfaces, say of different topology. 88 00:06:30,150 --> 00:06:33,070 So here, again, you sum of all possible surfaces 89 00:06:33,070 --> 00:06:35,160 with this one boundaries. 90 00:06:35,160 --> 00:06:37,766 So the simplest surface with one boundary is a disk. 91 00:06:41,446 --> 00:06:43,070 The difference with closed string case, 92 00:06:43,070 --> 00:06:46,960 now you have to sum over surfaces with boundaries. 93 00:06:46,960 --> 00:06:49,518 So the simplest one would be a disk. 94 00:06:49,518 --> 00:06:53,025 And the next would be annulus. 95 00:06:58,100 --> 00:07:01,196 Now you have two boundaries rather than one boundary. 96 00:07:01,196 --> 00:07:03,445 And exactly you can consider more and more complicated 97 00:07:03,445 --> 00:07:04,169 diagrams. 98 00:07:04,169 --> 00:07:06,335 You can consider more and more complicated diagrams. 99 00:07:09,730 --> 00:07:13,410 And the way to weight those surfaces exactly the same 100 00:07:13,410 --> 00:07:17,970 as before is that you have this g string 101 00:07:17,970 --> 00:07:20,065 than to the power minus chi, and the chi 102 00:07:20,065 --> 00:07:24,860 is the Euler number, just exactly as we described before. 103 00:07:24,860 --> 00:07:29,740 And Euler number now to apply surfaces with boundaries 104 00:07:29,740 --> 00:07:33,524 would be-- so previously, Euler number is 2 minus 2h. 105 00:07:33,524 --> 00:07:39,279 h is number of genuses, or number of holes. 106 00:07:39,279 --> 00:07:41,695 But now you also need to include the number of boundaries, 107 00:07:41,695 --> 00:07:44,660 which are called b. 108 00:07:44,660 --> 00:07:46,190 So when you include the boundaries, 109 00:07:46,190 --> 00:07:48,890 then that will change your Euler number, 110 00:07:48,890 --> 00:07:53,150 and also change the weight for each diagram. 111 00:07:56,230 --> 00:07:58,140 You can also add handles here. 112 00:07:58,140 --> 00:07:59,760 You can also add handles here. 113 00:07:59,760 --> 00:08:01,755 You can also add genuses to the disk. 114 00:08:05,010 --> 00:08:07,030 You can also add h here. 115 00:08:10,114 --> 00:08:11,780 So according to this counting, then this 116 00:08:11,780 --> 00:08:13,900 would count as gs minus 1. 117 00:08:18,290 --> 00:08:21,200 So this one has zero holes and one boundaries. 118 00:08:21,200 --> 00:08:22,740 So this is 2 minus 1. 119 00:08:22,740 --> 00:08:23,840 So this is 1. 120 00:08:23,840 --> 00:08:26,840 So this is gs to the power negative 1. 121 00:08:26,840 --> 00:08:30,350 And this one has low hole but two boundaries. 122 00:08:30,350 --> 00:08:31,680 2 minus 2 is 0. 123 00:08:31,680 --> 00:08:35,990 This is 1 to the power gs to the 0. 124 00:08:35,990 --> 00:08:38,230 And then you have higher diagrams. 125 00:08:38,230 --> 00:08:43,409 You have higher surfaces with all positive powers, gs. 126 00:08:43,409 --> 00:08:44,344 Yes? 127 00:08:44,344 --> 00:08:46,044 AUDIENCE: What about like a Mobius strip 128 00:08:46,044 --> 00:08:46,960 that has one boundary? 129 00:08:46,960 --> 00:08:48,020 HONG LIU: Right. 130 00:08:48,020 --> 00:08:50,480 So the Mobius strip is a very good question. 131 00:08:50,480 --> 00:08:53,320 A Mobius strip is unoriented surface. 132 00:08:53,320 --> 00:08:55,890 So here, we can see the oriented string. 133 00:08:55,890 --> 00:08:59,750 You can consider unoriented string. 134 00:08:59,750 --> 00:09:04,120 But most of what we said applies to that case. 135 00:09:04,120 --> 00:09:06,900 It's just we have to worry a little bit about orientation, 136 00:09:06,900 --> 00:09:08,520 so we don't go there. 137 00:09:08,520 --> 00:09:10,330 AUDIENCE: Will that really contribute 138 00:09:10,330 --> 00:09:12,089 to the vector [INAUDIBLE]? 139 00:09:12,089 --> 00:09:12,630 HONG LIU: Hm? 140 00:09:12,630 --> 00:09:16,240 AUDIENCE: Will the Mobius strip contribute to the-- 141 00:09:16,240 --> 00:09:17,580 HONG LIU: Yeah, yeah, it will. 142 00:09:17,580 --> 00:09:24,880 In the case when you have unoriented string. 143 00:09:24,880 --> 00:09:27,750 AUDIENCE: But you make a restriction and say, 144 00:09:27,750 --> 00:09:31,985 on this D-brane, we have or not have unoriented-- 145 00:09:31,985 --> 00:09:34,110 HONG LIU: Here, we only consider oriented surfaces. 146 00:09:34,110 --> 00:09:35,860 We only consider oriented strings. 147 00:09:35,860 --> 00:09:39,940 We have not talked about unoriented strings. 148 00:09:39,940 --> 00:09:42,610 AUDIENCE: But no restriction in principle. 149 00:09:42,610 --> 00:09:43,360 HONG LIU: You can. 150 00:09:46,060 --> 00:09:48,750 It's actually a technical complication 151 00:09:48,750 --> 00:09:51,454 I don't want to go into right now. 152 00:09:51,454 --> 00:09:52,340 Yes? 153 00:09:52,340 --> 00:09:54,960 AUDIENCE: So we think of vertical axis as time. 154 00:09:54,960 --> 00:09:57,062 So the disk would be a string kind 155 00:09:57,062 --> 00:09:58,760 of nucleating and propagating-- 156 00:09:58,760 --> 00:10:00,720 HONG LIU: No. 157 00:10:00,720 --> 00:10:02,820 This is a Euclidean and you can think of time 158 00:10:02,820 --> 00:10:04,214 as whatever you want. 159 00:10:04,214 --> 00:10:04,880 AUDIENCE: Right. 160 00:10:04,880 --> 00:10:07,540 But still, so the disk would be like a nucleating open string 161 00:10:07,540 --> 00:10:10,300 that propagates and then disappears, right? 162 00:10:10,300 --> 00:10:12,340 HONG LIU: Yeah. 163 00:10:12,340 --> 00:10:14,530 For example, this open string you can consider. 164 00:10:19,386 --> 00:10:21,010 Heuristically, you may be able to think 165 00:10:21,010 --> 00:10:26,025 of some kind of a single string, just rotate, for example. 166 00:10:26,025 --> 00:10:27,875 AUDIENCE: Like forever? 167 00:10:27,875 --> 00:10:28,500 HONG LIU: Yeah. 168 00:10:28,500 --> 00:10:30,180 For example, I just said a single. 169 00:10:30,180 --> 00:10:33,620 I'm just saying it's hard to interpret as a time now. 170 00:10:33,620 --> 00:10:37,489 But the time in this direction would be periodic time 171 00:10:37,489 --> 00:10:39,030 if you think from that point of view. 172 00:10:42,862 --> 00:10:44,320 But the good thing is that when you 173 00:10:44,320 --> 00:10:50,700 go to Euclidean, what you call time and the spatial direction 174 00:10:50,700 --> 00:10:53,330 then becomes obscured. 175 00:10:53,330 --> 00:10:54,640 It depends on your convenience. 176 00:10:57,440 --> 00:11:00,989 AUDIENCE: In the center, is that a genus? 177 00:11:00,989 --> 00:11:01,530 HONG LIU: No. 178 00:11:01,530 --> 00:11:05,515 The center is completely smooth. 179 00:11:05,515 --> 00:11:07,339 AUDIENCE: It's not like a torus? 180 00:11:07,339 --> 00:11:07,880 HONG LIU: No. 181 00:11:07,880 --> 00:11:09,010 A disk is a disk. 182 00:11:09,010 --> 00:11:11,120 A disk is not a torus. 183 00:11:11,120 --> 00:11:13,000 AUDIENCE: But there's a-- 184 00:11:13,000 --> 00:11:14,740 HONG LIU: You're talking about this one? 185 00:11:14,740 --> 00:11:15,365 AUDIENCE: Yeah. 186 00:11:15,365 --> 00:11:16,570 HONG LIU: Oh. 187 00:11:16,570 --> 00:11:19,630 This one is annulus. 188 00:11:19,630 --> 00:11:20,620 This guy is annulus. 189 00:11:20,620 --> 00:11:21,645 This is a flat surface. 190 00:11:24,255 --> 00:11:26,590 AUDIENCE: So the inside and outside are the-- 191 00:11:26,590 --> 00:11:27,865 HONG LIU: They are different. 192 00:11:27,865 --> 00:11:31,330 If you identify this and that, then they become a torus. 193 00:11:36,154 --> 00:11:37,820 When you identify this one and that one, 194 00:11:37,820 --> 00:11:39,985 then they become a torus. 195 00:11:39,985 --> 00:11:41,617 Then you get rid of the boundaries. 196 00:11:41,617 --> 00:11:43,950 When you identify them, then there's no boundary anymore 197 00:11:43,950 --> 00:11:45,935 because they become a circle. 198 00:11:50,070 --> 00:11:51,810 Good? 199 00:11:51,810 --> 00:12:01,110 So that means if I have weak coupling, that 200 00:12:01,110 --> 00:12:05,920 means when gs is much smaller than 1, which is the cases we 201 00:12:05,920 --> 00:12:09,420 can only consider because if you have gs more than 1, then 202 00:12:09,420 --> 00:12:11,620 you have some infinite number of diagrams. 203 00:12:11,620 --> 00:12:15,590 And we don't know how to deal with this. 204 00:12:15,590 --> 00:12:20,840 So weak coupling, when gs more than 1, then the brane tension, 205 00:12:20,840 --> 00:12:25,360 then the D-brane will be always scaled 206 00:12:25,360 --> 00:12:28,510 with string coupling at 1 over gs because of that, 207 00:12:28,510 --> 00:12:32,390 because this term will dominate. 208 00:12:32,390 --> 00:12:33,880 This term will dominate. 209 00:12:33,880 --> 00:12:35,630 And then the energy should be 1 over gs. 210 00:12:39,790 --> 00:12:45,060 This is a very important result. It's a very important result. 211 00:12:45,060 --> 00:12:48,170 The mass of the D-brane is actually 1 over gs. 212 00:12:52,010 --> 00:13:03,680 So on dimensional ground, you can just essentially write down 213 00:13:03,680 --> 00:13:06,180 what's the tension of the D-brane 214 00:13:06,180 --> 00:13:08,765 because the only dimensional parameter is alpha prime. 215 00:13:11,640 --> 00:13:13,250 So the dimension of the D-brane should 216 00:13:13,250 --> 00:13:18,160 be-- so this is mass per unit volume. 217 00:13:18,160 --> 00:13:20,780 So you have a p dimension of volume, then 218 00:13:20,780 --> 00:13:22,830 that would be p plus 1. 219 00:13:22,830 --> 00:13:26,900 So the mass dimension of the tension would be p plus 1. 220 00:13:26,900 --> 00:13:30,045 So just on dimension ground, I can write gs 221 00:13:30,045 --> 00:13:37,000 because it's 1 over s, alpha prime 1/2 p plus 1. 222 00:13:37,000 --> 00:13:39,940 So that gives you the right dimension. 223 00:13:39,940 --> 00:13:43,210 And then you can have some numerical constant 224 00:13:43,210 --> 00:13:46,106 which you need to determine. 225 00:13:46,106 --> 00:13:54,500 You have some numerical constant which 226 00:13:54,500 --> 00:14:01,871 you can determine string theory by doing that path integral. 227 00:14:04,760 --> 00:14:06,880 Any questions about this? 228 00:14:06,880 --> 00:14:07,839 Yes? 229 00:14:07,839 --> 00:14:10,284 AUDIENCE: Do these open string vacuum diagrams 230 00:14:10,284 --> 00:14:12,718 have any interpretations like half of a closed string 231 00:14:12,718 --> 00:14:13,218 diagram? 232 00:14:13,218 --> 00:14:15,180 So if you put two disks together, you have a sphere. 233 00:14:15,180 --> 00:14:15,910 HONG LIU: Right. 234 00:14:15,910 --> 00:14:18,950 We will talk about this in a minute. 235 00:14:18,950 --> 00:14:19,742 Yes? 236 00:14:19,742 --> 00:14:22,700 AUDIENCE: With the logical decomposition 237 00:14:22,700 --> 00:14:26,080 of the powers of g, why is the disk g minus 1. 238 00:14:26,080 --> 00:14:30,155 HONG LIU: This comes from this formula. 239 00:14:30,155 --> 00:14:33,760 As we discussed before, the weight of different topology 240 00:14:33,760 --> 00:14:38,820 is always weighted by some constant to the power of Euler 241 00:14:38,820 --> 00:14:40,060 number. 242 00:14:40,060 --> 00:14:42,560 Now, the Euler number, if you have surfaces with boundaries, 243 00:14:42,560 --> 00:14:46,910 then the Euler number depends on the number of boundaries. 244 00:14:46,910 --> 00:14:48,990 And then you can just work it out. 245 00:14:51,830 --> 00:14:55,650 So this is based on simple topology. 246 00:14:55,650 --> 00:14:58,316 Any other questions? 247 00:14:58,316 --> 00:14:58,815 Yes? 248 00:14:58,815 --> 00:14:59,560 AUDIENCE: I'm sorry. 249 00:14:59,560 --> 00:15:01,809 I'm just a little bit confused about the vacuum energy 250 00:15:01,809 --> 00:15:05,200 here as the one-- remember when you calculate 251 00:15:05,200 --> 00:15:06,421 the mass of the string. 252 00:15:06,421 --> 00:15:08,480 You know, we have a naught term there. 253 00:15:08,480 --> 00:15:11,234 There is no excitation. 254 00:15:11,234 --> 00:15:11,900 HONG LIU: Sorry. 255 00:15:11,900 --> 00:15:12,441 Say it again. 256 00:15:12,441 --> 00:15:15,125 AUDIENCE: So when we calculate the mass of the open string 257 00:15:15,125 --> 00:15:18,030 and there is a0 term, which is completely different. 258 00:15:18,030 --> 00:15:20,680 HONG LIU: That's completely different. 259 00:15:20,680 --> 00:15:25,270 So that a0, we considered before, it's 260 00:15:25,270 --> 00:15:27,770 the zero point energy for the oscillation 261 00:15:27,770 --> 00:15:29,510 modes on the screen. 262 00:15:32,380 --> 00:15:37,740 So that a0 is that we are considering 263 00:15:37,740 --> 00:15:45,300 this string, and the zero point energy for the oscillation 264 00:15:45,300 --> 00:15:47,200 mode on this string. 265 00:15:47,200 --> 00:15:49,770 But here, we are considering the zero point 266 00:15:49,770 --> 00:15:50,970 energy not of the string. 267 00:15:50,970 --> 00:15:54,790 We are considering the zero point energy of the D-brane. 268 00:15:54,790 --> 00:15:56,820 And the zero point energy of the D-brane 269 00:15:56,820 --> 00:16:01,760 would be to write down the vacuum energy of all the fields 270 00:16:01,760 --> 00:16:03,350 living on the D-brane. 271 00:16:03,350 --> 00:16:05,700 And all the fields living on the D-brane corresponding 272 00:16:05,700 --> 00:16:09,600 to all the-- now, each string excitation 273 00:16:09,600 --> 00:16:11,390 becomes a field on the D-brane. 274 00:16:11,390 --> 00:16:13,540 And so that's corresponding to sum of that. 275 00:16:13,540 --> 00:16:15,280 And that, then, in turn corresponding 276 00:16:15,280 --> 00:16:16,795 to sum of these kind of surfaces. 277 00:16:19,896 --> 00:16:21,380 Any other questions? 278 00:16:24,350 --> 00:16:26,230 So there's an alternative way to think 279 00:16:26,230 --> 00:16:30,115 about how to compute the D-brane mass 280 00:16:30,115 --> 00:16:38,960 or energy as follows, which is actually extremely instructive. 281 00:16:38,960 --> 00:16:40,985 There is an alternative way of doing this. 282 00:16:47,430 --> 00:16:50,280 So let's consider just D-brane. 283 00:16:54,410 --> 00:16:57,060 So consider the interaction between the two D-branes. 284 00:16:57,060 --> 00:17:00,580 So let's consider two D-branes separated by some distance. 