1 00:00:00,090 --> 00:00:02,430 The following content is provided under a Creative 2 00:00:02,430 --> 00:00:03,820 Commons license. 3 00:00:03,820 --> 00:00:06,060 Your support will help MIT OpenCourseWare 4 00:00:06,060 --> 00:00:10,150 continue to offer high quality educational resources for free. 5 00:00:10,150 --> 00:00:12,690 To make a donation, or to view additional materials 6 00:00:12,690 --> 00:00:16,600 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,600 --> 00:00:17,255 at ocw.mit.edu. 8 00:00:21,980 --> 00:00:23,470 HONG LIU: OK let's start. 9 00:00:23,470 --> 00:00:25,790 So let me just mention brief-- first 10 00:00:25,790 --> 00:00:29,250 let me just mention briefly, this story regarding 11 00:00:29,250 --> 00:00:33,740 the D-brane and the Higgs mechanism, 12 00:00:33,740 --> 00:00:37,950 which we discussed at the end of last lecture. 13 00:00:37,950 --> 00:00:41,740 So there was a P-set problem which was related to this. 14 00:00:45,290 --> 00:00:47,770 Yes, I expected maybe the P-set problem was a little bit 15 00:00:47,770 --> 00:00:50,700 tough because we did not go to many details. 16 00:00:50,700 --> 00:00:52,225 We did not go to many details. 17 00:00:55,180 --> 00:01:03,459 Yeah, so in grading we will be very flexible because we-- 18 00:01:03,459 --> 00:01:05,459 So let me just say a few more things about that, 19 00:01:05,459 --> 00:01:06,885 just to make sure-- 20 00:01:10,250 --> 00:01:14,020 So here is the action. 21 00:01:14,020 --> 00:01:15,900 If you have n-- say if you supposedly 22 00:01:15,900 --> 00:01:20,884 have N D-branes together-- and then in such case-- 23 00:01:20,884 --> 00:01:22,300 so this is a low energy defractive 24 00:01:22,300 --> 00:01:24,690 action for the massless degree of freedom 25 00:01:24,690 --> 00:01:27,450 on the D-brane for which you have a gauge field, which 26 00:01:27,450 --> 00:01:30,690 is a matrix, and then you also have a scalar 27 00:01:30,690 --> 00:01:34,130 field, a number of scalar fields, 28 00:01:34,130 --> 00:01:36,500 which are also n-by-n matrices. 29 00:01:36,500 --> 00:01:37,990 And the number of scalar fields is 30 00:01:37,990 --> 00:01:41,050 the same as the number of transverse dimensions. 31 00:01:41,050 --> 00:01:46,430 And then we also discussed that the scalar fields 32 00:01:46,430 --> 00:01:51,030 can be considered as describing the transverse dynamics 33 00:01:51,030 --> 00:01:52,280 of the brain. 34 00:01:52,280 --> 00:01:55,400 Say, let's consider you have-- let's 35 00:01:55,400 --> 00:01:59,339 consider you have a single brane and there's only 36 00:01:59,339 --> 00:02:00,380 one transverse direction. 37 00:02:00,380 --> 00:02:04,590 Say let's call it the phi direction. 38 00:02:04,590 --> 00:02:07,070 There's only one direction, say phi 1 direction. 39 00:02:11,610 --> 00:02:15,420 Then essentially the location of the brane then essentially 40 00:02:15,420 --> 00:02:17,500 the expectation value of phi 1 can 41 00:02:17,500 --> 00:02:21,660 be considered-- say suppose this is the direction of x1, 42 00:02:21,660 --> 00:02:24,030 and the corresponding scalar field is phi 1. 43 00:02:24,030 --> 00:02:26,420 And then the expectation value of phi 1 44 00:02:26,420 --> 00:02:28,340 can be considered as the location 45 00:02:28,340 --> 00:02:32,880 of the brane and this phi is the string which 46 00:02:32,880 --> 00:02:34,835 ending on this brain and come back to itself. 47 00:02:37,510 --> 00:02:40,680 So now let's consider if you have two branes-- now consider 48 00:02:40,680 --> 00:02:46,370 you have n-branes-- again, b is only one transverse direction. 49 00:02:46,370 --> 00:02:49,740 I now have n-branes with one transverse direction, 50 00:02:49,740 --> 00:02:52,030 which again is x1. 51 00:02:52,030 --> 00:02:54,584 And now you can just separate them. 52 00:02:54,584 --> 00:02:56,750 So now let's consider the process which you separate 53 00:02:56,750 --> 00:03:02,310 these n-branes into-- say completely separate them 54 00:03:02,310 --> 00:03:04,830 in different locations. 55 00:03:04,830 --> 00:03:08,450 So this is x1, x2, to xn. 56 00:03:14,480 --> 00:03:17,570 So this situation can be considered as a situation. 57 00:03:17,570 --> 00:03:20,703 So in this case, the phi is the-- phi 58 00:03:20,703 --> 00:03:22,930 is the m-by-n matrices. 59 00:03:22,930 --> 00:03:25,780 And then this can be considered as phi having an expectation 60 00:03:25,780 --> 00:03:36,930 value which is only in the diagonal entry, 61 00:03:36,930 --> 00:03:40,090 and each that diagonal entry describes 62 00:03:40,090 --> 00:03:43,400 the location of each D-brane. 63 00:03:43,400 --> 00:03:44,880 And it's natural to think that why 64 00:03:44,880 --> 00:03:47,700 this is the diagonal entry because the diagonal entry 65 00:03:47,700 --> 00:03:52,480 describes the string, which ending on the brane itself. 66 00:03:52,480 --> 00:03:55,650 So this is a one-one string, and this is 67 00:03:55,650 --> 00:03:57,970 a two-two string, and n-string. 68 00:03:57,970 --> 00:04:01,560 And so the diagonal string is the excitation 69 00:04:01,560 --> 00:04:02,857 on the brane itself. 70 00:04:02,857 --> 00:04:04,440 And the off-diagonal degree of freedom 71 00:04:04,440 --> 00:04:07,160 corresponding to, essentially corresponding to the excitation 72 00:04:07,160 --> 00:04:09,519 between the string between the branes. 73 00:04:09,519 --> 00:04:11,060 And so it's natural that the location 74 00:04:11,060 --> 00:04:13,090 of the branes, and then just corresponding 75 00:04:13,090 --> 00:04:16,779 to the diagonal entries. 76 00:04:16,779 --> 00:04:23,680 And so now we can generalize to the general dimensions. 77 00:04:23,680 --> 00:04:29,660 Now suppose there are two transverse dimensions-- 78 00:04:29,660 --> 00:04:33,300 let me call them x1 and x2-- and then 79 00:04:33,300 --> 00:04:35,950 now you have two scalar fields. 80 00:04:35,950 --> 00:04:39,840 So now the story is a little bit more complicated 81 00:04:39,840 --> 00:04:42,060 because when you have two transverse directions, 82 00:04:42,060 --> 00:04:44,226 say if you have more than two transverse directions, 83 00:04:44,226 --> 00:04:47,320 now you have nontrivial potential term. 84 00:04:47,320 --> 00:04:50,870 And under the law of the configuration, 85 00:04:50,870 --> 00:04:53,580 say suppose it's five, have low space time-dependent 86 00:04:53,580 --> 00:04:58,571 on the law of configuration, say the minimum with the potential, 87 00:04:58,571 --> 00:05:00,570 the mass was corresponding to the configuration, 88 00:05:00,570 --> 00:05:03,320 which the commutative phi a and phi b to be zero. 89 00:05:06,370 --> 00:05:11,000 Right now, say we have two transverse directions, 90 00:05:11,000 --> 00:05:15,480 so we only have two, but you can generalize to any number 91 00:05:15,480 --> 00:05:18,070 of transverse directions. 92 00:05:18,070 --> 00:05:22,100 So that means that the phi, or the phi 93 00:05:22,100 --> 00:05:23,670 equals 1, or the scalar field, they 94 00:05:23,670 --> 00:05:25,045 have to converge with each other. 95 00:05:27,950 --> 00:05:31,420 That means they can be simultaneously diagonalized. 96 00:05:31,420 --> 00:05:34,840 That means they simultaneously diagonalized. 97 00:05:34,840 --> 00:05:41,190 That means we can choose a basis, say if I write 98 00:05:41,190 --> 00:05:50,880 the phi, the collection of all the D-branes, 99 00:05:50,880 --> 00:05:55,110 then I can simultaneously diagonalize them, and then 100 00:05:55,110 --> 00:05:56,140 have the following form. 101 00:06:00,620 --> 00:06:04,070 So each x1 goes one into-- so each vector goes one 102 00:06:04,070 --> 00:06:08,810 into-- for example here, is x1-- I should 103 00:06:08,810 --> 00:06:13,940 use a different notation, so let me call the spacetime y1, y2-- 104 00:06:13,940 --> 00:06:17,950 and say this is x2, et cetera. 105 00:06:17,950 --> 00:06:19,640 So this defines the configuration, 106 00:06:19,640 --> 00:06:21,940 which you have one D-brane here and one D-brane 107 00:06:21,940 --> 00:06:26,990 at this point, and at x3, and et cetera. 108 00:06:26,990 --> 00:06:30,360 And again, it's only the diagonal entry 109 00:06:30,360 --> 00:06:33,595 that describes the location of the D-branes. 110 00:06:33,595 --> 00:06:35,970 The diagonal entry describe the location of the D-branes. 111 00:06:35,970 --> 00:06:41,227 And these, because they are all diagonal and the commutator 112 00:06:41,227 --> 00:06:43,435 is zero, and these are the law of the configurations. 113 00:06:46,902 --> 00:06:48,360 So this is a very important lesson. 114 00:06:48,360 --> 00:06:52,590 It tells you that even you have a potential, still the D-brane, 115 00:06:52,590 --> 00:06:54,780 you can put it anywhere you want. 116 00:06:54,780 --> 00:06:57,220 And you have the freedom to put the D-brane-- you 117 00:06:57,220 --> 00:06:58,620 don't break translations image. 118 00:07:03,540 --> 00:07:04,510 Any questions on this? 119 00:07:14,926 --> 00:07:21,110 Of course when you say, when the scalar field develops 120 00:07:21,110 --> 00:07:24,360 expectation value and then your original un gauge symmetry's 121 00:07:24,360 --> 00:07:29,070 broken, and then broken into the just 122 00:07:29,070 --> 00:07:32,790 in the standard story to whatever 123 00:07:32,790 --> 00:07:35,700 [INAUDIBLE] group allowed by this-- 124 00:07:35,700 --> 00:07:41,470 so if all the x1-- if all of them are different, 125 00:07:41,470 --> 00:07:44,440 then the remaining just be 1 to the power n. 126 00:07:44,440 --> 00:07:46,950 Because now you just have each brane separated, 127 00:07:46,950 --> 00:07:49,062 then you just have 1 to the power n. 128 00:07:54,780 --> 00:07:55,280 Good? 129 00:07:59,030 --> 00:08:03,680 So not let's move to new stuff. 130 00:08:03,680 --> 00:08:06,000 So now let's say I little bit about the D-branes 131 00:08:06,000 --> 00:08:07,740 in super string theory. 132 00:08:07,740 --> 00:08:09,700 So what we said so far actually applies 133 00:08:09,700 --> 00:08:13,200 to D-branes in bosonic string and in superstring. 134 00:08:15,780 --> 00:08:17,960 Now let's say a little bit more above the D-branes. 135 00:08:30,970 --> 00:08:34,970 So as we discussed before, when we quantized the open string, 136 00:08:34,970 --> 00:08:47,200 the D-branes in bosonic string always 137 00:08:47,200 --> 00:09:02,010 have a tachyon-- always have open string tachyon, 138 00:09:02,010 --> 00:09:06,430 no matter what D-brane you have because as we discussed before, 139 00:09:06,430 --> 00:09:08,720 the quantization, the zero-point energy on the string 140 00:09:08,720 --> 00:09:11,680 is the same no matter which dimension of the brane 141 00:09:11,680 --> 00:09:12,819 is et cetera. 142 00:09:12,819 --> 00:09:14,610 And so you always have open string tachyon. 143 00:09:21,340 --> 00:09:24,450 So that means, as we said before, tachyon 144 00:09:24,450 --> 00:09:30,953 means that you are seeking-- if you have a scalar field which 145 00:09:30,953 --> 00:09:34,730 have a negative mass squared, that means you're essentially 146 00:09:34,730 --> 00:09:37,245 sitting at the top of the potential-- in the top 147 00:09:37,245 --> 00:09:39,130 of some potential. 