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 --> 00:00:06.050 Your support will help MIT OpenCourseWare 00:00:06.050 --> 00:00:10.150 continue to offer high quality educational resources for free. 00:00:10.150 --> 00:00:12.690 To make a donation or to view additional materials 00:00:12.690 --> 00:00:16.122 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.122 --> 00:00:17.084 at ocw.mit.edu. 00:00:21.625 --> 00:00:22.750 HONG LIU: And let us start. 00:00:25.940 --> 00:00:31.040 So let me again start by reminding you 00:00:31.040 --> 00:00:31.995 what we did before. 00:00:35.320 --> 00:00:42.170 So consider, say, we have a scalar field in AdS 00:00:42.170 --> 00:00:45.170 So let me just summarize what we did before. 00:00:49.900 --> 00:00:59.360 So consider, say, a scalar field, 00:00:59.360 --> 00:01:09.350 phi, in AdS and with mass times square, so with mass 00:01:09.350 --> 00:01:13.130 squared given by m squared, then we know, 00:01:13.130 --> 00:01:15.725 when you go to the boundary of AdS-- so let's 00:01:15.725 --> 00:01:22.010 take z goes to 0-- then phi will have 00:01:22.010 --> 00:01:26.080 the following asymptotic behavior. 00:01:26.080 --> 00:01:33.120 So these are the two independent modes of phi [? to ?] infinity. 00:01:46.060 --> 00:01:48.760 Of course, there are higher order corrections in z. 00:01:51.470 --> 00:01:59.170 And this delta is given by 1/2 d plus nu. 00:01:59.170 --> 00:02:02.310 And nu is given by this square divided 00:02:02.310 --> 00:02:05.326 by 4 plus m squared r squared. 00:02:10.315 --> 00:02:11.314 So this is the behavior. 00:02:16.210 --> 00:02:24.625 And so this phi is assumed to be dual to some boundary, scalar 00:02:24.625 --> 00:02:25.125 operator. 00:02:28.290 --> 00:02:29.960 And then all of those quantities, 00:02:29.960 --> 00:02:32.010 on the gravity side, they all have counterparts 00:02:32.010 --> 00:02:34.050 on field theory side. 00:02:34.050 --> 00:02:39.340 For example, the delta, which, essentially, 00:02:39.340 --> 00:02:42.470 is a function of mass, then, essentially, it's 00:02:42.470 --> 00:02:51.410 the scaling dimension of O. It's a scaling dimension of O. 00:02:51.410 --> 00:02:54.730 And this Ax essentially translates 00:02:54.730 --> 00:02:58.000 into the source for phi. 00:02:58.000 --> 00:03:02.330 Or you can consider the momentum space version, say, you 00:03:02.330 --> 00:03:06.080 can do Fourier transform, Ak, then go to the phi k. 00:03:09.630 --> 00:03:16.410 And then we also showed, last time, that the 2 nu times Bx 00:03:16.410 --> 00:03:23.420 is actually related to the expectation value of O. 00:03:23.420 --> 00:03:25.815 And again, there's a momentum space version of it. 00:03:25.815 --> 00:03:27.440 So you can just Fourier transform, then 00:03:27.440 --> 00:03:28.910 become O k and Bk. 00:03:34.300 --> 00:03:36.160 And in particular, in the example, 00:03:36.160 --> 00:03:37.760 we can see the scalar field example. 00:03:37.760 --> 00:03:39.759 In particular, in the example, you can see that. 00:03:52.510 --> 00:03:57.850 So we derived this relation from a free scalar field. 00:04:00.959 --> 00:04:03.500 but you can actually show this relation is actually generally 00:04:03.500 --> 00:04:05.560 true at non-linear level. 00:04:05.560 --> 00:04:09.330 So if you also include the high orders, et cetera, so including 00:04:09.330 --> 00:04:12.260 non-linear dependence on phi, actually this relation 00:04:12.260 --> 00:04:15.305 remains true. 00:04:15.305 --> 00:04:19.670 But due to the time, we will not go into that. 00:04:19.670 --> 00:04:24.410 The proof is actually not very difficult. 00:04:24.410 --> 00:04:26.440 So you can see that the free scalar examples-- 00:04:26.440 --> 00:04:28.273 so, in the example, we can see that actually 00:04:28.273 --> 00:04:35.360 the B-- so in momentum space, the B is actually 00:04:35.360 --> 00:04:44.485 proportional to A. If you remember what we did last time. 00:04:44.485 --> 00:04:46.210 So in other words, this one-point 00:04:46.210 --> 00:04:49.890 function-- so B is related to the one-point function. 00:04:49.890 --> 00:04:52.080 So in other words, the one-point function 00:04:52.080 --> 00:04:55.090 it's proportional to the source. 00:04:55.090 --> 00:04:56.540 So this is the standard story. 00:05:00.630 --> 00:05:04.040 Say all x equal to 0 when phi equal to 0. 00:05:06.820 --> 00:05:09.540 So if you don't have the source, then this one-point function 00:05:09.540 --> 00:05:10.940 is 0. 00:05:10.940 --> 00:05:17.830 So you don't have a spontaneous expectation value for the O. 00:05:17.830 --> 00:05:23.640 And now the fact that B is proportional to A, 00:05:23.640 --> 00:05:36.640 so this is just reflects that, in the presence of the source, 00:05:36.640 --> 00:05:40.430 then, of course, the expectation value-- so 00:05:40.430 --> 00:05:44.020 let me write it in momentum space for simplicity. 00:05:44.020 --> 00:05:45.400 Then we're no longer [INAUDIBLE]. 00:05:50.530 --> 00:05:54.470 Assume the source is small, then you can expand the expectation 00:05:54.470 --> 00:05:57.880 value in power series of phi. 00:05:57.880 --> 00:06:00.590 So to the 0, so that would be phi, and then 00:06:00.590 --> 00:06:01.610 phi squared, et cetera. 00:06:06.440 --> 00:06:10.420 So at linearized level, say we just find 00:06:10.420 --> 00:06:12.226 O k is proportional to phi k. 00:06:20.040 --> 00:06:24.870 We just find that O k is proportional to phi k. 00:06:24.870 --> 00:06:27.740 And the proportional constant is, in fact, 00:06:27.740 --> 00:06:28.740 the two-point function. 00:06:36.090 --> 00:06:38.979 So at linear level, so O k would be just proportional to phi k, 00:06:38.979 --> 00:06:41.520 but the proportional constant is just the two-point function. 00:06:45.782 --> 00:06:46.990 So the reason is very simple. 00:06:52.576 --> 00:06:54.200 First, let me write down the definition 00:06:54.200 --> 00:06:57.120 of this two-point function. 00:06:57.120 --> 00:06:59.560 So GEx in coordinate space is defined 00:06:59.560 --> 00:07:06.115 to be O x of 0, because of translational symmetry, 00:07:06.115 --> 00:07:08.050 so I can put the other point to be 0. 00:07:08.050 --> 00:07:12.007 So this is only the function of one variable, x. 00:07:12.007 --> 00:07:13.465 And then you can Fourier transform, 00:07:13.465 --> 00:07:15.196 and then you get GEk. 00:07:15.196 --> 00:07:16.570 So this is how we define the GEk. 00:07:20.050 --> 00:07:24.920 So to see that this is true, you need 00:07:24.920 --> 00:07:28.800 to remember that the GEk is given 00:07:28.800 --> 00:07:39.290 from taking the generating functional, delta phi k 00:07:39.290 --> 00:07:44.570 and delta phi minus k, then you set phi equal to 0. 00:07:44.570 --> 00:07:50.002 So this is the definition of the two-point function. 00:07:50.002 --> 00:07:52.376 And then next, you can see that the [? one derivative ?] 00:07:52.376 --> 00:07:59.110 of phi minus k-- so you can see that this takes 1 derivative 00:07:59.110 --> 00:07:59.860 on phi minus k. 00:07:59.860 --> 00:08:04.610 That just gives you the one-point function 00:08:04.610 --> 00:08:07.650 in the presence of phi and then delta phi k. 00:08:10.440 --> 00:08:13.060 And take phi equal to 0. 00:08:13.060 --> 00:08:15.140 So I just evaluated this derivative. 00:08:15.140 --> 00:08:17.679 So if you take this generating functional, 00:08:17.679 --> 00:08:19.845 take one derivative, we just get one-point function. 00:08:19.845 --> 00:08:23.620 And then this is the other point function. 00:08:23.620 --> 00:08:27.650 Then at the linear level-- so if O k 00:08:27.650 --> 00:08:29.980 is linear in phi k, of course, taking the derivative 00:08:29.980 --> 00:08:32.960 is just the same as dividing it. 00:08:32.960 --> 00:08:33.970 So this is the same. 00:08:39.409 --> 00:08:42.100 So continue over. 00:08:42.100 --> 00:08:49.290 This is the same just as to the phi k divided by phi k 00:08:49.290 --> 00:08:50.610 at the linearized level. 00:08:50.610 --> 00:08:54.062 And then so this tells you that this pre-factor is just 00:08:54.062 --> 00:08:55.020 the two-point function. 00:09:02.340 --> 00:09:06.160 And so we can now write a two-point function explicitly 00:09:06.160 --> 00:09:08.760 in terms of B and A, so this is just given, 00:09:08.760 --> 00:09:12.690 because this is 2 nu Bk. 00:09:12.690 --> 00:09:17.080 So this is just given by that. 00:09:22.080 --> 00:09:23.830 So this is just a quick summary of what we 00:09:23.830 --> 00:09:25.670 did at the end of last lecture. 00:09:30.000 --> 00:09:33.870 So that's how you can find the two-point function. 