285 00:17:05,760 --> 00:17:08,838 And then they have a mass. 286 00:17:08,838 --> 00:17:13,126 Then they will interact gravitationally. 287 00:17:13,126 --> 00:17:14,750 In particular, in a weak coupling limit 288 00:17:14,750 --> 00:17:16,490 they're pretty massive. 289 00:17:16,490 --> 00:17:19,650 They're very massive because it's 1 over the g string. 290 00:17:19,650 --> 00:17:24,670 So when g string is small, which is the only regime 291 00:17:24,670 --> 00:17:26,890 we're working with, so the D-brane is very heavy. 292 00:17:29,590 --> 00:17:31,970 And so you can ask, what is the gravitational attraction 293 00:17:31,970 --> 00:17:33,830 between the two? 294 00:17:33,830 --> 00:17:37,170 What is the interaction between the two? 295 00:17:37,170 --> 00:17:41,960 And we know that at low energies, 296 00:17:41,960 --> 00:17:44,170 say if the two D-branes are not excited, 297 00:17:44,170 --> 00:17:48,400 if their distance are very far apart, 298 00:17:48,400 --> 00:17:51,770 then the leading interaction between them 299 00:17:51,770 --> 00:17:53,710 just comes from the massless mode 300 00:17:53,710 --> 00:17:57,540 because only massless mode mediates normal interactions. 301 00:17:57,540 --> 00:18:00,210 And so interaction between them just 302 00:18:00,210 --> 00:18:06,630 comes from graviton or this [INAUDIBLE], 303 00:18:06,630 --> 00:18:11,040 essentially just corresponding to small number of massless 304 00:18:11,040 --> 00:18:12,540 closed string modes. 305 00:18:12,540 --> 00:18:14,650 Only those massless modes will contribute 306 00:18:14,650 --> 00:18:16,410 because the massive mode only contributes short range 307 00:18:16,410 --> 00:18:16,951 interactions. 308 00:18:20,140 --> 00:18:25,540 AUDIENCE: Why not the vector mode in open string? 309 00:18:25,540 --> 00:18:27,660 HONG LIU: No. 310 00:18:27,660 --> 00:18:30,740 Vector mode of open string only lives on each brane. 311 00:18:30,740 --> 00:18:34,960 AUDIENCE: But we can have an open string like-- 312 00:18:34,960 --> 00:18:37,500 HONG LIU: I will talk about that separately. 313 00:18:37,500 --> 00:18:38,480 Just wait a little bit. 314 00:18:41,190 --> 00:18:45,154 So if I think purely from the alternative gravity 315 00:18:45,154 --> 00:18:47,320 point of view, not from string theory point of view, 316 00:18:47,320 --> 00:18:50,310 I have two massive objects. 317 00:18:50,310 --> 00:18:53,060 I want to look at the interaction between them. 318 00:18:53,060 --> 00:18:58,350 And then interaction will be proportional to gn, 319 00:18:58,350 --> 00:19:00,920 say their mass. 320 00:19:00,920 --> 00:19:04,312 So if I factor out the volume factor, 321 00:19:04,312 --> 00:19:08,030 it would be just GN TP squared. 322 00:19:08,030 --> 00:19:10,694 So this essentially is the gravitational interaction 323 00:19:10,694 --> 00:19:11,360 between the two. 324 00:19:14,070 --> 00:19:16,230 And from the string theory point of view, 325 00:19:16,230 --> 00:19:17,740 such a diagram corresponding to you 326 00:19:17,740 --> 00:19:19,580 exchange your closed string. 327 00:19:25,270 --> 00:19:27,380 So this diagram corresponding says, 328 00:19:27,380 --> 00:19:29,370 suppose you have brane one, brane two. 329 00:19:32,760 --> 00:19:37,950 So this picture that brane one will emit graviton 330 00:19:37,950 --> 00:19:39,940 absorbed by the other brane. 331 00:19:39,940 --> 00:19:45,440 And then that's how we measure, say, the newtons 332 00:19:45,440 --> 00:19:49,000 force between them. 333 00:19:49,000 --> 00:19:51,500 And it translates to the string theory picture. 334 00:19:51,500 --> 00:19:56,150 This corresponding to one D-brane emits a closed string, 335 00:19:56,150 --> 00:20:00,689 and then absorbed by the other D-brane. 336 00:20:00,689 --> 00:20:02,230 And when you go to [INAUDIBLE], which 337 00:20:02,230 --> 00:20:05,310 only massless mode matters, and then becomes this picture. 338 00:20:08,272 --> 00:20:10,480 So this is the string theory version of that diagram. 339 00:20:15,900 --> 00:20:17,330 So now essentially, what you need 340 00:20:17,330 --> 00:20:21,790 to do to calculate this thing in the string theory 341 00:20:21,790 --> 00:20:26,110 is to calculate this thing in the diagram. 342 00:20:26,110 --> 00:20:29,270 So form the string theory point of view, 343 00:20:29,270 --> 00:20:39,205 now what you need to consider is to do path integral 344 00:20:39,205 --> 00:20:43,220 on the topology of a cylinder with one 345 00:20:43,220 --> 00:20:46,420 boundary on the brane one and the other boundary 346 00:20:46,420 --> 00:20:47,040 on brane two. 347 00:20:50,270 --> 00:20:54,560 So this corresponds to exchange of a closed 348 00:20:54,560 --> 00:20:56,350 string in this direction. 349 00:21:03,299 --> 00:21:04,090 AUDIENCE: Question. 350 00:21:04,090 --> 00:21:06,860 HONG LIU: Yes? 351 00:21:06,860 --> 00:21:09,240 AUDIENCE: What's the mechanism for the D-brane emitting 352 00:21:09,240 --> 00:21:10,440 a closed string? 353 00:21:10,440 --> 00:21:13,960 Or equivalently, on the other picture, 354 00:21:13,960 --> 00:21:16,500 why can it emit a graviton? 355 00:21:16,500 --> 00:21:18,830 HONG LIU: It's coupled to graviton. 356 00:21:18,830 --> 00:21:21,469 AUDIENCE: So how did we introduce the coupling? 357 00:21:21,469 --> 00:21:22,010 HONG LIU: Hm? 358 00:21:22,010 --> 00:21:23,130 AUDIENCE: How did we introduce the coupling? 359 00:21:23,130 --> 00:21:24,740 I mean, we introduced them as boundary conditions 360 00:21:24,740 --> 00:21:25,620 for open strings. 361 00:21:25,620 --> 00:21:26,570 HONG LIU: Yeah. 362 00:21:26,570 --> 00:21:29,445 AUDIENCE: So does that naturally introduce coupling? 363 00:21:29,445 --> 00:21:30,320 HONG LIU: No, no, no. 364 00:21:30,320 --> 00:21:32,410 This is what I'm writing here. 365 00:21:32,410 --> 00:21:36,040 And this diagram, you emit from a closed string corresponding 366 00:21:36,040 --> 00:21:38,950 to look at the cylinder. 367 00:21:38,950 --> 00:21:47,390 One boundary of the cylinder on the location with one D-brane, 368 00:21:47,390 --> 00:21:49,680 and the other boundary of the cylinder on the location 369 00:21:49,680 --> 00:21:51,140 of the other D-brane. 370 00:21:51,140 --> 00:21:54,290 And then you just integrate over this surface. 371 00:21:54,290 --> 00:22:00,235 And then that will give you the graviton exchange. 372 00:22:00,235 --> 00:22:01,735 That will give you the closed string 373 00:22:01,735 --> 00:22:03,254 exchange between the two. 374 00:22:03,254 --> 00:22:05,920 AUDIENCE: So the coupling of the closed string to the brane kind 375 00:22:05,920 --> 00:22:07,860 of naturally arises? 376 00:22:07,860 --> 00:22:10,860 HONG LIU: In the boundary condition imposed here. 377 00:22:10,860 --> 00:22:17,630 So you impose this closed string to initiate it 378 00:22:17,630 --> 00:22:20,260 from-- so you impose the boundary condition here 379 00:22:20,260 --> 00:22:25,140 so that this closed string starts from brane one 380 00:22:25,140 --> 00:22:27,440 and then ends on brane two. 381 00:22:27,440 --> 00:22:30,200 And then you integrate over all surfaces 382 00:22:30,200 --> 00:22:31,880 this cylinder topology. 383 00:22:34,410 --> 00:22:36,690 AUDIENCE: We know the interaction constant 384 00:22:36,690 --> 00:22:41,620 for closed string is g closed, but is it the same here? 385 00:22:41,620 --> 00:22:42,390 HONG LIU: No. 386 00:22:42,390 --> 00:22:45,427 That's what I'm going to talk about. 387 00:22:45,427 --> 00:22:46,010 Is this clear? 388 00:22:49,090 --> 00:22:49,590 OK. 389 00:22:49,590 --> 00:22:52,080 So now I'm going to mention two things. 390 00:22:52,080 --> 00:22:54,340 First, as I said before, whenever 391 00:22:54,340 --> 00:22:57,850 we do some calculations, we often do analytic integration 392 00:22:57,850 --> 00:23:00,420 to the Euclidean signature. 393 00:23:00,420 --> 00:23:04,240 So now it will be the same. 394 00:23:04,240 --> 00:23:08,550 When we do this calculation, we have a closed string start 395 00:23:08,550 --> 00:23:11,610 at location of the brane one, and move forward 396 00:23:11,610 --> 00:23:14,740 in time to end on the brane two. 397 00:23:14,740 --> 00:23:17,130 So this is the simplest diagram. 398 00:23:17,130 --> 00:23:20,660 You can also add some holes here. 399 00:23:20,660 --> 00:23:22,390 You can also add some holes here. 400 00:23:22,390 --> 00:23:25,110 And then that corresponds to higher order diagram structure. 401 00:23:25,110 --> 00:23:27,650 So now don't worry about that. 402 00:23:27,650 --> 00:23:29,030 Just the simplest diagram. 403 00:23:29,030 --> 00:23:29,998 Yes? 404 00:23:29,998 --> 00:23:34,260 AUDIENCE: Was it time dimension inside brane 405 00:23:34,260 --> 00:23:37,592 to define time dimension as one of the dimensions 406 00:23:37,592 --> 00:23:39,083 that lived inside brane? 407 00:23:39,083 --> 00:23:40,080 HONG LIU: Yeah. 408 00:23:40,080 --> 00:23:46,280 But this is not a space time [INAUDIBLE]. 409 00:23:46,280 --> 00:23:48,380 This is virtual time. 410 00:23:48,380 --> 00:23:52,590 This is virtual time associated with this graviton. 411 00:23:52,590 --> 00:23:55,740 So essentially, I do create the closed string here 412 00:23:55,740 --> 00:23:58,670 on this brane and then propagate then absorbed 413 00:23:58,670 --> 00:24:00,370 by the other D-brane. 414 00:24:03,150 --> 00:24:04,900 AUDIENCE: Can you also think about this as 415 00:24:04,900 --> 00:24:06,180 like an open string-- 416 00:24:06,180 --> 00:24:07,440 HONG LIU: Yeah. 417 00:24:07,440 --> 00:24:09,330 One second. 418 00:24:09,330 --> 00:24:10,230 I'm going to explain. 419 00:24:14,130 --> 00:24:18,200 So now there are two remarkable things about this diagram. 420 00:24:18,200 --> 00:24:20,270 The two remarkable things about this diagram. 421 00:24:20,270 --> 00:24:24,640 First, this is a cylinder diagram. 422 00:24:24,640 --> 00:24:27,476 And this is a diagram with two boundaries 423 00:24:27,476 --> 00:24:31,180 because we have to emit a closed string from here. 424 00:24:31,180 --> 00:24:32,520 And so you have one boundary. 425 00:24:32,520 --> 00:24:34,250 You have an initial closed string, 426 00:24:34,250 --> 00:24:36,260 then you have a final closed string. 427 00:24:36,260 --> 00:24:41,148 So this is a surface with two boundaries with no holes. 428 00:24:41,148 --> 00:24:45,190 And if you calculate the chi, so this would be zero. 429 00:24:45,190 --> 00:24:51,190 Then that means this diagram is gs to power of 0. 430 00:24:51,190 --> 00:24:53,779 And then from here, we know that then 431 00:24:53,779 --> 00:24:55,320 this from string theory point of view 432 00:24:55,320 --> 00:24:57,400 will be gs to the power of 0. 433 00:24:57,400 --> 00:25:01,050 And so this is another way to see that the TP should 434 00:25:01,050 --> 00:25:03,096 be 1 over g string. 435 00:25:08,080 --> 00:25:11,990 Because we said before that the G Newton-- 436 00:25:11,990 --> 00:25:14,690 so we explained before G Newton would be order of gs squared. 437 00:25:18,880 --> 00:25:22,470 G Newton is the g string squared. 438 00:25:22,470 --> 00:25:25,374 So do you remember G Newton is g string squared? 439 00:25:25,374 --> 00:25:25,874 Good. 440 00:25:29,700 --> 00:25:34,860 But something remarkable about this diagram 441 00:25:34,860 --> 00:25:37,870 is that you can also view this diagram. 442 00:25:37,870 --> 00:25:40,229 So right now, we see it from this direction. 443 00:25:40,229 --> 00:25:42,270 Now we can also view it from the other direction. 444 00:25:47,420 --> 00:25:49,035 Viewed from this direction, so now 445 00:25:49,035 --> 00:25:50,410 try to think about this direction 446 00:25:50,410 --> 00:25:56,160 as the time and this direction as the sigma. 447 00:25:56,160 --> 00:25:59,770 So right now, we are seeing this as virtual time 448 00:25:59,770 --> 00:26:01,860 and this as sigma. 449 00:26:01,860 --> 00:26:04,990 This is a closed string. 450 00:26:04,990 --> 00:26:09,860 So now think about this direction as sigma 451 00:26:09,860 --> 00:26:12,180 and this direction as time. 452 00:26:12,180 --> 00:26:19,260 And then this is an open string with one end ending 453 00:26:19,260 --> 00:26:24,160 on brane two and the other end ending on brane one, 454 00:26:24,160 --> 00:26:27,440 and then going in the loop. 455 00:26:27,440 --> 00:26:29,600 So this is the one loop open string. 456 00:26:29,600 --> 00:26:31,350 So even though this is tree-level exchange 457 00:26:31,350 --> 00:26:33,270 in closed string. 458 00:26:33,270 --> 00:26:36,910 So here is the tree-level exchange in closed string. 459 00:26:46,380 --> 00:26:50,130 So here is one loop in open string. 460 00:26:55,820 --> 00:26:57,875 So this tells you the same process. 461 00:27:01,640 --> 00:27:03,760 You can really view it from two perspectives. 462 00:27:08,480 --> 00:27:12,810 From one perspective, it's the standard point of view, 463 00:27:12,810 --> 00:27:16,280 is that we exchange some closed strings. 464 00:27:16,280 --> 00:27:19,440 So we exchange some gravitons, some massless particles, 465 00:27:19,440 --> 00:27:22,240 some particles between these two. 466 00:27:22,240 --> 00:27:29,220 But there's another way to think about it is we say, 467 00:27:29,220 --> 00:27:32,514 because we have two D-branes here 468 00:27:32,514 --> 00:27:34,930 and because D-branes correspond in two places, open string 469 00:27:34,930 --> 00:27:44,940 can end, then I have open string connect between them. 470 00:27:44,940 --> 00:27:47,650 And this one loop open string is essentially 471 00:27:47,650 --> 00:27:51,745 corresponding to the vacuum diagram of those open strings 472 00:27:51,745 --> 00:27:53,775 connect between them. 473 00:27:53,775 --> 00:27:55,150 So when you add the vacuum energy 474 00:27:55,150 --> 00:27:59,480 of all those open strings ending between them, 475 00:27:59,480 --> 00:28:02,700 then you're effectively calculating the interaction 476 00:28:02,700 --> 00:28:06,770 between the two D-branes. 477 00:28:06,770 --> 00:28:09,730 So we have two completely different perspectives 478 00:28:09,730 --> 00:28:12,970 to look at the same process. 479 00:28:12,970 --> 00:28:18,527 And this is very, very deep and profound. 480 00:28:18,527 --> 00:28:19,235 Deep is profound. 