148 00:09:39,130 --> 00:09:42,220 And so that's why you get the negative mass squared. 149 00:09:42,220 --> 00:09:43,852 And so that means you have-- this 150 00:09:43,852 --> 00:09:45,060 is an unstable configuration. 151 00:09:49,630 --> 00:09:54,190 Because if you, say, give the tachyon a little bit back, 152 00:09:54,190 --> 00:09:56,447 then the tachyon want to roll down, 153 00:09:56,447 --> 00:09:58,405 and then you move away from this configuration. 154 00:10:01,930 --> 00:10:05,600 You move away from this configuration. 155 00:10:05,600 --> 00:10:17,240 So when you go to superstring-- so we also discussed before, 156 00:10:17,240 --> 00:10:20,940 for the bosonic string, the close string factor also 157 00:10:20,940 --> 00:10:24,790 has a tachyon, and then you can get rid of those tachyons 158 00:10:24,790 --> 00:10:27,540 by going to the superstring. 159 00:10:27,540 --> 00:10:35,010 So similarly, going into the superstring, the D-branes 160 00:10:35,010 --> 00:10:42,250 of certain dimensions-- not for all dimensions, 161 00:10:42,250 --> 00:10:46,900 but only for some dimensions-- do not have tachyons. 162 00:10:55,020 --> 00:10:55,850 So they are stable. 163 00:10:59,890 --> 00:11:03,604 So their lowest modes are just massless modes. 164 00:11:03,604 --> 00:11:05,395 They don't have negative mass square modes. 165 00:11:15,246 --> 00:11:21,120 In particular-- so this I will-- this is a long story. 166 00:11:21,120 --> 00:11:24,660 This is a somewhat long story which I will not explain here, 167 00:11:24,660 --> 00:11:27,450 but I will just make the statement. 168 00:11:27,450 --> 00:11:32,630 In particular you can show that for those stable D-branes 169 00:11:32,630 --> 00:11:36,040 in superstring, they always carry a conserved charge. 170 00:11:45,290 --> 00:11:48,080 So I will explain where does this conserved charge come 171 00:11:48,080 --> 00:11:51,730 from, and they always carry some conserved charged. 172 00:11:51,730 --> 00:11:54,920 And which is also a reason why they are stable. 173 00:11:54,920 --> 00:12:05,270 And the second feature is that the worldvolume theory 174 00:12:05,270 --> 00:12:06,020 is supersymmetric. 175 00:12:13,570 --> 00:12:19,860 So in addition to this phi-- so in addition to this 176 00:12:19,860 --> 00:12:34,880 say A alpha phi we are looking at here, A alpha and phi a, 177 00:12:34,880 --> 00:12:37,407 which we already see in the bosonic string, 178 00:12:37,407 --> 00:12:38,740 they are also massless fermions. 179 00:12:46,460 --> 00:12:49,120 So the open string excitations also include massless fermions. 180 00:12:53,190 --> 00:12:55,000 They also include massless fermions. 181 00:12:55,000 --> 00:12:59,180 And in particular, the low energy theories, 182 00:12:59,180 --> 00:13:02,920 all of them together, is the supersymmetric version of this. 183 00:13:02,920 --> 00:13:05,621 It's a super-Yang-Mills theory, supersymmetric version of this. 184 00:13:05,621 --> 00:13:07,537 And it's some kind of super-Yang-Mills theory. 185 00:13:21,620 --> 00:13:27,810 So now let me elaborate a little bit on the first point. 186 00:13:27,810 --> 00:13:31,210 They carry a conserved charged. 187 00:13:31,210 --> 00:13:33,560 Let me elaborate a little bit on the first point. 188 00:13:37,612 --> 00:13:41,800 So maybe go to the one, elaborate more on one. 189 00:13:41,800 --> 00:13:51,890 So let me just remind you again, the bosonic part of the 190 00:13:51,890 --> 00:14:03,170 of massless closed superstring spectrum, 191 00:14:03,170 --> 00:14:09,110 which we briefly mentioned before-- 192 00:14:09,110 --> 00:14:12,700 so this is so-called type II string, so-called type II 193 00:14:12,700 --> 00:14:13,200 string. 194 00:14:13,200 --> 00:14:15,533 When I say the superstring I always mean type II string. 195 00:14:24,120 --> 00:14:26,460 So if you quantize the superstring, 196 00:14:26,460 --> 00:14:31,260 then you find that they are again, you find the metric, 197 00:14:31,260 --> 00:14:33,540 you find the graviton, then you find 198 00:14:33,540 --> 00:14:35,760 that there are this metric tensor, 199 00:14:35,760 --> 00:14:38,320 and you find that there are scalar fields just 200 00:14:38,320 --> 00:14:42,080 as what we did in the bosonic string. 201 00:14:42,080 --> 00:14:46,990 But they have some additional massless modes. 202 00:14:46,990 --> 00:14:49,020 Depending on your quantization or how 203 00:14:49,020 --> 00:14:52,730 you treat the fermions in the superstring, then there can be, 204 00:14:52,730 --> 00:14:57,970 say IIA or IIB, there are two types of superstring. 205 00:15:00,680 --> 00:15:04,810 And for IIA you have one additional 1-form field 206 00:15:04,810 --> 00:15:06,490 and additional 3-form field. 207 00:15:12,520 --> 00:15:16,880 And for IIB, you have additional scalar fields, 208 00:15:16,880 --> 00:15:21,290 and additional 2-form fields, and additional 4-form fields. 209 00:15:28,980 --> 00:15:34,690 In particular, this is so-called self-dual. 210 00:15:34,690 --> 00:15:37,260 I will explain that in a little bit. 211 00:15:37,260 --> 00:15:39,100 In the IIB, contain a self-dual 4-form. 212 00:15:42,490 --> 00:15:47,590 And this additional field are called the Ramond-Ramond field, 213 00:15:47,590 --> 00:15:48,420 typically. 214 00:15:48,420 --> 00:15:51,836 Just a name, it's called the Ramond-Ramond fields. 215 00:15:51,836 --> 00:15:56,048 And so this called the Ramond-Ramond fields. 216 00:15:56,048 --> 00:15:58,800 AUDIENCE: C1 and C3? 217 00:15:58,800 --> 00:16:00,450 HONG LIU: C1 and C3, that's right. 218 00:16:00,450 --> 00:16:02,320 They're all called Ramond-Ramond fields. 219 00:16:02,320 --> 00:16:02,820 Yeah. 220 00:16:06,420 --> 00:16:09,060 So the story is essentially the same, 221 00:16:09,060 --> 00:16:13,290 very similar as to what we did for the bosonic string. 222 00:16:13,290 --> 00:16:17,270 For the bosonic string you only have x on one sheet, 223 00:16:17,270 --> 00:16:19,850 when you quantize x then you find the graviton, 224 00:16:19,850 --> 00:16:23,860 you find those B, you find the phi, and in the superstring 225 00:16:23,860 --> 00:16:26,380 you pull some additional fermions on the world-sheets, 226 00:16:26,380 --> 00:16:28,580 and those fermions give you additional modes, 227 00:16:28,580 --> 00:16:32,920 and then you will have those additional-- they will also 228 00:16:32,920 --> 00:16:37,390 give rise to additional massless modes in the spacetime, 229 00:16:37,390 --> 00:16:39,830 and those are this Ramond-Ramond fields. 230 00:16:43,090 --> 00:16:45,870 They are also massless fermions, which I will not write here. 231 00:16:45,870 --> 00:16:46,650 Yes? 232 00:16:46,650 --> 00:16:50,450 AUDIENCE: So why is it that type IIA and type IIB string theory, 233 00:16:50,450 --> 00:16:53,840 they appear to be very, very different, which seems bad. 234 00:16:53,840 --> 00:16:56,256 It just seems like there's this arbitrary choice about how 235 00:16:56,256 --> 00:16:57,672 you go about doing the mathematics 236 00:16:57,672 --> 00:17:00,370 and you get this spectrum of particles or this spectrum. 237 00:17:00,370 --> 00:17:04,384 So who do you get around this problem? 238 00:17:04,384 --> 00:17:05,800 HONG LIU: This is not the problem. 239 00:17:05,800 --> 00:17:06,829 This is a fact of life. 240 00:17:09,339 --> 00:17:12,369 This is just what you find. 241 00:17:12,369 --> 00:17:13,700 This is just what you find. 242 00:17:13,700 --> 00:17:15,108 AUDIENCE: So which one is right? 243 00:17:23,400 --> 00:17:25,430 HONG LIU: They both can be right, 244 00:17:25,430 --> 00:17:27,300 and they both can be wrong. 245 00:17:27,300 --> 00:17:33,840 And our goal is just to find all possible theories are there. 246 00:17:33,840 --> 00:17:37,020 And this just allow the quantizations. 247 00:17:37,020 --> 00:17:40,830 A lot consists in the quantizations. 248 00:17:40,830 --> 00:17:45,920 And both of them give you consistent quantum gravity. 249 00:17:45,920 --> 00:17:51,890 And if I have time, if this is a string theory class, if this 250 00:17:51,890 --> 00:17:53,840 were a string theory class, then I 251 00:17:53,840 --> 00:17:59,452 would explain that actually secretly they are equivalent, 252 00:17:59,452 --> 00:18:00,660 secretly they are equivalent. 253 00:18:03,980 --> 00:18:06,090 So in some sense, they are not that different. 254 00:18:12,420 --> 00:18:17,170 Anyway, for your purpose, at the perturbative level, 255 00:18:17,170 --> 00:18:20,320 when we can see that the string coupling to be small, 256 00:18:20,320 --> 00:18:24,060 then they appear to be different, 257 00:18:24,060 --> 00:18:28,100 and yeah, we have two types of strings here. 258 00:18:28,100 --> 00:18:32,550 And so let me mention a little bit about those forms. 259 00:18:32,550 --> 00:18:36,050 All of these are anti-symmetric potentials, 260 00:18:36,050 --> 00:18:48,310 so these anti-symmetric potentials 261 00:18:48,310 --> 00:19:05,880 are generalizations of the Maxwell fields, of the Maxwell 262 00:19:05,880 --> 00:19:06,590 field A mu. 263 00:19:09,830 --> 00:19:13,110 So mathematically, you can write-- say for example, 264 00:19:13,110 --> 00:19:16,540 mathematically you can write the gauge field A mu 265 00:19:16,540 --> 00:19:21,925 as the so-called 1-form, so-called 1-form, and then 266 00:19:21,925 --> 00:19:25,660 A mu, it just coefficients of this 1-form. 267 00:19:25,660 --> 00:19:29,120 And then the field strings is just so the derivative, 268 00:19:29,120 --> 00:19:33,480 it's just exterior derivative of this 1-form. 269 00:19:33,480 --> 00:19:36,910 And then the Lagrangian will be just 270 00:19:36,910 --> 00:19:40,730 minus one quarter F mu mu, F mu mu 271 00:19:40,730 --> 00:19:44,590 construct from this procedure. 272 00:19:44,590 --> 00:19:46,380 So you can just general-- mathematically 273 00:19:46,380 --> 00:19:48,210 straightforward to generalize this you 274 00:19:48,210 --> 00:19:49,770 have dimension of forms. 275 00:19:49,770 --> 00:19:54,010 So for example, we can consider n-form, 276 00:19:54,010 --> 00:20:00,730 which is the n-component tensor, but all indices 277 00:20:00,730 --> 00:20:02,565 are fully anti-symmetric with each other. 278 00:20:11,550 --> 00:20:15,194 So all the indices are fully anti-symmetric with each other. 279 00:20:15,194 --> 00:20:16,610 So these are fully anti-symmetric. 280 00:20:23,764 --> 00:20:25,180 If they depend on the conventions, 281 00:20:25,180 --> 00:20:27,346 sometimes we also put the one over n-factorial here. 282 00:20:31,659 --> 00:20:34,075 So you can define such objects, such as the generalization 283 00:20:34,075 --> 00:20:38,960 of A, and the corresponding field strings for this C, 284 00:20:38,960 --> 00:20:46,080 we also called F. So the field strings for C 285 00:20:46,080 --> 00:20:51,330 is just a n plus 1-form, which is exterior derivative of dC. 286 00:20:54,710 --> 00:20:56,530 And then this is n plus 1-form. 287 00:20:56,530 --> 00:21:00,310 Again, this is a fully anti-symmetric tensor 288 00:21:00,310 --> 00:21:03,390 now with n plus 1 indices. 