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HONG LIU: So last time we
talked about what corresponding

9
00:00:25,670 --> 00:00:35,790
to external fundamental
quark in [INAUDIBLE] series.

10
00:00:42,330 --> 00:00:50,970
So you can see there [? m ?]
plus 1 is [? rebrain. ?] Then

11
00:00:50,970 --> 00:00:52,410
you separate one of them.

12
00:00:58,919 --> 00:00:59,960
You separate one of them.

13
00:01:03,020 --> 00:01:10,496
And then you can see there
one string connect between,

14
00:01:10,496 --> 00:01:12,370
then you can see that
the open string connect

15
00:01:12,370 --> 00:01:14,035
between this [? m ?] and this 1.

16
00:01:16,730 --> 00:01:21,960
And such object from
the point of view,

17
00:01:21,960 --> 00:01:25,040
because this end
have a unique index.

18
00:01:25,040 --> 00:01:27,360
And this end have indices.

19
00:01:27,360 --> 00:01:29,950
So this will transform
on the fundamental

20
00:01:29,950 --> 00:01:34,330
[? representation ?]
of this SU(N).

21
00:01:34,330 --> 00:01:38,940
Because when you're reshuffling
the indices of this [? n, ?]

22
00:01:38,940 --> 00:01:40,650
and this will
transform as a vector,

23
00:01:40,650 --> 00:01:46,190
because there's only one
end [INAUDIBLE] on it.

24
00:01:46,190 --> 00:01:49,215
So this is the field
series picture.

25
00:01:49,215 --> 00:01:52,760
Then in the gravity
picture, when

26
00:01:52,760 --> 00:01:56,090
you go to AdS, so when
you take a [INAUDIBLE]

27
00:01:56,090 --> 00:01:59,390
in the gravity side, then
this M-Brain just disappeared.

28
00:02:03,850 --> 00:02:07,580
And then you have
a boundary, say

29
00:02:07,580 --> 00:02:11,090
i equal to infinity,
or z equal to 0.

30
00:02:11,090 --> 00:02:14,170
And then you find in
a suitable [INAUDIBLE]

31
00:02:14,170 --> 00:02:15,970
that the single brain
still remains there.

32
00:02:18,910 --> 00:02:20,460
And then this string
become a string

33
00:02:20,460 --> 00:02:25,280
just an ending on this
brain, but extend all the way

34
00:02:25,280 --> 00:02:28,260
to i equal to 0.

35
00:02:28,260 --> 00:02:30,910
Because all the other
end, this [? rebrain ?]

36
00:02:30,910 --> 00:02:33,730
not disappeared, but
replaced by the geometry.

37
00:02:33,730 --> 00:02:37,850
So now what you find
in the gravity side,

38
00:02:37,850 --> 00:02:40,040
this just become
a single D-Brain,

39
00:02:40,040 --> 00:02:46,250
single [? D ?] [? rebrain ?] now
in the AdS 5 Now in the AdS 5.

40
00:02:46,250 --> 00:02:47,930
And there's one point on the S5.

41
00:02:47,930 --> 00:02:50,250
So depending on the direction.

42
00:02:50,250 --> 00:02:52,530
So this will be extended
in some direction,

43
00:02:52,530 --> 00:02:55,200
in the transverse direction,
to this [? rebrain. ?]

44
00:02:55,200 --> 00:02:58,010
And so that direction
determines the point on the S5.

45
00:02:58,010 --> 00:03:00,960
So we're relying on the
single point on the S5.

46
00:03:00,960 --> 00:03:05,700
And all this special
dimension is parallel

47
00:03:05,700 --> 00:03:07,310
to the boundary direction.

48
00:03:07,310 --> 00:03:10,700
So this, is the geometry clear?

49
00:03:10,700 --> 00:03:12,450
Good.

50
00:03:12,450 --> 00:03:14,870
And now if we want, you can
see the external quark which

51
00:03:14,870 --> 00:03:17,390
have infinite
mass, then you want

52
00:03:17,390 --> 00:03:19,760
to put this brain
to the boundary,

53
00:03:19,760 --> 00:03:22,730
because the mass of this quark
is controlled by the length.

54
00:03:22,730 --> 00:03:24,490
So you want this
length go to infinite.

55
00:03:24,490 --> 00:03:26,070
Then you put it to the boundary.

56
00:03:26,070 --> 00:03:28,040
And, yeah.

57
00:03:28,040 --> 00:03:29,480
Essentially.

58
00:03:29,480 --> 00:03:42,310
So this picture tells
you that the external

59
00:03:42,310 --> 00:03:51,670
quark in [INAUDIBLE] theory.

60
00:03:54,630 --> 00:04:03,110
Then you just mapped to a
string ending on the boundary.

61
00:04:06,690 --> 00:04:10,080
A string ending on the boundary.

62
00:04:10,080 --> 00:04:12,150
Do you have any questions
regarding this picture?

63
00:04:17,630 --> 00:04:21,329
And the mass of the string,
as we discussed before,

64
00:04:21,329 --> 00:04:29,950
either from this picture
the mass of the string

65
00:04:29,950 --> 00:04:34,218
would be the length divided
by 2 pi alpha prime.

66
00:04:37,210 --> 00:04:40,800
And that this mass getting
[INAUDIBLE] to this picture.

67
00:04:40,800 --> 00:04:44,140
And then you can check it
as a self-consistency check

68
00:04:44,140 --> 00:04:48,280
that the straight string
extend all the way say

69
00:04:48,280 --> 00:04:51,060
from some radius,
from some location r,

70
00:04:51,060 --> 00:04:54,160
to the i equal to 0, and
precisely have this mass.

71
00:05:03,600 --> 00:05:05,175
So now let me make a note.

72
00:05:12,160 --> 00:05:15,700
So as we said before, if
you say an ordinary [? K ?]

73
00:05:15,700 --> 00:05:17,650
[? series ?], if you
take an external quark,

74
00:05:17,650 --> 00:05:21,140
say moving in the fundamental
[? representation. ?] And when

75
00:05:21,140 --> 00:05:26,220
you parallel transport that
quark say along some loop,

76
00:05:26,220 --> 00:05:28,460
then you get the Wilson loop.

77
00:05:28,460 --> 00:05:30,320
Then you get the Wilson loop.

78
00:05:30,320 --> 00:05:33,940
But in this case, the
parallel transport

79
00:05:33,940 --> 00:05:52,116
of this m of such
an external quark

80
00:05:52,116 --> 00:05:53,490
actually gives
something which is

81
00:05:53,490 --> 00:05:57,950
slightly different from
the standard Wilson loop.

82
00:05:57,950 --> 00:06:00,405
So at the standard Wilson
loop is the following.

83
00:06:07,230 --> 00:06:11,270
It's you take the trace,
[? your ?] path order,

84
00:06:11,270 --> 00:06:17,610
and the exponential--
say i, A mu,

85
00:06:17,610 --> 00:06:24,120
say supposed S parameterize
the length along the path.

86
00:06:24,120 --> 00:06:25,930
And then you have A mu.

87
00:06:25,930 --> 00:06:29,101
And then you have dx mu ds.

88
00:06:29,101 --> 00:06:30,600
So this is the
standard Wilson loop.

89
00:06:32,785 --> 00:06:34,660
But in this case, actually
you have something

90
00:06:34,660 --> 00:06:38,094
more in the exponential
for the following reason.

91
00:06:40,880 --> 00:06:45,300
So in the [INAUDIBLE]
series, you'll

92
00:06:45,300 --> 00:06:46,477
not only have a gate field.

93
00:06:46,477 --> 00:06:47,810
You'll also have a scalar field.

94
00:06:50,339 --> 00:06:52,130
As we [? stated ?]
before, the scalar field

95
00:06:52,130 --> 00:06:57,890
parameterizes the
external fluctuation

96
00:06:57,890 --> 00:07:02,100
in the transverse
direction of the D-Brains.

97
00:07:02,100 --> 00:07:09,180
So now if you imagine you have a
string ending on such D-Brains,

98
00:07:09,180 --> 00:07:12,550
just by mechanical thinking,
really mechanical thinking,

99
00:07:12,550 --> 00:07:15,190
if you have something
ending on the D-Brain,

100
00:07:15,190 --> 00:07:17,360
and if this thing
have a mass, and it

101
00:07:17,360 --> 00:07:20,690
got to pull this D-Brain
in the direction.

102
00:07:20,690 --> 00:07:23,490
Because this thing have a
[? finite ?] [? tension. ?] So

103
00:07:23,490 --> 00:07:25,130
you want to pull the D-Brain.

104
00:07:25,130 --> 00:07:28,520
So that means that
the string actually

105
00:07:28,520 --> 00:07:31,790
will deform as a shape
of the D-Brain around

106
00:07:31,790 --> 00:07:33,270
at the ending point.

107
00:07:33,270 --> 00:07:36,340
So what that means is
that the string should

108
00:07:36,340 --> 00:07:39,320
couple to the scalar
field on the D-Brain.

109
00:07:39,320 --> 00:07:41,826
Such open string should
couple to the scalar

110
00:07:41,826 --> 00:07:42,700
field on the D-Brain.

111
00:07:42,700 --> 00:07:45,535
The scalar field paramertize
the transfer direction

112
00:07:45,535 --> 00:07:47,440
and the transfer motion.

113
00:07:47,440 --> 00:07:49,820
And the string should
pull the D-Brain.

114
00:07:53,340 --> 00:07:57,880
So this open string got to
couple to the scalar fields.

115
00:07:57,880 --> 00:07:59,620
So this is a rough
[? infusion. ?]

116
00:07:59,620 --> 00:08:04,520
But you can work
this out explicitly.

117
00:08:04,520 --> 00:08:06,110
You just take this series.

118
00:08:06,110 --> 00:08:10,580
So this is SU(N)
[INAUDIBLE] series, N + 1,

119
00:08:10,580 --> 00:08:11,580
your [INAUDIBLE] series.

120
00:08:11,580 --> 00:08:14,690
And then you break
to SU (N) times u(1).

121
00:08:17,910 --> 00:08:19,300
So this open string
goes bounding

122
00:08:19,300 --> 00:08:21,980
to the off diagonal
degree of freedom,

123
00:08:21,980 --> 00:08:25,960
of going into this breaking
between the N and the 1,

124
00:08:25,960 --> 00:08:28,299
and corresponding to explicit
field in this series.

125
00:08:28,299 --> 00:08:30,840
And you can just work out the
[? dynamically ?] of the field.

126
00:08:30,840 --> 00:08:32,770
You can usually check
that actually this

127
00:08:32,770 --> 00:08:33,980
coupled to the scalar field.

128
00:08:33,980 --> 00:08:36,740
So I will not go
into detail there.

129
00:08:36,740 --> 00:08:40,960
But you can work
out more explicitly.

130
00:08:40,960 --> 00:08:45,430
So the bottom line
is that such objects,

131
00:08:45,430 --> 00:08:48,210
we are not only coupled
to the gate field,

132
00:08:48,210 --> 00:08:51,430
but also coupled to
the scalar fields.

133
00:08:51,430 --> 00:08:54,190
And if you work
it out explicit;y,

134
00:08:54,190 --> 00:08:56,550
it will couple to the scalar
field in the following way.

135
00:09:00,500 --> 00:09:05,374
So a limit you not [? define. ?]
So you have six scalar fields.

136
00:09:05,374 --> 00:09:06,665
Let me denote them as a vector.

137
00:09:15,690 --> 00:09:16,190
Yeah.

138
00:09:16,190 --> 00:09:19,760
So now let me explain
this notation.

139
00:09:19,760 --> 00:09:22,770
So this n essentially the
direction of the string.

140
00:09:22,770 --> 00:09:33,430
So n is a unit
vector on S 5, which

141
00:09:33,430 --> 00:09:35,550
is a location of
that D-Brain, which

142
00:09:35,550 --> 00:09:37,185
is enabled the location
of this D-Brain

143
00:09:37,185 --> 00:09:42,170
is 5, because it's in
some transverse direction.

144
00:09:42,170 --> 00:09:49,550
And the 5 is just the
six scalar fields,

145
00:09:49,550 --> 00:09:53,920
which I writed the vector, six
fields of [INAUDIBLE] series.

146
00:09:58,290 --> 00:10:02,090
And then the dot x squared is
just the modulus of this guy.

147
00:10:04,690 --> 00:10:06,350
You have any questions on this?

148
00:10:06,350 --> 00:10:11,550
So now, because this is the
Wilson loop corresponding

149
00:10:11,550 --> 00:10:16,540
to such an external quark.

150
00:10:16,540 --> 00:10:21,540
And actually, when you
consider the string dynamics

151
00:10:21,540 --> 00:10:23,286
on the gravity
side, that we will

152
00:10:23,286 --> 00:10:25,160
relate to the expectation
value, for example,

153
00:10:25,160 --> 00:10:26,970
of this Wilson loop.

154
00:10:26,970 --> 00:10:28,152
Not the standard one.

155
00:10:33,800 --> 00:10:56,385
So now let's consider such
a quark traverses some path

156
00:10:56,385 --> 00:11:02,500
C, some loop C on the boundary.

157
00:11:13,220 --> 00:11:15,377
So I will make two statements.

158
00:11:15,377 --> 00:11:16,960
And from these two
statements, then we

159
00:11:16,960 --> 00:11:19,150
will be able to
guess what the Wilson

160
00:11:19,150 --> 00:11:22,660
loop will be corresponding
to on the gravity sides.

161
00:11:22,660 --> 00:11:32,388
So the first statement
is that this quark,

162
00:11:32,388 --> 00:11:47,110
lies under the endpoint
of a string in AdS.

163
00:11:54,740 --> 00:12:02,470
So that means the C-- so the
loop formed by this quark

164
00:12:02,470 --> 00:12:21,750
trajectory, there must be the
boundary of the corresponding

165
00:12:21,750 --> 00:12:29,570
string [? worksheet. ?] So let
me call this [? worksheet ?]

166
00:12:29,570 --> 00:12:32,070
sigma.

167
00:12:32,070 --> 00:12:36,380
So in other words, the
C must be corresponding

168
00:12:36,380 --> 00:12:37,660
to the boundary of the sigma.

169
00:12:49,920 --> 00:12:56,760
So the second statement is
that the expectation value

170
00:12:56,760 --> 00:13:01,360
of the Wilson loop-- so this can
see considered as the partition

171
00:13:01,360 --> 00:13:13,560
function of this quark system.

172
00:13:22,384 --> 00:13:24,800
So you can think about the
expectation value of the Wilson

173
00:13:24,800 --> 00:13:26,280
loop as the following.

174
00:13:26,280 --> 00:13:28,830
So you take the quark, and
write down the path integral

175
00:13:28,830 --> 00:13:30,720
for this quark.

176
00:13:30,720 --> 00:13:34,630
Write down the path of
the quark along this loop,

177
00:13:34,630 --> 00:13:43,550
and then just integrate all
of the dynamics of the quark.

178
00:13:43,550 --> 00:13:51,020
And then what you remaining
would be just the Wilson loop.

179
00:13:51,020 --> 00:13:54,982
So you can just imagine this
Wilson loop corresponding

180
00:13:54,982 --> 00:13:56,940
to the partition function
of this quark system.

181
00:14:00,110 --> 00:14:08,330
So now from our experience
with the duality, now

182
00:14:08,330 --> 00:14:11,459
we can just make a guess.

183
00:14:11,459 --> 00:14:13,250
We can just make a
guess, because we always

184
00:14:13,250 --> 00:14:15,583
identify one partition function
with the other partition

185
00:14:15,583 --> 00:14:16,500
function.

186
00:14:16,500 --> 00:14:18,380
They should be the same object.

187
00:14:18,380 --> 00:14:23,660
So now we should be able to
identify the expectation value

188
00:14:23,660 --> 00:14:29,315
of the Wilson loop with the
string [? Cartesian ?] function

189
00:14:29,315 --> 00:14:34,030
of this particular string,
which ending at the boundary,

190
00:14:34,030 --> 00:14:40,580
the string partition
function, and the host whose

191
00:14:40,580 --> 00:14:50,360
[? worksheet ?] ending
at the boundary as at C.

192
00:14:50,360 --> 00:14:54,350
So this is just, you can
just make a natural guess.

193
00:14:59,162 --> 00:15:00,995
You have any questions
regarding this guess?

194
00:15:09,281 --> 00:15:09,780
Good.

195
00:15:14,969 --> 00:15:17,010
So now let me may say a
few words about this guy.

196
00:15:25,700 --> 00:15:35,840
So suppose an [INAUDIBLE] just
parameterize say M sigma alpha

197
00:15:35,840 --> 00:15:47,100
to parameterize the
[? worksheet. ?] Say sigma

198
00:15:47,100 --> 00:15:51,120
alpha is the
worksheet coordinates,

199
00:15:51,120 --> 00:15:55,460
and [? XM ?] are the
coordinate on the AdS.

200
00:15:55,460 --> 00:15:58,860
And then this parameterized
a string moving AdS.

201
00:16:01,420 --> 00:16:07,070
And then the string
partition function,

202
00:16:07,070 --> 00:16:11,010
you can just write it as DX.

203
00:16:11,010 --> 00:16:17,950
You integrate of all possible
X. And then the string action

204
00:16:17,950 --> 00:16:21,790
with the boundary condition that
the worksheet of the boundary

205
00:16:21,790 --> 00:16:27,780
should ending on C.
So this essentially

206
00:16:27,780 --> 00:16:28,820
is the right-hand side.

