WEBVTT 00:00:00.080 --> 00:00:02.430 The following content is provided under a Creative 00:00:02.430 --> 00:00:03.810 Commons license. 00:00:03.810 --> 00:00:06.050 Your support will help MIT OpenCourseWare 00:00:06.050 --> 00:00:10.150 continue to offer high quality educational resources for free. 00:00:10.150 --> 00:00:12.690 To make a donation or to view additional materials 00:00:12.690 --> 00:00:16.610 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.610 --> 00:00:17.266 at ocw.mit.edu. 00:00:21.590 --> 00:00:27.310 HONG LIU: So last time, we talked about-- we introduced 00:00:27.310 --> 00:00:29.950 the concept of D-branes. 00:00:29.950 --> 00:00:34.720 And then we quantized open strings on the D-branes. 00:00:34.720 --> 00:00:40.880 And we see the massive spectrum on the D-branes 00:00:40.880 --> 00:00:45.520 includes, say, massless gauge field, 00:00:45.520 --> 00:00:50.030 and also some massless scalar fields. 00:00:50.030 --> 00:00:58.030 And then I described that one can interpolate 00:00:58.030 --> 00:00:59.770 the dynamics of the scalar fields 00:00:59.770 --> 00:01:02.500 actually as the motion of the D-branes. 00:01:02.500 --> 00:01:04.860 So in other words, at the beginning, 00:01:04.860 --> 00:01:08.700 even when we quantize the open string, 00:01:08.700 --> 00:01:11.230 we started with a rigid boundary condition, 00:01:11.230 --> 00:01:13.550 so we started with a rigid ring. 00:01:13.550 --> 00:01:19.470 But now, after you quantize it, then you get the fluctuations. 00:01:19.470 --> 00:01:21.291 And because of those fluctuations, 00:01:21.291 --> 00:01:22.540 the D-branes become dynamical. 00:01:25.200 --> 00:01:27.000 Those fluctuations of the D-branes 00:01:27.000 --> 00:01:29.750 make it into a dynamic object in principle 00:01:29.750 --> 00:01:33.140 to make it move, or et cetera. 00:01:33.140 --> 00:01:35.630 So now let's say a little bit regarding 00:01:35.630 --> 00:01:38.120 the math of a D-brane. 00:01:38.120 --> 00:01:42.680 In the gravitational theory, anything gravitates. 00:01:42.680 --> 00:01:44.080 D-brane will have energy. 00:01:44.080 --> 00:01:46.440 It will have a mass, et cetera. 00:01:46.440 --> 00:01:50.210 so let's talk about what should be the mass of a D-brane 00:01:50.210 --> 00:01:51.435 if it's a dynamic object. 00:02:07.760 --> 00:02:12.340 So here, there's a very simple and intuitive answer. 00:02:12.340 --> 00:02:18.450 So on the D-brane, there are many open strings. 00:02:18.450 --> 00:02:20.600 In principle, there are an infinite number 00:02:20.600 --> 00:02:25.970 of open string excitations live on the D-brane. 00:02:25.970 --> 00:02:27.750 And each of them can be considered, say, 00:02:27.750 --> 00:02:31.070 as a space time field living on the D-brane, et cetera. 00:02:31.070 --> 00:02:42.990 So actual definition for the mass of a D-brane 00:02:42.990 --> 00:02:48.010 is that this should be the energy of a D-brane, which 00:02:48.010 --> 00:02:50.930 essentially should be the ground state, the energy of the ground 00:02:50.930 --> 00:02:52.280 state of the D-brane. 00:02:52.280 --> 00:02:54.220 The energy of the ground state of a D-brane 00:02:54.220 --> 00:02:57.150 could be corresponding to the energy of the D-brane none 00:02:57.150 --> 00:03:00.300 of those strings are excited. 00:03:00.300 --> 00:03:08.656 That should correspond to the vacuum energy 00:03:08.656 --> 00:03:17.006 of open strings living on it. 00:03:24.290 --> 00:03:29.440 So this is very intuitive definition 00:03:29.440 --> 00:03:31.520 and obviously makes sense. 00:03:31.520 --> 00:03:35.800 So we can write the mass of a D-brane, DP-brane, 00:03:35.800 --> 00:03:38.890 as the tension, which is the mass per unit 00:03:38.890 --> 00:03:42.490 volume times the total volume. 00:03:42.490 --> 00:03:45.080 And this should be equal to the vacuum 00:03:45.080 --> 00:03:47.590 energy of all the open strings. 00:03:50.890 --> 00:03:55.170 So say each open string excitation corresponding 00:03:55.170 --> 00:03:55.680 to a field. 00:03:55.680 --> 00:03:56.785 You have a tachyon. 00:03:56.785 --> 00:03:58.196 You have gauge field. 00:03:58.196 --> 00:03:59.820 You have massive scalar field, and also 00:03:59.820 --> 00:04:01.420 infinite number of massive fields. 00:04:01.420 --> 00:04:04.460 All those fields, they have vacuum energy. 00:04:04.460 --> 00:04:07.235 So you need to sum all of them together. 00:04:09.910 --> 00:04:12.400 The sum of those vacuum energies would 00:04:12.400 --> 00:04:14.174 be the mass of the D-brane. 00:04:16.959 --> 00:04:20.560 So this can be achieved just by doing the vacuum 00:04:20.560 --> 00:04:23.980 diagram of open strings. 00:04:23.980 --> 00:04:26.660 So we describe in the closed stream case, 00:04:26.660 --> 00:04:29.340 if you want to find the vacuum energy in the closed string, 00:04:29.340 --> 00:04:34.010 you just sum of all possible. 00:04:34.010 --> 00:04:35.910 So this will be just the vacuum diagram. 00:04:51.036 --> 00:04:54.580 In other words, so the difference 00:04:54.580 --> 00:04:57.840 between the open string is that open strings have boundary 00:04:57.840 --> 00:05:02.810 and closed strings are closed. 00:05:02.810 --> 00:05:14.450 So that means the sum of all two dimensional surfaces 00:05:14.450 --> 00:05:32.610 with this one boundaries but no external open strings. 00:05:35.440 --> 00:05:40.180 This is the natural definition of the vacuum diagram, 00:05:40.180 --> 00:05:42.860 as we would do. 00:05:42.860 --> 00:05:44.890 And we will do in the Euclidean path integral. 00:05:44.890 --> 00:05:47.222 So you can do this in the Euclidean path integral. 00:05:53.980 --> 00:05:56.480 You sum over all surfaces. 00:05:56.480 --> 00:06:00.770 In the case when you need the sum of all surfaces, 00:06:00.770 --> 00:06:03.460 in some sense, the only way we know how to define such a sum 00:06:03.460 --> 00:06:05.270 is to do the Euclidean path integral. 00:06:13.930 --> 00:06:21.820 And this sum-- so previously, we talked about the vacuum 00:06:21.820 --> 00:06:24.280 energy of the closed stream, the sum 00:06:24.280 --> 00:06:30.150 of all possible closed surfaces, say of different topology. 00:06:30.150 --> 00:06:33.070 So here, again, you sum of all possible surfaces 00:06:33.070 --> 00:06:35.160 with this one boundaries. 00:06:35.160 --> 00:06:37.766 So the simplest surface with one boundary is a disk. 00:06:41.446 --> 00:06:43.070 The difference with closed string case, 00:06:43.070 --> 00:06:46.960 now you have to sum over surfaces with boundaries. 00:06:46.960 --> 00:06:49.518 So the simplest one would be a disk. 00:06:49.518 --> 00:06:53.025 And the next would be annulus. 00:06:58.100 --> 00:07:01.196 Now you have two boundaries rather than one boundary. 00:07:01.196 --> 00:07:03.445 And exactly you can consider more and more complicated 00:07:03.445 --> 00:07:04.169 diagrams. 00:07:04.169 --> 00:07:06.335 You can consider more and more complicated diagrams. 00:07:09.730 --> 00:07:13.410 And the way to weight those surfaces exactly the same 00:07:13.410 --> 00:07:17.970 as before is that you have this g string 00:07:17.970 --> 00:07:20.065 than to the power minus chi, and the chi 00:07:20.065 --> 00:07:24.860 is the Euler number, just exactly as we described before. 00:07:24.860 --> 00:07:29.740 And Euler number now to apply surfaces with boundaries 00:07:29.740 --> 00:07:33.524 would be-- so previously, Euler number is 2 minus 2h. 00:07:33.524 --> 00:07:39.279 h is number of genuses, or number of holes. 00:07:39.279 --> 00:07:41.695 But now you also need to include the number of boundaries, 00:07:41.695 --> 00:07:44.660 which are called b. 00:07:44.660 --> 00:07:46.190 So when you include the boundaries, 00:07:46.190 --> 00:07:48.890 then that will change your Euler number, 00:07:48.890 --> 00:07:53.150 and also change the weight for each diagram. 00:07:56.230 --> 00:07:58.140 You can also add handles here. 00:07:58.140 --> 00:07:59.760 You can also add handles here. 00:07:59.760 --> 00:08:01.755 You can also add genuses to the disk. 00:08:05.010 --> 00:08:07.030 You can also add h here. 00:08:10.114 --> 00:08:11.780 So according to this counting, then this 00:08:11.780 --> 00:08:13.900 would count as gs minus 1. 00:08:18.290 --> 00:08:21.200 So this one has zero holes and one boundaries. 00:08:21.200 --> 00:08:22.740 So this is 2 minus 1. 00:08:22.740 --> 00:08:23.840 So this is 1. 00:08:23.840 --> 00:08:26.840 So this is gs to the power negative 1. 00:08:26.840 --> 00:08:30.350 And this one has low hole but two boundaries. 00:08:30.350 --> 00:08:31.680 2 minus 2 is 0. 00:08:31.680 --> 00:08:35.990 This is 1 to the power gs to the 0. 00:08:35.990 --> 00:08:38.230 And then you have higher diagrams. 00:08:38.230 --> 00:08:43.409 You have higher surfaces with all positive powers, gs. 00:08:43.409 --> 00:08:44.344 Yes? 00:08:44.344 --> 00:08:46.044 AUDIENCE: What about like a Mobius strip 00:08:46.044 --> 00:08:46.960 that has one boundary? 00:08:46.960 --> 00:08:48.020 HONG LIU: Right. 00:08:48.020 --> 00:08:50.480 So the Mobius strip is a very good question. 00:08:50.480 --> 00:08:53.320 A Mobius strip is unoriented surface. 00:08:53.320 --> 00:08:55.890 So here, we can see the oriented string. 00:08:55.890 --> 00:08:59.750 You can consider unoriented string. 00:08:59.750 --> 00:09:04.120 But most of what we said applies to that case. 00:09:04.120 --> 00:09:06.900 It's just we have to worry a little bit about orientation, 00:09:06.900 --> 00:09:08.520 so we don't go there. 00:09:08.520 --> 00:09:10.330 AUDIENCE: Will that really contribute 00:09:10.330 --> 00:09:12.089 to the vector [INAUDIBLE]? 00:09:12.089 --> 00:09:12.630 HONG LIU: Hm? 00:09:12.630 --> 00:09:16.240 AUDIENCE: Will the Mobius strip contribute to the-- 00:09:16.240 --> 00:09:17.580 HONG LIU: Yeah, yeah, it will. 00:09:17.580 --> 00:09:24.880 In the case when you have unoriented string. 00:09:24.880 --> 00:09:27.750 AUDIENCE: But you make a restriction and say, 00:09:27.750 --> 00:09:31.985 on this D-brane, we have or not have unoriented-- 00:09:31.985 --> 00:09:34.110 HONG LIU: Here, we only consider oriented surfaces. 00:09:34.110 --> 00:09:35.860 We only consider oriented strings. 00:09:35.860 --> 00:09:39.940 We have not talked about unoriented strings. 00:09:39.940 --> 00:09:42.610 AUDIENCE: But no restriction in principle. 00:09:42.610 --> 00:09:43.360 HONG LIU: You can. 00:09:46.060 --> 00:09:48.750 It's actually a technical complication 00:09:48.750 --> 00:09:51.454 I don't want to go into right now. 00:09:51.454 --> 00:09:52.340 Yes? 00:09:52.340 --> 00:09:54.960 AUDIENCE: So we think of vertical axis as time. 00:09:54.960 --> 00:09:57.062 So the disk would be a string kind 00:09:57.062 --> 00:09:58.760 of nucleating and propagating-- 00:09:58.760 --> 00:10:00.720 HONG LIU: No. 00:10:00.720 --> 00:10:02.820 This is a Euclidean and you can think of time 00:10:02.820 --> 00:10:04.214 as whatever you want. 00:10:04.214 --> 00:10:04.880 AUDIENCE: Right. 00:10:04.880 --> 00:10:07.540 But still, so the disk would be like a nucleating open string 00:10:07.540 --> 00:10:10.300 that propagates and then disappears, right? 00:10:10.300 --> 00:10:12.340 HONG LIU: Yeah. 00:10:12.340 --> 00:10:14.530 For example, this open string you can consider. 00:10:19.386 --> 00:10:21.010 Heuristically, you may be able to think 00:10:21.010 --> 00:10:26.025 of some kind of a single string, just rotate, for example. 00:10:26.025 --> 00:10:27.875 AUDIENCE: Like forever? 00:10:27.875 --> 00:10:28.500 HONG LIU: Yeah. 00:10:28.500 --> 00:10:30.180 For example, I just said a single. 00:10:30.180 --> 00:10:33.620 I'm just saying it's hard to interpret as a time now. 00:10:33.620 --> 00:10:37.489 But the time in this direction would be periodic time 00:10:37.489 --> 00:10:39.030 if you think from that point of view. 00:10:42.862 --> 00:10:44.320 But the good thing is that when you 00:10:44.320 --> 00:10:50.700 go to Euclidean, what you call time and the spatial direction 00:10:50.700 --> 00:10:53.330 then becomes obscured. 00:10:53.330 --> 00:10:54.640 It depends on your convenience. 00:10:57.440 --> 00:11:00.989 AUDIENCE: In the center, is that a genus? 00:11:00.989 --> 00:11:01.530 HONG LIU: No. 00:11:01.530 --> 00:11:05.515 The center is completely smooth. 00:11:05.515 --> 00:11:07.339 AUDIENCE: It's not like a torus? 00:11:07.339 --> 00:11:07.880 HONG LIU: No. 00:11:07.880 --> 00:11:09.010 A disk is a disk. 00:11:09.010 --> 00:11:11.120 A disk is not a torus. 00:11:11.120 --> 00:11:13.000 AUDIENCE: But there's a-- 00:11:13.000 --> 00:11:14.740 HONG LIU: You're talking about this one? 00:11:14.740 --> 00:11:15.365 AUDIENCE: Yeah. 00:11:15.365 --> 00:11:16.570 HONG LIU: Oh. 00:11:16.570 --> 00:11:19.630 This one is annulus. 00:11:19.630 --> 00:11:20.620 This guy is annulus. 00:11:20.620 --> 00:11:21.645 This is a flat surface. 00:11:24.255 --> 00:11:26.590 AUDIENCE: So the inside and outside are the-- 00:11:26.590 --> 00:11:27.865 HONG LIU: They are different. 00:11:27.865 --> 00:11:31.330 If you identify this and that, then they become a torus. 00:11:36.154 --> 00:11:37.820 When you identify this one and that one, 00:11:37.820 --> 00:11:39.985 then they become a torus. 00:11:39.985 --> 00:11:41.617 Then you get rid of the boundaries. 00:11:41.617 --> 00:11:43.950 When you identify them, then there's no boundary anymore 00:11:43.950 --> 00:11:45.935 because they become a circle. 00:11:50.070 --> 00:11:51.810 Good? 00:11:51.810 --> 00:12:01.110 So that means if I have weak coupling, that 00:12:01.110 --> 00:12:05.920 means when gs is much smaller than 1, which is the cases we 00:12:05.920 --> 00:12:09.420 can only consider because if you have gs more than 1, then 00:12:09.420 --> 00:12:11.620 you have some infinite number of diagrams. 00:12:11.620 --> 00:12:15.590 And we don't know how to deal with this. 00:12:15.590 --> 00:12:20.840 So weak coupling, when gs more than 1, then the brane tension, 00:12:20.840 --> 00:12:25.360 then the D-brane will be always scaled 00:12:25.360 --> 00:12:28.510 with string coupling at 1 over gs because of that, 00:12:28.510 --> 00:12:32.390 because this term will dominate. 