289 00:21:03,390 --> 00:21:07,590 And then you can write down your Lagrangian similarly 290 00:21:07,590 --> 00:21:10,880 by generalizing these, you can write s1 291 00:21:10,880 --> 00:21:16,880 over-- let me write it here-- you can write the Lagrangian s 292 00:21:16,880 --> 00:21:23,470 minus 1 over 1/2 times n factorial-- 2 times 293 00:21:23,470 --> 00:21:29,520 n factorial, not 2n factorial-- 2 times n factorial F mu1, 294 00:21:29,520 --> 00:21:36,090 mu n plus 1, F mu1 mu n plus 1. 295 00:21:36,090 --> 00:21:42,370 So F is n plus 1-form, so it's a tensor of n plus 1 indices, 296 00:21:42,370 --> 00:21:45,830 fully anti-symmetric fully anti-symmetric. 297 00:21:51,720 --> 00:21:56,550 So those fields essentially have this kind of structure, 298 00:21:56,550 --> 00:21:58,620 essentially have this kind of structure. 299 00:21:58,620 --> 00:22:03,120 And say the 1-form is very similar to our Maxwell. 300 00:22:03,120 --> 00:22:07,650 So this is a 3-form and this is a scalar, then just 301 00:22:07,650 --> 00:22:10,550 a massless scalar, then this is a 2-form 302 00:22:10,550 --> 00:22:12,790 whose field string will be a 3-form, 303 00:22:12,790 --> 00:22:14,760 and then this is a 4-form factor. 304 00:22:18,440 --> 00:22:23,550 And then just by definition, just like here, 305 00:22:23,550 --> 00:22:28,160 there's a gauge symmetry because F equal to dA, here, 306 00:22:28,160 --> 00:22:30,720 because this is F equal to dC. 307 00:22:30,720 --> 00:22:38,500 So there's a gauge symmetry because I 308 00:22:38,500 --> 00:22:45,470 can add C n, the total derivative, of any n plus 1, 309 00:22:45,470 --> 00:22:48,270 n minus 1-form. 310 00:22:48,270 --> 00:22:51,560 So I can make a gauge symmetry, so lambda n minus 1 311 00:22:51,560 --> 00:22:52,915 can be any n minus 1-form. 312 00:22:59,710 --> 00:23:07,040 And because of this square equal to zero, now F is invariant. 313 00:23:07,040 --> 00:23:08,476 So F n plus 1 is invariant. 314 00:23:13,780 --> 00:23:18,280 So if you put this into here, and because this is already 315 00:23:18,280 --> 00:23:21,396 a total derivative, and so F is invariant. 316 00:23:30,440 --> 00:23:31,950 So there's a great symmetry, then 317 00:23:31,950 --> 00:23:37,207 there will be a conserved charge associated with those n-forms. 318 00:23:37,207 --> 00:23:39,790 You can define conserved charge associated with those n-forms. 319 00:23:46,811 --> 00:23:47,810 So so far any questions? 320 00:23:50,641 --> 00:23:51,140 Yes. 321 00:23:51,140 --> 00:23:53,770 AUDIENCE: And maybe, so there's a C field that they 322 00:23:53,770 --> 00:23:56,390 denoted fermions of bosons. 323 00:23:56,390 --> 00:23:58,500 HONG LIU: Bosons, it's all bosons. 324 00:23:58,500 --> 00:24:04,670 AUDIENCE: But it looks-- so this anti-symmetric has 325 00:24:04,670 --> 00:24:06,490 nothing to do with fermions? 326 00:24:06,490 --> 00:24:10,080 HONG LIU: No, this is just the indices. 327 00:24:10,080 --> 00:24:14,080 This just indices which are anti-symmetric. 328 00:24:14,080 --> 00:24:17,730 This is not the location, spacetime location. 329 00:24:17,730 --> 00:24:19,970 This is just indices. 330 00:24:19,970 --> 00:24:22,510 It's the same thing, I think, F mu mu. 331 00:24:22,510 --> 00:24:24,580 F mu mu, the indices are anti-symmetric. 332 00:24:24,580 --> 00:24:25,825 It's the same thing here. 333 00:24:29,740 --> 00:24:32,060 Are the people comfortable with this differential form 334 00:24:32,060 --> 00:24:34,930 notation? 335 00:24:34,930 --> 00:24:36,174 Yes. 336 00:24:36,174 --> 00:24:37,098 AUDIENCE: [INAUDIBLE]? 337 00:24:40,959 --> 00:24:42,500 HONG LIU: Bosons, they're all bosons. 338 00:24:47,704 --> 00:24:49,620 AUDIENCE: And where are the partners of those? 339 00:24:49,620 --> 00:24:50,830 HONG LIU: Sorry? 340 00:24:50,830 --> 00:24:56,870 AUDIENCE: Hb and phi, where are their partners? 341 00:24:56,870 --> 00:24:59,030 HONG LIU: So far I did not write them down. 342 00:24:59,030 --> 00:25:00,420 So they are also fermionics. 343 00:25:00,420 --> 00:25:01,753 They are also massless fermions. 344 00:25:09,180 --> 00:25:10,970 We will not need to worry about them. 345 00:25:10,970 --> 00:25:12,825 So they are massless fermions in this. 346 00:25:12,825 --> 00:25:13,741 AUDIENCE: [INAUDIBLE]? 347 00:25:25,294 --> 00:25:25,960 HONG LIU: Right. 348 00:25:25,960 --> 00:25:27,281 AUDIENCE: With superstrings. 349 00:25:27,281 --> 00:25:28,780 HONG LIU: Yeah those just come from, 350 00:25:28,780 --> 00:25:31,130 because when you go to the superstring, 351 00:25:31,130 --> 00:25:33,600 you have additional fermions on the world-sheet. 352 00:25:33,600 --> 00:25:36,720 And then they can give you additional massless modes 353 00:25:36,720 --> 00:25:39,400 essentially come from them. 354 00:25:39,400 --> 00:25:44,250 The story, the real story is a little bit more complicated 355 00:25:44,250 --> 00:25:46,210 because actually in the superstring, 356 00:25:46,210 --> 00:25:50,530 even those come from the fermions, world-sheet fermions. 357 00:25:50,530 --> 00:25:54,070 Yeah but anyway vehicle you have at the formula 358 00:25:54,070 --> 00:25:57,595 for the world-sheet, then you have more possibilities then 359 00:25:57,595 --> 00:25:59,920 that give rise to those modes. 360 00:25:59,920 --> 00:26:01,990 That's the long story. 361 00:26:01,990 --> 00:26:05,830 AUDIENCE: Those modes are also for bosons, not for fermions? 362 00:26:05,830 --> 00:26:12,030 HONG LIU: No, these are all of the spacetime bosonic fields, 363 00:26:12,030 --> 00:26:15,000 which are low energy excitations of the strings. 364 00:26:15,000 --> 00:26:18,690 And how they arise on the world-sheet, 365 00:26:18,690 --> 00:26:21,140 whether they come from fermions or from bosons 366 00:26:21,140 --> 00:26:25,680 on the world-sheet, it's a separate issue. 367 00:26:25,680 --> 00:26:27,930 And, in fact, all of them actually 368 00:26:27,930 --> 00:26:33,941 arise from world-sheet fermions, but they are spacetime bosons. 369 00:26:33,941 --> 00:26:34,440 Yes. 370 00:26:34,440 --> 00:26:40,205 AUDIENCE: [INAUDIBLE] of these Ramond-Ramond fields? 371 00:26:40,205 --> 00:26:47,920 HONG LIU: Yeah, you just workout their repetitions 372 00:26:47,920 --> 00:26:50,585 under their say, Lorentz symmetry. 373 00:26:54,120 --> 00:26:55,820 They're all integer spin-- they're 374 00:26:55,820 --> 00:26:59,680 all integer spin-- they're all integer spin repetitions 375 00:26:59,680 --> 00:27:01,600 of the Lorentz group. 376 00:27:01,600 --> 00:27:04,840 And not like fermions, then will be half integer. 377 00:27:08,730 --> 00:27:14,160 They are generating repetitions of the rotational group. 378 00:27:14,160 --> 00:27:18,460 AUDIENCE: So they are like spin 2, spin-- 379 00:27:18,460 --> 00:27:22,060 HONG LIU: No, normally we don't call them spin 2 or spin three. 380 00:27:22,060 --> 00:27:26,730 When we call spin 2, we mean a fully symmetric indices, 381 00:27:26,730 --> 00:27:27,730 symmetric and traceless. 382 00:27:27,730 --> 00:27:30,063 AUDIENCE: But anti-symmetric, we would never call them-- 383 00:27:30,063 --> 00:27:32,600 HONG LIU: Yeah, we don't call them spin. 384 00:27:32,600 --> 00:27:34,960 Just the repetitions of the Lorentz group, 385 00:27:34,960 --> 00:27:38,000 it's-- it's just some repetitions of it's integer. 386 00:27:41,470 --> 00:27:43,330 Good? 387 00:27:43,330 --> 00:27:52,280 And in the case of the Maxwell, we know that for the 1-form, 388 00:27:52,280 --> 00:27:59,422 for the case of the Maxwell, its source, where for A, its source 389 00:27:59,422 --> 00:28:00,255 is a point particle. 390 00:28:08,000 --> 00:28:14,270 So we can couple the-- so the point particle 391 00:28:14,270 --> 00:28:17,720 interact say with this extender vector 392 00:28:17,720 --> 00:28:19,280 potential in the following way. 393 00:28:23,140 --> 00:28:27,530 So suppose a particle following some trajectory described 394 00:28:27,530 --> 00:28:32,170 by x mu tau, and then this particle, 395 00:28:32,170 --> 00:28:37,420 we are intact with this vector potential in the following way. 396 00:28:37,420 --> 00:28:41,700 This is familiar from E and M. So you just 397 00:28:41,700 --> 00:28:47,370 integrate essentially with A along the trajectory 398 00:28:47,370 --> 00:28:48,770 of the particle. 399 00:28:48,770 --> 00:28:52,830 And you can write this in the more compact mathematical form. 400 00:28:52,830 --> 00:28:56,800 You just say you integrate along the trajectory 401 00:28:56,800 --> 00:29:00,680 and A is a 1-form, and you just integrate this 1-form 402 00:29:00,680 --> 00:29:02,910 along the trajectory. 403 00:29:02,910 --> 00:29:17,154 And so this is so-called pull back of this 1-form A to C. 404 00:29:17,154 --> 00:29:18,695 So this is the mathematical language. 405 00:29:21,250 --> 00:29:26,150 Similarly, we can generalize to these higher dimensional forms. 406 00:29:28,680 --> 00:29:33,650 So 1-form, naturally coupled to a point particle-- 407 00:29:33,650 --> 00:29:38,250 which can be considered as a zero-dimensional object. 408 00:29:38,250 --> 00:29:54,880 So a p plus 1-form, then naturally-- I should say 409 00:29:54,880 --> 00:29:55,700 is fermion. 410 00:29:55,700 --> 00:30:08,360 Then a p-dimensional object that naturally couples-- just 411 00:30:08,360 --> 00:30:17,090 as a generalization of this-- to a p plus 1-form. 412 00:30:22,490 --> 00:30:29,150 As follows, you can just-- the worldvolume 413 00:30:29,150 --> 00:30:33,880 of a p-dimensional object is a surface of p plus 1 dimension. 414 00:30:36,440 --> 00:30:40,520 So the worldvolume of p-dimensional objects-- 415 00:30:40,520 --> 00:30:43,000 p plus 1 dimensions because there's also time. 416 00:30:43,000 --> 00:30:45,810 You also move in the-- it's a p-dimensional object moving 417 00:30:45,810 --> 00:30:48,550 the time, then in the p plus one dimension. 418 00:30:48,550 --> 00:30:52,350 So let me call that worldvolume sigma. 419 00:30:52,350 --> 00:30:55,090 And then the naturalization of this 420 00:30:55,090 --> 00:31:02,690 will be this, the coupling between a p-dimensional object 421 00:31:02,690 --> 00:31:05,240 to a p plus 1-form will be like this. 422 00:31:05,240 --> 00:31:08,990 Just you integrate this p plus 1-form 423 00:31:08,990 --> 00:31:12,920 on the worldvolume of this p-dimensional object. 424 00:31:15,950 --> 00:31:25,400 And again, this is the pull back of C 425 00:31:25,400 --> 00:31:32,790 to sigma, C plus 1 to sigma. 426 00:31:32,790 --> 00:31:37,410 And let me just write it in a more explicit form. 427 00:31:37,410 --> 00:31:40,730 So here I use tau to parametrize the trajectory 428 00:31:40,730 --> 00:31:42,010 along the [INAUDIBLE]. 429 00:31:42,010 --> 00:31:45,440 So here, let me use psi to parametrize 430 00:31:45,440 --> 00:31:49,600 the worldvolume coordinate of this brane. 431 00:31:49,600 --> 00:31:59,910 Then this pull back means mu 1, mu p plus 1, partial x mu 432 00:31:59,910 --> 00:32:08,980 1, partial psi 1, partial x mu p plus 1, partial psi p plus 1. 