00:09:33.870 --> 00:09:37.480 So you work out what B is, what the A is, 00:09:37.480 --> 00:09:39.770 and then the rest of them is the two-point function. 00:09:42.350 --> 00:09:47.527 And the way we work out the B and A is the following. 00:09:47.527 --> 00:09:50.110 So at the linearized level, when you solve the equation motion 00:09:50.110 --> 00:09:54.580 for phi, then you impose the boundary conditions 00:09:54.580 --> 00:09:59.604 at infinity, A should be equal to phi. 00:09:59.604 --> 00:10:01.020 But then there's another condition 00:10:01.020 --> 00:10:03.290 being proposed to make sure the phi is actually 00:10:03.290 --> 00:10:05.320 regular in the interior. 00:10:05.320 --> 00:10:08.700 So the regularity condition, there's still 00:10:08.700 --> 00:10:12.900 a relation between A and B. And then 00:10:12.900 --> 00:10:17.950 that leads to this two-point function. 00:10:17.950 --> 00:10:18.870 Any questions on this? 00:10:21.630 --> 00:10:24.060 Is everything clear? 00:10:24.060 --> 00:10:25.364 Good. 00:10:25.364 --> 00:10:27.405 So now let's consider the higher-point functions. 00:10:50.260 --> 00:10:54.610 So this is, in principle, straightforward to do. 00:10:54.610 --> 00:11:05.110 Because, recall, that the log CFT phi, this generating 00:11:05.110 --> 00:11:09.700 functional, in the field theory, is just 00:11:09.700 --> 00:11:11.640 given by the classical action on the gravity 00:11:11.640 --> 00:11:16.770 side, which is the boundary condition given by phi. 00:11:19.719 --> 00:11:21.260 You're solving the classical equation 00:11:21.260 --> 00:11:23.170 motion, the initial action. 00:11:26.730 --> 00:11:29.640 Then you just, as we did earlier, 00:11:29.640 --> 00:11:32.590 you just solve the equation motion for phi 00:11:32.590 --> 00:11:35.110 to the [? linear ?] order. 00:11:35.110 --> 00:11:37.940 And then you validated the initial action. 00:11:41.890 --> 00:11:44.890 Then you can express, for example, this SE 00:11:44.890 --> 00:11:46.700 in power series of phi. 00:11:46.700 --> 00:11:52.950 And then the quotient for the power series of this phi 00:11:52.950 --> 00:11:56.540 then just gives you correlation functions. 00:11:56.540 --> 00:11:59.620 So the principle of this is straightforward to do. 00:11:59.620 --> 00:12:02.620 We know how to solve the classical equation motion. 00:12:02.620 --> 00:12:06.100 We know how to plug back into the action, 00:12:06.100 --> 00:12:08.950 and then you are done. 00:12:08.950 --> 00:12:11.522 So just to be able to be more specific, 00:12:11.522 --> 00:12:12.605 let's consider an example. 00:12:21.660 --> 00:12:25.671 Let's just consider two examples. 00:12:25.671 --> 00:12:27.545 Again, let we just consider the scalar field. 00:12:35.450 --> 00:12:36.590 Now, I add the linear term. 00:12:43.670 --> 00:12:47.780 For illustration, let me give you two examples. 00:12:47.780 --> 00:12:52.870 And so a very important thing is that in our set-up, the lambda, 00:12:52.870 --> 00:12:57.440 this cubic quotient is proportional to kappa. 00:12:57.440 --> 00:12:59.830 And then if you translate into the field series side, 00:12:59.830 --> 00:13:02.180 this is 1/n. 00:13:02.180 --> 00:13:06.300 So remember, for anything in the gravity action, 00:13:06.300 --> 00:13:08.990 you always have the 1 over kappa squared, 1 00:13:08.990 --> 00:13:12.370 over G Newton before [INAUDIBLE] action. 00:13:12.370 --> 00:13:16.870 And then when we try to change phi into canonical 00:13:16.870 --> 00:13:20.370 normalized, and then you rescale phi. 00:13:20.370 --> 00:13:24.560 And the consequence is that this cubic term or quartic term, 00:13:24.560 --> 00:13:28.820 then they will have quotients proportionate to kappa. 00:13:28.820 --> 00:13:32.080 And according to this rule, if you have a quartic term, then 00:13:32.080 --> 00:13:35.070 that will be proportional to kappa squared, et cetera. 00:13:35.070 --> 00:13:39.774 And so the key is that as far as [? phi is ?] order 1, 00:13:39.774 --> 00:13:41.940 then you can actually treat those higher-order terms 00:13:41.940 --> 00:13:44.760 by perturbation because the quotient is small. 00:13:51.667 --> 00:13:53.750 And of course, you can also consider the situation 00:13:53.750 --> 00:13:55.208 in which phi is large, and then you 00:13:55.208 --> 00:13:57.072 have to solve the full nonlinear problem. 00:13:57.072 --> 00:13:59.930 Then you cannot do a perturbative expansion. 00:13:59.930 --> 00:14:04.080 You have to do the problem. 00:14:04.080 --> 00:14:05.740 So for our purpose, if you want to find 00:14:05.740 --> 00:14:08.080 the higher-point function for phi, 00:14:08.080 --> 00:14:09.790 then you need to solve this equation. 00:14:14.940 --> 00:14:16.700 So box phi is just a Laplace operator. 00:14:19.228 --> 00:14:20.860 I just save time. 00:14:25.650 --> 00:14:28.660 So you need to solve this equation. 00:14:28.660 --> 00:14:44.090 And with the boundary condition still the boundary condition, 00:14:44.090 --> 00:14:46.810 the boundary value of capital phi to equal to this phi x. 00:14:50.110 --> 00:14:53.900 And let me call this equation star. 00:14:53.900 --> 00:14:56.690 So as I said, because this lambda is small, 00:14:56.690 --> 00:15:00.744 you can actually try to solve this equation just 00:15:00.744 --> 00:15:01.785 perturbatively in lambda. 00:15:04.320 --> 00:15:08.720 And essentially, you're solving it perturbatively in this phi, 00:15:08.720 --> 00:15:11.440 because everything, in the end, will be expressed in this phi. 00:15:23.360 --> 00:15:34.850 So one can solve star perturbatively 00:15:34.850 --> 00:15:40.760 in phi, which is the boundary value of the field. 00:15:40.760 --> 00:15:47.822 And say phi c, you can expand it as phi 1 plus phi 2, et cetera, 00:15:47.822 --> 00:15:49.280 so the classical solution expanded. 00:15:49.280 --> 00:15:50.350 And this linear in phi. 00:15:53.780 --> 00:16:01.010 And this is quadratic in phi-- and et cetera. 00:16:03.990 --> 00:16:10.130 And then when you plug back this into the action, 00:16:10.130 --> 00:16:13.240 then you find on the initial action 00:16:13.240 --> 00:16:16.310 can also be expanded in power series of this phi. 00:16:16.310 --> 00:16:22.365 So you will have S2 phi plus-- started with quadratic order-- 00:16:22.365 --> 00:16:27.030 and 3 phi, et cetera. 00:16:27.030 --> 00:16:31.270 And then the quotient here gives you the two-point function. 00:16:31.270 --> 00:16:33.999 And then quotient here gives you the three-point function. 00:16:33.999 --> 00:16:36.540 And the quotient here gives a four-point function, et cetera. 00:16:46.432 --> 00:16:48.045 I hope the procedure is clear. 00:16:56.207 --> 00:16:57.415 Any questions regarding this? 00:17:06.068 --> 00:17:07.859 So even though this is very straightforward 00:17:07.859 --> 00:17:09.800 to do, conceptually, but in practice 00:17:09.800 --> 00:17:16.720 that it not what we would do, because this is a very tedious. 00:17:16.720 --> 00:17:18.859 It's very tedious. 00:17:18.859 --> 00:17:21.770 And our knowledge of quantum field theory 00:17:21.770 --> 00:17:26.290 gives us something much simpler to deal with. 00:17:26.290 --> 00:17:31.410 Because this, essentially, if you think about it, 00:17:31.410 --> 00:17:35.790 is not very different from doing a correlation function 00:17:35.790 --> 00:17:39.130 calculation, just in the flat-space quantum field 00:17:39.130 --> 00:17:40.320 theory. 00:17:40.320 --> 00:17:42.430 And there, whey you calculate the Green function 00:17:42.430 --> 00:17:46.380 in flat-space-- in flat-space quantum field 00:17:46.380 --> 00:17:48.810 theory, what do you do? 00:17:48.810 --> 00:17:51.509 Do we try to solve the classical equation motion and you iterate 00:17:51.509 --> 00:17:52.050 the equation? 00:17:52.050 --> 00:17:52.907 What do you do? 00:17:52.907 --> 00:17:54.100 AUDIENCE: Draw a diagram. 00:17:54.100 --> 00:17:56.270 HONG LIU: Yeah, you use Feynman diagrams. 00:17:56.270 --> 00:17:58.680 So here, it's exactly the same thing. 00:17:58.680 --> 00:18:04.720 So here it's much easier if you just do the Feynman diagrams. 00:18:04.720 --> 00:18:08.810 So let me first remind you what we do in the flat-space. 00:18:13.440 --> 00:18:15.740 So let's now go to the field theory 00:18:15.740 --> 00:18:24.040 one, standard, ordinary flat-space QFT. 00:18:24.040 --> 00:18:29.570 So now let's consider the lambda phi cubed theory in flat-space 00:18:29.570 --> 00:18:33.040 Just the same theory now using the flat-space. 00:18:36.710 --> 00:18:42.131 Let me call this star-star. 00:18:42.131 --> 00:18:44.005 So essentially, let's just consider star-star 00:18:44.005 --> 00:18:51.486 in flat-space, now in flat Euclidean space. 