WEBVTT 00:00:00.080 --> 00:00:02.430 The following content is provided under a Creative 00:00:02.430 --> 00:00:03.820 Commons license. 00:00:03.820 --> 00:00:06.060 Your support will help MIT OpenCourseWare 00:00:06.060 --> 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 --> 00:00:16.600 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.600 --> 00:00:17.255 at ocw.mit.edu. 00:00:22.220 --> 00:00:24.130 HONG LIU: OK, let's start. 00:00:24.130 --> 00:00:27.490 So first let me just remind you what we 00:00:27.490 --> 00:00:34.380 did at the end of last lecture. 00:00:34.380 --> 00:00:39.700 So we see that the large N expansion of gauge theory 00:00:39.700 --> 00:00:45.460 have essentially exactly the same mathematical structure 00:00:45.460 --> 00:00:50.980 with, say, the mathematics of the [? N string ?] scattering. 00:00:50.980 --> 00:00:54.250 And so here the observable is a correlation function 00:00:54.250 --> 00:00:57.020 of gauging [? invariant ?] operators. 00:00:57.020 --> 00:01:01.520 And then these have a large N expansion as follows. 00:01:01.520 --> 00:01:04.290 And on this side you have just an N string scattering 00:01:04.290 --> 00:01:04.930 amplitude. 00:01:04.930 --> 00:01:09.800 Just imagine you have some kind of scattering of strings, 00:01:09.800 --> 00:01:12.036 with total number of N strings. 00:01:12.036 --> 00:01:13.410 And then this also have expansion 00:01:13.410 --> 00:01:18.460 in terms of the string counting in this form. 00:01:18.460 --> 00:01:27.330 So now, if we identify-- so if we can identify the g 00:01:27.330 --> 00:01:32.810 string as 1/N. So if we identify g string with 1/N, 00:01:32.810 --> 00:01:37.040 then these two are essentially the same kind of expansion, OK? 00:01:37.040 --> 00:01:45.050 And you also can identify these external strings, 00:01:45.050 --> 00:01:51.230 string states, within the large N 00:01:51.230 --> 00:01:56.285 theory which we called the glueball states 00:01:56.285 --> 00:01:57.410 for single-trace operators. 00:02:05.150 --> 00:02:07.715 And then each case is corresponding to [? sum ?] 00:02:07.715 --> 00:02:08.465 over the topology. 00:02:11.830 --> 00:02:14.120 It's an expansion [? in ?] terms of the topology. 00:02:14.120 --> 00:02:29.516 So here is the topology of the worldsheet string. 00:02:29.516 --> 00:02:31.390 And here is the topology of Feynman diagrams. 00:02:43.010 --> 00:02:45.160 Here is the topology of the Feynman diagrams. 00:02:49.800 --> 00:02:53.690 So still at this stage, it's just 00:02:53.690 --> 00:02:56.359 like a mathematical correspondence. 00:02:56.359 --> 00:02:58.400 We're looking at two completely different things. 00:03:01.370 --> 00:03:06.940 But probably there's no-- yeah, no obvious connection 00:03:06.940 --> 00:03:11.060 between these two objects we are discussing. 00:03:14.446 --> 00:03:18.740 Yeah, we just have a precise mathematical structure. 00:03:18.740 --> 00:03:26.760 But one can actually argue that, actually, they also describe 00:03:26.760 --> 00:03:32.025 the same physical structure once you realize that when 00:03:32.025 --> 00:03:35.740 you sum over all possible Feynman diagrams. 00:03:35.740 --> 00:03:46.830 So once you realize that each Feynman diagram, say, 00:03:46.830 --> 00:04:02.880 of genus-h can be considered as a partition, 00:04:02.880 --> 00:04:10.340 or in other words, triangulization 00:04:10.340 --> 00:04:19.055 over genus-h surfaces, [? 2D ?] surfaces. 00:04:19.055 --> 00:04:19.555 OK. 00:04:26.920 --> 00:04:35.810 So if you write more explicitly this fh, 00:04:35.810 --> 00:04:45.787 so if we write explicitly this fh, then this fh, this fnh, 00:04:45.787 --> 00:04:47.370 then will be corresponding to your sum 00:04:47.370 --> 00:04:58.290 of all Feynman diagrams of genus-h. 00:05:06.170 --> 00:05:12.425 Suppose G is the expression for each Feynman diagram. 00:05:16.596 --> 00:05:17.470 Say for each diagram. 00:05:26.780 --> 00:05:31.220 And then I can just rewrite this. 00:05:31.220 --> 00:05:46.170 In some sense, I [? accept ?] all possible triangulation 00:05:46.170 --> 00:05:50.140 of [? a genus-g ?] surface. 00:05:56.620 --> 00:06:06.355 Say there will be some weight G. And summing over 00:06:06.355 --> 00:06:12.030 all possible triangulations of a surface 00:06:12.030 --> 00:06:16.720 is essentially-- so this is essentially 00:06:16.720 --> 00:06:29.180 the same as this sum over all possible surfaces. 00:06:34.980 --> 00:06:38.290 So this is a discrete version. 00:06:46.880 --> 00:06:50.150 So sum all possible triangulations 00:06:50.150 --> 00:06:54.365 of some genus-g surfaces, or translations 00:06:54.365 --> 00:06:58.107 of genus-g surfaces. 00:06:58.107 --> 00:07:00.440 Then they can be considered as a discrete version of sum 00:07:00.440 --> 00:07:04.112 over all possible surfaces, OK? 00:07:04.112 --> 00:07:06.070 AUDIENCE: So you're saying it's like a sum over 00:07:06.070 --> 00:07:08.000 [? syntheses, ?] like a simple [? x? ?] 00:07:08.000 --> 00:07:09.120 HONG LIU: Exactly. 00:07:09.120 --> 00:07:11.350 Exactly. 00:07:11.350 --> 00:07:21.080 Yeah, because, say, imagine when you sum over surfaces, 00:07:21.080 --> 00:07:23.590 so you sum over all possible metric. 00:07:23.590 --> 00:07:25.330 You can put [INAUDIBLE]. 00:07:25.330 --> 00:07:27.440 And that's the same way as you sum 00:07:27.440 --> 00:07:32.270 over different discretizations of that surface 00:07:32.270 --> 00:07:37.260 once you have defined the unit for that discretization. 00:07:37.260 --> 00:07:43.140 So if we can identify-- so for now record this Fh. 00:07:46.440 --> 00:07:58.060 So this Fh, this Fnh is the path integral 00:07:58.060 --> 00:08:06.584 over all genus-h surfaces with some string action, 00:08:06.584 --> 00:08:07.875 weighted by some string action. 00:08:18.050 --> 00:08:35.545 So if we can, say, identify this G 00:08:35.545 --> 00:08:47.710 with some string action-- the exponential of some string 00:08:47.710 --> 00:08:48.210 action. 00:08:53.050 --> 00:08:55.980 Then we would have-- then one can 00:08:55.980 --> 00:09:08.580 conclude that large N gauge theory is just a string 00:09:08.580 --> 00:09:09.750 theory, OK? 00:09:15.772 --> 00:09:17.855 That large N gauge theory is just a string theory, 00:09:17.855 --> 00:09:19.015 if you can do that. 00:09:22.510 --> 00:09:30.130 In particular, the large N limits-- 00:09:30.130 --> 00:09:34.690 so large N limit here, as we discussed before, 00:09:34.690 --> 00:09:38.590 can considered as a classical theory of glueballs. 00:09:41.300 --> 00:09:44.610 Or a classical theory of the single-trace operators. 00:09:44.610 --> 00:09:47.670 So this would be matched to the classical string theory. 00:09:55.590 --> 00:09:58.610 So as we mentioned last time, so I was mentioning before, 00:09:58.610 --> 00:10:07.090 this expression-- so just as in the as we discussed 00:10:07.090 --> 00:10:09.037 [INAUDIBLE], the [? expansion ?] in g 00:10:09.037 --> 00:10:14.850 string the same as expansion in the topology. 00:10:14.850 --> 00:10:16.690 And the expansion in the topology 00:10:16.690 --> 00:10:18.380 can also be considered as the expansion 00:10:18.380 --> 00:10:20.300 of the groups of a string. 00:10:20.300 --> 00:10:24.020 Because whenever you add a hole to the genus-- 00:10:24.020 --> 00:10:26.650 when you add the genus, and you actually add the string hole, 00:10:26.650 --> 00:10:28.655 you add the string loop diagram. 00:10:33.050 --> 00:10:35.550 So in this sense, you can [? integrate ?] all these 00:10:35.550 --> 00:10:39.300 higher order corrections, as the quantum 00:10:39.300 --> 00:10:43.337 correction to this classical string behavior. 00:10:43.337 --> 00:10:45.420 So this is just a tree-level amplitude for string. 00:10:45.420 --> 00:10:48.340 And this [? goes ?] one into the loops. 00:10:48.340 --> 00:10:50.290 Whenever you add this thing, you add the loop. 00:10:50.290 --> 00:10:50.790 OK. 00:10:50.790 --> 00:10:52.002 Is this clear? 00:10:52.002 --> 00:10:53.975 Now, remember what we discussed for the torus. 00:10:57.120 --> 00:10:59.940 If you've got a torus, then correspondingly 00:10:59.940 --> 00:11:02.960 you have a string split and joined together. 00:11:02.960 --> 00:11:07.280 And this split and join process you can also 00:11:07.280 --> 00:11:12.250 consider as a string loop, a single string going 00:11:12.250 --> 00:11:17.075 around a loop, [? just like ?] in the particle case, OK? 00:11:17.075 --> 00:11:19.790 In the standard field theory case. 00:11:19.790 --> 00:11:28.450 And so the large N limit, which is the leading order term here, 00:11:28.450 --> 00:11:31.034 would map to a leading order in the string scattering. 00:11:31.034 --> 00:11:33.450 And the leading order in the string scattering-- they only 00:11:33.450 --> 00:11:35.860 consider tree-level [? skin ?] scatterings, 00:11:35.860 --> 00:11:40.710 and then corresponding to classical string theory. 00:11:40.710 --> 00:11:46.830 And also the single-trace operator 00:11:46.830 --> 00:11:53.685 here can be mapped to the string states. 00:11:53.685 --> 00:11:55.577 Yeah, can be mapped to the string states. 00:12:08.010 --> 00:12:14.780 But this is only-- this is a very nice picture. 00:12:14.780 --> 00:12:18.410 But for many years, this was just a dream. 00:12:18.410 --> 00:12:25.840 And because this guy looks very different from this guy, 00:12:25.840 --> 00:12:29.310 but this is difficult. So this has 00:12:29.310 --> 00:12:44.670 some [? identification is ?] difficult for the following 00:12:44.670 --> 00:12:45.170 reasons. 00:12:52.020 --> 00:12:58.680 So first, so this G just-- say your Feynman diagrams, 00:12:58.680 --> 00:13:01.380 amplitude for particular Feynman diagram. 00:13:01.380 --> 00:13:14.220 So G is typically expressed as product 00:13:14.220 --> 00:13:19.300 of field theory propagators. 00:13:24.280 --> 00:13:27.645 So imagine how you evaluate the Feynman diagram. 00:13:27.645 --> 00:13:29.020 The Feynman diagram, essentially, 00:13:29.020 --> 00:13:32.540 is just a product of the [? propagators. ?] 00:13:32.540 --> 00:13:38.657 And then you integrate it [INAUDIBLE] 00:13:38.657 --> 00:13:39.740 integrated over spacetime. 00:13:46.430 --> 00:13:49.200 So they just take the Yang-Mills theory. 00:13:49.200 --> 00:13:55.540 And if you look at the expression for this diagram, 00:13:55.540 --> 00:13:58.920 of course, it looks nothing. 00:13:58.920 --> 00:14:15.820 So they look nothing like-- OK. 00:14:23.000 --> 00:14:26.840 So let me make a few comments about this thing. 00:14:26.840 --> 00:14:30.270 Because if you want to match, say if I gave you a Yang-Mills 00:14:30.270 --> 00:14:33.712 theory, so I gave you a QCD, then you can write down-- then 00:14:33.712 --> 00:14:35.920 you can go to large N. You can write down expressions 00:14:35.920 --> 00:14:38.304 for the common diagrams. 00:14:38.304 --> 00:14:39.720 But if you say, I want to write it 00:14:39.720 --> 00:14:44.720 as a string theory, the first thing you have to say, 00:14:44.720 --> 00:14:46.710 what string theory do you want to compare? 00:14:51.850 --> 00:14:54.070 So first you have to ask yourself 00:14:54.070 --> 00:14:57.350 what string action do you want to compare. 00:14:57.350 --> 00:15:06.480 So the string action, as we discussed last time, 00:15:06.480 --> 00:15:11.000 this describes the embedding of the worldsheet 00:15:11.000 --> 00:15:13.430 into some spacetime. 00:15:13.430 --> 00:15:21.600 OK, so this is worldsheet into a spacetime. 00:15:27.510 --> 00:15:30.780 So this is also sometimes called the target space. 00:15:34.730 --> 00:15:36.290 So this is a spacetime. 00:15:36.290 --> 00:15:38.590 This string moves. 00:15:38.590 --> 00:15:41.240 And the mathematical of this is just the-- 00:15:41.240 --> 00:15:47.640 this is encoded in this mapping X mu sigma tau. 00:15:47.640 --> 00:15:50.350 OK, X mu is the coordinate for M. 00:15:50.350 --> 00:15:51.925 And then sigma tau is the coordinate 00:15:51.925 --> 00:15:55.470 as you parameterize your worldsheet. 00:15:55.470 --> 00:15:59.080 So in order to write down action, of course, 00:15:59.080 --> 00:16:06.354 you have to choice of space manifold. 00:16:06.354 --> 00:16:07.770 You have to choose your spacetime. 00:16:11.250 --> 00:16:14.890 And also you have to-- when you fix the spacetime, 00:16:14.890 --> 00:16:18.710 you don't have a choice. 00:16:18.710 --> 00:16:23.435 And sometimes the way to write down such kind of embedding 00:16:23.435 --> 00:16:25.210 is not unique. 00:16:25.210 --> 00:16:28.010 The action for such [? finding ?] is unique, 00:16:28.010 --> 00:16:35.220 so you only need to choose what action you include. 00:16:35.220 --> 00:16:41.700 And also often, in addition to this embedding, 00:16:41.700 --> 00:16:51.560 sometimes you can have additional internal degrees 00:16:51.560 --> 00:16:52.060 freedom. 00:16:58.505 --> 00:16:59.380 living on worldsheet. 00:17:04.160 --> 00:17:08.790 For example, you can have some fermions. 00:17:08.790 --> 00:17:10.449 Say if you have a superstring, then 00:17:10.449 --> 00:17:12.630 you can have some additional fermions 00:17:12.630 --> 00:17:17.389 are living on the worldsheet, in addition to this embedding. 00:17:22.680 --> 00:17:30.100 So in other words, the choice of this guy in some sense 00:17:30.100 --> 00:17:32.410 is infinite. 00:17:32.410 --> 00:17:35.845 And without any clue-- so you need 00:17:35.845 --> 00:17:39.230 some clue to know what to compare the gauge theory to. 00:17:39.230 --> 00:17:43.740 And otherwise, even if this works, 00:17:43.740 --> 00:17:46.590 you're searching for some needle in the big ocean. 00:17:53.040 --> 00:17:57.250 And then there's another very important reason 00:17:57.250 --> 00:18:09.605 why this is difficult, is that this string theory 00:18:09.605 --> 00:18:11.140 is formulated in a continuum. 00:18:19.804 --> 00:18:21.095 It's formulated in a continuum. 00:18:24.080 --> 00:18:32.800 And these Feynman diagrams, even if they're 00:18:32.800 --> 00:18:36.160 corresponding to some kind of string theory, 00:18:36.160 --> 00:18:38.260 they correspond into a discrete version of that. 00:18:43.820 --> 00:18:45.875 So at best, it's a discrete version. 00:18:54.170 --> 00:19:11.810 So we expect such a geometric picture for G, 00:19:11.810 --> 00:19:20.840 for these Feynman diagrams, to emerge only 00:19:20.840 --> 00:19:21.990 at strong couplings. 00:19:29.320 --> 00:19:30.545 OK? 00:19:30.545 --> 00:19:32.920 Emerge only at strong couplings for the following reason. 00:19:36.710 --> 00:19:40.520 So if you look at the Feynman diagram-- 00:19:40.520 --> 00:19:43.505 so the simplest Feynman diagram we draw before, 00:19:43.505 --> 00:19:46.585 say for example just this diagram. 