433 00:32:08,980 --> 00:32:19,995 And x mu psi one, psi zero to psi p 434 00:32:19,995 --> 00:32:21,870 describe the embedding of sigma in spacetime. 435 00:32:33,570 --> 00:32:34,956 Any questions on this? 436 00:32:38,430 --> 00:32:43,140 So it tells you that-- so this just tells you that the 437 00:32:43,140 --> 00:32:46,544 AUDIENCE: There is [INAUDIBLE] p plus 1 back 438 00:32:46,544 --> 00:32:52,236 there could say zero because of p because of notation. 439 00:32:52,236 --> 00:32:53,718 HONG LIU: Let me just do zero. 440 00:32:58,164 --> 00:33:01,660 AUDIENCE: Yes, so here the p-dimensional object 441 00:33:01,660 --> 00:33:04,060 in the simple case is just like the point particle 442 00:33:04,060 --> 00:33:05,340 is a zero-dimensional object. 443 00:33:05,340 --> 00:33:06,612 So that's what the object is. 444 00:33:06,612 --> 00:33:07,570 HONG LIU: That's right. 445 00:33:11,260 --> 00:33:14,390 The zero-dimensional object naturally couple to 1-form 446 00:33:14,390 --> 00:33:15,250 in this way. 447 00:33:15,250 --> 00:33:17,760 And then a p-dimensional object naturally couple to a p 448 00:33:17,760 --> 00:33:19,323 plus 1 dimensional this way. 449 00:33:34,440 --> 00:33:47,070 So let me just make, in the Maxwell case, 450 00:33:47,070 --> 00:33:53,060 the gate symmetry here implies an electric charge is 451 00:33:53,060 --> 00:33:57,290 conserved, OK, the electrical charge is conserved. 452 00:33:57,290 --> 00:34:01,060 So for a p-dimensional object coupled 453 00:34:01,060 --> 00:34:04,980 to a p plus 1 dimensional form, and because 454 00:34:04,980 --> 00:34:09,755 of this gate symmetry and then this charge is also conserved. 455 00:34:15,790 --> 00:34:18,780 In particular, the object with minimal charge 456 00:34:18,780 --> 00:34:22,190 must be a stable object because there's nothing to decay to. 457 00:34:27,070 --> 00:34:35,000 So again, after generalization of this case, so 458 00:34:35,000 --> 00:34:39,540 in the standard story, in the standard electromagnetism, 459 00:34:39,540 --> 00:34:42,101 we can have an electrically-charged particle, 460 00:34:42,101 --> 00:34:45,409 we can also have a magnetically-charged particle. 461 00:34:45,409 --> 00:34:49,540 We can have a magnetically-charged particle. 462 00:34:49,540 --> 00:34:57,270 In particular, you can talk about the magnetic monopole 463 00:34:57,270 --> 00:34:59,497 in quantum mechanics and et cetera, 464 00:34:59,497 --> 00:35:01,080 even though we have not observed them. 465 00:35:04,570 --> 00:35:08,400 So similarly one can generalize that concept 466 00:35:08,400 --> 00:35:10,360 to higher dimensions. 467 00:35:10,360 --> 00:35:13,910 So let me first just remind you of how 468 00:35:13,910 --> 00:35:17,710 we define the magmatic-- the mathematical way of how 469 00:35:17,710 --> 00:35:22,060 we define the magnetically-charged object EM. 470 00:35:22,060 --> 00:35:29,030 So EM-- so because F is a 2-form, 471 00:35:29,030 --> 00:35:34,500 we can dualize F-- so this is Hodge dual. 472 00:35:34,500 --> 00:35:37,015 We can define another form, which 473 00:35:37,015 --> 00:35:43,950 I call F tilde, which is related to original F by Hodge dual. 474 00:35:43,950 --> 00:35:45,560 So this is a 2-form, the Hodge dual 475 00:35:45,560 --> 00:35:49,650 of a 2-form in the four dimension is another 2-form. 476 00:35:49,650 --> 00:35:55,850 And then I can write it as the total derivative 477 00:35:55,850 --> 00:35:59,360 with another 1-form potential. 478 00:35:59,360 --> 00:36:05,590 And then the object-- so you know that the under this dual, 479 00:36:05,590 --> 00:36:10,680 so F 0,1 is mapped to F tilde 2, 3. 480 00:36:10,680 --> 00:36:13,520 So the electric field is mapped to the magnetic field, 481 00:36:13,520 --> 00:36:16,410 and similarly, the magnetic field here 482 00:36:16,410 --> 00:36:20,980 is mapped to the electric field here, so under this Hodge dual, 483 00:36:20,980 --> 00:36:22,790 under this Hodge dual. 484 00:36:22,790 --> 00:36:25,290 I should not just write this. 485 00:36:25,290 --> 00:36:28,760 So after some minus sign we'll not keep track of, 486 00:36:28,760 --> 00:36:31,370 and under this Hodge dual, you essentially 487 00:36:31,370 --> 00:36:36,640 say the electric field now I call magnetic field. 488 00:36:36,640 --> 00:36:48,780 So now object coupled to a tilde, 489 00:36:48,780 --> 00:36:50,270 then it's magnetically charged. 490 00:37:03,140 --> 00:37:08,230 Because in terms of A tilde, such objects 491 00:37:08,230 --> 00:37:11,870 they generate electric fields, and from our original F point 492 00:37:11,870 --> 00:37:14,082 there is a magnetic fields. 493 00:37:14,082 --> 00:37:16,040 So there will be a magnetically-charged object. 494 00:37:18,570 --> 00:37:21,325 So essentially that's how we think about the magnetic moment 495 00:37:21,325 --> 00:37:24,560 pole, EM. 496 00:37:24,560 --> 00:37:29,462 So this can also be generalized to this n-form case. 497 00:37:29,462 --> 00:37:36,200 So one can also dualize an n-form. 498 00:37:43,260 --> 00:37:47,670 Is that we introduce use a lot of C tilde, which 499 00:37:47,670 --> 00:37:50,390 is the dual of dC. 500 00:37:50,390 --> 00:37:56,330 So dC is the field strength for C. So this is n-form. 501 00:37:56,330 --> 00:38:00,580 And the dC is the field strength of C, 502 00:38:00,580 --> 00:38:09,070 and then I do a Hodge dual, and then I define to be dC tilde. 503 00:38:09,070 --> 00:38:10,920 I define to be dC tilde. 504 00:38:10,920 --> 00:38:16,640 And the dC tilde-- so this is n-form. 505 00:38:16,640 --> 00:38:21,140 When you do a field strength become n plus 1-form. 506 00:38:21,140 --> 00:38:30,652 And when you do the Hodge dual, when you take the Hodg-- dual, 507 00:38:30,652 --> 00:38:35,330 the Hodge dual of n-form is D minus n, OK? 508 00:38:35,330 --> 00:38:39,330 So this is D min n minus 2. 509 00:38:42,390 --> 00:38:46,440 So this should be a D minus n minus 2-form 510 00:38:46,440 --> 00:38:48,850 because when you take the D, then that 511 00:38:48,850 --> 00:38:52,160 makes into D minus n minus 1-form, 512 00:38:52,160 --> 00:38:54,920 and this Hodge dual take this n plus 1-form 513 00:38:54,920 --> 00:38:59,230 into D minus n minus 1 form. 514 00:38:59,230 --> 00:39:01,930 So you map your n-form to a D minus n minus 2-form. 515 00:39:05,380 --> 00:39:07,230 So essentially, it's the integration 516 00:39:07,230 --> 00:39:15,150 of the electric-magnetic dual of E and M. 517 00:39:15,150 --> 00:39:34,230 So now you can couple a D minus n minus 3-dimensional object 518 00:39:34,230 --> 00:39:35,265 to C tilde. 519 00:39:42,880 --> 00:39:50,430 And in terms of C, in terms of regional C-- 520 00:39:50,430 --> 00:39:52,560 so this is a magnetic object, OK? 521 00:40:02,470 --> 00:40:07,070 Just the same as we define magnetic object 522 00:40:07,070 --> 00:40:14,910 or magnetic monopole for E and M. So is the clear? 523 00:40:14,910 --> 00:40:15,936 Yes. 524 00:40:15,936 --> 00:40:18,652 AUDIENCE: In order to write F tilde and A tilde, 525 00:40:18,652 --> 00:40:22,700 do we have to assume that there are no electric charges? 526 00:40:22,700 --> 00:40:24,780 HONG LIU: That's a good point. 527 00:40:24,780 --> 00:40:31,010 No, you don't have to-- so you do this, 528 00:40:31,010 --> 00:40:33,040 you have to assume that the equation of motion 529 00:40:33,040 --> 00:40:33,719 is satisfied. 530 00:40:33,719 --> 00:40:36,010 You have to assume the equation of motion is satisfied, 531 00:40:36,010 --> 00:40:37,426 then you can write this procedure. 532 00:40:40,540 --> 00:40:46,550 AUDIENCE: So if dF is equal to J, then that is non-zero. 533 00:40:46,550 --> 00:40:48,047 So if there are-- 534 00:40:48,047 --> 00:40:50,130 HONG LIU: So you have the exact the same situation 535 00:40:50,130 --> 00:40:52,490 with E and M. It's just identical situation 536 00:40:52,490 --> 00:40:55,170 with E and M. It's just whatever you do in E and M 537 00:40:55,170 --> 00:40:57,122 you do it here. 538 00:40:57,122 --> 00:40:58,830 AUDIENCE: So I thought that we could only 539 00:40:58,830 --> 00:41:02,138 write F equals to dA because there 540 00:41:02,138 --> 00:41:04,940 were no magnetic monopole? 541 00:41:04,940 --> 00:41:07,470 HONG LIU: No, so you can do that, 542 00:41:07,470 --> 00:41:10,950 now you have to introduce some similarity, 543 00:41:10,950 --> 00:41:14,520 so that's why the magnetic monopole have a Dirac string. 544 00:41:14,520 --> 00:41:16,729 That's why the magnetic monopole have a Dirac string, 545 00:41:16,729 --> 00:41:18,728 then you have this Dirac quantization condition, 546 00:41:18,728 --> 00:41:19,290 et cetera. 547 00:41:19,290 --> 00:41:22,870 So as you would then, say either in E and M, 548 00:41:22,870 --> 00:41:26,800 or in the quantum mechanics, say in the graduate quantum 549 00:41:26,800 --> 00:41:32,730 mechanics, or E and M. 550 00:41:32,730 --> 00:41:35,670 So are people familiar with the concept of magnetic monopole 551 00:41:35,670 --> 00:41:38,440 and the Dirac-- so-called Dirac string and Dirac quantization 552 00:41:38,440 --> 00:41:40,750 condition? 553 00:41:40,750 --> 00:41:44,140 Yeah, I saw some nodding heads. 554 00:41:44,140 --> 00:41:47,730 So you're not familiar with it, then look at, 555 00:41:47,730 --> 00:41:52,380 for example, in one chapter of Jackson 556 00:41:52,380 --> 00:41:57,240 have a very detailed discussion of magnetic monopoles. 557 00:41:57,240 --> 00:41:59,890 But the best, actually, was Dirac's original paper. 558 00:41:59,890 --> 00:42:03,300 It was really beautiful, very, very beautiful. 559 00:42:03,300 --> 00:42:07,090 But Jackson has a rather pedagogical discussion. 560 00:42:07,090 --> 00:42:10,210 Jackson's in actual dynamics, that's a rather-- 561 00:42:10,210 --> 00:42:12,584 AUDIENCE: Does he discuss the differential forms, though, 562 00:42:12,584 --> 00:42:13,175 or no? 563 00:42:13,175 --> 00:42:14,300 HONG LIU: I don't remember. 564 00:42:14,300 --> 00:42:15,060 Oh, you mean Jackson? 565 00:42:15,060 --> 00:42:15,790 AUDIENCE: Yeah. 566 00:42:15,790 --> 00:42:16,915 HONG LIU: I don't remember. 567 00:42:20,150 --> 00:42:24,660 Yeah, it's the fact that you can allow the magnetic monopole 568 00:42:24,660 --> 00:42:28,620 then if you allow the certain singularities in the vector 569 00:42:28,620 --> 00:42:29,967 potential. 570 00:42:29,967 --> 00:42:31,550 And that's the so-called Dirac string. 571 00:42:34,550 --> 00:42:36,170 So you don't have a thing to do. 572 00:42:36,170 --> 00:42:40,680 So the idea is that you-- what you were saying is right, 573 00:42:40,680 --> 00:42:42,400 the fact that you could write F as dA 574 00:42:42,400 --> 00:42:44,540 means there's no magmatic source, 575 00:42:44,540 --> 00:42:46,470 but you can introduce a magnet source 576 00:42:46,470 --> 00:42:48,510 if you allow certain singularities, 577 00:42:48,510 --> 00:42:52,235 and that's what Dirac realized. 