481 00:28:22,510 --> 00:28:27,770 Because that means the process that you 482 00:28:27,770 --> 00:28:30,340 can think from closed string perspective 483 00:28:30,340 --> 00:28:34,920 can be fully understood in a different way 484 00:28:34,920 --> 00:28:37,400 from the open string perspective, 485 00:28:37,400 --> 00:28:38,940 in a completely equivalent way. 486 00:28:41,636 --> 00:28:43,635 And this is normally called the channel duality. 487 00:28:50,050 --> 00:28:55,200 So it's a very simple geometric fact 488 00:28:55,200 --> 00:28:57,320 about two dimensional surfaces. 489 00:28:57,320 --> 00:29:00,222 But physical significance is very profound. 490 00:29:00,222 --> 00:29:01,756 AUDIENCE: I have a question. 491 00:29:01,756 --> 00:29:02,630 HONG LIU: One second. 492 00:29:02,630 --> 00:29:04,430 Let me finish. 493 00:29:04,430 --> 00:29:10,840 And this is precisely the string theory 494 00:29:10,840 --> 00:29:15,140 origin of the holographic duality or the idea, say, of t 495 00:29:15,140 --> 00:29:18,860 we are going to see in a couple of lectures. 496 00:29:18,860 --> 00:29:23,150 Just because of this simple geometric picture. 497 00:29:28,510 --> 00:29:36,610 This side is gravity, and the D-brane is about gauge theory. 498 00:29:36,610 --> 00:29:42,140 And then we see gravity to be equivalent to gauge theory. 499 00:29:42,140 --> 00:29:45,770 That's something we are going to see later. 500 00:29:45,770 --> 00:29:48,200 Yeah? 501 00:29:48,200 --> 00:29:52,030 AUDIENCE: So you said the dynamics of closed string 502 00:29:52,030 --> 00:29:54,880 can be fully understood by open string. 503 00:29:54,880 --> 00:29:57,290 Is that why you say it's a closed surface 504 00:29:57,290 --> 00:29:58,870 formed by a closed string? 505 00:29:58,870 --> 00:30:00,686 How can you interpret? 506 00:30:00,686 --> 00:30:01,310 HONG LIU: Yeah. 507 00:30:01,310 --> 00:30:03,967 I'm talking about this particular diagram. 508 00:30:03,967 --> 00:30:06,050 I'm just saying this gives you a hint of something 509 00:30:06,050 --> 00:30:07,272 very profound. 510 00:30:07,272 --> 00:30:08,480 AUDIENCE: And one more thing. 511 00:30:08,480 --> 00:30:11,879 Why is the gs 0? 512 00:30:11,879 --> 00:30:12,420 HONG LIU: No. 513 00:30:12,420 --> 00:30:16,290 This is the surface of two boundaries. 514 00:30:16,290 --> 00:30:18,539 AUDIENCE: But that is for open string. 515 00:30:18,539 --> 00:30:19,080 HONG LIU: No. 516 00:30:19,080 --> 00:30:21,140 Chi is for everything. 517 00:30:21,140 --> 00:30:22,030 Chi is everything. 518 00:30:22,030 --> 00:30:23,830 Doesn't matter open or closed string. 519 00:30:23,830 --> 00:30:26,030 This is the universal formula. 520 00:30:26,030 --> 00:30:26,572 This is open. 521 00:30:26,572 --> 00:30:29,071 The open string just means we imposed the boundary condition 522 00:30:29,071 --> 00:30:29,970 on the open string. 523 00:30:29,970 --> 00:30:31,600 The topology is the same. 524 00:30:31,600 --> 00:30:34,260 We understand the topology is the same. 525 00:30:34,260 --> 00:30:34,955 Yes? 526 00:30:34,955 --> 00:30:36,470 AUDIENCE: Why do you call it one loop string? 527 00:30:36,470 --> 00:30:36,830 Where is the loop? 528 00:30:36,830 --> 00:30:38,910 HONG LIU: Because this is the open string. 529 00:30:38,910 --> 00:30:39,960 So this is the open. 530 00:30:39,960 --> 00:30:41,210 Think from this point of view. 531 00:30:41,210 --> 00:30:44,990 This is the open string on brane one and brane two. 532 00:30:44,990 --> 00:30:48,620 And then you go around once, go around in circle. 533 00:30:48,620 --> 00:30:51,905 So this is one loop. 534 00:30:51,905 --> 00:30:54,722 AUDIENCE: What's the free momentum? 535 00:30:54,722 --> 00:30:55,680 HONG LIU: What is this? 536 00:30:58,400 --> 00:31:00,270 What is this? 537 00:31:00,270 --> 00:31:02,590 This is one loop if you have a particle. 538 00:31:02,590 --> 00:31:03,670 And so you have a string. 539 00:31:03,670 --> 00:31:04,920 Then you go around the circle. 540 00:31:04,920 --> 00:31:05,950 This is one loop. 541 00:31:09,840 --> 00:31:14,960 And indeed, when you sum over such surfaces, 542 00:31:14,960 --> 00:31:18,000 you will lead to sum of all possible momentums, et cetera. 543 00:31:18,000 --> 00:31:19,520 So the field theory momentum is one 544 00:31:19,520 --> 00:31:21,710 of the modes you have to sum over 545 00:31:21,710 --> 00:31:25,530 when you do past integral over surface of such topology. 546 00:31:25,530 --> 00:31:26,080 Yes? 547 00:31:26,080 --> 00:31:29,384 AUDIENCE: Once you go to strings connecting different branes, 548 00:31:29,384 --> 00:31:33,380 your quantization conditions change. 549 00:31:33,380 --> 00:31:34,890 HONG LIU: We will talk about that. 550 00:31:34,890 --> 00:31:37,230 Quantization condition almost does not change. 551 00:31:37,230 --> 00:31:39,570 We will talk about that. 552 00:31:39,570 --> 00:31:41,310 We'll talk about that in a little bit. 553 00:31:41,310 --> 00:31:42,810 But right now, it's just intuitively 554 00:31:42,810 --> 00:31:45,520 clear you have open string connect between them 555 00:31:45,520 --> 00:31:47,526 and they just go around. 556 00:31:47,526 --> 00:31:49,275 You have open string connect between them. 557 00:31:49,275 --> 00:31:51,340 You can just go around the circle. 558 00:31:51,340 --> 00:31:52,765 AUDIENCE: But this correspondence 559 00:31:52,765 --> 00:31:57,040 doesn't count the tree-level around the open string. 560 00:31:57,040 --> 00:31:58,005 HONG LIU: Count what? 561 00:31:58,005 --> 00:32:02,955 AUDIENCE: If we generalize the tree-level open string. 562 00:32:02,955 --> 00:32:03,830 HONG LIU: No, no, no. 563 00:32:06,380 --> 00:32:10,160 This doesn't have to account for the tree-level open string. 564 00:32:10,160 --> 00:32:13,120 AUDIENCE: Tree-level open string contributes to the interaction? 565 00:32:13,120 --> 00:32:13,661 HONG LIU: No. 566 00:32:19,260 --> 00:32:21,150 Here, I'm only talking about this diagram 567 00:32:21,150 --> 00:32:23,410 and just say this hints that there 568 00:32:23,410 --> 00:32:26,120 are certain closed string processes can be completely 569 00:32:26,120 --> 00:32:27,386 described by the open string. 570 00:32:31,820 --> 00:32:37,100 So what want to extrapolate is that open string 571 00:32:37,100 --> 00:32:39,480 is a more fundamental description 572 00:32:39,480 --> 00:32:42,110 because the tree-level diagram in closed string 573 00:32:42,110 --> 00:32:44,110 can be described by the open string. 574 00:32:44,110 --> 00:32:47,580 And now if you can generalize that maybe everything closed 575 00:32:47,580 --> 00:32:49,570 string can be described by open string. 576 00:32:49,570 --> 00:32:51,660 But you don't want to do in the opposite way. 577 00:32:51,660 --> 00:32:53,302 Open string is open string. 578 00:32:57,616 --> 00:32:58,115 Good? 579 00:33:00,810 --> 00:33:03,160 Any other questions? 580 00:33:03,160 --> 00:33:05,782 AUDIENCE: Once you go higher dimensions, 581 00:33:05,782 --> 00:33:09,897 when you leave tree-level closed string can be distinct? 582 00:33:09,897 --> 00:33:10,480 HONG LIU: Yes. 583 00:33:10,480 --> 00:33:11,771 Things become more complicated. 584 00:33:14,340 --> 00:33:16,800 Things become more complicated. 585 00:33:16,800 --> 00:33:19,720 But the similar picture will exist. 586 00:33:24,160 --> 00:33:27,230 But nobody has made it work. 587 00:33:27,230 --> 00:33:31,000 Nobody has made it work at full string theory level 588 00:33:31,000 --> 00:33:33,740 to construct the whole closed string theory 589 00:33:33,740 --> 00:33:35,110 from the open string theory. 590 00:33:35,110 --> 00:33:36,780 Nobody has made it work. 591 00:33:36,780 --> 00:33:41,870 But there are many such kind of indications from the geometry 592 00:33:41,870 --> 00:33:44,053 of the surface point of view. 593 00:33:47,290 --> 00:33:54,314 So now let's talk about relaxing the strength of open string 594 00:33:54,314 --> 00:33:54,855 interactions. 595 00:34:06,110 --> 00:34:10,199 Actually, before I do that, now is a good place 596 00:34:10,199 --> 00:34:14,440 to go back to examine what we discussed 597 00:34:14,440 --> 00:34:18,010 at the end of last lecture. 598 00:34:18,010 --> 00:34:23,090 Now is a good time to go back to talk about what we did 599 00:34:23,090 --> 00:34:25,810 at the end of the last lecture. 600 00:34:25,810 --> 00:34:33,429 So in the last lecture, at the end, 601 00:34:33,429 --> 00:34:35,540 we described that one can work out 602 00:34:35,540 --> 00:34:38,469 the low energy effective action of the massless modes 603 00:34:38,469 --> 00:34:40,320 on the D-branes. 604 00:34:40,320 --> 00:34:42,469 So the massless modes on the D-branes on the gauge 605 00:34:42,469 --> 00:34:49,159 fields along the D-brane, so A alpha from 0, 1, to p. 606 00:34:49,159 --> 00:34:54,754 And then the scalar field and a label all the perpendicular 607 00:34:54,754 --> 00:34:55,254 directions. 608 00:35:03,670 --> 00:35:08,020 So you can write down effective action for them. 609 00:35:08,020 --> 00:35:12,300 I mentioned if you work out things carefully, 610 00:35:12,300 --> 00:35:15,950 then you find the prefactor is actually 611 00:35:15,950 --> 00:35:19,250 just the brane tension. 612 00:35:19,250 --> 00:35:25,590 And if last thing is excited, p plus 1. 613 00:35:25,590 --> 00:35:28,690 If last thing is excited, then you just 614 00:35:28,690 --> 00:35:32,390 have the vacuum energy, so you will have a one. 615 00:35:32,390 --> 00:35:37,240 So this is just the brane mass, the total brane mass. 616 00:35:37,240 --> 00:35:37,950 This is et. 617 00:35:41,800 --> 00:35:44,380 So if last thing is excited, then you just 618 00:35:44,380 --> 00:35:46,180 have the zero point energy, which 619 00:35:46,180 --> 00:35:50,950 is just tp times the volume. 620 00:35:50,950 --> 00:35:55,140 But now, if you also have gauge field, then 621 00:35:55,140 --> 00:35:59,380 based on general argument, you must have the Maxwell. 622 00:35:59,380 --> 00:36:03,620 And if the scalar field is excited, 623 00:36:03,620 --> 00:36:13,890 then you also have the action for massless scalar field. 624 00:36:27,760 --> 00:36:30,760 And then we mentioned that for example, you 625 00:36:30,760 --> 00:36:33,250 can consider special case. 626 00:36:33,250 --> 00:36:42,810 Suppose A alpha is not excited but the brane, rather than 627 00:36:42,810 --> 00:36:46,010 a scalar field that moves in a coherent way 628 00:36:46,010 --> 00:36:56,670 the same at all points on the D-brane, just phi a, 629 00:36:56,670 --> 00:37:01,220 is a function of t rather than x. 630 00:37:01,220 --> 00:37:02,165 So phi, in general. 631 00:37:04,780 --> 00:37:09,860 Suppose the brane coordinates our x0 and the p. 632 00:37:09,860 --> 00:37:15,760 So in general, A alpha is the function of x0, xp. 633 00:37:15,760 --> 00:37:20,340 And phi a is x0 and xp. 634 00:37:20,340 --> 00:37:22,545 So they describe you can have arbitrary profile 635 00:37:22,545 --> 00:37:23,415 on the world-volume. 636 00:37:25,930 --> 00:37:28,910 But suppose, say, let me consider the uniform situation 637 00:37:28,910 --> 00:37:35,180 which I only consider every point has 638 00:37:35,180 --> 00:37:38,380 the same behavior for phi. 639 00:37:38,380 --> 00:37:46,130 And then this s just becomes 1/2, dt just becomes dt. 640 00:37:48,800 --> 00:37:57,070 And D-brane plus 1/2 and D-brane phi dot squared. 641 00:38:06,340 --> 00:38:13,840 So this is precisely the motion of just a massive object. 642 00:38:13,840 --> 00:38:23,120 OK And m is just this guy. 643 00:38:23,120 --> 00:38:26,040 So if last thing depends on the spatial coordinate, 644 00:38:26,040 --> 00:38:28,190 then the integration of the spatial coordinate 645 00:38:28,190 --> 00:38:29,590 becomes the volume. 646 00:38:29,590 --> 00:38:32,080 Combine the volume with that, becomes the mass, 647 00:38:32,080 --> 00:38:33,326 and then just becomes that. 648 00:38:42,340 --> 00:38:43,650 I think this is minus sign. 649 00:38:43,650 --> 00:38:45,060 This is plus sign. 650 00:38:51,310 --> 00:38:53,970 So as I mentioned, this is another way 651 00:38:53,970 --> 00:38:56,940 to see that the D-brane becomes dynamical, 652 00:38:56,940 --> 00:38:58,690 and that in particular, this phi describes 653 00:38:58,690 --> 00:38:59,773 the motion of the D-brane. 654 00:39:05,370 --> 00:39:11,170 So in fact, this result can be much, much strengthened. 655 00:39:11,170 --> 00:39:15,140 But I will only quote the result. 656 00:39:15,140 --> 00:39:18,440 I will only quote the result. 657 00:39:18,440 --> 00:39:28,520 It turns out for D-brane with constant. 658 00:39:28,520 --> 00:39:35,170 So as opposed to the D-brane move with constant velocity, 659 00:39:35,170 --> 00:39:38,020 so now you can also have a motion 660 00:39:38,020 --> 00:39:40,560 in the spatial direction. 661 00:39:40,560 --> 00:39:45,855 You have a constant of this and F alpha beta. 662 00:39:49,480 --> 00:39:51,180 You can also excite the gauge field, 663 00:39:51,180 --> 00:39:52,685 but the field strength is constant. 664 00:39:55,950 --> 00:40:00,010 Or this quantity is small. 665 00:40:00,010 --> 00:40:01,980 They don't have to be strictly constant, 666 00:40:01,980 --> 00:40:05,110 but at least their derivatives are small. 667 00:40:11,880 --> 00:40:16,120 In such a situation, one can actually 668 00:40:16,120 --> 00:40:22,650 sum all the higher order terms from string theory corrections. 669 00:40:22,650 --> 00:40:25,420 So this is just a low energy, just like field theory. 670 00:40:25,420 --> 00:40:28,850 And in such a situation, you can actually 671 00:40:28,850 --> 00:40:32,160 sum over infinite number of higher order terms. 672 00:40:32,160 --> 00:40:33,110 And what do you find? 673 00:40:33,110 --> 00:40:35,700 You find so-called wave sum of all infinite number higher 674 00:40:35,700 --> 00:40:37,560 terms. 675 00:40:37,560 --> 00:40:39,720 You find so-called Dirac-Born-Infeld action. 676 00:40:43,340 --> 00:40:45,590 You find that this effective action becomes like this. 677 00:40:50,150 --> 00:40:55,327 This is very [INAUDIBLE] result. So I just want to mention it. 678 00:41:12,400 --> 00:41:15,710 You can actually sum into this form. 679 00:41:15,710 --> 00:41:20,200 So you can sum into this form. 680 00:41:20,200 --> 00:41:25,925 And this g alpha beta is the [INAUDIBLE]. 