00:19:49.220 --> 00:19:52.420 And if you draw it on the sphere, 00:19:52.420 --> 00:19:58.380 it separated the sphere into three parts, OK? 00:19:58.380 --> 00:20:02.140 So this [? discretizes ?] a sphere into three parts. 00:20:02.140 --> 00:20:04.210 And essentially, just as the sphere just becomes 00:20:04.210 --> 00:20:06.126 three points, because each particle is wanting 00:20:06.126 --> 00:20:09.290 to-- when you're trying to [INAUDIBLE] each part, 00:20:09.290 --> 00:20:11.130 you approximate it by one point. 00:20:11.130 --> 00:20:14.410 So essentially, in this diagram, you 00:20:14.410 --> 00:20:17.220 approximate the whole sphere essentially by three points. 00:20:17.220 --> 00:20:17.720 OK. 00:20:20.177 --> 00:20:22.510 And of course, it's hard to see your [? magic ?] picture 00:20:22.510 --> 00:20:23.980 from here. 00:20:23.980 --> 00:20:27.212 And your [? magic picture ?] you expect to emerge, 00:20:27.212 --> 00:20:30.350 but your Feynman diagrams become very complicated. 00:20:30.350 --> 00:20:34.096 For example, if you have this kind of diagram, 00:20:34.096 --> 00:20:35.470 because of the four-point vertex. 00:20:35.470 --> 00:20:37.445 In principle, you can have all these diagrams. 00:20:37.445 --> 00:20:39.070 And then this [INAUDIBLE] [? wanting ?] 00:20:39.070 --> 00:20:41.790 to discretize-- yes, I suppose this is on the torus. 00:20:44.380 --> 00:20:46.580 Suppose you have a-- for example, 00:20:46.580 --> 00:20:50.074 this could be a Feynman diagram on the torus, OK? 00:20:50.074 --> 00:20:51.240 For the vacuum [? energy. ?] 00:20:54.860 --> 00:20:59.150 And now this is next some kind of proper discretization. 00:20:59.150 --> 00:21:01.610 And this will go to a continuum limit, 00:21:01.610 --> 00:21:04.587 say when the number of these box go to infinity. 00:21:04.587 --> 00:21:06.170 When the number of box go to infinity, 00:21:06.170 --> 00:21:08.390 then you need a number of propagators, 00:21:08.390 --> 00:21:11.770 and the number of vertices goes to infinity, OK? 00:21:11.770 --> 00:21:14.560 So in order for continuum, a picture 00:21:14.560 --> 00:21:17.920 to emerge, so you want those complicated 00:21:17.920 --> 00:21:20.960 diagrams-- it's not your number of vertices or large number 00:21:20.960 --> 00:21:24.012 of propagators that dominate. 00:21:24.012 --> 00:21:25.470 And for those things that dominate, 00:21:25.470 --> 00:21:27.240 then you need the strong coupling. 00:21:27.240 --> 00:21:31.030 Because with this coupling, this is the leading order diagram. 00:21:31.030 --> 00:21:33.750 And there's no geometry from here, OK? 00:21:33.750 --> 00:21:36.150 So in order to have the geometry, 00:21:36.150 --> 00:21:39.130 you want the diagram are very, very complicated, 00:21:39.130 --> 00:21:42.470 so that they really-- [INAUDIBLE] 00:21:42.470 --> 00:21:45.510 a triangulation of a surface. 00:21:45.510 --> 00:21:48.670 A weak coupled diagram with small number of lines will 00:21:48.670 --> 00:21:54.940 cause [? one ?] [? and two ?] are very close triangulization 00:21:54.940 --> 00:21:56.970 of a surface. 00:21:56.970 --> 00:22:01.050 So we expect this only appears in strong couplings, OK? 00:22:08.010 --> 00:22:08.690 Yeah. 00:22:08.690 --> 00:22:12.340 AUDIENCE: By the cases like we have to sum over all the 00:22:12.340 --> 00:22:13.020 [INAUDIBLE]. 00:22:13.020 --> 00:22:15.200 HONG LIU: Yeah, sum over the [INAUDIBLE] diagram. 00:22:15.200 --> 00:22:16.220 AUDIENCE: Including those simple ones. 00:22:16.220 --> 00:22:17.803 HONG LIU: Including those simple ones. 00:22:17.803 --> 00:22:20.620 So that's why you want to-- so if you're 00:22:20.620 --> 00:22:28.390 in a weak coupling, then the simple ones-- 00:22:28.390 --> 00:22:29.615 so we sum all those diagrams. 00:22:29.615 --> 00:22:31.698 And each diagram you can associate with a coupling 00:22:31.698 --> 00:22:32.970 power. 00:22:32.970 --> 00:22:36.120 So at weak coupling, then the lowest order term 00:22:36.120 --> 00:22:37.690 would just dominate. 00:22:37.690 --> 00:22:40.670 And the lowest order term have a very simple diagrams. 00:22:40.670 --> 00:22:42.794 And then that's because [? one ?] and [? two ?] are 00:22:42.794 --> 00:22:46.050 very crude triangulization over the surface. 00:22:46.050 --> 00:22:50.150 But if you have a strong coupling-- in particular, 00:22:50.150 --> 00:22:53.650 if you have an infinite coupling-- the diagrams, 00:22:53.650 --> 00:22:55.914 the infinite number of vertices will dominate. 00:22:55.914 --> 00:22:58.080 And then that's because [? one ?] and [? two ?] have 00:22:58.080 --> 00:23:00.110 very fine triangulization over the surface. 00:23:00.110 --> 00:23:02.334 And then that can go to the [INAUDIBLE]. 00:23:02.334 --> 00:23:05.738 AUDIENCE: [INAUDIBLE] interaction a coupling constant 00:23:05.738 --> 00:23:08.529 has been [? dragging ?] out from-- 00:23:08.529 --> 00:23:09.070 HONG LIU: No. 00:23:09.070 --> 00:23:10.397 That's just N dragged out. 00:23:10.397 --> 00:23:11.230 AUDIENCE: Oh, I see. 00:23:11.230 --> 00:23:13.355 HONG LIU: No, there's what we call this [INAUDIBLE] 00:23:13.355 --> 00:23:14.950 still remaining. 00:23:14.950 --> 00:23:16.860 By coupling, it's only [? N. ?] 00:23:16.860 --> 00:23:19.800 AUDIENCE: [INAUDIBLE] 00:23:19.800 --> 00:23:22.639 HONG LIU: No, no, this isn't to [? hold ?] coupling. 00:23:22.639 --> 00:23:24.180 In coupling we mean that [INAUDIBLE]. 00:23:24.180 --> 00:23:26.555 So example we talk about, [? because one ?] [? and two ?] 00:23:26.555 --> 00:23:27.200 [INAUDIBLE]. 00:23:27.200 --> 00:23:30.870 Yeah, and then we make more precise. 00:23:30.870 --> 00:23:33.820 So in the [? toy ?] example we talked about before. 00:23:33.820 --> 00:23:36.870 So previously we talked about this example, 00:23:36.870 --> 00:23:44.410 N divided by lambda, trace, say 1/2 partial phi squared, 00:23:44.410 --> 00:23:49.210 plus 1/4 phi to the power 4. 00:23:49.210 --> 00:23:51.300 And strong coupling means the lambda large. 00:23:54.724 --> 00:23:58.360 Because of the N I've already factored out, 00:23:58.360 --> 00:23:59.890 so you're coupling just lambda. 00:23:59.890 --> 00:24:01.255 AUDIENCE: Oh, I see. 00:24:04.900 --> 00:24:05.740 HONG LIU: Yes. 00:24:05.740 --> 00:24:11.108 AUDIENCE: So in these [INAUDIBLE] 00:24:11.108 --> 00:24:14.524 the propagator in that version would become 00:24:14.524 --> 00:24:16.964 the spacetime integration? 00:24:16.964 --> 00:24:17.940 HONG LIU: Hm? 00:24:17.940 --> 00:24:19.892 AUDIENCE: I was just wondering how 00:24:19.892 --> 00:24:23.308 the propagator can [? agree, ?] can match 00:24:23.308 --> 00:24:24.780 to the spacetime [INAUDIBLE]. 00:24:24.780 --> 00:24:25.654 HONG LIU: Yeah, yeah. 00:24:31.290 --> 00:24:34.660 So the propogator-- yeah, propagator you 00:24:34.660 --> 00:24:35.760 do in the standard way. 00:24:35.760 --> 00:24:38.230 You just write down your propagator, 00:24:38.230 --> 00:24:40.350 and then you try to repackage that. 00:24:40.350 --> 00:24:44.210 As the question, you said, whatever your rule, 00:24:44.210 --> 00:24:46.592 Feynman rule is we just do that Feynman rule. 00:24:46.592 --> 00:24:48.050 And you write down this expression. 00:24:48.050 --> 00:24:49.970 It's something very complicated. 00:24:49.970 --> 00:24:52.390 And then you say, can I find some geometric interpretation 00:24:52.390 --> 00:24:54.120 of that? 00:24:54.120 --> 00:24:58.230 Yeah, what I'm saying is that doing from this perspective 00:24:58.230 --> 00:25:05.140 is very hard because you don't know what thing to compare. 00:25:05.140 --> 00:25:09.620 And further, in the second, you expect 00:25:09.620 --> 00:25:11.540 that your [INAUDIBLE] would emerge only 00:25:11.540 --> 00:25:13.810 in those very complicated diagrams. 00:25:13.810 --> 00:25:18.040 And those complicated diagrams we don't know how to deal with. 00:25:18.040 --> 00:25:20.315 Because they only emerge in the strong coupling limit, 00:25:20.315 --> 00:25:21.690 but in the strong coupling limit, 00:25:21.690 --> 00:25:24.430 we don't know how to deal with that. 00:25:24.430 --> 00:25:27.720 And so that's why it's also difficult. 00:25:27.720 --> 00:25:32.605 But [? nevertheless, ?] for some very simple theories, say, 00:25:32.605 --> 00:25:34.720 if you don't consider the Yang-Mills theory, 00:25:34.720 --> 00:25:36.220 you don't consider the gauge theory. 00:25:36.220 --> 00:25:40.180 But suppose you do consider some matrix integrals. 00:25:40.180 --> 00:25:55.705 Say, for very simple systems, like a matrix integral. 00:26:02.170 --> 00:26:06.170 So this structure emphasizes-- this structure only 00:26:06.170 --> 00:26:10.130 have to do with you have a matrices, OK? 00:26:10.130 --> 00:26:13.730 And then you can have matrix-valued fields [? or ?] 00:26:13.730 --> 00:26:15.960 this structure will emerge. 00:26:15.960 --> 00:26:18.191 Or you only have a matrix integral. 00:26:18.191 --> 00:26:20.440 So there no field at all, just have a matrix integral. 00:26:20.440 --> 00:26:23.070 That same structure will also emerge. 00:26:23.070 --> 00:26:23.970 For example. 00:26:23.970 --> 00:26:29.950 I can consider theory-- have a theory like this. 00:26:39.270 --> 00:26:40.210 Something like this. 00:26:43.880 --> 00:26:46.620 And have a theory like this, OK? 00:26:46.620 --> 00:26:50.210 And M is just some [INAUDIBLE] matrices. 00:26:50.210 --> 00:26:52.012 So this is just integral. 00:26:52.012 --> 00:26:53.720 And the same structure will emerge, also, 00:26:53.720 --> 00:26:56.270 in this series when we do large N expansion. 00:26:59.510 --> 00:27:02.740 So that structure have nothing to do-- yeah, you can do it. 00:27:05.356 --> 00:27:07.730 So matrix integral is much simpler than [INAUDIBLE] field 00:27:07.730 --> 00:27:10.980 theory because you have much less degrees freedom. 00:27:10.980 --> 00:27:14.960 So for simple systems like, say, your matrix integral or matrix 00:27:14.960 --> 00:27:24.540 quantum mechanics, actually, you can 00:27:24.540 --> 00:27:27.630 guess the corresponding string theory. 00:27:36.999 --> 00:27:38.790 Because also the string theory in that case 00:27:38.790 --> 00:27:40.710 is also very simple. 00:27:40.710 --> 00:27:44.160 You can guess where is simple string theory. 00:27:44.160 --> 00:27:47.460 But it's not possible for field theory. 00:27:47.460 --> 00:27:48.961 It's not possible for field theory. 00:27:48.961 --> 00:27:49.460 Yes. 00:27:49.460 --> 00:27:53.005 AUDIENCE: So what do you mean by matrix quantum mechanics? 00:27:53.005 --> 00:27:54.400 Like that, OK. 00:27:54.400 --> 00:27:57.670 HONG LIU: So this is a matrix integral. 00:27:57.670 --> 00:27:59.880 And I can make it a little bit more complicated. 00:27:59.880 --> 00:28:03.839 So I make this M to depend on t, and then this 00:28:03.839 --> 00:28:05.255 become a matrix quantum mechanics. 00:28:08.560 --> 00:28:15.767 Say trace M dot squared plus M squared plus M4. 00:28:15.767 --> 00:28:17.600 Then this become a matrix quantum mechanics, 00:28:17.600 --> 00:28:20.510 because it only have time. 00:28:20.510 --> 00:28:22.740 And then I can make it more complicated. 00:28:22.740 --> 00:28:26.310 I can make M be t, x. 00:28:26.310 --> 00:28:29.130 Then this becomes one plus one dimension of field theory. 00:28:29.130 --> 00:28:32.020 AUDIENCE: So in what context is this matrix quantum mechanics 00:28:32.020 --> 00:28:33.540 [? conflicted? ?] 00:28:33.540 --> 00:28:35.550 HONG LIU: Just at some [? toy ?] model. 00:28:35.550 --> 00:28:41.570 I just say, and this is a very difficult question. 00:28:41.570 --> 00:28:44.690 You said, I don't know how to deal with field theories. 00:28:44.690 --> 00:28:46.780 Then this [? part of it's ?] a simple system. 00:28:46.780 --> 00:28:48.970 And then just try to use this philosophy, 00:28:48.970 --> 00:28:51.450 can see whether it can do it for simple system. 00:28:51.450 --> 00:28:54.510 And then you can show that this philosophy actually 00:28:54.510 --> 00:28:58.350 works if you do a matrix integral or matrix quantum 00:28:58.350 --> 00:28:59.980 mechanics. 00:28:59.980 --> 00:29:03.840 Simple enough, matrix integral and matrix quantum mechanics. 00:29:03.840 --> 00:29:06.110 OK. 00:29:06.110 --> 00:29:07.860 And if you want references, I can give you 00:29:07.860 --> 00:29:10.180 references regarding these. 00:29:10.180 --> 00:29:13.230 There's a huge, huge amount of works, thousands 00:29:13.230 --> 00:29:19.400 of papers, written on this subject in the late '80s 00:29:19.400 --> 00:29:20.110 and early '90s. 00:29:26.330 --> 00:29:29.050 So those [? toy ?] examples just to show actually 00:29:29.050 --> 00:29:31.100 this philosophy works. 00:29:31.100 --> 00:29:33.110 I just showed this philosophy works, OK? 00:29:37.940 --> 00:29:42.050 But it's not possible if we want to go to higher dimensions. 00:29:42.050 --> 00:29:45.320 Actually, there's one paper-- let me just write it here. 00:29:45.320 --> 00:29:49.200 So this one paper explains the philosophy. 00:29:49.200 --> 00:29:51.890 So here I did not gave you many details, 00:29:51.890 --> 00:29:55.000 say, how you write this G down, how you in principle 00:29:55.000 --> 00:29:58.310 can match with this thing. 00:29:58.310 --> 00:30:00.045 With [? another ?] maybe [INAUDIBLE] 00:30:00.045 --> 00:30:04.590 you can make this discussion a little bit more explicit, 00:30:04.590 --> 00:30:06.040 but I don't have time. 00:30:06.040 --> 00:30:10.610 But if you want, you can take a look at this paper. 00:30:10.610 --> 00:30:15.145 So this paper discusses the story for the matrix quantum 00:30:15.145 --> 00:30:15.645 mechanics. 00:30:18.740 --> 00:30:21.252 But in the section 2 of this paper-- so this 00:30:21.252 --> 00:30:22.210 is a paper by Klebanov. 00:30:29.260 --> 00:30:32.360 So in the section 2 of this paper, 00:30:32.360 --> 00:30:37.240 it explains this mapping of Feynman diagrams 00:30:37.240 --> 00:30:39.420 to the string action. 00:30:39.420 --> 00:30:43.020 And this discretization picture give you 00:30:43.020 --> 00:30:48.080 a nice summary of that philosophy with more details 00:30:48.080 --> 00:30:49.800 than I have given to you. 00:30:49.800 --> 00:30:51.760 So you can take a look at that. 00:30:51.760 --> 00:30:54.430 And this paper also has some other references 00:30:54.430 --> 00:30:57.200 if you want to take a look at it. 00:30:57.200 --> 00:30:57.700 OK. 00:31:02.067 --> 00:31:02.650 Any questions? 00:31:07.320 --> 00:31:08.561 Yes. 00:31:08.561 --> 00:31:10.602 AUDIENCE: Sorry, but who was the first to realize 00:31:10.602 --> 00:31:12.102 this connection between the surfaces 00:31:12.102 --> 00:31:13.494 in topology of Feynman diagrams? 