00:19:03.080 --> 00:19:05.680 So if we are given a field theory like that, 00:19:05.680 --> 00:19:08.790 so how would you calculate the following correlation function? 00:19:08.790 --> 00:19:15.600 Say, suppose I want to calculate phi x1, phi xn. 00:19:15.600 --> 00:19:19.495 So now this x1, xn, they're just flat-space coordinates. 00:19:23.180 --> 00:19:25.150 To distinguish them, let me just call y1, 00:19:25.150 --> 00:19:30.470 yn, just not to confuse about the AdS coordinate. 00:19:30.470 --> 00:19:37.120 So in flat-space with y, y mu, OK? 00:19:37.120 --> 00:19:40.854 In flat-space with coordinate y mu. 00:19:40.854 --> 00:19:43.020 So now suppose you want to calculate the correlation 00:19:43.020 --> 00:19:43.770 function for this. 00:19:48.260 --> 00:19:49.110 What do you do? 00:19:53.980 --> 00:19:55.330 So what do you do? 00:19:55.330 --> 00:19:55.990 It's easy. 00:19:55.990 --> 00:20:05.980 What you do is that this theory has a propagator, G, 00:20:05.980 --> 00:20:07.530 [? with ?] [? arrow ?] propagator. 00:20:07.530 --> 00:20:11.270 And also, this theory have a interaction vertex, 00:20:11.270 --> 00:20:14.320 which is controlled by lambda. 00:20:14.320 --> 00:20:21.280 And then, when you calculate such endpoint functions, 00:20:21.280 --> 00:20:24.310 you just fix your endpoint. 00:20:24.310 --> 00:20:26.680 So now remember, this is slightly different than what 00:20:26.680 --> 00:20:30.549 you would normally do in field theory one, 00:20:30.549 --> 00:20:32.340 because, here, we're doing coordinate space 00:20:32.340 --> 00:20:34.360 rather than momentum space. 00:20:34.360 --> 00:20:37.800 In field theory one, you're more used to doing momentum space. 00:20:37.800 --> 00:20:46.360 So here what you do is that you fix y1, y2 y3, extends to yn. 00:20:46.360 --> 00:20:51.930 So this is essentially the location of your insertion, 00:20:51.930 --> 00:20:53.820 of your field insertion. 00:20:53.820 --> 00:20:56.240 So essentially, imagine there's a source there. 00:20:56.240 --> 00:21:01.230 And then you just connect all the extended points 00:21:01.230 --> 00:21:09.250 by propagators and with these kind of interaction vertices. 00:21:09.250 --> 00:21:19.940 So for example, you can have something like this, et cetera. 00:21:19.940 --> 00:21:23.830 And this connects to some boundary point, 00:21:23.830 --> 00:21:26.870 some other point, et cetera-- say y's. 00:21:29.590 --> 00:21:31.940 And this also have lines. 00:21:31.940 --> 00:21:34.810 We just draw all possible diagrams, 00:21:34.810 --> 00:21:37.840 and you calculate that diagram. 00:21:37.840 --> 00:21:42.320 You calculate those diagrams, and they give you this thing. 00:21:42.320 --> 00:21:44.846 And essentially, the diagrams automatically 00:21:44.846 --> 00:21:46.220 give you the iteration procedure. 00:21:49.490 --> 00:21:51.400 So now, back to AdS, essentially, we 00:21:51.400 --> 00:21:53.000 can just do exactly the same thing. 00:22:00.989 --> 00:22:02.530 Just remind yourself of the procedure 00:22:02.530 --> 00:22:06.510 of calculating this and what we are going to do here, 00:22:06.510 --> 00:22:10.790 and you easily convince yourself. 00:22:10.790 --> 00:22:13.620 So it requires a couple of minutes' thinking, 00:22:13.620 --> 00:22:16.820 but we'll leave it to yourself. 00:22:16.820 --> 00:22:20.250 Actually, that procedure is no different from just doing 00:22:20.250 --> 00:22:21.380 the calculation like this. 00:22:29.150 --> 00:22:30.975 Of course, the difference is that now we 00:22:30.975 --> 00:22:33.804 are in the curved spacetime rather than flat spacetime. 00:22:33.804 --> 00:22:35.345 But there's another major difference. 00:22:45.910 --> 00:22:49.800 So here, the source all lies in the interior of your spacetime, 00:22:49.800 --> 00:22:52.560 where you inserted the operator. 00:22:52.560 --> 00:22:56.470 But here, all the source, phi x, they lie at the boundary. 00:23:07.157 --> 00:23:08.865 That's, essentially, the only difference. 00:23:12.520 --> 00:23:16.630 So if I just schematically-- so this is AdS. 00:23:16.630 --> 00:23:19.184 This is the boundary. 00:23:19.184 --> 00:23:21.600 And essentially, all the points are lying on the boundary, 00:23:21.600 --> 00:23:25.020 like we have x1, x2. 00:23:25.020 --> 00:23:29.720 So let's consider four-point function. 00:23:29.720 --> 00:23:34.770 Because each one of them is labeled by the boundary point. 00:23:34.770 --> 00:23:39.720 And then for the four-point function, you just connect. 00:23:39.720 --> 00:23:43.810 So you can connect the diagram like this. 00:23:43.810 --> 00:23:45.960 Again, you have the vertices, which 00:23:45.960 --> 00:23:48.180 is the [? cube ?] of vertices. 00:23:48.180 --> 00:23:51.690 But your endpoint is all lying on the boundary. 00:23:57.160 --> 00:23:59.260 So this structure also tells you, actually, 00:23:59.260 --> 00:24:02.070 there's two types of propagators. 00:24:02.070 --> 00:24:06.120 Because one type of propagator connects the two bulk points. 00:24:06.120 --> 00:24:08.150 And then there's this kind of propagator 00:24:08.150 --> 00:24:12.150 which connects the bulk point into the boundary points. 00:24:12.150 --> 00:24:14.870 So this is the difference, one major difference, 00:24:14.870 --> 00:24:16.360 from the flat-space case. 00:24:16.360 --> 00:24:20.330 So here, all the propagators are the same. 00:24:20.330 --> 00:24:23.491 So here, the difference is now, you also 00:24:23.491 --> 00:24:25.740 have propagators which come from boundary to the bulk. 00:24:28.376 --> 00:24:30.500 So in this case, there are two kind of propagators. 00:24:44.770 --> 00:24:50.990 Here, you have two type of propagators. 00:24:58.710 --> 00:25:02.420 So one type of propagator is what we call 00:25:02.420 --> 00:25:03.770 the bulk-to-bulk propagator. 00:25:11.345 --> 00:25:13.350 Somehow this is going up and up. 00:25:21.330 --> 00:25:23.435 OK, I lost some blackboard space. 00:25:33.510 --> 00:25:38.090 So this the complete analog, this 00:25:38.090 --> 00:25:45.750 is the precise counterpart of the flat-space propagator. 00:25:45.750 --> 00:25:51.875 So this connects two points, z, x, z prime, x prime. 00:25:51.875 --> 00:25:55.684 So I still use the notation that z is the bulk 00:25:55.684 --> 00:25:56.600 [INAUDIBLE] direction. 00:25:56.600 --> 00:25:58.349 And the x is along the boundary direction. 00:26:04.030 --> 00:26:07.180 And this bulk-to-bulk will satisfy the standard Laplace 00:26:07.180 --> 00:26:14.260 equation just as in flat-space. 00:26:14.260 --> 00:26:15.970 So this gives you the delta function. 00:26:22.140 --> 00:26:35.550 Let me just express it, z minus z prime delta x minus x prime. 00:26:35.550 --> 00:26:39.780 So this is a complete analog of the standard flat-space 00:26:39.780 --> 00:26:40.790 propagator. 00:26:40.790 --> 00:26:43.160 It's just now in Anti-de Sitter space. 00:26:51.210 --> 00:27:07.278 So this is a counterpart in AdS standard flat-space propagator. 00:27:12.530 --> 00:27:27.420 So in particular, so as a propagator, 00:27:27.420 --> 00:27:56.207 this should be normalizable in either z or z prime, 00:27:56.207 --> 00:27:57.790 so when you take them to the boundary. 00:28:04.060 --> 00:28:10.970 So more explicitly, for example, the z at z prime, 00:28:10.970 --> 00:28:15.775 x prime, should scale as z prime to the power delta, when 00:28:15.775 --> 00:28:17.190 you take z prime goes to 0. 00:28:23.639 --> 00:28:25.430 And of course, this should also be regular. 00:28:28.220 --> 00:28:31.940 We should not have singularities as z goes to infinity. 00:28:35.100 --> 00:28:38.184 They should also be regular. 00:28:38.184 --> 00:28:40.100 So essentially, these condition will precisely 00:28:40.100 --> 00:28:41.950 define those propagators. 00:28:51.590 --> 00:28:55.940 But the idea is we also, because the source 00:28:55.940 --> 00:29:00.010 lies on the boundary, because of this boundary condition, 00:29:00.010 --> 00:29:02.225 we also have so-called boundary-to-bulk propagators. 00:29:46.440 --> 00:29:48.730 Actually, the standard in flat-space one, 00:29:48.730 --> 00:29:51.720 it propagates the field from one point to the other point. 00:29:51.720 --> 00:29:54.220 So that's why there's a delta function here. 00:29:54.220 --> 00:29:56.520 So the standard story, there's a delta function here. 00:29:56.520 --> 00:30:02.020 And you propagate the field starting 00:30:02.020 --> 00:30:05.040 from this point to that point. 00:30:05.040 --> 00:30:12.590 So this boundary-to-bulk propagator normally is written 00:30:12.590 --> 00:30:18.470 as k z,x, but the second index only has x prime, 00:30:18.470 --> 00:30:20.970 because this is when you take the boundary point to a bulk 00:30:20.970 --> 00:30:21.470 point. 