681 00:41:40,600 --> 00:41:49,524 So let me just explain a little bit the physics 682 00:41:49,524 --> 00:41:50,190 of this formula. 683 00:41:59,860 --> 00:42:05,050 So let's consider the case of the phi 684 00:42:05,050 --> 00:42:07,660 is not excited at all, just phi for the constant, 685 00:42:07,660 --> 00:42:08,760 say, for example. 686 00:42:08,760 --> 00:42:10,620 Then this term vanishes. 687 00:42:10,620 --> 00:42:15,630 Then this g alpha beta just becomes eta alpha beta. 688 00:42:15,630 --> 00:42:17,750 And now you just have a square root, 689 00:42:17,750 --> 00:42:21,430 say, your Minkowski metric plus F alpha beta. 690 00:42:21,430 --> 00:42:24,580 Forget about this 2 pi alpha prime. 691 00:42:24,580 --> 00:42:27,310 This is just some dimensional factor. 692 00:42:27,310 --> 00:42:30,910 And then you just have eta alpha beta plus F alpha beta. 693 00:42:30,910 --> 00:42:34,370 And then you write the determinant. 694 00:42:34,370 --> 00:42:38,700 And suppose when F alpha beta is small, when alpha prime times F 695 00:42:38,700 --> 00:42:41,005 alpha beta is small, then you can expand it 696 00:42:41,005 --> 00:42:44,440 in powers of F alpha beta. 697 00:42:44,440 --> 00:42:48,800 It's a simple exercise but instructive exercise. 698 00:42:48,800 --> 00:42:55,050 You see that precisely reproduces the Maxwell term. 699 00:42:55,050 --> 00:42:58,540 But this will give rise to higher nonlinear terms. 700 00:42:58,540 --> 00:43:02,060 There will be higher order nonlinear terms. 701 00:43:02,060 --> 00:43:05,800 So this can be considered as a nonlinear generalization 702 00:43:05,800 --> 00:43:08,494 of the Maxwell theory. 703 00:43:08,494 --> 00:43:09,910 It turns out actually, this theory 704 00:43:09,910 --> 00:43:17,295 was considered in the '30s by this guy, Born and Infeld. 705 00:43:21,210 --> 00:43:23,610 Actually, maybe '30s or '40s. 706 00:43:23,610 --> 00:43:25,965 Anyway, prehistory. 707 00:43:29,430 --> 00:43:38,510 They invented as a way to avoid-- they 708 00:43:38,510 --> 00:43:41,500 want to avoid the similarity of the Maxwell theory. 709 00:43:41,500 --> 00:43:44,700 So in the Maxwell theory, if you have a charged particle, 710 00:43:44,700 --> 00:43:47,154 and then the location of the charged particle, 711 00:43:47,154 --> 00:43:48,570 then the field due to that charged 712 00:43:48,570 --> 00:43:52,580 particle is singular and the location of that particle. 713 00:43:52,580 --> 00:43:54,890 And so they want to avoid that singular behavior, 714 00:43:54,890 --> 00:43:57,890 so they invented this Born-Infeld action. 715 00:43:57,890 --> 00:44:02,800 And for many years, this action does not have any applications. 716 00:44:02,800 --> 00:44:07,210 But if you invent something nice, it will find its use. 717 00:44:07,210 --> 00:44:11,040 Just like in this movie, Jurassic Park, 718 00:44:11,040 --> 00:44:12,220 life finds a way. 719 00:44:15,010 --> 00:44:16,320 Life always finds a way. 720 00:44:19,360 --> 00:44:20,810 So that's Born-Infeld. 721 00:44:24,420 --> 00:44:28,670 Now let's set F equal to 0. 722 00:44:28,670 --> 00:44:30,805 Let's just look at g alpha beta. 723 00:44:34,390 --> 00:44:37,460 Now let's look at g alpha beta. 724 00:44:37,460 --> 00:44:40,160 So g alpha beta, we can write it in a slightly more transparent 725 00:44:40,160 --> 00:44:40,660 form. 726 00:44:45,040 --> 00:44:49,590 We can write a form which makes it a bit more transparent. 727 00:44:49,590 --> 00:44:52,510 I can write it as the following. 728 00:44:52,510 --> 00:44:56,345 So even mu, remember, is the Minkowski metric 729 00:44:56,345 --> 00:44:59,650 of the full space time. 730 00:44:59,650 --> 00:45:02,290 And I can write this as following. 731 00:45:02,290 --> 00:45:12,640 x mu beta x mu x mu with x alpha equal to x alpha, 732 00:45:12,640 --> 00:45:17,420 which is along the brane direction, and xa to be phi a. 733 00:45:20,450 --> 00:45:24,780 So if you look at this formula, you 734 00:45:24,780 --> 00:45:31,930 can see this is an induced metric for some brane embedded 735 00:45:31,930 --> 00:45:33,920 in the full Minkowski space time. 736 00:45:33,920 --> 00:45:36,710 And the x describes such embedding. 737 00:45:43,250 --> 00:45:45,550 This generalization of this induces the metric formula 738 00:45:45,550 --> 00:45:47,464 we encountered before for the string. 739 00:45:47,464 --> 00:45:48,880 But right now, the only difference 740 00:45:48,880 --> 00:45:51,260 is that now alpha beta, they run all 741 00:45:51,260 --> 00:45:53,610 in the world-volume direction of the D-brane. 742 00:45:53,610 --> 00:45:55,330 And then this becomes an induced metric 743 00:45:55,330 --> 00:45:59,670 on the D-brane when it's embedded in the space time. 744 00:45:59,670 --> 00:46:02,040 And this x alpha equal to x alpha, 745 00:46:02,040 --> 00:46:04,180 it just means that when we embed it, 746 00:46:04,180 --> 00:46:06,770 and we choose the world-volume direction 747 00:46:06,770 --> 00:46:09,480 to be the same as the space time direction 748 00:46:09,480 --> 00:46:11,430 along the brane direction. 749 00:46:11,430 --> 00:46:15,320 And in the perpendicular direction, this is just phi a. 750 00:46:15,320 --> 00:46:19,760 And if you look at this, this is exactly that. 751 00:46:19,760 --> 00:46:23,680 It's exactly that because x alpha equal to x alpha. 752 00:46:23,680 --> 00:46:26,500 And then you just get the eta alpha beta. 753 00:46:26,500 --> 00:46:31,160 And then for the other direction, you get this one. 754 00:46:31,160 --> 00:46:33,410 Is this clear? 755 00:46:33,410 --> 00:46:50,850 So when f equal to 0, so this S just becomes eta g alpha beta. 756 00:46:50,850 --> 00:47:07,960 So this is precisely the volume element of DB-brane. 757 00:47:07,960 --> 00:47:10,439 Because this is the induced metric, and then this 758 00:47:10,439 --> 00:47:12,480 is just the total volume element of the DP-brane. 759 00:47:16,800 --> 00:47:18,810 And we see this is precisely is the relativistic 760 00:47:18,810 --> 00:47:19,434 generalization. 761 00:47:23,330 --> 00:47:26,730 So this is just a generalization of the [INAUDIBLE] action 762 00:47:26,730 --> 00:47:30,720 we wrote earlier, which is for a string. 763 00:47:30,720 --> 00:47:34,540 Then this would be a two dimensional area. 764 00:47:34,540 --> 00:47:37,450 And here, you just integrate over the volume element 765 00:47:37,450 --> 00:47:39,830 of the whole D-brane. 766 00:47:39,830 --> 00:47:42,725 So we see that this Born-Infeld corresponding 767 00:47:42,725 --> 00:47:47,519 to really describes the relativistic motion of a p 768 00:47:47,519 --> 00:47:48,310 dimensional object. 769 00:47:52,250 --> 00:47:55,810 Describes the relativistic motion of a p dimensional 770 00:47:55,810 --> 00:47:58,150 object. 771 00:47:58,150 --> 00:48:01,810 And this Dirac-Born-Infeld, when you combine these two together, 772 00:48:01,810 --> 00:48:07,660 it magically combines these two things into a single thing. 773 00:48:07,660 --> 00:48:09,484 Yes? 774 00:48:09,484 --> 00:48:11,150 AUDIENCE: So I recall you saying earlier 775 00:48:11,150 --> 00:48:16,130 that-- you said that people have played 776 00:48:16,130 --> 00:48:17,930 with this idea of thinking about branes 777 00:48:17,930 --> 00:48:20,940 instead of just generically higher dimensional objects 778 00:48:20,940 --> 00:48:23,250 and strings but no one really understood the theory 779 00:48:23,250 --> 00:48:25,525 of these things because the topology and geometry 780 00:48:25,525 --> 00:48:26,710 were too complicated. 781 00:48:26,710 --> 00:48:29,430 So it seems to me that wouldn't you run into that same problem 782 00:48:29,430 --> 00:48:31,630 right here if it's indeed some generalization 783 00:48:31,630 --> 00:48:32,750 of the [INAUDIBLE] action? 784 00:48:32,750 --> 00:48:36,370 HONG LIU: Yeah, but we don't try to quantize it. 785 00:48:36,370 --> 00:48:38,770 At least we don't try to quantize this action. 786 00:48:38,770 --> 00:48:41,080 And we know how to quantize this action. 787 00:48:41,080 --> 00:48:46,520 And this is just our ordinary field theory. 788 00:48:46,520 --> 00:48:48,060 AUDIENCE: I have a question. 789 00:48:48,060 --> 00:48:56,770 Here, we must impose the big x as the coordinates 790 00:48:56,770 --> 00:48:58,640 in the target space. 791 00:48:58,640 --> 00:49:00,070 HONG LIU: That's right. 792 00:49:00,070 --> 00:49:05,530 AUDIENCE: So phi a must be kind of a constant? 793 00:49:05,530 --> 00:49:09,200 I mean, why there should be a constant part on-- 794 00:49:09,200 --> 00:49:11,370 HONG LIU: No. 795 00:49:11,370 --> 00:49:14,920 If the derivative of those things are not small, 796 00:49:14,920 --> 00:49:18,210 then there will be many other terms. 797 00:49:18,210 --> 00:49:19,660 This will not be the only action. 798 00:49:19,660 --> 00:49:21,572 AUDIENCE: Given a constant. 799 00:49:21,572 --> 00:49:22,882 HONG LIU: Sorry? 800 00:49:22,882 --> 00:49:24,340 AUDIENCE: You said with a constant. 801 00:49:24,340 --> 00:49:26,506 HONG LIU: The partial alpha phi a equal to constant. 802 00:49:26,506 --> 00:49:28,240 AUDIENCE: Oh, equals a constant. 803 00:49:28,240 --> 00:49:29,190 So that means-- 804 00:49:29,190 --> 00:49:30,580 HONG LIU: No. 805 00:49:30,580 --> 00:49:33,020 What I'm saying is that if these are constants, then 806 00:49:33,020 --> 00:49:36,920 this is our exact string theory action. 807 00:49:36,920 --> 00:49:38,512 And when these are not constants, 808 00:49:38,512 --> 00:49:40,220 and then this is a leading approximation, 809 00:49:40,220 --> 00:49:41,860 there will be higher order terms which 810 00:49:41,860 --> 00:49:46,790 depend on their derivatives. 811 00:49:46,790 --> 00:49:47,684 Yes? 812 00:49:47,684 --> 00:49:48,350 AUDIENCE: Sorry. 813 00:49:48,350 --> 00:49:49,808 One thing I just don't understand-- 814 00:49:49,808 --> 00:49:52,230 why is it that we don't want to try to quantize anything? 815 00:49:52,230 --> 00:49:55,200 Shouldn't it be quantized in principle? 816 00:49:55,200 --> 00:49:57,249 These are the sort of classical analogs 817 00:49:57,249 --> 00:49:58,640 of things you want to quantize. 818 00:49:58,640 --> 00:49:59,717 HONG LIU: Yeah. 819 00:49:59,717 --> 00:50:01,550 AUDIENCE: So this is to say that this object 820 00:50:01,550 --> 00:50:03,258 that we don't really know how to quantize 821 00:50:03,258 --> 00:50:05,997 is-- we just don't do it because we don't know how. 822 00:50:05,997 --> 00:50:06,622 HONG LIU: Yeah. 823 00:50:06,622 --> 00:50:07,095 AUDIENCE: I see. 824 00:50:07,095 --> 00:50:07,595 OK. 825 00:50:07,595 --> 00:50:09,495 It's not because-- fair enough. 826 00:50:09,495 --> 00:50:10,120 HONG LIU: Yeah. 827 00:50:10,120 --> 00:50:12,960 You should try anything. 828 00:50:12,960 --> 00:50:17,460 And only those people who have succeeded in the history books. 829 00:50:21,670 --> 00:50:24,194 Only a few have won the battle in the history books. 830 00:50:24,194 --> 00:50:26,860 And if you just fail the battle, you're not in the history book. 831 00:50:30,964 --> 00:50:32,630 So people have tried this but failed it, 832 00:50:32,630 --> 00:50:34,740 but that won't be written in the books. 833 00:50:34,740 --> 00:50:36,620 AUDIENCE: Sure. 834 00:50:36,620 --> 00:50:37,410 AUDIENCE: Sorry. 835 00:50:37,410 --> 00:50:39,540 So you did the square root by summing over 836 00:50:39,540 --> 00:50:43,000 all the massive terms in the-- 837 00:50:43,000 --> 00:50:43,680 HONG LIU: Sorry? 838 00:50:43,680 --> 00:50:45,380 AUDIENCE: You get the square root term 839 00:50:45,380 --> 00:50:49,070 by summing the series, including the massive fields? 840 00:50:49,070 --> 00:50:49,640 HONG LIU: No. 841 00:50:49,640 --> 00:50:51,554 This is still the gauge field and the phi. 842 00:50:51,554 --> 00:50:52,220 AUDIENCE: Right. 843 00:50:52,220 --> 00:50:53,850 So how do you get it? 844 00:50:53,850 --> 00:50:55,919 What's the series that you're summing? 845 00:50:55,919 --> 00:50:56,460 HONG LIU: Hm? 846 00:50:56,460 --> 00:50:57,650 AUDIENCE: What's the series that you're summing? 847 00:50:57,650 --> 00:50:57,800 HONG LIU: Oh. 848 00:50:57,800 --> 00:50:59,341 I'm just saying in the string theory, 849 00:50:59,341 --> 00:51:01,700 typically you don't start by f squared. 850 00:51:01,700 --> 00:51:04,220 You have f cubed, f four, et cetera. 851 00:51:04,220 --> 00:51:08,010 You can sum all of them together. 852 00:51:08,010 --> 00:51:12,520 Even for the massless mode, these are just leading terms. 853 00:51:12,520 --> 00:51:15,910 And these terms would be the smallest number of derivatives, 854 00:51:15,910 --> 00:51:20,250 and so they dominate at low energies. 855 00:51:20,250 --> 00:51:24,950 But in general, even just for the effective serial 856 00:51:24,950 --> 00:51:28,150 massless mode, you can have many, many other terms. 857 00:51:28,150 --> 00:51:29,790 AUDIENCE: I have a question. 858 00:51:29,790 --> 00:51:33,050 If we assume that partial alpha phi a is a constant, 859 00:51:33,050 --> 00:51:37,135 then we can solve out the phi a is proportional to x alpha. 860 00:51:37,135 --> 00:51:40,220 But how can you assume they're just 861 00:51:40,220 --> 00:51:42,402 the coordinate in target space? 862 00:51:42,402 --> 00:51:43,110 HONG LIU: No, no. 863 00:51:43,110 --> 00:51:45,780 This is a function of alpha. 864 00:51:45,780 --> 00:51:47,660 AUDIENCE: Yes. 865 00:51:47,660 --> 00:51:51,040 But since it's a function of x alpha, 866 00:51:51,040 --> 00:51:56,157 why it can be regarded as the coordinate in target space? 867 00:51:56,157 --> 00:51:56,823 HONG LIU: Sorry. 868 00:51:56,823 --> 00:51:58,230 I don't understand. 869 00:51:58,230 --> 00:51:59,730 AUDIENCE: All the coordinates should 870 00:51:59,730 --> 00:52:02,124 be independent in target space. 871 00:52:02,124 --> 00:52:02,790 HONG LIU: Sorry. 872 00:52:02,790 --> 00:52:03,550 I don't understand. 873 00:52:03,550 --> 00:52:04,425 They are independent. 874 00:52:07,640 --> 00:52:10,240 These are the virtual coordinates. 875 00:52:10,240 --> 00:52:13,640 These are the volume coordinates of D-brane. 