00:31:13.494 --> 00:31:14.458 HONG LIU: Sorry? 00:31:14.458 --> 00:31:16.208 AUDIENCE: Who first realized this relation 00:31:16.208 --> 00:31:17.350 between topology and-- 00:31:17.350 --> 00:31:20.840 HONG LIU: So of course, already when 00:31:20.840 --> 00:31:25.040 't Hooft invented this large N expansion, 00:31:25.040 --> 00:31:29.920 he already noticed that this is similar to string theory. 00:31:29.920 --> 00:31:32.640 So he already commented on that. 00:31:32.640 --> 00:31:35.000 And he already commented on that. 00:31:35.000 --> 00:31:40.960 And for many years people did not make progress. 00:31:40.960 --> 00:31:44.650 For many years, people did not make progress. 00:31:44.650 --> 00:31:48.770 But in the late '80s-- in the mid to late '80s, 00:31:48.770 --> 00:31:52.740 people started thinking about the question 00:31:52.740 --> 00:31:56.290 from this perspective, not from that perspective. 00:31:56.290 --> 00:31:59.970 So they started to think about the order 00:31:59.970 --> 00:32:00.890 from this perspective. 00:32:00.890 --> 00:32:06.050 Because just typical string theory are hard to solve, 00:32:06.050 --> 00:32:06.800 et cetera. 00:32:06.800 --> 00:32:08.630 So people think, maybe we can actually 00:32:08.630 --> 00:32:11.980 understand or generalize our understanding of string theory 00:32:11.980 --> 00:32:13.625 by discretize the worldsheets. 00:32:17.080 --> 00:32:20.515 And then they just integrate over 00:32:20.515 --> 00:32:22.960 all possible triangulization, et cetera. 00:32:22.960 --> 00:32:26.030 And then they realized that that thing actually 00:32:26.030 --> 00:32:28.180 is like something over Feynman diagrams. 00:32:28.180 --> 00:32:30.770 And then for the very simple situations, 00:32:30.770 --> 00:32:34.810 say like if you have only a matrix integral, actually 00:32:34.810 --> 00:32:38.330 you can make the connection explicit. 00:32:38.330 --> 00:32:40.230 So that was in the late '80s. 00:32:40.230 --> 00:32:44.210 So people like [? McDowell ?] or [? Kazakov ?] et cetera that 00:32:44.210 --> 00:32:46.150 were trying to explore that. 00:32:53.790 --> 00:32:55.374 Other questions? 00:32:55.374 --> 00:32:57.162 AUDIENCE: I'm having trouble seeing 00:32:57.162 --> 00:33:01.918 how the sum over all triangulations [INAUDIBLE] each 00:33:01.918 --> 00:33:02.844 surfaces. 00:33:02.844 --> 00:33:05.962 How does that correspond to the discrete version of summing 00:33:05.962 --> 00:33:06.940 over all [INAUDIBLE]? 00:33:06.940 --> 00:33:08.310 HONG LIU: Right. 00:33:08.310 --> 00:33:11.250 AUDIENCE: That's the discrete sum over all possible 00:33:11.250 --> 00:33:12.580 [? genus-h, ?] right? 00:33:12.580 --> 00:33:14.850 HONG LIU: Yeah. 00:33:14.850 --> 00:33:17.100 I think this is the example. 00:33:17.100 --> 00:33:19.780 Yeah, let's consider torus. 00:33:19.780 --> 00:33:24.200 So a torus is a box with this identified with this, 00:33:24.200 --> 00:33:25.890 and this identified with that. 00:33:25.890 --> 00:33:26.930 OK. 00:33:26.930 --> 00:33:30.420 And let me first just draw the simplest partition here. 00:33:30.420 --> 00:33:31.390 Just draw like that. 00:33:36.150 --> 00:33:36.650 Yeah. 00:33:39.620 --> 00:33:43.300 Let me just look at these two things. 00:33:43.300 --> 00:33:54.700 So suppose I give each box-- so if I specify each box, 00:33:54.700 --> 00:33:57.620 say, give a unit area. 00:33:57.620 --> 00:33:58.870 OK? 00:33:58.870 --> 00:34:02.190 And I do this one, I do that one, 00:34:02.190 --> 00:34:07.190 or I do some other ways to triangulize it. 00:34:07.190 --> 00:34:10.429 Then because [? one and two ?] give a different symmetric 00:34:10.429 --> 00:34:11.844 to the surface. 00:34:11.844 --> 00:34:13.260 And then because [? one and two ?] 00:34:13.260 --> 00:34:16.029 integrate over all possible metric on this surface. 00:34:16.029 --> 00:34:17.820 And they integrate over all possible metric 00:34:17.820 --> 00:34:19.489 on this surface, you can integrate [INAUDIBLE] 00:34:19.489 --> 00:34:20.460 all possible surfaces. 00:34:22.519 --> 00:34:24.560 AUDIENCE: In the case of the strings for example, 00:34:24.560 --> 00:34:26.920 [? we put some ?] over the torus here 00:34:26.920 --> 00:34:28.982 and the torus and the torus there. 00:34:28.982 --> 00:34:29.690 HONG LIU: No, no. 00:34:29.690 --> 00:34:31.909 You only sum over a single torus. 00:34:31.909 --> 00:34:34.616 Now, what do you mean by summing over torus here, torus there? 00:34:34.616 --> 00:34:36.532 AUDIENCE: I thought like in the path integral, 00:34:36.532 --> 00:34:39.376 in the case of the string theory-- 00:34:39.376 --> 00:34:42.087 HONG LIU: No, you're only summing over a single torus. 00:34:42.087 --> 00:34:44.060 You're only summing over a single surface, 00:34:44.060 --> 00:34:46.570 but all possible ways to write-- all possible ways 00:34:46.570 --> 00:34:47.550 to draw that surface. 00:34:54.860 --> 00:34:57.030 So what you said about summing torus here, 00:34:57.030 --> 00:34:59.510 summing torus there, because [INAUDIBLE] what we call 00:34:59.510 --> 00:35:01.840 the disconnected amplitudes. 00:35:01.840 --> 00:35:05.100 And then you don't need to consider them in physically 00:35:05.100 --> 00:35:06.100 disconnected amplitude. 00:35:06.100 --> 00:35:07.760 You can just [? exponentiate ?] what 00:35:07.760 --> 00:35:09.770 we call by connected amplitude. 00:35:09.770 --> 00:35:11.580 And you don't need to do that separately. 00:35:11.580 --> 00:35:13.640 So once you know how to do a single one, 00:35:13.640 --> 00:35:16.051 and the disconnected one just automatically obtained 00:35:16.051 --> 00:35:17.050 by [? exponentiation. ?] 00:35:20.006 --> 00:35:20.881 AUDIENCE: [INAUDIBLE] 00:35:31.074 --> 00:35:31.740 HONG LIU: Sorry? 00:35:31.740 --> 00:35:32.790 No, no. 00:35:32.790 --> 00:35:38.575 Here the metric matters, the geometry matters. 00:35:38.575 --> 00:35:41.465 It's not just the topology. 00:35:41.465 --> 00:35:49.445 AUDIENCE: [INAUDIBLE] Feynman diagram [INAUDIBLE]? 00:35:49.445 --> 00:35:50.070 HONG LIU: Yeah. 00:35:50.070 --> 00:35:52.403 Yeah, just the key is that the propagator of the Feynman 00:35:52.403 --> 00:35:54.775 diagram essentially [? encodes ?] the geometries. 00:35:54.775 --> 00:36:00.710 And in encoding a very indirect way. 00:36:00.710 --> 00:36:01.210 Yeah. 00:36:01.210 --> 00:36:02.770 Just read this part. 00:36:02.770 --> 00:36:05.240 This section only have a few pages, 00:36:05.240 --> 00:36:08.350 but contain a little bit more details on what I have here. 00:36:08.350 --> 00:36:13.800 It requires maybe one more hour to explain this in more detail. 00:36:13.800 --> 00:36:15.070 Yeah, this is just that. 00:36:15.070 --> 00:36:16.695 I just want to explain this philosophy. 00:36:16.695 --> 00:36:19.380 I don't want to go through the details of how you do this. 00:36:22.360 --> 00:36:23.525 OK, good. 00:36:23.525 --> 00:36:25.816 So now let me just mention a couple of generalizations. 00:36:40.540 --> 00:36:42.220 So the first thing you already asked 00:36:42.220 --> 00:36:45.870 before, I think maybe both you have asked. 00:36:48.692 --> 00:36:50.066 Let me just mention them quickly. 00:36:53.650 --> 00:36:57.830 And if you are interested, I can certainly 00:36:57.830 --> 00:37:02.400 give you a reference for you to read about them, 00:37:02.400 --> 00:37:05.070 or I can put it in [? your P ?] sets. 00:37:05.070 --> 00:37:17.100 And so, so far, it's all matrix-valued fields, OK? 00:37:17.100 --> 00:37:20.350 But if you can see the theory-- or in other words, 00:37:20.350 --> 00:37:22.314 in the mathematical language, say, 00:37:22.314 --> 00:37:23.605 it's an adjoint representation. 00:37:27.330 --> 00:37:31.030 It's an adjoint representation of the-- 00:37:31.030 --> 00:37:34.261 because our symmetries are UN, it's a UN gauge group. 00:37:38.500 --> 00:37:40.360 OK? 00:37:40.360 --> 00:37:43.490 UN gauge group. 00:37:43.490 --> 00:37:47.745 But you can also, for example, in QCD, you also have quarks. 00:37:47.745 --> 00:37:51.850 So you also have field in the fundamental representations. 00:37:51.850 --> 00:37:59.220 So it can also include field in the fundamental representation. 00:37:59.220 --> 00:38:03.762 So rather than matrix-valued, they're N vector. 00:38:03.762 --> 00:38:07.160 OK, they're N [? vectors. ?] 00:38:07.160 --> 00:38:11.950 So for quarks, of course, for the standard QCD N will be 3, 00:38:11.950 --> 00:38:13.380 so you have three quarks. 00:38:13.380 --> 00:38:15.265 You have three different colored quarks. 00:38:18.850 --> 00:38:22.570 And so then your Feynman diagrams, 00:38:22.570 --> 00:38:25.850 in addition to have those matrix [? lines, ?] 00:38:25.850 --> 00:38:29.080 which you have a double line. 00:38:29.080 --> 00:38:32.790 And now here you only have a single index, OK? 00:38:32.790 --> 00:38:34.606 And then you only have a single line. 00:38:34.606 --> 00:38:35.980 So the propagator of those quarks 00:38:35.980 --> 00:38:37.440 will just have a single line. 00:38:37.440 --> 00:38:38.981 And then also in your Feynman diagram 00:38:38.981 --> 00:38:41.790 you can have loops over the quarks, et cetera. 00:38:41.790 --> 00:38:43.266 So you can again work this out. 00:38:43.266 --> 00:38:45.515 And then you find it is a very nice large N expansion. 00:38:53.540 --> 00:39:02.260 And then you find the diagrams, the Feynman diagrams. 00:39:02.260 --> 00:39:04.310 Now you find in this case the Feynman 00:39:04.310 --> 00:39:25.345 diagrams can be classified by 2D surfaces with boundaries. 00:39:32.990 --> 00:39:38.400 So essentially, you have-- and let me just say, for example, 00:39:38.400 --> 00:39:43.690 this is the vacuum diagrams, for all the vacuum process. 00:39:43.690 --> 00:39:48.660 Then you can [INAUDIBLE] or the vacuum diagrams. 00:39:52.620 --> 00:39:55.460 And then they can all be [? collectified. ?] 00:39:55.460 --> 00:40:00.150 So previously, we have a matrix-valued field. 00:40:00.150 --> 00:40:04.445 Then all your vacuum diagrams, they 00:40:04.445 --> 00:40:07.550 are corresponding closed surfaces-- so sphere, 00:40:07.550 --> 00:40:09.130 torus, et cetera. 00:40:09.130 --> 00:40:10.900 But now if you include the quarks, 00:40:10.900 --> 00:40:13.310 then those surfaces can have boundaries. 00:40:13.310 --> 00:40:16.170 And then [INAUDIBLE] into the quark groups, et cetera. 00:40:16.170 --> 00:40:21.270 And then they [? cannot ?] be classified. 00:40:21.270 --> 00:40:26.250 And so these also have a counterpart 00:40:26.250 --> 00:40:28.970 if you try to map to the string theory. 00:40:28.970 --> 00:40:31.345 So this [INAUDIBLE] [? one and ?] [? two, ?] string 00:40:31.345 --> 00:40:31.845 theory. 00:40:35.230 --> 00:40:47.280 There's string theory with both closed and open strings. 00:40:55.150 --> 00:40:57.610 And so essentially those boundaries 00:40:57.610 --> 00:40:59.360 give rise to the open strings. 00:40:59.360 --> 00:41:01.820 So here, it's all closed strings. 00:41:01.820 --> 00:41:02.820 It's all closed surface. 00:41:02.820 --> 00:41:05.670 Well, now you can, by adding the open strings, 00:41:05.670 --> 00:41:10.840 and then you can, again, have the correspondence 00:41:10.840 --> 00:41:12.900 between the two. 00:41:12.900 --> 00:41:15.250 OK. 00:41:15.250 --> 00:41:17.420 So all the discussion is very similar to what 00:41:17.420 --> 00:41:18.330 we discussed before. 00:41:18.330 --> 00:41:22.720 We just apply all this the same philosophy to the quarks. 00:41:22.720 --> 00:41:23.690 Yes. 00:41:23.690 --> 00:41:27.570 AUDIENCE: [INAUDIBLE] do the same trick on string theory 00:41:27.570 --> 00:41:30.480 and find some sort of expression which 00:41:30.480 --> 00:41:34.360 then will map to some higher order surfaces, [INAUDIBLE]? 00:41:37.220 --> 00:41:38.553 HONG LIU: Sorry, say that again? 00:41:38.553 --> 00:41:41.511 AUDIENCE: [INAUDIBLE] Feynman diagrams 00:41:41.511 --> 00:41:44.469 we move to string theory for surfaces. 00:41:44.469 --> 00:41:48.070 Is there some [INAUDIBLE] from surfaces just they go 00:41:48.070 --> 00:41:49.670 one more [? step up? ?] 00:41:49.670 --> 00:41:53.460 HONG LIU: You mean higher dimensions, not strings. 00:41:53.460 --> 00:41:58.260 Yeah, that will become-- of course, 00:41:58.260 --> 00:42:00.885 that's a [? lateral ?] idea. 00:42:00.885 --> 00:42:03.510 So that will [INAUDIBLE] you can consider [? rather ?] strings, 00:42:03.510 --> 00:42:06.320 you can consider two-dimensional surface, 00:42:06.320 --> 00:42:09.610 a two-dimensional surface moving in spacetime. 00:42:09.610 --> 00:42:12.650 And then [INAUDIBLE] into [? so-called ?] the membrane 00:42:12.650 --> 00:42:14.000 theory. 00:42:14.000 --> 00:42:17.850 But let's say where it turns out to be-- turns out 00:42:17.850 --> 00:42:20.380 string is a nice balance. 00:42:20.380 --> 00:42:22.880 It's not too complicated or not too simple. 00:42:22.880 --> 00:42:24.340 And it give you lots of structure. 00:42:24.340 --> 00:42:26.131 But when you go to membrane, then the story 00:42:26.131 --> 00:42:28.060 become too complicated, and nobody knows 00:42:28.060 --> 00:42:30.621 how to quantize that theory. 00:42:50.615 --> 00:42:58.590 So the second remark is that here we consider UN. 00:42:58.590 --> 00:43:00.640 So here our symmetry group is UN. 00:43:04.950 --> 00:43:11.450 Because our phi-- phi there is [? commission. ?] So when you 00:43:11.450 --> 00:43:14.070 have a [? commission ?] matrix, then there's a difference 00:43:14.070 --> 00:43:20.820 between the two indices, so we put one up and one down. 00:43:20.820 --> 00:43:24.370 So they are propagators that lead 00:43:24.370 --> 00:43:38.580 to-- so it leads to the lines with arrows, 00:43:38.580 --> 00:43:42.550 because we need to distinguish upper and lower indices. 00:43:42.550 --> 00:43:43.180 OK? 00:43:43.180 --> 00:43:45.810 Between the two indices. 00:43:45.810 --> 00:43:52.040 But you can also consider, for example, 00:43:52.040 --> 00:43:55.870 phi is a symmetric matrix. 00:43:55.870 --> 00:43:58.780 Say it's a real symmetric matrix. 00:43:58.780 --> 00:44:04.490 It's a real symmetric, or real anti-symmetric. 00:44:11.490 --> 00:44:15.260 In those cases, then there's no difference 00:44:15.260 --> 00:44:19.050 between the two indices. 00:44:19.050 --> 00:44:25.820 And then when you draw a propagator-- so in this case 00:44:25.820 --> 00:44:34.320 the symmetry group would be, say, 00:44:34.320 --> 00:44:37.500 SON, say, or SPN, et cetera. 00:44:41.840 --> 00:44:48.980 And then the propagators, they will no longer 00:44:48.980 --> 00:44:51.680 have orientations. 00:44:51.680 --> 00:44:52.180 OK? 00:44:52.180 --> 00:44:54.900 They will no longer have orientations. 