00:12:32.390 --> 00:12:33.880 This term will dominate. 00:12:33.880 --> 00:12:35.630 And then the energy should be 1 over gs. 00:12:39.790 --> 00:12:45.060 This is a very important result. It's a very important result. 00:12:45.060 --> 00:12:48.170 The mass of the D-brane is actually 1 over gs. 00:12:52.010 --> 00:13:03.680 So on dimensional ground, you can just essentially write down 00:13:03.680 --> 00:13:06.180 what's the tension of the D-brane 00:13:06.180 --> 00:13:08.765 because the only dimensional parameter is alpha prime. 00:13:11.640 --> 00:13:13.250 So the dimension of the D-brane should 00:13:13.250 --> 00:13:18.160 be-- so this is mass per unit volume. 00:13:18.160 --> 00:13:20.780 So you have a p dimension of volume, then 00:13:20.780 --> 00:13:22.830 that would be p plus 1. 00:13:22.830 --> 00:13:26.900 So the mass dimension of the tension would be p plus 1. 00:13:26.900 --> 00:13:30.045 So just on dimension ground, I can write gs 00:13:30.045 --> 00:13:37.000 because it's 1 over s, alpha prime 1/2 p plus 1. 00:13:37.000 --> 00:13:39.940 So that gives you the right dimension. 00:13:39.940 --> 00:13:43.210 And then you can have some numerical constant 00:13:43.210 --> 00:13:46.106 which you need to determine. 00:13:46.106 --> 00:13:54.500 You have some numerical constant which 00:13:54.500 --> 00:14:01.871 you can determine string theory by doing that path integral. 00:14:04.760 --> 00:14:06.880 Any questions about this? 00:14:06.880 --> 00:14:07.839 Yes? 00:14:07.839 --> 00:14:10.284 AUDIENCE: Do these open string vacuum diagrams 00:14:10.284 --> 00:14:12.718 have any interpretations like half of a closed string 00:14:12.718 --> 00:14:13.218 diagram? 00:14:13.218 --> 00:14:15.180 So if you put two disks together, you have a sphere. 00:14:15.180 --> 00:14:15.910 HONG LIU: Right. 00:14:15.910 --> 00:14:18.950 We will talk about this in a minute. 00:14:18.950 --> 00:14:19.742 Yes? 00:14:19.742 --> 00:14:22.700 AUDIENCE: With the logical decomposition 00:14:22.700 --> 00:14:26.080 of the powers of g, why is the disk g minus 1. 00:14:26.080 --> 00:14:30.155 HONG LIU: This comes from this formula. 00:14:30.155 --> 00:14:33.760 As we discussed before, the weight of different topology 00:14:33.760 --> 00:14:38.820 is always weighted by some constant to the power of Euler 00:14:38.820 --> 00:14:40.060 number. 00:14:40.060 --> 00:14:42.560 Now, the Euler number, if you have surfaces with boundaries, 00:14:42.560 --> 00:14:46.910 then the Euler number depends on the number of boundaries. 00:14:46.910 --> 00:14:48.990 And then you can just work it out. 00:14:51.830 --> 00:14:55.650 So this is based on simple topology. 00:14:55.650 --> 00:14:58.316 Any other questions? 00:14:58.316 --> 00:14:58.815 Yes? 00:14:58.815 --> 00:14:59.560 AUDIENCE: I'm sorry. 00:14:59.560 --> 00:15:01.809 I'm just a little bit confused about the vacuum energy 00:15:01.809 --> 00:15:05.200 here as the one-- remember when you calculate 00:15:05.200 --> 00:15:06.421 the mass of the string. 00:15:06.421 --> 00:15:08.480 You know, we have a naught term there. 00:15:08.480 --> 00:15:11.234 There is no excitation. 00:15:11.234 --> 00:15:11.900 HONG LIU: Sorry. 00:15:11.900 --> 00:15:12.441 Say it again. 00:15:12.441 --> 00:15:15.125 AUDIENCE: So when we calculate the mass of the open string 00:15:15.125 --> 00:15:18.030 and there is a0 term, which is completely different. 00:15:18.030 --> 00:15:20.680 HONG LIU: That's completely different. 00:15:20.680 --> 00:15:25.270 So that a0, we considered before, it's 00:15:25.270 --> 00:15:27.770 the zero point energy for the oscillation 00:15:27.770 --> 00:15:29.510 modes on the screen. 00:15:32.380 --> 00:15:37.740 So that a0 is that we are considering 00:15:37.740 --> 00:15:45.300 this string, and the zero point energy for the oscillation 00:15:45.300 --> 00:15:47.200 mode on this string. 00:15:47.200 --> 00:15:49.770 But here, we are considering the zero point 00:15:49.770 --> 00:15:50.970 energy not of the string. 00:15:50.970 --> 00:15:54.790 We are considering the zero point energy of the D-brane. 00:15:54.790 --> 00:15:56.820 And the zero point energy of the D-brane 00:15:56.820 --> 00:16:01.760 would be to write down the vacuum energy of all the fields 00:16:01.760 --> 00:16:03.350 living on the D-brane. 00:16:03.350 --> 00:16:05.700 And all the fields living on the D-brane corresponding 00:16:05.700 --> 00:16:09.600 to all the-- now, each string excitation 00:16:09.600 --> 00:16:11.390 becomes a field on the D-brane. 00:16:11.390 --> 00:16:13.540 And so that's corresponding to sum of that. 00:16:13.540 --> 00:16:15.280 And that, then, in turn corresponding 00:16:15.280 --> 00:16:16.795 to sum of these kind of surfaces. 00:16:19.896 --> 00:16:21.380 Any other questions? 00:16:24.350 --> 00:16:26.230 So there's an alternative way to think 00:16:26.230 --> 00:16:30.115 about how to compute the D-brane mass 00:16:30.115 --> 00:16:38.960 or energy as follows, which is actually extremely instructive. 00:16:38.960 --> 00:16:40.985 There is an alternative way of doing this. 00:16:47.430 --> 00:16:50.280 So let's consider just D-brane. 00:16:54.410 --> 00:16:57.060 So consider the interaction between the two D-branes. 00:16:57.060 --> 00:17:00.580 So let's consider two D-branes separated by some distance. 00:17:05.760 --> 00:17:08.838 And then they have a mass. 00:17:08.838 --> 00:17:13.126 Then they will interact gravitationally. 00:17:13.126 --> 00:17:14.750 In particular, in a weak coupling limit 00:17:14.750 --> 00:17:16.490 they're pretty massive. 00:17:16.490 --> 00:17:19.650 They're very massive because it's 1 over the g string. 00:17:19.650 --> 00:17:24.670 So when g string is small, which is the only regime 00:17:24.670 --> 00:17:26.890 we're working with, so the D-brane is very heavy. 00:17:29.590 --> 00:17:31.970 And so you can ask, what is the gravitational attraction 00:17:31.970 --> 00:17:33.830 between the two? 00:17:33.830 --> 00:17:37.170 What is the interaction between the two? 00:17:37.170 --> 00:17:41.960 And we know that at low energies, 00:17:41.960 --> 00:17:44.170 say if the two D-branes are not excited, 00:17:44.170 --> 00:17:48.400 if their distance are very far apart, 00:17:48.400 --> 00:17:51.770 then the leading interaction between them 00:17:51.770 --> 00:17:53.710 just comes from the massless mode 00:17:53.710 --> 00:17:57.540 because only massless mode mediates normal interactions. 00:17:57.540 --> 00:18:00.210 And so interaction between them just 00:18:00.210 --> 00:18:06.630 comes from graviton or this [INAUDIBLE], 00:18:06.630 --> 00:18:11.040 essentially just corresponding to small number of massless 00:18:11.040 --> 00:18:12.540 closed string modes. 00:18:12.540 --> 00:18:14.650 Only those massless modes will contribute 00:18:14.650 --> 00:18:16.410 because the massive mode only contributes short range 00:18:16.410 --> 00:18:16.951 interactions. 00:18:20.140 --> 00:18:25.540 AUDIENCE: Why not the vector mode in open string? 00:18:25.540 --> 00:18:27.660 HONG LIU: No. 00:18:27.660 --> 00:18:30.740 Vector mode of open string only lives on each brane. 00:18:30.740 --> 00:18:34.960 AUDIENCE: But we can have an open string like-- 00:18:34.960 --> 00:18:37.500 HONG LIU: I will talk about that separately. 00:18:37.500 --> 00:18:38.480 Just wait a little bit. 00:18:41.190 --> 00:18:45.154 So if I think purely from the alternative gravity 00:18:45.154 --> 00:18:47.320 point of view, not from string theory point of view, 00:18:47.320 --> 00:18:50.310 I have two massive objects. 00:18:50.310 --> 00:18:53.060 I want to look at the interaction between them. 00:18:53.060 --> 00:18:58.350 And then interaction will be proportional to gn, 00:18:58.350 --> 00:19:00.920 say their mass. 00:19:00.920 --> 00:19:04.312 So if I factor out the volume factor, 00:19:04.312 --> 00:19:08.030 it would be just GN TP squared. 00:19:08.030 --> 00:19:10.694 So this essentially is the gravitational interaction 00:19:10.694 --> 00:19:11.360 between the two. 00:19:14.070 --> 00:19:16.230 And from the string theory point of view, 00:19:16.230 --> 00:19:17.740 such a diagram corresponding to you 00:19:17.740 --> 00:19:19.580 exchange your closed string. 00:19:25.270 --> 00:19:27.380 So this diagram corresponding says, 00:19:27.380 --> 00:19:29.370 suppose you have brane one, brane two. 00:19:32.760 --> 00:19:37.950 So this picture that brane one will emit graviton 00:19:37.950 --> 00:19:39.940 absorbed by the other brane. 00:19:39.940 --> 00:19:45.440 And then that's how we measure, say, the newtons 00:19:45.440 --> 00:19:49.000 force between them. 00:19:49.000 --> 00:19:51.500 And it translates to the string theory picture. 00:19:51.500 --> 00:19:56.150 This corresponding to one D-brane emits a closed string, 00:19:56.150 --> 00:20:00.689 and then absorbed by the other D-brane. 00:20:00.689 --> 00:20:02.230 And when you go to [INAUDIBLE], which 00:20:02.230 --> 00:20:05.310 only massless mode matters, and then becomes this picture. 00:20:08.272 --> 00:20:10.480 So this is the string theory version of that diagram. 00:20:15.900 --> 00:20:17.330 So now essentially, what you need 00:20:17.330 --> 00:20:21.790 to do to calculate this thing in the string theory 00:20:21.790 --> 00:20:26.110 is to calculate this thing in the diagram. 00:20:26.110 --> 00:20:29.270 So form the string theory point of view, 00:20:29.270 --> 00:20:39.205 now what you need to consider is to do path integral 00:20:39.205 --> 00:20:43.220 on the topology of a cylinder with one 00:20:43.220 --> 00:20:46.420 boundary on the brane one and the other boundary 00:20:46.420 --> 00:20:47.040 on brane two. 00:20:50.270 --> 00:20:54.560 So this corresponds to exchange of a closed 00:20:54.560 --> 00:20:56.350 string in this direction. 00:21:03.299 --> 00:21:04.090 AUDIENCE: Question. 00:21:04.090 --> 00:21:06.860 HONG LIU: Yes? 00:21:06.860 --> 00:21:09.240 AUDIENCE: What's the mechanism for the D-brane emitting 00:21:09.240 --> 00:21:10.440 a closed string? 00:21:10.440 --> 00:21:13.960 Or equivalently, on the other picture, 00:21:13.960 --> 00:21:16.500 why can it emit a graviton? 00:21:16.500 --> 00:21:18.830 HONG LIU: It's coupled to graviton. 00:21:18.830 --> 00:21:21.469 AUDIENCE: So how did we introduce the coupling? 00:21:21.469 --> 00:21:22.010 HONG LIU: Hm? 00:21:22.010 --> 00:21:23.130 AUDIENCE: How did we introduce the coupling? 00:21:23.130 --> 00:21:24.740 I mean, we introduced them as boundary conditions 00:21:24.740 --> 00:21:25.620 for open strings. 00:21:25.620 --> 00:21:26.570 HONG LIU: Yeah. 00:21:26.570 --> 00:21:29.445 AUDIENCE: So does that naturally introduce coupling? 00:21:29.445 --> 00:21:30.320 HONG LIU: No, no, no. 00:21:30.320 --> 00:21:32.410 This is what I'm writing here. 00:21:32.410 --> 00:21:36.040 And this diagram, you emit from a closed string corresponding 00:21:36.040 --> 00:21:38.950 to look at the cylinder. 00:21:38.950 --> 00:21:47.390 One boundary of the cylinder on the location with one D-brane, 00:21:47.390 --> 00:21:49.680 and the other boundary of the cylinder on the location 00:21:49.680 --> 00:21:51.140 of the other D-brane. 00:21:51.140 --> 00:21:54.290 And then you just integrate over this surface. 00:21:54.290 --> 00:22:00.235 And then that will give you the graviton exchange. 00:22:00.235 --> 00:22:01.735 That will give you the closed string 00:22:01.735 --> 00:22:03.254 exchange between the two. 00:22:03.254 --> 00:22:05.920 AUDIENCE: So the coupling of the closed string to the brane kind 00:22:05.920 --> 00:22:07.860 of naturally arises? 00:22:07.860 --> 00:22:10.860 HONG LIU: In the boundary condition imposed here. 00:22:10.860 --> 00:22:17.630 So you impose this closed string to initiate it 00:22:17.630 --> 00:22:20.260 from-- so you impose the boundary condition here 00:22:20.260 --> 00:22:25.140 so that this closed string starts from brane one 00:22:25.140 --> 00:22:27.440 and then ends on brane two. 00:22:27.440 --> 00:22:30.200 And then you integrate over all surfaces 00:22:30.200 --> 00:22:31.880 this cylinder topology. 00:22:34.410 --> 00:22:36.690 AUDIENCE: We know the interaction constant 00:22:36.690 --> 00:22:41.620 for closed string is g closed, but is it the same here? 00:22:41.620 --> 00:22:42.390 HONG LIU: No. 00:22:42.390 --> 00:22:45.427 That's what I'm going to talk about. 00:22:45.427 --> 00:22:46.010 Is this clear? 00:22:49.090 --> 00:22:49.590 OK. 00:22:49.590 --> 00:22:52.080 So now I'm going to mention two things. 00:22:52.080 --> 00:22:54.340 First, as I said before, whenever 00:22:54.340 --> 00:22:57.850 we do some calculations, we often do analytic integration 00:22:57.850 --> 00:23:00.420 to the Euclidean signature. 00:23:00.420 --> 00:23:04.240 So now it will be the same. 00:23:04.240 --> 00:23:08.550 When we do this calculation, we have a closed string start 00:23:08.550 --> 00:23:11.610 at location of the brane one, and move forward 00:23:11.610 --> 00:23:14.740 in time to end on the brane two. 00:23:14.740 --> 00:23:17.130 So this is the simplest diagram. 00:23:17.130 --> 00:23:20.660 You can also add some holes here. 00:23:20.660 --> 00:23:22.390 You can also add some holes here. 00:23:22.390 --> 00:23:25.110 And then that corresponds to higher order diagram structure. 00:23:25.110 --> 00:23:27.650 So now don't worry about that. 00:23:27.650 --> 00:23:29.030 Just the simplest diagram. 00:23:29.030 --> 00:23:29.998 Yes? 00:23:29.998 --> 00:23:34.260 AUDIENCE: Was it time dimension inside brane 00:23:34.260 --> 00:23:37.592 to define time dimension as one of the dimensions 00:23:37.592 --> 00:23:39.083 that lived inside brane? 00:23:39.083 --> 00:23:40.080 HONG LIU: Yeah. 00:23:40.080 --> 00:23:46.280 But this is not a space time [INAUDIBLE]. 00:23:46.280 --> 00:23:48.380 This is virtual time. 00:23:48.380 --> 00:23:52.590 This is virtual time associated with this graviton. 