207
00:16:31,450 --> 00:16:33,390
So according to
your convenience,

208
00:16:33,390 --> 00:16:37,040
you can either choose
the string action

209
00:16:37,040 --> 00:16:46,730
say to be a lambigoto or
[? Poliakoff ?] as you want.

210
00:16:46,730 --> 00:16:47,230
A Poliakoff.

211
00:16:50,000 --> 00:16:52,700
For example, let me just write
down the remind you what is

212
00:16:52,700 --> 00:16:58,450
the [? lambigoto. ?] For the
[? lambigoto, ?] you just 2 pi

213
00:16:58,450 --> 00:17:06,929
alpha prime delta h.

214
00:17:06,929 --> 00:17:09,790
And the h is the
induced the matrix on

215
00:17:09,790 --> 00:17:23,934
the [? worksheet. ?]
H is induced matrix on

216
00:17:23,934 --> 00:17:29,520
the [? worksheet. ?] Yeah.

217
00:17:29,520 --> 00:17:38,240
So in principle we have
a precise mathematical

218
00:17:38,240 --> 00:17:41,664
formulation we can work with.

219
00:17:41,664 --> 00:17:43,330
Then you can check
whether this proposal

220
00:17:43,330 --> 00:17:44,680
makes sense, et cetera.

221
00:17:52,546 --> 00:17:56,496
Of course, again, this object
is very hard to calculate.

222
00:17:56,496 --> 00:18:01,320
Yeah, I should say you may also
include the fermions, depending

223
00:18:01,320 --> 00:18:04,100
on superstring [INAUDIBLE]
you may have other things.

224
00:18:04,100 --> 00:18:11,940
But this is the
most basic object,

225
00:18:11,940 --> 00:18:13,840
So again, this
object in principle

226
00:18:13,840 --> 00:18:19,420
very difficult to calculate,
but then become simple

227
00:18:19,420 --> 00:18:23,304
if we can see the g string
goes to the 0 limit,

228
00:18:23,304 --> 00:18:24,970
and the alpha prime
goes to the 0 limit.

229
00:18:29,530 --> 00:18:31,380
So in the g string
goes to 0 limit,

230
00:18:31,380 --> 00:18:39,210
you remember, then we connect
a different topologies.

231
00:18:39,210 --> 00:18:42,230
You can just stick to
the lowest topology.

232
00:18:49,080 --> 00:18:52,430
So in other words, you can,
when g string goes to 0,

233
00:18:52,430 --> 00:18:57,690
you can neglect splitting
or joining on the string.

234
00:19:14,560 --> 00:19:17,110
So you can just can see that
the worksheet with the simplest

235
00:19:17,110 --> 00:19:17,610
topology.

236
00:19:21,020 --> 00:19:24,570
And in alpha prime
goes to 0 limit.

237
00:19:24,570 --> 00:19:27,510
So alpha prime appears
as a prefactor.

238
00:19:27,510 --> 00:19:30,010
1 over alpha prime
appears as a prefactor

239
00:19:30,010 --> 00:19:31,717
in this [? worksheet ?] series.

240
00:19:31,717 --> 00:19:33,550
Whether you do the
[? Poliakoff ?] or you do

241
00:19:33,550 --> 00:19:35,770
the [? lambogotto, ?]
it's the same thing.

242
00:19:35,770 --> 00:19:38,610
So alpha prime essentially
plays to the coupling

243
00:19:38,610 --> 00:19:41,650
constant on the [? worksheet. ?]

244
00:19:41,650 --> 00:19:44,260
So the alpha prime
goes to 0 limit.

245
00:19:44,260 --> 00:19:47,530
So alpha prime is
essentially the h bar

246
00:19:47,530 --> 00:19:49,660
on the [? worksheet, ?]
is the [? factive ?] h bar

247
00:19:49,660 --> 00:19:53,520
on the [? worksheets. ?] So
when alpha prime goes to 0,

248
00:19:53,520 --> 00:19:59,540
you can evaluate, then
the path integral.

249
00:20:02,950 --> 00:20:05,110
Just LIKE in the
gravity formulation,

250
00:20:05,110 --> 00:20:06,880
we can evaluate
the path integral

251
00:20:06,880 --> 00:20:08,730
when the g Newton goes to 0.

252
00:20:08,730 --> 00:20:11,287
So here we can evaluate path
integral using the sadale point

253
00:20:11,287 --> 00:20:11,870
approximation.

254
00:20:18,832 --> 00:20:19,540
The sadale point.

255
00:20:22,550 --> 00:20:26,120
So this is [INAUDIBLE].

256
00:20:26,120 --> 00:20:29,290
It implies, in the
sadale point especially,

257
00:20:29,290 --> 00:20:33,330
essentially you just evaluate
the classical action.

258
00:20:33,330 --> 00:20:35,550
Just find the
classical solution,

259
00:20:35,550 --> 00:20:38,840
classical string solution.

260
00:20:38,840 --> 00:20:43,000
And then evaluate the action
on that classical solution.

261
00:20:43,000 --> 00:20:48,361
And so this is
corresponding to [INAUDIBLE]

262
00:20:48,361 --> 00:20:49,860
fluctuation of the
[? worksheets. ?]

263
00:20:59,500 --> 00:21:02,840
So as I said, this
alpha prime appears

264
00:21:02,840 --> 00:21:05,610
in a [? worksheet ?]
action just as the place

265
00:21:05,610 --> 00:21:07,989
which the h bar appears.

266
00:21:07,989 --> 00:21:09,530
And so when you take
alpha prime goes

267
00:21:09,530 --> 00:21:11,160
to 0 limit, then
corresponding to your,

268
00:21:11,160 --> 00:21:13,285
you neglect the fluctuation
of the [? worksheet. ?]

269
00:21:13,285 --> 00:21:16,685
Then you can just can see that
the classical string dynamics.

270
00:21:20,770 --> 00:21:26,030
Then in this limit, then
we can do this easily.

271
00:21:26,030 --> 00:21:29,800
So that means in the g
string goes to the 0 limit,

272
00:21:29,800 --> 00:21:31,610
and the alpha prime
goes to 0 limit, which

273
00:21:31,610 --> 00:21:35,500
again translate on
the field series side

274
00:21:35,500 --> 00:21:37,920
to include infinity.

275
00:21:37,920 --> 00:21:41,330
And the lambda goes
to infinite limit.

276
00:21:41,330 --> 00:21:47,590
Then we can write the
expectation value of the Wilson

277
00:21:47,590 --> 00:21:58,569
loop just as the classical
action-- the action

278
00:21:58,569 --> 00:22:00,610
for the string evaluated
as a classical solution.

279
00:22:11,640 --> 00:22:26,770
So S cl is the action evaluated
at a classical solution.

280
00:22:35,980 --> 00:22:45,090
So essentially you just have a
[? regions ?] [? worksheet. ?]

281
00:22:45,090 --> 00:22:46,240
Any questions on this?

282
00:22:55,865 --> 00:22:57,490
So now we can talk
about some examples.

283
00:23:10,680 --> 00:23:15,197
So the simplest example
is that it's just

284
00:23:15,197 --> 00:23:16,280
considered a static quark.

285
00:23:22,550 --> 00:23:24,540
A static quark does not move.

286
00:23:24,540 --> 00:23:26,756
So if this is a
time direction, then

287
00:23:26,756 --> 00:23:28,130
the [INAUDIBLE]
line of the quark

288
00:23:28,130 --> 00:23:32,030
is just a straight line
moving in the time direction.

289
00:23:35,310 --> 00:23:37,330
And say from minus
infinity, plus infinity.

290
00:23:37,330 --> 00:23:40,390
And then you can imagine
close to the loop at infinity.

291
00:23:40,390 --> 00:23:43,410
Anyway, this is just
a straight line.

292
00:23:43,410 --> 00:23:45,550
So this is a simple situation.

293
00:23:45,550 --> 00:23:49,640
And so a level is convenient
to give a length to this line,

294
00:23:49,640 --> 00:23:51,060
the time direction.

295
00:23:51,060 --> 00:23:58,270
So they record the total
length as T-- capital T.

296
00:23:58,270 --> 00:24:01,570
And then without
calculating, essentially

297
00:24:01,570 --> 00:24:06,390
by definition the expectation
value of the Wilson loop

298
00:24:06,390 --> 00:24:10,720
should be just minus i MT.

299
00:24:10,720 --> 00:24:15,110
So this is just the phase factor
associated with [INAUDIBLE].

300
00:24:15,110 --> 00:24:19,140
This [? thing ?] under
the quark does not move.

301
00:24:19,140 --> 00:24:21,630
And the M is the total
energy of the quark,

302
00:24:21,630 --> 00:24:23,080
the mass of the quark.

303
00:24:23,080 --> 00:24:25,230
And then you just have
the standard [INAUDIBLE]

304
00:24:25,230 --> 00:24:25,730
[? i ET. ?]

305
00:24:37,040 --> 00:24:40,442
So now let's try to calculate
on the gravity side.

306
00:24:40,442 --> 00:24:42,900
Now let's try to calculate what
is the corresponding object

307
00:24:42,900 --> 00:24:44,140
to this on the gravity side.

308
00:24:47,930 --> 00:24:57,600
So now let me just remind
you the AdS [? metric. ?]

309
00:24:57,600 --> 00:25:00,340
So we can use the
two difference way,

310
00:25:00,340 --> 00:25:11,760
say whether using
this R coordinate,

311
00:25:11,760 --> 00:25:13,030
or using this Z coordinate.

312
00:25:23,500 --> 00:25:25,730
And they're related by
a very simple coordinate

313
00:25:25,730 --> 00:25:26,355
transformation.

314
00:25:29,840 --> 00:25:31,649
Just 1 over.

315
00:25:31,649 --> 00:25:32,440
Remind you of that.

316
00:25:36,240 --> 00:25:43,440
So on the gravity side, so
let's now draw our boundary.

317
00:25:43,440 --> 00:25:46,555
So this is r equal to
infinity, or z equal to 0.

318
00:25:49,410 --> 00:25:53,425
So the r increase in this
direction, and the z increase

319
00:25:53,425 --> 00:25:54,175
in that direction.

320
00:25:59,080 --> 00:26:01,030
So such objects on
the gravity side,

321
00:26:01,030 --> 00:26:04,222
just from the description
we said earlier,

322
00:26:04,222 --> 00:26:05,680
was just corresponding
to a string.

323
00:26:05,680 --> 00:26:08,765
Does not move at all.

324
00:26:08,765 --> 00:26:11,015
Extends from the boundary
all the way to the interior.

325
00:26:14,070 --> 00:26:16,880
And because this is just a
single quark and does not move.

326
00:26:16,880 --> 00:26:18,440
This is just our
previous picture.

327
00:26:24,740 --> 00:26:27,690
So we can parameterize it.

328
00:26:31,760 --> 00:26:33,520
So as we discussed
before, this action

329
00:26:33,520 --> 00:26:40,850
is reparameterization
environment.

330
00:26:40,850 --> 00:26:44,070
So that means we can choose
the sigma and the tau

331
00:26:44,070 --> 00:26:45,910
at this 2 sigma coordinate.

332
00:26:45,910 --> 00:26:48,970
So sigma alpha.

333
00:26:48,970 --> 00:26:51,901
And we'll call the
tau on the sigma.

334
00:26:51,901 --> 00:26:53,400
So I can choose the
tau on the sigma

335
00:26:53,400 --> 00:26:55,855
as we want according
to the convenience.

336
00:26:59,380 --> 00:27:04,880
So for this one, let
me choose tau equal

337
00:27:04,880 --> 00:27:08,820
just t at the real space time t.

338
00:27:08,820 --> 00:27:11,030
So that's t.

339
00:27:11,030 --> 00:27:16,560
And then take the sigma along
the radial direction, this r.

340
00:27:23,010 --> 00:27:26,170
So this straight string
just corresponding

341
00:27:26,170 --> 00:27:33,700
to the solution, which x
i t r equal to constant.

342
00:27:33,700 --> 00:27:36,600
So this is a string
which does not move.

343
00:27:36,600 --> 00:27:41,730
So the other directions,
it just a constant.

344
00:27:41,730 --> 00:27:43,980
If you put it at some
point, it stays there.

345
00:27:47,930 --> 00:27:50,210
So this is the obvious
solution of that configuration.

346
00:27:55,782 --> 00:27:56,490
So you can check.

347
00:27:56,490 --> 00:27:59,056
This is indeed a solution to
the [? lambogoto ?] action.

348
00:27:59,056 --> 00:28:00,680
I will not check
there, because this is

349
00:28:00,680 --> 00:28:02,910
the essentially started with.

350
00:28:02,910 --> 00:28:05,100
So you can immediately tell
this must be a solution.

351
00:28:08,710 --> 00:28:12,490
So low let's try to find the
classical action corresponding

352
00:28:12,490 --> 00:28:13,775
to this solution.

353
00:28:16,770 --> 00:28:19,620
So for this purpose, we
have to work the [INAUDIBLE]

354
00:28:19,620 --> 00:28:25,590
[? metric. ?] So the easiest
way to work out this the reduced

355
00:28:25,590 --> 00:28:28,020
[? metric ?] as follows.

356
00:28:28,020 --> 00:28:33,510
So the [? worksheet ?]
[? metric ?] would be h alpha

357
00:28:33,510 --> 00:28:35,220
beta.

358
00:28:35,220 --> 00:28:38,870
So it is sigma
alpha to sigma beta.

359
00:28:38,870 --> 00:28:40,700
And h alpha beta is
given by this guy.

360
00:28:44,330 --> 00:28:50,140
And the easiest way to work
out is that you just write down

361
00:28:50,140 --> 00:29:06,230
the AdS [? metric. ?] And now
you think that [? ht ?] and X

362
00:29:06,230 --> 00:29:09,480
and r as a function
of the sigma tau.

363
00:29:09,480 --> 00:29:13,490
And you just substitute
the functions in.

364
00:29:13,490 --> 00:29:14,912
So t just equal to tau.

365
00:29:14,912 --> 00:29:16,370
So this just becomes
t tau squared.

366
00:29:20,140 --> 00:29:23,530
But X is independent of t and r.

367
00:29:23,530 --> 00:29:27,850
So when I write the space
time coordinates, I use x.

368
00:29:27,850 --> 00:29:29,540
But when I write
it as a coordinate

369
00:29:29,540 --> 00:29:31,620
as a function of
the [? worksheet ?],

370
00:29:31,620 --> 00:29:33,240
then I use the capital.

371
00:29:33,240 --> 00:29:37,310
So this is just to emphasize,
this is a function.

372
00:29:37,310 --> 00:29:40,640
Anyway, and here I replace by
that X which is independent

373
00:29:40,640 --> 00:29:41,170
of [? ht ?].

374
00:29:41,170 --> 00:29:43,670
So there's nothing here.

375
00:29:43,670 --> 00:29:46,760
And now the R square
become sigma squared.

376
00:29:50,050 --> 00:29:55,129
So it become sigma squared,
because i just equal to sigma.

377
00:29:55,129 --> 00:29:57,170
And then this is my
[? worksheet ?] [? metric. ?]

378
00:29:57,170 --> 00:30:06,660
Sigma square, r square minus d
tau square plus r square sigma

379
00:30:06,660 --> 00:30:11,090
square [? d sigma ?] squared.

380
00:30:11,090 --> 00:30:14,110
So this is my induced
[? worksheet ?] [? metric ?]

381
00:30:14,110 --> 00:30:16,675
for this special worksheet.

382
00:30:16,675 --> 00:30:18,550
And if you imagine,
there's a time direction,

383
00:30:18,550 --> 00:30:20,165
which is this string
just translate

384
00:30:20,165 --> 00:30:21,227
in the time direction.

385
00:30:21,227 --> 00:30:21,810
Does not move.

386
00:30:24,420 --> 00:30:28,830
So now we can easily work
out what is the determinant.

387
00:30:28,830 --> 00:30:31,850
This is diagonal [? metric. ?]
You see these two cancel.

388
00:30:31,850 --> 00:30:33,420
So the determinant
just equal to one.

389
00:30:40,734 --> 00:30:42,900
So now we can write down
the [? worksheet ?] action.

390
00:30:47,650 --> 00:30:48,445
Let me erase here.

391
00:30:55,980 --> 00:31:07,960
So the [? worksheet ?] action S
N [? g ?] will be just minus 1

392
00:31:07,960 --> 00:31:09,402
over 2 pi alpha prime.

393
00:31:12,000 --> 00:31:15,060
Now the theta h is
just equal to 1.

394
00:31:15,060 --> 00:31:17,590
Then I just have
integration of dt.

395
00:31:17,590 --> 00:31:21,340
Then I have integration of
dr, because tau equal to t

396
00:31:21,340 --> 00:31:22,315
and sigma equal to r.

397
00:31:28,030 --> 00:31:29,620
And r should be
from 0 to infinity.

398
00:31:36,950 --> 00:31:39,725
So this integration of dt just
give us a capital T factor.

399
00:31:48,485 --> 00:31:49,610
And then this is divergent.

400
00:31:53,280 --> 00:31:56,765
So normally when we see
something divergent,

401
00:31:56,765 --> 00:32:03,220
as we always do, we just
say this is not at infinity.

402
00:32:03,220 --> 00:32:06,375
Let's put a small cutoff
here at i equal to lambda.

403
00:32:10,280 --> 00:32:17,175
So if I do that, and then
this here become lambda.

404
00:32:23,210 --> 00:32:27,080
So now if you read
from here, this

405
00:32:27,080 --> 00:32:30,500
should be identified with wc.

406
00:32:30,500 --> 00:32:33,900
I times of that object
should be identified with wc.