00:44:54.900 --> 00:44:57.990 Because you can no longer-- yeah. 00:44:57.990 --> 00:45:02.675 So this will give rise-- so let me write it closer. 00:45:06.300 --> 00:45:13.810 So this will give rise to unorientable surfaces. 00:45:19.220 --> 00:45:22.166 Say, for example, to classify the diagrams, 00:45:22.166 --> 00:45:24.290 you can no longer just use the orientable surfaces. 00:45:24.290 --> 00:45:26.620 You also have to include the non-orientable surfaces 00:45:26.620 --> 00:45:29.470 to classify the diagrams. 00:45:29.470 --> 00:45:35.520 And the [INAUDIBLE] this also have a precise counterpart 00:45:35.520 --> 00:45:40.100 into unorientable strings. 00:45:40.100 --> 00:45:41.660 No, non-orientable strings. 00:45:47.070 --> 00:45:56.220 Yeah, I think non-orientable, non-orientable surfaces. 00:45:56.220 --> 00:45:57.410 Also non-orientable strings. 00:46:17.300 --> 00:46:18.620 Good. 00:46:18.620 --> 00:46:20.780 So I'm emphasizing how difficult it 00:46:20.780 --> 00:46:29.430 is if, say, we want to start with QCD 00:46:29.430 --> 00:46:35.640 and then try to find the string theory description. 00:46:35.640 --> 00:46:41.470 But this still, [? none of ?] this tries-- I just try. 00:46:51.930 --> 00:47:14.660 OK, so let's just consider, just take large N generalization 00:47:14.660 --> 00:47:16.450 of QCD. 00:47:16.450 --> 00:47:18.340 So this, again, will be some UN gauge theory, 00:47:18.340 --> 00:47:26.611 UN Yang-Mills theory, say, in 3 plus 1 dimensional Minkowski 00:47:26.611 --> 00:47:27.110 spacetime. 00:47:31.340 --> 00:47:41.770 And can we say anything about its string theory description? 00:47:55.800 --> 00:47:57.380 So [INAUDIBLE]. 00:47:57.380 --> 00:48:00.430 So maybe it's difficult, but let's try to guess it. 00:48:00.430 --> 00:48:02.340 OK. 00:48:02.340 --> 00:48:05.280 So in physics, in many situations, 00:48:05.280 --> 00:48:08.990 a seemingly difficult problem, if you know how to guess it, 00:48:08.990 --> 00:48:11.800 actually you can get the answer. 00:48:11.800 --> 00:48:14.140 On, for example, quantum hole effects, 00:48:14.140 --> 00:48:17.220 fractional quantum hole effects, you can just 00:48:17.220 --> 00:48:20.490 guess the wave function. 00:48:20.490 --> 00:48:23.840 So of course, the simplest guess-- 00:48:23.840 --> 00:48:28.140 so this is some gauge theory in 3 plus 1 dimensional Minkowski 00:48:28.140 --> 00:48:29.800 spacetime. 00:48:29.800 --> 00:48:32.800 So now we say this is a string theory. 00:48:32.800 --> 00:48:43.450 So natural guess is that this maybe 00:48:43.450 --> 00:48:51.160 is a string theory, again, in the 3 plus 1 00:48:51.160 --> 00:48:54.890 dimensional Minkowski spacetime. 00:48:54.890 --> 00:48:56.840 OK? 00:48:56.840 --> 00:49:00.360 So we just take what-- so these will, of course, 00:49:00.360 --> 00:49:09.860 run into a string, propagating in this spacetime, OK? 00:49:09.860 --> 00:49:13.590 As I said, when you write down the string theory, 00:49:13.590 --> 00:49:16.450 you first have to specify your target space, which, 00:49:16.450 --> 00:49:20.680 as the string moves, the larger question would be just, 00:49:20.680 --> 00:49:24.240 should it be the gauge theory's Minkowski spacetime. 00:49:24.240 --> 00:49:25.930 Maybe this string theory should be. 00:49:25.930 --> 00:49:27.720 OK? 00:49:27.720 --> 00:49:31.160 And then this. 00:49:31.160 --> 00:49:36.110 Then you can just try to-- then you can just 00:49:36.110 --> 00:49:38.546 write down the simplest action. 00:49:38.546 --> 00:49:48.780 So maybe say Nambu-Goto action, which we wrote last time, OK? 00:49:48.780 --> 00:49:55.550 Or the [? old ?] Polyakov action. 00:49:55.550 --> 00:50:03.510 So this Nambu-Goto action will result [INAUDIBLE] Polyakov. 00:50:03.510 --> 00:50:06.200 And let me not worry about that. 00:50:06.200 --> 00:50:08.510 For example, you can just guess, say, maybe 00:50:08.510 --> 00:50:12.360 this is a string theory also in the Minkowski spacetime. 00:50:12.360 --> 00:50:15.440 Say, consider the simplest action. 00:50:21.680 --> 00:50:24.666 Or the equivalent of this, OK? 00:50:36.890 --> 00:50:38.720 Then at least what you could try-- now 00:50:38.720 --> 00:50:47.554 you actually have an action. 00:50:47.554 --> 00:50:49.220 Now you think that you have this object. 00:50:49.220 --> 00:50:51.160 Now you think you can compare. 00:50:51.160 --> 00:50:52.890 OK, now you can essentially compare. 00:50:52.890 --> 00:50:56.110 Say, in QCD you calculated your Feynman diagrams, 00:50:56.110 --> 00:50:58.215 and now just compare. 00:50:58.215 --> 00:51:00.090 But of course, you still have the difficulty. 00:51:00.090 --> 00:51:01.923 Of course, you have to go to strong coupling 00:51:01.923 --> 00:51:04.090 to see the geometric limit, et cetera. 00:51:04.090 --> 00:51:07.950 But in principle, it's something you can do. 00:51:07.950 --> 00:51:10.490 But this actually does not work. 00:51:10.490 --> 00:51:13.140 OK? 00:51:13.140 --> 00:51:17.280 This does not work, for the following simple reason. 00:51:21.280 --> 00:51:30.430 Firstly, that such a string theory-- 00:51:30.430 --> 00:51:32.930 so a string theory, actually the remarkable thing 00:51:32.930 --> 00:51:37.050 about the string is that if you have a particle, 00:51:37.050 --> 00:51:40.800 you can put the particle in any spacetime. 00:51:40.800 --> 00:51:43.360 But strings are very picky. 00:51:43.360 --> 00:51:46.280 You cannot put them in any spacetime. 00:51:46.280 --> 00:51:50.145 And they can only propagate consistently, 00:51:50.145 --> 00:51:54.220 quantum mechanically consistently, in some spacetime 00:51:54.220 --> 00:51:57.160 but not in others. 00:51:57.160 --> 00:52:00.055 So for example, if you want to put the string 00:52:00.055 --> 00:52:02.180 to propagate in this 3 plus 1 dimensional Minkowski 00:52:02.180 --> 00:52:06.929 spacetime, then you actually find that the theory 00:52:06.929 --> 00:52:08.220 is mathematically inconsistent. 00:52:10.760 --> 00:52:16.050 So such a string theory is inconsistent. 00:52:18.570 --> 00:52:21.200 It's mathematically inconsistent. 00:52:21.200 --> 00:52:31.640 Except for the D equal to 26 or 10. 00:52:31.640 --> 00:52:32.490 OK? 00:52:32.490 --> 00:52:37.410 So 26 if you just purely have the theory, and 10 00:52:37.410 --> 00:52:38.690 if you also add some fermion. 00:52:43.260 --> 00:52:45.610 So such a string theory does not exist mathematically. 00:52:48.640 --> 00:52:50.990 So you say, oh, OK. 00:52:50.990 --> 00:52:55.240 You say, I'm a smart fellow. 00:52:55.240 --> 00:52:57.904 I can go around this. 00:52:57.904 --> 00:52:59.570 Because we want the Minkowski spacetime. 00:52:59.570 --> 00:53:01.740 Because those gauge theory propagating the Minkowski 00:53:01.740 --> 00:53:06.200 spacetime, so this Minkowski [INAUDIBLE] must be somewhere. 00:53:06.200 --> 00:53:10.860 They cannot go away, because all these glueballs [INAUDIBLE] 00:53:10.860 --> 00:53:13.450 in this 3 plus 1 dimensional Minkowski spacetime. 00:53:13.450 --> 00:53:15.950 And if we want to identify the strings with those glueballs, 00:53:15.950 --> 00:53:17.741 those strings must at least [? know ?] some 00:53:17.741 --> 00:53:20.780 of this Minkowski spacetime. 00:53:20.780 --> 00:53:22.970 And then you say, oh, suppose you tell me 00:53:22.970 --> 00:53:25.530 that this string theory is only consistent in 10 dimension. 00:53:28.120 --> 00:53:39.880 But then let me take a string theory in 10 dimensions, which 00:53:39.880 --> 00:53:42.180 itself consistent. 00:53:42.180 --> 00:53:45.970 But I take this 10-dimensional spacetime to have 00:53:45.970 --> 00:53:48.120 the form of a 3 plus 1 dimensional 00:53:48.120 --> 00:53:49.550 Minkowski spacetime. 00:53:49.550 --> 00:53:52.880 And the [? time, ?] some compact manifold, OK? 00:53:58.360 --> 00:53:59.290 Some compact manifold. 00:54:02.840 --> 00:54:06.510 And in such case-- so if this is a compact manifold, 00:54:06.510 --> 00:54:08.760 then the symmetry of this spacetime, 00:54:08.760 --> 00:54:11.430 so the spacetime symmetry still only have 00:54:11.430 --> 00:54:15.890 the 3 plus 1 dimensional, [? say, ?] Poincare symmetry. 00:54:15.890 --> 00:54:19.010 Because if you want to describe the QCD in 3 plus 1 dimension, 00:54:19.010 --> 00:54:21.180 QCD has the Poincare symmetry. 00:54:21.180 --> 00:54:22.910 You can do Lorentz transformation, 00:54:22.910 --> 00:54:24.450 and then you can do rotation. 00:54:24.450 --> 00:54:27.010 Or you can do translation. 00:54:27.010 --> 00:54:30.110 The string theory should not have more symmetries or less 00:54:30.110 --> 00:54:31.110 symmetries than QCD. 00:54:31.110 --> 00:54:32.609 They should have the same symmetries 00:54:32.609 --> 00:54:35.260 because they are supposed to be the same description. 00:54:35.260 --> 00:54:39.360 But if you take the 10-dimensional Minkowski space, 00:54:39.360 --> 00:54:40.530 of course, it's not right. 00:54:40.530 --> 00:54:42.030 Because the 10-dimensional Minkowski 00:54:42.030 --> 00:54:43.930 space have 10-dimensional translation 00:54:43.930 --> 00:54:47.450 and 10-dimensional Lorentz symmetry. 00:54:47.450 --> 00:54:52.030 But what you can do is that you take this 10-dimensional space 00:54:52.030 --> 00:54:54.660 to be a form of the 3 plus 1 dimensional Minkowski 00:54:54.660 --> 00:54:58.590 spacetime and times some additional compact manifold, 00:54:58.590 --> 00:55:02.890 and then you have solved the symmetry problem. 00:55:02.890 --> 00:55:06.510 But except this still does not work 00:55:06.510 --> 00:55:13.950 because the string theory, as we know, always contain gravity. 00:55:13.950 --> 00:55:18.820 And if you put a string theory on such a compact space N, 00:55:18.820 --> 00:55:20.195 [? there would be ?] always leads 00:55:20.195 --> 00:55:26.800 to a massless spin-2 particle in this 3 plus 1 dimensional part. 00:55:26.800 --> 00:55:31.090 But from Weinberg-Witten theorem we talked in the first lecture, 00:55:31.090 --> 00:55:34.090 in the QCD you are not supposed to have 00:55:34.090 --> 00:55:39.450 a 3 plus 1 dimensional massless spin-2 particle, OK? 00:55:39.450 --> 00:55:41.570 And so this won't work. 00:55:41.570 --> 00:55:44.012 So this won't work. 00:55:44.012 --> 00:55:46.610 Because this contains-- 00:56:19.160 --> 00:56:21.260 In 3 plus 1 dimensional [? Minkowski space, ?] 00:56:21.260 --> 00:56:28.840 which does not have-- OK? 00:56:28.840 --> 00:56:31.890 Or in the large N [INAUDIBLE]. 00:56:31.890 --> 00:56:35.200 So this does not work. 00:56:35.200 --> 00:56:36.220 So what to do? 00:56:36.220 --> 00:56:36.820 Yes? 00:56:36.820 --> 00:56:39.740 AUDIENCE: So does this just mean that it's mathematically 00:56:39.740 --> 00:56:40.820 inconsistent? 00:56:40.820 --> 00:56:41.705 HONG LIU: No, no. 00:56:41.705 --> 00:56:43.913 This does not mean it is mathematically inconsistent. 00:56:43.913 --> 00:56:47.300 It just means this string theory cannot not correspond 00:56:47.300 --> 00:56:49.590 to the string theory [? describe ?] QCD. 00:56:49.590 --> 00:56:55.530 The string theory description-- the equivalent string theory 00:56:55.530 --> 00:56:58.150 for QCD cannot have this feature. 00:56:58.150 --> 00:57:00.720 Yeah, just say this cannot be the right answer for that 00:57:00.720 --> 00:57:01.760 string theory. 00:57:01.760 --> 00:57:03.850 This string theory is consistent. 00:57:03.850 --> 00:57:04.391 Yes. 00:57:04.391 --> 00:57:05.890 AUDIENCE: So is that you were saying 00:57:05.890 --> 00:57:08.882 if there is a massless spin-2 particle in that string theory, 00:57:08.882 --> 00:57:10.840 there has to be a [? counterpart in the ?] QCD. 00:57:10.840 --> 00:57:11.798 HONG LIU: That's right. 00:57:11.798 --> 00:57:15.155 AUDIENCE: If there is not a [INAUDIBLE], that won't work 00:57:15.155 --> 00:57:15.780 HONG LIU: Yeah. 00:57:15.780 --> 00:57:17.910 This cannot be a description of that. 00:57:17.910 --> 00:57:21.250 From Weinberg-Witten theorem, we know in QCD there's 00:57:21.250 --> 00:57:22.650 no massless spin-2 particle. 00:57:25.280 --> 00:57:25.940 Yes. 00:57:25.940 --> 00:57:30.487 AUDIENCE: I thought we have talked about maybe we can do 00:57:30.487 --> 00:57:35.260 strings to [? find ?] QCD in a different dimension [? in ?] 00:57:35.260 --> 00:57:35.760 space. 00:57:35.760 --> 00:57:37.145 HONG LIU: We will go into that. 00:57:37.145 --> 00:57:38.770 But now they are in the same dimension, 00:57:38.770 --> 00:57:43.070 because this Minkowski 4, this will 00:57:43.070 --> 00:57:50.370 have-- because this is a compact [? part, ?] it doesn't matter. 00:57:50.370 --> 00:57:57.260 So in this part, [? there are ?] massless spin-2 particles. 00:57:57.260 --> 00:57:58.782 This does not [? apply ?] in QCD. 00:58:02.880 --> 00:58:03.980 So what can you do? 00:58:03.980 --> 00:58:07.650 So most people just give up. 00:58:07.650 --> 00:58:09.360 Most people give up. 00:58:09.360 --> 00:58:15.900 So other than give up, the option is say maybe 00:58:15.900 --> 00:58:18.580 this action is too simple. 00:58:18.580 --> 00:58:20.600 Maybe you have to look at more exotic action. 00:58:26.980 --> 00:58:28.270 OK. 00:58:28.270 --> 00:58:30.790 So this is one possibility. 00:58:30.790 --> 00:58:32.900 And the second possibility is that maybe you 00:58:32.900 --> 00:58:34.610 need to look for some other target space. 00:58:41.031 --> 00:58:41.530 OK. 00:58:44.080 --> 00:58:45.950 But now, what if you go away from here? 00:58:49.459 --> 00:58:51.250 Once you go away from here, everything else 00:58:51.250 --> 00:58:57.649 is now becoming such little in the ocean, 00:58:57.649 --> 00:58:59.690 because then you don't have much clue what to do. 00:58:59.690 --> 00:59:04.330 We just say, your basic guess just could not work. 00:59:04.330 --> 00:59:07.689 So for many years, even though this is a very intriguing idea, 00:59:07.689 --> 00:59:08.980 people could not make progress. 00:59:14.900 --> 00:59:18.207 But now we have hindsight. 00:59:18.207 --> 00:59:19.290 But now we have hindsight. 00:59:25.110 --> 00:59:30.460 So we know that even this maybe cannot be described 00:59:30.460 --> 00:59:39.250 by a four-dimensional-- so even though this cannot have 00:59:39.250 --> 00:59:47.820 a-- so this cannot have a massless spin-2 particle 00:59:47.820 --> 00:59:51.650 in this 3 plus 1 dimension of Minkowski spacetime. 00:59:51.650 --> 00:59:55.330 Maybe you can still have some kind of graviton 00:59:55.330 --> 00:59:57.330 in some kind of a five-dimensional spacetime. 01:00:00.040 --> 01:00:03.095 You have some five dimensions, in a different dimension. 01:00:06.220 --> 01:00:09.940 So there were some rough hints. 01:00:09.940 --> 01:00:16.156 Maybe you can consider there's a five-dimensional string theory. 