00:23:52.590 --> 00:23:55.740 So essentially, I do create the closed string here 00:23:55.740 --> 00:23:58.670 on this brane and then propagate then absorbed 00:23:58.670 --> 00:24:00.370 by the other D-brane. 00:24:03.150 --> 00:24:04.900 AUDIENCE: Can you also think about this as 00:24:04.900 --> 00:24:06.180 like an open string-- 00:24:06.180 --> 00:24:07.440 HONG LIU: Yeah. 00:24:07.440 --> 00:24:09.330 One second. 00:24:09.330 --> 00:24:10.230 I'm going to explain. 00:24:14.130 --> 00:24:18.200 So now there are two remarkable things about this diagram. 00:24:18.200 --> 00:24:20.270 The two remarkable things about this diagram. 00:24:20.270 --> 00:24:24.640 First, this is a cylinder diagram. 00:24:24.640 --> 00:24:27.476 And this is a diagram with two boundaries 00:24:27.476 --> 00:24:31.180 because we have to emit a closed string from here. 00:24:31.180 --> 00:24:32.520 And so you have one boundary. 00:24:32.520 --> 00:24:34.250 You have an initial closed string, 00:24:34.250 --> 00:24:36.260 then you have a final closed string. 00:24:36.260 --> 00:24:41.148 So this is a surface with two boundaries with no holes. 00:24:41.148 --> 00:24:45.190 And if you calculate the chi, so this would be zero. 00:24:45.190 --> 00:24:51.190 Then that means this diagram is gs to power of 0. 00:24:51.190 --> 00:24:53.779 And then from here, we know that then 00:24:53.779 --> 00:24:55.320 this from string theory point of view 00:24:55.320 --> 00:24:57.400 will be gs to the power of 0. 00:24:57.400 --> 00:25:01.050 And so this is another way to see that the TP should 00:25:01.050 --> 00:25:03.096 be 1 over g string. 00:25:08.080 --> 00:25:11.990 Because we said before that the G Newton-- 00:25:11.990 --> 00:25:14.690 so we explained before G Newton would be order of gs squared. 00:25:18.880 --> 00:25:22.470 G Newton is the g string squared. 00:25:22.470 --> 00:25:25.374 So do you remember G Newton is g string squared? 00:25:25.374 --> 00:25:25.874 Good. 00:25:29.700 --> 00:25:34.860 But something remarkable about this diagram 00:25:34.860 --> 00:25:37.870 is that you can also view this diagram. 00:25:37.870 --> 00:25:40.229 So right now, we see it from this direction. 00:25:40.229 --> 00:25:42.270 Now we can also view it from the other direction. 00:25:47.420 --> 00:25:49.035 Viewed from this direction, so now 00:25:49.035 --> 00:25:50.410 try to think about this direction 00:25:50.410 --> 00:25:56.160 as the time and this direction as the sigma. 00:25:56.160 --> 00:25:59.770 So right now, we are seeing this as virtual time 00:25:59.770 --> 00:26:01.860 and this as sigma. 00:26:01.860 --> 00:26:04.990 This is a closed string. 00:26:04.990 --> 00:26:09.860 So now think about this direction as sigma 00:26:09.860 --> 00:26:12.180 and this direction as time. 00:26:12.180 --> 00:26:19.260 And then this is an open string with one end ending 00:26:19.260 --> 00:26:24.160 on brane two and the other end ending on brane one, 00:26:24.160 --> 00:26:27.440 and then going in the loop. 00:26:27.440 --> 00:26:29.600 So this is the one loop open string. 00:26:29.600 --> 00:26:31.350 So even though this is tree-level exchange 00:26:31.350 --> 00:26:33.270 in closed string. 00:26:33.270 --> 00:26:36.910 So here is the tree-level exchange in closed string. 00:26:46.380 --> 00:26:50.130 So here is one loop in open string. 00:26:55.820 --> 00:26:57.875 So this tells you the same process. 00:27:01.640 --> 00:27:03.760 You can really view it from two perspectives. 00:27:08.480 --> 00:27:12.810 From one perspective, it's the standard point of view, 00:27:12.810 --> 00:27:16.280 is that we exchange some closed strings. 00:27:16.280 --> 00:27:19.440 So we exchange some gravitons, some massless particles, 00:27:19.440 --> 00:27:22.240 some particles between these two. 00:27:22.240 --> 00:27:29.220 But there's another way to think about it is we say, 00:27:29.220 --> 00:27:32.514 because we have two D-branes here 00:27:32.514 --> 00:27:34.930 and because D-branes correspond in two places, open string 00:27:34.930 --> 00:27:44.940 can end, then I have open string connect between them. 00:27:44.940 --> 00:27:47.650 And this one loop open string is essentially 00:27:47.650 --> 00:27:51.745 corresponding to the vacuum diagram of those open strings 00:27:51.745 --> 00:27:53.775 connect between them. 00:27:53.775 --> 00:27:55.150 So when you add the vacuum energy 00:27:55.150 --> 00:27:59.480 of all those open strings ending between them, 00:27:59.480 --> 00:28:02.700 then you're effectively calculating the interaction 00:28:02.700 --> 00:28:06.770 between the two D-branes. 00:28:06.770 --> 00:28:09.730 So we have two completely different perspectives 00:28:09.730 --> 00:28:12.970 to look at the same process. 00:28:12.970 --> 00:28:18.527 And this is very, very deep and profound. 00:28:18.527 --> 00:28:19.235 Deep is profound. 00:28:22.510 --> 00:28:27.770 Because that means the process that you 00:28:27.770 --> 00:28:30.340 can think from closed string perspective 00:28:30.340 --> 00:28:34.920 can be fully understood in a different way 00:28:34.920 --> 00:28:37.400 from the open string perspective, 00:28:37.400 --> 00:28:38.940 in a completely equivalent way. 00:28:41.636 --> 00:28:43.635 And this is normally called the channel duality. 00:28:50.050 --> 00:28:55.200 So it's a very simple geometric fact 00:28:55.200 --> 00:28:57.320 about two dimensional surfaces. 00:28:57.320 --> 00:29:00.222 But physical significance is very profound. 00:29:00.222 --> 00:29:01.756 AUDIENCE: I have a question. 00:29:01.756 --> 00:29:02.630 HONG LIU: One second. 00:29:02.630 --> 00:29:04.430 Let me finish. 00:29:04.430 --> 00:29:10.840 And this is precisely the string theory 00:29:10.840 --> 00:29:15.140 origin of the holographic duality or the idea, say, of t 00:29:15.140 --> 00:29:18.860 we are going to see in a couple of lectures. 00:29:18.860 --> 00:29:23.150 Just because of this simple geometric picture. 00:29:28.510 --> 00:29:36.610 This side is gravity, and the D-brane is about gauge theory. 00:29:36.610 --> 00:29:42.140 And then we see gravity to be equivalent to gauge theory. 00:29:42.140 --> 00:29:45.770 That's something we are going to see later. 00:29:45.770 --> 00:29:48.200 Yeah? 00:29:48.200 --> 00:29:52.030 AUDIENCE: So you said the dynamics of closed string 00:29:52.030 --> 00:29:54.880 can be fully understood by open string. 00:29:54.880 --> 00:29:57.290 Is that why you say it's a closed surface 00:29:57.290 --> 00:29:58.870 formed by a closed string? 00:29:58.870 --> 00:30:00.686 How can you interpret? 00:30:00.686 --> 00:30:01.310 HONG LIU: Yeah. 00:30:01.310 --> 00:30:03.967 I'm talking about this particular diagram. 00:30:03.967 --> 00:30:06.050 I'm just saying this gives you a hint of something 00:30:06.050 --> 00:30:07.272 very profound. 00:30:07.272 --> 00:30:08.480 AUDIENCE: And one more thing. 00:30:08.480 --> 00:30:11.879 Why is the gs 0? 00:30:11.879 --> 00:30:12.420 HONG LIU: No. 00:30:12.420 --> 00:30:16.290 This is the surface of two boundaries. 00:30:16.290 --> 00:30:18.539 AUDIENCE: But that is for open string. 00:30:18.539 --> 00:30:19.080 HONG LIU: No. 00:30:19.080 --> 00:30:21.140 Chi is for everything. 00:30:21.140 --> 00:30:22.030 Chi is everything. 00:30:22.030 --> 00:30:23.830 Doesn't matter open or closed string. 00:30:23.830 --> 00:30:26.030 This is the universal formula. 00:30:26.030 --> 00:30:26.572 This is open. 00:30:26.572 --> 00:30:29.071 The open string just means we imposed the boundary condition 00:30:29.071 --> 00:30:29.970 on the open string. 00:30:29.970 --> 00:30:31.600 The topology is the same. 00:30:31.600 --> 00:30:34.260 We understand the topology is the same. 00:30:34.260 --> 00:30:34.955 Yes? 00:30:34.955 --> 00:30:36.470 AUDIENCE: Why do you call it one loop string? 00:30:36.470 --> 00:30:36.830 Where is the loop? 00:30:36.830 --> 00:30:38.910 HONG LIU: Because this is the open string. 00:30:38.910 --> 00:30:39.960 So this is the open. 00:30:39.960 --> 00:30:41.210 Think from this point of view. 00:30:41.210 --> 00:30:44.990 This is the open string on brane one and brane two. 00:30:44.990 --> 00:30:48.620 And then you go around once, go around in circle. 00:30:48.620 --> 00:30:51.905 So this is one loop. 00:30:51.905 --> 00:30:54.722 AUDIENCE: What's the free momentum? 00:30:54.722 --> 00:30:55.680 HONG LIU: What is this? 00:30:58.400 --> 00:31:00.270 What is this? 00:31:00.270 --> 00:31:02.590 This is one loop if you have a particle. 00:31:02.590 --> 00:31:03.670 And so you have a string. 00:31:03.670 --> 00:31:04.920 Then you go around the circle. 00:31:04.920 --> 00:31:05.950 This is one loop. 00:31:09.840 --> 00:31:14.960 And indeed, when you sum over such surfaces, 00:31:14.960 --> 00:31:18.000 you will lead to sum of all possible momentums, et cetera. 00:31:18.000 --> 00:31:19.520 So the field theory momentum is one 00:31:19.520 --> 00:31:21.710 of the modes you have to sum over 00:31:21.710 --> 00:31:25.530 when you do past integral over surface of such topology. 00:31:25.530 --> 00:31:26.080 Yes? 00:31:26.080 --> 00:31:29.384 AUDIENCE: Once you go to strings connecting different branes, 00:31:29.384 --> 00:31:33.380 your quantization conditions change. 00:31:33.380 --> 00:31:34.890 HONG LIU: We will talk about that. 00:31:34.890 --> 00:31:37.230 Quantization condition almost does not change. 00:31:37.230 --> 00:31:39.570 We will talk about that. 00:31:39.570 --> 00:31:41.310 We'll talk about that in a little bit. 00:31:41.310 --> 00:31:42.810 But right now, it's just intuitively 00:31:42.810 --> 00:31:45.520 clear you have open string connect between them 00:31:45.520 --> 00:31:47.526 and they just go around. 00:31:47.526 --> 00:31:49.275 You have open string connect between them. 00:31:49.275 --> 00:31:51.340 You can just go around the circle. 00:31:51.340 --> 00:31:52.765 AUDIENCE: But this correspondence 00:31:52.765 --> 00:31:57.040 doesn't count the tree-level around the open string. 00:31:57.040 --> 00:31:58.005 HONG LIU: Count what? 00:31:58.005 --> 00:32:02.955 AUDIENCE: If we generalize the tree-level open string. 00:32:02.955 --> 00:32:03.830 HONG LIU: No, no, no. 00:32:06.380 --> 00:32:10.160 This doesn't have to account for the tree-level open string. 00:32:10.160 --> 00:32:13.120 AUDIENCE: Tree-level open string contributes to the interaction? 00:32:13.120 --> 00:32:13.661 HONG LIU: No. 00:32:19.260 --> 00:32:21.150 Here, I'm only talking about this diagram 00:32:21.150 --> 00:32:23.410 and just say this hints that there 00:32:23.410 --> 00:32:26.120 are certain closed string processes can be completely 00:32:26.120 --> 00:32:27.386 described by the open string. 00:32:31.820 --> 00:32:37.100 So what want to extrapolate is that open string 00:32:37.100 --> 00:32:39.480 is a more fundamental description 00:32:39.480 --> 00:32:42.110 because the tree-level diagram in closed string 00:32:42.110 --> 00:32:44.110 can be described by the open string. 00:32:44.110 --> 00:32:47.580 And now if you can generalize that maybe everything closed 00:32:47.580 --> 00:32:49.570 string can be described by open string. 00:32:49.570 --> 00:32:51.660 But you don't want to do in the opposite way. 00:32:51.660 --> 00:32:53.302 Open string is open string. 00:32:57.616 --> 00:32:58.115 Good? 00:33:00.810 --> 00:33:03.160 Any other questions? 00:33:03.160 --> 00:33:05.782 AUDIENCE: Once you go higher dimensions, 00:33:05.782 --> 00:33:09.897 when you leave tree-level closed string can be distinct? 00:33:09.897 --> 00:33:10.480 HONG LIU: Yes. 00:33:10.480 --> 00:33:11.771 Things become more complicated. 00:33:14.340 --> 00:33:16.800 Things become more complicated. 00:33:16.800 --> 00:33:19.720 But the similar picture will exist. 00:33:24.160 --> 00:33:27.230 But nobody has made it work. 00:33:27.230 --> 00:33:31.000 Nobody has made it work at full string theory level 00:33:31.000 --> 00:33:33.740 to construct the whole closed string theory 00:33:33.740 --> 00:33:35.110 from the open string theory. 00:33:35.110 --> 00:33:36.780 Nobody has made it work. 00:33:36.780 --> 00:33:41.870 But there are many such kind of indications from the geometry 00:33:41.870 --> 00:33:44.053 of the surface point of view. 00:33:47.290 --> 00:33:54.314 So now let's talk about relaxing the strength of open string 00:33:54.314 --> 00:33:54.855 interactions. 00:34:06.110 --> 00:34:10.199 Actually, before I do that, now is a good place 00:34:10.199 --> 00:34:14.440 to go back to examine what we discussed 00:34:14.440 --> 00:34:18.010 at the end of last lecture. 00:34:18.010 --> 00:34:23.090 Now is a good time to go back to talk about what we did 00:34:23.090 --> 00:34:25.810 at the end of the last lecture. 00:34:25.810 --> 00:34:33.429 So in the last lecture, at the end, 00:34:33.429 --> 00:34:35.540 we described that one can work out 00:34:35.540 --> 00:34:38.469 the low energy effective action of the massless modes 00:34:38.469 --> 00:34:40.320 on the D-branes. 00:34:40.320 --> 00:34:42.469 So the massless modes on the D-branes on the gauge 00:34:42.469 --> 00:34:49.159 fields along the D-brane, so A alpha from 0, 1, to p. 00:34:49.159 --> 00:34:54.754 And then the scalar field and a label all the perpendicular 00:34:54.754 --> 00:34:55.254 directions. 00:35:03.670 --> 00:35:08.020 So you can write down effective action for them. 00:35:08.020 --> 00:35:12.300 I mentioned if you work out things carefully, 00:35:12.300 --> 00:35:15.950 then you find the prefactor is actually 00:35:15.950 --> 00:35:19.250 just the brane tension. 00:35:19.250 --> 00:35:25.590 And if last thing is excited, p plus 1. 00:35:25.590 --> 00:35:28.690 If last thing is excited, then you just 00:35:28.690 --> 00:35:32.390 have the vacuum energy, so you will have a one. 00:35:32.390 --> 00:35:37.240 So this is just the brane mass, the total brane mass. 00:35:37.240 --> 00:35:37.950 This is et. 00:35:41.800 --> 00:35:44.380 So if last thing is excited, then you just 00:35:44.380 --> 00:35:46.180 have the zero point energy, which 00:35:46.180 --> 00:35:50.950 is just tp times the volume. 00:35:50.950 --> 00:35:55.140 But now, if you also have gauge field, then 00:35:55.140 --> 00:35:59.380 based on general argument, you must have the Maxwell. 