578 00:42:55,810 --> 00:42:56,980 Good. 579 00:42:56,980 --> 00:43:03,630 So now, this other gauge fields, in type IIA and type 580 00:43:03,630 --> 00:43:08,370 IIB string, and they can give rise 581 00:43:08,370 --> 00:43:11,600 to extended objects which couple to them. 582 00:43:11,600 --> 00:43:15,010 And it turns out the extended object 583 00:43:15,010 --> 00:43:18,080 cover to those Ramond-Ramond fields, are precisely D-branes. 584 00:43:22,580 --> 00:43:26,160 And they are precisely D-branes, and because they 585 00:43:26,160 --> 00:43:31,090 couple to those gauge fields, and they are stable objects. 586 00:43:31,090 --> 00:43:35,000 At least the minimally-charged ones are stable objects. 587 00:43:35,000 --> 00:43:37,440 So this is related to the statement that actually 588 00:43:37,440 --> 00:43:40,660 in superstring you actually find some stable branes. 589 00:43:40,660 --> 00:43:42,620 And the other dimensions-- so let 590 00:43:42,620 --> 00:43:56,030 me just list them just to-- So with this preparation, 591 00:43:56,030 --> 00:44:05,380 then in IIA, so we can have the following electric and fully 592 00:44:05,380 --> 00:44:08,480 magnetic object. 593 00:44:08,480 --> 00:44:11,442 See in the IIA we have a C mu1. 594 00:44:11,442 --> 00:44:12,150 We have a 1-form. 595 00:44:14,740 --> 00:44:20,490 So that means there's a natural object coupled to it-- 596 00:44:20,490 --> 00:44:21,990 electric object coupled to it. 597 00:44:21,990 --> 00:44:23,198 Turns out this is a D0-brane. 598 00:44:25,520 --> 00:44:30,130 A zero-dimension object coupled to a 1-form, this is D0-brane. 599 00:44:30,130 --> 00:44:34,320 And as a 1-form in 10 dimension-- 600 00:44:34,320 --> 00:44:40,080 so in superstring we have 10 dimension-- is dual to-- 601 00:44:40,080 --> 00:44:53,430 so D is 10 minus 1 is 9, minus 3 is 6, 602 00:44:53,430 --> 00:44:55,610 and a then the magnetic object will be a D6-brane. 603 00:44:58,240 --> 00:45:04,300 So similarly we have a 3-form, and then this 604 00:45:04,300 --> 00:45:07,660 gave an electric D2-brane, and a magmatic D4-brane. 605 00:45:11,270 --> 00:45:14,520 So if you just follow this rule. 606 00:45:14,520 --> 00:45:25,930 So for the type IIB, so let me now first start with this C2. 607 00:45:25,930 --> 00:45:29,740 So this gives you a D-string one-dimensional object 608 00:45:29,740 --> 00:45:32,740 coupled to it for the D-string. 609 00:45:32,740 --> 00:45:35,292 And then the mathematical object will 610 00:45:35,292 --> 00:45:36,500 be a five-dimensional object. 611 00:45:36,500 --> 00:45:38,460 It's called a D5-brane. 612 00:45:38,460 --> 00:45:41,410 So this string is a D1-brane. 613 00:45:41,410 --> 00:45:44,010 So now, let's look at this 4-form. 614 00:45:48,740 --> 00:45:52,270 So the so-called self-dual-- so sometimes we put a plus here 615 00:45:52,270 --> 00:45:54,100 to indicate this is a self-dual. 616 00:45:54,100 --> 00:45:57,860 So let me now explain what this self-dual means. 617 00:45:57,860 --> 00:46:07,090 So self-dual means that from C4, as we construct an F5-- 618 00:46:07,090 --> 00:46:13,640 and construct an F5 from this C4-- 619 00:46:13,640 --> 00:46:16,920 to be a self-dual form means that this forms satisfies 620 00:46:16,920 --> 00:46:21,970 the condition that F5 equal to F5 dual. 621 00:46:21,970 --> 00:46:24,260 So this is a self-dual. 622 00:46:24,260 --> 00:46:28,379 Just means this is a constraint that this C has to satisfy. 623 00:46:28,379 --> 00:46:29,920 So this is called a self-dual 4-form. 624 00:46:36,950 --> 00:46:39,230 If you look at repetitions of the Lorentz symmetry, 625 00:46:39,230 --> 00:46:42,330 you'll find that this is actually a repetition. 626 00:46:42,330 --> 00:46:46,780 And so for this C, you have to satisfy 627 00:46:46,780 --> 00:46:49,590 this self-duality condition. 628 00:46:49,590 --> 00:46:54,010 Then now, for object which coupled to the C, 629 00:46:54,010 --> 00:46:57,450 would be a naturally D3-brane because 630 00:46:57,450 --> 00:47:01,130 the three-dimensional object coupled to a 4-form, 631 00:47:01,130 --> 00:47:03,900 so it would be a D3-brane. 632 00:47:03,900 --> 00:47:09,330 And this D3-brane, because of this self-duality condition, 633 00:47:09,330 --> 00:47:17,410 we have to be both electrically and magnetically-charged 634 00:47:17,410 --> 00:47:18,990 because this is a self-dual form. 635 00:47:26,540 --> 00:47:29,060 So this must source both electric flux 636 00:47:29,060 --> 00:47:32,600 and the magnetic flux to be consistent with self-duality. 637 00:47:39,600 --> 00:47:42,730 And then finally, you can also have 638 00:47:42,730 --> 00:47:47,080 a scalar field in the type IIB. 639 00:47:47,080 --> 00:47:49,330 And the scalar field, now eventually you 640 00:47:49,330 --> 00:47:53,620 will couple to so-called the D minus 1 dimensional brane. 641 00:47:53,620 --> 00:47:56,550 Because scalar field, by itself, is zero dimension. 642 00:47:56,550 --> 00:47:59,610 It's already zero dimensional form. 643 00:47:59,610 --> 00:48:01,110 So following this convention we will 644 00:48:01,110 --> 00:48:04,350 couple to a D minus 1 dimensional brane. 645 00:48:04,350 --> 00:48:08,410 And that obviously not make much sense, 646 00:48:08,410 --> 00:48:11,750 but actually can make sense when you go to Euclidean signature, 647 00:48:11,750 --> 00:48:15,920 and then it turns out this scalar fields 648 00:48:15,920 --> 00:48:18,210 couple to something called the instanton, 649 00:48:18,210 --> 00:48:19,700 which I will not go into there. 650 00:48:19,700 --> 00:48:20,908 Something called D-instanton. 651 00:48:23,580 --> 00:48:28,710 So the bottom line is that the object, 652 00:48:28,710 --> 00:48:30,880 they are charged under [INAUDIBLE] fields, 653 00:48:30,880 --> 00:48:32,325 they are all D-branes. 654 00:48:32,325 --> 00:48:38,480 They are all D-branes, and they come into those dimensions. 655 00:48:38,480 --> 00:48:41,090 And any D-brane-- of course you can define 656 00:48:41,090 --> 00:48:43,220 D-branes with other dimensions. 657 00:48:43,220 --> 00:48:47,290 So you can consider in IIA a D3-brane. 658 00:48:47,290 --> 00:48:50,510 But in IIA, a D3-brane will not be a stable object 659 00:48:50,510 --> 00:48:53,520 because there's not conserved charge for you to couple to. 660 00:48:53,520 --> 00:48:57,080 So even though in IIA you can consider a D3-brane, 661 00:48:57,080 --> 00:49:00,120 but it's not a stable object, and actually, 662 00:49:00,120 --> 00:49:04,286 when you quantized the spectrum on the D3-brane in IIA, then 663 00:49:04,286 --> 00:49:05,410 you find there's a tachyon. 664 00:49:05,410 --> 00:49:08,040 Again there's a tachyon in the worldvolume that indicates 665 00:49:08,040 --> 00:49:10,320 that's not a stable object. 666 00:49:10,320 --> 00:49:14,280 But those branes don't have a tachyon. 667 00:49:16,950 --> 00:49:18,500 And in particular, on those branes, 668 00:49:18,500 --> 00:49:23,100 these are all supersymmetric field series. 669 00:49:23,100 --> 00:49:25,680 For example, the most important one 670 00:49:25,680 --> 00:49:28,350 is this D3-brane because then you 671 00:49:28,350 --> 00:49:30,217 have four dimensions of worldvolume, 672 00:49:30,217 --> 00:49:32,050 then you actually have four-dimension theory 673 00:49:32,050 --> 00:49:36,160 on this D3-brane, and that's what 674 00:49:36,160 --> 00:49:39,565 gives you so-called super-Yang-Mills theory. 675 00:49:39,565 --> 00:49:45,980 So on these branes, on these stable branes, 676 00:49:45,980 --> 00:49:56,490 all of the theories-- super-Yang-Mills theory, 677 00:49:56,490 --> 00:50:01,270 in particular for D3, it's given by a so-called n 678 00:50:01,270 --> 00:50:06,230 equal to four super-Yang-Mills theory in four dimensions. 679 00:50:14,310 --> 00:50:17,144 AUDIENCE: So what you're saying is for any sign table, does 680 00:50:17,144 --> 00:50:19,004 it mean dissolving-- 681 00:50:19,004 --> 00:50:21,090 HONG LIU: Yeah, it can decay. 682 00:50:21,090 --> 00:50:21,800 It can decay. 683 00:50:21,800 --> 00:50:23,591 AUDIENCE: But you lower dimension of brane? 684 00:50:26,286 --> 00:50:27,910 HONG LIU: It's possible for it to decay 685 00:50:27,910 --> 00:50:30,140 into lower dimensional brane, and it can also 686 00:50:30,140 --> 00:50:32,850 for it just to decay into closed string modes, 687 00:50:32,850 --> 00:50:36,296 radiated closed string modes. 688 00:50:36,296 --> 00:50:38,848 AUDIENCE: But, I mean the dimensions of these two objects 689 00:50:38,848 --> 00:50:40,060 are different. 690 00:50:40,060 --> 00:50:42,850 It sounds like if branes decay to a string, 691 00:50:42,850 --> 00:50:44,250 it will become infinite string. 692 00:50:44,250 --> 00:50:46,625 HONG LIU: Yeah, that's right, ud type density of strings. 693 00:50:53,630 --> 00:50:55,770 It's actually a beautiful subject 694 00:50:55,770 --> 00:50:57,870 to discuss those branes, how they decay. 695 00:51:04,060 --> 00:51:08,550 Yeah, it's a nice subject, but it's way out of our discussion 696 00:51:08,550 --> 00:51:09,050 here. 697 00:51:12,570 --> 00:51:13,750 Any questions? 698 00:51:13,750 --> 00:51:14,380 Yes. 699 00:51:14,380 --> 00:51:17,820 AUDIENCE: As to four, if we consider stacks of such branes, 700 00:51:17,820 --> 00:51:19,572 do we get higher-ranked age groups? 701 00:51:19,572 --> 00:51:20,780 HONG LIU: Yeah, that's right. 702 00:51:20,780 --> 00:51:21,460 Exactly. 703 00:51:21,460 --> 00:51:23,070 It always give you n. 704 00:51:27,570 --> 00:51:37,220 Good, so that concludes our discussion of the D-branes 705 00:51:37,220 --> 00:51:40,750 from the point of view as the rich boundary conditions. 706 00:51:48,220 --> 00:51:52,284 So now, I want to take a slightly different perspective 707 00:51:52,284 --> 00:51:52,950 on the D-branes. 708 00:51:56,060 --> 00:52:02,030 From this perspective, from the perspective 709 00:52:02,030 --> 00:52:06,180 the object, which are charged under those 710 00:52:06,180 --> 00:52:09,510 generalized gauge fields. 711 00:52:09,510 --> 00:52:12,000 And now I want to view those D-branes. 712 00:52:12,000 --> 00:52:14,420 I have some objects, some solitons, 713 00:52:14,420 --> 00:52:16,290 if you want some solitons, which are 714 00:52:16,290 --> 00:52:18,940 charged under those generalized gauge fields. 715 00:52:21,740 --> 00:52:23,365 I want elaborate from this perspective. 716 00:52:41,600 --> 00:52:45,620 I talk about the different perspective. 717 00:53:15,760 --> 00:53:20,250 So D-brane has math, has tension. 718 00:53:20,250 --> 00:53:22,510 Also D-brane carry-- so now we only 719 00:53:22,510 --> 00:53:24,440 consider those stable D-branes. 720 00:53:24,440 --> 00:53:26,830 So they have a mass, they also carry those charges, 721 00:53:26,830 --> 00:53:28,320 conserved charges. 722 00:53:28,320 --> 00:53:30,610 So that means there will be flux coming out-- 723 00:53:30,610 --> 00:53:33,380 electrical or magnetic flux coming out of those branes. 724 00:53:36,440 --> 00:53:41,470 So that will deform the spacetime 725 00:53:41,470 --> 00:53:43,940 because the gravity will deform spacetime. 726 00:53:48,220 --> 00:53:49,670 And so now, we'd like to find out 727 00:53:49,670 --> 00:53:52,860 what are the spacetime around those D-branes. 