01:00:24.640 --> 01:00:27.195 So let me emphasize when we say five or four, 01:00:27.195 --> 01:00:32.710 I always mention the non-compact part. 01:00:32.710 --> 01:00:37.300 So the compact part, it doesn't count because compact part just 01:00:37.300 --> 01:00:40.640 goes for the ride. 01:00:40.640 --> 01:00:42.720 What determines the properties, say, 01:00:42.720 --> 01:00:44.300 of a massless particle, et cetera, 01:00:44.300 --> 01:00:47.319 is the uncompact of the spacetime. 01:00:47.319 --> 01:00:49.360 Yeah, because this is a 10-dimensional spacetime. 01:00:49.360 --> 01:00:50.693 This is already not [INAUDIBLE]. 01:00:50.693 --> 01:01:01.890 So maybe we [? change ?] for string theory 01:01:01.890 --> 01:01:05.300 in five-dimensional uncompact. 01:01:05.300 --> 01:01:07.180 AUDIENCE: Five, so in 4 plus 1? 01:01:07.180 --> 01:01:08.470 HONG LIU: Yeah. 01:01:08.470 --> 01:01:13.010 In 4 plus 1 uncompact spacetime. 01:01:28.450 --> 01:01:29.376 Yes. 01:01:29.376 --> 01:01:32.657 AUDIENCE: [INAUDIBLE] compactors. 01:01:32.657 --> 01:01:34.031 When you say compact, do you mean 01:01:34.031 --> 01:01:35.490 the mathematical definition of compactness? 01:01:35.490 --> 01:01:36.698 HONG LIU: Yeah, that's right. 01:01:36.698 --> 01:01:40.050 Yeah, I just say there is a finite volume. 01:01:40.050 --> 01:01:42.610 Just for our purpose here, we can do it simply. 01:01:42.610 --> 01:01:48.500 Just let's imagine-- yeah, compact always 01:01:48.500 --> 01:01:51.000 has a finite volume, for example. 01:01:51.000 --> 01:01:51.767 Yes? 01:01:51.767 --> 01:01:54.509 AUDIENCE: Why can we just ignore the compact dimensions? 01:01:54.509 --> 01:01:57.073 Is there any condition on how big they're allowed 01:01:57.073 --> 01:02:00.620 to be or something, like limit? 01:02:00.620 --> 01:02:03.270 HONG LIU: Yeah, just when you have-- so 01:02:03.270 --> 01:02:05.140 if you know a little bit about this thing 01:02:05.140 --> 01:02:07.050 called the Kaluza-Klein theory. 01:02:07.050 --> 01:02:16.820 And you know that the compact part-- the thing 01:02:16.820 --> 01:02:18.640 is that if you have a theory [? based ?] 01:02:18.640 --> 01:02:20.880 on uncompact and the compact part, 01:02:20.880 --> 01:02:22.630 and then most of the physical properties 01:02:22.630 --> 01:02:25.800 is controlled by the physics of uncompact parts. 01:02:25.800 --> 01:02:27.680 And this will determine some details 01:02:27.680 --> 01:02:30.280 like the detailed spectrum, et cetera. 01:02:30.280 --> 01:02:31.980 But the kind of thing we worry about, 01:02:31.980 --> 01:02:34.460 whether you have this massless spin-2 particle, et cetera, 01:02:34.460 --> 01:02:36.387 will not be determined by this kind of thing. 01:02:36.387 --> 01:02:38.004 AUDIENCE: Is there any volume limit 01:02:38.004 --> 01:02:39.535 on the compact part, like maximum? 01:02:39.535 --> 01:02:41.535 HONG LIU: No, it's fine to have a finite volume. 01:02:41.535 --> 01:02:44.897 AUDIENCE: Just finite, but can it be large? 01:02:44.897 --> 01:02:46.980 HONG LIU: No matter how large, this have infinite. 01:02:46.980 --> 01:02:48.604 It's always much smaller than this one. 01:02:51.450 --> 01:02:54.390 Yeah, but now it's just always relative. 01:02:54.390 --> 01:02:55.650 It's always relative. 01:02:55.650 --> 01:02:56.160 Yes. 01:02:56.160 --> 01:02:57.535 AUDIENCE: Tracking back a little, 01:02:57.535 --> 01:03:00.740 is there any quick explanation for 26 and 10 are special, 01:03:00.740 --> 01:03:02.010 or is it very complicated? 01:03:02.010 --> 01:03:03.931 HONG LIU: Um. 01:03:03.931 --> 01:03:07.830 [LAUGHTER] 01:03:07.830 --> 01:03:09.609 No, it's not complicated. 01:03:09.609 --> 01:03:11.650 Actually, we were going to do it in next lecture. 01:03:14.980 --> 01:03:18.730 Yeah, next lecture we will see 26, but maybe not 10. 01:03:18.730 --> 01:03:21.290 10 is little bit more complicated. 01:03:21.290 --> 01:03:23.820 Most people voted for my option one, 01:03:23.820 --> 01:03:28.010 so that means you will be able to see the 26. 01:03:28.010 --> 01:03:28.910 Right. 01:03:28.910 --> 01:03:31.226 AUDIENCE: Who [? discovered ?] 26 and 10? 01:03:31.226 --> 01:03:33.892 I mean, they are specific for this [INAUDIBLE] action 01:03:33.892 --> 01:03:35.975 rate, so for other action would be something else. 01:03:35.975 --> 01:03:42.560 HONG LIU: Specifically for the Nambu-Goto action is 26. 01:03:42.560 --> 01:03:45.920 And for the 10, you need to add some additional fermions 01:03:45.920 --> 01:03:49.010 and make it into a so-called superstring, then become 10. 01:03:49.010 --> 01:03:53.800 And even this 26 one is not completely self-consistent. 01:03:53.800 --> 01:03:56.070 And anyway, there's still some little, tiny problems 01:03:56.070 --> 01:03:56.990 with this. 01:03:56.990 --> 01:03:58.264 Anyway, so normally we use 10. 01:04:02.140 --> 01:04:08.420 OK so now, then there's some tantalizing hints 01:04:08.420 --> 01:04:13.317 for the-- say, maybe you cannot do it with the 3 plus 1 01:04:13.317 --> 01:04:14.650 dimensional uncompact spacetime. 01:04:14.650 --> 01:04:18.160 Maybe you can do a 4 plus 1 dimensional uncompact. 01:04:18.160 --> 01:04:22.554 So the first is the holographic principle, 01:04:22.554 --> 01:04:23.470 where you have length. 01:04:28.330 --> 01:04:31.130 Holographic principle we have learned because there we say, 01:04:31.130 --> 01:04:34.190 if you want to describe a theory with gravity, 01:04:34.190 --> 01:04:36.650 then this gravity should be able to be described 01:04:36.650 --> 01:04:39.760 by something on its boundary. 01:04:39.760 --> 01:04:42.956 And the string theory is a theory with gravity. 01:04:42.956 --> 01:04:44.330 So if the string theory should be 01:04:44.330 --> 01:04:49.010 equivalent to some kind of QCD, some kind of gauge theory 01:04:49.010 --> 01:04:52.820 without gravity, and then from holographic principle, 01:04:52.820 --> 01:04:57.080 this field theory maybe should be one lower dimension, OK? 01:04:57.080 --> 01:04:58.588 In one lower dimension. 01:05:01.330 --> 01:05:02.480 Is the logic here clear? 01:05:02.480 --> 01:05:05.520 AUDIENCE: Wait, can you say that again? 01:05:05.520 --> 01:05:08.480 HONG LIU: So here we want to equate large N 01:05:08.480 --> 01:05:11.670 QCD with some string theory. 01:05:11.670 --> 01:05:14.660 But string theory we know contains gravity. 01:05:14.660 --> 01:05:19.550 A list of all our experience contain gravity. 01:05:19.550 --> 01:05:22.240 But if you believe that the gravity should satisfy 01:05:22.240 --> 01:05:27.580 holographic principle, then the gravity should be equivalent, 01:05:27.580 --> 01:05:30.010 according to holographic principle, 01:05:30.010 --> 01:05:34.400 gravity in, say, D dimensional spacetime 01:05:34.400 --> 01:05:37.490 can be described by something on its boundary, 01:05:37.490 --> 01:05:40.879 something one dimension lower. 01:05:40.879 --> 01:05:42.920 AUDIENCE: But I thought the holographic principle 01:05:42.920 --> 01:05:44.560 was a statement about entropy. 01:05:44.560 --> 01:05:46.550 HONG LIU: No, it's a state started 01:05:46.550 --> 01:05:48.760 from a statement about entropy. 01:05:48.760 --> 01:05:51.330 But then you do a little bit of leap. 01:05:51.330 --> 01:05:54.140 So what I call it little bit of a conceptual leap 01:05:54.140 --> 01:05:58.410 is that the-- or [? little ?] leap of faith 01:05:58.410 --> 01:06:01.490 is that you promote that into the statement that 01:06:01.490 --> 01:06:03.200 said the number of degrees freedom 01:06:03.200 --> 01:06:07.060 you needed to describe the whole system. 01:06:07.060 --> 01:06:09.280 Yeah, so the holographic principle 01:06:09.280 --> 01:06:13.260 is that for any region, even the quantum gravity theory, 01:06:13.260 --> 01:06:16.960 for any region, you should be able to describe it 01:06:16.960 --> 01:06:21.204 by the degrees of freedom living on the boundary of that region. 01:06:21.204 --> 01:06:23.620 And degrees freedom living on the boundary of that region, 01:06:23.620 --> 01:06:25.130 then it's one dimensional lower. 01:06:25.130 --> 01:06:28.920 AUDIENCE: Wait, so can I ask one question about that? 01:06:28.920 --> 01:06:31.160 If I have some region, some volume in space, 01:06:31.160 --> 01:06:33.450 some closed ball or something. 01:06:33.450 --> 01:06:37.334 And I live in a universe which is, for example, a closed-- 01:06:37.334 --> 01:06:39.000 like maybe they live on some hypersphere 01:06:39.000 --> 01:06:40.160 or something like this. 01:06:40.160 --> 01:06:44.190 Then how do I know whether I'm-- how do I know that 01:06:44.190 --> 01:06:46.129 the information is encoded? 01:06:46.129 --> 01:06:47.920 How do I know whether I'm inside the sphere 01:06:47.920 --> 01:06:49.450 or outside of the sphere? 01:06:49.450 --> 01:06:51.380 For example, we see that the entropy that 01:06:51.380 --> 01:06:53.100 has to do with the sphere basically 01:06:53.100 --> 01:06:54.933 tells you about how much information can you 01:06:54.933 --> 01:06:56.810 contain inside the sphere. 01:06:56.810 --> 01:07:00.054 But if you live in a universe which is closed or something, 01:07:00.054 --> 01:07:01.470 then you don't know whether you're 01:07:01.470 --> 01:07:03.040 inside or outside the sphere. 01:07:03.040 --> 01:07:05.770 HONG LIU: Yeah, but that's a difficult question. 01:07:05.770 --> 01:07:08.040 Yeah, if you talk about closed universe here, 01:07:08.040 --> 01:07:09.840 we are not talking about closed universe. 01:07:09.840 --> 01:07:11.427 AUDIENCE: I see. 01:07:11.427 --> 01:07:12.010 HONG LIU: Yes. 01:07:12.010 --> 01:07:13.884 AUDIENCE: I thought the holographic principle 01:07:13.884 --> 01:07:17.130 is that the number of degrees freedom inside the region 01:07:17.130 --> 01:07:18.914 is actually bounded by the area. 01:07:18.914 --> 01:07:20.330 HONG LIU: Right, it's bounded by-- 01:07:20.330 --> 01:07:21.871 AUDIENCE: Yeah, but why is it that we 01:07:21.871 --> 01:07:25.169 use that degree of freedom living on the boundary? 01:07:25.169 --> 01:07:27.210 HONG LIU: There are several formulations of that. 01:07:27.210 --> 01:07:29.251 First is that the total number of degrees freedom 01:07:29.251 --> 01:07:32.150 in this region is bounded by the area. 01:07:32.150 --> 01:07:34.720 And then you can go to the next step, which is maybe 01:07:34.720 --> 01:07:37.510 the whole region can be just described 01:07:37.510 --> 01:07:40.170 by these degrees of freedom living 01:07:40.170 --> 01:07:41.915 on the boundary on that region. 01:07:41.915 --> 01:07:46.070 AUDIENCE: Is that because, say, the state of density 01:07:46.070 --> 01:07:50.840 on the boundary [INAUDIBLE] the state on the boundary 01:07:50.840 --> 01:07:53.720 is proportional to the area of the boundary? 01:07:53.720 --> 01:07:54.440 HONG LIU: Yeah. 01:07:54.440 --> 01:07:54.970 Exactly. 01:07:54.970 --> 01:07:56.896 That's right. 01:07:56.896 --> 01:08:01.020 AUDIENCE: So here our goal is to recover the large N theory in 3 01:08:01.020 --> 01:08:02.740 plus 1 dimensions without gravity. 01:08:02.740 --> 01:08:04.090 So we have no gravity. 01:08:04.090 --> 01:08:05.260 You can't 3 plus 1. 01:08:05.260 --> 01:08:06.710 HONG LIU: Right. 01:08:06.710 --> 01:08:09.240 Yeah, so if that is supposed to be equivalent to the gravity 01:08:09.240 --> 01:08:10.740 theory, and the gravity [? theory ?] 01:08:10.740 --> 01:08:13.440 to find the holographic principle, 01:08:13.440 --> 01:08:16.510 and then the natural guess is that 01:08:16.510 --> 01:08:18.740 this non-gravitational field theory should 01:08:18.740 --> 01:08:22.100 live in one dimensional lower. 01:08:22.100 --> 01:08:22.917 OK? 01:08:22.917 --> 01:08:23.750 So this is one hint. 01:08:26.899 --> 01:08:31.220 And the second is actually from the consistency 01:08:31.220 --> 01:08:32.300 of string theory itself. 01:08:45.380 --> 01:08:48.020 So this is a little bit technical. 01:08:48.020 --> 01:08:50.260 Again, we will only be able to explain it 01:08:50.260 --> 01:08:54.460 a little bit later, when we talk about more details about string 01:08:54.460 --> 01:08:55.580 theory. 01:08:55.580 --> 01:08:58.979 You can [? tell, ?] even though the string theory in this space 01:08:58.979 --> 01:09:02.386 is inconsistent. 01:09:05.090 --> 01:09:09.370 But there's a simple way. 01:09:09.370 --> 01:09:13.279 This is-- it's not a simple way. 01:09:13.279 --> 01:09:14.990 So what's happening is the following. 01:09:14.990 --> 01:09:18.290 So if you consider, say, a string 01:09:18.290 --> 01:09:26.450 propagating in this spacetime, and there are some symmetries 01:09:26.450 --> 01:09:28.750 on the worldsheet. 01:09:28.750 --> 01:09:30.990 And only in the 10 and 26 dimension, 01:09:30.990 --> 01:09:35.680 those symmetries are satisfied quantum mechanically. 01:09:35.680 --> 01:09:38.191 And in other dimensions, those symmetries, somehow, 01:09:38.191 --> 01:09:39.649 even though classically it's there, 01:09:39.649 --> 01:09:41.930 but quantum mechanically it's gone. 01:09:41.930 --> 01:09:43.680 And those symmetries become-- because they 01:09:43.680 --> 01:09:45.193 are gone quantum mechanically, then 01:09:45.193 --> 01:09:46.359 it leads to inconsistencies. 01:09:48.880 --> 01:09:54.189 And it turns out that there's some other way 01:09:54.189 --> 01:09:58.410 you can make that consistent, to make that symmetry still 01:09:58.410 --> 01:10:01.941 to be valid, is by adding some new degrees of freedom. 01:10:01.941 --> 01:10:02.440 OK? 01:10:02.440 --> 01:10:03.856 It's just there's some new degrees 01:10:03.856 --> 01:10:05.785 freedom dynamically generated. 01:10:05.785 --> 01:10:07.160 And then that new degrees freedom 01:10:07.160 --> 01:10:10.830 turned out to behave like an additional dimension. 01:10:10.830 --> 01:10:12.200 OK. 01:10:12.200 --> 01:10:13.970 Yeah, this will make no sense to you. 01:10:13.970 --> 01:10:18.970 I'm just saying a consistency of string theory actually 01:10:18.970 --> 01:10:21.060 sometimes can give you one additional dimension. 01:10:21.060 --> 01:10:23.684 AUDIENCE: What is the difference between these inconsistencies, 01:10:23.684 --> 01:10:25.630 talking about anomalies and-- 01:10:25.630 --> 01:10:27.380 HONG LIU: It is anomalies. 01:10:27.380 --> 01:10:30.210 But here it's called gauge anomalies. 01:10:30.210 --> 01:10:34.970 It's gauge anomalies is at the local symmetry anomalies, 01:10:34.970 --> 01:10:35.965 which is inconsistent. 01:10:35.965 --> 01:10:37.506 AUDIENCE: So just-- maybe this is not 01:10:37.506 --> 01:10:39.960 the time to ask this-- but are the degrees of freedom 01:10:39.960 --> 01:10:43.530 that you need to save you from this inconsistency problem. 