00:35:59.380 --> 00:36:03.620 And if the scalar field is excited, 00:36:03.620 --> 00:36:13.890 then you also have the action for massless scalar field. 00:36:27.760 --> 00:36:30.760 And then we mentioned that for example, you 00:36:30.760 --> 00:36:33.250 can consider special case. 00:36:33.250 --> 00:36:42.810 Suppose A alpha is not excited but the brane, rather than 00:36:42.810 --> 00:36:46.010 a scalar field that moves in a coherent way 00:36:46.010 --> 00:36:56.670 the same at all points on the D-brane, just phi a, 00:36:56.670 --> 00:37:01.220 is a function of t rather than x. 00:37:01.220 --> 00:37:02.165 So phi, in general. 00:37:04.780 --> 00:37:09.860 Suppose the brane coordinates our x0 and the p. 00:37:09.860 --> 00:37:15.760 So in general, A alpha is the function of x0, xp. 00:37:15.760 --> 00:37:20.340 And phi a is x0 and xp. 00:37:20.340 --> 00:37:22.545 So they describe you can have arbitrary profile 00:37:22.545 --> 00:37:23.415 on the world-volume. 00:37:25.930 --> 00:37:28.910 But suppose, say, let me consider the uniform situation 00:37:28.910 --> 00:37:35.180 which I only consider every point has 00:37:35.180 --> 00:37:38.380 the same behavior for phi. 00:37:38.380 --> 00:37:46.130 And then this s just becomes 1/2, dt just becomes dt. 00:37:48.800 --> 00:37:57.070 And D-brane plus 1/2 and D-brane phi dot squared. 00:38:06.340 --> 00:38:13.840 So this is precisely the motion of just a massive object. 00:38:13.840 --> 00:38:23.120 OK And m is just this guy. 00:38:23.120 --> 00:38:26.040 So if last thing depends on the spatial coordinate, 00:38:26.040 --> 00:38:28.190 then the integration of the spatial coordinate 00:38:28.190 --> 00:38:29.590 becomes the volume. 00:38:29.590 --> 00:38:32.080 Combine the volume with that, becomes the mass, 00:38:32.080 --> 00:38:33.326 and then just becomes that. 00:38:42.340 --> 00:38:43.650 I think this is minus sign. 00:38:43.650 --> 00:38:45.060 This is plus sign. 00:38:51.310 --> 00:38:53.970 So as I mentioned, this is another way 00:38:53.970 --> 00:38:56.940 to see that the D-brane becomes dynamical, 00:38:56.940 --> 00:38:58.690 and that in particular, this phi describes 00:38:58.690 --> 00:38:59.773 the motion of the D-brane. 00:39:05.370 --> 00:39:11.170 So in fact, this result can be much, much strengthened. 00:39:11.170 --> 00:39:15.140 But I will only quote the result. 00:39:15.140 --> 00:39:18.440 I will only quote the result. 00:39:18.440 --> 00:39:28.520 It turns out for D-brane with constant. 00:39:28.520 --> 00:39:35.170 So as opposed to the D-brane move with constant velocity, 00:39:35.170 --> 00:39:38.020 so now you can also have a motion 00:39:38.020 --> 00:39:40.560 in the spatial direction. 00:39:40.560 --> 00:39:45.855 You have a constant of this and F alpha beta. 00:39:49.480 --> 00:39:51.180 You can also excite the gauge field, 00:39:51.180 --> 00:39:52.685 but the field strength is constant. 00:39:55.950 --> 00:40:00.010 Or this quantity is small. 00:40:00.010 --> 00:40:01.980 They don't have to be strictly constant, 00:40:01.980 --> 00:40:05.110 but at least their derivatives are small. 00:40:11.880 --> 00:40:16.120 In such a situation, one can actually 00:40:16.120 --> 00:40:22.650 sum all the higher order terms from string theory corrections. 00:40:22.650 --> 00:40:25.420 So this is just a low energy, just like field theory. 00:40:25.420 --> 00:40:28.850 And in such a situation, you can actually 00:40:28.850 --> 00:40:32.160 sum over infinite number of higher order terms. 00:40:32.160 --> 00:40:33.110 And what do you find? 00:40:33.110 --> 00:40:35.700 You find so-called wave sum of all infinite number higher 00:40:35.700 --> 00:40:37.560 terms. 00:40:37.560 --> 00:40:39.720 You find so-called Dirac-Born-Infeld action. 00:40:43.340 --> 00:40:45.590 You find that this effective action becomes like this. 00:40:50.150 --> 00:40:55.327 This is very [INAUDIBLE] result. So I just want to mention it. 00:41:12.400 --> 00:41:15.710 You can actually sum into this form. 00:41:15.710 --> 00:41:20.200 So you can sum into this form. 00:41:20.200 --> 00:41:25.925 And this g alpha beta is the [INAUDIBLE]. 00:41:40.600 --> 00:41:49.524 So let me just explain a little bit the physics 00:41:49.524 --> 00:41:50.190 of this formula. 00:41:59.860 --> 00:42:05.050 So let's consider the case of the phi 00:42:05.050 --> 00:42:07.660 is not excited at all, just phi for the constant, 00:42:07.660 --> 00:42:08.760 say, for example. 00:42:08.760 --> 00:42:10.620 Then this term vanishes. 00:42:10.620 --> 00:42:15.630 Then this g alpha beta just becomes eta alpha beta. 00:42:15.630 --> 00:42:17.750 And now you just have a square root, 00:42:17.750 --> 00:42:21.430 say, your Minkowski metric plus F alpha beta. 00:42:21.430 --> 00:42:24.580 Forget about this 2 pi alpha prime. 00:42:24.580 --> 00:42:27.310 This is just some dimensional factor. 00:42:27.310 --> 00:42:30.910 And then you just have eta alpha beta plus F alpha beta. 00:42:30.910 --> 00:42:34.370 And then you write the determinant. 00:42:34.370 --> 00:42:38.700 And suppose when F alpha beta is small, when alpha prime times F 00:42:38.700 --> 00:42:41.005 alpha beta is small, then you can expand it 00:42:41.005 --> 00:42:44.440 in powers of F alpha beta. 00:42:44.440 --> 00:42:48.800 It's a simple exercise but instructive exercise. 00:42:48.800 --> 00:42:55.050 You see that precisely reproduces the Maxwell term. 00:42:55.050 --> 00:42:58.540 But this will give rise to higher nonlinear terms. 00:42:58.540 --> 00:43:02.060 There will be higher order nonlinear terms. 00:43:02.060 --> 00:43:05.800 So this can be considered as a nonlinear generalization 00:43:05.800 --> 00:43:08.494 of the Maxwell theory. 00:43:08.494 --> 00:43:09.910 It turns out actually, this theory 00:43:09.910 --> 00:43:17.295 was considered in the '30s by this guy, Born and Infeld. 00:43:21.210 --> 00:43:23.610 Actually, maybe '30s or '40s. 00:43:23.610 --> 00:43:25.965 Anyway, prehistory. 00:43:29.430 --> 00:43:38.510 They invented as a way to avoid-- they 00:43:38.510 --> 00:43:41.500 want to avoid the similarity of the Maxwell theory. 00:43:41.500 --> 00:43:44.700 So in the Maxwell theory, if you have a charged particle, 00:43:44.700 --> 00:43:47.154 and then the location of the charged particle, 00:43:47.154 --> 00:43:48.570 then the field due to that charged 00:43:48.570 --> 00:43:52.580 particle is singular and the location of that particle. 00:43:52.580 --> 00:43:54.890 And so they want to avoid that singular behavior, 00:43:54.890 --> 00:43:57.890 so they invented this Born-Infeld action. 00:43:57.890 --> 00:44:02.800 And for many years, this action does not have any applications. 00:44:02.800 --> 00:44:07.210 But if you invent something nice, it will find its use. 00:44:07.210 --> 00:44:11.040 Just like in this movie, Jurassic Park, 00:44:11.040 --> 00:44:12.220 life finds a way. 00:44:15.010 --> 00:44:16.320 Life always finds a way. 00:44:19.360 --> 00:44:20.810 So that's Born-Infeld. 00:44:24.420 --> 00:44:28.670 Now let's set F equal to 0. 00:44:28.670 --> 00:44:30.805 Let's just look at g alpha beta. 00:44:34.390 --> 00:44:37.460 Now let's look at g alpha beta. 00:44:37.460 --> 00:44:40.160 So g alpha beta, we can write it in a slightly more transparent 00:44:40.160 --> 00:44:40.660 form. 00:44:45.040 --> 00:44:49.590 We can write a form which makes it a bit more transparent. 00:44:49.590 --> 00:44:52.510 I can write it as the following. 00:44:52.510 --> 00:44:56.345 So even mu, remember, is the Minkowski metric 00:44:56.345 --> 00:44:59.650 of the full space time. 00:44:59.650 --> 00:45:02.290 And I can write this as following. 00:45:02.290 --> 00:45:12.640 x mu beta x mu x mu with x alpha equal to x alpha, 00:45:12.640 --> 00:45:17.420 which is along the brane direction, and xa to be phi a. 00:45:20.450 --> 00:45:24.780 So if you look at this formula, you 00:45:24.780 --> 00:45:31.930 can see this is an induced metric for some brane embedded 00:45:31.930 --> 00:45:33.920 in the full Minkowski space time. 00:45:33.920 --> 00:45:36.710 And the x describes such embedding. 00:45:43.250 --> 00:45:45.550 This generalization of this induces the metric formula 00:45:45.550 --> 00:45:47.464 we encountered before for the string. 00:45:47.464 --> 00:45:48.880 But right now, the only difference 00:45:48.880 --> 00:45:51.260 is that now alpha beta, they run all 00:45:51.260 --> 00:45:53.610 in the world-volume direction of the D-brane. 00:45:53.610 --> 00:45:55.330 And then this becomes an induced metric 00:45:55.330 --> 00:45:59.670 on the D-brane when it's embedded in the space time. 00:45:59.670 --> 00:46:02.040 And this x alpha equal to x alpha, 00:46:02.040 --> 00:46:04.180 it just means that when we embed it, 00:46:04.180 --> 00:46:06.770 and we choose the world-volume direction 00:46:06.770 --> 00:46:09.480 to be the same as the space time direction 00:46:09.480 --> 00:46:11.430 along the brane direction. 00:46:11.430 --> 00:46:15.320 And in the perpendicular direction, this is just phi a. 00:46:15.320 --> 00:46:19.760 And if you look at this, this is exactly that. 00:46:19.760 --> 00:46:23.680 It's exactly that because x alpha equal to x alpha. 00:46:23.680 --> 00:46:26.500 And then you just get the eta alpha beta. 00:46:26.500 --> 00:46:31.160 And then for the other direction, you get this one. 00:46:31.160 --> 00:46:33.410 Is this clear? 00:46:33.410 --> 00:46:50.850 So when f equal to 0, so this S just becomes eta g alpha beta. 00:46:50.850 --> 00:47:07.960 So this is precisely the volume element of DB-brane. 00:47:07.960 --> 00:47:10.439 Because this is the induced metric, and then this 00:47:10.439 --> 00:47:12.480 is just the total volume element of the DP-brane. 00:47:16.800 --> 00:47:18.810 And we see this is precisely is the relativistic 00:47:18.810 --> 00:47:19.434 generalization. 00:47:23.330 --> 00:47:26.730 So this is just a generalization of the [INAUDIBLE] action 00:47:26.730 --> 00:47:30.720 we wrote earlier, which is for a string. 00:47:30.720 --> 00:47:34.540 Then this would be a two dimensional area. 00:47:34.540 --> 00:47:37.450 And here, you just integrate over the volume element 00:47:37.450 --> 00:47:39.830 of the whole D-brane. 00:47:39.830 --> 00:47:42.725 So we see that this Born-Infeld corresponding 00:47:42.725 --> 00:47:47.519 to really describes the relativistic motion of a p 00:47:47.519 --> 00:47:48.310 dimensional object. 00:47:52.250 --> 00:47:55.810 Describes the relativistic motion of a p dimensional 00:47:55.810 --> 00:47:58.150 object. 00:47:58.150 --> 00:48:01.810 And this Dirac-Born-Infeld, when you combine these two together, 00:48:01.810 --> 00:48:07.660 it magically combines these two things into a single thing. 00:48:07.660 --> 00:48:09.484 Yes? 00:48:09.484 --> 00:48:11.150 AUDIENCE: So I recall you saying earlier 00:48:11.150 --> 00:48:16.130 that-- you said that people have played 00:48:16.130 --> 00:48:17.930 with this idea of thinking about branes 00:48:17.930 --> 00:48:20.940 instead of just generically higher dimensional objects 00:48:20.940 --> 00:48:23.250 and strings but no one really understood the theory 00:48:23.250 --> 00:48:25.525 of these things because the topology and geometry 00:48:25.525 --> 00:48:26.710 were too complicated. 00:48:26.710 --> 00:48:29.430 So it seems to me that wouldn't you run into that same problem 00:48:29.430 --> 00:48:31.630 right here if it's indeed some generalization 00:48:31.630 --> 00:48:32.750 of the [INAUDIBLE] action? 00:48:32.750 --> 00:48:36.370 HONG LIU: Yeah, but we don't try to quantize it. 00:48:36.370 --> 00:48:38.770 At least we don't try to quantize this action. 00:48:38.770 --> 00:48:41.080 And we know how to quantize this action. 00:48:41.080 --> 00:48:46.520 And this is just our ordinary field theory. 00:48:46.520 --> 00:48:48.060 AUDIENCE: I have a question. 00:48:48.060 --> 00:48:56.770 Here, we must impose the big x as the coordinates 00:48:56.770 --> 00:48:58.640 in the target space. 00:48:58.640 --> 00:49:00.070 HONG LIU: That's right. 00:49:00.070 --> 00:49:05.530 AUDIENCE: So phi a must be kind of a constant? 00:49:05.530 --> 00:49:09.200 I mean, why there should be a constant part on-- 00:49:09.200 --> 00:49:11.370 HONG LIU: No. 00:49:11.370 --> 00:49:14.920 If the derivative of those things are not small, 00:49:14.920 --> 00:49:18.210 then there will be many other terms. 00:49:18.210 --> 00:49:19.660 This will not be the only action. 00:49:19.660 --> 00:49:21.572 AUDIENCE: Given a constant. 00:49:21.572 --> 00:49:22.882 HONG LIU: Sorry? 00:49:22.882 --> 00:49:24.340 AUDIENCE: You said with a constant. 00:49:24.340 --> 00:49:26.506 HONG LIU: The partial alpha phi a equal to constant. 00:49:26.506 --> 00:49:28.240 AUDIENCE: Oh, equals a constant. 00:49:28.240 --> 00:49:29.190 So that means-- 00:49:29.190 --> 00:49:30.580 HONG LIU: No. 00:49:30.580 --> 00:49:33.020 What I'm saying is that if these are constants, then 00:49:33.020 --> 00:49:36.920 this is our exact string theory action. 00:49:36.920 --> 00:49:38.512 And when these are not constants, 00:49:38.512 --> 00:49:40.220 and then this is a leading approximation, 00:49:40.220 --> 00:49:41.860 there will be higher order terms which 00:49:41.860 --> 00:49:46.790 depend on their derivatives. 00:49:46.790 --> 00:49:47.684 Yes? 00:49:47.684 --> 00:49:48.350 AUDIENCE: Sorry. 00:49:48.350 --> 00:49:49.808 One thing I just don't understand-- 00:49:49.808 --> 00:49:52.230 why is it that we don't want to try to quantize anything? 00:49:52.230 --> 00:49:55.200 Shouldn't it be quantized in principle? 00:49:55.200 --> 00:49:57.249 These are the sort of classical analogs 00:49:57.249 --> 00:49:58.640 of things you want to quantize. 00:49:58.640 --> 00:49:59.717 HONG LIU: Yeah. 00:49:59.717 --> 00:50:01.550 AUDIENCE: So this is to say that this object 00:50:01.550 --> 00:50:03.258 that we don't really know how to quantize 00:50:03.258 --> 00:50:05.997 is-- we just don't do it because we don't know how. 00:50:05.997 --> 00:50:06.622 HONG LIU: Yeah. 00:50:06.622 --> 00:50:07.095 AUDIENCE: I see. 00:50:07.095 --> 00:50:07.595 OK. 00:50:07.595 --> 00:50:09.495 It's not because-- fair enough. 