728 00:53:52,860 --> 00:53:55,640 How do they deform spacetime? 729 00:53:55,640 --> 00:54:00,060 So for example, say it consider charge 730 00:54:00,060 --> 00:54:02,190 as a total example of this. 731 00:54:02,190 --> 00:54:05,520 Say, let's consider just again in the Maxwell theory, 732 00:54:05,520 --> 00:54:08,540 in Einstein-Maxwell theory, consider, for example, 733 00:54:08,540 --> 00:54:25,130 a charged particle sitting at the origin 734 00:54:25,130 --> 00:54:33,395 of a four-dimensional spacetime, Minkowski spacetime, 735 00:54:33,395 --> 00:54:34,280 for example. 736 00:54:39,180 --> 00:54:43,110 So if you include both GR and E and M, 737 00:54:43,110 --> 00:54:45,080 then the dynamics of this series controlled 738 00:54:45,080 --> 00:54:47,650 by so-called Einstein-Maxwell-- so you 739 00:54:47,650 --> 00:54:54,900 will have a Einstein theory and the Maxwell. 740 00:54:58,470 --> 00:55:00,560 So the dynamics of such particles 741 00:55:00,560 --> 00:55:02,000 should be captured by them. 742 00:55:02,000 --> 00:55:05,101 And so this captures the gravity due to the particle. 743 00:55:05,101 --> 00:55:06,850 This captures E and M due to the particle. 744 00:55:11,960 --> 00:55:17,560 So from here, you can work out how does this particle deform 745 00:55:17,560 --> 00:55:22,170 spacetime from the Einstein equation? 746 00:55:22,170 --> 00:55:24,810 So the equation motion of this system 747 00:55:24,810 --> 00:55:31,950 would be, you have a standard Maxwell-- 748 00:55:31,950 --> 00:55:37,490 so I will be careful about the minus sign-- and also 749 00:55:37,490 --> 00:55:46,587 the Einstein equation, the stress tensor. 750 00:55:53,330 --> 00:56:03,680 So j is just the-- if I have a charged particle sitting there, 751 00:56:03,680 --> 00:56:06,010 there's no current, there's only a charged density. 752 00:56:09,600 --> 00:56:15,920 And the stress tensor contains two parts. 753 00:56:15,920 --> 00:56:20,610 It's the stress tensor due to the particle 754 00:56:20,610 --> 00:56:25,000 and also the stress due to the electromagnetic fields. 755 00:56:25,000 --> 00:56:29,490 Because this excited the electrogmagnetic fields. 756 00:56:29,490 --> 00:56:36,670 And the particle part only have a zero component, 757 00:56:36,670 --> 00:56:38,580 so you only have a mass. 758 00:56:38,580 --> 00:56:40,310 There's no momentum. 759 00:56:40,310 --> 00:56:46,850 So this is just a-- maybe I should write r. 760 00:56:46,850 --> 00:56:48,790 I think I'm using the notation of r. 761 00:56:51,580 --> 00:56:55,550 So the particle part of the stress tensor 762 00:56:55,550 --> 00:56:57,850 would be just the data function. 763 00:56:57,850 --> 00:57:02,000 The only non-zero component are zero-zero component. 764 00:57:02,000 --> 00:57:04,450 You only have energy density and just given 765 00:57:04,450 --> 00:57:07,070 by the data function of the mass. 766 00:57:09,660 --> 00:57:13,610 So from here, by solving those equations 767 00:57:13,610 --> 00:57:17,170 we see those forces then you can work out 768 00:57:17,170 --> 00:57:19,800 how a charged particle deforms your spacetime. 769 00:57:47,980 --> 00:57:51,879 So let me just say I'm a bit further, 770 00:57:51,879 --> 00:57:54,170 for example you can easily solve the Einstein equation, 771 00:57:54,170 --> 00:57:57,010 we all know the solutions, so you just 772 00:57:57,010 --> 00:58:06,800 have a coulomb potential, and in other words 773 00:58:06,800 --> 00:58:10,670 you have electric flux surrounding 774 00:58:10,670 --> 00:58:16,050 the s2 around the point particle, you go to q. 775 00:58:19,630 --> 00:58:24,470 Again, this is the dual of F. You have electric field, when 776 00:58:24,470 --> 00:58:27,820 you dualize that hyperspatial component, then that gives you 777 00:58:27,820 --> 00:58:30,830 the-- so this a Gauss node, OK? 778 00:58:30,830 --> 00:58:32,060 This is a Gauss node. 779 00:58:38,300 --> 00:58:43,730 And you can also work out the metric surrounding 780 00:58:43,730 --> 00:58:45,780 this charged particles. 781 00:58:45,780 --> 00:58:50,446 So because of the spacetime symmetric, spherical symmetric, 782 00:58:50,446 --> 00:58:53,070 then you can actually write down the spherical symmetric ansatz 783 00:58:53,070 --> 00:58:54,610 for the metric around the particle. 784 00:59:10,570 --> 00:59:19,415 So from the Einstein equation, you can find out what's-- you 785 00:59:19,415 --> 00:59:22,610 can work out what is f(r) and h(r). 786 00:59:25,230 --> 00:59:29,860 Some of you may have already done this exercise before. 787 00:59:29,860 --> 00:59:34,920 So if your q is zero, what answer would you get? 788 00:59:34,920 --> 00:59:35,960 That's right. 789 00:59:35,960 --> 00:59:41,930 So if the q is zero, then you just 790 00:59:41,930 --> 00:59:43,840 get the Schwarzschild metric. 791 00:59:43,840 --> 00:59:46,550 And it's the exactly the metric say produced 792 00:59:46,550 --> 00:59:48,170 by a song, et cetera, far away. 793 00:59:51,630 --> 00:59:53,520 So here we have a charge, then what 794 00:59:53,520 --> 00:59:57,120 you get is something called the Reissner-Nordstrom metric. 795 00:59:57,120 --> 01:00:02,770 And when you have a charge having electromagnetic fields. 796 01:00:02,770 --> 01:00:15,540 So you will get so-called Reissner-Nordstrom metric. 797 01:00:19,412 --> 01:00:21,370 AUDIENCE: So that's if q equals 0? 798 01:00:21,370 --> 01:00:22,356 If q equals 0? 799 01:00:22,356 --> 01:00:26,460 HONG LIU: No, q larger equal to zero is the Reissner-Nordstrom. 800 01:00:26,460 --> 01:00:28,300 And when q equal to zero, then you just 801 01:00:28,300 --> 01:00:36,772 get the Schwarzschild That's how you calculate the precession. 802 01:00:36,772 --> 01:00:39,105 That's what you use to calculate the precession of, say, 803 01:00:39,105 --> 01:00:39,730 of the mercury. 804 01:00:43,320 --> 01:00:45,510 Similarly, you can also do this for 805 01:00:45,510 --> 01:00:47,730 magnetically-charged particle. 806 01:00:47,730 --> 01:01:00,170 So the only difference for magnetically-charged particle, 807 01:01:00,170 --> 01:01:03,150 the difference is that this equation now 808 01:01:03,150 --> 01:01:08,150 becomes just directly magnetic flux 809 01:01:08,150 --> 01:01:21,145 equal to G. Say for example, if the magnetic charge is G, 810 01:01:21,145 --> 01:01:24,130 and then you will get this. 811 01:01:24,130 --> 01:01:27,150 Then instead you have this, and then you 812 01:01:27,150 --> 01:01:32,490 have yeah-- We have a magnetic flux. 813 01:01:32,490 --> 01:01:35,545 And again, you could plug into any equation 814 01:01:35,545 --> 01:01:40,030 you solve, you get the same Reissner-Nordstrom metric. 815 01:01:40,030 --> 01:01:50,790 It's also a Dirac quantization telling you 816 01:01:50,790 --> 01:01:59,680 that the qg should be quantized in integers of 2 pi n-- so n 817 01:01:59,680 --> 01:02:01,560 equal one, two, et cetera. 818 01:02:10,410 --> 01:02:12,720 So this is the familiar story in the GR, 819 01:02:12,720 --> 01:02:19,040 so if you want to look at how the gravity and the E and M 820 01:02:19,040 --> 01:02:23,210 surrounding the charged objects, and so you can generalize 821 01:02:23,210 --> 01:02:27,480 this story immediately to those branes, which 822 01:02:27,480 --> 01:02:29,930 are charged-- which have their own mass 823 01:02:29,930 --> 01:02:34,560 and charged under some generalized gauge fields. 824 01:02:34,560 --> 01:02:37,940 So you can generalize that immediately. 825 01:02:37,940 --> 01:02:39,874 So you can generalize procedure immediately 826 01:02:39,874 --> 01:02:41,040 to higher-dimension objects. 827 01:02:46,780 --> 01:02:53,260 So you can just, as a single exercise in GR, 828 01:02:53,260 --> 01:02:56,690 you can just carry out this procedure, 829 01:02:56,690 --> 01:03:01,540 just replace this to some higher-dimensional form, 830 01:03:01,540 --> 01:03:03,970 under the source is some higher-dimensional object, 831 01:03:03,970 --> 01:03:05,469 then you can work out the solutions. 832 01:03:10,070 --> 01:03:14,890 And that was a simple exercise which actually Horowitz 833 01:03:14,890 --> 01:03:17,310 and Strominger-Dietz, in the early '90s, 834 01:03:17,310 --> 01:03:21,600 in 1991, it's a trivial generalization you can do, 835 01:03:21,600 --> 01:03:22,895 why not do it? 836 01:03:22,895 --> 01:03:27,640 But turns out that's a very, very important step 837 01:03:27,640 --> 01:03:34,310 to do for the reason which I will explain. 838 01:03:34,310 --> 01:03:37,010 So for simplicity, let me just talk a little bit 839 01:03:37,010 --> 01:03:39,980 about doing this for this D3-brane, 840 01:03:39,980 --> 01:03:42,620 also for definite entities, you could do it for any of this. 841 01:03:42,620 --> 01:03:47,752 And let me just say it for D3-brane. 842 01:03:47,752 --> 01:03:56,450 So now consider the D3-brane-- so D3-brane is part of the type 843 01:03:56,450 --> 01:04:03,350 IIB theory, so of course, we don't know to really do 844 01:04:03,350 --> 01:04:04,910 this in the full type IIB string, 845 01:04:04,910 --> 01:04:08,280 so we will do the low energy limit of the type IIB sting. 846 01:04:08,280 --> 01:04:11,010 So we will do this for the D3-brane. 847 01:04:11,010 --> 01:04:14,422 In the low energy limit of type IIB stream, 848 01:04:14,422 --> 01:04:16,130 which is so-called type IIB supergravity. 849 01:04:19,937 --> 01:04:22,020 So it doesn't matter that you don't learn anything 850 01:04:22,020 --> 01:04:22,670 about type IIB. 851 01:04:22,670 --> 01:04:25,310 Supergravity, and essentially, that's 852 01:04:25,310 --> 01:04:28,020 just some generalization of this equation. 853 01:04:28,020 --> 01:04:29,980 Type IIB supergravity is some generalization 854 01:04:29,980 --> 01:04:33,090 of this equation. 855 01:04:33,090 --> 01:04:35,550 But nevertheless let me just introduce some notations just 856 01:04:35,550 --> 01:04:47,590 to be-- So type IIB supergravity-- 857 01:04:47,590 --> 01:04:52,180 we will do this in the law of limit of the IIB string. 858 01:04:52,180 --> 01:05:03,235 So this is the law energy effective theory 859 01:05:03,235 --> 01:05:11,960 for massless modes of IIB string. 860 01:05:20,080 --> 01:05:24,880 So it has the form as what we had before. 861 01:05:24,880 --> 01:05:28,230 You have 1 over 6 pi, T pi GN. 862 01:05:28,230 --> 01:05:30,980 But not this is a 10-dimensional Newton constant, 863 01:05:30,980 --> 01:05:36,050 so this is a 10-dimensional theory equal to 10. 864 01:05:36,050 --> 01:05:40,300 And you have the Einstein term, and then you 865 01:05:40,300 --> 01:05:45,610 have forms associated with all of this flux, generalized gauge 866 01:05:45,610 --> 01:05:47,665 fields, et cetera, fermions, et cetera. 867 01:05:50,470 --> 01:05:55,600 And as we said before, the relation with string theory, 868 01:05:55,600 --> 01:06:02,700 the Newton constant should be proportional to gs squared. 869 01:06:02,700 --> 01:06:06,690 So you can actually work out the prefactor precisely. 870 01:06:06,690 --> 01:06:08,900 So let me just write down the prefactor. 