01:10:43.530 --> 01:10:47.280 So do they have to be extra dimensions of space? 01:10:47.280 --> 01:10:51.080 Or what I'm saying is that if we need to do string theory in 10 01:10:51.080 --> 01:10:53.390 dimensions, is it really four dimensions plus six 01:10:53.390 --> 01:10:54.350 degrees of freedom? 01:10:54.350 --> 01:10:57.590 Or are they actually six bona fide spatial dimensions? 01:10:57.590 --> 01:11:00.180 HONG LIU: Oh, this is a very good question. 01:11:00.180 --> 01:11:04.920 So if you have-- yeah, this something 01:11:04.920 --> 01:11:08.772 we would be a little bit more clear just even in-- oh, it's 01:11:08.772 --> 01:11:10.180 very late. 01:11:10.180 --> 01:11:12.550 Even the second part of this lecture 01:11:12.550 --> 01:11:15.190 is that here you have four degrees of freedom, 01:11:15.190 --> 01:11:17.440 you have six degrees of freedom. 01:11:17.440 --> 01:11:20.632 But turns out, if you only consider this guy, 01:11:20.632 --> 01:11:23.090 then this four degrees freedom by itself is not consistent. 01:11:23.090 --> 01:11:24.990 It's [? its own ?] violation of the symmetry at the quantum 01:11:24.990 --> 01:11:25.900 level. 01:11:25.900 --> 01:11:28.320 And then you need to add more, and then one more, 01:11:28.320 --> 01:11:31.050 because of course one and two have extra dimension. 01:11:31.050 --> 01:11:35.710 Anyway, we can make it more explicit in next lecture. 01:11:35.710 --> 01:11:39.920 Here I just throw a remark here. 01:11:39.920 --> 01:11:46.610 Anyway, this guy-- this is purely hindsight. 01:11:46.610 --> 01:11:50.580 Nobody have realized this point, this first point, 01:11:50.580 --> 01:11:53.610 nobody have realized it before this holographic duality 01:11:53.610 --> 01:11:55.410 was discovered. 01:11:55.410 --> 01:11:56.900 Nobody really made this connection. 01:11:59.950 --> 01:12:03.200 And at this point, saying there should 01:12:03.200 --> 01:12:04.790 be a five-dimensional string theory 01:12:04.790 --> 01:12:11.990 describing gauge theory, that was made just 01:12:11.990 --> 01:12:13.860 before the discovery. 01:12:13.860 --> 01:12:15.485 I will mention that a little bit later. 01:12:18.940 --> 01:12:23.560 Anyway, so now let's-- let me just maybe finish this, 01:12:23.560 --> 01:12:24.550 and we have a break. 01:12:30.890 --> 01:12:32.690 So now let's consider-- suppose there is 01:12:32.690 --> 01:12:36.340 a five-dimensional spacetime, string theory 01:12:36.340 --> 01:12:39.170 in some five-dimensional spacetime, 01:12:39.170 --> 01:12:47.685 say 4 plus 1 dimensional spacetime that describes QCD. 01:12:50.360 --> 01:12:53.410 Then what should be the property of this Y? 01:12:53.410 --> 01:12:58.550 So this Y denotes some manifold Y. OK? 01:12:58.550 --> 01:13:03.850 So as I mentioned, it must have at least all the symmetries 01:13:03.850 --> 01:13:06.010 of the QCD, but not more. 01:13:06.010 --> 01:13:10.180 Should have exactly the same amount of symmetries. 01:13:10.180 --> 01:13:12.430 So that means it must have the translation and Lorentz 01:13:12.430 --> 01:13:14.440 symmetries of QCD. 01:13:14.440 --> 01:13:16.420 OK? 01:13:16.420 --> 01:13:22.730 So that means the only metric I can write down 01:13:22.730 --> 01:13:23.972 must be of this form. 01:13:35.660 --> 01:13:38.570 The only metric I can write down, 01:13:38.570 --> 01:13:40.190 the metric must be have this form. 01:13:43.130 --> 01:13:45.480 So this az just some function. 01:13:45.480 --> 01:13:49.060 And z is the extra dimension to a Minkowski spacetime. 01:13:49.060 --> 01:13:52.170 And this is some Minkowski metric for 3 plus 1 dimension. 01:13:54.810 --> 01:13:58.170 AUDIENCE: You mean it's like a prototype to four dimension, 01:13:58.170 --> 01:14:01.525 we have to get the Minkowski space. 01:14:01.525 --> 01:14:02.150 HONG LIU: Yeah. 01:14:02.150 --> 01:14:05.040 Just say whatever this space, whatever 01:14:05.040 --> 01:14:09.400 is the symmetry of this-- so the symmetry of this spacetime 01:14:09.400 --> 01:14:11.770 must have the Poincare-- must have 01:14:11.770 --> 01:14:14.190 all the symmetry of the 3 plus 1 dimensional 01:14:14.190 --> 01:14:15.770 Minkowski spacetime. 01:14:15.770 --> 01:14:18.550 Then the simplest way, you're saying that the only way 01:14:18.550 --> 01:14:21.040 to do it is just you put the Minkowski spacetime there 01:14:21.040 --> 01:14:23.140 as a subspace. 01:14:23.140 --> 01:14:26.130 And then you have one additional space, 01:14:26.130 --> 01:14:30.240 and then you can have one additional dimension. 01:14:30.240 --> 01:14:33.530 And then, because you have to maintain 01:14:33.530 --> 01:14:37.170 the symmetries and [INAUDIBLE] to be thinking then 01:14:37.170 --> 01:14:40.796 you can convince yourself that the only additional degrees 01:14:40.796 --> 01:14:42.170 freedom in the metric [INAUDIBLE] 01:14:42.170 --> 01:14:44.060 is the overall function. 01:14:44.060 --> 01:14:48.720 So the function of this z, and nothing else. 01:14:48.720 --> 01:14:50.510 OK. 01:14:50.510 --> 01:14:53.500 AUDIENCE: Can that be part of kind 01:14:53.500 --> 01:14:55.440 of a scalar in Minkowski space? 01:14:55.440 --> 01:14:57.110 HONG LIU: Yeah. 01:14:57.110 --> 01:15:00.820 Let me just say, this is most general 01:15:00.820 --> 01:15:13.160 metric, consistent with four dimensional, 3 plus 1 01:15:13.160 --> 01:15:18.951 dimensional, Poincare symmetries. 01:15:30.510 --> 01:15:34.590 AUDIENCE: Why this additional dimension always in a space 01:15:34.590 --> 01:15:35.290 part? 01:15:35.290 --> 01:15:37.685 Can it be in a time-like part? 01:15:37.685 --> 01:15:39.665 Like a 3 plus 2? 01:15:46.820 --> 01:15:49.406 HONG LIU: Both arguments suggest it's a space part. 01:15:49.406 --> 01:15:51.530 So because this is just the boundary of some region 01:15:51.530 --> 01:15:53.970 there's a spatial dimension [? reduction ?], not time. 01:16:01.130 --> 01:16:04.280 So is this clear to you? 01:16:04.280 --> 01:16:07.140 Because you won't have a Minkowski spacetime, 01:16:07.140 --> 01:16:09.850 so you must have a Minkowski here. 01:16:09.850 --> 01:16:11.970 And then in the prefactor of the Minkowski, 01:16:11.970 --> 01:16:14.390 you can multiply by anything, any function, 01:16:14.390 --> 01:16:17.880 but this function cannot depend on the X. 01:16:17.880 --> 01:16:20.130 It can only depend on this extra dimension. 01:16:20.130 --> 01:16:23.400 Because if you have anything which depend on capital X, 01:16:23.400 --> 01:16:26.671 then you have violated the Poincare symmetry. 01:16:26.671 --> 01:16:28.420 You have violated the translation [? X. ?] 01:16:31.250 --> 01:16:35.290 So the only function you can put before this Minkowski spacetime 01:16:35.290 --> 01:16:38.949 is a function of this additional dimension. 01:16:38.949 --> 01:16:40.990 And then by redefining this additional dimension, 01:16:40.990 --> 01:16:44.360 I can always put this overall factor in the front. 01:16:44.360 --> 01:16:46.280 Yeah, so this tells you that this 01:16:46.280 --> 01:16:48.070 is the most general metric. 01:16:48.070 --> 01:16:49.697 OK? 01:16:49.697 --> 01:16:52.030 So if it's not clear to you, think about it a little bit 01:16:52.030 --> 01:16:52.530 afterwards. 01:16:55.420 --> 01:16:58.250 So these are the most as you can do. 01:17:01.100 --> 01:17:01.850 So that's the end. 01:17:01.850 --> 01:17:06.520 So you say, you cannot determine az, et cetera. 01:17:06.520 --> 01:17:12.840 So this is as most you can say for the QCD. 01:17:12.840 --> 01:17:33.950 But if the theory, if the field theory is scale invariant, 01:17:33.950 --> 01:17:39.850 say, conformal field theory, that normally we call CFT, OK? 01:17:39.850 --> 01:17:41.675 So conformal field theory. 01:17:44.590 --> 01:17:47.800 Then we can show this metric. 01:17:47.800 --> 01:17:49.387 So let me call this equation 1. 01:17:52.970 --> 01:17:59.605 Then 1 must be [INAUDIBLE] spacetime. 01:18:03.770 --> 01:18:05.893 AUDIENCE: [INAUDIBLE] symmetry on the boundary 01:18:05.893 --> 01:18:08.190 as well, [INAUDIBLE]? 01:18:08.190 --> 01:18:10.720 HONG LIU: Yeah, I'm going to show that. 01:18:10.720 --> 01:18:15.860 So if the field theory is scale invariant, that 01:18:15.860 --> 01:18:18.600 means that the fields theory have some additional symmetry, 01:18:18.600 --> 01:18:21.470 should be satisfied by this metric. 01:18:21.470 --> 01:18:24.590 And then I will show that this additional scaling symmetry 01:18:24.590 --> 01:18:27.780 will make this to precisely a so-called anti-de Sitter 01:18:27.780 --> 01:18:29.660 spacetime. 01:18:29.660 --> 01:18:31.760 AUDIENCE: Field theory, and then the 3 plus 1. 01:18:31.760 --> 01:18:32.610 HONG LIU: Yeah. 01:18:32.610 --> 01:18:33.110 Right. 01:18:33.110 --> 01:18:39.400 If the field theory, say the-- QCD does not have a scale. 01:18:39.400 --> 01:18:41.944 It's not scaling right, so I do not say a QCD anymore. 01:18:41.944 --> 01:18:43.360 Just say, suppose some other field 01:18:43.360 --> 01:18:46.020 theory, which have large N expansion, which 01:18:46.020 --> 01:18:48.270 is also scale invariant. 01:18:48.270 --> 01:18:51.290 And then the corresponding string theory 01:18:51.290 --> 01:18:53.864 must be in anti-de Sitter spacetime. 01:18:53.864 --> 01:18:55.864 AUDIENCE: Are we ever going to come back to QCD, 01:18:55.864 --> 01:18:56.810 or is that a-- 01:18:56.810 --> 01:18:59.480 HONG LIU: No, that's it. 01:18:59.480 --> 01:19:04.310 Maybe we'll come back to QCD, but in a somewhat indirect way. 01:19:04.310 --> 01:19:09.571 Yeah, not to your real-life, beloved QCD. 01:19:09.571 --> 01:19:11.570 AUDIENCE: So no one's solved that problem still? 01:19:11.570 --> 01:19:14.300 HONG LIU: Yeah, no one's solved that problem yet. 01:19:14.300 --> 01:19:15.425 So you still have a chance. 01:19:19.590 --> 01:19:21.040 So that remains very simple. 01:19:21.040 --> 01:19:24.925 So let me just say, then we will have a break. 01:19:24.925 --> 01:19:25.800 Then we will be done. 01:19:29.510 --> 01:19:32.830 I think I'm going very slowly today. 01:19:32.830 --> 01:19:57.550 So scale invariant theory-- is invariant under the scaling 01:19:57.550 --> 01:20:02.940 for any constant, constant lambda. 01:20:09.780 --> 01:20:12.200 So scale invariant theory should be 01:20:12.200 --> 01:20:14.390 invariant under such a scaling. 01:20:14.390 --> 01:20:17.270 And then now we want to require this metric also 01:20:17.270 --> 01:20:19.100 have this scaling. 01:20:19.100 --> 01:20:20.550 OK? 01:20:20.550 --> 01:20:28.068 So now, we require 1 also have such scaling. 01:20:32.040 --> 01:20:33.152 That's scaling symmetry. 01:20:41.520 --> 01:20:46.940 OK, so we just do a scaling X mu go to lambda X mu. 01:20:50.600 --> 01:20:55.320 And then this term will give me additional lambda squared. 01:20:55.320 --> 01:20:57.920 So we see, in order for this to be the same as before, the z 01:20:57.920 --> 01:21:01.530 should scale the same, OK? 01:21:01.530 --> 01:21:07.026 So in order for this to be-- so we need z to scale as the same, 01:21:07.026 --> 01:21:10.780 in order I can scale this lambda out. 01:21:10.780 --> 01:21:13.220 After I scale this lambda out, I also 01:21:13.220 --> 01:21:25.080 need that a lambda z should be equal to 1 over lambda az, OK? 01:21:25.080 --> 01:21:27.320 So the scaling symmetry of that equation 01:21:27.320 --> 01:21:28.810 requires these two conditions. 01:21:28.810 --> 01:21:32.980 So on the scaling of z, this a lambda z 01:21:32.980 --> 01:21:34.470 should satisfy this condition. 01:21:34.470 --> 01:21:36.400 Then the lambda will cancel. 01:21:36.400 --> 01:21:38.750 So this condition is important because we 01:21:38.750 --> 01:21:40.160 did scale them homogeneously. 01:21:40.160 --> 01:21:42.930 Otherwise, of course, lambda will not drop out. 01:21:42.930 --> 01:21:47.120 And the second condition just makes sure lambda is canceled. 01:21:47.120 --> 01:21:49.240 OK, is it clear? 01:21:49.240 --> 01:21:58.470 So now this condition just determined 01:21:58.470 --> 01:22:06.130 that az must be a simple power, must 01:22:06.130 --> 01:22:09.525 be written as R divided by z. 01:22:09.525 --> 01:22:10.952 See, R is some constant. 01:22:21.790 --> 01:22:24.765 And now we can write down the full metric. 01:22:27.540 --> 01:22:29.535 So now I've determined this function up 01:22:29.535 --> 01:22:30.535 to our overall constant. 01:22:33.040 --> 01:22:38.650 So the full metric is dS square equal to R squared divided 01:22:38.650 --> 01:22:48.390 by z squared dz squared plus eta mu, mu, dX mu, dX mu. 01:22:48.390 --> 01:22:52.090 And this is precisely AdS metric, 01:22:52.090 --> 01:22:55.500 written in certain coordinates. 01:22:55.500 --> 01:22:59.064 And then this R, then you adjust the curvature radius of AdS. 01:23:02.940 --> 01:23:05.120 So if you don't know about anti-de Sitter spacetime, 01:23:05.120 --> 01:23:07.320 it doesn't matter. 01:23:07.320 --> 01:23:11.320 So this is the metric, and the name of this metric 01:23:11.320 --> 01:23:14.180 is anti-de Sitter. 01:23:14.180 --> 01:23:19.400 And later we will explain the properties 01:23:19.400 --> 01:23:23.080 of the anti-de Sitter spacetime. 01:23:23.080 --> 01:23:28.660 So now we find, so now we reach a conclusion, 01:23:28.660 --> 01:23:32.850 is that if I have a large N conformal field 01:23:32.850 --> 01:23:37.600 theory in Minkowski D-dimensional space, time. 01:23:37.600 --> 01:23:40.040 So this can be applied to any dimensional. 01:23:40.040 --> 01:23:44.610 It's not necessary [? to be ?] 3 plus 1. 01:23:44.610 --> 01:23:53.220 In D-- so this, if it can be described by a string theory, 01:23:53.220 --> 01:23:58.710 should be string theory in AdS d plus 1. 01:24:01.270 --> 01:24:05.380 And in particular, the 1/N here is related to the g strings 01:24:05.380 --> 01:24:08.300 here, the string coupling here. 01:24:08.300 --> 01:24:10.140 So this is what we concluded. 01:24:20.220 --> 01:24:21.020 Yes? 01:24:21.020 --> 01:24:22.920 AUDIENCE: So all we've shown is that there 01:24:22.920 --> 01:24:25.834 is no obvious inconsistency with that correspondence. 01:24:25.834 --> 01:24:27.750 HONG LIU: What do you mean there's no obvious? 01:24:27.750 --> 01:24:29.960 AUDIENCE: As in, we didn't illustrate any way that they-- 01:24:29.960 --> 01:24:32.340 HONG LIU: Sure, I'm just saying this is a necessary condition. 01:24:32.340 --> 01:24:34.370 AUDIENCE: Right, so at least that is necessary. 01:24:34.370 --> 01:24:36.410 HONG LIU: Yeah, this is a necessary condition. 01:24:36.410 --> 01:24:42.860 So if you can describe a large N CFT by our string theory-- 01:24:42.860 --> 01:24:47.660 and it should be a string theory-- yeah, 01:24:47.660 --> 01:24:49.450 this proposal works. 