876 00:52:13,640 --> 00:52:16,860 And these are the target space coordinates. 877 00:52:16,860 --> 00:52:18,710 I'm just choosing the function of the target 878 00:52:18,710 --> 00:52:19,683 space coordinates. 879 00:52:23,140 --> 00:52:25,530 I'm choosing here just to be identical 880 00:52:25,530 --> 00:52:27,502 to the world-volume coordinate. 881 00:52:27,502 --> 00:52:29,210 And this one I choose to be some function 882 00:52:29,210 --> 00:52:30,633 of the world-volume coordinate. 883 00:52:30,633 --> 00:52:32,112 Of course I can do that. 884 00:52:35,467 --> 00:52:36,300 Any other questions? 885 00:52:46,420 --> 00:52:50,580 So again, this highlights that D-brane is really 886 00:52:50,580 --> 00:52:52,550 a dynamical object. 887 00:52:52,550 --> 00:53:00,900 In fact, at low energies, they move like [INAUDIBLE] motion. 888 00:53:00,900 --> 00:53:04,090 And they actually move relativistically 889 00:53:04,090 --> 00:53:06,180 if you give enough velocity, et cetera. 890 00:53:11,170 --> 00:53:13,290 Because of the fluctuations of the D-brane, 891 00:53:13,290 --> 00:53:18,920 they become a really full dynamical object. 892 00:53:18,920 --> 00:53:19,810 They have a mass. 893 00:53:19,810 --> 00:53:21,380 They can move around. 894 00:53:21,380 --> 00:53:23,775 And now you can deform their shape. 895 00:53:23,775 --> 00:53:27,212 If you have enough energy, you can bend them, et cetera. 896 00:53:27,212 --> 00:53:28,420 You can do whatever you want. 897 00:53:37,290 --> 00:53:42,150 So let me mention one last thing. 898 00:53:42,150 --> 00:53:50,060 So you may ask, why somehow those 899 00:53:50,060 --> 00:53:53,500 fields which describe the motion of the D-brane, 900 00:53:53,500 --> 00:53:56,330 they're corresponding to the massless modes 901 00:53:56,330 --> 00:54:04,390 on the world-volume of the D-branes? 902 00:54:04,390 --> 00:54:08,730 Whether this is a coincidence, or why it somehow 903 00:54:08,730 --> 00:54:14,710 happens to be the massless mode on the D-brane 904 00:54:14,710 --> 00:54:18,820 which describes the motion of the D-brane. 905 00:54:18,820 --> 00:54:21,710 So this is not an accident. 906 00:54:21,710 --> 00:54:23,490 This is not an accident. 907 00:54:23,490 --> 00:54:42,910 The reason is that-- so why modes 908 00:54:42,910 --> 00:55:04,000 describing motions of D-branes appear as massless modes? 909 00:55:04,000 --> 00:55:05,560 So this is not accident. 910 00:55:05,560 --> 00:55:13,630 So underlying reason, it's because the underlying 911 00:55:13,630 --> 00:55:27,980 Minkowski space is translation invariant. 912 00:55:31,390 --> 00:55:37,330 So that means that no matter where you put the D-brane, 913 00:55:37,330 --> 00:55:40,950 it should be a well-defined configuration. 914 00:55:40,950 --> 00:55:43,810 Should be a well-defined configuration no matter where 915 00:55:43,810 --> 00:55:45,226 you put on the D-brane. 916 00:55:45,226 --> 00:55:50,990 Then that means that the [INAUDIBLE] action for the phi 917 00:55:50,990 --> 00:55:53,620 cannot contain a potential term. 918 00:55:53,620 --> 00:55:54,880 They cannot be potential term. 919 00:55:54,880 --> 00:55:57,420 They should not be, say, somewhere is the minimum, 920 00:55:57,420 --> 00:55:59,330 somewhere is the maximum, cannot happen. 921 00:55:59,330 --> 00:56:01,030 Everywhere must be the same. 922 00:56:01,030 --> 00:56:05,175 So it means the dependence on y can only be derivative. 923 00:56:05,175 --> 00:56:07,370 Can only depend on derivatives. 924 00:56:07,370 --> 00:56:11,140 And of course, at low energies, if you have derivatives, 925 00:56:11,140 --> 00:56:12,860 then can only be the massless particle. 926 00:56:15,940 --> 00:56:18,880 So translation invariant. 927 00:56:18,880 --> 00:56:26,090 So this means that any phi a equal to constant 928 00:56:26,090 --> 00:56:31,080 should be allowed configurations. 929 00:56:42,320 --> 00:56:47,790 That means cannot have potential. 930 00:56:50,470 --> 00:56:52,475 So max term is like a potential for phi. 931 00:56:59,560 --> 00:57:03,140 To say in the fancy words of Quantum Field Theory 932 00:57:03,140 --> 00:57:11,920 II or Quantum Field Theory III, that the phi a, in other words, 933 00:57:11,920 --> 00:57:29,451 phi a are the Goldstone bosons for breaking 934 00:57:29,451 --> 00:57:30,437 translation symmetries. 935 00:57:46,740 --> 00:57:51,566 So previously, Minkowski space is translation invariant. 936 00:57:51,566 --> 00:57:53,210 And now if you put a D-brane there, 937 00:57:53,210 --> 00:57:55,161 then you break that translation symmetry. 938 00:57:55,161 --> 00:57:57,285 The location of the D-brane breaks that translation 939 00:57:57,285 --> 00:57:58,535 symmetry. 940 00:57:58,535 --> 00:58:01,700 If I even put the D-brane anywhere, 941 00:58:01,700 --> 00:58:04,260 then that means the modes, the dynamics 942 00:58:04,260 --> 00:58:07,130 control the location of the D-brane 943 00:58:07,130 --> 00:58:10,805 must not have any potential, can only have derivative terms. 944 00:58:14,470 --> 00:58:17,390 So in other words, when you put the D-brane in, 945 00:58:17,390 --> 00:58:23,740 you spontaneously break the translation symmetry 946 00:58:23,740 --> 00:58:25,102 on the line in Minkowski space. 947 00:58:33,634 --> 00:58:34,800 So let me mention one thing. 948 00:58:34,800 --> 00:58:35,880 Then we can have a break. 949 00:58:39,920 --> 00:58:41,270 I'll mention one quick thing. 950 00:58:58,440 --> 00:59:05,619 So let me say a few words on the strength of the open string 951 00:59:05,619 --> 00:59:06,160 interactions. 952 00:59:27,160 --> 00:59:34,500 So previously, we described that the closed strings, they 953 00:59:34,500 --> 00:59:40,350 interact by such joining and splitting procedure. 954 00:59:45,770 --> 00:59:47,390 And the strength here is capped by gs. 955 00:59:50,104 --> 00:59:52,270 So the closed string coupling is essentially the gs, 956 00:59:52,270 --> 00:59:55,187 which describes such a process. 957 00:59:55,187 --> 00:59:57,020 So if you have an open string, of course you 958 00:59:57,020 --> 01:00:00,840 have a similar process, just string ends joined together. 959 01:00:03,360 --> 01:00:05,272 You can join string ends together. 960 01:00:08,190 --> 01:00:11,760 So here, you really just have open string. 961 01:00:11,760 --> 01:00:16,200 So now these lines are the boundaries of open string, 962 01:00:16,200 --> 01:00:18,680 or the endpoints of open string. 963 01:00:18,680 --> 01:00:21,210 So you have two open strings joined together 964 01:00:21,210 --> 01:00:23,050 from another one. 965 01:00:23,050 --> 01:00:25,940 So let's call this interaction go 966 01:00:25,940 --> 01:00:28,508 describing the interaction of the open string. 967 01:00:31,200 --> 01:00:33,685 So the question is, how is this go related to gs? 968 01:00:36,820 --> 01:00:38,940 And there's a single way we can figure it out. 969 01:00:42,430 --> 01:00:44,780 So let's consider the simplest situation. 970 01:00:44,780 --> 01:00:50,390 Just have open string propagate in time. 971 01:00:50,390 --> 01:00:52,680 Again, this is two boundaries open string. 972 01:00:52,680 --> 01:00:53,970 We just propagate in time. 973 01:01:02,020 --> 01:01:05,300 Now let's consider a more complicated process. 974 01:01:05,300 --> 01:01:07,600 So the open string propagates in time. 975 01:01:07,600 --> 01:01:15,030 So this is just a simple surface with one initial open string 976 01:01:15,030 --> 01:01:17,756 and one final open string. 977 01:01:17,756 --> 01:01:19,990 And you can see the complicated process 978 01:01:19,990 --> 01:01:23,210 because we have a hole in the middle. 979 01:01:23,210 --> 01:01:27,770 So now the string worksheet is like this, just this part. 980 01:01:27,770 --> 01:01:29,440 We have a hole in the middle. 981 01:01:29,440 --> 01:01:34,250 And this is another configuration 982 01:01:34,250 --> 01:01:36,050 to have some initial open string propagate 983 01:01:36,050 --> 01:01:39,230 to some final open string. 984 01:01:39,230 --> 01:01:49,150 So now we know, by counting we did before described here. 985 01:01:49,150 --> 01:01:52,240 So here, we're adding one boundary. 986 01:01:52,240 --> 01:01:55,390 We are adding one boundary. 987 01:01:55,390 --> 01:01:56,480 So this adds a boundary. 988 01:02:01,580 --> 01:02:06,880 So that means we must add a factor of gs. 989 01:02:10,310 --> 01:02:13,200 Because from this formula, this is g minus chi. 990 01:02:13,200 --> 01:02:16,120 And chi is minus b. 991 01:02:16,120 --> 01:02:18,370 So if we increase one boundary, then you 992 01:02:18,370 --> 01:02:19,460 increase a factor of gs. 993 01:02:22,240 --> 01:02:28,100 But we can also view this diagram 994 01:02:28,100 --> 01:02:30,500 as a single open string comes in. 995 01:02:35,330 --> 01:02:39,350 The opposite of this splits into two open strings 996 01:02:39,350 --> 01:02:42,667 and then they close together. 997 01:02:42,667 --> 01:02:48,950 So one split operation, and the one join operation. 998 01:02:51,930 --> 01:02:56,160 So that should correspond to go squared. 999 01:02:56,160 --> 01:02:58,910 So then this means that we conclude that gs must 1000 01:02:58,910 --> 01:03:01,920 be proportional to go squared. 1001 01:03:01,920 --> 01:03:06,320 The open string coupling strings is the square root 1002 01:03:06,320 --> 01:03:08,880 of the closed string interaction. 1003 01:03:13,201 --> 01:03:13,700 Yes? 1004 01:03:13,700 --> 01:03:15,985 AUDIENCE: What's the strength of the process when 1005 01:03:15,985 --> 01:03:17,770 closed string becomes open string and vice versa? 1006 01:03:17,770 --> 01:03:18,035 HONG LIU: Sorry? 1007 01:03:18,035 --> 01:03:20,415 AUDIENCE: What's the strength of the process when closed 1008 01:03:20,415 --> 01:03:21,706 string becomes and open string? 1009 01:03:21,706 --> 01:03:22,390 HONG LIU: Right. 1010 01:03:22,390 --> 01:03:24,210 Yeah, you can consider such process. 1011 01:03:27,020 --> 01:03:33,620 Again, you can just do it by counting 1012 01:03:33,620 --> 01:03:35,462 the topology of the surface. 1013 01:03:35,462 --> 01:03:36,420 Such process can exist. 1014 01:03:44,510 --> 01:03:45,010 OK. 1015 01:03:45,010 --> 01:03:46,260 Then let's have a short break. 1016 01:03:48,940 --> 01:03:53,130 So what time is it? 1017 01:03:53,130 --> 01:03:53,740 It's 38. 1018 01:03:57,390 --> 01:04:02,030 When should we start again? 1019 01:04:02,030 --> 01:04:03,380 41? 1020 01:04:03,380 --> 01:04:03,880 OK, 41. 1021 01:04:03,880 --> 01:04:06,370 Let's start at 41. 1022 01:04:06,370 --> 01:04:08,980 So we have talked about D-branes, et cetera. 1023 01:04:12,480 --> 01:04:14,710 And we have already seen some remarkable aspects 1024 01:04:14,710 --> 01:04:16,380 of the D-brane, including this channel 1025 01:04:16,380 --> 01:04:23,530 duality between the closed string exchange 1026 01:04:23,530 --> 01:04:26,744 can be considered open string loop. 1027 01:04:26,744 --> 01:04:29,160 And now we are going to see a lot of magic of the D-brane. 1028 01:04:35,081 --> 01:04:37,330 And this comes when you put several D-branes together. 1029 01:05:06,400 --> 01:05:18,160 So normally, our conventional intuition 1030 01:05:18,160 --> 01:05:26,050 says if you find some particle, say in this case, 1031 01:05:26,050 --> 01:05:27,320 you find the D-brane. 1032 01:05:27,320 --> 01:05:29,980 So if you put two particles together, 1033 01:05:29,980 --> 01:05:31,710 nothing much really changes. 1034 01:05:31,710 --> 01:05:33,170 It's two particles. 1035 01:05:33,170 --> 01:05:35,590 Put three particles together. 1036 01:05:35,590 --> 01:05:36,436 Not much changes. 1037 01:05:36,436 --> 01:05:37,310 It's three particles. 1038 01:05:39,940 --> 01:05:43,370 But when you put multiple D-branes together, 1039 01:05:43,370 --> 01:05:48,690 things change a lot in a very profound way. 1040 01:05:48,690 --> 01:05:53,640 So now let's consider just two D-branes. 1041 01:05:53,640 --> 01:05:55,010 So let's consider two D-branes. 1042 01:06:23,310 --> 01:06:26,380 Let's consider example. 1043 01:06:26,380 --> 01:06:29,366 So let me first just tell you the naive intuition. 1044 01:06:33,590 --> 01:06:35,840 Suppose you have D-brane one, D-brane two. 1045 01:06:38,990 --> 01:06:42,850 So for this one, we have a u1 gauge field. 1046 01:06:42,850 --> 01:06:46,840 For this one, you have a u1 gauge field. 1047 01:06:46,840 --> 01:06:50,510 Because each one, we have a gauge field, a Maxwell. 1048 01:06:50,510 --> 01:06:53,420 When you put together, from conventional wisdom, 1049 01:06:53,420 --> 01:06:55,290 you say, maybe I just have two Maxwell. 1050 01:06:57,890 --> 01:06:59,850 From conventional wisdom, you two Maxwell. 1051 01:06:59,850 --> 01:07:02,600 Naively, if I put them together, I just have two Maxwells. 1052 01:07:02,600 --> 01:07:05,683 1 plus 1 equal to 2. 1053 01:07:10,780 --> 01:07:15,800 But in string theory, 1 plus 1 equal to 4. 1054 01:07:15,800 --> 01:07:17,370 It's actually equal to 2. 1055 01:07:17,370 --> 01:07:21,810 It's also equal to 2, depending on how you think about it. 1056 01:07:21,810 --> 01:07:26,050 Anyway, one way to think about it is 1 plus 1 becomes 4. 1057 01:07:26,050 --> 01:07:31,410 So to see 1 plus 1 become 4 is very easy. 1058 01:07:31,410 --> 01:07:34,636 So let's consider these two branes on top of each other. 1059 01:07:34,636 --> 01:07:36,510 But in order to distinguish these two branes, 1060 01:07:36,510 --> 01:07:38,350 I just separate them a little bit. 1061 01:07:38,350 --> 01:07:42,550 But you should really think of them on top of each other. 1062 01:07:42,550 --> 01:07:47,040 And so now you have four types of strings. 1063 01:07:47,040 --> 01:07:52,970 You can have string going to 1, 1, going to 2, 2, 1064 01:07:52,970 --> 01:07:56,180 then going to 1, 2, going to 2, 1. 1065 01:07:56,180 --> 01:08:00,080 So 2, 1 and 1, 2 are different because the oriented string. 1066 01:08:00,080 --> 01:08:02,850 So I put arrow there. 1067 01:08:02,850 --> 01:08:04,880 So this is from 1 to 2. 1068 01:08:04,880 --> 01:08:05,740 This is from 2 to 1. 1069 01:08:09,720 --> 01:08:15,769 So we have four types of strings-- 1, 1, 1, 2, 2, 1, 1070 01:08:15,769 --> 01:08:16,269 2, 2. 