00:30:21.470 --> 00:30:24.180 A boundary point, of course, there's no z anymore. 00:30:24.180 --> 00:30:26.100 So z prime is 0 here. 00:30:26.100 --> 00:30:29.270 You don't need to write it. 00:30:29.270 --> 00:30:33.560 And this boundary-to-bulk propagator, 00:30:33.560 --> 00:30:35.695 again, should satisfy the Laplace equation. 00:30:40.800 --> 00:30:44.750 But now the key point is that, on the right-hand side, 00:30:44.750 --> 00:30:47.320 because there's no bulk source, so the right-hand side 00:30:47.320 --> 00:30:49.790 should be 0 rather than the delta function. 00:30:49.790 --> 00:30:54.070 So that delta function is due to the bulk source. 00:30:54.070 --> 00:30:57.560 But we have to make sure this k have 00:30:57.560 --> 00:30:59.360 the right non-normalizable boundary 00:30:59.360 --> 00:31:01.950 conditions, so you have the right boundary 00:31:01.950 --> 00:31:04.050 condition to the boundary. 00:31:04.050 --> 00:31:16.260 So this when it approach to the boundary, as z goes to 0, 00:31:16.260 --> 00:31:22.680 should be like z d delta, delta x minus x prime. 00:31:29.010 --> 00:31:35.340 So the k satisfies the Laplace equation without a source. 00:31:35.340 --> 00:31:43.520 But when you take this propagator to the boundary, 00:31:43.520 --> 00:31:46.660 then it should give you the non-normalizable boundary 00:31:46.660 --> 00:31:49.540 conditions, because you should approach a source. 00:31:49.540 --> 00:31:55.690 And the delta function is put here. 00:31:55.690 --> 00:31:59.800 So if you write, say, a bulk field, 00:31:59.800 --> 00:32:07.570 which is sourced by some boundary source, say k z, x, 00:32:07.570 --> 00:32:15.450 x prime times phi x prime, and then this 00:32:15.450 --> 00:32:23.330 will have right boundary condition 00:32:23.330 --> 00:32:32.155 that the phi x, z goes to z d minus delta phi x. 00:32:42.650 --> 00:32:46.110 So this boundary-to-bulk propagator 00:32:46.110 --> 00:32:49.460 does not have a source in the bulk 00:32:49.460 --> 00:32:52.690 but does have a source in the boundary, 00:32:52.690 --> 00:32:55.825 non-normalizable off in the boundary. 00:32:59.260 --> 00:33:01.780 So now with this two propagator, then the story 00:33:01.780 --> 00:33:04.750 is just actually as in the standard way 00:33:04.750 --> 00:33:08.420 you obtain the Green functions. 00:33:08.420 --> 00:33:09.750 It's just really no different. 00:33:15.850 --> 00:33:20.565 So let me just summarize what we have discussed. 00:33:39.640 --> 00:33:47.830 The boundary endpoint function can 00:33:47.830 --> 00:33:54.260 be obtained by, essentially, endpoint function of the phi 00:33:54.260 --> 00:34:04.340 field in the gravity side-- let me write it here-- just 00:34:04.340 --> 00:34:08.389 related to the endpoint function of the gravity field 00:34:08.389 --> 00:34:18.310 on the gravity side, with all these points, x1, xn, 00:34:18.310 --> 00:34:19.850 they're lying on the boundary. 00:34:28.780 --> 00:34:31.800 This side is treated as ordinary field theory. 00:34:31.800 --> 00:34:35.170 Then compute this endpoint function of your field 00:34:35.170 --> 00:34:37.320 but just put the field on the boundary. 00:34:45.150 --> 00:34:48.110 And then that would just give you the AdS correlation 00:34:48.110 --> 00:34:49.830 functions. 00:34:49.830 --> 00:34:53.050 Yeah, and then you just need to distinguish 00:34:53.050 --> 00:34:57.000 two types of propagators, a bulk-to-bulk propagator 00:34:57.000 --> 00:35:00.000 and bulk-to-boundary propagator. 00:35:00.000 --> 00:35:01.763 So any questions regarding this? 00:35:01.763 --> 00:35:06.230 AUDIENCE: Now, here, [INAUDIBLE] d minus delta. 00:35:06.230 --> 00:35:11.035 Right there, the G goes to [? d ?] phi in the delta. 00:35:11.035 --> 00:35:11.660 HONG LIU: Yeah. 00:35:14.558 --> 00:35:18.262 AUDIENCE: But why aren't they different? 00:35:18.262 --> 00:35:20.220 HONG LIU: It's because this must have a source. 00:35:20.220 --> 00:35:24.290 So the boundary-to-bulk propagator-- 00:35:24.290 --> 00:35:27.120 you propagate the source, from the boundary to the bulk, 00:35:27.120 --> 00:35:29.730 so that means that field must have the right boundary 00:35:29.730 --> 00:35:30.230 condition. 00:35:33.310 --> 00:35:34.990 So that's the reason for this. 00:35:34.990 --> 00:35:38.530 We must have non-normalizable boundary conditions. 00:35:38.530 --> 00:35:42.300 And that propagator is just the standard flat-space propagator. 00:35:42.300 --> 00:35:44.620 Of course, it's always normalizable. 00:35:44.620 --> 00:35:47.480 And in the flat-space, when you construct 00:35:47.480 --> 00:35:49.084 a propagator, of course, it's always 00:35:49.084 --> 00:35:50.625 used in a normalizable wave function. 00:35:53.355 --> 00:35:55.655 AUDIENCE: This is that's non-normalizable? 00:35:55.655 --> 00:35:57.280 HONG LIU: No, this is non-normalizable. 00:35:57.280 --> 00:36:01.190 This is designed so that you have the right boundary 00:36:01.190 --> 00:36:01.690 condition. 00:36:04.470 --> 00:36:08.390 So this is designed so that you can propagate 00:36:08.390 --> 00:36:11.280 just as in flat-space. 00:36:11.280 --> 00:36:14.070 If you have phi at this point, and then 00:36:14.070 --> 00:36:16.480 you can propagate to the other point by convolution. 00:36:19.310 --> 00:36:23.000 And of course, when you write down the bulk field phi, 00:36:23.000 --> 00:36:28.050 in general, you only construct them out 00:36:28.050 --> 00:36:29.340 of the normalizable modes. 00:36:33.542 --> 00:36:34.125 Any questions? 00:36:40.771 --> 00:36:41.270 Good. 00:36:44.727 --> 00:36:46.060 So now let me make some remarks. 00:36:55.970 --> 00:37:00.000 So here, this procedure of iterating 00:37:00.000 --> 00:37:02.000 the classical equation of motion, 00:37:02.000 --> 00:37:03.780 solving the classical equation of motion, 00:37:03.780 --> 00:37:07.910 then iterating the action, of course, 00:37:07.910 --> 00:37:11.391 again, from our experience with quantum field theory, that 00:37:11.391 --> 00:37:13.015 only gives you the tree-level diagrams. 00:37:16.110 --> 00:37:20.440 So if I really write the bulk path integral, 00:37:20.440 --> 00:37:25.450 so Z CFT phi, then, in the same classical limit, 00:37:25.450 --> 00:37:30.590 it can be written as Z gravity, which 00:37:30.590 --> 00:37:33.720 is phi with the right boundary condition. 00:37:33.720 --> 00:37:37.775 And then on the gravity side, at semi-classical level, 00:37:37.775 --> 00:37:42.139 we can actually write down the path integral for this gravity 00:37:42.139 --> 00:37:42.930 partition function. 00:37:48.430 --> 00:37:50.560 So that's what we discussed before. 00:37:50.560 --> 00:37:57.230 And then at the leading order, in performing this path 00:37:57.230 --> 00:38:14.310 integral, of course, you just get 00:38:14.310 --> 00:38:17.170 what we have been doing so far. 00:38:17.170 --> 00:38:19.470 You just evaluate it at the classical solutions. 00:38:22.219 --> 00:38:24.385 But in principle, you can also see the fluctuations. 00:38:27.850 --> 00:38:36.330 So this is the fluctuations around, say, 00:38:36.330 --> 00:38:39.510 the classical solution, phi c. 00:38:39.510 --> 00:38:45.040 And then the action for the fluctuations 00:38:45.040 --> 00:38:48.050 is just given by phi c plus phi. 00:38:48.050 --> 00:38:51.550 The minus, of course, is the classical action. 00:38:51.550 --> 00:38:53.900 So we just expand around the phi c. 00:38:59.040 --> 00:39:02.140 So if we expand this, the leading order 00:39:02.140 --> 00:39:04.680 would be quadratic in small phi. 00:39:04.680 --> 00:39:06.960 I should now quote small phi. 00:39:06.960 --> 00:39:12.540 Maybe let me quote chi, because that is the same as that phi. 00:39:12.540 --> 00:39:16.320 So this chi is the small fluctuations around the phi c. 00:39:18.930 --> 00:39:22.670 So if you expanded this in chi, the linear order 00:39:22.670 --> 00:39:26.180 is 0, because phi c satisfies the equation of motion. 00:39:26.180 --> 00:39:29.610 So this will start in quadratic order and then cubic order, et 00:39:29.610 --> 00:39:30.150 cetera. 00:39:30.150 --> 00:39:33.500 So you validated the quadratic order as a Gaussian. 00:39:33.500 --> 00:39:35.610 And then that's the one-loop diagram. 00:39:35.610 --> 00:39:37.490 And then if you go to higher orders, 00:39:37.490 --> 00:39:39.310 then it'll give you higher-order diagrams. 00:39:39.310 --> 00:39:45.960 So if you now include the fluctuations, 00:39:45.960 --> 00:39:51.900 so the SE phi c, which is a classical action. 00:39:54.630 --> 00:39:56.520 This encodes only tree-level diagrams. 