01:24:49.450 --> 01:24:51.650 This proposal passed the minimal test. 01:24:53.644 --> 01:24:54.810 AUDIENCE: I have a question. 01:24:54.810 --> 01:24:57.482 So when Maldacena presumably actually did figure this out, 01:24:57.482 --> 01:24:59.940 you said that this resulted from the holographic principle, 01:24:59.940 --> 01:25:02.420 like it was just figured out right before he did it. 01:25:02.420 --> 01:25:04.330 Was he aware of the holographic-- 01:25:04.330 --> 01:25:06.610 HONG LIU: No, here is what I'm going to talk. 01:25:06.610 --> 01:25:21.240 So Maldacena, in 1997, Maldacena found precisely-- in 1997, 01:25:21.240 --> 01:25:25.205 Maldacena found a few examples of this, precisely realized 01:25:25.205 --> 01:25:26.220 this. 01:25:26.220 --> 01:25:31.310 And not using this mass or using some completely indirect way, 01:25:31.310 --> 01:25:32.455 which we will explain next. 01:25:35.450 --> 01:25:39.490 So he found this through some very indirect way. 01:25:39.490 --> 01:25:43.440 But in principle, one could have realized this 01:25:43.440 --> 01:25:47.010 if one kept those things in mind. 01:25:47.010 --> 01:25:49.470 So now let me tell you a little bit of the history, 01:25:49.470 --> 01:25:52.500 and then we will have a break. 01:25:52.500 --> 01:25:53.490 Then we can go home. 01:25:57.764 --> 01:25:59.680 It depends on whether you want a break or not. 01:25:59.680 --> 01:26:01.989 Maybe you don't want a break. 01:26:01.989 --> 01:26:03.905 Yeah, let me tell you a little bit of history. 01:26:07.280 --> 01:26:12.740 So yeah, just to save time, let me not write it down, just 01:26:12.740 --> 01:26:14.650 say it. 01:26:14.650 --> 01:26:18.830 So in the late '60s to early '70s, so 01:26:18.830 --> 01:26:24.740 string theory was developed to understand strong interactions. 01:26:24.740 --> 01:26:30.650 So understanding strong interactions was the problem. 01:26:30.650 --> 01:26:34.190 At the time, people were developing string theory 01:26:34.190 --> 01:26:37.350 to try to understand strong interactions. 01:26:37.350 --> 01:26:44.190 So in 1971, our friend Frank, Frank Wilczek, 01:26:44.190 --> 01:26:47.070 and other people, they discover the asymptotic freedom. 01:26:47.070 --> 01:26:50.727 And they established the Yang-Mills theory 01:26:50.727 --> 01:26:52.310 as a description of strong interaction 01:26:52.310 --> 01:26:55.180 which now have our QCD. 01:26:55.180 --> 01:26:58.845 And so that's essentially eliminated the hope of string 01:26:58.845 --> 01:27:02.690 theory to describe QCD. 01:27:02.690 --> 01:27:05.160 Because the QCD seems to be very different. 01:27:05.160 --> 01:27:07.030 You [? need ?] the help of string theory 01:27:07.030 --> 01:27:10.320 to describe strong interaction because the QCD 01:27:10.320 --> 01:27:13.650 [INAUDIBLE] gauge theory, it's very different from the string 01:27:13.650 --> 01:27:15.480 theory. 01:27:15.480 --> 01:27:19.330 So people soon abandoned the string theory. 01:27:19.330 --> 01:27:23.050 So now we go to 1974. 01:27:23.050 --> 01:27:30.310 So 1974, a big number of things were discovered in 1974. 01:27:30.310 --> 01:27:32.900 So 1974 was a golden year. 01:27:32.900 --> 01:27:36.580 So first is 't Hooft realized his large N expansion 01:27:36.580 --> 01:27:38.080 and then realized that this actually 01:27:38.080 --> 01:27:40.490 looks like string theory. 01:27:40.490 --> 01:27:44.690 And then completely independently, 01:27:44.690 --> 01:27:46.710 Scherk, Schwarz, and [? Yoneya, ?] 01:27:46.710 --> 01:27:48.630 they realized that string theory should 01:27:48.630 --> 01:27:51.200 considered a theory of gravity, rather than 01:27:51.200 --> 01:27:54.660 a theory of strong interaction. 01:27:54.660 --> 01:27:58.890 So they realized actually-- it's ironic, 01:27:58.890 --> 01:28:01.920 people started doing string theory in the '60s and '70s, et 01:28:01.920 --> 01:28:02.420 cetera. 01:28:02.420 --> 01:28:04.610 But only in 1974 people realized, 01:28:04.610 --> 01:28:06.616 ah, string theory always have a gravity 01:28:06.616 --> 01:28:08.490 and should be considered a theory of gravity. 01:28:08.490 --> 01:28:10.480 Anyway, so in 1974, they realized 01:28:10.480 --> 01:28:14.210 the string theory should be considered as a gravity. 01:28:14.210 --> 01:28:18.270 So that was a very, very exciting realization, 01:28:18.270 --> 01:28:21.020 because then you can have [? quantum ?] gravity. 01:28:21.020 --> 01:28:24.620 But by that time, people had given up on string theory. 01:28:24.620 --> 01:28:28.490 So nobody cared about this important observation. 01:28:28.490 --> 01:28:31.290 Nobody cared about this important observation. 01:28:31.290 --> 01:28:33.680 So, also in the same year, in 1974, 01:28:33.680 --> 01:28:36.420 Hawking discovered his Hawking radiation. 01:28:36.420 --> 01:28:40.245 And they established that black hole mechanics is really 01:28:40.245 --> 01:28:41.580 a thermodynamics. 01:28:41.580 --> 01:28:43.440 Then really established that the black hole 01:28:43.440 --> 01:28:46.260 is a thermodynamic object, 01:28:46.260 --> 01:28:51.070 And in 1974 there's also a lot of important discovery-- 01:28:51.070 --> 01:28:52.730 which is related to MIT, so that's 01:28:52.730 --> 01:28:57.890 why I'm mentioning it-- is that people first really saw quarks 01:28:57.890 --> 01:29:02.770 experimentally, is that, again, our friend, colleague Samuel 01:29:02.770 --> 01:29:06.720 Ting at Brookhaven, which they discovered 01:29:06.720 --> 01:29:10.820 a so-called charmonium, which is a bounce state of the charm 01:29:10.820 --> 01:29:12.830 quark and the anti-charm quark. 01:29:12.830 --> 01:29:14.970 And because the charm quark is very heavy, 01:29:14.970 --> 01:29:16.760 so they form a hydrogen-like structure. 01:29:16.760 --> 01:29:21.070 So in some sense, the charmonium is the first-- you first 01:29:21.070 --> 01:29:23.590 directly see the quarks. 01:29:23.590 --> 01:29:27.280 And actually, even after the 1971, after asymptotic freedom, 01:29:27.280 --> 01:29:29.590 many people do not believe QCD. 01:29:29.590 --> 01:29:31.010 They did not believe in quarks. 01:29:31.010 --> 01:29:33.930 They say, if there's quarks, why don't we see them? 01:29:33.930 --> 01:29:43.360 And then in 1974, Samuel Ting discovered this charmonium 01:29:43.360 --> 01:29:45.690 in October. 01:29:45.690 --> 01:29:47.750 And so people call it the October Revolution. 01:29:47.750 --> 01:29:51.110 [LAUGHTER] 01:29:54.120 --> 01:29:56.818 Do you know why they laugh? 01:29:56.818 --> 01:29:57.318 OK. 01:30:01.190 --> 01:30:02.260 Anyway. 01:30:02.260 --> 01:30:02.760 Yeah. 01:30:09.330 --> 01:30:11.400 Yeah, because I saw your emotions, 01:30:11.400 --> 01:30:15.080 I think you have very good composure. 01:30:15.080 --> 01:30:20.900 Anyway, in the same year, in 1974, Wilson 01:30:20.900 --> 01:30:23.790 proposed what we now call the lattice QCD, 01:30:23.790 --> 01:30:25.760 so he put the QCD on the lattice. 01:30:25.760 --> 01:30:30.070 And then he invented, and then he 01:30:30.070 --> 01:30:31.570 developed a very beautiful technique 01:30:31.570 --> 01:30:36.520 to show from this putting QCD on the lattice 01:30:36.520 --> 01:30:41.900 that, actually, the quark can be confined through the strings. 01:30:41.900 --> 01:30:45.750 So the quarks in QCD can be confined through the strings. 01:30:45.750 --> 01:30:48.850 And that essentially revived the idea maybe 01:30:48.850 --> 01:30:52.480 the QCD can be a string theory, because the quarks are 01:30:52.480 --> 01:30:54.550 confined through the strings. 01:30:54.550 --> 01:30:58.460 And this all happened in 1974. 01:30:58.460 --> 01:31:03.206 So then I mentioned the same, in the late '80s 01:31:03.206 --> 01:31:04.830 and the early '90s, people were looking 01:31:04.830 --> 01:31:07.290 at these so-called matrix models, the matrix 01:31:07.290 --> 01:31:08.500 integrals, et cetera. 01:31:08.500 --> 01:31:10.910 Then they showed they related to lower dimensional string 01:31:10.910 --> 01:31:12.280 theory. 01:31:12.280 --> 01:31:16.360 But nobody-- yeah, they showed this related 01:31:16.360 --> 01:31:19.060 to some kind of lower dimensional string theory. 01:31:19.060 --> 01:31:24.740 And then in 1993 and 1994, then 't Hooft 01:31:24.740 --> 01:31:28.060 had this crazy idea of this holographic principle. 01:31:28.060 --> 01:31:32.080 And he said maybe, things about the quantum gravity 01:31:32.080 --> 01:31:35.220 can be described by things living on the boundary. 01:31:35.220 --> 01:31:36.770 And again, it's a crazy idea. 01:31:36.770 --> 01:31:39.970 Very few people paid attention to it. 01:31:39.970 --> 01:31:45.880 But the only person who picked it up is Leonard Susskind. 01:31:45.880 --> 01:31:51.520 And then he tried to come up with some sort of experiments 01:31:51.520 --> 01:31:53.800 to show that that idea is not so crazy. 01:31:57.220 --> 01:32:00.080 Actually, Susskind wrote a very sexy name for his paper. 01:32:00.080 --> 01:32:02.660 It's called "The World As a Hologram." 01:32:02.660 --> 01:32:08.890 And so that paper received some attention, 01:32:08.890 --> 01:32:13.990 but still, still, people did not know what to make of it. 01:32:13.990 --> 01:32:16.920 And then in 1995, Polchinski discovers so-called D-branes. 01:32:23.990 --> 01:32:27.280 And then we go to 1997. 01:32:27.280 --> 01:32:32.910 So in 1997, first in June, so as I said, 01:32:32.910 --> 01:32:35.560 that QCD may be some kind of string theory. 01:32:35.560 --> 01:32:38.000 This idea is a long idea, starting 01:32:38.000 --> 01:32:42.160 from the 't Hooft and large N expansion, and also 01:32:42.160 --> 01:32:44.630 from the Wilson's picture of confining strings 01:32:44.630 --> 01:32:47.320 from the lattice QCD, etc. 01:32:47.320 --> 01:32:48.870 But it's just a very hard problem. 01:32:48.870 --> 01:32:51.927 If from QCD, how can you come up with a string theory? 01:32:51.927 --> 01:32:52.760 It's just very hard. 01:32:52.760 --> 01:32:54.680 Very few people are working on it. 01:32:54.680 --> 01:32:58.640 So in 1997, in June, Polyakov finally, he 01:32:58.640 --> 01:33:01.389 said, had a breakthrough. 01:33:01.389 --> 01:33:02.930 He said that this consistent [? of ?] 01:33:02.930 --> 01:33:06.700 string theory give you one extra dimension, 01:33:06.700 --> 01:33:08.910 you should consider a five-dimensional string 01:33:08.910 --> 01:33:12.220 theory rather than a four-dimensional string theory. 01:33:12.220 --> 01:33:17.160 And then he gave up some arguments, anyway. 01:33:17.160 --> 01:33:23.837 And he almost always actually write down this metric 01:33:23.837 --> 01:33:26.420 And maybe he already wrote down this metric, I don't remember. 01:33:26.420 --> 01:33:28.510 Anyway, he was very close to that. 01:33:28.510 --> 01:33:30.710 But then in November, then Maldacena 01:33:30.710 --> 01:33:33.660 came up with this idea of CFT. 01:33:33.660 --> 01:33:35.700 And then he provided [? explicit ?] 01:33:35.700 --> 01:33:40.020 examples of certain large N gauge theories, which 01:33:40.020 --> 01:33:42.600 is scale invariant and some string 01:33:42.600 --> 01:33:45.640 theory in certain anti-de Sitter spacetime. 01:33:45.640 --> 01:33:49.691 And as I said, through the understanding 01:33:49.691 --> 01:33:50.440 of these D-branes. 01:33:53.200 --> 01:33:54.952 But even Maldacena's paper, he did 01:33:54.952 --> 01:33:59.840 not-- he was still thinking from the picture of large N gauge 01:33:59.840 --> 01:34:02.010 theory corresponding to some string theory. 01:34:02.010 --> 01:34:06.680 He did not make the connection to the holographic principle. 01:34:06.680 --> 01:34:10.430 He did not make a connection to the holographic principle. 01:34:10.430 --> 01:34:14.640 But very soon, in February 1998, Witten wrote the paper, 01:34:14.640 --> 01:34:16.020 and he made the connection. 01:34:16.020 --> 01:34:19.730 He said, ah, this is precisely the holographic principle. 01:34:19.730 --> 01:34:22.910 And this example, he said, ah, this example is precisely 01:34:22.910 --> 01:34:25.860 the holographic principle Susskind 01:34:25.860 --> 01:34:27.410 and 't Hooft was talking about. 01:34:31.520 --> 01:34:34.620 So that's a brief history of how people actually 01:34:34.620 --> 01:34:36.970 reached this point. 01:34:36.970 --> 01:34:40.870 So the next stage, what we are going to do 01:34:40.870 --> 01:34:47.430 is to try to derive [INAUDIBLE]. 01:34:47.430 --> 01:34:51.530 So now we can-- as I said, we have two options. 01:34:51.530 --> 01:34:54.330 We can just start from here, assuming 01:34:54.330 --> 01:34:59.180 there is CFT [? that's ?] equivalent to some string 01:34:59.180 --> 01:35:00.150 theory. 01:35:00.150 --> 01:35:04.080 And then we can see how we can develop this further. 01:35:04.080 --> 01:35:06.570 And this is one option we can take. 01:35:06.570 --> 01:35:09.620 And our other option is to really see 01:35:09.620 --> 01:35:12.550 how this relation actually arises from string theory. 01:35:12.550 --> 01:35:17.190 And many people voted for the second option, 01:35:17.190 --> 01:35:18.980 which in my [? email ?] is option one. 01:35:18.980 --> 01:35:23.860 So you want to see how this is actually 01:35:23.860 --> 01:35:26.140 deduced from string theory. 01:35:26.140 --> 01:35:31.750 So now we will do that, OK? 01:35:31.750 --> 01:35:35.910 But I should warn you, there will be some technicality 01:35:35.910 --> 01:35:38.440 you have to tolerate. 01:35:38.440 --> 01:35:41.445 You wanted to see how this is derived, OK? 01:35:44.000 --> 01:35:48.000 So we do a lot of [? 20 ?] minutes today? 01:35:48.000 --> 01:35:49.881 Without break? 01:35:49.881 --> 01:35:50.380 Good. 01:35:50.380 --> 01:35:50.980 OK. 01:35:50.980 --> 01:35:52.730 Yeah, next time, I will remember to break. 01:36:11.910 --> 01:36:12.410 OK. 01:36:15.870 --> 01:36:17.970 So now we are going to derive this. 01:36:17.970 --> 01:36:23.410 So first just as a preparation, I 01:36:23.410 --> 01:36:26.580 need to tell you a little bit more about string theory. 01:36:26.580 --> 01:36:31.070 In particular, the spectrum of closed strings, 01:36:31.070 --> 01:36:32.285 closed and open strings. 01:36:39.580 --> 01:36:43.180 And so this is where the gravity-- 01:36:43.180 --> 01:36:45.440 and from a closed string you will see the gravity, 01:36:45.440 --> 01:36:47.773 and from the open string, you will see the gauge theory. 01:36:52.420 --> 01:36:52.920 OK. 01:36:56.240 --> 01:36:59.560 We will see gravity and gauge theory. 01:36:59.560 --> 01:37:03.130 So these are the first things we will do. 01:37:03.130 --> 01:37:12.600 So the second thing we will do-- so the second thing we will do 01:37:12.600 --> 01:37:14.410 is to understand the physics of D-branes. 