407
00:32:33,900 --> 00:32:37,630
And that should be
equal to minus iMT.

408
00:32:37,630 --> 00:32:46,290
So we conclude that
the mass must be

409
00:32:46,290 --> 00:32:48,615
lambda over 2 pi alpha prime.

410
00:32:55,640 --> 00:32:58,680
But this is actually what I
just advertised earlier, said

411
00:32:58,680 --> 00:33:02,120
you can check it explicitly,
that the mass of the object

412
00:33:02,120 --> 00:33:06,610
is precisely just the radial
location divided by r, divided

413
00:33:06,610 --> 00:33:07,720
by 2 pi alpha prime.

414
00:33:10,590 --> 00:33:14,569
And I just confirmed
that expectation.

415
00:33:14,569 --> 00:33:15,985
I just confirmed
that expectation.

416
00:33:21,090 --> 00:33:27,270
So this is infinite
by design, because I

417
00:33:27,270 --> 00:33:29,620
want to have an infinite
mass quark so that I

418
00:33:29,620 --> 00:33:30,760
don't have a fluctuation.

419
00:33:34,270 --> 00:33:36,300
And so in principle
I can take lambda.

420
00:33:36,300 --> 00:33:38,130
It's not just I want.

421
00:33:38,130 --> 00:33:40,010
But actually I can
show the lambda.

422
00:33:40,010 --> 00:33:42,010
So it actually make sense.

423
00:33:42,010 --> 00:33:44,926
So physically you
should think there's

424
00:33:44,926 --> 00:33:46,300
some kind of
[? disequilibrium ?]

425
00:33:46,300 --> 00:33:48,566
here which I can
show the location.

426
00:33:48,566 --> 00:33:50,440
And in the end, I take
lambda go to infinity,

427
00:33:50,440 --> 00:33:52,250
because I want the
mass go to infinity.

428
00:34:00,060 --> 00:34:03,050
So it's also instructive to
rewrite this in terms of the z.

429
00:34:10,050 --> 00:34:16,400
So in terms of the d, and
the z is related to capital R

430
00:34:16,400 --> 00:34:18,520
by this way, So in
this corresponding

431
00:34:18,520 --> 00:34:20,650
to a small thing in z.

432
00:34:20,650 --> 00:34:23,310
So z will be R squared.

433
00:34:23,310 --> 00:34:27,330
So this corresponding to
a cut off z at epsilon.

434
00:34:30,630 --> 00:34:31,239
At epsilon.

435
00:34:33,780 --> 00:34:39,431
So we can write this
M in terms of z.

436
00:34:39,431 --> 00:34:43,545
Then equal to become
2 pi alpha prime R

437
00:34:43,545 --> 00:34:44,670
squared divided by epsilon.

438
00:34:55,489 --> 00:34:57,930
So now let me do one more step.

439
00:34:57,930 --> 00:35:03,590
So as we said before, that
in terms of z-coordinate,

440
00:35:03,590 --> 00:35:05,090
so you'll also have
checked yourself

441
00:35:05,090 --> 00:35:08,480
by doing this holographic
bound exciter.

442
00:35:08,480 --> 00:35:10,710
So this epsilon can be
conceived as a short distance

443
00:35:10,710 --> 00:35:29,220
cutoff because of the
boundary, [INAUDIBLE]

444
00:35:29,220 --> 00:35:32,280
cut off at boundary.

445
00:35:32,280 --> 00:35:35,800
So this can also be conceived
as the mass of the quark.

446
00:35:35,800 --> 00:35:40,340
And due to that, you put
some short distance cutoff.

447
00:35:40,340 --> 00:35:44,270
Say epsilon distance
away from the quark.

448
00:35:44,270 --> 00:35:51,800
And so now let's remember the
dictionary R square divided

449
00:35:51,800 --> 00:35:55,410
by alpha prime is related to
on the gravity side to what?

450
00:35:55,410 --> 00:35:57,590
Do you remember?

451
00:35:57,590 --> 00:35:58,465
Yeah.

452
00:35:58,465 --> 00:35:59,090
A limit, right?

453
00:35:59,090 --> 00:36:00,305
The big box here.

454
00:36:03,920 --> 00:36:08,340
So R to the power of 4
divided by alpha prime

455
00:36:08,340 --> 00:36:10,210
squared equal to what?

456
00:36:10,210 --> 00:36:13,180
Equal to lambda, which
is the g [INAUDIBLE]

457
00:36:13,180 --> 00:36:21,960
square N [INAUDIBLE]
g square N. And then

458
00:36:21,960 --> 00:36:25,250
the five-dimensional
Newton constant divided

459
00:36:25,250 --> 00:36:29,286
by R cubed, that is related
to the N. So this is-- I

460
00:36:29,286 --> 00:36:32,760
think it's pi divided
by 2 N squared.

461
00:36:38,650 --> 00:36:41,530
You don't need to remember
the precise prefactors.

462
00:36:41,530 --> 00:36:45,890
But you should remember that R
squared divided by alpha prime

463
00:36:45,890 --> 00:36:46,765
is related to lambda.

464
00:36:52,100 --> 00:36:54,070
So lambda is the root coupling.

465
00:36:54,070 --> 00:36:58,140
So now we can rewrite this as
the square root lambda divided

466
00:36:58,140 --> 00:37:01,139
by 2 pi, then 1 over epsilon.

467
00:37:09,522 --> 00:37:11,105
So there's something
interesting here.

468
00:37:14,370 --> 00:37:18,090
He said there's a
square root lambda.

469
00:37:18,090 --> 00:37:20,200
It's the square root
lambda appearing here.

470
00:37:20,200 --> 00:37:23,820
And this is a square
root, g squared times N.

471
00:37:23,820 --> 00:37:27,040
So if you remember
in the [? QED ?],

472
00:37:27,040 --> 00:37:29,380
so if you try to calculate
what the [? self ?] energy

473
00:37:29,380 --> 00:37:33,130
of the electron in
the most naive way.

474
00:37:33,130 --> 00:37:36,170
So electron naively have an
infinite potential energy,

475
00:37:36,170 --> 00:37:38,760
because if you go
closer to the electron,

476
00:37:38,760 --> 00:37:41,880
and then the energy
will blow up.

477
00:37:41,880 --> 00:37:44,357
But if you put a small
cutoff around the electron,

478
00:37:44,357 --> 00:37:45,440
then what would be energy.

479
00:37:45,440 --> 00:37:47,587
You would you get?

480
00:37:47,587 --> 00:37:48,462
AUDIENCE: [INAUDIBLE]

481
00:37:51,240 --> 00:37:52,370
HONG LIU: Yep.

482
00:37:52,370 --> 00:37:54,950
Say if you put two electrons.

483
00:37:54,950 --> 00:37:57,530
Two electrons.

484
00:37:57,530 --> 00:37:58,810
AUDIENCE: [INAUDIBLE]

485
00:37:58,810 --> 00:37:59,435
HONG LIU: Yeah.

486
00:37:59,435 --> 00:38:00,560
You use a lot of electrons.

487
00:38:00,560 --> 00:38:03,390
You probe it.

488
00:38:03,390 --> 00:38:06,110
Then you would get e square.

489
00:38:06,110 --> 00:38:07,890
You would get e square.

490
00:38:07,890 --> 00:38:09,630
But here, interesting
thing is that it's

491
00:38:09,630 --> 00:38:12,620
proportional to the square
root of the e square.

492
00:38:12,620 --> 00:38:16,560
This is proportional to the
square root of e square.

493
00:38:16,560 --> 00:38:19,261
So this square root lambda is
essentially the strong coupling

494
00:38:19,261 --> 00:38:19,760
prediction.

495
00:38:27,300 --> 00:38:28,620
We will say more about this.

496
00:38:42,030 --> 00:38:43,530
So we have finished.

497
00:38:43,530 --> 00:38:47,130
This simple example
seems to make sense.

498
00:38:47,130 --> 00:38:50,550
So this just corresponding
to a mass of the quark.

499
00:38:50,550 --> 00:38:53,640
So now let's do something
a little bit more

500
00:38:53,640 --> 00:38:55,400
[? nontrivial ?].

501
00:38:55,400 --> 00:39:12,230
So let's consider the static
potential between a quark

502
00:39:12,230 --> 00:39:13,210
and the anti-quark.

503
00:39:26,780 --> 00:39:28,169
So from the field
series side, we

504
00:39:28,169 --> 00:39:29,585
can see that such
a configuration.

505
00:39:34,530 --> 00:39:38,920
So suppose this is
some x 1 direction.

506
00:39:38,920 --> 00:39:40,810
So this is a time direction.

507
00:39:40,810 --> 00:39:43,180
So we can see
there's such a loop.

508
00:39:43,180 --> 00:39:47,800
So this is L in
the x 1 direction.

509
00:39:47,800 --> 00:39:49,880
Say this is L divided by 2.

510
00:39:49,880 --> 00:39:52,240
And this is minus
L divided by 2.

511
00:39:52,240 --> 00:39:54,840
And in the T direction,
again the length

512
00:39:54,840 --> 00:39:59,960
is capital T. The
length is capital T.

513
00:39:59,960 --> 00:40:02,620
And we always take the T
to be much, much greater

514
00:40:02,620 --> 00:40:07,410
than L. Essentially you can
imagine this T as infinite.

515
00:40:10,330 --> 00:40:16,660
So as we mentioned last time, if
you can see the loop like this,

516
00:40:16,660 --> 00:40:18,710
when the T become much,
much greater than L,

517
00:40:18,710 --> 00:40:21,010
you can forget about the
beginning part and the end

518
00:40:21,010 --> 00:40:21,509
part.

519
00:40:21,509 --> 00:40:23,580
Essentially, you have
two parallel line.

520
00:40:23,580 --> 00:40:26,211
And the parallel lines in the
opposite direction in time,

521
00:40:26,211 --> 00:40:28,210
then you can see that
this is this corresponding

522
00:40:28,210 --> 00:40:31,320
to a quark, this corresponding
to the anti-quark moving

523
00:40:31,320 --> 00:40:34,020
parallel in time.

524
00:40:34,020 --> 00:40:38,820
And then again, the
expectation of the Wilson loop

525
00:40:38,820 --> 00:40:41,740
should give you-- so
this is a static system.

526
00:40:45,950 --> 00:40:49,380
This is a system with
translation variance in time.

527
00:40:49,380 --> 00:40:53,290
If I take T go to infinite, you
can imagine as a translation

528
00:40:53,290 --> 00:40:55,780
variance in time.

529
00:40:55,780 --> 00:40:58,710
And then due to this
translation variance in time,

530
00:40:58,710 --> 00:41:03,040
you can write it to [INAUDIBLE].

531
00:41:03,040 --> 00:41:07,490
It must have the falling
form, just on general ground.

532
00:41:07,490 --> 00:41:11,850
And e totally is just the
total energy of the system.

533
00:41:11,850 --> 00:41:13,180
Just in quantum mechanics.

534
00:41:13,180 --> 00:41:16,180
In quantum mechanics, any
translation variance system,

535
00:41:16,180 --> 00:41:20,080
the whole thing just minus iE T.

536
00:41:20,080 --> 00:41:23,300
And the total
should corresponding

537
00:41:23,300 --> 00:41:28,950
to the mass over the
two external quark,

538
00:41:28,950 --> 00:41:34,377
then plus the potential energy
between them [INAUDIBLE]

539
00:41:34,377 --> 00:41:35,210
energy between them.

540
00:41:38,300 --> 00:41:43,510
So if you calculate this
Wilson loop in say [? QCD. ?]

541
00:41:43,510 --> 00:41:46,448
So that will give you the
force between the quark

542
00:41:46,448 --> 00:41:47,697
and the anti-quark, et cetera.

543
00:41:50,840 --> 00:41:55,110
But in general, if you have
a strongly coupled [? gate ?]

544
00:41:55,110 --> 00:42:00,160
series, we don't know how
to do such a calculation.

545
00:42:00,160 --> 00:42:02,340
It's very hard to do.

546
00:42:02,340 --> 00:42:11,650
But in this case, equipped with
this, equipped by this, then

547
00:42:11,650 --> 00:42:15,090
the gravity will help us
to do this calculation.

548
00:42:15,090 --> 00:42:29,875
So the gravity can help us
find V(L) and strong coupling.

549
00:42:42,090 --> 00:42:45,290
So now we want to calculate
the string [? worksheets ?]

550
00:42:45,290 --> 00:42:46,790
corresponding to
such configuration.

551
00:42:56,150 --> 00:42:58,182
Do you have any questions
regarding our set-up?

552
00:43:11,490 --> 00:43:12,810
Good.

553
00:43:12,810 --> 00:43:21,210
So now let's write.

554
00:43:21,210 --> 00:43:23,240
You can see that the
gravity side again, this

555
00:43:23,240 --> 00:43:24,335
is the boundary of AdS.

556
00:43:27,220 --> 00:43:32,450
And now let's imagine
this is our x 1 direction.

557
00:43:32,450 --> 00:43:36,990
Then we have a quark
here at L divided by 2.

558
00:43:36,990 --> 00:43:41,860
And then have anti-quark
here at minus L divided by 2.

559
00:43:41,860 --> 00:43:44,235
So you can see that these two
lines, it's one from quark,

560
00:43:44,235 --> 00:43:46,190
one from anti-quark.

561
00:43:46,190 --> 00:43:48,820
And now this is
translates in time.

562
00:43:53,820 --> 00:43:59,260
So this the r equal to
infinite, or z equal to 0.

563
00:43:59,260 --> 00:44:03,030
So if we just have two
isolated quark, if each of them

564
00:44:03,030 --> 00:44:05,300
are isolated, then the
string [? worksheet ?],

565
00:44:05,300 --> 00:44:07,860
we are just as we said before,
just two straight line.

566
00:44:10,570 --> 00:44:12,100
Just two straight line.

567
00:44:12,100 --> 00:44:15,510
And that's corresponding
to an isolated quark.

568
00:44:15,510 --> 00:44:20,765
But now these two
quark can interact.

569
00:44:20,765 --> 00:44:22,760
Can interact.

570
00:44:22,760 --> 00:44:25,980
So the magic expectation is
that this string [? worksheet ?]

571
00:44:25,980 --> 00:44:29,600
will get deformed, and the
[INAUDIBLE] expectations

572
00:44:29,600 --> 00:44:31,900
that they will just
connect to each other.

573
00:44:41,724 --> 00:44:43,390
So you expect the
string [? worksheet ?]

574
00:44:43,390 --> 00:44:45,334
will be like this.

575
00:44:45,334 --> 00:44:47,250
So if the string
[? worksheet ?] is like this,

576
00:44:47,250 --> 00:44:48,770
that means there's no
interaction between these two

577
00:44:48,770 --> 00:44:49,269
quarks.

578
00:44:51,660 --> 00:44:59,817
And now if the two string,
yeah, so this is the quark.

579
00:44:59,817 --> 00:45:01,150
And then this is the anti-quark.

580
00:45:01,150 --> 00:45:04,520
So they should have
opposite orientation.

581
00:45:04,520 --> 00:45:06,570
And then the string
[? worksheet ?],

582
00:45:06,570 --> 00:45:09,190
we have orientation like this.

583
00:45:09,190 --> 00:45:11,890
So it will go off like this.

584
00:45:11,890 --> 00:45:18,011
So now you expect that
the [? worksheet ?]

585
00:45:18,011 --> 00:45:18,760
will be like this.

586
00:45:21,489 --> 00:45:23,280
And again, it's transition
variant in time.

587
00:45:25,800 --> 00:45:28,135
Any questions on this?

588
00:45:28,135 --> 00:45:29,010
Is this clear to you?

589
00:45:32,740 --> 00:45:34,500
So geometrically,
this is only way

590
00:45:34,500 --> 00:45:38,480
which these two quark can have
interactions is by joining

591
00:45:38,480 --> 00:45:40,442
their [? worksheet ?] together.

592
00:45:40,442 --> 00:45:42,900
AUDIENCE: So generally, we
would be doing the path integral

593
00:45:42,900 --> 00:45:47,070
over all the closed strings that
have those boundary emissions.

594
00:45:47,070 --> 00:45:47,570
Sorry.

595
00:45:47,570 --> 00:45:48,234
Open string.

596
00:45:48,234 --> 00:45:48,859
HONG LIU: Yeah.

597
00:45:48,859 --> 00:45:49,058
That's right.

598
00:45:49,058 --> 00:45:49,554
That's right.

599
00:45:49,554 --> 00:45:50,054
Yeah.

600
00:45:53,030 --> 00:45:54,680
Good.

601
00:45:54,680 --> 00:46:00,010
So now with this
realization, then the rest

602
00:46:00,010 --> 00:46:01,691
become mechanical.

603
00:46:01,691 --> 00:46:02,565
Not quite mechanical.

604
00:46:07,830 --> 00:46:08,330
Yeah.

605
00:46:08,330 --> 00:46:15,390
So we can choose, as
before, tau equal to T.

606
00:46:15,390 --> 00:46:17,650
So T equals infinity.

607
00:46:17,650 --> 00:46:20,410
So we can consider the system
translation variant in time.

608
00:46:30,050 --> 00:46:35,040
So let's choose again tau
equal to T. So we can also

609
00:46:35,040 --> 00:46:38,150
chose the sigma
still equal to R.

610
00:46:38,150 --> 00:46:40,540
But now let me do it
slightly differently.

611
00:46:40,540 --> 00:46:43,230
So let me do it
sigma equal to x 1.

612
00:46:43,230 --> 00:46:44,090
So you can do both.