00:40:04.870 --> 00:40:11.560 And when you include the fluctuations, phi, 00:40:11.560 --> 00:40:15.780 the fluctuations, chi, and then this corresponding 00:40:15.780 --> 00:40:16.640 to include loops. 00:40:23.300 --> 00:40:27.856 So again, the whole procedure can be captured 00:40:27.856 --> 00:40:28.855 by the Feynman diagrams. 00:40:32.990 --> 00:40:34.657 So the advantage of the Feynman diagram 00:40:34.657 --> 00:40:36.990 is actually there's a natural generalization to the loop 00:40:36.990 --> 00:40:38.190 diagrams. 00:40:38.190 --> 00:40:39.930 And then just include the loops here. 00:40:47.700 --> 00:40:52.655 For example, here, you can do something like this. 00:40:52.655 --> 00:40:54.405 Now you can also have something like this. 00:40:59.287 --> 00:41:01.120 So that's, of course, one and two, including 00:41:01.120 --> 00:41:05.810 the fluctuations, are under the saddle point, 00:41:05.810 --> 00:41:08.538 are including the fluctuation around the saddle point. 00:41:13.722 --> 00:41:15.180 Any questions regarding this point? 00:41:19.990 --> 00:41:20.490 Good. 00:41:30.340 --> 00:41:31.130 Yes? 00:41:31.130 --> 00:41:33.890 AUDIENCE: So when people draw these like so-called Witten 00:41:33.890 --> 00:41:35.825 diagrams, where they draw up like this circle. 00:41:35.825 --> 00:41:36.630 HONG LIU: Right. 00:41:36.630 --> 00:41:38.030 AUDIENCE: Is that basically what we're doing? 00:41:38.030 --> 00:41:38.390 HONG LIU: Yeah. 00:41:38.390 --> 00:41:39.000 Yeah, right. 00:41:39.000 --> 00:41:41.300 Yeah, so let me just explain one thing. 00:41:41.300 --> 00:41:46.130 In the Euclidean, I have been drawing the ideas like this. 00:41:46.130 --> 00:41:50.040 So this is z equal to 0, et cetera. 00:41:50.040 --> 00:41:54.800 So in the Euclidean signature, as you will do in your PSet, 00:41:54.800 --> 00:41:59.240 then this z equal to infinity, you 00:41:59.240 --> 00:42:02.780 can argue is actually a single point. 00:42:02.780 --> 00:42:05.580 So let me just write down the metric. 00:42:05.580 --> 00:42:12.002 So the metric is R square, z square. 00:42:12.002 --> 00:42:13.460 When you go to Euclidean signature, 00:42:13.460 --> 00:42:15.000 you just have dx squared. 00:42:15.000 --> 00:42:19.900 So this is a full Euclidean metric. 00:42:19.900 --> 00:42:22.289 So there's something interesting about the difference 00:42:22.289 --> 00:42:23.830 between the Euclidean and Lorentzian, 00:42:23.830 --> 00:42:27.700 even though, from this picture, it's roughly the same. 00:42:27.700 --> 00:42:31.600 So if it's equal to 0, you go to z notch. 00:42:31.600 --> 00:42:35.580 And then in the Euclidean case, you can actually 00:42:35.580 --> 00:42:38.920 show that z equal infinity, even though, 00:42:38.920 --> 00:42:43.224 [? naively, ?] you still have a full Euclidean plane, 00:42:43.224 --> 00:42:45.390 but you can argue, actually, that the whole thing is 00:42:45.390 --> 00:42:46.940 actually a point. 00:42:46.940 --> 00:42:50.570 And you can roughly see, when you go to z equals infinity, 00:42:50.570 --> 00:42:54.560 the overall factor goes to 0. 00:42:54.560 --> 00:43:00.160 So essentially, the whole space shrinks to a point. 00:43:00.160 --> 00:43:02.280 The whole space shrinks to a point. 00:43:02.280 --> 00:43:05.500 So in Euclidean, if it's equal to infinity, 00:43:05.500 --> 00:43:07.220 it's essentially a point. 00:43:07.220 --> 00:43:12.100 And then topologically, this is equivalent to having a disk. 00:43:16.780 --> 00:43:20.360 Topologically, the whole space is just disk. 00:43:20.360 --> 00:43:24.670 And the z equal to infinity is one point on the boundary. 00:43:24.670 --> 00:43:27.125 And then the other is z equal to 0. 00:43:27.125 --> 00:43:28.890 And so this is the interior of the space. 00:43:33.140 --> 00:43:33.640 Yes? 00:43:33.640 --> 00:43:36.265 AUDIENCE: So in other words, the entire perimeter of the circle 00:43:36.265 --> 00:43:37.054 is identi-- 00:43:37.054 --> 00:43:38.220 HONG LIU: It's the boundary. 00:43:38.220 --> 00:43:38.770 AUDIENCE: [INAUDIBLE]. 00:43:38.770 --> 00:43:40.311 HONG LIU: Yeah, this is the boundary. 00:43:43.290 --> 00:43:48.727 Essentially, you can imagine, each point here is R to d. 00:43:48.727 --> 00:43:50.185 Then you add the point at infinity, 00:43:50.185 --> 00:43:51.514 and then it becomes a sphere. 00:43:51.514 --> 00:43:52.430 AUDIENCE: [INAUDIBLE]. 00:43:52.430 --> 00:43:54.610 HONG LIU: Yeah, it becomes a sphere. 00:43:54.610 --> 00:43:57.240 And then the whole thing becomes like a disk. 00:44:03.700 --> 00:44:07.046 So normally, when people will draw-- so topologically, 00:44:07.046 --> 00:44:08.670 you're creating a disk, just like this. 00:44:08.670 --> 00:44:10.680 Then people will just draw diagrams like this. 00:44:15.400 --> 00:44:18.434 Yeah, I draw diagrams like this. 00:44:18.434 --> 00:44:21.200 AUDIENCE: If you add a single point to the boundary, 00:44:21.200 --> 00:44:25.110 that single point is equal to infinity to the boundary, which 00:44:25.110 --> 00:44:27.010 is z equals 0? 00:44:27.010 --> 00:44:28.220 HONG LIU: Sorry? 00:44:28.220 --> 00:44:31.440 AUDIENCE: You mean that you have a point that [INAUDIBLE]. 00:44:31.440 --> 00:44:32.210 HONG LIU: Yeah. 00:44:32.210 --> 00:44:35.310 Yeah, this point turns out, it can actually 00:44:35.310 --> 00:44:36.780 lie on the boundary. 00:44:36.780 --> 00:44:40.215 AUDIENCE: But then, on the boundary, the value of t 00:44:40.215 --> 00:44:42.470 is not continuous. 00:44:42.470 --> 00:44:43.730 HONG LIU: Hm? 00:44:43.730 --> 00:44:44.230 Yeah. 00:44:52.450 --> 00:44:54.430 So this is due to the coordinate choice. 00:44:54.430 --> 00:44:59.200 It's just this coordinate becomes singular at that point. 00:44:59.200 --> 00:45:01.437 You can rewrite it. 00:45:01.437 --> 00:45:04.020 Yeah, the simplest thing is that you rewrite these [INAUDIBLE] 00:45:04.020 --> 00:45:06.900 coordinates to the coordinate which has a parameter that has 00:45:06.900 --> 00:45:08.610 a boundary really as a sphere. 00:45:08.610 --> 00:45:09.680 And then you will see it. 00:45:09.680 --> 00:45:12.860 Because when you write the sphere as a plane, 00:45:12.860 --> 00:45:15.220 then this one point becomes singular. 00:45:15.220 --> 00:45:17.634 This is just the standard story. 00:45:17.634 --> 00:45:19.300 Just that coordinate choice is singular. 00:45:23.047 --> 00:45:23.880 Any other questions? 00:45:32.830 --> 00:45:34.450 Good. 00:45:34.450 --> 00:45:37.410 So this is the first remark. 00:45:37.410 --> 00:45:44.960 The secondary remark is that this 00:45:44.960 --> 00:45:47.260 is a little bit of an unusual correlation function. 00:45:47.260 --> 00:45:50.010 So this is the bulk correlation functions, 00:45:50.010 --> 00:45:52.830 but with all points lying on the boundary. 00:46:01.680 --> 00:46:10.650 So the counterpart or just the exact analog 00:46:10.650 --> 00:46:33.110 of standard flat-space correlation functions in AdS-- 00:46:33.110 --> 00:46:46.882 you can see the end bulk point and look 00:46:46.882 --> 00:46:48.090 at its correlation functions. 00:46:52.050 --> 00:46:57.345 So this then will be really just bring you to the standard QFT 00:46:57.345 --> 00:46:59.530 one correlation functions. 00:46:59.530 --> 00:47:07.620 Then you have endpoints, z1, x1, or in the bulk, z2, x2, et 00:47:07.620 --> 00:47:08.320 cetera. 00:47:08.320 --> 00:47:14.380 And then you just draw a diagram between them, et cetera. 00:47:14.380 --> 00:47:16.000 The area propagator here would be 00:47:16.000 --> 00:47:17.250 just bulk-to-bulk propagator. 00:47:17.250 --> 00:47:19.819 There's no boundary-to-bulk propagator. 00:47:19.819 --> 00:47:21.610 There's no boundary-to-bulk propagator just 00:47:21.610 --> 00:47:24.767 all bulk-to-bulk propagator. 00:47:32.560 --> 00:47:36.876 So this is a complete analog of your ordinary flat-space Green 00:47:36.876 --> 00:47:37.375 functions. 00:47:39.910 --> 00:47:46.420 But we can easily imagine that this correlation function must 00:47:46.420 --> 00:47:49.634 be related to those correlations functions 00:47:49.634 --> 00:47:51.550 if you just take those points to the boundary. 00:48:20.260 --> 00:48:33.670 So it's natural to expect, say, those correlations functions, 00:48:33.670 --> 00:48:38.150 since they're at the boundary, must 00:48:38.150 --> 00:48:43.340 be related to those kind of standard correlation 00:48:43.340 --> 00:48:48.550 functions, in which you would take the point to the boundary. 