00:50:09.495 --> 00:50:10.120 HONG LIU: Yeah. 00:50:10.120 --> 00:50:12.960 You should try anything. 00:50:12.960 --> 00:50:17.460 And only those people who have succeeded in the history books. 00:50:21.670 --> 00:50:24.194 Only a few have won the battle in the history books. 00:50:24.194 --> 00:50:26.860 And if you just fail the battle, you're not in the history book. 00:50:30.964 --> 00:50:32.630 So people have tried this but failed it, 00:50:32.630 --> 00:50:34.740 but that won't be written in the books. 00:50:34.740 --> 00:50:36.620 AUDIENCE: Sure. 00:50:36.620 --> 00:50:37.410 AUDIENCE: Sorry. 00:50:37.410 --> 00:50:39.540 So you did the square root by summing over 00:50:39.540 --> 00:50:43.000 all the massive terms in the-- 00:50:43.000 --> 00:50:43.680 HONG LIU: Sorry? 00:50:43.680 --> 00:50:45.380 AUDIENCE: You get the square root term 00:50:45.380 --> 00:50:49.070 by summing the series, including the massive fields? 00:50:49.070 --> 00:50:49.640 HONG LIU: No. 00:50:49.640 --> 00:50:51.554 This is still the gauge field and the phi. 00:50:51.554 --> 00:50:52.220 AUDIENCE: Right. 00:50:52.220 --> 00:50:53.850 So how do you get it? 00:50:53.850 --> 00:50:55.919 What's the series that you're summing? 00:50:55.919 --> 00:50:56.460 HONG LIU: Hm? 00:50:56.460 --> 00:50:57.650 AUDIENCE: What's the series that you're summing? 00:50:57.650 --> 00:50:57.800 HONG LIU: Oh. 00:50:57.800 --> 00:50:59.341 I'm just saying in the string theory, 00:50:59.341 --> 00:51:01.700 typically you don't start by f squared. 00:51:01.700 --> 00:51:04.220 You have f cubed, f four, et cetera. 00:51:04.220 --> 00:51:08.010 You can sum all of them together. 00:51:08.010 --> 00:51:12.520 Even for the massless mode, these are just leading terms. 00:51:12.520 --> 00:51:15.910 And these terms would be the smallest number of derivatives, 00:51:15.910 --> 00:51:20.250 and so they dominate at low energies. 00:51:20.250 --> 00:51:24.950 But in general, even just for the effective serial 00:51:24.950 --> 00:51:28.150 massless mode, you can have many, many other terms. 00:51:28.150 --> 00:51:29.790 AUDIENCE: I have a question. 00:51:29.790 --> 00:51:33.050 If we assume that partial alpha phi a is a constant, 00:51:33.050 --> 00:51:37.135 then we can solve out the phi a is proportional to x alpha. 00:51:37.135 --> 00:51:40.220 But how can you assume they're just 00:51:40.220 --> 00:51:42.402 the coordinate in target space? 00:51:42.402 --> 00:51:43.110 HONG LIU: No, no. 00:51:43.110 --> 00:51:45.780 This is a function of alpha. 00:51:45.780 --> 00:51:47.660 AUDIENCE: Yes. 00:51:47.660 --> 00:51:51.040 But since it's a function of x alpha, 00:51:51.040 --> 00:51:56.157 why it can be regarded as the coordinate in target space? 00:51:56.157 --> 00:51:56.823 HONG LIU: Sorry. 00:51:56.823 --> 00:51:58.230 I don't understand. 00:51:58.230 --> 00:51:59.730 AUDIENCE: All the coordinates should 00:51:59.730 --> 00:52:02.124 be independent in target space. 00:52:02.124 --> 00:52:02.790 HONG LIU: Sorry. 00:52:02.790 --> 00:52:03.550 I don't understand. 00:52:03.550 --> 00:52:04.425 They are independent. 00:52:07.640 --> 00:52:10.240 These are the virtual coordinates. 00:52:10.240 --> 00:52:13.640 These are the volume coordinates of D-brane. 00:52:13.640 --> 00:52:16.860 And these are the target space coordinates. 00:52:16.860 --> 00:52:18.710 I'm just choosing the function of the target 00:52:18.710 --> 00:52:19.683 space coordinates. 00:52:23.140 --> 00:52:25.530 I'm choosing here just to be identical 00:52:25.530 --> 00:52:27.502 to the world-volume coordinate. 00:52:27.502 --> 00:52:29.210 And this one I choose to be some function 00:52:29.210 --> 00:52:30.633 of the world-volume coordinate. 00:52:30.633 --> 00:52:32.112 Of course I can do that. 00:52:35.467 --> 00:52:36.300 Any other questions? 00:52:46.420 --> 00:52:50.580 So again, this highlights that D-brane is really 00:52:50.580 --> 00:52:52.550 a dynamical object. 00:52:52.550 --> 00:53:00.900 In fact, at low energies, they move like [INAUDIBLE] motion. 00:53:00.900 --> 00:53:04.090 And they actually move relativistically 00:53:04.090 --> 00:53:06.180 if you give enough velocity, et cetera. 00:53:11.170 --> 00:53:13.290 Because of the fluctuations of the D-brane, 00:53:13.290 --> 00:53:18.920 they become a really full dynamical object. 00:53:18.920 --> 00:53:19.810 They have a mass. 00:53:19.810 --> 00:53:21.380 They can move around. 00:53:21.380 --> 00:53:23.775 And now you can deform their shape. 00:53:23.775 --> 00:53:27.212 If you have enough energy, you can bend them, et cetera. 00:53:27.212 --> 00:53:28.420 You can do whatever you want. 00:53:37.290 --> 00:53:42.150 So let me mention one last thing. 00:53:42.150 --> 00:53:50.060 So you may ask, why somehow those 00:53:50.060 --> 00:53:53.500 fields which describe the motion of the D-brane, 00:53:53.500 --> 00:53:56.330 they're corresponding to the massless modes 00:53:56.330 --> 00:54:04.390 on the world-volume of the D-branes? 00:54:04.390 --> 00:54:08.730 Whether this is a coincidence, or why it somehow 00:54:08.730 --> 00:54:14.710 happens to be the massless mode on the D-brane 00:54:14.710 --> 00:54:18.820 which describes the motion of the D-brane. 00:54:18.820 --> 00:54:21.710 So this is not an accident. 00:54:21.710 --> 00:54:23.490 This is not an accident. 00:54:23.490 --> 00:54:42.910 The reason is that-- so why modes 00:54:42.910 --> 00:55:04.000 describing motions of D-branes appear as massless modes? 00:55:04.000 --> 00:55:05.560 So this is not accident. 00:55:05.560 --> 00:55:13.630 So underlying reason, it's because the underlying 00:55:13.630 --> 00:55:27.980 Minkowski space is translation invariant. 00:55:31.390 --> 00:55:37.330 So that means that no matter where you put the D-brane, 00:55:37.330 --> 00:55:40.950 it should be a well-defined configuration. 00:55:40.950 --> 00:55:43.810 Should be a well-defined configuration no matter where 00:55:43.810 --> 00:55:45.226 you put on the D-brane. 00:55:45.226 --> 00:55:50.990 Then that means that the [INAUDIBLE] action for the phi 00:55:50.990 --> 00:55:53.620 cannot contain a potential term. 00:55:53.620 --> 00:55:54.880 They cannot be potential term. 00:55:54.880 --> 00:55:57.420 They should not be, say, somewhere is the minimum, 00:55:57.420 --> 00:55:59.330 somewhere is the maximum, cannot happen. 00:55:59.330 --> 00:56:01.030 Everywhere must be the same. 00:56:01.030 --> 00:56:05.175 So it means the dependence on y can only be derivative. 00:56:05.175 --> 00:56:07.370 Can only depend on derivatives. 00:56:07.370 --> 00:56:11.140 And of course, at low energies, if you have derivatives, 00:56:11.140 --> 00:56:12.860 then can only be the massless particle. 00:56:15.940 --> 00:56:18.880 So translation invariant. 00:56:18.880 --> 00:56:26.090 So this means that any phi a equal to constant 00:56:26.090 --> 00:56:31.080 should be allowed configurations. 00:56:42.320 --> 00:56:47.790 That means cannot have potential. 00:56:50.470 --> 00:56:52.475 So max term is like a potential for phi. 00:56:59.560 --> 00:57:03.140 To say in the fancy words of Quantum Field Theory 00:57:03.140 --> 00:57:11.920 II or Quantum Field Theory III, that the phi a, in other words, 00:57:11.920 --> 00:57:29.451 phi a are the Goldstone bosons for breaking 00:57:29.451 --> 00:57:30.437 translation symmetries. 00:57:46.740 --> 00:57:51.566 So previously, Minkowski space is translation invariant. 00:57:51.566 --> 00:57:53.210 And now if you put a D-brane there, 00:57:53.210 --> 00:57:55.161 then you break that translation symmetry. 00:57:55.161 --> 00:57:57.285 The location of the D-brane breaks that translation 00:57:57.285 --> 00:57:58.535 symmetry. 00:57:58.535 --> 00:58:01.700 If I even put the D-brane anywhere, 00:58:01.700 --> 00:58:04.260 then that means the modes, the dynamics 00:58:04.260 --> 00:58:07.130 control the location of the D-brane 00:58:07.130 --> 00:58:10.805 must not have any potential, can only have derivative terms. 00:58:14.470 --> 00:58:17.390 So in other words, when you put the D-brane in, 00:58:17.390 --> 00:58:23.740 you spontaneously break the translation symmetry 00:58:23.740 --> 00:58:25.102 on the line in Minkowski space. 00:58:33.634 --> 00:58:34.800 So let me mention one thing. 00:58:34.800 --> 00:58:35.880 Then we can have a break. 00:58:39.920 --> 00:58:41.270 I'll mention one quick thing. 00:58:58.440 --> 00:59:05.619 So let me say a few words on the strength of the open string 00:59:05.619 --> 00:59:06.160 interactions. 00:59:27.160 --> 00:59:34.500 So previously, we described that the closed strings, they 00:59:34.500 --> 00:59:40.350 interact by such joining and splitting procedure. 00:59:45.770 --> 00:59:47.390 And the strength here is capped by gs. 00:59:50.104 --> 00:59:52.270 So the closed string coupling is essentially the gs, 00:59:52.270 --> 00:59:55.187 which describes such a process. 00:59:55.187 --> 00:59:57.020 So if you have an open string, of course you 00:59:57.020 --> 01:00:00.840 have a similar process, just string ends joined together. 01:00:03.360 --> 01:00:05.272 You can join string ends together. 01:00:08.190 --> 01:00:11.760 So here, you really just have open string. 01:00:11.760 --> 01:00:16.200 So now these lines are the boundaries of open string, 01:00:16.200 --> 01:00:18.680 or the endpoints of open string. 01:00:18.680 --> 01:00:21.210 So you have two open strings joined together 01:00:21.210 --> 01:00:23.050 from another one. 01:00:23.050 --> 01:00:25.940 So let's call this interaction go 01:00:25.940 --> 01:00:28.508 describing the interaction of the open string. 01:00:31.200 --> 01:00:33.685 So the question is, how is this go related to gs? 01:00:36.820 --> 01:00:38.940 And there's a single way we can figure it out. 01:00:42.430 --> 01:00:44.780 So let's consider the simplest situation. 01:00:44.780 --> 01:00:50.390 Just have open string propagate in time. 01:00:50.390 --> 01:00:52.680 Again, this is two boundaries open string. 01:00:52.680 --> 01:00:53.970 We just propagate in time. 01:01:02.020 --> 01:01:05.300 Now let's consider a more complicated process. 01:01:05.300 --> 01:01:07.600 So the open string propagates in time. 01:01:07.600 --> 01:01:15.030 So this is just a simple surface with one initial open string 01:01:15.030 --> 01:01:17.756 and one final open string. 01:01:17.756 --> 01:01:19.990 And you can see the complicated process 01:01:19.990 --> 01:01:23.210 because we have a hole in the middle. 01:01:23.210 --> 01:01:27.770 So now the string worksheet is like this, just this part. 01:01:27.770 --> 01:01:29.440 We have a hole in the middle. 01:01:29.440 --> 01:01:34.250 And this is another configuration 01:01:34.250 --> 01:01:36.050 to have some initial open string propagate 01:01:36.050 --> 01:01:39.230 to some final open string. 01:01:39.230 --> 01:01:49.150 So now we know, by counting we did before described here. 01:01:49.150 --> 01:01:52.240 So here, we're adding one boundary. 01:01:52.240 --> 01:01:55.390 We are adding one boundary. 01:01:55.390 --> 01:01:56.480 So this adds a boundary. 01:02:01.580 --> 01:02:06.880 So that means we must add a factor of gs. 01:02:10.310 --> 01:02:13.200 Because from this formula, this is g minus chi. 01:02:13.200 --> 01:02:16.120 And chi is minus b. 01:02:16.120 --> 01:02:18.370 So if we increase one boundary, then you 01:02:18.370 --> 01:02:19.460 increase a factor of gs. 01:02:22.240 --> 01:02:28.100 But we can also view this diagram 01:02:28.100 --> 01:02:30.500 as a single open string comes in. 01:02:35.330 --> 01:02:39.350 The opposite of this splits into two open strings 01:02:39.350 --> 01:02:42.667 and then they close together. 01:02:42.667 --> 01:02:48.950 So one split operation, and the one join operation. 01:02:51.930 --> 01:02:56.160 So that should correspond to go squared. 01:02:56.160 --> 01:02:58.910 So then this means that we conclude that gs must 01:02:58.910 --> 01:03:01.920 be proportional to go squared. 01:03:01.920 --> 01:03:06.320 The open string coupling strings is the square root 01:03:06.320 --> 01:03:08.880 of the closed string interaction. 01:03:13.201 --> 01:03:13.700 Yes? 01:03:13.700 --> 01:03:15.985 AUDIENCE: What's the strength of the process when 01:03:15.985 --> 01:03:17.770 closed string becomes open string and vice versa? 01:03:17.770 --> 01:03:18.035 HONG LIU: Sorry? 01:03:18.035 --> 01:03:20.415 AUDIENCE: What's the strength of the process when closed 01:03:20.415 --> 01:03:21.706 string becomes and open string? 01:03:21.706 --> 01:03:22.390 HONG LIU: Right. 01:03:22.390 --> 01:03:24.210 Yeah, you can consider such process. 01:03:27.020 --> 01:03:33.620 Again, you can just do it by counting 01:03:33.620 --> 01:03:35.462 the topology of the surface. 01:03:35.462 --> 01:03:36.420 Such process can exist. 01:03:44.510 --> 01:03:45.010 OK. 01:03:45.010 --> 01:03:46.260 Then let's have a short break. 01:03:48.940 --> 01:03:53.130 So what time is it? 01:03:53.130 --> 01:03:53.740 It's 38. 01:03:57.390 --> 01:04:02.030 When should we start again? 01:04:02.030 --> 01:04:03.380 41? 01:04:03.380 --> 01:04:03.880 OK, 41. 01:04:03.880 --> 01:04:06.370 Let's start at 41. 01:04:06.370 --> 01:04:08.980 So we have talked about D-branes, et cetera. 01:04:12.480 --> 01:04:14.710 And we have already seen some remarkable aspects 01:04:14.710 --> 01:04:16.380 of the D-brane, including this channel 01:04:16.380 --> 01:04:23.530 duality between the closed string exchange 01:04:23.530 --> 01:04:26.744 can be considered open string loop. 01:04:26.744 --> 01:04:29.160 And now we are going to see a lot of magic of the D-brane. 01:04:35.081 --> 01:04:37.330 And this comes when you put several D-branes together. 01:05:06.400 --> 01:05:18.160 So normally, our conventional intuition 01:05:18.160 --> 01:05:26.050 says if you find some particle, say in this case, 01:05:26.050 --> 01:05:27.320 you find the D-brane. 