613
00:46:44,090 --> 00:46:47,592
But just to give you
a variety, [INAUDIBLE]

614
00:46:47,592 --> 00:46:49,800
sigma equals to x 1, because
x 1 [? direction also ?]

615
00:46:49,800 --> 00:46:52,510
become nontrivial.

616
00:46:52,510 --> 00:46:55,224
And let me choose
sigma equal to x 1.

617
00:46:55,224 --> 00:46:56,640
And then this
[? worksheet ?] then

618
00:46:56,640 --> 00:47:01,440
is parameterized
just by z, which

619
00:47:01,440 --> 00:47:02,695
is only a function of sigma.

620
00:47:05,330 --> 00:47:09,650
So sigma runs from
minus half L to L.

621
00:47:09,650 --> 00:47:12,150
And then this string would be
parameterized by some function

622
00:47:12,150 --> 00:47:13,704
z as a function of sigma.

623
00:47:13,704 --> 00:47:15,370
And because of the
translation variance,

624
00:47:15,370 --> 00:47:18,540
you cannot have dependence on t.

625
00:47:18,540 --> 00:47:20,840
Cannot have dependence on t.

626
00:47:20,840 --> 00:47:27,220
And also, all the other
directions are not equal to 1.

627
00:47:27,220 --> 00:47:33,100
They should be just
constant, because they're not

628
00:47:33,100 --> 00:47:34,980
moving in the other directions.

629
00:47:34,980 --> 00:47:36,560
So they should be
just constants.

630
00:47:36,560 --> 00:47:41,760
So the only non-trivial thing
would be in the x 1 direction.

631
00:47:41,760 --> 00:47:44,590
You can also choose
the sigma to be z,

632
00:47:44,590 --> 00:47:47,830
and the x 1 would
be a function of z,

633
00:47:47,830 --> 00:47:49,619
then the x 1 would be
a function of sigma.

634
00:47:49,619 --> 00:47:50,660
You can do it either way.

635
00:47:50,660 --> 00:47:52,890
Doesn't matter.

636
00:47:52,890 --> 00:47:59,180
So the boundary condition
would be z re-evaluated

637
00:47:59,180 --> 00:48:02,797
at plus minus 1/2 L.
Then this of course

638
00:48:02,797 --> 00:48:03,880
should be at the boundary.

639
00:48:03,880 --> 00:48:06,805
So it should be equal to 0.

640
00:48:06,805 --> 00:48:08,180
So this is the
boundary condition

641
00:48:08,180 --> 00:48:09,775
that this starting
from the boundary.

642
00:48:16,910 --> 00:48:20,789
So now let's try to write down
the [? worksheet ?] action.

643
00:48:20,789 --> 00:48:22,580
Let's try to write down
the [? worksheet ?]

644
00:48:22,580 --> 00:48:25,870
metric under the
[? worksheet ?] action.

645
00:48:25,870 --> 00:48:28,217
So the [? worksheet ?]
metric is again as

646
00:48:28,217 --> 00:48:29,300
what we were doing before.

647
00:48:29,300 --> 00:48:33,590
You can just write down
i equal to z square minus

648
00:48:33,590 --> 00:48:39,115
dt square plus dx
square plus tz square.

649
00:48:41,810 --> 00:48:44,900
So now t equal to tau.

650
00:48:44,900 --> 00:48:46,990
So dt squared just
become d tau square.

651
00:48:52,090 --> 00:48:54,632
And the [? dx 1 ?] square just
become [? d ?] sigma square.

652
00:48:54,632 --> 00:48:55,340
under the others.

653
00:48:55,340 --> 00:48:56,210
We don't need to care.

654
00:48:56,210 --> 00:48:57,668
So this just become
d sigma square.

655
00:49:00,840 --> 00:49:03,220
And the z is a
function of sigma.

656
00:49:03,220 --> 00:49:10,970
So this just become z prime
sigma square d sigma square.

657
00:49:14,070 --> 00:49:17,240
And then you just find
this is equal to R square

658
00:49:17,240 --> 00:49:24,855
z square d tau square plus
1 plus z prime squared

659
00:49:24,855 --> 00:49:26,980
d sigma square.

660
00:49:26,980 --> 00:49:31,992
So this is the induced metric
on the [? worksheet ?].

661
00:49:31,992 --> 00:49:34,450
And then you can now write down
the [? lambagoto ?] action.

662
00:49:37,330 --> 00:49:43,625
So it just 1 over 2 pi
alpha prime d 2 sigma.

663
00:49:43,625 --> 00:49:45,000
And then you take
the determinant

664
00:49:45,000 --> 00:49:47,460
of this metric,
which is easy to do.

665
00:49:47,460 --> 00:49:50,620
You get R square factor out.

666
00:49:50,620 --> 00:49:53,855
And then you get
1 over d square.

667
00:49:57,330 --> 00:50:00,465
And it just square root
of 1 plus z prime squared.

668
00:50:00,465 --> 00:50:02,090
So this is the square
root determinant.

669
00:50:09,260 --> 00:50:13,380
So we can do a little bit of
[? massage. ?] So this d 2

670
00:50:13,380 --> 00:50:18,460
sigma is dt d sigma.

671
00:50:18,460 --> 00:50:23,210
And the sigma is from minus L to
1/2 L, because this is a sigma,

672
00:50:23,210 --> 00:50:27,760
is x 1 from minus L to 1/2 L.

673
00:50:27,760 --> 00:50:30,170
So [INAUDIBLE] depend on t.

674
00:50:30,170 --> 00:50:38,470
So we can just replace it
by a factor of capital T.

675
00:50:38,470 --> 00:50:41,910
And obviously this system
have symmetry between the x

676
00:50:41,910 --> 00:50:45,015
and the minus x.

677
00:50:45,015 --> 00:50:48,980
And then I can just choose the
integration from half of it.

678
00:50:48,980 --> 00:50:53,560
So I do it from 0
to L divided by 2.

679
00:50:53,560 --> 00:50:56,360
And then I eliminated 2
on the [? downstairs. ?]

680
00:51:04,980 --> 00:51:08,310
So this is my action.

681
00:51:08,310 --> 00:51:10,560
Then what I need to do is
I need to [? extremize ?].

682
00:51:13,400 --> 00:51:18,380
This is action
respect to z sigma.

683
00:51:23,070 --> 00:51:24,590
Then you solve
for z sigma, which

684
00:51:24,590 --> 00:51:27,390
satisfy the boundary condition.

685
00:51:27,390 --> 00:51:34,630
And then solve for z sigma.

686
00:51:34,630 --> 00:51:38,650
And then find S
classical action.

687
00:51:50,400 --> 00:51:56,050
So now this reduce to
a freshmen problem--

688
00:51:56,050 --> 00:52:00,050
literally, little
cute freshman problem.

689
00:52:07,950 --> 00:52:13,640
So which is actually quite fun
to do with a little bit twist.

690
00:52:13,640 --> 00:52:15,060
With a little bit twist.

691
00:52:17,830 --> 00:52:19,330
So do you want to
do it immediately,

692
00:52:19,330 --> 00:52:22,190
or do want to have a break?

693
00:52:22,190 --> 00:52:26,860
So now let's try to do this.

694
00:52:26,860 --> 00:52:28,520
Try to do this.

695
00:52:28,520 --> 00:52:37,090
So first, just by
consistency we expect

696
00:52:37,090 --> 00:52:43,890
this thing must be divergent,
because this contains 2M.

697
00:52:43,890 --> 00:52:47,580
But 2M in our set-up,
if you start something

698
00:52:47,580 --> 00:52:50,480
from the boundary,
and the M is infinite,

699
00:52:50,480 --> 00:52:54,370
then we would expect this
thing to be divergent.

700
00:52:54,370 --> 00:52:57,380
We would expect something
to be divergent.

701
00:52:57,380 --> 00:53:01,120
And indeed, you see
the dangerous thing

702
00:53:01,120 --> 00:53:07,470
is that when the sigma
approach [INAUDIBLE] over 2,

703
00:53:07,470 --> 00:53:09,980
then you have a 1
over d square here,

704
00:53:09,980 --> 00:53:14,470
and at this z of 1
over 2, this is 0.

705
00:53:14,470 --> 00:53:16,970
So there is some potential
divergences here.

706
00:53:16,970 --> 00:53:18,940
But of course you
also have to look

707
00:53:18,940 --> 00:53:20,840
at the behavior of this term.

708
00:53:20,840 --> 00:53:24,790
By this you see there is some
potential divergence here.

709
00:53:24,790 --> 00:53:26,460
And this is expected,
because we know

710
00:53:26,460 --> 00:53:29,860
that this must contain the 2M.

711
00:53:29,860 --> 00:53:31,640
And if this whole
story is consistent,

712
00:53:31,640 --> 00:53:35,350
there must be a divergent.

713
00:53:35,350 --> 00:53:41,240
So let us just write
with that in mind.

714
00:53:43,790 --> 00:53:47,660
So let me just subtract
the 2 MT from here.

715
00:53:47,660 --> 00:53:49,390
So this [? UNG ?].

716
00:53:49,390 --> 00:53:54,880
So this is supposed
to be minus i E

717
00:53:54,880 --> 00:54:05,940
total T. So from
this expectation,

718
00:54:05,940 --> 00:54:09,455
we would write V(L).

719
00:54:09,455 --> 00:54:14,680
So now again I replace the R
square divided by alpha prime

720
00:54:14,680 --> 00:54:16,950
by square root lambda.

721
00:54:16,950 --> 00:54:18,695
So we have square
root lambda pi.

722
00:54:21,440 --> 00:54:29,680
Then L divided by 2 d
sigma 1 over z square 1

723
00:54:29,680 --> 00:54:33,820
plus z prime square.

724
00:54:33,820 --> 00:54:40,450
And the minus now
1 over epsilon.

725
00:54:40,450 --> 00:54:42,580
Because the [? twice ?]
the M is just

726
00:54:42,580 --> 00:54:45,370
the square root of pi divided by
the square root lambda divided

727
00:54:45,370 --> 00:54:47,100
by pi times 1 over epsilon.

728
00:54:47,100 --> 00:54:48,970
So I take out this factor.

729
00:54:48,970 --> 00:54:50,718
And so we expect the
V(L) [INAUDIBLE].

730
00:54:54,070 --> 00:55:00,990
So if this guess we have
said earlier is correct,

731
00:55:00,990 --> 00:55:05,164
that the partition
function. of the Wilson loop

732
00:55:05,164 --> 00:55:06,580
should be related
to the partition

733
00:55:06,580 --> 00:55:08,620
function of the string,
and then reduced

734
00:55:08,620 --> 00:55:13,810
to a classical action, et
cetera-- if that idea is right,

735
00:55:13,810 --> 00:55:16,880
we should find by
self consistency

736
00:55:16,880 --> 00:55:19,310
that this thing
should be finite.

737
00:55:19,310 --> 00:55:22,330
This thing should be
finite, and should also

738
00:55:22,330 --> 00:55:25,560
be [? inactive, ?] because
the quark and the anti-quark

739
00:55:25,560 --> 00:55:31,700
have an attractive force.

740
00:55:31,700 --> 00:55:36,270
So we should find this thing
to be finite and negative.

741
00:55:36,270 --> 00:55:39,360
You see, this guy is
actually positive.

742
00:55:39,360 --> 00:55:43,020
And no matter what is z, this
guy is actually positive.

743
00:55:43,020 --> 00:55:46,290
So that means that this guy must
be smaller than 1 over epsilon.

744
00:55:46,290 --> 00:55:49,170
And so you get something active.

745
00:55:49,170 --> 00:55:53,040
So this guy must be some 1
over epsilon lambda something.

746
00:55:56,250 --> 00:55:58,680
So now let me call this
to be my Lagrangian.

747
00:55:58,680 --> 00:56:01,575
So now we had to do a small
variation of the problem.

748
00:56:04,870 --> 00:56:05,600
Yeah.

749
00:56:05,600 --> 00:56:09,380
Maybe not confused by that
L. Let me put this script

750
00:56:09,380 --> 00:56:14,190
L. So this is my Lagrangian.

751
00:56:14,190 --> 00:56:19,130
And then this is a
one-dimensional Lagrangian

752
00:56:19,130 --> 00:56:20,966
problem.

753
00:56:20,966 --> 00:56:22,840
And this Lagrangian does
not depend on sigma.

754
00:56:25,740 --> 00:56:28,272
So we treat the sigma as time.

755
00:56:28,272 --> 00:56:29,980
So this is like a
one-dimensional problem

756
00:56:29,980 --> 00:56:31,660
with the sigma as time.

757
00:56:31,660 --> 00:56:35,180
So if the Lagrangian
does not depend on time,

758
00:56:35,180 --> 00:56:36,300
what's going to happen?

759
00:56:36,300 --> 00:56:37,550
AUDIENCE: Energy conservation.

760
00:56:37,550 --> 00:56:38,216
HONG LIU: Exact.

761
00:56:38,216 --> 00:56:39,750
The energy's conserved.

762
00:56:39,750 --> 00:56:40,830
The energy's conserved.

763
00:56:40,830 --> 00:56:43,890
So we can write down the energy
corresponding to this guy.

764
00:56:43,890 --> 00:56:51,690
And the energy is z
prime pi z minus L.

765
00:56:51,690 --> 00:56:54,110
So this must be constant.

766
00:56:54,110 --> 00:56:59,260
And the pi z, just the economic
momentum conjugates to the z.

767
00:57:05,180 --> 00:57:06,520
So all this is very elementary.

768
00:57:06,520 --> 00:57:10,915
You can immediately
write down this equation,

769
00:57:10,915 --> 00:57:12,040
[? h of 1 over ?] d square.

770
00:57:18,040 --> 00:57:20,260
So you find the
pi you plug here.

771
00:57:20,260 --> 00:57:21,600
Then you find this equation.

772
00:57:21,600 --> 00:57:23,389
This equation just become that.

773
00:57:31,220 --> 00:57:35,591
So now let's try to
parameterize this constant.

774
00:57:35,591 --> 00:57:37,840
So the way to do it is just
look for the special point

775
00:57:37,840 --> 00:57:39,820
on the trajectory.

776
00:57:39,820 --> 00:57:42,605
So the special point on the
trajectory is this point.

777
00:57:42,605 --> 00:57:44,230
By symmetry, it should
be corresponding

778
00:57:44,230 --> 00:57:45,390
to sigma equal to 0.

779
00:57:47,677 --> 00:57:49,260
So this point, it
should corresponding

780
00:57:49,260 --> 00:57:51,030
to sigma equal to zero.

781
00:57:51,030 --> 00:57:55,030
And so let me call
this point a z 0.

782
00:57:55,030 --> 00:58:00,408
So this is essentially how
deep this string are going

783
00:58:00,408 --> 00:58:04,980
into the [? back. ?] So
let's call that 2 [? bz ?] 0.

784
00:58:04,980 --> 00:58:09,580
So it's clear at sigma
equal to zero, the z

785
00:58:09,580 --> 00:58:13,390
prime should be equal to zero.

786
00:58:13,390 --> 00:58:15,880
So this should be just
equal to 1 over z 0 squared.

787
00:58:19,580 --> 00:58:22,370
Should be 1 over z 0 squared.

788
00:58:22,370 --> 00:58:25,030
And then you can just
immediately rewrite

789
00:58:25,030 --> 00:58:28,020
this as ordinary
differential equation.

790
00:58:43,020 --> 00:58:45,470
So now you can immediately
integrate this equation.

791
00:58:48,680 --> 00:59:01,230
And say z 0 is found, it's
determined at the boundary

792
00:59:01,230 --> 00:59:05,610
condition by requiring
z L equal to 0.

793
00:59:08,860 --> 00:59:15,130
So I will not do it here, but
you can do it in 30 seconds

794
00:59:15,130 --> 00:59:16,345
to find the z 0.

795
00:59:16,345 --> 00:59:18,220
You actually don't need
to actually integrate

796
00:59:18,220 --> 00:59:18,830
this equation.

797
00:59:18,830 --> 00:59:20,730
There's a simple trick to do it.

798
00:59:20,730 --> 00:59:24,410
Anyway so let me just
write down the answer.

799
00:59:24,410 --> 00:59:29,700
You can find the z 0 equal
to L. It's proportional to L,

800
00:59:29,700 --> 00:59:35,430
under the square root of
pi divided by 2 comma 1/4,

801
00:59:35,430 --> 00:59:37,471
and become 3/4.

802
00:59:37,471 --> 00:59:40,200
AUDIENCE: [INAUDIBLE]

803
00:59:40,200 --> 00:59:43,670
HONG LIU: Oh, this funny number.

804
00:59:43,670 --> 00:59:44,999
AUDIENCE: 30 seconds.

805
00:59:44,999 --> 00:59:45,624
HONG LIU: Yeah.

806
00:59:49,745 --> 00:59:50,245
Yeah.

807
00:59:50,245 --> 00:59:51,328
Mathematica will tell you.

808
00:59:55,410 --> 00:59:56,530
Take you 25 seconds.

809
00:59:56,530 --> 01:00:00,610
You type into the Mathematica,
and the four seconds

810
01:00:00,610 --> 01:00:03,290
to check the error,
and the Mathematica one

811
01:00:03,290 --> 01:00:04,540
second to give you the answer.

812
01:00:09,940 --> 01:00:11,180
So you find this.

813
01:00:16,530 --> 01:00:19,560
So we will comment on this
implication a little bit later.

814
01:00:19,560 --> 01:00:25,350
So now we can plug
this into this action.

815
01:00:25,350 --> 01:00:28,670
So let me see.