00:49:03.540 --> 00:49:07.091 So those, somehow, in the end, must be the same. 00:49:07.091 --> 00:49:08.840 So if I take those points to the boundary, 00:49:08.840 --> 00:49:10.230 I should recover that guy. 00:49:13.870 --> 00:49:20.040 So indeed, this relation is true. 00:49:20.040 --> 00:49:22.860 And you will work it out yourself in the PSet. 00:49:27.210 --> 00:49:29.510 You see, the only difference is the following. 00:49:29.510 --> 00:49:33.920 When you take those points to the boundary, what you get 00:49:33.920 --> 00:49:34.920 is the boundary limit. 00:49:37.530 --> 00:49:44.160 So now let's imagine you take those points to the boundary. 00:49:44.160 --> 00:49:47.106 Because here, it's all bulk propagator. 00:49:47.106 --> 00:49:49.230 So those propagator, which connect to the boundary, 00:49:49.230 --> 00:49:53.170 would be the boundary limit of the bulk propagator. 00:49:53.170 --> 00:49:55.940 And then, it just boils down to, what 00:49:55.940 --> 00:49:57.440 is the relation between the boundary 00:49:57.440 --> 00:50:00.742 limit of the bulk propagator and this boundary 00:50:00.742 --> 00:50:01.700 to the bulk propagator? 00:50:09.430 --> 00:50:15.157 So the above relation-- let me just-- I'd rather just 00:50:15.157 --> 00:50:18.630 call it a number. 00:50:18.630 --> 00:50:21.350 So this star-star-star just boils 00:50:21.350 --> 00:50:36.183 down to the relation between the k 00:50:36.183 --> 00:50:42.950 z, x, x prime, which is our boundary-to-bulk propagator, 00:50:42.950 --> 00:50:55.860 and limit z prime goes to 0, G x, z, x prime, z prime. 00:50:58.970 --> 00:51:02.420 So you will find that these two guys 00:51:02.420 --> 00:51:04.660 are proportional to each other. 00:51:04.660 --> 00:51:11.190 And once you extract out that proportional factor, 00:51:11.190 --> 00:51:15.220 then you just relate them. 00:51:15.220 --> 00:51:21.690 So this is just another way to calculate. 00:51:21.690 --> 00:51:24.690 Because this is the bounty correlation function. 00:51:24.690 --> 00:51:27.110 So this is the boundary correlation function. 00:51:27.110 --> 00:51:30.960 So this is related to the CFT correlation functions. 00:51:30.960 --> 00:51:33.970 And then you can also calculate the CFT correlation function 00:51:33.970 --> 00:51:37.205 just by taking the limit of the bulk correlation functions. 00:51:39.810 --> 00:51:44.080 So this you will work out a little bit. 00:51:44.080 --> 00:51:48.200 Yeah, you will work out the precise relation in your PSet. 00:51:48.200 --> 00:51:51.790 Maybe some of you have looked at it already. 00:51:51.790 --> 00:51:53.904 So any questions on this? 00:51:53.904 --> 00:51:56.264 AUDIENCE: What is the advantage of maybe 00:51:56.264 --> 00:51:59.580 using another to express it? 00:51:59.580 --> 00:52:02.510 HONG LIU: Yeah, just different ways of calculating things. 00:52:02.510 --> 00:52:07.510 And of course, normally, I have many different ways 00:52:07.510 --> 00:52:09.310 to calculate things. 00:52:09.310 --> 00:52:11.301 Some of them may be convenient this way. 00:52:11.301 --> 00:52:13.675 So of them may be convenient for that purpose, et cetera. 00:52:13.675 --> 00:52:14.230 It depends. 00:52:16.460 --> 00:52:17.960 Normally, we just directly are using 00:52:17.960 --> 00:52:21.196 the bulk-to-boundary propagator. 00:52:21.196 --> 00:52:23.820 If you just do the calculation, the bulk-to boundary propagator 00:52:23.820 --> 00:52:25.980 actually is simpler. 00:52:25.980 --> 00:52:29.130 But sometimes, for certain conceptual questions, 00:52:29.130 --> 00:52:30.520 this actually becomes simpler. 00:52:34.630 --> 00:52:39.518 So now let's look at Wilson loops. 00:52:45.210 --> 00:52:49.140 So how do you calculate the Wilson loops using gravity? 00:52:53.640 --> 00:53:02.140 So in the gauge theory-- so I assume 00:53:02.140 --> 00:53:05.490 you have already done the QFT II and the gauge theory? 00:53:10.645 --> 00:53:12.789 In the gauge theory, we also know 00:53:12.789 --> 00:53:15.080 it's essentially one of the most important observables. 00:53:18.715 --> 00:53:21.090 The Wilson loop is normally defined in the following way. 00:53:34.890 --> 00:53:36.352 So now I will explain my notation. 00:53:41.100 --> 00:53:44.400 So let me also add in subscript r here. 00:53:47.320 --> 00:53:48.910 So c is a closed path. 00:53:52.400 --> 00:53:54.320 So this is defined for closed paths. 00:53:57.130 --> 00:53:58.720 And A mu is a matrix. 00:53:58.720 --> 00:54:02.960 So we can see the [INAUDIBLE] in gauge theory. 00:54:02.960 --> 00:54:08.750 A mu is a matrix, writing the standard away, 00:54:08.750 --> 00:54:11.940 in terms of the generator, in some representations r. 00:54:14.920 --> 00:54:19.140 So this is in some representations. 00:54:19.140 --> 00:54:22.140 So this is just the generators of the gauge 00:54:22.140 --> 00:54:26.750 group in some representation, which we'll call r. 00:54:52.620 --> 00:54:54.550 And P is the path ordering. 00:55:05.700 --> 00:55:09.330 So in general, you can choose any representations you want. 00:55:09.330 --> 00:55:17.665 But often, we choose r, say, in fundamental representations. 00:55:20.700 --> 00:55:23.250 For example, that's normally what we do, in QCD, 00:55:23.250 --> 00:55:25.170 in fundamental representation. 00:55:25.170 --> 00:55:28.695 But you can choose it to be any representation. 00:55:35.130 --> 00:55:39.644 So the physical meaning of this, so this operator, 00:55:39.644 --> 00:55:41.060 by definition, is gauge invariant, 00:55:41.060 --> 00:55:42.930 because of this trace. 00:55:42.930 --> 00:55:45.840 And need the path ordering, because you have a matrix here. 00:55:45.840 --> 00:55:48.420 And the matrix, at different points, don't commute. 00:55:48.420 --> 00:55:50.550 So you need to specify ordering. 00:55:50.550 --> 00:55:53.900 And so you just specify it by the order of the path. 00:55:56.830 --> 00:56:05.180 And so the physical meaning of the Wilson loop, 00:56:05.180 --> 00:56:19.240 so this is essentially the phase factor associated 00:56:19.240 --> 00:56:38.640 with transporting an external particle, in a given 00:56:38.640 --> 00:56:44.220 representation, say, in r representation, along c. 00:56:50.080 --> 00:56:53.480 So you transport the particle along some path. 00:56:53.480 --> 00:56:57.600 And you come back to the same point. 00:56:57.600 --> 00:56:59.840 Then you find that that phase does not necessarily 00:56:59.840 --> 00:57:02.890 go back to 0. 00:57:02.890 --> 00:57:04.680 And so this is a nontrivial phase. 00:57:04.680 --> 00:57:07.346 It essentially tells you there's a nontrivial gauge field there. 00:57:11.120 --> 00:57:15.480 Yeah, so this provides a probe of the gauge fields. 00:57:15.480 --> 00:57:15.980 Yes? 00:57:15.980 --> 00:57:17.354 AUDIENCE: I know that people talk 00:57:17.354 --> 00:57:19.519 about-- I know that it's somehow and observable. 00:57:19.519 --> 00:57:20.810 But is it actually observable ? 00:57:20.810 --> 00:57:22.428 Can you actually do an experiment 00:57:22.428 --> 00:57:23.518 to measure this phase? 00:57:27.496 --> 00:57:28.620 HONG LIU: I don't think so. 00:57:28.620 --> 00:57:29.119 Ha. 00:57:39.800 --> 00:57:40.385 It depends. 00:57:46.450 --> 00:57:48.460 Yeah, in some situations, you can. 00:57:53.180 --> 00:57:56.900 Let me mention something else, then the answer 00:57:56.900 --> 00:57:58.250 to this question will be seen. 00:58:03.230 --> 00:58:06.840 So the simplest case would be the W c, just 00:58:06.840 --> 00:58:11.800 the single-point function of W c, say, in a vacuum. 00:58:11.800 --> 00:58:14.060 So this is the simplest observable. 00:58:14.060 --> 00:58:19.310 But of course, you can also consider the generic, say, 00:58:19.310 --> 00:58:21.700 a large number, several [? routes ?] in some general 00:58:21.700 --> 00:58:23.950 state, say, for example, [? finite ?] [? telemetry ?], 00:58:23.950 --> 00:58:24.450 et cetera. 00:58:27.199 --> 00:58:28.740 So these are the typical observables. 00:58:32.370 --> 00:58:34.330 So now let me emphasize this external. 00:58:37.410 --> 00:58:40.470 So this external is very important. 00:58:40.470 --> 00:58:44.880 So normally, by calling something external, 00:58:44.880 --> 00:58:46.950 we mean that this particle has infinite mass. 00:58:46.950 --> 00:58:49.290 It's infinitely heavy. 00:58:49.290 --> 00:58:52.550 And the reason we want it to be infinitely heavy 00:58:52.550 --> 00:58:57.500 is because only for heavy particles, 00:58:57.500 --> 00:59:00.130 in principle, you can localize the lower path. 00:59:00.130 --> 00:59:05.040 Then you can make mathematical sense of the precise path. 00:59:05.040 --> 00:59:07.530 Otherwise, if it's a fluctuating particle, 00:59:07.530 --> 00:59:14.004 then you can not specify a path-- yeah, ambiguously. 00:59:14.004 --> 00:59:15.920 So when we say, an external particle, I always 00:59:15.920 --> 00:59:18.294 say a particle which is assumed to be infinitely massive. 