01:05:27.320 --> 01:05:29.980 So if you put two particles together, 01:05:29.980 --> 01:05:31.710 nothing much really changes. 01:05:31.710 --> 01:05:33.170 It's two particles. 01:05:33.170 --> 01:05:35.590 Put three particles together. 01:05:35.590 --> 01:05:36.436 Not much changes. 01:05:36.436 --> 01:05:37.310 It's three particles. 01:05:39.940 --> 01:05:43.370 But when you put multiple D-branes together, 01:05:43.370 --> 01:05:48.690 things change a lot in a very profound way. 01:05:48.690 --> 01:05:53.640 So now let's consider just two D-branes. 01:05:53.640 --> 01:05:55.010 So let's consider two D-branes. 01:06:23.310 --> 01:06:26.380 Let's consider example. 01:06:26.380 --> 01:06:29.366 So let me first just tell you the naive intuition. 01:06:33.590 --> 01:06:35.840 Suppose you have D-brane one, D-brane two. 01:06:38.990 --> 01:06:42.850 So for this one, we have a u1 gauge field. 01:06:42.850 --> 01:06:46.840 For this one, you have a u1 gauge field. 01:06:46.840 --> 01:06:50.510 Because each one, we have a gauge field, a Maxwell. 01:06:50.510 --> 01:06:53.420 When you put together, from conventional wisdom, 01:06:53.420 --> 01:06:55.290 you say, maybe I just have two Maxwell. 01:06:57.890 --> 01:06:59.850 From conventional wisdom, you two Maxwell. 01:06:59.850 --> 01:07:02.600 Naively, if I put them together, I just have two Maxwells. 01:07:02.600 --> 01:07:05.683 1 plus 1 equal to 2. 01:07:10.780 --> 01:07:15.800 But in string theory, 1 plus 1 equal to 4. 01:07:15.800 --> 01:07:17.370 It's actually equal to 2. 01:07:17.370 --> 01:07:21.810 It's also equal to 2, depending on how you think about it. 01:07:21.810 --> 01:07:26.050 Anyway, one way to think about it is 1 plus 1 becomes 4. 01:07:26.050 --> 01:07:31.410 So to see 1 plus 1 become 4 is very easy. 01:07:31.410 --> 01:07:34.636 So let's consider these two branes on top of each other. 01:07:34.636 --> 01:07:36.510 But in order to distinguish these two branes, 01:07:36.510 --> 01:07:38.350 I just separate them a little bit. 01:07:38.350 --> 01:07:42.550 But you should really think of them on top of each other. 01:07:42.550 --> 01:07:47.040 And so now you have four types of strings. 01:07:47.040 --> 01:07:52.970 You can have string going to 1, 1, going to 2, 2, 01:07:52.970 --> 01:07:56.180 then going to 1, 2, going to 2, 1. 01:07:56.180 --> 01:08:00.080 So 2, 1 and 1, 2 are different because the oriented string. 01:08:00.080 --> 01:08:02.850 So I put arrow there. 01:08:02.850 --> 01:08:04.880 So this is from 1 to 2. 01:08:04.880 --> 01:08:05.740 This is from 2 to 1. 01:08:09.720 --> 01:08:15.769 So we have four types of strings-- 1, 1, 1, 2, 2, 1, 01:08:15.769 --> 01:08:16.269 2, 2. 01:08:21.760 --> 01:08:23.256 AUDIENCE: Why is oriented-- 01:08:23.256 --> 01:08:24.779 HONG LIU: Hm? 01:08:24.779 --> 01:08:28.609 AUDIENCE: Why the 1 to 2, 2 to 1 are different? 01:08:28.609 --> 01:08:32.550 HONG LIU: It's because for this string, sigma 0 here. 01:08:32.550 --> 01:08:33.899 For this one, sigma pi there. 01:08:37.240 --> 01:08:39.140 For this string, sigma 0 there. 01:08:39.140 --> 01:08:41.189 And for this one, sigma pi there. 01:08:46.149 --> 01:08:48.410 Let me just elaborate on this point. 01:08:48.410 --> 01:08:52.040 So suppose I have a string like this. 01:08:52.040 --> 01:08:55.819 Then this string is sigma 0 point ending on 1, 01:08:55.819 --> 01:08:58.170 sigma equal to pi ending on 2. 01:08:58.170 --> 01:08:59.880 But if I have a string like that, 01:08:59.880 --> 01:09:03.066 then there's a sigma 0 ending here and sigma pi ending there. 01:09:09.386 --> 01:09:13.370 So you have four types of open strings. 01:09:13.370 --> 01:09:15.120 And now if you think about how we quantize 01:09:15.120 --> 01:09:18.569 those strings, and the four types of open strings, 01:09:18.569 --> 01:09:21.370 they actually have identical spectrum. 01:09:21.370 --> 01:09:23.370 Because for all of them, the boundary conditions 01:09:23.370 --> 01:09:25.446 are exactly the same. 01:09:25.446 --> 01:09:26.946 Because the boundary conditions only 01:09:26.946 --> 01:09:29.359 know the location of the D-branes. 01:09:29.359 --> 01:09:31.810 So all four types of open strings 01:09:31.810 --> 01:09:35.140 have identical spectrum. 01:09:35.140 --> 01:09:46.890 So in other words, each string excitation-- 01:09:46.890 --> 01:09:53.750 say this is the state on the worksheet-- each state becomes 01:09:53.750 --> 01:09:58.270 four states because I can label IJ. 01:09:58.270 --> 01:10:02.170 Now suppose I use I and J to label 1 and the 2. 01:10:02.170 --> 01:10:05.850 I and J can be 1 and 2. 01:10:05.850 --> 01:10:08.590 So depending on whether this is 1, 1 string, or 1, 2 string, 01:10:08.590 --> 01:10:12.100 or 2, 1 string, or 2, 2 string. 01:10:12.100 --> 01:10:15.030 So this is what I said 1 plus 1 equal to 4. 01:10:15.030 --> 01:10:19.070 Because naively, you would say I have two massless modes. 01:10:19.070 --> 01:10:20.150 But now I have four. 01:10:24.680 --> 01:10:37.910 For example, the massless modes become four copies of them. 01:10:51.990 --> 01:11:19.815 In other words, you can think each open string excitation a 2 01:11:19.815 --> 01:11:20.380 by 2 matrix. 01:11:26.210 --> 01:11:28.616 I can use 2 index to label them. 01:11:32.320 --> 01:11:34.570 In particular, for example, the corresponding fields-- 01:11:34.570 --> 01:11:38.210 so each string excitation corresponding to some field. 01:11:38.210 --> 01:11:43.020 For example, the gauge field associated with this 01:11:43.020 --> 01:11:48.270 now has two index, I J. And similar with phi 01:11:48.270 --> 01:12:04.050 a I J. Of course, this generalizes immediately 01:12:04.050 --> 01:12:20.186 to if you have n branes, then just becomes n times 01:12:20.186 --> 01:12:20.920 n matrices. 01:12:27.200 --> 01:12:30.741 So 1 plus 1 plus 1 to n becomes n squared. 01:12:37.800 --> 01:12:40.740 So now let me give some remarks. 01:12:45.150 --> 01:12:48.330 So this basic structure turns out 01:12:48.330 --> 01:12:52.340 to be, again, very, very profound. 01:12:52.340 --> 01:12:54.234 Now let me give some remarks. 01:13:06.360 --> 01:13:09.070 So there's a reason I call-- so this 01:13:09.070 --> 01:13:11.135 is something with the 2 index. 01:13:16.010 --> 01:13:19.330 So of course, you naturally call it a matrix. 01:13:19.330 --> 01:13:22.010 But there's another reason to think about this really 01:13:22.010 --> 01:13:25.100 as a matrix. 01:13:25.100 --> 01:13:29.640 It's because the strings, as we were 01:13:29.640 --> 01:13:32.394 doing there, the open strings, they interact 01:13:32.394 --> 01:13:33.310 by joining their ends. 01:13:49.280 --> 01:13:54.270 So this naturally leads to-- when those strings interact 01:13:54.270 --> 01:13:59.150 with each other, and those parts naturally 01:13:59.150 --> 01:14:14.970 just emerges as a matrix product, I, J indices. 01:14:19.450 --> 01:14:22.990 So it's easy to see. 01:14:25.810 --> 01:14:27.410 So let me just draw that. 01:14:27.410 --> 01:14:30.730 Let me just do it here to save some time. 01:14:30.730 --> 01:14:37.320 Suppose this is I. This is J. So this is sigma equal to 0, 01:14:37.320 --> 01:14:38.740 sigma equal to pi. 01:14:38.740 --> 01:14:41.090 And the sigma equal to pi joins with stigma 0 01:14:41.090 --> 01:14:42.279 to end of the other one. 01:14:42.279 --> 01:14:44.320 But of course, if you want to join them together, 01:14:44.320 --> 01:14:46.570 their J's have to be the same. 01:14:46.570 --> 01:14:51.870 This is K. Then you go to I, K. And when they join together, 01:14:51.870 --> 01:14:54.390 then you sum of all possible J's. 01:14:54.390 --> 01:14:56.164 Then this is like a matrix product. 01:15:00.680 --> 01:15:05.450 So if I draw it in the diagram not very well. 01:15:05.450 --> 01:15:11.560 Now let me separate I, J, K to be three things. 01:15:11.560 --> 01:15:14.010 But they don't have to be separated. 01:15:14.010 --> 01:15:17.560 I, J, K can also be the same. 01:15:17.560 --> 01:15:20.410 But in order to emphasize this picture is 01:15:20.410 --> 01:15:28.700 that you have a string to go from I to J. Suppose this is I, 01:15:28.700 --> 01:15:31.890 this is J, this is K. Go from I to J, 01:15:31.890 --> 01:15:37.830 then from J to K. So sigma 0, sigma pi. 01:15:37.830 --> 01:15:40.280 And the pi end joins with the sigma 0 end. 01:15:40.280 --> 01:15:42.185 And then here, you get the string. 01:15:45.830 --> 01:15:49.290 So that diagram roughly can also be 01:15:49.290 --> 01:15:51.350 think of a diagram like this. 01:15:51.350 --> 01:15:57.350 Two strings join into one string with index I and K. 01:15:57.350 --> 01:15:58.850 And the I, K, K can all be the same. 01:15:58.850 --> 01:16:03.690 I just make them different to make it clear. 01:16:03.690 --> 01:16:09.950 And of course, when you join J together, 01:16:09.950 --> 01:16:14.860 you have the sum of them because they can be in principle all 01:16:14.860 --> 01:16:15.570 possible J's. 01:16:19.615 --> 01:16:23.810 So it naturally appears as a matrix product. 01:16:23.810 --> 01:16:28.720 Just follows by the nature of string interaction. 01:16:28.720 --> 01:16:32.180 And now there's another remarkable thing 01:16:32.180 --> 01:16:35.330 is that if you can see the phi a, so the same thing applies 01:16:35.330 --> 01:16:39.370 for a alpha applies to any field. 01:16:39.370 --> 01:16:43.240 It's that 1q-- so this corresponds 01:16:43.240 --> 01:16:49.849 to a string with sigma equal to 0 at 1 and sigma pi. 01:16:49.849 --> 01:16:51.390 So this corresponds to a 1, 2 string. 01:16:53.940 --> 01:16:59.875 And then we can also think about the 2, 1 string. 01:17:02.650 --> 01:17:06.500 So it turns out that these two can 01:17:06.500 --> 01:17:20.009 be considered as complex conjugates of each other 01:17:20.009 --> 01:17:21.050 for the following reason. 01:17:28.820 --> 01:17:34.910 Again, now let me just again separate this 1 and 2 01:17:34.910 --> 01:17:37.360 to make it clear. 01:17:37.360 --> 01:17:40.350 So this is a 1, 2 string. 01:17:40.350 --> 01:17:41.903 So this is a 2, 1 string. 01:17:45.980 --> 01:17:54.639 So I claim string interactions defined by this way 01:17:54.639 --> 01:17:55.805 have the following symmetry. 01:17:58.410 --> 01:18:06.507 So string interactions described by joining the ends 01:18:06.507 --> 01:18:08.590 or splitting the ends have the following symmetry. 01:18:22.280 --> 01:18:34.258 It's that I can associate each brane by a phase factor. 01:18:38.700 --> 01:18:43.099 So I explained to you i theta I. So I labeled to the brane 01:18:43.099 --> 01:18:45.265 and the theta can be some different-- can be a phase 01:18:45.265 --> 01:18:47.840 factor. 01:18:47.840 --> 01:18:51.010 And now the rule is that if the sigma is 01:18:51.010 --> 01:18:58.875 equal to 0 and on that brane, then 01:18:58.875 --> 01:19:04.250 I multiply it by the exponential of i theta I. 01:19:04.250 --> 01:19:14.335 And the sigma equal to pi ends on that brane, ends on I. 01:19:14.335 --> 01:19:15.585 Let me write it more explicit. 01:19:15.585 --> 01:19:23.692 So if sigma 0 ending on I, then I multiply 01:19:23.692 --> 01:19:26.880 by a phase factor, exponential i theta I. 01:19:26.880 --> 01:19:31.390 And if the sigma equal to pi factor ending on I, 01:19:31.390 --> 01:19:40.020 then I multiply by exponential minus i theta I. 01:19:40.020 --> 01:19:41.530 So let's consider this operation. 01:19:41.530 --> 01:19:44.850 And I claim this operation is the symmetry of the string 01:19:44.850 --> 01:19:46.050 interaction. 01:19:46.050 --> 01:19:50.090 So let's first consider if you just have a single brane. 01:19:50.090 --> 01:19:54.290 So if you have a single brane like this, then just 01:19:54.290 --> 01:19:55.790 nothing changes because you multiply 01:19:55.790 --> 01:19:59.190 one end by i theta and the other by exponential minus i theta, 01:19:59.190 --> 01:20:02.260 does not change. 01:20:02.260 --> 01:20:06.470 But now, if you have such kind of interactions, 01:20:06.470 --> 01:20:14.750 because the sigma 0 and sigma pi ends join together, 01:20:14.750 --> 01:20:18.280 and they can only join if they are ending on the same brane, 01:20:18.280 --> 01:20:21.640 then those factors always cancel each other. 01:20:21.640 --> 01:20:25.340 And so this would be a symmetry. 01:20:25.340 --> 01:20:26.170 Is this clear? 01:20:28.980 --> 01:20:41.280 So under this operation, phi a ending on the same brane 01:20:41.280 --> 01:20:41.880 is invariant. 01:20:49.268 --> 01:20:55.508 And phi a I J then transforms by a phase factor theta 01:20:55.508 --> 01:21:05.290 I minus theta J of phi a I J. And the phi a J 01:21:05.290 --> 01:21:17.890 I transforms as a factor minus I theta I minus theta J. 01:21:17.890 --> 01:21:22.039 So we can actually think of them as complex conjugates. 01:21:22.039 --> 01:21:24.580 So they're transforming opposite way under this phase change. 01:21:24.580 --> 01:21:25.020 Yes? 01:21:25.020 --> 01:21:26.978 AUDIENCE: And this is because we're considering 01:21:26.978 --> 01:21:28.000 them to be u1 branes? 01:21:28.000 --> 01:21:28.500 HONG LIU: Sorry? 01:21:28.500 --> 01:21:30.916 AUDIENCE: Is this because we're considering them to be u1? 01:21:30.916 --> 01:21:32.620 HONG LIU: No. 01:21:32.620 --> 01:21:34.350 This is a good point. 01:21:34.350 --> 01:21:36.330 I will talk about this more. 01:21:36.330 --> 01:21:38.455 Right now, let's think about each brane separately. 01:21:42.264 --> 01:21:43.430 AUDIENCE: I have a question. 01:21:43.430 --> 01:21:48.310 In principle, can we write the interaction of open string 01:21:48.310 --> 01:21:52.154 with the same note, J, J by random number? 01:21:52.154 --> 01:21:52.820 HONG LIU: Sorry. 01:21:52.820 --> 01:21:54.660 What do you mean by random number? 01:21:54.660 --> 01:21:57.840 AUDIENCE: Say it's I, J, K, L. 01:21:57.840 --> 01:21:58.800 HONG LIU: No, no, no. 01:21:58.800 --> 01:21:59.490 The strings can only join together 01:21:59.490 --> 01:22:01.329 if they're ending on the same brane. 01:22:01.329 --> 01:22:01.870 AUDIENCE: Oh. 01:22:01.870 --> 01:22:02.422 OK. 01:22:02.422 --> 01:22:04.130 HONG LIU: So the J's have to be the same. 01:22:06.179 --> 01:22:07.970 So this guarantees that this will symmetry. 