816
01:00:28,670 --> 01:00:30,460
Yeah.

817
01:00:30,460 --> 01:00:36,030
So now let me plug this equation
back into this equation.

818
01:00:43,712 --> 01:00:44,670
[INAUDIBLE] [? star ?].

819
01:00:47,560 --> 01:00:51,100
So there's again a
simple trick to do it.

820
01:00:51,100 --> 01:00:54,690
You don't need to know
the expressive form of z,

821
01:00:54,690 --> 01:00:59,320
because what you can do is
you can rewrite this V(L).

822
01:00:59,320 --> 01:01:03,490
You can rewrite this
integral as follows-- V(L),

823
01:01:03,490 --> 01:01:07,520
say this square root of
lambda divided by pi.

824
01:01:07,520 --> 01:01:11,910
So this d sigma, you can write
it as dz divided by z prime.

825
01:01:11,910 --> 01:01:13,180
So this is the d sigma.

826
01:01:13,180 --> 01:01:16,270
You can change the
variable from sigma to z.

827
01:01:16,270 --> 01:01:21,960
And then this just
integration become z 0 to z 0.

828
01:01:21,960 --> 01:01:30,030
And then you have the 1 over z
square, 1 plus z prime square.

829
01:01:30,030 --> 01:01:33,330
So now you can substitute
what is the z prime into here.

830
01:01:33,330 --> 01:01:35,920
And now this just
become integral over z,

831
01:01:35,920 --> 01:01:37,030
and just pure integral.

832
01:01:39,772 --> 01:01:41,230
We still need to
minus [INAUDIBLE].

833
01:01:44,160 --> 01:01:48,550
So if you plug into the
expression for the z prime,

834
01:01:48,550 --> 01:01:53,250
and then do a
scaling-- say take z

835
01:01:53,250 --> 01:01:56,620
equal to z 0 y--
because there you

836
01:01:56,620 --> 01:02:00,790
see this all have the
nice scaling form.

837
01:02:00,790 --> 01:02:03,800
So it's good to
scale the z 0 out.

838
01:02:03,800 --> 01:02:06,680
So let's do a scaling,
z equal to z 0 y.

839
01:02:06,680 --> 01:02:08,860
Then you could
rewrite that integral.

840
01:02:08,860 --> 01:02:10,375
Sorry this is no V(Z).

841
01:02:10,375 --> 01:02:11,840
This is V(L).

842
01:02:11,840 --> 01:02:15,995
So you can rewrite that integral
as V(L) equal to square root

843
01:02:15,995 --> 01:02:20,000
of pi, square root of lambda
divided by pi 1 over z 0.

844
01:02:27,380 --> 01:02:31,840
So now I have to put
the epsilon in here

845
01:02:31,840 --> 01:02:35,900
anticipating that this integral
will be divergent here.

846
01:02:35,900 --> 01:02:38,310
So this is the same
epsilon we put in there

847
01:02:38,310 --> 01:02:41,210
when we do the single quark.

848
01:02:43,980 --> 01:02:45,390
So we put it there.

849
01:02:49,110 --> 01:02:49,610
Yeah.

850
01:02:49,610 --> 01:02:52,000
Maybe let me just
write one more step

851
01:02:52,000 --> 01:02:55,530
so that you can see
it more explicitly.

852
01:02:55,530 --> 01:03:02,090
Yeah, so I just plug this
in, plug this into here.

853
01:03:02,090 --> 01:03:09,330
Then what you get square root
lambda z zero square epsilon.

854
01:03:09,330 --> 01:03:12,170
Then I put a cutoff
in the epsilon.

855
01:03:12,170 --> 01:03:13,890
And this becomes
[? tz ?] square.

856
01:03:25,670 --> 01:03:28,920
And now I can do this scaling
to express in terms of y.

857
01:03:32,450 --> 01:03:36,060
And then this becomes
square root lambda pi 1

858
01:03:36,060 --> 01:03:45,986
over z 0 t y over 1
all over y square.

859
01:03:49,914 --> 01:03:56,680
[? of ?] [? 4. ?] So
minus 1 over epsilon.

860
01:03:56,680 --> 01:03:59,860
For simplicity, I will slightly
rewrite this 1 over epsilon

861
01:03:59,860 --> 01:04:04,900
into the following
thing just as a contrast

862
01:04:04,900 --> 01:04:09,630
as from infinite, epsilon
divided by z zero dy divided

863
01:04:09,630 --> 01:04:10,410
by y squared.

864
01:04:15,980 --> 01:04:17,400
You can easily do this integral.

865
01:04:17,400 --> 01:04:20,697
This integral you
integrate as 1 over y.

866
01:04:20,697 --> 01:04:22,030
And the other end it give you 0.

867
01:04:22,030 --> 01:04:25,730
Lower end they give you
z 0 divided by epsilon.

868
01:04:25,730 --> 01:04:28,050
And then z 0 cancels with z 0.

869
01:04:28,050 --> 01:04:31,770
So you just get 1 over epsilon.

870
01:04:31,770 --> 01:04:35,860
But the reason to
write in this form

871
01:04:35,860 --> 01:04:39,780
is to see that this two
singularity indeed cancel.

872
01:04:39,780 --> 01:04:42,330
So these have a singularity.

873
01:04:42,330 --> 01:04:46,030
As y goes to 0, this
is non-integrable.

874
01:04:46,030 --> 01:04:47,840
And here you have
the same singularity.

875
01:04:47,840 --> 01:04:50,430
They cancel.

876
01:04:50,430 --> 01:04:52,850
So when [? d ?] over [? y ?]
equal to 0, they cancel,

877
01:04:52,850 --> 01:04:56,520
and the next order term come
from by expanding this term.

878
01:04:56,520 --> 01:04:59,850
Then you get the y to the power
4 when you expand to this term.

879
01:04:59,850 --> 01:05:02,870
[? In the upstair ?] you
get 1 over y power 4.

880
01:05:02,870 --> 01:05:05,530
[? Other ?] than that, will
give you something finite.

881
01:05:05,530 --> 01:05:07,400
So that just tells
you immediately

882
01:05:07,400 --> 01:05:10,530
see that the singularity
have canceled.

883
01:05:10,530 --> 01:05:14,440
So you will guaranteed
to get something finite.

884
01:05:14,440 --> 01:05:16,870
You are guaranteed to
get something finite.

885
01:05:16,870 --> 01:05:20,300
And another reason
to write in this form

886
01:05:20,300 --> 01:05:23,280
is that this is
[? something ?] integrate to 1,

887
01:05:23,280 --> 01:05:26,266
and this is something
integrate to infinity,

888
01:05:26,266 --> 01:05:27,890
even though the
difference between them

889
01:05:27,890 --> 01:05:32,530
is negative, because you have
agreed to do the calculation.

890
01:05:32,530 --> 01:05:37,970
But this is because integrand
is not exactly the same.

891
01:05:37,970 --> 01:05:39,970
But the fact that this
is going to infinite

892
01:05:39,970 --> 01:05:42,160
give you a chance that
this become active.

893
01:05:42,160 --> 01:05:45,740
And when you calculate
it, it indeed negative.

894
01:05:45,740 --> 01:05:50,050
And the [INAUDIBLE],
literally you can do it.

895
01:05:50,050 --> 01:05:51,976
Yeah.

896
01:05:51,976 --> 01:05:53,350
So you can write
down the answer.

897
01:05:53,350 --> 01:05:55,516
You find the answer is given
by something like this.

898
01:05:55,516 --> 01:05:58,390
It's minus square root
lambda divided by pi,

899
01:05:58,390 --> 01:06:02,421
then some numerical factor,
which is not very important,

900
01:06:02,421 --> 01:06:03,670
but let me just write it down.

901
01:06:08,621 --> 01:06:09,370
To the power of 4.

902
01:06:22,240 --> 01:06:24,890
So this answer is nice.

903
01:06:24,890 --> 01:06:27,880
So then now let me make
some final remarks.

904
01:06:27,880 --> 01:06:31,170
So this is finite and negative.

905
01:06:34,860 --> 01:06:39,009
So this pass the minimal
consistency check

906
01:06:39,009 --> 01:06:40,550
that we are doing
something sensible.

907
01:06:43,662 --> 01:06:44,995
We are doing something sensible.

908
01:06:44,995 --> 01:06:48,860
If we find something
infinite, then that proposal

909
01:06:48,860 --> 01:06:49,510
must be wrong.

910
01:06:49,510 --> 01:06:51,632
You can immediately rule it out.

911
01:06:51,632 --> 01:06:53,090
And if you find
something positive,

912
01:06:53,090 --> 01:06:55,640
then you can also
immediately rule it out.

913
01:06:55,640 --> 01:07:00,860
And so this is a
self-consistency check.

914
01:07:00,860 --> 01:07:06,350
Oh, I forgot the
1 over y at z 0.

915
01:07:06,350 --> 01:07:10,530
Oh, actually, if you
plug in the 1 over z 0,

916
01:07:10,530 --> 01:07:13,470
I think you just--
no no, no, no.

917
01:07:13,470 --> 01:07:14,555
I write it wrong.

918
01:07:14,555 --> 01:07:15,180
This is not pi.

919
01:07:15,180 --> 01:07:19,990
This is L. So I have
already plugged in z 0.

920
01:07:19,990 --> 01:07:21,860
So z 0 is given by this.

921
01:07:21,860 --> 01:07:28,770
It's proportionate to L.
And then this is 1 over L.

922
01:07:28,770 --> 01:07:32,226
So this 1 over L is also
makes sense physically,

923
01:07:32,226 --> 01:07:34,570
because 1 over L is just
the cooling potential.

924
01:07:39,300 --> 01:07:42,240
And we do expect the
cooling potential,

925
01:07:42,240 --> 01:07:46,580
because [INAUDIBLE]
as we said is a scale

926
01:07:46,580 --> 01:07:48,640
[? invariant ?] theory.

927
01:07:48,640 --> 01:07:51,800
And in four-dimensional
scale, [? invariant ?] series,

928
01:07:51,800 --> 01:07:54,740
potential just [INAUDIBLE] the
[? ground ?] it will be 1 over

929
01:07:54,740 --> 01:07:57,520
L, just by scaling symmetry.

930
01:07:57,520 --> 01:08:00,090
It just exactly
[INAUDIBLE] [? why ?]

931
01:08:00,090 --> 01:08:02,955
[? electromagnetism ?]
is 1 over L.

932
01:08:02,955 --> 01:08:08,167
And so this just from
the scale invariance.

933
01:08:08,167 --> 01:08:09,125
There's no other scale.

934
01:08:11,770 --> 01:08:14,360
So the dependence on
L must be 1 over L,

935
01:08:14,360 --> 01:08:17,850
because we have dimension mass.

936
01:08:17,850 --> 01:08:19,475
So they cannot have
other L dependence.

937
01:08:23,399 --> 01:08:27,859
So the second point is now we
again see this proportional

938
01:08:27,859 --> 01:08:28,775
to square root lambda.

939
01:08:31,670 --> 01:08:34,175
Again, this is a strong
coupling with that.

940
01:08:40,170 --> 01:08:52,130
Because at weak
coupling, we know

941
01:08:52,130 --> 01:08:56,720
it must be [? proportional ?]
to lambda divided by L.

942
01:08:56,720 --> 01:09:00,540
And lambda is just the analog
of the fine structure constant.

943
01:09:00,540 --> 01:09:03,399
And then this is the
cooling potential.

944
01:09:03,399 --> 01:09:14,920
So 1 over L is always
true, no matter

945
01:09:14,920 --> 01:09:17,010
the value of the coupling.

946
01:09:17,010 --> 01:09:22,080
So you see, when you [INAUDIBLE]
the coupling from lambda

947
01:09:22,080 --> 01:09:24,574
equal to 0, then it should
be linear in lambda.

948
01:09:24,574 --> 01:09:25,990
Then when lambda
goes to infinity,

949
01:09:25,990 --> 01:09:27,573
then become the
square root in lambda.

950
01:09:34,670 --> 01:09:35,670
So now the final remark.

951
01:09:39,060 --> 01:09:47,410
[INAUDIBLE] we see that z 0 is
proportional to L. So now let's

952
01:09:47,410 --> 01:09:51,229
look at the geometrical
meaning of this z 0.

953
01:09:51,229 --> 01:09:59,260
So this z 0 is how far a string
extends into the [? back. ?]

954
01:09:59,260 --> 01:10:05,055
So if you say L is
small, then z 0 is small.

955
01:10:10,970 --> 01:10:17,180
Now if L is large, if I make L
large, then z 0 become bigger.

956
01:10:17,180 --> 01:10:18,930
That means that the
string extends further

957
01:10:18,930 --> 01:10:23,200
into the [? back. ?] And when
you take L goes to infinity,

958
01:10:23,200 --> 01:10:26,865
and this go all
the way to the AdS.

959
01:10:29,970 --> 01:10:32,410
And again this is the
refraction of IR/UV.

960
01:10:35,290 --> 01:10:42,790
Refraction of IR/UV,
because L corresponding

961
01:10:42,790 --> 01:10:44,720
to whether you measure
UV physics or IR

962
01:10:44,720 --> 01:10:46,630
physics in the field series.

963
01:10:46,630 --> 01:10:48,430
So L small is a short
distance. [INAUDIBLE]

964
01:10:48,430 --> 01:10:50,470
is long-distance physics.

965
01:10:50,470 --> 01:10:55,140
And you see when you probe
long-distance physics,

966
01:10:55,140 --> 01:11:01,150
then you are probing much
deeper in the interior in AdS.

967
01:11:01,150 --> 01:11:03,190
And if you're looking
at short distance,

968
01:11:03,190 --> 01:11:05,810
then you're probing the region
very close to the boundary.

969
01:11:09,630 --> 01:11:11,820
Any questions on this?

970
01:11:11,820 --> 01:11:12,580
Yes.

971
01:11:12,580 --> 01:11:14,376
AUDIENCE: So are there any
interesting experimental

972
01:11:14,376 --> 01:11:15,917
implications of
these sorts of things

973
01:11:15,917 --> 01:11:18,540
that we couldn't get otherwise?

974
01:11:18,540 --> 01:11:22,426
Could we have known these
four facts basically

975
01:11:22,426 --> 01:11:24,050
without doing this
calculation by doing

976
01:11:24,050 --> 01:11:25,790
more standard techniques?

977
01:11:25,790 --> 01:11:27,040
HONG LIU: You mean this thing?

978
01:11:27,040 --> 01:11:27,800
AUDIENCE: Yeah.

979
01:11:27,800 --> 01:11:30,730
HONG LIU: No.

980
01:11:30,730 --> 01:11:33,344
Yeah, so this is really
strong coupling prediction.

981
01:11:33,344 --> 01:11:35,010
AUDIENCE: So what
would that correspond?

982
01:11:35,010 --> 01:11:36,684
What sort of thing
could you measure

983
01:11:36,684 --> 01:11:38,205
in nature that would sort of--?

984
01:11:38,205 --> 01:11:38,830
HONG LIU: Yeah.

985
01:11:38,830 --> 01:11:43,310
So then what you would
do in your p-set.

986
01:11:43,310 --> 01:11:45,090
So in your p-set,
in the last problem,

987
01:11:45,090 --> 01:11:48,080
you will calculate this thing
at the finite temperature,

988
01:11:48,080 --> 01:11:53,380
which we will talk of finite
temperature in a few minutes.

989
01:11:53,380 --> 01:11:55,630
You do it in the
finite temperature.

990
01:11:55,630 --> 01:11:59,780
And then you can essentially
calculate this potential

991
01:11:59,780 --> 01:12:02,024
at the finite temperature.

992
01:12:02,024 --> 01:12:03,690
And as you will do
in the p-set, there's

993
01:12:03,690 --> 01:12:05,065
a [? screening ?] length.

994
01:12:05,065 --> 01:12:07,440
And the finite temperature is
no longer scale invariance.

995
01:12:07,440 --> 01:12:10,800
Beyond a certain length, the
potential will go to zero.

996
01:12:10,800 --> 01:12:12,540
And you can measure
the screening length.

997
01:12:12,540 --> 01:12:14,500
And people on the
[? lattice, ?] [? QCD ?],

998
01:12:14,500 --> 01:12:16,080
that precisely match with that.

999
01:12:16,080 --> 01:12:18,160
So you can compare
AdS calculation

1000
01:12:18,160 --> 01:12:24,240
to the strongly coupled
[? QCD ?] calculation,

1001
01:12:24,240 --> 01:12:27,920
spend a huge amount of computing
time, large amount of money,

1002
01:12:27,920 --> 01:12:30,520
large amount of
manpower on the computer

1003
01:12:30,520 --> 01:12:33,580
to do that kind of Wilson loop
calculation on the computer.

1004
01:12:33,580 --> 01:12:37,938
And then you see the
answer is very similar.

1005
01:12:37,938 --> 01:12:38,910
Yeah.

1006
01:12:38,910 --> 01:12:41,826
AUDIENCE: That's interesting.

1007
01:12:41,826 --> 01:12:47,047
[INAUDIBLE] the strong
coupling I guess

1008
01:12:47,047 --> 01:12:48,630
is proportional to
square root lambda.

1009
01:12:48,630 --> 01:12:51,001
But the square
root of lambda was

1010
01:12:51,001 --> 01:12:52,375
what you said from
the beginning,

1011
01:12:52,375 --> 01:12:55,782
because [INAUDIBLE]
of the alpha prime.

1012
01:12:55,782 --> 01:12:57,630
It does not involve
the calculation at all.