871 01:06:08,900 --> 01:06:11,720 So the relation with the string is-- just 872 01:06:11,720 --> 01:06:14,480 on the dimensional ground, so this is the Newton 873 01:06:14,480 --> 01:06:16,680 constant in 10 dimensions, so this 874 01:06:16,680 --> 01:06:21,930 would have dimension eight in terms of the lens 875 01:06:21,930 --> 01:06:23,002 have dimension eight. 876 01:06:23,002 --> 01:06:25,460 So the right-hand side I must be upper prime to the power 4 877 01:06:25,460 --> 01:06:28,989 because this is only lens scale in string theory. 878 01:06:28,989 --> 01:06:30,530 And then the prefactor, it turned out 879 01:06:30,530 --> 01:06:35,680 to be-- so you can work out the prefactor, 880 01:06:35,680 --> 01:06:40,980 so turns out 16 pi GN is equal to 2 pi to the power 7. 881 01:06:46,709 --> 01:06:49,000 Say, when you work out the law energy effective theory, 882 01:06:49,000 --> 01:06:53,020 you can work out this precise factor. 883 01:06:53,020 --> 01:06:55,370 And the regime of the validity of this type IIB 884 01:06:55,370 --> 01:07:08,065 supergravity is that the gs is much, much smaller than 1. 885 01:07:08,065 --> 01:07:11,620 The string [INAUDIBLE] should be much, much smaller than 1. 886 01:07:11,620 --> 01:07:14,589 So that just tells-- this is a statement the string loop 887 01:07:14,589 --> 01:07:16,380 correction should be small because the when 888 01:07:16,380 --> 01:07:19,640 the string become big, the loop correction 889 01:07:19,640 --> 01:07:21,165 become more and more important. 890 01:07:26,170 --> 01:07:27,920 The loop correction would be more and more 891 01:07:27,920 --> 01:07:33,110 important from the point of view of gravity 892 01:07:33,110 --> 01:07:35,800 this quantum gravitational corrections. 893 01:07:35,800 --> 01:07:38,460 And so when G string become big, the quantum gravitational 894 01:07:38,460 --> 01:07:41,586 corrections-- so spacetime will fluctuate more. 895 01:07:41,586 --> 01:07:44,700 And then of course you cannot trust the classical gravity 896 01:07:44,700 --> 01:07:45,790 in that region. 897 01:07:45,790 --> 01:07:47,930 So if you want to trust the classical gravity, 898 01:07:47,930 --> 01:07:51,120 you want to have small coupling, or translate 899 01:07:51,120 --> 01:07:55,620 into this language, the Newton constant has to be small. 900 01:07:55,620 --> 01:07:56,120 OK? 901 01:07:56,120 --> 01:07:57,620 The Newton constant has to be small. 902 01:07:57,620 --> 01:07:58,990 That means the gravity's weak. 903 01:08:01,720 --> 01:08:06,240 But in real life, gravity's weak, so it's OK. 904 01:08:06,240 --> 01:08:13,586 And so this means the quantum correction is small, 905 01:08:13,586 --> 01:08:15,210 so you can trust the classical gravity. 906 01:08:18,520 --> 01:08:21,870 So you can trust classical gravity. 907 01:08:21,870 --> 01:08:26,450 And then we also want the energy-- no energy, 908 01:08:26,450 --> 01:08:28,290 so we want energy squared to be smaller 909 01:08:28,290 --> 01:08:31,580 then one over alpha prime. 910 01:08:31,580 --> 01:08:34,189 Because when energy squared becomes more than one 911 01:08:34,189 --> 01:08:36,446 over alpha prime, then the massless string modes 912 01:08:36,446 --> 01:08:38,279 become important because all of the massless 913 01:08:38,279 --> 01:08:41,729 string modes have mass square of one over alpha prime. 914 01:08:41,729 --> 01:08:43,430 And if we only want to concentrate 915 01:08:43,430 --> 01:08:46,250 on the massless modes, and then you don't want to excite them. 916 01:08:50,140 --> 01:08:53,319 And also you want the curvature of the spacetime-- here 917 01:08:53,319 --> 01:08:56,390 we're considering curve spacetime-- 918 01:08:56,390 --> 01:08:58,689 so you want the curvature to be much smaller than one 919 01:08:58,689 --> 01:08:59,740 over alpha prime. 920 01:09:04,813 --> 01:09:06,229 So this gravity, essentially, when 921 01:09:06,229 --> 01:09:07,854 you consider the mass of this particle, 922 01:09:07,854 --> 01:09:11,270 you consider the particle limit of the spacetime. 923 01:09:11,270 --> 01:09:13,590 So this can be considered-- as you 924 01:09:13,590 --> 01:09:16,300 can see, the spacetime curvature-- 925 01:09:16,300 --> 01:09:19,479 the spacetime curvature radius is much much-- so alpha prime 926 01:09:19,479 --> 01:09:25,363 is essentially it gives us the rough lens scale of a string. 927 01:09:25,363 --> 01:09:27,779 And then when the curvature is much, much smaller than one 928 01:09:27,779 --> 01:09:30,154 over alpha prime, that means that the sight of the string 929 01:09:30,154 --> 01:09:34,979 is tiny compared to the curvature radius, 930 01:09:34,979 --> 01:09:37,210 the typical size of your system. 931 01:09:37,210 --> 01:09:41,090 And that you can roughly treat them as point particles, 932 01:09:41,090 --> 01:09:43,840 and that's what the gravity theory does. 933 01:09:43,840 --> 01:09:46,060 In the supergravity theory it's all point particles. 934 01:09:49,819 --> 01:09:54,735 And so this means we decouple massless modes. 935 01:10:04,180 --> 01:10:05,884 So you can count on massless modes. 936 01:10:09,970 --> 01:10:13,010 So whenever you take a limit, you always 937 01:10:13,010 --> 01:10:14,881 have to take the dimensionless number small. 938 01:10:14,881 --> 01:10:16,380 So the dimensionless number is alpha 939 01:10:16,380 --> 01:10:20,360 prime times curvature, of alpha times energy squared. 940 01:10:20,360 --> 01:10:23,820 But in reality, we often-- when we work with a theory, 941 01:10:23,820 --> 01:10:25,950 we often just keep our curvature, 942 01:10:25,950 --> 01:10:29,299 or our energy scale as what we want. 943 01:10:29,299 --> 01:10:30,840 And then in talking about this limit, 944 01:10:30,840 --> 01:10:34,970 we just say, we take alpha prime goes to zero limit. 945 01:10:34,970 --> 01:10:37,061 We say we take alpha prime goes to zero limit. 946 01:10:37,061 --> 01:10:38,560 So that means a law of energy limit. 947 01:10:41,130 --> 01:10:44,170 Even though, legitimately, this is not the right thing 948 01:10:44,170 --> 01:10:47,880 to say because this is the dimensional prime integral. 949 01:10:47,880 --> 01:10:53,210 And so the regime where supergravity works, 950 01:10:53,210 --> 01:10:56,760 can be formulated, say, as alpha prime goes to zero 951 01:10:56,760 --> 01:10:57,940 and the g string goes to 0. 952 01:11:01,720 --> 01:11:06,470 So in this regime, now you can work with supergravity, 953 01:11:06,470 --> 01:11:14,105 and now you can now generalize this story to the branes. 954 01:11:14,105 --> 01:11:16,137 Now generalize this story for branes. 955 01:11:16,137 --> 01:11:18,720 So now let me say a few things about the D-branes-- D3-branes. 956 01:11:44,070 --> 01:11:53,100 So they D-brane, D3-brane's charged under C 4 957 01:11:53,100 --> 01:11:59,014 plus-- process and we said this C 4 plus 958 01:11:59,014 --> 01:12:00,180 is required to be self-dual. 959 01:12:02,740 --> 01:12:06,050 That means any source of C 4 plus 960 01:12:06,050 --> 01:12:08,630 must carry both electric and magnetic charge. 961 01:12:08,630 --> 01:12:15,520 So that means that for the D3-brane-- say 962 01:12:15,520 --> 01:12:23,060 if we define an electric charge to be-- So now, 963 01:12:23,060 --> 01:12:27,860 so the D3-brane is a three-dimensional object 964 01:12:27,860 --> 01:12:31,060 and the transverse direction is a six-dimensional because this 965 01:12:31,060 --> 01:12:34,240 is 3 plus 1 including time, so the transverse direction 966 01:12:34,240 --> 01:12:36,080 is six-dimensional. 967 01:12:36,080 --> 01:12:39,790 So we can similarly just think about the D3-brane. 968 01:12:39,790 --> 01:12:45,790 So if we ignore the spatial direction along the brane, then 969 01:12:45,790 --> 01:12:48,300 the D3-brane, the transverse direction 970 01:12:48,300 --> 01:12:52,560 then can be just considered a point in R6, OK? 971 01:12:52,560 --> 01:12:55,170 So a way to think about the D3-brane, the surrounding 972 01:12:55,170 --> 01:12:59,420 of the D3-brane, it's an analog of a point in R6, 973 01:12:59,420 --> 01:13:01,360 in the six-dimensional space. 974 01:13:01,360 --> 01:13:03,700 Is this clear? 975 01:13:03,700 --> 01:13:05,790 This is going to be very important. 976 01:13:05,790 --> 01:13:12,400 And then the surrounding this point, the sphere will S5. 977 01:13:12,400 --> 01:13:16,460 Just like in the electric case, just like the E and M case, 978 01:13:16,460 --> 01:13:19,220 this is the S2 surround the point particle, 979 01:13:19,220 --> 01:13:21,220 and this S5 is the sphere, which is 980 01:13:21,220 --> 01:13:23,720 surrounding the whole D3-brane. 981 01:13:23,720 --> 01:13:26,620 And then this D3 can have electric and magnetic flux 982 01:13:26,620 --> 01:13:29,580 through this S5. 983 01:13:29,580 --> 01:13:38,300 And so electric charge would be just the 5-form running 984 01:13:38,300 --> 01:13:38,910 through it. 985 01:13:41,430 --> 01:13:52,500 And the magnetic charge would be just-- right. 986 01:13:52,500 --> 01:13:55,610 Minus the dual of 5-form is 1-form-- 987 01:13:55,610 --> 01:13:58,270 just make the relation between these two. 988 01:14:01,090 --> 01:14:06,070 And now because of self-duality condition, 989 01:14:06,070 --> 01:14:07,980 then they must be the same. 990 01:14:07,980 --> 01:14:10,430 So that means for the D3-brane, the electric and magnetic 991 01:14:10,430 --> 01:14:11,430 charge must be the same. 992 01:14:14,980 --> 01:14:17,870 And also for higher-dimensional object-- 993 01:14:17,870 --> 01:14:20,510 AUDIENCE: Excuse me, why should they be the same? 994 01:14:20,510 --> 01:14:21,090 Why not-- 995 01:14:21,090 --> 01:14:24,930 HONG LIU: It's because of F5, you get F5 star. 996 01:14:24,930 --> 01:14:26,424 It's because of that. 997 01:14:26,424 --> 01:14:28,663 Because of that self-dual condition. 998 01:14:31,402 --> 01:14:33,360 So electric and magnetic charge of the D3-brane 999 01:14:33,360 --> 01:14:34,068 must be the same. 1000 01:14:34,068 --> 01:14:38,120 It must carry both electric and magnetic charge. 1001 01:14:38,120 --> 01:14:42,840 And now, this Dirac quantization condition 1002 01:14:42,840 --> 01:14:49,240 also works for higher-dimensional objects. 1003 01:14:49,240 --> 01:14:52,230 And again, it's a very simple exercise 1004 01:14:52,230 --> 01:14:57,290 to do it-- very simple exercise to generalize Dirac's 1005 01:14:57,290 --> 01:14:59,770 argument to higher dimension. 1006 01:14:59,770 --> 01:15:01,380 Again, it's just related to how you 1007 01:15:01,380 --> 01:15:04,660 treat those higher-dimensional forms, et cetera. 1008 01:15:04,660 --> 01:15:08,780 Again, it's a very simple step, but it's a very important step. 1009 01:15:08,780 --> 01:15:14,470 So that tells mu that's for D3-brane, which is self-dual, 1010 01:15:14,470 --> 01:15:18,170 which carry both electric and magnetic charge, 1011 01:15:18,170 --> 01:15:21,950 then you have to satisfy this quantization condition. 