01:37:20.670 --> 01:37:24.300 So D-brane is some object in string theory. 01:37:24.300 --> 01:37:29.490 And it turned out to play a very, very special role, 01:37:29.490 --> 01:37:33.480 to connect the gravity and the string theory. 01:37:33.480 --> 01:37:34.120 OK. 01:37:34.120 --> 01:37:37.010 Connect the gravity and the string theory. 01:37:37.010 --> 01:37:39.800 Because this is the connection between the gravity 01:37:39.800 --> 01:37:41.790 and the string theory. 01:37:41.790 --> 01:37:45.380 And in string theory, this [? object will ?] 01:37:45.380 --> 01:37:47.060 deeply and precisely play this role, 01:37:47.060 --> 01:37:49.101 which connects the gravity and the string theory. 01:37:49.101 --> 01:37:51.060 So that's why you can deduce such a relation. 01:37:51.060 --> 01:37:53.086 OK. 01:37:53.086 --> 01:37:54.710 Yeah, so this is the two things we will 01:37:54.710 --> 01:37:56.620 do before we can derive this. 01:38:01.000 --> 01:38:07.810 So this is, say, the rough plan we 01:38:07.810 --> 01:38:11.830 will do before we can derive this gravity. 01:38:11.830 --> 01:38:14.920 So first let's tell you a little bit more about string theory. 01:38:25.420 --> 01:38:28.930 So at beginning, just say some more general setup 01:38:28.930 --> 01:38:30.230 of string theory. 01:38:36.470 --> 01:38:56.320 So let's consider a string moving in a spacetime, which 01:38:56.320 --> 01:39:02.560 I denote by M, say, with the metric ds squared 01:39:02.560 --> 01:39:05.630 equal to g mu mu. 01:39:05.630 --> 01:39:11.670 And this can depend on X, dX mu, dX mu. 01:39:11.670 --> 01:39:13.440 OK? 01:39:13.440 --> 01:39:17.400 So you can imagine some general curved spacetime. 01:39:17.400 --> 01:39:22.075 Say mu and nu will go from 0 to 1, to D minus 1. 01:39:22.075 --> 01:39:31.090 So D is the total number of space dimensions for this M. 01:39:31.090 --> 01:39:49.380 So the motion of the string, as we said quite a few times now, 01:39:49.380 --> 01:39:54.750 is the embedding of the worldsheet to the spacetime. 01:39:54.750 --> 01:39:59.760 So this is in the form of X mu sigma tau. 01:39:59.760 --> 01:40:03.010 OK, you parameterize the worldsheet by two coordinates. 01:40:03.010 --> 01:40:08.170 So I will also write it as X mu sigma a. 01:40:08.170 --> 01:40:13.260 And the sigma a is equal to sigma 0, and the sigma 01:40:13.260 --> 01:40:15.500 1 is equal to tau sigma, OK? 01:40:18.715 --> 01:40:19.965 And we will use this notation. 01:40:24.300 --> 01:40:31.922 So now imagine a surface embedded in some spacetime. 01:40:31.922 --> 01:40:33.380 And this is the embedding equation. 01:40:33.380 --> 01:40:37.770 Because if you know those functions, 01:40:37.770 --> 01:40:40.810 then you know precisely how the surface are embedded, OK? 01:40:43.620 --> 01:40:47.870 And because the original spacetime have a metric, 01:40:47.870 --> 01:40:56.845 then this induced metric on the worldsheet. 01:40:59.780 --> 01:41:04.086 And this induced metric is very easy to write down. 01:41:04.086 --> 01:41:07.335 You just plug in this function into here. 01:41:09.920 --> 01:41:11.330 And when you take the derivative, 01:41:11.330 --> 01:41:15.080 you only worry that sigma and tau, 01:41:15.080 --> 01:41:18.820 because then that means you're restricted on the surface, 01:41:18.820 --> 01:41:20.980 when your only [? value is ?] sigma and tau. 01:41:20.980 --> 01:41:23.900 And then you can plug this into there. 01:41:23.900 --> 01:41:28.130 So you can get the metric, then can be written in this form. 01:41:28.130 --> 01:41:30.930 Here's sigma a and this sigma b. 01:41:30.930 --> 01:41:31.660 OK? 01:41:31.660 --> 01:41:35.440 So remember, sigma a and sigma b just tau and sigma. 01:41:35.440 --> 01:41:47.490 And this hab is just equal to g mu mu, X, partial a, X mu, 01:41:47.490 --> 01:41:50.230 partial b, X nu. 01:41:50.230 --> 01:41:51.937 OK? 01:41:51.937 --> 01:41:53.020 So this is trivial to see. 01:41:53.020 --> 01:41:57.960 Just plug this into there, to the variation with sigma 01:41:57.960 --> 01:42:01.000 and tau, you just get that, and it's that. 01:42:01.000 --> 01:42:02.200 OK? 01:42:02.200 --> 01:42:03.200 Is it clear? 01:42:12.200 --> 01:42:20.420 So this Nambu-Goto action is the tension-- tension 01:42:20.420 --> 01:42:26.765 we always write this 1 over 2 pi alpha prime-- dA. 01:42:26.765 --> 01:42:32.230 So alpha prime is the [INAUDIBLE] dimensions square. 01:42:32.230 --> 01:42:38.270 So we often also write alpha prime as ls square. 01:42:38.270 --> 01:42:40.520 So alpha prime, just a parameter, too. 01:42:40.520 --> 01:42:44.770 Parameterize to [? load ?] the tension of the string. 01:42:44.770 --> 01:42:48.410 So this area, of course, you can just 01:42:48.410 --> 01:42:55.350 write it as d squared sigma. 01:42:55.350 --> 01:42:58.210 So again, you use the notation d squared sigma 01:42:58.210 --> 01:43:01.140 just d sigma d tau. 01:43:01.140 --> 01:43:05.520 d squared sigma minus delta h. 01:43:05.520 --> 01:43:06.020 OK. 01:43:06.020 --> 01:43:07.603 So this is just the area, because this 01:43:07.603 --> 01:43:11.354 is the induced metric on the worldsheet. 01:43:11.354 --> 01:43:13.770 Then you take the determinant, and that give you the area. 01:43:13.770 --> 01:43:15.519 So this is the standard geometric formula. 01:43:19.240 --> 01:43:22.850 So now let me call this equation 1. 01:43:22.850 --> 01:43:25.237 So I have a [? lot ?] equation 1 before, 01:43:25.237 --> 01:43:26.320 but this is a new chapter. 01:43:31.783 --> 01:43:32.470 OK. 01:43:32.470 --> 01:43:36.570 So this is the explicit form of this Nambu-Goto action. 01:43:36.570 --> 01:43:40.160 But this action is a little bit awkward, 01:43:40.160 --> 01:43:43.720 because involving the square root. 01:43:43.720 --> 01:43:46.660 A square root, it's considered to be 01:43:46.660 --> 01:43:48.980 not a good thing in physics. 01:43:48.980 --> 01:43:51.030 Because when you write down action, 01:43:51.030 --> 01:43:52.280 because it's a non-polynomial. 01:43:54.400 --> 01:43:56.602 We typically like polynomial things. 01:43:56.602 --> 01:43:59.060 Because the only integral we can do is a Gaussian integral, 01:43:59.060 --> 01:44:00.351 and the Gaussian is polynomial. 01:44:03.590 --> 01:44:08.810 So this is inconvenient, so one can rewrite it a little bit. 01:44:08.810 --> 01:44:10.358 So you write down the answer. 01:44:17.850 --> 01:44:22.500 So we can rewrite it in the polynomial form. 01:44:22.500 --> 01:44:25.120 And this polynomial form is corresponding-- 01:44:25.120 --> 01:44:29.930 it's called the Polyakov action, so I call it SP, 01:44:29.930 --> 01:44:31.900 even though Polyakov had nothing to do with it. 01:44:36.270 --> 01:44:39.810 And this action can be written in the following form. 01:44:39.810 --> 01:44:41.600 And let me write down the answer. 01:44:41.600 --> 01:44:43.076 Then I will show the equivalent. 01:44:47.666 --> 01:44:51.390 AUDIENCE: Wasn't it invented by Leonard Susskind? 01:44:51.390 --> 01:44:53.340 HONG LIU: No, it's not Leonard Susskind. 01:44:53.340 --> 01:44:55.840 [INTERPOSING VOICES] 01:44:55.840 --> 01:44:57.600 AUDIENCE: Why is it called Polyakov-- 01:44:57.600 --> 01:44:59.900 HONG LIU: Polyakov-- yeah, actually Polyakov 01:44:59.900 --> 01:45:01.180 had something to do with it. 01:45:05.030 --> 01:45:08.060 Polyakov used it mostly [INAUDIBLE] first. 01:45:15.120 --> 01:45:19.110 OK, so you can rewrite it as that, in this form. 01:45:19.110 --> 01:45:22.030 And the gamma ab is a new variable introduced. 01:45:22.030 --> 01:45:25.090 It's a Lagrangian multiplier. 01:45:25.090 --> 01:45:25.590 OK. 01:45:28.320 --> 01:45:30.300 So let me point out a few things. 01:45:30.300 --> 01:45:33.980 So this structure is precisely just this hab. 01:45:33.980 --> 01:45:35.700 So that's if you look at this structure, 01:45:35.700 --> 01:45:41.010 so this structure is precisely what I called hab. 01:45:41.010 --> 01:45:44.690 So now the claim is adding [INAUDIBLE] 01:45:44.690 --> 01:45:48.770 to original variable with just X. Now 01:45:48.770 --> 01:45:52.520 I introduce a new variable, gamma. 01:45:52.520 --> 01:45:54.480 And gamma is like a Lagrangian multiplier, 01:45:54.480 --> 01:45:57.730 because there's no connected term for gamma. 01:45:57.730 --> 01:46:02.860 So if I eliminate gamma, then I will recover this. 01:46:02.860 --> 01:46:04.430 OK, so this is the claim. 01:46:04.430 --> 01:46:05.470 So now let me show that. 01:46:08.650 --> 01:46:12.980 This is very easy to see. 01:46:12.980 --> 01:46:19.410 Because if you just do the variation of gamma, 01:46:19.410 --> 01:46:22.400 do the variation of gamma ab. 01:46:22.400 --> 01:46:23.060 OK. 01:46:23.060 --> 01:46:26.730 So whenever I wrote in this is in [? upstairs, ?] 01:46:26.730 --> 01:46:27.940 it always means the inverse. 01:46:27.940 --> 01:46:31.352 OK, this is the standard notation for the metric. 01:46:31.352 --> 01:46:33.560 So if you look at the equation of motion, [INAUDIBLE] 01:46:33.560 --> 01:46:36.930 by variation of this gamma ab, then what you'll find 01:46:36.930 --> 01:46:39.929 is that the gamma ab-- just do the variation of that action. 01:46:39.929 --> 01:46:41.762 You find the equation of motion for gamma ab 01:46:41.762 --> 01:46:42.845 is given by the following. 01:46:45.950 --> 01:46:48.350 So hab, just that guy. 01:46:48.350 --> 01:46:51.374 And the lambda is arbitrary constant, or lambda 01:46:51.374 --> 01:46:52.856 is arbitrary function. 01:47:06.220 --> 01:47:09.480 So this I'm sure you can do. 01:47:09.480 --> 01:47:11.060 You just do the variation. 01:47:11.060 --> 01:47:13.060 You find that equation. 01:47:13.060 --> 01:47:16.940 So now we can just verify this actually works. 01:47:16.940 --> 01:47:22.360 When you substitute this into here, OK, into here. 01:47:22.360 --> 01:47:25.560 So this gamma ab, when you take the inverse, then [? cause ?] 01:47:25.560 --> 01:47:30.020 one into the inverse, hab, inverse hab contracted 01:47:30.020 --> 01:47:32.800 with this hab just give you 2. 01:47:32.800 --> 01:47:38.401 And that 2-- did I put that 2 in the right place? 01:47:41.390 --> 01:47:43.122 That gave you 2. 01:47:43.122 --> 01:47:49.350 And that have a 2 on-- yeah, I'm confused about 2 now. 01:47:49.350 --> 01:47:53.020 Oh, no, no, it's fine. 01:47:53.020 --> 01:47:56.310 Anyway, so this contracted with that, 01:47:56.310 --> 01:48:03.850 so gamma ab contracted with hab give you 2 divided by lambda, 01:48:03.850 --> 01:48:06.490 times 2. 01:48:06.490 --> 01:48:07.720 OK? 01:48:07.720 --> 01:48:11.690 Because you just invert this guy and invert the lambda and 2. 01:48:11.690 --> 01:48:17.190 And then square root of minus gamma give me 1/2 01:48:17.190 --> 01:48:21.460 lambda, square root minus h. 01:48:21.460 --> 01:48:21.960 OK? 01:48:24.650 --> 01:48:28.230 So sometimes I also approximate. 01:48:28.230 --> 01:48:30.275 I will not write this determinant explicitly. 01:48:30.275 --> 01:48:32.650 When I write [? less h, ?] it means the determinant of h. 01:48:32.650 --> 01:48:34.970 And the minus gamma, determinant of gamma. 01:48:34.970 --> 01:48:35.770 OK? 01:48:35.770 --> 01:48:39.350 So you multiply these two together, so these two cancel. 01:48:39.350 --> 01:48:43.240 And this two, multiply this 4 pi alpha prime, 01:48:43.240 --> 01:48:46.194 and then get back that, OK? 01:48:46.194 --> 01:48:47.110 So they're equivalent. 01:48:49.980 --> 01:48:51.930 Clear? 01:48:51.930 --> 01:48:54.264 So this gives you [? SNG. ?] 01:49:00.660 --> 01:49:10.350 So now the key-- so now if you look at this form, 01:49:10.350 --> 01:49:17.410 this really have a polynomial form for X, OK? 01:49:17.410 --> 01:49:21.940 So now let me call this equation 2. 01:49:21.940 --> 01:49:26.280 So equation 2, if you look at that expression, 01:49:26.280 --> 01:49:28.880 just has the form-- so this is just 01:49:28.880 --> 01:49:33.560 like a two-dimensional field theory-- has 01:49:33.560 --> 01:49:38.320 the form of a two-dimensional scalar field 01:49:38.320 --> 01:49:50.917 theory in the curved spacetime. 01:49:57.740 --> 01:50:03.030 Of course, the curved spacetime is just our worldsheet sigma 01:50:03.030 --> 01:50:10.298 with metric gamma ab, OK? 01:50:19.200 --> 01:50:26.400 So this is just like-- but the key here, 01:50:26.400 --> 01:50:29.225 so sometimes 2 is called the nonlinear sigma model, just 01:50:29.225 --> 01:50:32.480 traditionally, a theory of the form 01:50:32.480 --> 01:50:35.120 that equation 2 is called the nonlinear sigma model. 01:50:35.120 --> 01:50:38.640 Nonlinear because typically this metric can depend on X, 01:50:38.640 --> 01:50:40.550 and so dependence on X is nonlinear. 01:50:40.550 --> 01:50:43.802 So it's called nonlinear sigma model. 01:50:43.802 --> 01:50:51.328 But I would say it's both gamma ab and X are dynamical. 01:50:59.200 --> 01:51:00.570 Are dynamical variables. 01:51:08.310 --> 01:51:12.900 So that means when you do the path integral, 01:51:12.900 --> 01:51:19.926 so in the path integral quantization, 01:51:19.926 --> 01:51:21.800 you need to integrate over all possible gamma 01:51:21.800 --> 01:51:24.010 ab and all possible X mu. 01:51:24.010 --> 01:51:26.820 Not only integrate over all possible X mu, 01:51:26.820 --> 01:51:31.270 but also integrate all possible gamma ab with this action. 01:51:39.770 --> 01:51:41.860 OK. 01:51:41.860 --> 01:51:46.060 So this is a two-dimensional [? world ?] 01:51:46.060 --> 01:51:47.810 with some scalar field. 01:51:47.810 --> 01:51:52.060 And you integrate over all possible metric, 01:51:52.060 --> 01:51:56.100 so over all possible intrinsic metric in that [? world. ?] 01:51:56.100 --> 01:52:07.810 So this can also be considered as 2D gravity, 01:52:07.810 --> 01:52:16.538 two-dimensional gravity, coupled to D scalar fields. 01:52:25.570 --> 01:52:29.400 So now we see that when you rewrite anything 01:52:29.400 --> 01:52:36.120 in this polynomial form, in this Polyakov form, 01:52:36.120 --> 01:52:39.730 the problem of quantizing the string 01:52:39.730 --> 01:52:44.010 become the problem of quantizing two-dimensional gravity coupled 01:52:44.010 --> 01:52:47.050 to D scalar fields. 01:52:47.050 --> 01:52:48.020 OK. 01:52:48.020 --> 01:52:52.470 So this may look very scary, but it turns out actually 01:52:52.470 --> 01:52:55.150 two-dimensional gravity is very simple. 01:52:55.150 --> 01:52:56.490 So it's actually not scary. 01:52:56.490 --> 01:52:58.700 So in the end, for many situations, 01:52:58.700 --> 01:53:01.637 this just reduced to, say, a quantizing scalar 01:53:01.637 --> 01:53:03.220 field with a little bit of subtleties. 01:53:06.860 --> 01:53:08.782 So yeah, let's stop here.