01:22:12.150 --> 01:22:14.910 Good? 01:22:14.910 --> 01:22:18.570 So in this sense, they're complex conjugates. 01:22:18.570 --> 01:22:21.160 And now we can build on this a little bit further. 01:22:21.160 --> 01:22:22.729 So this actually works. 01:22:22.729 --> 01:22:24.270 Doesn't matter whether the branes are 01:22:24.270 --> 01:22:26.870 coincident or not coincident. 01:22:26.870 --> 01:22:28.596 This is a generally true. 01:22:28.596 --> 01:22:29.970 Now let's consider all the branes 01:22:29.970 --> 01:22:31.261 are coincident with each other. 01:22:36.801 --> 01:22:57.660 So for coincidental branes, since branes 01:22:57.660 --> 01:23:12.960 are indistinguishable from each other, 01:23:12.960 --> 01:23:15.010 so they are higher dimensional generalization 01:23:15.010 --> 01:23:19.330 of what we call identical particles. 01:23:19.330 --> 01:23:25.050 So if they're indistinguishable from each, 01:23:25.050 --> 01:23:31.274 we can shuffle their indices. 01:23:31.274 --> 01:23:33.690 We should have the symmetry to reshuffle their symmetries. 01:23:43.260 --> 01:23:47.515 So whether we call this 1, 1 or this 1, 2 or this 1, 01:23:47.515 --> 01:23:48.565 1 should not matter. 01:23:53.230 --> 01:23:58.810 So if I combine these two facts, two observations together, 01:23:58.810 --> 01:24:03.720 then when we have n coincidental branes, 01:24:03.720 --> 01:24:06.664 then there's in fact u(n) symmetry. 01:24:18.100 --> 01:24:23.510 If that whole string interaction is invariant, say 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, 01:24:39.200 --> 01:24:49.150 L. So back here on U just corresponding to I 01:24:49.150 --> 01:24:50.710 reshuffle all the indices. 01:24:50.710 --> 01:24:53.880 So I have to do the same to U. So I reshuffle the two indices 01:24:53.880 --> 01:24:55.000 in the same way. 01:24:55.000 --> 01:24:57.214 But I have a star here. 01:24:57.214 --> 01:24:58.380 It's because of this reason. 01:24:58.380 --> 01:25:02.870 Because in some sense, the sigma 0 and sigma pi, 01:25:02.870 --> 01:25:07.390 they're only symmetries if I multiply opposite phase factor. 01:25:07.390 --> 01:25:10.950 I can rewrite this as a matrix notation. 01:25:10.950 --> 01:25:13.010 If you think about each side as a matrix, 01:25:13.010 --> 01:25:14.634 then this is the symmetry corresponding 01:25:14.634 --> 01:25:17.288 to psi U psi dagger. 01:25:21.120 --> 01:25:29.840 And the U can be arbitrary unitary matrices. 01:25:36.210 --> 01:25:40.610 So in fact, when you have n coincidental branes together, 01:25:40.610 --> 01:25:43.810 there's u(n) symmetries for the string interactions. 01:25:52.320 --> 01:26:02.670 So to say it in a more fancy mathematical language 01:26:02.670 --> 01:26:15.140 is that each open string excitation 01:26:15.140 --> 01:26:30.510 transforms under the adjunct representation of this u(n). 01:26:34.950 --> 01:26:36.650 So this is like a join representation. 01:26:36.650 --> 01:26:38.496 If you have u(n) symmetry, then this 01:26:38.496 --> 01:26:40.222 is like a join representation. 01:26:47.130 --> 01:26:57.256 So on the string worksheet, this is really a global symmetry. 01:27:10.380 --> 01:27:14.090 So it's just a phase factor associated with each n. 01:27:14.090 --> 01:27:15.730 There's nothing. 01:27:15.730 --> 01:27:17.416 It's a global symmetry. 01:27:22.260 --> 01:27:28.850 But the remarkable thing is that in the space time, 01:27:28.850 --> 01:27:32.220 this becomes a gauge symmetry. 01:27:32.220 --> 01:27:36.050 So this tells you, because of the presence of this u(n) gauge 01:27:36.050 --> 01:27:39.140 symmetry and because of each mode 01:27:39.140 --> 01:27:44.170 transforms under a join representation of some u(n), 01:27:44.170 --> 01:27:48.480 that as a space-time field, they also 01:27:48.480 --> 01:27:53.890 must transform under a join representation of some u(n). 01:27:53.890 --> 01:27:57.445 And interpreting the space-time, then this u(n) 01:27:57.445 --> 01:28:01.120 must be a gauge symmetry. 01:28:01.120 --> 01:28:03.820 On the worksheet, it's a global symmetry. 01:28:03.820 --> 01:28:15.342 But in space-time-- so let me just write it down. 01:28:15.342 --> 01:28:22.110 In space-time-- or in other words, 01:28:22.110 --> 01:28:31.720 in the world volume of D-branes, this u(n) 01:28:31.720 --> 01:28:46.230 must be a gauge symmetry for the following reason. 01:28:53.760 --> 01:29:01.480 Because the only way we know-- for example, 01:29:01.480 --> 01:29:05.910 this gauge field becomes transformed 01:29:05.910 --> 01:29:12.380 under a join representation of some u(n) symmetry. 01:29:17.160 --> 01:29:21.030 And all the excitations will have this symmetry. 01:29:21.030 --> 01:29:22.650 And then the only way we can make 01:29:22.650 --> 01:29:26.970 sense the gauge field under such symmetry 01:29:26.970 --> 01:29:31.210 is that this is a gauge symmetry and this is the gauge 01:29:31.210 --> 01:29:34.520 boson for that symmetry. 01:29:34.520 --> 01:29:37.780 Is it clear? 01:29:37.780 --> 01:29:43.571 So let me just say it in words. 01:29:43.571 --> 01:29:45.943 It must be a gauge symmetry. 01:29:45.943 --> 01:30:16.096 And in particular, a alpha I J must be the corresponding gauge 01:30:16.096 --> 01:30:16.596 bosons. 01:30:22.392 --> 01:30:27.590 Because this is the only way we know how to make sense, 01:30:27.590 --> 01:30:29.160 because this is the only way we know 01:30:29.160 --> 01:30:32.260 can happen at low energies. 01:30:32.260 --> 01:30:37.710 Some gauge fields transformed as a matrix interact 01:30:37.710 --> 01:30:39.180 with each other. 01:30:39.180 --> 01:30:41.315 And this can only be Yang-Mills theory. 01:30:41.315 --> 01:30:42.690 And if this is Yang-Mills theory, 01:30:42.690 --> 01:30:45.721 then this must be a gauge symmetry. 01:30:45.721 --> 01:30:46.970 So this is the basic argument. 01:30:49.690 --> 01:31:04.992 And at low energies, we must have Yang-Mills theory. 01:31:10.800 --> 01:31:12.185 So something remarkable happens. 01:31:17.830 --> 01:31:25.420 So each D-brane, when we separate them. 01:31:25.420 --> 01:31:27.590 is a Maxwell theory. 01:31:27.590 --> 01:31:30.820 When you put them together, they become Yang-Mills theory. 01:31:33.400 --> 01:31:35.060 And somehow, they become non-Rubinian. 01:31:39.480 --> 01:31:44.790 And everything comes from, in the very trivial way 01:31:44.790 --> 01:31:50.762 in string theory, you just count the indicies. 01:31:50.762 --> 01:31:52.470 But the physical implication is profound. 01:31:55.780 --> 01:31:57.680 And this can be confirmed, again, 01:31:57.680 --> 01:31:59.509 by just starting the explicit string 01:31:59.509 --> 01:32:00.675 theory scattering amplitude. 01:32:04.530 --> 01:32:10.007 You can calculate the scattering of 3a in string theory. 01:32:10.007 --> 01:32:12.590 Then you find it's precisely-- at low energy, precisely given, 01:32:12.590 --> 01:32:18.960 but the same vertex as the Yang-Mills theory, et cetera. 01:32:18.960 --> 01:32:20.100 So you can do that. 01:32:20.100 --> 01:32:22.184 Then you find the low energy effective action can 01:32:22.184 --> 01:32:23.350 be written as the following. 01:32:25.980 --> 01:32:34.210 Some Yang-Mills coupling trace. 01:32:34.210 --> 01:32:36.444 Let me just write down the Yang-Mills theory. 01:33:00.020 --> 01:33:03.160 You find the low energy effective action 01:33:03.160 --> 01:33:05.910 can be written in this form. 01:33:05.910 --> 01:33:08.577 So this is standard Yang-Mills field strings 01:33:08.577 --> 01:33:10.035 because now everything is a matrix. 01:33:15.262 --> 01:33:16.220 Everything is a matrix. 01:33:29.240 --> 01:33:35.040 And both A alpha and phi a now are embedded in matrices. 01:33:55.300 --> 01:34:01.400 So this g Yang-Mills is the Yang-Mills coupling 01:34:01.400 --> 01:34:03.620 which describes how the gauge fields interact 01:34:03.620 --> 01:34:05.160 with each other. 01:34:05.160 --> 01:34:11.350 And obviously, this should be related to the open string 01:34:11.350 --> 01:34:17.590 interaction because-- I think I have already erased it. 01:34:17.590 --> 01:34:22.110 This kind of interaction, joining the string. 01:34:25.120 --> 01:34:26.700 This corresponding to two strings 01:34:26.700 --> 01:34:30.260 joined into one string, and this proportional to g0. 01:34:30.260 --> 01:34:33.690 And this, from the field theory point of view, 01:34:33.690 --> 01:34:37.590 it just controls the interaction of the three A's. 01:34:37.590 --> 01:34:39.600 So that must be the Yang-Mills coupling. 01:34:39.600 --> 01:34:41.810 So this must be the Yang-Mills coupling. 01:34:41.810 --> 01:34:44.365 And then this should be related to gs 01:34:44.365 --> 01:34:50.480 to the power 1/2, which we just derived slightly earlier. 01:34:50.480 --> 01:34:59.770 So now, on dimensional ground, the g Yang-Mills square 01:34:59.770 --> 01:35:01.590 can be written as-- let me write it here. 01:35:01.590 --> 01:35:03.930 This is an important formula. 01:35:03.930 --> 01:35:08.420 On dimensional grounds, the g Yang-Mills square, 01:35:08.420 --> 01:35:11.930 I should be able to write it as gs. 01:35:11.930 --> 01:35:17.240 So on the Yang-Mills coupling, we 01:35:17.240 --> 01:35:21.160 will use the standard convention that A 01:35:21.160 --> 01:35:24.450 has the dimension of mass. 01:35:24.450 --> 01:35:27.552 And then this is a dimension 4 object. 01:35:27.552 --> 01:35:30.402 And this is a dimension p plus 1. 01:35:30.402 --> 01:35:31.860 And so the Yang-Mills coupling, you 01:35:31.860 --> 01:35:38.370 can deduce the dimension of the Yang-Mills coupling there. 01:35:38.370 --> 01:35:47.790 And it just turns out to be p minus 3 dense-- again, 01:35:47.790 --> 01:35:53.000 because alpha prime in string theory is only length scale. 01:35:53.000 --> 01:35:57.200 So the alpha prime must come from here. 01:35:57.200 --> 01:35:59.170 And then times some constant. 01:35:59.170 --> 01:36:01.248 So again, dp is just some numerical constant. 01:36:09.650 --> 01:36:16.730 And you can see explicitly that when p equal to 3, 01:36:16.730 --> 01:36:18.760 then you have d3 brane. 01:36:18.760 --> 01:36:22.175 Then the world volume theory is four dimensional. 01:36:22.175 --> 01:36:24.790 Then we recall that in four dimensions, 01:36:24.790 --> 01:36:26.940 Yang-Mills theory is dimensionless. 01:36:26.940 --> 01:36:29.370 QCD [INAUDIBLE]. 01:36:37.200 --> 01:36:38.170 Good? 01:36:38.170 --> 01:36:39.740 Any questions? 01:36:39.740 --> 01:36:40.240 Yes? 01:36:40.240 --> 01:36:42.829 AUDIENCE: What to do with phi h? 01:36:42.829 --> 01:36:43.370 HONG LIU: Hm? 01:36:43.370 --> 01:36:44.770 AUDIENCE: What to do with phi h? 01:36:44.770 --> 01:36:45.770 HONG LIU: What to do with phi h? 01:36:45.770 --> 01:36:47.394 What do you mean what to do with phi h? 01:36:47.394 --> 01:36:49.970 AUDIENCE: Scalar color particles? 01:36:49.970 --> 01:36:52.560 HONG LIU: Yeah. 01:36:52.560 --> 01:36:55.510 I forgot to mention here is the standard derivative, covariant 01:36:55.510 --> 01:36:57.130 derivative. 01:36:57.130 --> 01:37:02.130 So the alpha phi a is just the standard particle phi 01:37:02.130 --> 01:37:08.250 a minus i a alpha phi a. 01:37:08.250 --> 01:37:15.420 So it's the standard gauge covariant derivative. 01:37:15.420 --> 01:37:19.510 AUDIENCE: Shouldn't phi be more fundamental presentation of-- 01:37:19.510 --> 01:37:20.230 HONG LIU: No. 01:37:20.230 --> 01:37:22.910 Everything is to the join because everything 01:37:22.910 --> 01:37:24.111 has two ends. 01:37:24.111 --> 01:37:26.152 AUDIENCE: So how to interpret phi, then, in our-- 01:37:26.152 --> 01:37:27.010 HONG LIU: Hm? 01:37:27.010 --> 01:37:29.765 AUDIENCE: From the present zoology of particles, 01:37:29.765 --> 01:37:32.400 where to put phi, the observed particles? 01:37:32.400 --> 01:37:33.795 HONG LIU: Sorry? 01:37:33.795 --> 01:37:36.944 AUDIENCE: If a is just the gauge boson [INAUDIBLE] 01:37:36.944 --> 01:37:38.800 or something, what is phi? 01:37:38.800 --> 01:37:40.590 HONG LIU: Phi is some scalar field 01:37:40.590 --> 01:37:45.180 transformed on the join representation of the gauge 01:37:45.180 --> 01:37:45.680 group. 01:37:45.680 --> 01:37:48.690 It's a matter field describing the motion of the brane. 01:37:51.372 --> 01:37:52.720 AUDIENCE: I have a question. 01:37:52.720 --> 01:37:55.920 There you say that we can reshuffle their indexes. 01:37:55.920 --> 01:37:59.489 So that symmetry should be permutation symmetry? 01:37:59.489 --> 01:38:00.030 HONG LIU: No. 01:38:00.030 --> 01:38:01.113 AUDIENCE: Why [INAUDIBLE]? 01:38:05.690 --> 01:38:07.590 HONG LIU: But I reshuffle because I say it 01:38:07.590 --> 01:38:10.420 in the more heuristic way. 01:38:10.420 --> 01:38:15.580 But normally, if the indices is not-- these are the states. 01:38:15.580 --> 01:38:17.580 And you can just swivel-post them. 01:38:17.580 --> 01:38:20.010 A different i, they should be the same thing. 01:38:20.010 --> 01:38:22.426 They're just corresponding to relabeling, I'm just saying. 01:38:24.689 --> 01:38:26.980 The lateral action of anything on the state, of course, 01:38:26.980 --> 01:38:28.230 is the unitary transformation. 01:38:35.300 --> 01:38:39.206 AUDIENCE: Is there anything to do with that? 01:38:41.639 --> 01:38:42.180 HONG LIU: No. 01:38:42.180 --> 01:38:44.010 That thing is related to this star, 01:38:44.010 --> 01:38:45.300 why we put this star here. 01:38:50.500 --> 01:38:53.030 This is the reason why we put the star there, 01:38:53.030 --> 01:38:55.250 because the two endpoints, they should 01:38:55.250 --> 01:38:56.400 transform in opposite way. 01:38:59.630 --> 01:39:00.950 Yes? 01:39:00.950 --> 01:39:03.840 AUDIENCE: Since the branes become dynamical, 01:39:03.840 --> 01:39:06.490 can't they fluctuate in different ways 01:39:06.490 --> 01:39:09.494 and so no longer coincide? 01:39:09.494 --> 01:39:10.160 HONG LIU: Sorry? 01:39:10.160 --> 01:39:13.210 AUDIENCE: Since the branes are dynamical, 01:39:13.210 --> 01:39:15.680 you put them all in one place at first. 