1012 01:15:21,950 --> 01:15:25,960 Then that means the minimal single D3-brane, 1013 01:15:25,960 --> 01:15:31,634 the minimally-charged object, the charge 1014 01:15:31,634 --> 01:15:32,800 must be square root of 2 pi. 1015 01:15:36,680 --> 01:15:45,586 And for n of them, it must D3 equal to q3 equal to 2 pi n. 1016 01:15:59,680 --> 01:16:01,850 So now we have to specify the analog 1017 01:16:01,850 --> 01:16:03,640 for E and M, those conditions. 1018 01:16:06,440 --> 01:16:10,150 Now we also have to specify the mass of the D3-brane. 1019 01:16:10,150 --> 01:16:12,040 So as we discussed before, tension 1020 01:16:12,040 --> 01:16:19,730 over the D3-brane, tension over D-brane 1021 01:16:19,730 --> 01:16:24,260 should be proportional to one over gs. 1022 01:16:24,260 --> 01:16:27,070 That you also did in your P-sets. 1023 01:16:27,070 --> 01:16:30,780 And again, by doing a precise calculation, 1024 01:16:30,780 --> 01:16:33,210 you can work out the precise prefactors. 1025 01:16:33,210 --> 01:16:38,500 It turns out actually to be the following form. 1026 01:16:38,500 --> 01:16:43,791 It's the q3 divided by 16 pi GN. 1027 01:16:47,850 --> 01:16:52,220 And the GN is just gs squared, so if you translate 1028 01:16:52,220 --> 01:16:58,170 into-- so for n D3-brane, that would be n divided 1029 01:16:58,170 --> 01:17:04,690 by 2 pi cubed gs alpha prime squared. 1030 01:17:04,690 --> 01:17:06,780 So this gs and alpha prime squared 1031 01:17:06,780 --> 01:17:08,140 we know on general grounds. 1032 01:17:08,140 --> 01:17:11,150 So gs we'd-- one over gs we know on general ground, 1033 01:17:11,150 --> 01:17:14,170 and alpha prime squared just from dimensional analysis. 1034 01:17:14,170 --> 01:17:17,010 Because this is a three-dimensional object, 1035 01:17:17,010 --> 01:17:18,790 the tension we'll have mass dimension 1036 01:17:18,790 --> 01:17:22,360 4, so that's why you have one over alpha prime squared here. 1037 01:17:22,360 --> 01:17:26,450 And then the other 2 pi cubed from precise calculations. 1038 01:17:26,450 --> 01:17:28,360 And you can also write it in this form 1039 01:17:28,360 --> 01:17:29,636 in terms of Newton constants. 1040 01:17:32,140 --> 01:17:34,120 And the reason I write this form because this 1041 01:17:34,120 --> 01:17:36,594 is a very special case. 1042 01:17:36,594 --> 01:17:39,010 And you see that the tension is actually precisely related 1043 01:17:39,010 --> 01:17:41,780 to the charge and divided by the square root of Newton's 1044 01:17:41,780 --> 01:17:42,890 constant. 1045 01:17:42,890 --> 01:17:47,060 And it turns out that only special objects 1046 01:17:47,060 --> 01:17:48,820 have this kind of property. 1047 01:17:48,820 --> 01:17:51,022 And this is so-called BPS objects. 1048 01:17:51,022 --> 01:17:51,980 I will not go to there. 1049 01:17:51,980 --> 01:17:54,188 It's related to those branes that are supersymmetric. 1050 01:17:57,160 --> 01:18:01,380 Anyway, so these are the mass over the-- essentially, 1051 01:18:01,380 --> 01:18:04,370 this is the mass of the brane, and these 1052 01:18:04,370 --> 01:18:06,846 are the charge of the brane. 1053 01:18:06,846 --> 01:18:10,560 When you specify the charge, than those fields, then 1054 01:18:10,560 --> 01:18:13,290 those flux that you uniquely determined, 1055 01:18:13,290 --> 01:18:17,720 and then the C4 potential uniquely determined, 1056 01:18:17,720 --> 01:18:20,024 up to gauge symmetry are determined. 1057 01:18:20,024 --> 01:18:22,065 And then now you can plug into the generalization 1058 01:18:22,065 --> 01:18:26,360 of the Einstein equation to work out the geometry. 1059 01:18:26,360 --> 01:18:31,440 So again, the symmetry here-- I won't 1060 01:18:31,440 --> 01:18:34,760 have time for the bottom line, so that's OK. 1061 01:18:34,760 --> 01:18:42,670 So the symmetry preserved by the D-brane, 1062 01:18:42,670 --> 01:18:47,080 by D3-brane as we said before, similar to all D-branes 1063 01:18:47,080 --> 01:18:51,020 is you have a Poincare symmetry along the brane direction, 1064 01:18:51,020 --> 01:18:52,697 and then you have rotational symmetry 1065 01:18:52,697 --> 01:18:54,530 in the transverse direction along the brane. 1066 01:18:54,530 --> 01:18:56,370 So the transverse we have six dimensions. 1067 01:18:56,370 --> 01:19:01,071 So there's SO(6) directions you can rotate in the-- 1068 01:19:01,071 --> 01:19:03,070 and then you have Poincare symmetry in the (1,3) 1069 01:19:03,070 --> 01:19:04,320 direction. 1070 01:19:04,320 --> 01:19:08,440 So based on these symmetries you can write down 1071 01:19:08,440 --> 01:19:10,280 what the metric looks like. 1072 01:19:10,280 --> 01:19:12,680 You can write down answers for the metric. 1073 01:19:12,680 --> 01:19:14,350 Just like right here, you can write down 1074 01:19:14,350 --> 01:19:18,100 the answers for the metric of a point particle based 1075 01:19:18,100 --> 01:19:19,310 on symmetry. 1076 01:19:19,310 --> 01:19:25,470 So here, if you can write down the answers for the metric 1077 01:19:25,470 --> 01:19:26,610 based on symmetry. 1078 01:19:26,610 --> 01:19:31,350 So first, you should have a part which 1079 01:19:31,350 --> 01:19:34,700 is preserved Poincare symmetry. 1080 01:19:38,460 --> 01:19:41,460 Means that this part can only be this form. 1081 01:19:41,460 --> 01:19:43,530 Say along the brane direction, can only 1082 01:19:43,530 --> 01:19:47,370 be this form times some other factor which 1083 01:19:47,370 --> 01:19:49,840 depends on r, so r is the transverse radius 1084 01:19:49,840 --> 01:19:51,600 from the brane. 1085 01:19:51,600 --> 01:19:58,530 And then, the other dimension must be spherical symmetric, 1086 01:19:58,530 --> 01:20:00,350 so you can have a lot of factors. 1087 01:20:00,350 --> 01:20:06,180 And then the r squared d omega 5 squared. 1088 01:20:06,180 --> 01:20:08,830 So this is s5 surrounding the brane, 1089 01:20:08,830 --> 01:20:10,690 and then this is the radial direction. 1090 01:20:10,690 --> 01:20:14,970 So this is the only form of the metric. 1091 01:20:14,970 --> 01:20:18,830 And then essentially, again, as in the charged particle case, 1092 01:20:18,830 --> 01:20:21,879 you need to determine these two functions. 1093 01:20:21,879 --> 01:20:23,670 Which you can plug in the Einstein equation 1094 01:20:23,670 --> 01:20:26,130 to determine these two functions. 1095 01:20:26,130 --> 01:20:28,380 So let me just write down the form of those functions, 1096 01:20:28,380 --> 01:20:32,758 then we discuss what they mean next time. 1097 01:20:38,750 --> 01:20:42,300 Some how, this year I'm consistently much, much slower 1098 01:20:42,300 --> 01:20:44,300 than what I did before. 1099 01:20:44,300 --> 01:20:45,800 Maybe I'm explaining physics better. 1100 01:20:50,090 --> 01:20:53,430 Anyway, so now it's just a mechanical exercise. 1101 01:20:53,430 --> 01:20:56,790 You plug in these answers into the Einstein equation, 1102 01:20:56,790 --> 01:21:01,840 then you can find the two equations for f and h, 1103 01:21:01,840 --> 01:21:03,030 then you can solve it. 1104 01:21:03,030 --> 01:21:07,850 In the old days, when Horowitz and Strominger did it-- 1105 01:21:07,850 --> 01:21:09,610 maybe they did not have Mathematica yet, 1106 01:21:09,610 --> 01:21:11,240 but nowadays you can do it. 1107 01:21:11,240 --> 01:21:14,110 Using Mathematica, if you have the right program, 1108 01:21:14,110 --> 01:21:16,340 you can do it in five minutes. 1109 01:21:16,340 --> 01:21:18,870 But in the old days-- they were friends with Wolfram, 1110 01:21:18,870 --> 01:21:21,670 maybe they already have the program at that time. 1111 01:21:21,670 --> 01:21:27,310 Anyway, so it was a [INAUDIBLE] exercise to do at the time. 1112 01:21:27,310 --> 01:21:30,240 So you find actually the answer is very simple. 1113 01:21:30,240 --> 01:21:34,230 Turns out, both f and h, they just-- 1114 01:21:34,230 --> 01:21:36,178 can be written in the following form. 1115 01:21:45,410 --> 01:21:47,435 H is just a harmonic function. 1116 01:21:54,100 --> 01:22:06,850 And R to the power 4 is given by some constant GN, 1117 01:22:06,850 --> 01:22:11,960 Newton's constant times the D3-brane tension times n. 1118 01:22:11,960 --> 01:22:16,840 And it can also be written as 4 pi gs N alpha prime squared. 1119 01:22:20,770 --> 01:22:23,080 So turns out that those functions are very simple. 1120 01:22:23,080 --> 01:22:26,610 They just square root of a harmonic function. 1121 01:22:26,610 --> 01:22:30,410 So this is a harmonic function for the transverse R6. 1122 01:22:30,410 --> 01:22:33,370 So if you solve a coulomb problem, 1123 01:22:33,370 --> 01:22:35,560 and that will be the solution. 1124 01:22:35,560 --> 01:22:38,150 And this is the harmonic-- so this is the generalization of 1 1125 01:22:38,150 --> 01:22:40,070 over R potential. 1126 01:22:40,070 --> 01:22:42,350 So this is just generalization of 1 over R potential. 1127 01:22:42,350 --> 01:22:46,460 And then f and g is very simply related to this by square root. 1128 01:22:46,460 --> 01:22:50,230 And this constant R to the power 4, so this is not curvature, 1129 01:22:50,230 --> 01:22:53,260 this is just some constant. 1130 01:22:53,260 --> 01:22:55,210 I'm using the standard notation. 1131 01:22:55,210 --> 01:22:57,770 And then this can be-- this is just related 1132 01:22:57,770 --> 01:23:00,830 to the mass of the brane. 1133 01:23:00,830 --> 01:23:03,200 Mass of the brane, of course, is proportional to n. 1134 01:23:06,870 --> 01:23:07,370 Yeah. 1135 01:23:07,370 --> 01:23:09,345 AUDIENCE: In this case, is the mass, 1136 01:23:09,345 --> 01:23:11,414 the charge is it massless? 1137 01:23:11,414 --> 01:23:12,080 HONG LIU: Sorry? 1138 01:23:12,080 --> 01:23:14,560 AUDIENCE: The charge is massless? 1139 01:23:14,560 --> 01:23:15,655 HONG LIU: Sorry? 1140 01:23:15,655 --> 01:23:19,280 AUDIENCE: Is the charge considered on the D3-brane 1141 01:23:19,280 --> 01:23:23,210 brane massless or massive? 1142 01:23:23,210 --> 01:23:24,660 HONG LIU: No. 1143 01:23:24,660 --> 01:23:25,920 Charge are just charge. 1144 01:23:25,920 --> 01:23:27,545 What do you mean by charge is massless? 1145 01:23:30,490 --> 01:23:34,210 This is just the charge carried by this D3-brane. 1146 01:23:34,210 --> 01:23:40,989 Anyway, so this R4 is just like GN times the tension, 1147 01:23:40,989 --> 01:23:41,780 N have the tension. 1148 01:23:41,780 --> 01:23:44,070 We have n objects. 1149 01:23:44,070 --> 01:23:46,580 So this is what you're familiar with the Schwarzschild 1150 01:23:46,580 --> 01:23:47,710 black hole. 1151 01:23:47,710 --> 01:23:50,600 Schwarzschild metric just using constant times mass, 1152 01:23:50,600 --> 01:23:52,490 so this is just generalization of that. 1153 01:23:52,490 --> 01:23:55,090 The Newton constant times the tension, and you have n-branes, 1154 01:23:55,090 --> 01:23:56,610 we just multiply n. 1155 01:23:56,610 --> 01:23:59,430 And you write it-- if you write everything from gs, 1156 01:23:59,430 --> 01:24:01,720 this is a proponent to gs squared, 1157 01:24:01,720 --> 01:24:03,490 this is a proponent to one over gs, 1158 01:24:03,490 --> 01:24:06,500 so and that, the thing's proportionate to gs. 1159 01:24:06,500 --> 01:24:07,700 OK. 1160 01:24:07,700 --> 01:24:10,150 OK we'll stop here.