1071 01:08:21,760 --> 01:08:23,256 AUDIENCE: Why is oriented-- 1072 01:08:23,256 --> 01:08:24,779 HONG LIU: Hm? 1073 01:08:24,779 --> 01:08:28,609 AUDIENCE: Why the 1 to 2, 2 to 1 are different? 1074 01:08:28,609 --> 01:08:32,550 HONG LIU: It's because for this string, sigma 0 here. 1075 01:08:32,550 --> 01:08:33,899 For this one, sigma pi there. 1076 01:08:37,240 --> 01:08:39,140 For this string, sigma 0 there. 1077 01:08:39,140 --> 01:08:41,189 And for this one, sigma pi there. 1078 01:08:46,149 --> 01:08:48,410 Let me just elaborate on this point. 1079 01:08:48,410 --> 01:08:52,040 So suppose I have a string like this. 1080 01:08:52,040 --> 01:08:55,819 Then this string is sigma 0 point ending on 1, 1081 01:08:55,819 --> 01:08:58,170 sigma equal to pi ending on 2. 1082 01:08:58,170 --> 01:08:59,880 But if I have a string like that, 1083 01:08:59,880 --> 01:09:03,066 then there's a sigma 0 ending here and sigma pi ending there. 1084 01:09:09,386 --> 01:09:13,370 So you have four types of open strings. 1085 01:09:13,370 --> 01:09:15,120 And now if you think about how we quantize 1086 01:09:15,120 --> 01:09:18,569 those strings, and the four types of open strings, 1087 01:09:18,569 --> 01:09:21,370 they actually have identical spectrum. 1088 01:09:21,370 --> 01:09:23,370 Because for all of them, the boundary conditions 1089 01:09:23,370 --> 01:09:25,446 are exactly the same. 1090 01:09:25,446 --> 01:09:26,946 Because the boundary conditions only 1091 01:09:26,946 --> 01:09:29,359 know the location of the D-branes. 1092 01:09:29,359 --> 01:09:31,810 So all four types of open strings 1093 01:09:31,810 --> 01:09:35,140 have identical spectrum. 1094 01:09:35,140 --> 01:09:46,890 So in other words, each string excitation-- 1095 01:09:46,890 --> 01:09:53,750 say this is the state on the worksheet-- each state becomes 1096 01:09:53,750 --> 01:09:58,270 four states because I can label IJ. 1097 01:09:58,270 --> 01:10:02,170 Now suppose I use I and J to label 1 and the 2. 1098 01:10:02,170 --> 01:10:05,850 I and J can be 1 and 2. 1099 01:10:05,850 --> 01:10:08,590 So depending on whether this is 1, 1 string, or 1, 2 string, 1100 01:10:08,590 --> 01:10:12,100 or 2, 1 string, or 2, 2 string. 1101 01:10:12,100 --> 01:10:15,030 So this is what I said 1 plus 1 equal to 4. 1102 01:10:15,030 --> 01:10:19,070 Because naively, you would say I have two massless modes. 1103 01:10:19,070 --> 01:10:20,150 But now I have four. 1104 01:10:24,680 --> 01:10:37,910 For example, the massless modes become four copies of them. 1105 01:10:51,990 --> 01:11:19,815 In other words, you can think each open string excitation a 2 1106 01:11:19,815 --> 01:11:20,380 by 2 matrix. 1107 01:11:26,210 --> 01:11:28,616 I can use 2 index to label them. 1108 01:11:32,320 --> 01:11:34,570 In particular, for example, the corresponding fields-- 1109 01:11:34,570 --> 01:11:38,210 so each string excitation corresponding to some field. 1110 01:11:38,210 --> 01:11:43,020 For example, the gauge field associated with this 1111 01:11:43,020 --> 01:11:48,270 now has two index, I J. And similar with phi 1112 01:11:48,270 --> 01:12:04,050 a I J. Of course, this generalizes immediately 1113 01:12:04,050 --> 01:12:20,186 to if you have n branes, then just becomes n times 1114 01:12:20,186 --> 01:12:20,920 n matrices. 1115 01:12:27,200 --> 01:12:30,741 So 1 plus 1 plus 1 to n becomes n squared. 1116 01:12:37,800 --> 01:12:40,740 So now let me give some remarks. 1117 01:12:45,150 --> 01:12:48,330 So this basic structure turns out 1118 01:12:48,330 --> 01:12:52,340 to be, again, very, very profound. 1119 01:12:52,340 --> 01:12:54,234 Now let me give some remarks. 1120 01:13:06,360 --> 01:13:09,070 So there's a reason I call-- so this 1121 01:13:09,070 --> 01:13:11,135 is something with the 2 index. 1122 01:13:16,010 --> 01:13:19,330 So of course, you naturally call it a matrix. 1123 01:13:19,330 --> 01:13:22,010 But there's another reason to think about this really 1124 01:13:22,010 --> 01:13:25,100 as a matrix. 1125 01:13:25,100 --> 01:13:29,640 It's because the strings, as we were 1126 01:13:29,640 --> 01:13:32,394 doing there, the open strings, they interact 1127 01:13:32,394 --> 01:13:33,310 by joining their ends. 1128 01:13:49,280 --> 01:13:54,270 So this naturally leads to-- when those strings interact 1129 01:13:54,270 --> 01:13:59,150 with each other, and those parts naturally 1130 01:13:59,150 --> 01:14:14,970 just emerges as a matrix product, I, J indices. 1131 01:14:19,450 --> 01:14:22,990 So it's easy to see. 1132 01:14:25,810 --> 01:14:27,410 So let me just draw that. 1133 01:14:27,410 --> 01:14:30,730 Let me just do it here to save some time. 1134 01:14:30,730 --> 01:14:37,320 Suppose this is I. This is J. So this is sigma equal to 0, 1135 01:14:37,320 --> 01:14:38,740 sigma equal to pi. 1136 01:14:38,740 --> 01:14:41,090 And the sigma equal to pi joins with stigma 0 1137 01:14:41,090 --> 01:14:42,279 to end of the other one. 1138 01:14:42,279 --> 01:14:44,320 But of course, if you want to join them together, 1139 01:14:44,320 --> 01:14:46,570 their J's have to be the same. 1140 01:14:46,570 --> 01:14:51,870 This is K. Then you go to I, K. And when they join together, 1141 01:14:51,870 --> 01:14:54,390 then you sum of all possible J's. 1142 01:14:54,390 --> 01:14:56,164 Then this is like a matrix product. 1143 01:15:00,680 --> 01:15:05,450 So if I draw it in the diagram not very well. 1144 01:15:05,450 --> 01:15:11,560 Now let me separate I, J, K to be three things. 1145 01:15:11,560 --> 01:15:14,010 But they don't have to be separated. 1146 01:15:14,010 --> 01:15:17,560 I, J, K can also be the same. 1147 01:15:17,560 --> 01:15:20,410 But in order to emphasize this picture is 1148 01:15:20,410 --> 01:15:28,700 that you have a string to go from I to J. Suppose this is I, 1149 01:15:28,700 --> 01:15:31,890 this is J, this is K. Go from I to J, 1150 01:15:31,890 --> 01:15:37,830 then from J to K. So sigma 0, sigma pi. 1151 01:15:37,830 --> 01:15:40,280 And the pi end joins with the sigma 0 end. 1152 01:15:40,280 --> 01:15:42,185 And then here, you get the string. 1153 01:15:45,830 --> 01:15:49,290 So that diagram roughly can also be 1154 01:15:49,290 --> 01:15:51,350 think of a diagram like this. 1155 01:15:51,350 --> 01:15:57,350 Two strings join into one string with index I and K. 1156 01:15:57,350 --> 01:15:58,850 And the I, K, K can all be the same. 1157 01:15:58,850 --> 01:16:03,690 I just make them different to make it clear. 1158 01:16:03,690 --> 01:16:09,950 And of course, when you join J together, 1159 01:16:09,950 --> 01:16:14,860 you have the sum of them because they can be in principle all 1160 01:16:14,860 --> 01:16:15,570 possible J's. 1161 01:16:19,615 --> 01:16:23,810 So it naturally appears as a matrix product. 1162 01:16:23,810 --> 01:16:28,720 Just follows by the nature of string interaction. 1163 01:16:28,720 --> 01:16:32,180 And now there's another remarkable thing 1164 01:16:32,180 --> 01:16:35,330 is that if you can see the phi a, so the same thing applies 1165 01:16:35,330 --> 01:16:39,370 for a alpha applies to any field. 1166 01:16:39,370 --> 01:16:43,240 It's that 1q-- so this corresponds 1167 01:16:43,240 --> 01:16:49,849 to a string with sigma equal to 0 at 1 and sigma pi. 1168 01:16:49,849 --> 01:16:51,390 So this corresponds to a 1, 2 string. 1169 01:16:53,940 --> 01:16:59,875 And then we can also think about the 2, 1 string. 1170 01:17:02,650 --> 01:17:06,500 So it turns out that these two can 1171 01:17:06,500 --> 01:17:20,009 be considered as complex conjugates of each other 1172 01:17:20,009 --> 01:17:21,050 for the following reason. 1173 01:17:28,820 --> 01:17:34,910 Again, now let me just again separate this 1 and 2 1174 01:17:34,910 --> 01:17:37,360 to make it clear. 1175 01:17:37,360 --> 01:17:40,350 So this is a 1, 2 string. 1176 01:17:40,350 --> 01:17:41,903 So this is a 2, 1 string. 1177 01:17:45,980 --> 01:17:54,639 So I claim string interactions defined by this way 1178 01:17:54,639 --> 01:17:55,805 have the following symmetry. 1179 01:17:58,410 --> 01:18:06,507 So string interactions described by joining the ends 1180 01:18:06,507 --> 01:18:08,590 or splitting the ends have the following symmetry. 1181 01:18:22,280 --> 01:18:34,258 It's that I can associate each brane by a phase factor. 1182 01:18:38,700 --> 01:18:43,099 So I explained to you i theta I. So I labeled to the brane 1183 01:18:43,099 --> 01:18:45,265 and the theta can be some different-- can be a phase 1184 01:18:45,265 --> 01:18:47,840 factor. 1185 01:18:47,840 --> 01:18:51,010 And now the rule is that if the sigma is 1186 01:18:51,010 --> 01:18:58,875 equal to 0 and on that brane, then 1187 01:18:58,875 --> 01:19:04,250 I multiply it by the exponential of i theta I. 1188 01:19:04,250 --> 01:19:14,335 And the sigma equal to pi ends on that brane, ends on I. 1189 01:19:14,335 --> 01:19:15,585 Let me write it more explicit. 1190 01:19:15,585 --> 01:19:23,692 So if sigma 0 ending on I, then I multiply 1191 01:19:23,692 --> 01:19:26,880 by a phase factor, exponential i theta I. 1192 01:19:26,880 --> 01:19:31,390 And if the sigma equal to pi factor ending on I, 1193 01:19:31,390 --> 01:19:40,020 then I multiply by exponential minus i theta I. 1194 01:19:40,020 --> 01:19:41,530 So let's consider this operation. 1195 01:19:41,530 --> 01:19:44,850 And I claim this operation is the symmetry of the string 1196 01:19:44,850 --> 01:19:46,050 interaction. 1197 01:19:46,050 --> 01:19:50,090 So let's first consider if you just have a single brane. 1198 01:19:50,090 --> 01:19:54,290 So if you have a single brane like this, then just 1199 01:19:54,290 --> 01:19:55,790 nothing changes because you multiply 1200 01:19:55,790 --> 01:19:59,190 one end by i theta and the other by exponential minus i theta, 1201 01:19:59,190 --> 01:20:02,260 does not change. 1202 01:20:02,260 --> 01:20:06,470 But now, if you have such kind of interactions, 1203 01:20:06,470 --> 01:20:14,750 because the sigma 0 and sigma pi ends join together, 1204 01:20:14,750 --> 01:20:18,280 and they can only join if they are ending on the same brane, 1205 01:20:18,280 --> 01:20:21,640 then those factors always cancel each other. 1206 01:20:21,640 --> 01:20:25,340 And so this would be a symmetry. 1207 01:20:25,340 --> 01:20:26,170 Is this clear? 1208 01:20:28,980 --> 01:20:41,280 So under this operation, phi a ending on the same brane 1209 01:20:41,280 --> 01:20:41,880 is invariant. 1210 01:20:49,268 --> 01:20:55,508 And phi a I J then transforms by a phase factor theta 1211 01:20:55,508 --> 01:21:05,290 I minus theta J of phi a I J. And the phi a J 1212 01:21:05,290 --> 01:21:17,890 I transforms as a factor minus I theta I minus theta J. 1213 01:21:17,890 --> 01:21:22,039 So we can actually think of them as complex conjugates. 1214 01:21:22,039 --> 01:21:24,580 So they're transforming opposite way under this phase change. 1215 01:21:24,580 --> 01:21:25,020 Yes? 1216 01:21:25,020 --> 01:21:26,978 AUDIENCE: And this is because we're considering 1217 01:21:26,978 --> 01:21:28,000 them to be u1 branes? 1218 01:21:28,000 --> 01:21:28,500 HONG LIU: Sorry? 1219 01:21:28,500 --> 01:21:30,916 AUDIENCE: Is this because we're considering them to be u1? 1220 01:21:30,916 --> 01:21:32,620 HONG LIU: No. 1221 01:21:32,620 --> 01:21:34,350 This is a good point. 1222 01:21:34,350 --> 01:21:36,330 I will talk about this more. 1223 01:21:36,330 --> 01:21:38,455 Right now, let's think about each brane separately. 1224 01:21:42,264 --> 01:21:43,430 AUDIENCE: I have a question. 1225 01:21:43,430 --> 01:21:48,310 In principle, can we write the interaction of open string 1226 01:21:48,310 --> 01:21:52,154 with the same note, J, J by random number? 1227 01:21:52,154 --> 01:21:52,820 HONG LIU: Sorry. 1228 01:21:52,820 --> 01:21:54,660 What do you mean by random number? 1229 01:21:54,660 --> 01:21:57,840 AUDIENCE: Say it's I, J, K, L. 1230 01:21:57,840 --> 01:21:58,800 HONG LIU: No, no, no. 1231 01:21:58,800 --> 01:21:59,490 The strings can only join together 1232 01:21:59,490 --> 01:22:01,329 if they're ending on the same brane. 1233 01:22:01,329 --> 01:22:01,870 AUDIENCE: Oh. 1234 01:22:01,870 --> 01:22:02,422 OK. 1235 01:22:02,422 --> 01:22:04,130 HONG LIU: So the J's have to be the same. 1236 01:22:06,179 --> 01:22:07,970 So this guarantees that this will symmetry. 1237 01:22:12,150 --> 01:22:14,910 Good? 1238 01:22:14,910 --> 01:22:18,570 So in this sense, they're complex conjugates. 1239 01:22:18,570 --> 01:22:21,160 And now we can build on this a little bit further. 1240 01:22:21,160 --> 01:22:22,729 So this actually works. 1241 01:22:22,729 --> 01:22:24,270 Doesn't matter whether the branes are 1242 01:22:24,270 --> 01:22:26,870 coincident or not coincident. 1243 01:22:26,870 --> 01:22:28,596 This is a generally true. 1244 01:22:28,596 --> 01:22:29,970 Now let's consider all the branes 1245 01:22:29,970 --> 01:22:31,261 are coincident with each other. 1246 01:22:36,801 --> 01:22:57,660 So for coincidental branes, since branes 1247 01:22:57,660 --> 01:23:12,960 are indistinguishable from each other, 1248 01:23:12,960 --> 01:23:15,010 so they are higher dimensional generalization 1249 01:23:15,010 --> 01:23:19,330 of what we call identical particles. 1250 01:23:19,330 --> 01:23:25,050 So if they're indistinguishable from each, 1251 01:23:25,050 --> 01:23:31,274 we can shuffle their indices. 1252 01:23:31,274 --> 01:23:33,690 We should have the symmetry to reshuffle their symmetries. 1253 01:23:43,260 --> 01:23:47,515 So whether we call this 1, 1 or this 1, 2 or this 1, 1254 01:23:47,515 --> 01:23:48,565 1 should not matter. 1255 01:23:53,230 --> 01:23:58,810 So if I combine these two facts, two observations together, 1256 01:23:58,810 --> 01:24:03,720 then when we have n coincidental branes, 1257 01:24:03,720 --> 01:24:06,664 then there's in fact u(n) symmetry. 1258 01:24:18,100 --> 01:24:23,510 If that whole string interaction is invariant, say 1259 01:24:23,510 --> 01:24:39,200 if I have psi, I, J goes to say U, I, K, U, J, L, star, psi, K, 1260 01:24:39,200 --> 01:24:49,150 L. So back here on U just corresponding to I 1261 01:24:49,150 --> 01:24:50,710 reshuffle all the indices. 