00:59:22.850 --> 00:59:29.480 And also, I've introduced terminology. 00:59:29.480 --> 00:59:33.290 When this r is in the fundamental representation-- 00:59:33.290 --> 00:59:34.740 yeah, maybe not right here. 00:59:34.740 --> 00:59:38.070 When this r is in the fundamental representation, 00:59:38.070 --> 00:59:43.060 we'll call the corresponding particle a quark. 00:59:43.060 --> 00:59:45.415 If it is in the fundamental representation, 00:59:45.415 --> 00:59:47.950 I will call it a quark. 00:59:47.950 --> 00:59:51.657 I will often just consider the fundamental representation. 00:59:51.657 --> 00:59:53.740 In the phase factor, where we're corresponding to, 00:59:53.740 --> 00:59:55.000 you transport a quark. 01:00:05.340 --> 01:00:18.930 So one of the very often used loops is a rectangular loop. 01:00:24.150 --> 01:00:25.920 So this is along some spatial direction, 01:00:25.920 --> 01:00:28.710 so this is the time direction. 01:00:28.710 --> 01:00:32.562 So let's say the spatial direction is L, 01:00:32.562 --> 01:00:34.270 and the time direction, using the length, 01:00:34.270 --> 01:00:39.020 is T. So imagine you have a loop like this. 01:01:11.970 --> 01:01:13.902 So normally, we can see that the T, say, 01:01:13.902 --> 01:01:15.777 the length of the loop in the time direction, 01:01:15.777 --> 01:01:19.200 is much, much greater than the L. 01:01:19.200 --> 01:01:26.620 So in such a limit, then, essentially, 01:01:26.620 --> 01:01:29.660 you can ignore the contribution from this side, 01:01:29.660 --> 01:01:30.970 from these two short sides. 01:01:30.970 --> 01:01:36.650 So in the limit, in which this T goes to infinity, 01:01:36.650 --> 01:01:40.085 to leading order, you can ignore the short side. 01:01:40.085 --> 01:01:43.930 And then you can think of this loop 01:01:43.930 --> 01:01:47.164 when it just has a particle-- so this is a particle moving 01:01:47.164 --> 01:01:47.830 forward in time. 01:01:47.830 --> 01:01:49.690 This is a particle going backward in time. 01:01:49.690 --> 01:01:53.520 Then you can consider this as a particle and an anti-particle 01:01:53.520 --> 01:01:54.670 moving forward in time. 01:01:58.160 --> 01:02:02.830 So in this case, we can argue, on general grounds, 01:02:02.830 --> 01:02:04.510 that this will just, to leading order, 01:02:04.510 --> 01:02:07.350 will give you something iET. 01:02:07.350 --> 01:02:10.320 And this e, it just can be interpreted 01:02:10.320 --> 01:02:31.680 as a potential energy between a single quark and an anti-quark. 01:02:41.260 --> 01:02:45.304 So in this example, this is the same. 01:02:45.304 --> 01:02:47.720 In this case, you may be able to measure this Wilson loop, 01:02:47.720 --> 01:02:51.095 because if you can measure-- experimentally, 01:02:51.095 --> 01:02:53.975 you may be able to measure this energy between the two 01:02:53.975 --> 01:02:54.475 particles. 01:02:59.389 --> 01:03:01.930 And then, essentially, you can say, I have measured the loop. 01:03:01.930 --> 01:03:04.305 And then you can compare with the prediction of the loop. 01:03:12.260 --> 01:03:16.574 So again, in order to make this interpretation make sense, 01:03:16.574 --> 01:03:18.865 this particle having infinite energy is very important. 01:03:22.200 --> 01:03:27.720 And again, only for the very heavy particle, 01:03:27.720 --> 01:03:29.420 you can really localize them, and then 01:03:29.420 --> 01:03:32.850 talk about the potential energy between these two particles. 01:03:32.850 --> 01:03:36.160 Otherwise, if they fluctuate a lot, 01:03:36.160 --> 01:03:39.815 then it's hard to make it precise. 01:03:39.815 --> 01:03:44.340 So now the question is how do we calculate this quantity in AdS? 01:04:11.420 --> 01:04:14.170 When I don't put an index here, I just mean that this 01:04:14.170 --> 01:04:15.573 the fundamental representation. 01:04:29.880 --> 01:04:33.000 The question is how we do that. 01:04:33.000 --> 01:04:35.850 So in order to answer this question, 01:04:35.850 --> 01:04:50.176 we first need to understand how to introduce 01:04:50.176 --> 01:05:07.685 a fundamental external particle, external quark in the N 01:05:07.685 --> 01:05:09.060 equals 4 super Yang-Mills theory. 01:05:18.980 --> 01:05:21.708 So how to introduce a fundamental external quark 01:05:21.708 --> 01:05:23.457 in the N equals 4 super Yang-Mills theory. 01:05:26.800 --> 01:05:30.600 So we know, as we discussed before, 01:05:30.600 --> 01:05:33.340 that everything in the N equals 4 super Yang-Mills theory, so 01:05:33.340 --> 01:05:36.295 all fundamental fields, all the fields in the N 01:05:36.295 --> 01:05:37.670 equals 4 super Yang-Mills theory, 01:05:37.670 --> 01:05:40.360 they are in a joint representation. 01:05:40.360 --> 01:05:43.090 There's nothing in the fundamental representation. 01:05:43.090 --> 01:05:46.530 Everything is in the joint representation. 01:05:46.530 --> 01:05:50.287 So first we need to introduce a fundamental object, 01:05:50.287 --> 01:05:52.620 an object we transform under fundamental representation. 01:05:56.780 --> 01:06:01.350 After we have done that, then we need to translate that quantity 01:06:01.350 --> 01:06:02.260 to the gravity side. 01:06:05.800 --> 01:06:19.082 And then we understand how the gravity description 01:06:19.082 --> 01:06:20.290 of such an external particle. 01:06:38.150 --> 01:06:41.870 So this is easy to do. 01:06:41.870 --> 01:06:43.355 This is easy to do. 01:06:43.355 --> 01:06:45.820 So let's first think about how to do this in the N 01:06:45.820 --> 01:06:48.050 equals 4 super Yang-Mills theory. 01:06:48.050 --> 01:06:51.640 And then this is a very intuitive by using 01:06:51.640 --> 01:06:54.860 this [INAUDIBLE] picture. 01:06:54.860 --> 01:07:01.180 So we know the N equals 4 super Yang-Mills 01:07:01.180 --> 01:07:04.860 theory comes from low-energy theory of N D3-branes. 01:07:07.834 --> 01:07:10.000 They come from the low-energy theory of N D3-branes. 01:07:24.100 --> 01:07:28.500 So how do we introduce an external particles? 01:07:28.500 --> 01:07:36.080 So now let's imagine, consider N plus 1 of them, 01:07:36.080 --> 01:07:38.780 and let me separate one from the rest. 01:07:43.440 --> 01:07:49.320 So this is still N, separate one from the rest. 01:07:49.320 --> 01:07:53.860 So then there can be open string connects between them. 01:07:53.860 --> 01:07:56.890 So suppose I separate them by some distance, r, 01:07:56.890 --> 01:07:59.080 in the transverse direction to the D3-brane. 01:08:04.580 --> 01:08:12.180 So as we described before, such as a separation breaks the SU N 01:08:12.180 --> 01:08:19.380 plus 1 symmetry of the D3-brane SU N 01:08:19.380 --> 01:08:25.439 and then a single U1 of this 1-brane. 01:08:32.930 --> 01:08:39.390 In particular, such an open string-- 01:08:39.390 --> 01:08:41.424 so let's look at the endpoint. 01:08:41.424 --> 01:08:43.715 Then the endpoint of this string can only have indices. 01:08:46.580 --> 01:08:49.102 Because this is a single stream, one 01:08:49.102 --> 01:08:50.310 end ending on the other side. 01:08:50.310 --> 01:08:52.180 There's only one index. 01:08:52.180 --> 01:08:54.760 And here, they can only have N possible index. 01:08:54.760 --> 01:09:09.279 So such a fundamental string transforms 01:09:09.279 --> 01:09:10.760 in the fundamental representation. 01:09:21.149 --> 01:09:23.569 So from the point of view of this SU N gauge theory, 01:09:23.569 --> 01:09:25.270 this is a quark. 01:09:25.270 --> 01:09:27.829 So this endpoint is a quark. 01:09:27.829 --> 01:09:28.620 So this is a quark. 01:09:36.109 --> 01:09:46.060 And the mass of this quark is equal to-- so let 01:09:46.060 --> 01:09:52.960 me call the mass capital M-- is equal to r, 01:09:52.960 --> 01:09:58.439 the length between them divided by tension of the string. 01:09:58.439 --> 01:09:59.730 So this we have derived before. 01:10:02.126 --> 01:10:04.000 So this just gives you the mass of the quark. 01:10:07.040 --> 01:10:10.390 So when you separate one D-brane from the other, 01:10:10.390 --> 01:10:13.840 so you have introduced this quark, with a mass. 01:10:21.750 --> 01:10:24.190 So now let's consider that there's no low-energy limit 01:10:24.190 --> 01:10:24.690 [INAUDIBLE]. 01:10:48.500 --> 01:10:56.960 So in the low-energy limit of [INAUDIBLE], 01:10:56.960 --> 01:11:03.944 so we want to keep this quark, so there, of course, 01:11:03.944 --> 01:11:05.485 you want to take alpha prime go to 0. 01:11:11.354 --> 01:11:13.770 But we want to keep this quark in the low-energy spectrum, 01:11:13.770 --> 01:11:15.350 because we want this to remain in N 01:11:15.350 --> 01:11:16.725 equals 4 super Yang-Mills theory, 01:11:16.725 --> 01:11:19.250 because we want to introduce this particle. 