1013
01:12:57,630 --> 01:13:00,200
So this [INAUDIBLE] produce
a weak coupling result.

1014
01:13:00,200 --> 01:13:00,950
HONG LIU: Oh sure.

1015
01:13:00,950 --> 01:13:01,780
Sure.

1016
01:13:01,780 --> 01:13:04,850
Because we are calculating
a strong coupling.

1017
01:13:04,850 --> 01:13:07,430
Because we are calculating
the strong coupling.

1018
01:13:07,430 --> 01:13:08,050
Yeah.

1019
01:13:08,050 --> 01:13:10,884
So this whole set-up is
for the strong coupling.

1020
01:13:10,884 --> 01:13:14,710
AUDIENCE: So this [INAUDIBLE].

1021
01:13:14,710 --> 01:13:15,357
HONG LIU: No.

1022
01:13:15,357 --> 01:13:17,440
But weak coupling you can
do [INAUDIBLE] diagrams.

1023
01:13:17,440 --> 01:13:19,480
You don't need this.

1024
01:13:19,480 --> 01:13:22,720
But strong coupling,
we have no other way

1025
01:13:22,720 --> 01:13:26,180
to do this kind of
calculation, other than doing

1026
01:13:26,180 --> 01:13:29,320
massive computer simulation.

1027
01:13:29,320 --> 01:13:33,100
And for some things, you cannot
even do computer calculation.

1028
01:13:33,100 --> 01:13:33,600
Good.

1029
01:13:33,600 --> 01:13:35,551
Any other questions?

1030
01:13:35,551 --> 01:13:36,050
Yeah.

1031
01:13:36,050 --> 01:13:37,910
So this is a good question.

1032
01:13:37,910 --> 01:13:41,280
So this actually
a very good point.

1033
01:13:41,280 --> 01:13:42,869
So this tells you
on the gravity side,

1034
01:13:42,869 --> 01:13:44,910
essentially anything
involving these [? always ?]

1035
01:13:44,910 --> 01:13:46,743
[? pointing ?] to the
square root of lambda,

1036
01:13:46,743 --> 01:13:48,350
just related one of alpha prime.

1037
01:13:48,350 --> 01:13:48,850
Yeah.

1038
01:13:48,850 --> 01:13:52,385
It just come from
the kinematics.

1039
01:13:52,385 --> 01:13:57,752
Yeah, but [INAUDIBLE] the system
the very nontrivial prediction

1040
01:13:57,752 --> 01:13:59,085
of the strong coupling behavior.

1041
01:14:02,980 --> 01:14:03,480
Good.

1042
01:14:03,480 --> 01:14:09,270
We need to go to the
finite temperature.

1043
01:14:09,270 --> 01:14:12,990
Because I want you to be
able to do your p-set.

1044
01:14:12,990 --> 01:14:18,300
So now we have talked
about how to calculate

1045
01:14:18,300 --> 01:14:20,337
the various observables.

1046
01:14:20,337 --> 01:14:22,920
Now we can pause a little bit
to talk about other generations.

1047
01:14:25,690 --> 01:14:28,770
So the first generalization
is [INAUDIBLE],

1048
01:14:28,770 --> 01:14:31,530
let's do finite temperature,
do a [? langio ?] temperature.

1049
01:14:39,900 --> 01:14:43,160
So so far, we, can
see that so far we

1050
01:14:43,160 --> 01:14:46,970
discussed AdS 5 times [? AdS ?]
5 string series AdS [INAUDIBLE]

1051
01:14:46,970 --> 01:14:49,430
is 5 string.

1052
01:14:52,480 --> 01:14:56,010
In AdS 5 times [? s ?] 5 is
related to [INAUDIBLE] series.

1053
01:15:00,470 --> 01:15:05,980
So we talked about this in
terms of small perturbations.

1054
01:15:05,980 --> 01:15:21,470
The normalizable mode on this
side should map to some state.

1055
01:15:21,470 --> 01:15:24,270
Say you have some
normalizable mode,

1056
01:15:24,270 --> 01:15:27,100
should do [INAUDIBLE]
some states.

1057
01:15:27,100 --> 01:15:28,950
And for example, the pure AdS 5.

1058
01:15:32,066 --> 01:15:33,190
So there's nothing excited.

1059
01:15:33,190 --> 01:15:37,140
This is example of
this [INAUDIBLE] kind

1060
01:15:37,140 --> 01:15:40,610
of normalizable mode.

1061
01:15:40,610 --> 01:15:43,349
So this is should be
due to the vacuum,

1062
01:15:43,349 --> 01:15:45,390
because nothing excited
on the field series side.

1063
01:15:45,390 --> 01:15:48,770
So this must be
due to the vacuum.

1064
01:15:48,770 --> 01:15:52,590
So now start with
this pure AdS 5.

1065
01:15:52,590 --> 01:15:56,430
Now as you start putting things
inside, excited the system,

1066
01:15:56,430 --> 01:16:01,760
then you will say in some sense
excite some normalizable modes.

1067
01:16:01,760 --> 01:16:06,560
Then you go to excited states
in your [INAUDIBLE] series.

1068
01:16:06,560 --> 01:16:09,325
And then one of the
obvious excited states

1069
01:16:09,325 --> 01:16:12,150
is the finite
temperature states.

1070
01:16:12,150 --> 01:16:14,945
So the obvious question is
what does the thermal state

1071
01:16:14,945 --> 01:16:15,564
correspond to?

1072
01:16:26,030 --> 01:16:26,530
Yeah.

1073
01:16:26,530 --> 01:16:27,863
corresponding to just save time.

1074
01:16:32,300 --> 01:16:34,549
So the gravity description
should have satisfied

1075
01:16:34,549 --> 01:16:35,590
the following conditions.

1076
01:16:46,770 --> 01:16:49,447
To satisfy the
following features.

1077
01:16:49,447 --> 01:16:51,030
So now we want to
guess what should be

1078
01:16:51,030 --> 01:16:53,620
the thing on the gravity side.

1079
01:16:53,620 --> 01:16:58,416
First he said it should
be asymptotic AdS 5.

1080
01:17:02,230 --> 01:17:04,565
That means that there should
be only normalizable modes.

1081
01:17:09,850 --> 01:17:11,650
There should be only
normalizable mode,

1082
01:17:11,650 --> 01:17:14,730
because the finite temperature
state is a specific state

1083
01:17:14,730 --> 01:17:16,410
of the field series.

1084
01:17:16,410 --> 01:17:21,120
So there should be no
non-normalizable modes excited.

1085
01:17:24,320 --> 01:17:28,300
And of course if the
field series have

1086
01:17:28,300 --> 01:17:30,300
a finite temperature,
the other side

1087
01:17:30,300 --> 01:17:32,680
must also have a
finite temperature.

1088
01:17:32,680 --> 01:17:35,705
So it must have a
finite temperature.

1089
01:17:39,340 --> 01:17:41,280
So now I use T means
the temperature.

1090
01:17:41,280 --> 01:17:44,690
Now the capital T
means the temperature.

1091
01:17:44,690 --> 01:17:49,425
And in particular,
this should satisfy

1092
01:17:49,425 --> 01:17:50,550
all laws of thermodynamics.

1093
01:18:03,451 --> 01:18:04,950
And the third one
is that you should

1094
01:18:04,950 --> 01:18:08,320
have of course all
of the symmetries

1095
01:18:08,320 --> 01:18:10,070
of the finite
temperature system.

1096
01:18:10,070 --> 01:18:12,910
So you should have translation
variance, rotational symmetry,

1097
01:18:12,910 --> 01:18:15,200
et cetera.

1098
01:18:15,200 --> 01:18:18,810
Of course, a Lorentz symmetry,
Lorentz boost would be broken.

1099
01:18:18,810 --> 01:18:24,023
So you still should have
translation symmetry

1100
01:18:24,023 --> 01:18:25,273
and the rotational symmetries.

1101
01:18:29,560 --> 01:18:31,604
But the boost will be
broken, because now when

1102
01:18:31,604 --> 01:18:34,020
you add finite temperature,
you no longer have the Lorentz

1103
01:18:34,020 --> 01:18:35,820
boost.

1104
01:18:35,820 --> 01:18:38,300
So you need to find some
kind of gravity solution

1105
01:18:38,300 --> 01:18:44,015
which will satisfy
those criteria.

1106
01:18:44,015 --> 01:18:45,515
So there are two
obvious candidates.

1107
01:19:06,104 --> 01:19:07,270
So there are two candidates.

1108
01:19:16,094 --> 01:19:17,760
So the first is just
you can see there's

1109
01:19:17,760 --> 01:19:24,680
some kind of thermal gas in AdS.

1110
01:19:28,610 --> 01:19:32,435
So just imagine I have
some thermal gas in AdS.

1111
01:19:32,435 --> 01:19:34,810
And the second possibility is
that you have a black hole.

1112
01:19:38,030 --> 01:19:41,539
Because we also know a black
hole is a thermal object.

1113
01:19:41,539 --> 01:19:43,372
We also know black hole
is a thermal object.

1114
01:19:51,730 --> 01:19:55,110
So it turns out so
we will only consider

1115
01:19:55,110 --> 01:19:56,735
the [INAUDIBLE]
[? patch. ?] So we only

1116
01:19:56,735 --> 01:20:03,280
consider the boundaries is
really the r [INAUDIBLE]

1117
01:20:03,280 --> 01:20:05,550
rather than the [INAUDIBLE].

1118
01:20:05,550 --> 01:20:06,650
So the thermal gas.

1119
01:20:06,650 --> 01:20:09,360
So let me first describe the
thermal gas, a description.

1120
01:20:14,909 --> 01:20:16,700
So normally in the
field series, how do you

1121
01:20:16,700 --> 01:20:17,950
describe a thermal system?

1122
01:20:21,600 --> 01:20:25,640
So how would you describe
a thermal system?

1123
01:20:25,640 --> 01:20:27,560
AUDIENCE: [INAUDIBLE]

1124
01:20:27,560 --> 01:20:29,583
HONG LIU: In field series.

1125
01:20:29,583 --> 01:20:30,940
AUDIENCE: [INAUDIBLE]

1126
01:20:30,940 --> 01:20:33,270
HONG LIU: Yeah.

1127
01:20:33,270 --> 01:20:36,965
A partition function is one
observable of a thermal system.

1128
01:20:36,965 --> 01:20:39,590
Say if you want to describe the
thermal system in field series,

1129
01:20:39,590 --> 01:20:40,907
what do you do?

1130
01:20:40,907 --> 01:20:42,990
Say if you want to calculate
correlation function,

1131
01:20:42,990 --> 01:20:45,390
you want to calculate
many other things?

1132
01:20:45,390 --> 01:20:47,230
AUDIENCE: [INAUDIBLE]
Euclidean [INAUDIBLE]?

1133
01:20:47,230 --> 01:20:48,430
HONG LIU: Exactly.

1134
01:20:48,430 --> 01:20:51,760
So normally in the field
series, what you do

1135
01:20:51,760 --> 01:20:55,070
is that you go to Euclidean
signature, and then

1136
01:20:55,070 --> 01:20:57,060
periodically identify time.

1137
01:20:57,060 --> 01:21:00,450
And the period of time is
the inverse temperature.

1138
01:21:00,450 --> 01:21:01,340
So we can do it here.

1139
01:21:01,340 --> 01:21:04,530
The thermal gassing idea is
essentially is what we do.

1140
01:21:04,530 --> 01:21:06,660
You say we go to the
Euclidean signature.

1141
01:21:13,860 --> 01:21:17,190
Yeah, let me call it t E.
We go to Euclidean time.

1142
01:21:22,930 --> 01:21:25,880
We go to Euclidean time.

1143
01:21:25,880 --> 01:21:32,270
And then we identify
Euclidean time periodically

1144
01:21:32,270 --> 01:21:34,595
with inverse temperature,
which we always call beta.

1145
01:21:37,440 --> 01:21:41,190
And so in particular, the
fermions is anti-periodic.

1146
01:21:45,857 --> 01:21:46,440
Anti-periodic.

1147
01:21:53,980 --> 01:21:56,520
That means anti-periodic.

1148
01:21:56,520 --> 01:22:00,150
Of course, the
Boson is periodic.

1149
01:22:00,150 --> 01:22:04,710
But actually, this metric is
bad for the following reason.

1150
01:22:08,320 --> 01:22:16,900
So in gravity as a
general rule, whenever

1151
01:22:16,900 --> 01:22:22,060
you see a compact
space go to 0 sites,

1152
01:22:22,060 --> 01:22:26,160
then there's a danger of
having a curvature singularity.

1153
01:22:26,160 --> 01:22:28,620
It's almost always you have
a curvature singularity.

1154
01:22:28,620 --> 01:22:32,440
Only for some special
situations it does not have it.

1155
01:22:32,440 --> 01:22:36,800
And so this is bad,
because now when

1156
01:22:36,800 --> 01:22:38,510
you compactify this
Euclidean time,

1157
01:22:38,510 --> 01:22:40,200
this become a compact direction.

1158
01:22:40,200 --> 01:22:41,910
This is a circle.

1159
01:22:41,910 --> 01:22:44,550
And when you go to
z goes to infinity,

1160
01:22:44,550 --> 01:22:47,099
this circle goes to zero sides.

1161
01:22:47,099 --> 01:22:48,640
You can check that
actually this have

1162
01:22:48,640 --> 01:22:51,530
more curvature singularity.

1163
01:22:51,530 --> 01:22:53,320
So this is actually bad.

1164
01:22:53,320 --> 01:22:56,260
So when t is
uncompact, then this

1165
01:22:56,260 --> 01:22:58,550
is a coordinate singularity.

1166
01:22:58,550 --> 01:23:01,200
But when t become
compact, and then this

1167
01:23:01,200 --> 01:23:02,863
become a [? general ?]
singularity.

1168
01:23:02,863 --> 01:23:04,446
This become [? general ?]
singularity.

1169
01:23:08,730 --> 01:23:12,560
So first this metric's bad,
because I have singularity.

1170
01:23:18,772 --> 01:23:19,730
At z equal to infinity.

1171
01:23:22,730 --> 01:23:26,200
At deep in the interior of AdS.

1172
01:23:26,200 --> 01:23:29,140
There's also a second reason,
which is more stringent reason,

1173
01:23:29,140 --> 01:23:31,890
which you don't need to care.

1174
01:23:31,890 --> 01:23:36,400
Let me just write it
down for completeness.

1175
01:23:36,400 --> 01:23:38,280
So if you actually
analyze a string theory

1176
01:23:38,280 --> 01:23:41,380
in such kind of space time.

1177
01:23:41,380 --> 01:23:47,150
So also there's a generic
rule in string theory

1178
01:23:47,150 --> 01:23:50,430
is that if you have a
very small circle, then

1179
01:23:50,430 --> 01:23:53,570
the string can wind
around the circle.

1180
01:23:53,570 --> 01:23:56,440
The string can wind
around the circle.

1181
01:23:56,440 --> 01:23:59,890
And then when that circle
become very, very small,

1182
01:23:59,890 --> 01:24:02,430
then it takes less and
less energy for string

1183
01:24:02,430 --> 01:24:03,817
to wind around that circle.

1184
01:24:07,170 --> 01:24:10,420
And then normally, so if
you have supersymmetry,

1185
01:24:10,420 --> 01:24:13,800
and also you have a
string wind around circle,

1186
01:24:13,800 --> 01:24:16,430
there's a [? casmir energy. ?]
And the [? casmir ?] energy is

1187
01:24:16,430 --> 01:24:18,300
typically is inactive
for the Boson,

1188
01:24:18,300 --> 01:24:19,657
and the fermion is positive.

1189
01:24:19,657 --> 01:24:22,240
And if you have a supersymmetry,
then that [? casmir ?] energy

1190
01:24:22,240 --> 01:24:23,220
cancels.

1191
01:24:23,220 --> 01:24:26,590
And then when the circle
shrink to zero size,

1192
01:24:26,590 --> 01:24:28,550
then the string become massless.

1193
01:24:28,550 --> 01:24:31,210
So that's typical
in the superstring.

1194
01:24:31,210 --> 01:24:35,090
But if you in the situation
where you just saw a symmetry's

1195
01:24:35,090 --> 01:24:37,100
broken, then
typically then there's

1196
01:24:37,100 --> 01:24:40,060
an active
[? casmir image. ?] And then

1197
01:24:40,060 --> 01:24:45,977
when the circle side
becomes smaller and smaller,

1198
01:24:45,977 --> 01:24:48,060
and then it take lower and
lower energy for string

1199
01:24:48,060 --> 01:24:49,540
to wrap around circle.

1200
01:24:49,540 --> 01:24:52,719
And then anyway,
if you include the

1201
01:24:52,719 --> 01:24:55,010
[? casmir image, et cetera, ?]
then you find the string

1202
01:24:55,010 --> 01:24:57,370
actually become
[? negative image. ?]

1203
01:24:57,370 --> 01:25:11,540
And you find that the
strings winding around t E

1204
01:25:11,540 --> 01:25:16,980
actually develops
negative mass square.

1205
01:25:21,760 --> 01:25:24,660
So as we discussed before,
this is called [INAUDIBLE],

1206
01:25:24,660 --> 01:25:26,770
and signals instability.

1207
01:25:26,770 --> 01:25:31,062
And here there's a circle
which become a small size.

1208
01:25:31,062 --> 01:25:33,020
And here there's no
supersymmetry, because when

1209
01:25:33,020 --> 01:25:35,519
you have a finite temperature,
your supersymmetry is broken,

1210
01:25:35,519 --> 01:25:40,690
because both on fermion have
different boundary conditions.