1262 01:24:50,710 --> 01:24:53,880 So I have to do the same to U. So I reshuffle the two indices 1263 01:24:53,880 --> 01:24:55,000 in the same way. 1264 01:24:55,000 --> 01:24:57,214 But I have a star here. 1265 01:24:57,214 --> 01:24:58,380 It's because of this reason. 1266 01:24:58,380 --> 01:25:02,870 Because in some sense, the sigma 0 and sigma pi, 1267 01:25:02,870 --> 01:25:07,390 they're only symmetries if I multiply opposite phase factor. 1268 01:25:07,390 --> 01:25:10,950 I can rewrite this as a matrix notation. 1269 01:25:10,950 --> 01:25:13,010 If you think about each side as a matrix, 1270 01:25:13,010 --> 01:25:14,634 then this is the symmetry corresponding 1271 01:25:14,634 --> 01:25:17,288 to psi U psi dagger. 1272 01:25:21,120 --> 01:25:29,840 And the U can be arbitrary unitary matrices. 1273 01:25:36,210 --> 01:25:40,610 So in fact, when you have n coincidental branes together, 1274 01:25:40,610 --> 01:25:43,810 there's u(n) symmetries for the string interactions. 1275 01:25:52,320 --> 01:26:02,670 So to say it in a more fancy mathematical language 1276 01:26:02,670 --> 01:26:15,140 is that each open string excitation 1277 01:26:15,140 --> 01:26:30,510 transforms under the adjunct representation of this u(n). 1278 01:26:34,950 --> 01:26:36,650 So this is like a join representation. 1279 01:26:36,650 --> 01:26:38,496 If you have u(n) symmetry, then this 1280 01:26:38,496 --> 01:26:40,222 is like a join representation. 1281 01:26:47,130 --> 01:26:57,256 So on the string worksheet, this is really a global symmetry. 1282 01:27:10,380 --> 01:27:14,090 So it's just a phase factor associated with each n. 1283 01:27:14,090 --> 01:27:15,730 There's nothing. 1284 01:27:15,730 --> 01:27:17,416 It's a global symmetry. 1285 01:27:22,260 --> 01:27:28,850 But the remarkable thing is that in the space time, 1286 01:27:28,850 --> 01:27:32,220 this becomes a gauge symmetry. 1287 01:27:32,220 --> 01:27:36,050 So this tells you, because of the presence of this u(n) gauge 1288 01:27:36,050 --> 01:27:39,140 symmetry and because of each mode 1289 01:27:39,140 --> 01:27:44,170 transforms under a join representation of some u(n), 1290 01:27:44,170 --> 01:27:48,480 that as a space-time field, they also 1291 01:27:48,480 --> 01:27:53,890 must transform under a join representation of some u(n). 1292 01:27:53,890 --> 01:27:57,445 And interpreting the space-time, then this u(n) 1293 01:27:57,445 --> 01:28:01,120 must be a gauge symmetry. 1294 01:28:01,120 --> 01:28:03,820 On the worksheet, it's a global symmetry. 1295 01:28:03,820 --> 01:28:15,342 But in space-time-- so let me just write it down. 1296 01:28:15,342 --> 01:28:22,110 In space-time-- or in other words, 1297 01:28:22,110 --> 01:28:31,720 in the world volume of D-branes, this u(n) 1298 01:28:31,720 --> 01:28:46,230 must be a gauge symmetry for the following reason. 1299 01:28:53,760 --> 01:29:01,480 Because the only way we know-- for example, 1300 01:29:01,480 --> 01:29:05,910 this gauge field becomes transformed 1301 01:29:05,910 --> 01:29:12,380 under a join representation of some u(n) symmetry. 1302 01:29:17,160 --> 01:29:21,030 And all the excitations will have this symmetry. 1303 01:29:21,030 --> 01:29:22,650 And then the only way we can make 1304 01:29:22,650 --> 01:29:26,970 sense the gauge field under such symmetry 1305 01:29:26,970 --> 01:29:31,210 is that this is a gauge symmetry and this is the gauge 1306 01:29:31,210 --> 01:29:34,520 boson for that symmetry. 1307 01:29:34,520 --> 01:29:37,780 Is it clear? 1308 01:29:37,780 --> 01:29:43,571 So let me just say it in words. 1309 01:29:43,571 --> 01:29:45,943 It must be a gauge symmetry. 1310 01:29:45,943 --> 01:30:16,096 And in particular, a alpha I J must be the corresponding gauge 1311 01:30:16,096 --> 01:30:16,596 bosons. 1312 01:30:22,392 --> 01:30:27,590 Because this is the only way we know how to make sense, 1313 01:30:27,590 --> 01:30:29,160 because this is the only way we know 1314 01:30:29,160 --> 01:30:32,260 can happen at low energies. 1315 01:30:32,260 --> 01:30:37,710 Some gauge fields transformed as a matrix interact 1316 01:30:37,710 --> 01:30:39,180 with each other. 1317 01:30:39,180 --> 01:30:41,315 And this can only be Yang-Mills theory. 1318 01:30:41,315 --> 01:30:42,690 And if this is Yang-Mills theory, 1319 01:30:42,690 --> 01:30:45,721 then this must be a gauge symmetry. 1320 01:30:45,721 --> 01:30:46,970 So this is the basic argument. 1321 01:30:49,690 --> 01:31:04,992 And at low energies, we must have Yang-Mills theory. 1322 01:31:10,800 --> 01:31:12,185 So something remarkable happens. 1323 01:31:17,830 --> 01:31:25,420 So each D-brane, when we separate them. 1324 01:31:25,420 --> 01:31:27,590 is a Maxwell theory. 1325 01:31:27,590 --> 01:31:30,820 When you put them together, they become Yang-Mills theory. 1326 01:31:33,400 --> 01:31:35,060 And somehow, they become non-Rubinian. 1327 01:31:39,480 --> 01:31:44,790 And everything comes from, in the very trivial way 1328 01:31:44,790 --> 01:31:50,762 in string theory, you just count the indicies. 1329 01:31:50,762 --> 01:31:52,470 But the physical implication is profound. 1330 01:31:55,780 --> 01:31:57,680 And this can be confirmed, again, 1331 01:31:57,680 --> 01:31:59,509 by just starting the explicit string 1332 01:31:59,509 --> 01:32:00,675 theory scattering amplitude. 1333 01:32:04,530 --> 01:32:10,007 You can calculate the scattering of 3a in string theory. 1334 01:32:10,007 --> 01:32:12,590 Then you find it's precisely-- at low energy, precisely given, 1335 01:32:12,590 --> 01:32:18,960 but the same vertex as the Yang-Mills theory, et cetera. 1336 01:32:18,960 --> 01:32:20,100 So you can do that. 1337 01:32:20,100 --> 01:32:22,184 Then you find the low energy effective action can 1338 01:32:22,184 --> 01:32:23,350 be written as the following. 1339 01:32:25,980 --> 01:32:34,210 Some Yang-Mills coupling trace. 1340 01:32:34,210 --> 01:32:36,444 Let me just write down the Yang-Mills theory. 1341 01:33:00,020 --> 01:33:03,160 You find the low energy effective action 1342 01:33:03,160 --> 01:33:05,910 can be written in this form. 1343 01:33:05,910 --> 01:33:08,577 So this is standard Yang-Mills field strings 1344 01:33:08,577 --> 01:33:10,035 because now everything is a matrix. 1345 01:33:15,262 --> 01:33:16,220 Everything is a matrix. 1346 01:33:29,240 --> 01:33:35,040 And both A alpha and phi a now are embedded in matrices. 1347 01:33:55,300 --> 01:34:01,400 So this g Yang-Mills is the Yang-Mills coupling 1348 01:34:01,400 --> 01:34:03,620 which describes how the gauge fields interact 1349 01:34:03,620 --> 01:34:05,160 with each other. 1350 01:34:05,160 --> 01:34:11,350 And obviously, this should be related to the open string 1351 01:34:11,350 --> 01:34:17,590 interaction because-- I think I have already erased it. 1352 01:34:17,590 --> 01:34:22,110 This kind of interaction, joining the string. 1353 01:34:25,120 --> 01:34:26,700 This corresponding to two strings 1354 01:34:26,700 --> 01:34:30,260 joined into one string, and this proportional to g0. 1355 01:34:30,260 --> 01:34:33,690 And this, from the field theory point of view, 1356 01:34:33,690 --> 01:34:37,590 it just controls the interaction of the three A's. 1357 01:34:37,590 --> 01:34:39,600 So that must be the Yang-Mills coupling. 1358 01:34:39,600 --> 01:34:41,810 So this must be the Yang-Mills coupling. 1359 01:34:41,810 --> 01:34:44,365 And then this should be related to gs 1360 01:34:44,365 --> 01:34:50,480 to the power 1/2, which we just derived slightly earlier. 1361 01:34:50,480 --> 01:34:59,770 So now, on dimensional ground, the g Yang-Mills square 1362 01:34:59,770 --> 01:35:01,590 can be written as-- let me write it here. 1363 01:35:01,590 --> 01:35:03,930 This is an important formula. 1364 01:35:03,930 --> 01:35:08,420 On dimensional grounds, the g Yang-Mills square, 1365 01:35:08,420 --> 01:35:11,930 I should be able to write it as gs. 1366 01:35:11,930 --> 01:35:17,240 So on the Yang-Mills coupling, we 1367 01:35:17,240 --> 01:35:21,160 will use the standard convention that A 1368 01:35:21,160 --> 01:35:24,450 has the dimension of mass. 1369 01:35:24,450 --> 01:35:27,552 And then this is a dimension 4 object. 1370 01:35:27,552 --> 01:35:30,402 And this is a dimension p plus 1. 1371 01:35:30,402 --> 01:35:31,860 And so the Yang-Mills coupling, you 1372 01:35:31,860 --> 01:35:38,370 can deduce the dimension of the Yang-Mills coupling there. 1373 01:35:38,370 --> 01:35:47,790 And it just turns out to be p minus 3 dense-- again, 1374 01:35:47,790 --> 01:35:53,000 because alpha prime in string theory is only length scale. 1375 01:35:53,000 --> 01:35:57,200 So the alpha prime must come from here. 1376 01:35:57,200 --> 01:35:59,170 And then times some constant. 1377 01:35:59,170 --> 01:36:01,248 So again, dp is just some numerical constant. 1378 01:36:09,650 --> 01:36:16,730 And you can see explicitly that when p equal to 3, 1379 01:36:16,730 --> 01:36:18,760 then you have d3 brane. 1380 01:36:18,760 --> 01:36:22,175 Then the world volume theory is four dimensional. 1381 01:36:22,175 --> 01:36:24,790 Then we recall that in four dimensions, 1382 01:36:24,790 --> 01:36:26,940 Yang-Mills theory is dimensionless. 1383 01:36:26,940 --> 01:36:29,370 QCD [INAUDIBLE]. 1384 01:36:37,200 --> 01:36:38,170 Good? 1385 01:36:38,170 --> 01:36:39,740 Any questions? 1386 01:36:39,740 --> 01:36:40,240 Yes? 1387 01:36:40,240 --> 01:36:42,829 AUDIENCE: What to do with phi h? 1388 01:36:42,829 --> 01:36:43,370 HONG LIU: Hm? 1389 01:36:43,370 --> 01:36:44,770 AUDIENCE: What to do with phi h? 1390 01:36:44,770 --> 01:36:45,770 HONG LIU: What to do with phi h? 1391 01:36:45,770 --> 01:36:47,394 What do you mean what to do with phi h? 1392 01:36:47,394 --> 01:36:49,970 AUDIENCE: Scalar color particles? 1393 01:36:49,970 --> 01:36:52,560 HONG LIU: Yeah. 1394 01:36:52,560 --> 01:36:55,510 I forgot to mention here is the standard derivative, covariant 1395 01:36:55,510 --> 01:36:57,130 derivative. 1396 01:36:57,130 --> 01:37:02,130 So the alpha phi a is just the standard particle phi 1397 01:37:02,130 --> 01:37:08,250 a minus i a alpha phi a. 1398 01:37:08,250 --> 01:37:15,420 So it's the standard gauge covariant derivative. 1399 01:37:15,420 --> 01:37:19,510 AUDIENCE: Shouldn't phi be more fundamental presentation of-- 1400 01:37:19,510 --> 01:37:20,230 HONG LIU: No. 1401 01:37:20,230 --> 01:37:22,910 Everything is to the join because everything 1402 01:37:22,910 --> 01:37:24,111 has two ends. 1403 01:37:24,111 --> 01:37:26,152 AUDIENCE: So how to interpret phi, then, in our-- 1404 01:37:26,152 --> 01:37:27,010 HONG LIU: Hm? 1405 01:37:27,010 --> 01:37:29,765 AUDIENCE: From the present zoology of particles, 1406 01:37:29,765 --> 01:37:32,400 where to put phi, the observed particles? 1407 01:37:32,400 --> 01:37:33,795 HONG LIU: Sorry? 1408 01:37:33,795 --> 01:37:36,944 AUDIENCE: If a is just the gauge boson [INAUDIBLE] 1409 01:37:36,944 --> 01:37:38,800 or something, what is phi? 1410 01:37:38,800 --> 01:37:40,590 HONG LIU: Phi is some scalar field 1411 01:37:40,590 --> 01:37:45,180 transformed on the join representation of the gauge 1412 01:37:45,180 --> 01:37:45,680 group. 1413 01:37:45,680 --> 01:37:48,690 It's a matter field describing the motion of the brane. 1414 01:37:51,372 --> 01:37:52,720 AUDIENCE: I have a question. 1415 01:37:52,720 --> 01:37:55,920 There you say that we can reshuffle their indexes. 1416 01:37:55,920 --> 01:37:59,489 So that symmetry should be permutation symmetry? 1417 01:37:59,489 --> 01:38:00,030 HONG LIU: No. 1418 01:38:00,030 --> 01:38:01,113 AUDIENCE: Why [INAUDIBLE]? 1419 01:38:05,690 --> 01:38:07,590 HONG LIU: But I reshuffle because I say it 1420 01:38:07,590 --> 01:38:10,420 in the more heuristic way. 1421 01:38:10,420 --> 01:38:15,580 But normally, if the indices is not-- these are the states. 1422 01:38:15,580 --> 01:38:17,580 And you can just swivel-post them. 1423 01:38:17,580 --> 01:38:20,010 A different i, they should be the same thing. 1424 01:38:20,010 --> 01:38:22,426 They're just corresponding to relabeling, I'm just saying. 1425 01:38:24,689 --> 01:38:26,980 The lateral action of anything on the state, of course, 1426 01:38:26,980 --> 01:38:28,230 is the unitary transformation. 1427 01:38:35,300 --> 01:38:39,206 AUDIENCE: Is there anything to do with that? 1428 01:38:41,639 --> 01:38:42,180 HONG LIU: No. 1429 01:38:42,180 --> 01:38:44,010 That thing is related to this star, 1430 01:38:44,010 --> 01:38:45,300 why we put this star here. 1431 01:38:50,500 --> 01:38:53,030 This is the reason why we put the star there, 1432 01:38:53,030 --> 01:38:55,250 because the two endpoints, they should 1433 01:38:55,250 --> 01:38:56,400 transform in opposite way. 1434 01:38:59,630 --> 01:39:00,950 Yes? 1435 01:39:00,950 --> 01:39:03,840 AUDIENCE: Since the branes become dynamical, 1436 01:39:03,840 --> 01:39:06,490 can't they fluctuate in different ways 1437 01:39:06,490 --> 01:39:09,494 and so no longer coincide? 1438 01:39:09,494 --> 01:39:10,160 HONG LIU: Sorry? 1439 01:39:10,160 --> 01:39:13,210 AUDIENCE: Since the branes are dynamical, 1440 01:39:13,210 --> 01:39:15,680 you put them all in one place at first. 1441 01:39:15,680 --> 01:39:18,870 But can't they now start fluctuating and separate? 1442 01:39:18,870 --> 01:39:20,844 HONG LIU: They can certainly. 1443 01:39:20,844 --> 01:39:23,330 AUDIENCE: They're no longer symmetric. 1444 01:39:23,330 --> 01:39:24,900 They become distinguishable. 1445 01:39:24,900 --> 01:39:26,622 HONG LIU: They can certainly start 1446 01:39:26,622 --> 01:39:32,030 moving apart if you give them some initial motion. 1447 01:39:32,030 --> 01:39:38,530 But the fluctuation, the system is isotropic. 1448 01:39:38,530 --> 01:39:41,315 There's no place for them to-- they will fluctuate, 1449 01:39:41,315 --> 01:39:43,190 but they will still at that point on average. 1450 01:39:46,176 --> 01:39:47,800 There's no preferred direction for them 1451 01:39:47,800 --> 01:39:50,730 to go unless you give them a direction. 1452 01:39:50,730 --> 01:39:53,019 You say, I want them to go in that direction. 