01:11:19.250 --> 01:11:23.950 So in this low-energy limit, when we take off a prime, 01:11:23.950 --> 01:11:26.900 go to 0 limit, we also want to take, at the same time, 01:11:26.900 --> 01:11:33.740 r goes to 0 limit, so that this mass term, r divided by alpha 01:11:33.740 --> 01:11:35.605 prime, will remain finite. 01:11:40.616 --> 01:11:44.920 Then that means this will remain, 01:11:44.920 --> 01:11:48.450 this particle, such kind of fundamental representation, 01:11:48.450 --> 01:11:51.438 will remain in the N equals 4 super Yang-Mills theory. 01:11:58.430 --> 01:12:00.311 Because other modes are massive. 01:12:00.311 --> 01:12:02.060 And then when alpha prime goes to 0 limit, 01:12:02.060 --> 01:12:03.530 it goes to infinity. 01:12:03.530 --> 01:12:06.065 And under those modes, we will remain with the N 01:12:06.065 --> 01:12:07.440 equals 4 super Yang-Mills theory. 01:12:09.950 --> 01:12:12.345 So now let's see what happens on the gravity side. 01:12:15.120 --> 01:12:19.250 So here, you will need to fill in some details yourself. 01:12:22.310 --> 01:12:23.860 I'll only tell you the answer. 01:12:36.760 --> 01:12:40.720 On the gravity side, you have this N D3-brane. 01:12:40.720 --> 01:12:44.680 They have one brane which [? is ?] [? r ?] separated from 01:12:44.680 --> 01:12:46.580 this N D3-brane. 01:12:46.580 --> 01:12:48.420 And they're separation is such that, when 01:12:48.420 --> 01:12:50.230 you take off alpha prime and go to 0 limit, 01:12:50.230 --> 01:12:51.320 their ratio is finite. 01:12:54.894 --> 01:12:56.310 Now, in the gravity side, you have 01:12:56.310 --> 01:12:59.580 to take this so-called near-horizon limit, 01:12:59.580 --> 01:13:02.220 with taking into account. 01:13:05.110 --> 01:13:13.130 So what you find in the gravity side is that, 01:13:13.130 --> 01:13:34.110 as before, that N D3-brane now disappeared into r equal to 0. 01:13:34.110 --> 01:13:38.460 Remember, r equal to 0 is that infinite proper distance away. 01:13:38.460 --> 01:13:42.060 And now we are using this r coordinate. 01:13:42.060 --> 01:13:44.380 And when we take this low limit, we're 01:13:44.380 --> 01:13:47.040 using this r coordinate rather than this z coordinate. 01:13:47.040 --> 01:13:50.180 So in that case, you have i equal infinity. 01:13:50.180 --> 01:13:53.950 Then the brane, essentially, goes to r equal to 0. 01:13:57.964 --> 01:13:59.380 which has infinite proper distance 01:13:59.380 --> 01:14:04.510 away from any finite r. 01:14:04.510 --> 01:14:10.310 So now this is an exercise for yourself just 01:14:10.310 --> 01:14:14.700 to repeat our previous argument. 01:14:14.700 --> 01:14:18.050 Then you will find that in this regime, when 01:14:18.050 --> 01:14:21.300 you take alpha prime go to 0, this r finite, 01:14:21.300 --> 01:14:23.640 this D3-brane does not disappear. 01:14:23.640 --> 01:14:29.900 The single D3-brane will remain in the AdS. 01:14:29.900 --> 01:14:34.270 So the difference, from our previous story, 01:14:34.270 --> 01:14:42.450 is that, in addition to the AdS, now 01:14:42.450 --> 01:14:48.450 you have an additional D3-brane at some point in S5. 01:14:48.450 --> 01:14:51.170 Because D3-brane is in some transverse direction, 01:14:51.170 --> 01:14:52.910 which now become S5. 01:14:52.910 --> 01:14:55.760 So you have a D3-brane, which is parallel 01:14:55.760 --> 01:15:02.240 to the boundary coordinates, but sitting at one point in S5. 01:15:02.240 --> 01:15:05.370 And I think this picture is reasonable, 01:15:05.370 --> 01:15:07.420 but you should check yourself. 01:15:07.420 --> 01:15:10.960 Because in this regime, this D3-brane does not go away. 01:15:14.310 --> 01:15:19.800 And a remarkable thing you will check yourself 01:15:19.800 --> 01:15:22.571 is that when you take the [INAUDIBLE] limit, 01:15:22.571 --> 01:15:24.070 in this picture, this is flat-space. 01:15:28.330 --> 01:15:31.300 Then you have this formula. 01:15:31.300 --> 01:15:35.130 And now, when you take this limit, this now become AdS. 01:15:35.130 --> 01:15:44.210 Now, this AdS at some radius r, controlled by this r. 01:15:44.210 --> 01:15:45.890 And now you can check one nice thing. 01:15:55.670 --> 01:15:59.390 In AdS, if you consider a string, straight string, 01:15:59.390 --> 01:16:03.420 from this D3-brane all the way to r equal to 0, 01:16:03.420 --> 01:16:06.010 which is this string, this string 01:16:06.010 --> 01:16:07.510 has exactly the same mass. 01:16:10.450 --> 01:16:15.250 So the mass is equal to r divided by 2 pi alpha prime-- 01:16:15.250 --> 01:16:18.590 also in AdS5. 01:16:21.800 --> 01:16:23.290 So this is a consistency check. 01:16:25.940 --> 01:16:28.580 Because when you take the low-energy limits, of course, 01:16:28.580 --> 01:16:30.670 you should not change the mass. 01:16:30.670 --> 01:16:33.055 The mass of the string will not change. 01:16:33.055 --> 01:16:35.430 The only question is whether it will stay in the spectrum 01:16:35.430 --> 01:16:37.920 or will not stay in the low-energy spectrum. 01:16:37.920 --> 01:16:39.350 The mass will not change. 01:16:39.350 --> 01:16:41.540 And you will see that when you take that limit, 01:16:41.540 --> 01:16:45.130 you find this D3-brane remains here. 01:16:45.130 --> 01:16:50.410 And this string, which now will connect some r to r equal to 0, 01:16:50.410 --> 01:16:54.140 which is now just some AdS. 01:16:54.140 --> 01:16:56.240 This is just some interior or AdS, infinite proper 01:16:56.240 --> 01:16:57.460 distance away. 01:16:57.460 --> 01:16:59.250 And then you can check that, in the AdS 01:16:59.250 --> 01:17:11.660 metric, the image of this brane, viewed from the boundary time-- 01:17:11.660 --> 01:17:14.900 you have to this ratio factor, et cetera-- 01:17:14.900 --> 01:17:18.022 remains exactly equal to this. 01:17:18.022 --> 01:17:19.980 So this is an important self-consistency check, 01:17:19.980 --> 01:17:23.390 which I will not do here. 01:17:23.390 --> 01:17:24.860 So now, the story becomes simple. 01:17:29.620 --> 01:17:34.180 Now we generate such a string, such a fundamental particle 01:17:34.180 --> 01:17:35.880 with such a mass. 01:17:35.880 --> 01:17:50.660 But in order to introduce an external particle-- in order 01:17:50.660 --> 01:17:55.070 to have an external particle take a mass to go to infinity, 01:17:55.070 --> 01:17:58.070 so that means we need to take r goes to infinity. 01:17:58.070 --> 01:18:00.435 So that means we want to this D3-brane 01:18:00.435 --> 01:18:02.400 to lie on the boundary of AdS. 01:18:06.110 --> 01:18:13.765 So to summarize, now we want to move this D3-brane-- in order 01:18:13.765 --> 01:18:15.744 for the external particle, we want 01:18:15.744 --> 01:18:17.410 to take M goes to infinity, then we want 01:18:17.410 --> 01:18:18.600 to take r goes to infinity. 01:18:18.600 --> 01:18:20.610 So we want this D3-brane to actually lie 01:18:20.610 --> 01:18:23.590 on the boundary of AdS. 01:18:23.590 --> 01:18:25.423 So let us just conclude, summarize. 01:18:32.530 --> 01:18:45.250 An external quark with infinite mass, 01:18:45.250 --> 01:18:51.550 in N equals 4 super Yang-Mills theory, 01:18:51.550 --> 01:18:57.300 is described by a string ending on the boundary of AdS. 01:18:59.920 --> 01:19:01.590 So this is r equal to infinity. 01:19:04.360 --> 01:19:06.777 Because now, we have put in this D3-brane to the boundary, 01:19:06.777 --> 01:19:08.901 and then we'll, of course, run into a string ending 01:19:08.901 --> 01:19:09.882 on the boundary of AdS. 01:19:14.220 --> 01:19:16.950 A external quark with [INAUDIBLE] AdS, of course, 01:19:16.950 --> 01:19:31.516 run into a string ending at boundary of AdS. 01:19:31.516 --> 01:19:33.390 And in particular, the location of the string 01:19:33.390 --> 01:19:36.470 can be mapped to the location of the quark. 01:19:36.470 --> 01:19:45.931 So then endpoint of the string is the location of the quark. 01:19:54.120 --> 01:19:58.550 So now we have found a very nice picture that introduces 01:19:58.550 --> 01:20:01.530 a fundamental quark, in N equals 4 super Yang-Mills theory, 01:20:01.530 --> 01:20:02.490 because [? what you ?] [? need to ?] [? do with ?] 01:20:02.490 --> 01:20:03.073 such a string. 01:20:06.410 --> 01:20:08.740 And now, when you do a Wilson loop, 01:20:08.740 --> 01:20:17.440 then corresponding to transport this string around some path, 01:20:17.440 --> 01:20:19.700 then endpoint of the strings runs 01:20:19.700 --> 01:20:21.160 on a path in the field theory. 01:20:27.460 --> 01:20:29.910 I thought we would stop here.