1211
01:25:40,690 --> 01:25:46,090
Anyway, so that means that this
option is gone, cannot work.

1212
01:25:46,090 --> 01:25:48,090
So the only thing we know
is to do a black hole.

1213
01:25:50,570 --> 01:25:53,070
So now we need to find a black
hole with the right symmetry.

1214
01:26:08,410 --> 01:26:10,820
So the reason I emphasized
with the right symmetry is

1215
01:26:10,820 --> 01:26:14,120
because when we say
black hole, it's

1216
01:26:14,120 --> 01:26:17,500
because a black hole is
a hole, is a [INAUDIBLE]

1217
01:26:17,500 --> 01:26:20,390
symmetric hole.

1218
01:26:20,390 --> 01:26:26,000
But here we need to have
a translation symmetry,

1219
01:26:26,000 --> 01:26:29,320
because this is a [INAUDIBLE].

1220
01:26:29,320 --> 01:26:32,990
So here now we need
to have a black brain,

1221
01:26:32,990 --> 01:26:34,209
black the whole horizon.

1222
01:26:34,209 --> 01:26:35,250
It's actually not a hole.

1223
01:26:35,250 --> 01:26:37,940
It's actually a plane.

1224
01:26:37,940 --> 01:26:43,260
So I did the black
hole in this topology

1225
01:26:43,260 --> 01:26:46,855
for the horizon topology.

1226
01:26:50,260 --> 01:26:56,320
Topology should be r to the d
minus 1, rather than a sphere.

1227
01:27:03,050 --> 01:27:07,040
Once you've realized that,
then it's easy to do.

1228
01:27:07,040 --> 01:27:10,080
Once you realize
that, it's easy to do.

1229
01:27:10,080 --> 01:27:14,130
So you can just write down the
answers, because this system

1230
01:27:14,130 --> 01:27:16,854
have lots of symmetries.

1231
01:27:16,854 --> 01:27:18,520
And you can just write
down the answers,

1232
01:27:18,520 --> 01:27:23,242
and then just plug into the
Einstein equation to solve it.

1233
01:27:23,242 --> 01:27:25,200
Say for example you can
write down the answers.

1234
01:27:27,860 --> 01:27:30,540
So you can write
down say r square.

1235
01:27:30,540 --> 01:27:33,640
So you can write down
answer's ds square equal

1236
01:27:33,640 --> 01:27:37,250
to r squared
divided by z square.

1237
01:27:37,250 --> 01:27:42,930
So now you still get
translation symmetry,

1238
01:27:42,930 --> 01:27:46,080
so you can put something
before dt square.

1239
01:27:46,080 --> 01:27:48,170
But you still have
rotation symmetry.

1240
01:27:48,170 --> 01:27:52,780
Yeah, so anyway, you can
write down answers like this.

1241
01:27:52,780 --> 01:27:54,310
Put in some two functions here.

1242
01:28:00,070 --> 01:28:01,420
Put in some two functions here.

1243
01:28:01,420 --> 01:28:03,860
So this is a most
general solution,

1244
01:28:03,860 --> 01:28:06,450
which is consistent with the
translation symmetry in the t

1245
01:28:06,450 --> 01:28:09,221
and x direction, and the
rotational symmetry in the t

1246
01:28:09,221 --> 01:28:09,930
and x directions.

1247
01:28:12,720 --> 01:28:15,240
And then you have
just two functions

1248
01:28:15,240 --> 01:28:16,500
which can only depend on z.

1249
01:28:16,500 --> 01:28:18,720
You cannot depend on t and x.

1250
01:28:18,720 --> 01:28:20,619
Otherwise you'll break
translation symmetry.

1251
01:28:20,619 --> 01:28:22,660
So these are the most
[? general ?] [? metric ?],

1252
01:28:22,660 --> 01:28:24,820
you have two
undetermined functions.

1253
01:28:24,820 --> 01:28:26,830
Then you plug into
Einstein equation.

1254
01:28:26,830 --> 01:28:27,640
Then you solve it.

1255
01:28:37,760 --> 01:28:44,510
So you find in fact h
actually equal to 1 over g

1256
01:28:44,510 --> 01:28:47,900
equal to something like this.

1257
01:28:47,900 --> 01:28:49,860
It's in d-dimension.

1258
01:28:49,860 --> 01:28:53,710
In d-dimension, say if
I have AdS d plus, 1

1259
01:28:53,710 --> 01:28:56,250
and then you find a very
simple [? metric ?] like this.

1260
01:29:02,980 --> 01:29:08,130
So let me again give
you a slight quiz.

1261
01:29:08,130 --> 01:29:09,622
So why I don't put
a function here?

1262
01:29:15,090 --> 01:29:20,557
So why in principle, I can also
put a function here, right?

1263
01:29:20,557 --> 01:29:22,890
So why I only put a function
here and the function here,

1264
01:29:22,890 --> 01:29:24,330
but not put the function here?

1265
01:29:24,330 --> 01:29:26,330
AUDIENCE: [INAUDIBLE]
[? Because it redefines ?]

1266
01:29:26,330 --> 01:29:26,950
[? z. ?]

1267
01:29:26,950 --> 01:29:28,260
HONG LIU: Yeah, exactly.

1268
01:29:28,260 --> 01:29:34,000
Because you can imagine this
just as a definition of z.

1269
01:29:34,000 --> 01:29:37,150
But then I don't have
freedom to choose this z.

1270
01:29:37,150 --> 01:29:39,240
Or I can choose here
to be 1, and then I

1271
01:29:39,240 --> 01:29:40,611
need to put the function there.

1272
01:29:43,910 --> 01:29:53,960
So in this metric, the horizon
we set z is equal to z 0.

1273
01:29:53,960 --> 01:29:55,840
z 0 is some constant.

1274
01:29:55,840 --> 01:29:57,645
So z 0 is some constant.

1275
01:30:01,210 --> 01:30:02,940
Zero some constant.

1276
01:30:02,940 --> 01:30:03,440
Right.

1277
01:30:03,440 --> 01:30:04,250
Is z 0.

1278
01:30:04,250 --> 01:30:09,320
And indeed I have a
topology of the R t minus 1,

1279
01:30:09,320 --> 01:30:13,550
because [? the d, ?] z equal to
0, the topology is at this r.

1280
01:30:13,550 --> 01:30:14,600
It's not at the sphere.

1281
01:30:17,560 --> 01:30:24,400
And also you see that when you
go to infinity, the f and the g

1282
01:30:24,400 --> 01:30:30,089
approach 1 with a very fast
fall-off with d to the power d.

1283
01:30:30,089 --> 01:30:31,255
So this is the normalizable.

1284
01:30:33,880 --> 01:30:35,110
And this is normalizable.

1285
01:30:35,110 --> 01:30:37,616
So this behavior
is normalizable.

1286
01:30:48,810 --> 01:30:54,690
So this is metric satisfy
all our criterions.

1287
01:30:54,690 --> 01:30:57,570
And under the standard
black hole thermodynamics,

1288
01:30:57,570 --> 01:30:59,620
we are sure this is a
thermodynamical system.

1289
01:31:02,430 --> 01:31:09,317
So this must be [? due ?] to our
finite temperature [INAUDIBLE]

1290
01:31:09,317 --> 01:31:09,817
series.

1291
01:31:14,180 --> 01:31:18,310
So now let's work out the
temperature in terms of z 0.

1292
01:31:18,310 --> 01:31:23,190
So it's just using
the standard trick

1293
01:31:23,190 --> 01:31:32,225
going to the Euclidean signature
which we have discussed before.

1294
01:31:35,160 --> 01:31:42,664
And the required the metric
is regular at the horizon.

1295
01:31:42,664 --> 01:31:44,080
When you go to
Euclidean signature

1296
01:31:44,080 --> 01:31:46,640
require the metric is
regular at the horizon.

1297
01:31:46,640 --> 01:31:51,060
And then you can
deduce that this should

1298
01:31:51,060 --> 01:31:54,710
have some specific periodicity.

1299
01:31:54,710 --> 01:31:58,070
Anyway, I hope you
still remember that.

1300
01:31:58,070 --> 01:32:03,720
So you can deduce a
temperature of the black hole.

1301
01:32:03,720 --> 01:32:10,100
So beta is equal to 4
pi to the d times z 0.

1302
01:32:13,370 --> 01:32:16,340
So you can deduce.

1303
01:32:16,340 --> 01:32:19,065
So this you remember, right?

1304
01:32:19,065 --> 01:32:19,565
OK.

1305
01:32:19,565 --> 01:32:20,065
Good.

1306
01:32:25,750 --> 01:32:30,496
So emphasize this is the
temperature matched in t.

1307
01:32:30,496 --> 01:32:32,954
Because whenever, as we
emphasized many times before,

1308
01:32:32,954 --> 01:32:34,620
but whenever you talk
about temperature,

1309
01:32:34,620 --> 01:32:38,280
you have to refer to
a time, time units

1310
01:32:38,280 --> 01:32:40,200
to define the temperature.

1311
01:32:40,200 --> 01:32:44,240
And this temperature is defined
with respect to this time.

1312
01:32:44,240 --> 01:32:46,940
So in other words, this
is boundary temperature.

1313
01:32:55,280 --> 01:32:57,920
So in the [? back ?], of course,
depending on the value of z,

1314
01:32:57,920 --> 01:33:00,270
the local temperature
is different.

1315
01:33:00,270 --> 01:33:02,978
And this is the temperature
corresponding to the boundary.

1316
01:33:09,670 --> 01:33:13,080
So this relation is
again very interesting.

1317
01:33:17,550 --> 01:33:30,530
So this is z equal
to 0 boundary.

1318
01:33:30,530 --> 01:33:34,080
And now on the gravity side
now there's a horizon at

1319
01:33:34,080 --> 01:33:40,000
z equal to 0, z equal to z 0.

1320
01:33:40,000 --> 01:33:43,100
And this horizon
according to this formula

1321
01:33:43,100 --> 01:33:45,026
is proportional to 1 over t.

1322
01:33:53,000 --> 01:33:55,980
It's proportional to 1 over t.

1323
01:33:55,980 --> 01:33:58,915
Again, this is the refraction
of the IR/UV connection.

1324
01:34:02,080 --> 01:34:08,090
Because if you can see
the very high temperature,

1325
01:34:08,090 --> 01:34:12,740
it means you are probing
very high energy process.

1326
01:34:12,740 --> 01:34:14,935
Then the z 0 becomes small.

1327
01:34:14,935 --> 01:34:16,560
Then the horizon
moves to the boundary.

1328
01:34:20,240 --> 01:34:22,230
And the horizon moves
to the boundary.

1329
01:34:22,230 --> 01:34:25,290
And when t becomes small,
then z 0 becomes big,

1330
01:34:25,290 --> 01:34:26,430
then the horizon move down.

1331
01:34:29,720 --> 01:34:32,090
So that means that
depending on the value of t,

1332
01:34:32,090 --> 01:34:35,260
you can probe different
regions in the gravity side.

1333
01:34:35,260 --> 01:34:37,100
So for very high
temperature, you

1334
01:34:37,100 --> 01:34:40,200
probe very tiny region
near the infinity.

1335
01:34:40,200 --> 01:34:43,440
And for a small t, you can
probe one or more regions.

1336
01:34:43,440 --> 01:34:44,940
Again this is a
refraction of IR/UV.

1337
01:34:49,960 --> 01:34:52,320
Any questions on this?

1338
01:34:52,320 --> 01:34:53,759
Now with this black
hole solution,

1339
01:34:53,759 --> 01:34:55,550
we can work out
[INAUDIBLE] thermodynamics.

1340
01:34:59,440 --> 01:35:05,392
And then we should be able to
work essentially by definition.

1341
01:35:05,392 --> 01:35:07,100
Then, for example,
the black hole entropy

1342
01:35:07,100 --> 01:35:10,580
should be identified with
the field series entropy,

1343
01:35:10,580 --> 01:35:12,070
or black hole free
energy should be

1344
01:35:12,070 --> 01:35:16,120
identified with the free field
series for energy, et cetera.

1345
01:35:16,120 --> 01:35:18,970
It just essentially,
if there's a relation,

1346
01:35:18,970 --> 01:35:22,420
if black hole describe the field
series at finite temperature,

1347
01:35:22,420 --> 01:35:24,630
then you should
be able to equate

1348
01:35:24,630 --> 01:35:28,040
the thermodynamic quantities.

1349
01:35:28,040 --> 01:35:30,460
And in this way, we can
actually use the black hole

1350
01:35:30,460 --> 01:35:34,190
thermodynamics now to calculate
the thermodynamical behavior

1351
01:35:34,190 --> 01:35:38,150
of a strongly coupled
[INAUDIBLE] series.

1352
01:35:38,150 --> 01:35:45,705
So we can now obtain
thermodynamical behavior,

1353
01:35:45,705 --> 01:36:02,280
thermodynamical quantities
at strong coupling

1354
01:36:02,280 --> 01:36:06,520
in the [INAUDIBLE]
series in a black hole.

1355
01:36:12,440 --> 01:36:13,980
So now let's do that.

1356
01:36:13,980 --> 01:36:17,950
So now let's specify
to d equal to 4.

1357
01:36:17,950 --> 01:36:20,910
So that's corresponding to the
four-dimensional boundaries,

1358
01:36:20,910 --> 01:36:24,300
and that's [? 2t ?] equal to 4.

1359
01:36:24,300 --> 01:36:27,140
So now let's first do
the black hole entropy,

1360
01:36:27,140 --> 01:36:29,730
which is the
easiest thing to do.

1361
01:36:29,730 --> 01:36:33,810
So the black hole entropy,
just the area of the horizon

1362
01:36:33,810 --> 01:36:36,580
divided by Newton constant.

1363
01:36:36,580 --> 01:36:39,160
So we have to use the factor
of five-dimensional Newton

1364
01:36:39,160 --> 01:36:40,130
constants.

1365
01:36:40,130 --> 01:36:41,790
So we work with AdS 5.

1366
01:36:41,790 --> 01:36:43,990
So we dimension
reduction on the S 5.

1367
01:36:43,990 --> 01:36:48,530
So we work with the
five-dimension Newton constant.

1368
01:36:48,530 --> 01:36:55,790
And then the area here
is just infinite volume.

1369
01:36:55,790 --> 01:37:00,424
It is just dx 1, dx 2, dx 3,
because this is just parallel

1370
01:37:00,424 --> 01:37:01,090
to the boundary.

1371
01:37:01,090 --> 01:37:02,720
It doesn't matter.

1372
01:37:02,720 --> 01:37:05,770
It doesn't matter.

1373
01:37:05,770 --> 01:37:07,890
So what this defines,
or [INAUDIBLE]

1374
01:37:07,890 --> 01:37:17,070
times R cubed to the
z 0 cubed, because you

1375
01:37:17,070 --> 01:37:20,570
have to multiply the
area at the horizon.

1376
01:37:20,570 --> 01:37:24,520
So you essentially compute the
determinant of the spatial part

1377
01:37:24,520 --> 01:37:27,630
at the horizon, you
evaluate it at the z 0.

1378
01:37:27,630 --> 01:37:29,470
You evaluate it at z 0.

1379
01:37:29,470 --> 01:37:31,010
So you have this factor.

1380
01:37:31,010 --> 01:37:33,640
And then you have
this infinite factor.

1381
01:37:33,640 --> 01:37:35,560
But this infinite factor,
it doesn't matter.

1382
01:37:35,560 --> 01:37:37,100
We can just take it out.

1383
01:37:37,100 --> 01:37:40,010
Then what you get is
the entropy density.

1384
01:37:40,010 --> 01:37:48,920
So then we find the entropy
density is essentially

1385
01:37:48,920 --> 01:37:52,490
just given by this R
cubed divided by z 0

1386
01:37:52,490 --> 01:37:55,430
to the cube 4G 5.

1387
01:38:02,150 --> 01:38:05,550
So now you can just
plug in the numbers.

1388
01:38:05,550 --> 01:38:08,890
So now you have to
use this equation.

1389
01:38:08,890 --> 01:38:12,370
G 5 divided by R
cubed is related

1390
01:38:12,370 --> 01:38:18,390
to the [INAUDIBLE] related
to pi divided by 2 N square.

1391
01:38:18,390 --> 01:38:24,860
And then you plug in this
relation to relate z 0 to t.

1392
01:38:24,860 --> 01:38:30,170
And then, after a
tiny bit of algebra,

1393
01:38:30,170 --> 01:38:33,160
so you find N squared T cubed.

1394
01:38:38,860 --> 01:38:41,870
Again this answer makes sense.

1395
01:38:41,870 --> 01:38:45,891
Again, you can do some very
minimal consistency check.

1396
01:38:45,891 --> 01:38:47,390
So first thing you
said is that this

1397
01:38:47,390 --> 01:38:50,230
is proportional to T cubed.

1398
01:38:50,230 --> 01:38:52,110
It makes sense,
because we are working

1399
01:38:52,110 --> 01:38:55,500
with a four-dimensional
scale invariant series.

1400
01:38:55,500 --> 01:38:59,520
And entropy by itself
does not have a dimension.

1401
01:38:59,520 --> 01:39:04,840
And entropy density should
have dimension cubed

1402
01:39:04,840 --> 01:39:06,380
because of the volume.

1403
01:39:06,380 --> 01:39:08,410
You divide it by the volume.

1404
01:39:08,410 --> 01:39:09,409
Should be 1 over volume.

1405
01:39:15,370 --> 01:39:16,410
Do you follow?