01:39:15.680 --> 01:39:18.870 But can't they now start fluctuating and separate? 01:39:18.870 --> 01:39:20.844 HONG LIU: They can certainly. 01:39:20.844 --> 01:39:23.330 AUDIENCE: They're no longer symmetric. 01:39:23.330 --> 01:39:24.900 They become distinguishable. 01:39:24.900 --> 01:39:26.622 HONG LIU: They can certainly start 01:39:26.622 --> 01:39:32.030 moving apart if you give them some initial motion. 01:39:32.030 --> 01:39:38.530 But the fluctuation, the system is isotropic. 01:39:38.530 --> 01:39:41.315 There's no place for them to-- they will fluctuate, 01:39:41.315 --> 01:39:43.190 but they will still at that point on average. 01:39:46.176 --> 01:39:47.800 There's no preferred direction for them 01:39:47.800 --> 01:39:50.730 to go unless you give them a direction. 01:39:50.730 --> 01:39:53.019 You say, I want them to go in that direction. 01:39:53.019 --> 01:39:53.810 Then you push them. 01:39:53.810 --> 01:39:55.185 AUDIENCE: But you can still think 01:39:55.185 --> 01:39:57.300 of them fluctuating kind of separately, 01:39:57.300 --> 01:39:59.090 separating in their fluctuations? 01:39:59.090 --> 01:40:00.500 HONG LIU: Sure. 01:40:00.500 --> 01:40:01.760 This a are their fluctuations. 01:40:01.760 --> 01:40:05.250 Phi are their fluctuations. 01:40:05.250 --> 01:40:07.950 AUDIENCE: Is this commutative term belonging to the effective 01:40:07.950 --> 01:40:11.735 action an interaction between the D-branes or is it-- 01:40:11.735 --> 01:40:12.360 HONG LIU: Yeah. 01:40:15.870 --> 01:40:17.850 Good point. 01:40:17.850 --> 01:40:21.940 Let me just comment on this term. 01:40:21.940 --> 01:40:28.160 So if you think about it, as we said before, 01:40:28.160 --> 01:40:33.490 a D-brane, no matter what dimension 01:40:33.490 --> 01:40:35.490 of the D-brane, the story when we quantize them, 01:40:35.490 --> 01:40:36.967 they're almost the same. 01:40:36.967 --> 01:40:37.925 It's the same spectrum. 01:40:42.120 --> 01:40:43.920 If you have a space-feeling brane 01:40:43.920 --> 01:40:46.156 that everything is A alpha, and then 01:40:46.156 --> 01:40:48.840 if you have some lower dimensional brane, some of them 01:40:48.840 --> 01:40:50.580 become scalar field, et cetera. 01:40:50.580 --> 01:40:54.710 So essentially, they all have the same dynamics. 01:40:54.710 --> 01:40:59.540 So this interaction can be considered just come from here. 01:40:59.540 --> 01:41:02.110 So you can start with a space time feeling brane, 01:41:02.110 --> 01:41:03.750 and then you go to lower dimensions. 01:41:03.750 --> 01:41:08.100 And then this just can be considered to come from there. 01:41:08.100 --> 01:41:10.720 It's part of the gauge theory. 01:41:10.720 --> 01:41:13.144 AUDIENCE: So the whole low energy actions 01:41:13.144 --> 01:41:17.016 are the action of supergravity plus this D-brane interaction, 01:41:17.016 --> 01:41:20.900 you can think of the whole low energy action? 01:41:20.900 --> 01:41:23.356 HONG LIU: You mean if you include the closed string? 01:41:23.356 --> 01:41:23.980 AUDIENCE: Yeah. 01:41:23.980 --> 01:41:24.722 HONG LIU: Yeah. 01:41:24.722 --> 01:41:26.180 That's right, if you have D-branes. 01:41:26.180 --> 01:41:26.950 That's right. 01:41:26.950 --> 01:41:30.270 This is effective action on the D-brane. 01:41:30.270 --> 01:41:32.860 This is the effective action on the D-brane. 01:41:32.860 --> 01:41:37.690 AUDIENCE: I mean if you extend D-brane [INAUDIBLE], 01:41:37.690 --> 01:41:41.426 then where does this action come from? 01:41:41.426 --> 01:41:43.664 HONG LIU: No, that's what you said. 01:41:43.664 --> 01:41:46.165 AUDIENCE: It's just two actions crossed together? 01:41:46.165 --> 01:41:46.790 HONG LIU: Yeah. 01:41:49.640 --> 01:41:50.140 Good? 01:41:53.469 --> 01:41:54.435 AUDIENCE: Excuse me. 01:41:54.435 --> 01:41:56.430 How come it has the term in action that's 01:41:56.430 --> 01:41:59.869 proportional to phi without derivative? 01:41:59.869 --> 01:42:00.410 HONG LIU: No. 01:42:00.410 --> 01:42:01.270 They have derivative. 01:42:01.270 --> 01:42:02.519 This is covariant derivatives. 01:42:02.519 --> 01:42:03.859 AUDIENCE: The commutator. 01:42:03.859 --> 01:42:05.900 HONG LIU: No, but phi have covariant derivatives. 01:42:05.900 --> 01:42:06.830 AUDIENCE: But the next time. 01:42:06.830 --> 01:42:07.454 HONG LIU: Yeah? 01:42:07.454 --> 01:42:10.692 AUDIENCE: That thing appears to be proportional to phi 01:42:10.692 --> 01:42:11.664 without derivative. 01:42:11.664 --> 01:42:13.122 HONG LIU: Yes? 01:42:13.122 --> 01:42:15.630 AUDIENCE: But this is the Minkowski space or something. 01:42:15.630 --> 01:42:16.369 HONG LIU: Yeah. 01:42:16.369 --> 01:42:17.410 That's a very good point. 01:42:17.410 --> 01:42:19.740 I'm going to mention that point. 01:42:19.740 --> 01:42:24.266 But the key is that this particular potential-- this 01:42:24.266 --> 01:42:25.140 is a very good point. 01:42:25.140 --> 01:42:27.394 I'm going to mention that. 01:42:27.394 --> 01:42:28.935 I will mention that in a few minutes. 01:42:31.910 --> 01:42:35.770 I do because I do want you to do your p-set. 01:42:48.750 --> 01:42:52.065 So now let's consider separating the branes. 01:42:55.599 --> 01:42:57.390 Now we will consider separating the branes. 01:43:10.782 --> 01:43:12.240 Again, let's consider the situation 01:43:12.240 --> 01:43:13.495 we just have two of them. 01:43:16.460 --> 01:43:18.680 So at the beginning, they're coincidental. 01:43:18.680 --> 01:43:21.690 And then now let's separate them in some direction. 01:43:21.690 --> 01:43:24.312 Let's call that direction x. 01:43:24.312 --> 01:43:26.270 So let's say they separate by some distance, d. 01:43:28.940 --> 01:43:30.748 This is 1, this is 2. 01:43:34.180 --> 01:43:36.690 So of course, this 1, 1 and 2, 2 string, nothing 01:43:36.690 --> 01:43:43.570 changes because they're just still ending on the same brane. 01:43:43.570 --> 01:43:45.129 But those strings are now different. 01:43:49.720 --> 01:43:51.690 So now 1, 2 and 2, 1 strings become different. 01:43:58.420 --> 01:44:01.750 So 1, 1, 2, 2, exactly the same as before. 01:44:11.070 --> 01:44:12.130 And the 1, 2 string. 01:44:12.130 --> 01:44:13.963 For example, let's consider the 1, 2 string. 01:44:16.440 --> 01:44:18.780 Then the boundary condition changes 01:44:18.780 --> 01:44:22.940 in the way I have sigma equal to 0, say, in the x direction. 01:44:22.940 --> 01:44:23.770 Sigma equal to 0. 01:44:26.692 --> 01:44:28.239 Tau, say, is at some location. 01:44:28.239 --> 01:44:29.030 Let's call this x0. 01:44:33.190 --> 01:44:39.640 And then x sigma equal to pi tau then becomes x0 plus d. 01:44:43.290 --> 01:44:46.480 So now it means when you contact this string, 01:44:46.480 --> 01:44:51.560 you have to do a slightly modified boundary condition. 01:44:51.560 --> 01:44:54.000 So you have start with x equal to 0. 01:44:54.000 --> 01:44:56.768 Now you must include a term depend on sigma. 01:45:08.140 --> 01:45:11.020 And again, as before, we take the xL 01:45:11.020 --> 01:45:14.080 to be minus xR and periodic, et cetera. 01:45:17.730 --> 01:45:20.295 So for sigma equal to 0, then this boundary condition 01:45:20.295 --> 01:45:21.570 is satisfied. 01:45:21.570 --> 01:45:24.410 But in order to satisfy the boundary condition at pi, 01:45:24.410 --> 01:45:28.130 then you now need this w equal to d divided by pi. 01:45:32.020 --> 01:45:36.490 You need to develop d divided by pi. 01:45:36.490 --> 01:45:37.770 So now, you have a sigma term. 01:45:37.770 --> 01:45:41.419 So that will change your [INAUDIBLE] condition. 01:45:41.419 --> 01:45:42.960 So remember the [INAUDIBLE] condition 01:45:42.960 --> 01:45:48.190 we had before is something like this, q alpha prime p minus, 01:45:48.190 --> 01:45:53.770 say, is p plus 4 pi alpha prime. 01:45:53.770 --> 01:45:54.770 Say for the open string. 01:45:54.770 --> 01:45:56.270 I'm writing the open string version, 01:45:56.270 --> 01:46:00.085 which is, up to some numerical factor, the same as closed 01:46:00.085 --> 01:46:01.948 string we wrote down before. 01:46:10.890 --> 01:46:15.440 So now, because of this term, then there's 01:46:15.440 --> 01:46:19.630 additional contribution on the right hand side. 01:46:19.630 --> 01:46:22.030 It makes sense because now string 01:46:22.030 --> 01:46:24.490 has to be stretched over some distance. 01:46:24.490 --> 01:46:26.580 That costs energy. 01:46:26.580 --> 01:46:30.630 So take one minute to do it yourselves. 01:46:30.630 --> 01:46:31.820 Just plug this into here. 01:46:31.820 --> 01:46:35.270 You only need to look at the behavior of this term. 01:46:35.270 --> 01:46:36.330 Just plug in there. 01:46:36.330 --> 01:46:40.170 Take you, say, five seconds. 01:46:40.170 --> 01:46:44.900 You find that the massless condition now 01:46:44.900 --> 01:46:54.720 has one more term plus the rest, as before. 01:46:54.720 --> 01:47:04.920 And that means now all the previously massless particle, 01:47:04.920 --> 01:47:08.315 this here with the corresponding a alpha and phi a, 01:47:08.315 --> 01:47:13.660 are no longer massless. 01:47:15.997 --> 01:47:17.580 Because previously, they were massless 01:47:17.580 --> 01:47:19.520 because those terms are 0. 01:47:19.520 --> 01:47:22.210 But now you have one more term. 01:47:22.210 --> 01:47:27.240 And they have a mass given by M divided 01:47:27.240 --> 01:47:31.070 by d divided by 2 alpha prime. 01:47:31.070 --> 01:47:33.790 Just take the square root of that. 01:47:33.790 --> 01:47:37.925 And because this is precisely the d times the string tension, 01:47:37.925 --> 01:47:40.280 because 1 over 2 pi alpha prime is the string tension. 01:47:40.280 --> 01:47:42.720 So this is exactly the energy we expect, just 01:47:42.720 --> 01:47:44.527 from a classical picture. 01:47:44.527 --> 01:47:46.360 You have a string stretched between the two. 01:47:46.360 --> 01:47:50.860 Then this is the tension times the length of the string. 01:47:59.310 --> 01:48:01.060 So now you can also easily understand 01:48:01.060 --> 01:48:05.170 what's going on from field theory point of view. 01:48:05.170 --> 01:48:11.930 And now you only have two sets of massless modes 01:48:11.930 --> 01:48:14.370 now rather than four. 01:48:14.370 --> 01:48:22.840 So what's happening is that now, the gauge symmetry is broken 01:48:22.840 --> 01:48:25.330 from u(2) to u(1) times u(1). 01:48:34.780 --> 01:48:45.210 So the separation of the branes essentially 01:48:45.210 --> 01:48:55.310 corresponding to the Higgs mechanism 01:48:55.310 --> 01:48:56.355 in the following sense. 01:49:01.160 --> 01:49:05.240 When you separate the brane, I said 01:49:05.240 --> 01:49:10.420 before phi should be interpreted as the [INAUDIBLE] 01:49:10.420 --> 01:49:12.120 location of the brane. 01:49:12.120 --> 01:49:13.630 But you separate the brane. 01:49:13.630 --> 01:49:15.720 That means one of the five fields corresponding 01:49:15.720 --> 01:49:18.307 to the x-- the x is in the transverse direction 01:49:18.307 --> 01:49:18.890 for the brane. 01:49:18.890 --> 01:49:21.889 One of the five fields corresponding to the x 01:49:21.889 --> 01:49:23.180 must develop expectation value. 01:49:26.970 --> 01:49:31.808 And that expectation value then gives rise to this mass. 01:49:31.808 --> 01:49:34.203 So this is precisely a Higgs mechanism. 01:49:38.040 --> 01:49:45.650 So now let's go back to the question 01:49:45.650 --> 01:49:50.310 what this potential term means, whether we can actually, 01:49:50.310 --> 01:49:53.910 as we said before, because of the translation symmetry, 01:49:53.910 --> 01:49:57.940 in principle, we can pull the brane anywhere. 01:49:57.940 --> 01:50:01.070 And now, I'm out of time. 01:50:01.070 --> 01:50:07.830 So let me just say, if you look at this potential, 01:50:07.830 --> 01:50:12.726 this becomes 0 precisely when phi a and phi b all 01:50:12.726 --> 01:50:14.220 become commutes. 01:50:17.590 --> 01:50:20.470 So that means we can diagonalize the phi corresponding to all 01:50:20.470 --> 01:50:23.429 the transverse directions. 01:50:23.429 --> 01:50:24.970 You can diagonalize phi corresponding 01:50:24.970 --> 01:50:27.234 to all the transverse directions. 01:50:27.234 --> 01:50:28.900 And then they correspond to the location 01:50:28.900 --> 01:50:31.780 you put all your branes. 01:50:31.780 --> 01:50:37.190 Go do your p-set, and you will see a more explicit discussion 01:50:37.190 --> 01:50:38.917 of this. 01:50:38.917 --> 01:50:40.000 And then one final remark. 01:50:42.700 --> 01:50:47.100 At the beginning, you started with n branes together. 01:50:47.100 --> 01:50:50.900 And then because of this, we can separate them. 01:50:50.900 --> 01:50:54.670 So we can find a solution which, say they commute. 01:50:54.670 --> 01:51:01.825 We separate them into different stacks, n1, n2, n3, et cetera. 01:51:05.410 --> 01:51:09.760 So this corresponds to the configuration of phi. 01:51:16.190 --> 01:51:19.340 So let me just write all phi as a vector. 01:51:19.340 --> 01:51:23.385 Say there's a1, n1 of them at location a1, 01:51:23.385 --> 01:51:27.820 and n2 of them at location a2, and ak. 01:51:31.580 --> 01:51:36.980 If I separate them into n stacks, then will be like this. 01:51:36.980 --> 01:51:42.710 So n1 of them at location a1, and n2 of them at location a2. 01:51:42.710 --> 01:51:45.090 You can check. 01:51:45.090 --> 01:51:53.206 So this is n1 times n1 identity matrix. 01:51:53.206 --> 01:51:56.565 This is n2 times n2. 01:51:56.565 --> 01:52:00.480 And you can check that such a configuration does satisfy 01:52:00.480 --> 01:52:03.290 this condition so that you actually 01:52:03.290 --> 01:52:05.890 can separate the brane into such configurations. 01:52:05.890 --> 01:52:07.830 And in this case, then the gauge symmetry 01:52:07.830 --> 01:52:14.320 is broken into u n1 times u n2 times 01:52:14.320 --> 01:52:18.620 u nk, because only those points [INAUDIBLE] parts survives. 01:52:21.390 --> 01:52:23.930 And all the other strings between them become massive. 01:52:23.930 --> 01:52:24.430 OK. 01:52:24.430 --> 01:52:26.312 Let's stop here.