1453 01:39:53,019 --> 01:39:53,810 Then you push them. 1454 01:39:53,810 --> 01:39:55,185 AUDIENCE: But you can still think 1455 01:39:55,185 --> 01:39:57,300 of them fluctuating kind of separately, 1456 01:39:57,300 --> 01:39:59,090 separating in their fluctuations? 1457 01:39:59,090 --> 01:40:00,500 HONG LIU: Sure. 1458 01:40:00,500 --> 01:40:01,760 This a are their fluctuations. 1459 01:40:01,760 --> 01:40:05,250 Phi are their fluctuations. 1460 01:40:05,250 --> 01:40:07,950 AUDIENCE: Is this commutative term belonging to the effective 1461 01:40:07,950 --> 01:40:11,735 action an interaction between the D-branes or is it-- 1462 01:40:11,735 --> 01:40:12,360 HONG LIU: Yeah. 1463 01:40:15,870 --> 01:40:17,850 Good point. 1464 01:40:17,850 --> 01:40:21,940 Let me just comment on this term. 1465 01:40:21,940 --> 01:40:28,160 So if you think about it, as we said before, 1466 01:40:28,160 --> 01:40:33,490 a D-brane, no matter what dimension 1467 01:40:33,490 --> 01:40:35,490 of the D-brane, the story when we quantize them, 1468 01:40:35,490 --> 01:40:36,967 they're almost the same. 1469 01:40:36,967 --> 01:40:37,925 It's the same spectrum. 1470 01:40:42,120 --> 01:40:43,920 If you have a space-feeling brane 1471 01:40:43,920 --> 01:40:46,156 that everything is A alpha, and then 1472 01:40:46,156 --> 01:40:48,840 if you have some lower dimensional brane, some of them 1473 01:40:48,840 --> 01:40:50,580 become scalar field, et cetera. 1474 01:40:50,580 --> 01:40:54,710 So essentially, they all have the same dynamics. 1475 01:40:54,710 --> 01:40:59,540 So this interaction can be considered just come from here. 1476 01:40:59,540 --> 01:41:02,110 So you can start with a space time feeling brane, 1477 01:41:02,110 --> 01:41:03,750 and then you go to lower dimensions. 1478 01:41:03,750 --> 01:41:08,100 And then this just can be considered to come from there. 1479 01:41:08,100 --> 01:41:10,720 It's part of the gauge theory. 1480 01:41:10,720 --> 01:41:13,144 AUDIENCE: So the whole low energy actions 1481 01:41:13,144 --> 01:41:17,016 are the action of supergravity plus this D-brane interaction, 1482 01:41:17,016 --> 01:41:20,900 you can think of the whole low energy action? 1483 01:41:20,900 --> 01:41:23,356 HONG LIU: You mean if you include the closed string? 1484 01:41:23,356 --> 01:41:23,980 AUDIENCE: Yeah. 1485 01:41:23,980 --> 01:41:24,722 HONG LIU: Yeah. 1486 01:41:24,722 --> 01:41:26,180 That's right, if you have D-branes. 1487 01:41:26,180 --> 01:41:26,950 That's right. 1488 01:41:26,950 --> 01:41:30,270 This is effective action on the D-brane. 1489 01:41:30,270 --> 01:41:32,860 This is the effective action on the D-brane. 1490 01:41:32,860 --> 01:41:37,690 AUDIENCE: I mean if you extend D-brane [INAUDIBLE], 1491 01:41:37,690 --> 01:41:41,426 then where does this action come from? 1492 01:41:41,426 --> 01:41:43,664 HONG LIU: No, that's what you said. 1493 01:41:43,664 --> 01:41:46,165 AUDIENCE: It's just two actions crossed together? 1494 01:41:46,165 --> 01:41:46,790 HONG LIU: Yeah. 1495 01:41:49,640 --> 01:41:50,140 Good? 1496 01:41:53,469 --> 01:41:54,435 AUDIENCE: Excuse me. 1497 01:41:54,435 --> 01:41:56,430 How come it has the term in action that's 1498 01:41:56,430 --> 01:41:59,869 proportional to phi without derivative? 1499 01:41:59,869 --> 01:42:00,410 HONG LIU: No. 1500 01:42:00,410 --> 01:42:01,270 They have derivative. 1501 01:42:01,270 --> 01:42:02,519 This is covariant derivatives. 1502 01:42:02,519 --> 01:42:03,859 AUDIENCE: The commutator. 1503 01:42:03,859 --> 01:42:05,900 HONG LIU: No, but phi have covariant derivatives. 1504 01:42:05,900 --> 01:42:06,830 AUDIENCE: But the next time. 1505 01:42:06,830 --> 01:42:07,454 HONG LIU: Yeah? 1506 01:42:07,454 --> 01:42:10,692 AUDIENCE: That thing appears to be proportional to phi 1507 01:42:10,692 --> 01:42:11,664 without derivative. 1508 01:42:11,664 --> 01:42:13,122 HONG LIU: Yes? 1509 01:42:13,122 --> 01:42:15,630 AUDIENCE: But this is the Minkowski space or something. 1510 01:42:15,630 --> 01:42:16,369 HONG LIU: Yeah. 1511 01:42:16,369 --> 01:42:17,410 That's a very good point. 1512 01:42:17,410 --> 01:42:19,740 I'm going to mention that point. 1513 01:42:19,740 --> 01:42:24,266 But the key is that this particular potential-- this 1514 01:42:24,266 --> 01:42:25,140 is a very good point. 1515 01:42:25,140 --> 01:42:27,394 I'm going to mention that. 1516 01:42:27,394 --> 01:42:28,935 I will mention that in a few minutes. 1517 01:42:31,910 --> 01:42:35,770 I do because I do want you to do your p-set. 1518 01:42:48,750 --> 01:42:52,065 So now let's consider separating the branes. 1519 01:42:55,599 --> 01:42:57,390 Now we will consider separating the branes. 1520 01:43:10,782 --> 01:43:12,240 Again, let's consider the situation 1521 01:43:12,240 --> 01:43:13,495 we just have two of them. 1522 01:43:16,460 --> 01:43:18,680 So at the beginning, they're coincidental. 1523 01:43:18,680 --> 01:43:21,690 And then now let's separate them in some direction. 1524 01:43:21,690 --> 01:43:24,312 Let's call that direction x. 1525 01:43:24,312 --> 01:43:26,270 So let's say they separate by some distance, d. 1526 01:43:28,940 --> 01:43:30,748 This is 1, this is 2. 1527 01:43:34,180 --> 01:43:36,690 So of course, this 1, 1 and 2, 2 string, nothing 1528 01:43:36,690 --> 01:43:43,570 changes because they're just still ending on the same brane. 1529 01:43:43,570 --> 01:43:45,129 But those strings are now different. 1530 01:43:49,720 --> 01:43:51,690 So now 1, 2 and 2, 1 strings become different. 1531 01:43:58,420 --> 01:44:01,750 So 1, 1, 2, 2, exactly the same as before. 1532 01:44:11,070 --> 01:44:12,130 And the 1, 2 string. 1533 01:44:12,130 --> 01:44:13,963 For example, let's consider the 1, 2 string. 1534 01:44:16,440 --> 01:44:18,780 Then the boundary condition changes 1535 01:44:18,780 --> 01:44:22,940 in the way I have sigma equal to 0, say, in the x direction. 1536 01:44:22,940 --> 01:44:23,770 Sigma equal to 0. 1537 01:44:26,692 --> 01:44:28,239 Tau, say, is at some location. 1538 01:44:28,239 --> 01:44:29,030 Let's call this x0. 1539 01:44:33,190 --> 01:44:39,640 And then x sigma equal to pi tau then becomes x0 plus d. 1540 01:44:43,290 --> 01:44:46,480 So now it means when you contact this string, 1541 01:44:46,480 --> 01:44:51,560 you have to do a slightly modified boundary condition. 1542 01:44:51,560 --> 01:44:54,000 So you have start with x equal to 0. 1543 01:44:54,000 --> 01:44:56,768 Now you must include a term depend on sigma. 1544 01:45:08,140 --> 01:45:11,020 And again, as before, we take the xL 1545 01:45:11,020 --> 01:45:14,080 to be minus xR and periodic, et cetera. 1546 01:45:17,730 --> 01:45:20,295 So for sigma equal to 0, then this boundary condition 1547 01:45:20,295 --> 01:45:21,570 is satisfied. 1548 01:45:21,570 --> 01:45:24,410 But in order to satisfy the boundary condition at pi, 1549 01:45:24,410 --> 01:45:28,130 then you now need this w equal to d divided by pi. 1550 01:45:32,020 --> 01:45:36,490 You need to develop d divided by pi. 1551 01:45:36,490 --> 01:45:37,770 So now, you have a sigma term. 1552 01:45:37,770 --> 01:45:41,419 So that will change your [INAUDIBLE] condition. 1553 01:45:41,419 --> 01:45:42,960 So remember the [INAUDIBLE] condition 1554 01:45:42,960 --> 01:45:48,190 we had before is something like this, q alpha prime p minus, 1555 01:45:48,190 --> 01:45:53,770 say, is p plus 4 pi alpha prime. 1556 01:45:53,770 --> 01:45:54,770 Say for the open string. 1557 01:45:54,770 --> 01:45:56,270 I'm writing the open string version, 1558 01:45:56,270 --> 01:46:00,085 which is, up to some numerical factor, the same as closed 1559 01:46:00,085 --> 01:46:01,948 string we wrote down before. 1560 01:46:10,890 --> 01:46:15,440 So now, because of this term, then there's 1561 01:46:15,440 --> 01:46:19,630 additional contribution on the right hand side. 1562 01:46:19,630 --> 01:46:22,030 It makes sense because now string 1563 01:46:22,030 --> 01:46:24,490 has to be stretched over some distance. 1564 01:46:24,490 --> 01:46:26,580 That costs energy. 1565 01:46:26,580 --> 01:46:30,630 So take one minute to do it yourselves. 1566 01:46:30,630 --> 01:46:31,820 Just plug this into here. 1567 01:46:31,820 --> 01:46:35,270 You only need to look at the behavior of this term. 1568 01:46:35,270 --> 01:46:36,330 Just plug in there. 1569 01:46:36,330 --> 01:46:40,170 Take you, say, five seconds. 1570 01:46:40,170 --> 01:46:44,900 You find that the massless condition now 1571 01:46:44,900 --> 01:46:54,720 has one more term plus the rest, as before. 1572 01:46:54,720 --> 01:47:04,920 And that means now all the previously massless particle, 1573 01:47:04,920 --> 01:47:08,315 this here with the corresponding a alpha and phi a, 1574 01:47:08,315 --> 01:47:13,660 are no longer massless. 1575 01:47:15,997 --> 01:47:17,580 Because previously, they were massless 1576 01:47:17,580 --> 01:47:19,520 because those terms are 0. 1577 01:47:19,520 --> 01:47:22,210 But now you have one more term. 1578 01:47:22,210 --> 01:47:27,240 And they have a mass given by M divided 1579 01:47:27,240 --> 01:47:31,070 by d divided by 2 alpha prime. 1580 01:47:31,070 --> 01:47:33,790 Just take the square root of that. 1581 01:47:33,790 --> 01:47:37,925 And because this is precisely the d times the string tension, 1582 01:47:37,925 --> 01:47:40,280 because 1 over 2 pi alpha prime is the string tension. 1583 01:47:40,280 --> 01:47:42,720 So this is exactly the energy we expect, just 1584 01:47:42,720 --> 01:47:44,527 from a classical picture. 1585 01:47:44,527 --> 01:47:46,360 You have a string stretched between the two. 1586 01:47:46,360 --> 01:47:50,860 Then this is the tension times the length of the string. 1587 01:47:59,310 --> 01:48:01,060 So now you can also easily understand 1588 01:48:01,060 --> 01:48:05,170 what's going on from field theory point of view. 1589 01:48:05,170 --> 01:48:11,930 And now you only have two sets of massless modes 1590 01:48:11,930 --> 01:48:14,370 now rather than four. 1591 01:48:14,370 --> 01:48:22,840 So what's happening is that now, the gauge symmetry is broken 1592 01:48:22,840 --> 01:48:25,330 from u(2) to u(1) times u(1). 1593 01:48:34,780 --> 01:48:45,210 So the separation of the branes essentially 1594 01:48:45,210 --> 01:48:55,310 corresponding to the Higgs mechanism 1595 01:48:55,310 --> 01:48:56,355 in the following sense. 1596 01:49:01,160 --> 01:49:05,240 When you separate the brane, I said 1597 01:49:05,240 --> 01:49:10,420 before phi should be interpreted as the [INAUDIBLE] 1598 01:49:10,420 --> 01:49:12,120 location of the brane. 1599 01:49:12,120 --> 01:49:13,630 But you separate the brane. 1600 01:49:13,630 --> 01:49:15,720 That means one of the five fields corresponding 1601 01:49:15,720 --> 01:49:18,307 to the x-- the x is in the transverse direction 1602 01:49:18,307 --> 01:49:18,890 for the brane. 1603 01:49:18,890 --> 01:49:21,889 One of the five fields corresponding to the x 1604 01:49:21,889 --> 01:49:23,180 must develop expectation value. 1605 01:49:26,970 --> 01:49:31,808 And that expectation value then gives rise to this mass. 1606 01:49:31,808 --> 01:49:34,203 So this is precisely a Higgs mechanism. 1607 01:49:38,040 --> 01:49:45,650 So now let's go back to the question 1608 01:49:45,650 --> 01:49:50,310 what this potential term means, whether we can actually, 1609 01:49:50,310 --> 01:49:53,910 as we said before, because of the translation symmetry, 1610 01:49:53,910 --> 01:49:57,940 in principle, we can pull the brane anywhere. 1611 01:49:57,940 --> 01:50:01,070 And now, I'm out of time. 1612 01:50:01,070 --> 01:50:07,830 So let me just say, if you look at this potential, 1613 01:50:07,830 --> 01:50:12,726 this becomes 0 precisely when phi a and phi b all 1614 01:50:12,726 --> 01:50:14,220 become commutes. 1615 01:50:17,590 --> 01:50:20,470 So that means we can diagonalize the phi corresponding to all 1616 01:50:20,470 --> 01:50:23,429 the transverse directions. 1617 01:50:23,429 --> 01:50:24,970 You can diagonalize phi corresponding 1618 01:50:24,970 --> 01:50:27,234 to all the transverse directions. 1619 01:50:27,234 --> 01:50:28,900 And then they correspond to the location 1620 01:50:28,900 --> 01:50:31,780 you put all your branes. 1621 01:50:31,780 --> 01:50:37,190 Go do your p-set, and you will see a more explicit discussion 1622 01:50:37,190 --> 01:50:38,917 of this. 1623 01:50:38,917 --> 01:50:40,000 And then one final remark. 1624 01:50:42,700 --> 01:50:47,100 At the beginning, you started with n branes together. 1625 01:50:47,100 --> 01:50:50,900 And then because of this, we can separate them. 1626 01:50:50,900 --> 01:50:54,670 So we can find a solution which, say they commute. 1627 01:50:54,670 --> 01:51:01,825 We separate them into different stacks, n1, n2, n3, et cetera. 1628 01:51:05,410 --> 01:51:09,760 So this corresponds to the configuration of phi. 1629 01:51:16,190 --> 01:51:19,340 So let me just write all phi as a vector. 1630 01:51:19,340 --> 01:51:23,385 Say there's a1, n1 of them at location a1, 1631 01:51:23,385 --> 01:51:27,820 and n2 of them at location a2, and ak. 1632 01:51:31,580 --> 01:51:36,980 If I separate them into n stacks, then will be like this. 1633 01:51:36,980 --> 01:51:42,710 So n1 of them at location a1, and n2 of them at location a2. 1634 01:51:42,710 --> 01:51:45,090 You can check. 1635 01:51:45,090 --> 01:51:53,206 So this is n1 times n1 identity matrix. 1636 01:51:53,206 --> 01:51:56,565 This is n2 times n2. 1637 01:51:56,565 --> 01:52:00,480 And you can check that such a configuration does satisfy 1638 01:52:00,480 --> 01:52:03,290 this condition so that you actually 1639 01:52:03,290 --> 01:52:05,890 can separate the brane into such configurations. 1640 01:52:05,890 --> 01:52:07,830 And in this case, then the gauge symmetry 1641 01:52:07,830 --> 01:52:14,320 is broken into u n1 times u n2 times 1642 01:52:14,320 --> 01:52:18,620 u nk, because only those points [INAUDIBLE] parts survives. 1643 01:52:21,390 --> 01:52:23,930 And all the other strings between them become massive. 1644 01:52:23,930 --> 01:52:24,430 OK. 1645 01:52:24,430 --> 01:52:26,312 Let's stop here.