1406
01:39:16,410 --> 01:39:18,290
So entropy density
should have dimension.

1407
01:39:20,920 --> 01:39:25,810
Entropy density should
have dimension M d minus 1.

1408
01:39:25,810 --> 01:39:28,940
Because at the d minus 1 come
from because you divide it

1409
01:39:28,940 --> 01:39:32,140
by entropy by the volume,
and the volume have,

1410
01:39:32,140 --> 01:39:34,980
yeah, 1 over volume have
dimension M d minus 1.

1411
01:39:38,900 --> 01:39:40,350
But here in the
scale environment

1412
01:39:40,350 --> 01:39:42,040
you don't have any
of the scale factor.

1413
01:39:42,040 --> 01:39:46,440
You only have T. So it must
come from T to the d minus 1.

1414
01:39:46,440 --> 01:39:48,720
So you get T cubed.

1415
01:39:48,720 --> 01:39:51,340
And also, this is a
proponent to N square.

1416
01:39:51,340 --> 01:39:53,100
And of course in the
[INAUDIBLE] series

1417
01:39:53,100 --> 01:39:54,516
should be [INAUDIBLE]
to N square.

1418
01:39:57,830 --> 01:40:01,710
So now you can also obtain, say
for example, the energy density

1419
01:40:01,710 --> 01:40:06,240
and the pressure, et cetera,
by calculating the T m

1420
01:40:06,240 --> 01:40:09,120
[INAUDIBLE] mu.

1421
01:40:09,120 --> 01:40:11,240
So you can calculate say
one [? point ?] function

1422
01:40:11,240 --> 01:40:12,790
of the stress
tensor Then you can

1423
01:40:12,790 --> 01:40:15,590
read what is the energy
density, and what

1424
01:40:15,590 --> 01:40:17,700
is the pressure, et cetera.

1425
01:40:17,700 --> 01:40:20,560
And so this is the
same as what we did.

1426
01:40:20,560 --> 01:40:24,920
It's very similar procedure
as what we did before

1427
01:40:24,920 --> 01:40:26,810
for the scalar field series.

1428
01:40:26,810 --> 01:40:31,680
The one point function
of the stress tensor,

1429
01:40:31,680 --> 01:40:33,670
you can read it
from the counterpart

1430
01:40:33,670 --> 01:40:37,870
of the [? B ?] modes, a
counterpart of the [INAUDIBLE]

1431
01:40:37,870 --> 01:40:39,150
modes.

1432
01:40:39,150 --> 01:40:43,610
So here the [INAUDIBLE] mode
is just this z to the power d.

1433
01:40:43,610 --> 01:40:46,830
And the [? corruption ?]
is 1 over 0, 1 over z 0

1434
01:40:46,830 --> 01:40:48,310
to the power d.

1435
01:40:48,310 --> 01:40:52,710
And so that tells you you should
be proportional to 1 over z

1436
01:40:52,710 --> 01:40:54,560
to the 0 to the 4.

1437
01:40:58,610 --> 01:41:00,460
Because that's an
analog of this [? B ?]

1438
01:41:00,460 --> 01:41:05,460
mode-- the normalizable mode--
to the 1 over z to the 0 4.

1439
01:41:05,460 --> 01:41:07,480
And again, this will
be proportional to T

1440
01:41:07,480 --> 01:41:08,380
to the power of 4.

1441
01:41:08,380 --> 01:41:11,580
Again, this is consistent
on dimensional ground,

1442
01:41:11,580 --> 01:41:15,280
because energy density have
dimension m to the power d.

1443
01:41:15,280 --> 01:41:18,440
And so it's T to the power 4.

1444
01:41:18,440 --> 01:41:24,730
But you actually work out the
prefactor request [INAUDIBLE]

1445
01:41:24,730 --> 01:41:27,260
effort.

1446
01:41:27,260 --> 01:41:31,930
Yeah, in a scalar case, we
say it's 2 [? mu ?] times b.

1447
01:41:31,930 --> 01:41:34,827
So in the gravity case,
in the [? metric ?] case,

1448
01:41:34,827 --> 01:41:36,410
it gets a little bit
more complicated.

1449
01:41:36,410 --> 01:41:42,330
Anyway, but you
can read the scale.

1450
01:41:42,330 --> 01:41:47,120
I will not calculate the
prefactor using this method.

1451
01:41:47,120 --> 01:41:49,799
So to calculate the
prefactor, it's easier

1452
01:41:49,799 --> 01:41:51,090
just to use the thermodynamics.

1453
01:41:58,820 --> 01:42:10,550
So because S just equal to
the free energy divided by T.

1454
01:42:10,550 --> 01:42:12,590
So now everything
here, we can see

1455
01:42:12,590 --> 01:42:15,090
that the density is
S is entropy density,

1456
01:42:15,090 --> 01:42:18,540
and F is the free energy
density [? exciter. ?]

1457
01:42:18,540 --> 01:42:21,010
So we know now the entropy.

1458
01:42:21,010 --> 01:42:24,394
You can just integrate
the free energy density.

1459
01:42:32,590 --> 01:42:37,670
So this is equal to pi square
divided by 8 N squared T

1460
01:42:37,670 --> 01:42:40,340
to the 4.

1461
01:42:40,340 --> 01:42:46,190
And then you can find out the
energy density of plus TS.

1462
01:42:46,190 --> 01:42:50,990
So you find this 3
pi square [INAUDIBLE]

1463
01:42:50,990 --> 01:42:51,950
squared [INAUDIBLE].

1464
01:43:00,457 --> 01:43:02,290
Now if you work hard,
the one point function

1465
01:43:02,290 --> 01:43:05,385
of the stress tensor carefully
of the [? free ?] factor, you

1466
01:43:05,385 --> 01:43:06,110
will find this.

1467
01:43:09,100 --> 01:43:13,074
So this is, as a result, at
lambda equal to infinity.

1468
01:43:13,074 --> 01:43:14,740
So these are the
strong coupling we got.

1469
01:43:14,740 --> 01:43:16,448
All these are the
strong coupling we got.

1470
01:43:25,027 --> 01:43:25,902
AUDIENCE: [INAUDIBLE]

1471
01:43:29,585 --> 01:43:31,210
HONG LIU: The
thermodynamical behavior?

1472
01:43:31,210 --> 01:43:33,020
AUDIENCE: [INAUDIBLE]
How do you get N?

1473
01:43:33,020 --> 01:43:33,561
HONG LIU: No.

1474
01:43:33,561 --> 01:43:35,980
You just integrate
this equation.

1475
01:43:35,980 --> 01:43:36,480
Yeah.

1476
01:43:36,480 --> 01:43:39,260
Just integrate that equation.

1477
01:43:39,260 --> 01:43:42,510
So this is the behavior,
that infinite lambda.

1478
01:43:42,510 --> 01:43:47,710
So now let's compare
with the free series.

1479
01:43:55,250 --> 01:43:57,500
So this is the undergraduate
thermodynamics.

1480
01:44:01,550 --> 01:44:05,734
So let's just calculate
the entropy density

1481
01:44:05,734 --> 01:44:06,525
with zero coupling.

1482
01:44:09,710 --> 01:44:13,480
So I have to remind
you one thing.

1483
01:44:13,480 --> 01:44:18,630
It said in each massless
[INAUDIBLE] degree of freedom

1484
01:44:18,630 --> 01:44:22,520
contribute to the entropy
density 2 pi square divided

1485
01:44:22,520 --> 01:44:25,610
by 45 to the power T cube.

1486
01:44:25,610 --> 01:44:28,640
I don't know whether you
still remember this thing.

1487
01:44:28,640 --> 01:44:32,130
So in four dimension, a single
massless degree of freedom

1488
01:44:32,130 --> 01:44:35,110
contribute to the
entropy density.

1489
01:44:35,110 --> 01:44:38,490
Yeah, this come from-- doing
a single massless [INAUDIBLE]

1490
01:44:38,490 --> 01:44:41,850
degree of freedom contributed
entropy like this.

1491
01:44:44,450 --> 01:44:47,340
This is a good
thing to remember.

1492
01:44:47,340 --> 01:44:50,850
And now in the
[INAUDIBLE] series

1493
01:44:50,850 --> 01:44:53,170
we have eight Boson,
as we said before.

1494
01:44:53,170 --> 01:44:55,430
We have eight massless
Boson degrees of freedom.

1495
01:44:59,120 --> 01:45:03,820
And then we have eight fermionic
massless degrees of freedom.

1496
01:45:03,820 --> 01:45:08,930
And each fermionic degrees of
freedom contribute to how much

1497
01:45:08,930 --> 01:45:09,935
of the Boson?

1498
01:45:09,935 --> 01:45:13,230
Do anybody know?

1499
01:45:13,230 --> 01:45:15,820
A massless fermion compared
to the massless Boson.

1500
01:45:15,820 --> 01:45:17,871
What is the ratio for
each degrees of freedom

1501
01:45:17,871 --> 01:45:20,120
contributed to the entropy
density in the free series,

1502
01:45:20,120 --> 01:45:21,152
free particle?

1503
01:45:21,152 --> 01:45:22,300
AUDIENCE: One half?

1504
01:45:22,300 --> 01:45:24,170
HONG LIU: No [INAUDIBLE]

1505
01:45:24,170 --> 01:45:24,949
AUDIENCE: One.

1506
01:45:24,949 --> 01:45:25,490
HONG LIU: No.

1507
01:45:25,490 --> 01:45:25,990
7/8.

1508
01:45:30,940 --> 01:45:34,650
And now in the SU(N)
[? Gate ?] series,

1509
01:45:34,650 --> 01:45:36,870
we have N squared minus 1.

1510
01:45:36,870 --> 01:45:40,240
We have N squared minus 1 in
the join [? representation. ?]

1511
01:45:40,240 --> 01:45:43,800
So now you can calculate this
is a [INAUDIBLE] free series.

1512
01:45:43,800 --> 01:45:47,780
So let's take [INAUDIBLE],
forget about the 1.

1513
01:45:47,780 --> 01:45:53,090
So then you get 2
pi over 3, 2/3 pi

1514
01:45:53,090 --> 01:45:56,579
squared N squared T
to the power cubed.

1515
01:45:56,579 --> 01:45:58,120
So now you find
something remarkable.

1516
01:46:01,960 --> 01:46:03,710
So now you find
something remarkable.

1517
01:46:03,710 --> 01:46:07,210
You find S lambda
equal to infinity

1518
01:46:07,210 --> 01:46:12,950
divided by S lambda equal to 0
is equal to 1/2 divided by 2/3.

1519
01:46:12,950 --> 01:46:14,434
So it becomes 3/4.

1520
01:46:21,850 --> 01:46:26,822
So you go from zero coupling
just to infinite coupling,

1521
01:46:26,822 --> 01:46:28,490
and the entropy change by 3/4.

1522
01:46:31,660 --> 01:46:33,290
So there's a long story.

1523
01:46:33,290 --> 01:46:35,230
There are several
long story I can tell,

1524
01:46:35,230 --> 01:46:37,480
a good story I can
tell about this thing.

1525
01:46:37,480 --> 01:46:41,560
But now I run out of time.

1526
01:46:41,560 --> 01:46:49,046
But let me mention a couple
words on your p-set problem.

1527
01:46:49,046 --> 01:46:50,420
So in the p-set
problem, you want

1528
01:46:50,420 --> 01:46:53,160
to compute that this Wilson
loop, the same Wilson

1529
01:46:53,160 --> 01:46:58,480
loop, the L and the T at
the finite temperature.

1530
01:46:58,480 --> 01:47:01,290
So now instead of
the pure AdS, you

1531
01:47:01,290 --> 01:47:03,660
use this black hole geometry.

1532
01:47:03,660 --> 01:47:05,035
You use this black
hole geometry.

1533
01:47:07,570 --> 01:47:09,510
Use this black hole geometry.

1534
01:47:09,510 --> 01:47:16,960
So what you will find,
so physical expectation,

1535
01:47:16,960 --> 01:47:19,790
physically the V(L).

1536
01:47:23,780 --> 01:47:25,820
So you expect the
potential between the quark

1537
01:47:25,820 --> 01:47:30,680
and anti-quark will go
to 0 when L is large.

1538
01:47:37,990 --> 01:47:42,030
So physically for
this reason, so here

1539
01:47:42,030 --> 01:47:44,580
this is the same thing
happens even in the [? QED ?].

1540
01:47:47,500 --> 01:47:49,849
So this is just the
story of the screening.

1541
01:47:49,849 --> 01:47:51,640
Suppose you have
electrons, anti-electrons.

1542
01:47:51,640 --> 01:47:54,100
So of course they have
interaction between them.

1543
01:47:54,100 --> 01:47:55,850
But now if you have a
plasma between them,

1544
01:47:55,850 --> 01:47:57,560
you have many positive
and [INAUDIBLE]

1545
01:47:57,560 --> 01:48:00,870
charge of particles
[INAUDIBLE] running around.

1546
01:48:00,870 --> 01:48:03,635
And then they interacting
will be screened.

1547
01:48:03,635 --> 01:48:05,010
The interaction
will be screened,

1548
01:48:05,010 --> 01:48:06,970
because there will be
slightly more active particle

1549
01:48:06,970 --> 01:48:09,080
here, and slightly more
positive particle here.

1550
01:48:09,080 --> 01:48:10,251
Then they will be screened.

1551
01:48:10,251 --> 01:48:12,500
Similarly, if you have a
quark and an anti-quark which

1552
01:48:12,500 --> 01:48:17,450
[INAUDIBLE] object, and we input
them at the finite temperature,

1553
01:48:17,450 --> 01:48:21,430
in the finite temperature, you
can cite many other massless

1554
01:48:21,430 --> 01:48:23,010
[INAUDIBLE] degrees of freedom.

1555
01:48:23,010 --> 01:48:24,080
Then they running
around [INAUDIBLE]

1556
01:48:24,080 --> 01:48:25,121
quark and the anti-quark.

1557
01:48:25,121 --> 01:48:26,550
Then they will screen them.

1558
01:48:26,550 --> 01:48:30,300
So when they take the
distance sufficiently large,

1559
01:48:30,300 --> 01:48:35,575
then you will find the potential
between them goes to 0.

1560
01:48:35,575 --> 01:48:44,370
So on the gravity side,
what you will see--

1561
01:48:44,370 --> 01:48:50,570
so now you cannot do
everything analytically.

1562
01:48:50,570 --> 01:48:52,480
But you can derive the
qualitative behavior

1563
01:48:52,480 --> 01:48:53,750
analytically.

1564
01:48:53,750 --> 01:48:59,680
It said if the L is small-- so
suppose here is the horizon.

1565
01:48:59,680 --> 01:49:02,360
When the L is small,
then of course

1566
01:49:02,360 --> 01:49:03,660
we just connect a string.

1567
01:49:03,660 --> 01:49:05,790
We're close-by.

1568
01:49:05,790 --> 01:49:06,980
Nothing changes.

1569
01:49:06,980 --> 01:49:11,567
Nothing much changes from our
zero temperature calculation.

1570
01:49:11,567 --> 01:49:13,650
And this makes sense,
because at a short distance,

1571
01:49:13,650 --> 01:49:16,940
you probe the UV physics.

1572
01:49:16,940 --> 01:49:19,010
And the probe, the
UV physics, of course

1573
01:49:19,010 --> 01:49:21,300
you don't see the temperature.

1574
01:49:21,300 --> 01:49:26,120
So for L is much, much smaller
than 1 over temperature.

1575
01:49:26,120 --> 01:49:29,200
Then you don't see the
effect of the temperature,

1576
01:49:29,200 --> 01:49:33,060
and [INAUDIBLE] will be
just like small corrections

1577
01:49:33,060 --> 01:49:35,880
to the vacuum [? we got. ?]

1578
01:49:35,880 --> 01:49:37,480
But now when L
become of [? all ?]

1579
01:49:37,480 --> 01:49:40,600
the 1 over the
temperature, then you

1580
01:49:40,600 --> 01:49:43,270
find something
interesting happens.

1581
01:49:43,270 --> 01:49:49,810
Then the L, then
this [? worksheet ?]

1582
01:49:49,810 --> 01:49:52,690
will roughly go to the horizon.

1583
01:49:52,690 --> 01:49:54,200
Then what we will
find in the end

1584
01:49:54,200 --> 01:49:58,240
when the L become say
larger than some value say

1585
01:49:58,240 --> 01:50:06,120
of all the 1 over t, the
only solution you can find

1586
01:50:06,120 --> 01:50:08,820
is that this string just
two string heading down.

1587
01:50:08,820 --> 01:50:10,830
That's the only solution.

1588
01:50:10,830 --> 01:50:14,440
And then that means that the
quark, anti-quark no longer

1589
01:50:14,440 --> 01:50:17,540
interact with each other.

1590
01:50:17,540 --> 01:50:20,180
And then they are screened.

1591
01:50:20,180 --> 01:50:21,570
Then they are screened.

1592
01:50:21,570 --> 01:50:23,890
So a short distance is
the same as in the vacuum.

1593
01:50:23,890 --> 01:50:26,375
But when you go to
long distance, of all

1594
01:50:26,375 --> 01:50:28,740
the 1 over temperature, and
then eventually it would get

1595
01:50:28,740 --> 01:50:29,910
screened.

1596
01:50:29,910 --> 01:50:38,240
So this is another
cute calculus problem

1597
01:50:38,240 --> 01:50:39,490
which I'm sure you will enjoy.

1598
01:50:43,080 --> 01:50:45,720
Yeah.

1599
01:50:45,720 --> 01:50:47,570
That's all for today.