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HONG LIU: OK.

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00:00:22,380 --> 00:00:28,260
So last time, we talked
about this IR-UV connection

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00:00:28,260 --> 00:00:31,120
between the bulk
and the boundary.

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00:00:31,120 --> 00:00:34,810
So the key thing is-- let me
write down the metric here,

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00:00:34,810 --> 00:00:38,230
the Ads metric in this
Poincare coordinate.

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00:00:46,420 --> 00:00:51,190
So this is the boundaries
are equal to zero.

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00:00:51,190 --> 00:00:55,300
And when you increase
z, you go to interior.

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00:00:55,300 --> 00:00:58,650
And then for each
constant, z, you

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00:00:58,650 --> 00:01:01,240
have a Minkowski space time.

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00:01:01,240 --> 00:01:02,880
You have a Minkowski d.

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00:01:02,880 --> 00:01:06,670
So this is Ads d plus 1.

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00:01:06,670 --> 00:01:11,520
So each slice is a d
dimension Minkowski space.

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00:01:11,520 --> 00:01:16,230
And due to this redshift
factor, the more

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00:01:16,230 --> 00:01:20,610
you go to the interior
of the space time,

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00:01:20,610 --> 00:01:22,390
then corresponding
to the lower energy

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00:01:22,390 --> 00:01:24,750
process when viewed
form the field theory.

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00:01:28,660 --> 00:01:32,280
So here, the same
process is happening here

25
00:01:32,280 --> 00:01:35,370
compared to happening here,
and here corresponding

26
00:01:35,370 --> 00:01:38,350
to the IR process, and
the [INAUDIBLE] boundary

27
00:01:38,350 --> 00:01:41,980
corresponding to the UV process.

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00:01:41,980 --> 00:01:49,610
So this is a key
relation between the bulk

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00:01:49,610 --> 00:01:51,500
and the boundary theories.

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00:01:51,500 --> 00:01:54,480
And also, this gives you
an intuitive understanding

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00:01:54,480 --> 00:01:58,070
where does this actual dimension
come from from the field theory

32
00:01:58,070 --> 00:01:59,540
point of view.

33
00:01:59,540 --> 00:02:02,030
Then from the field
theory perspective,

34
00:02:02,030 --> 00:02:06,820
this actual dimension can be
considered as a geometrization

35
00:02:06,820 --> 00:02:10,720
of the energy scale.

36
00:02:10,720 --> 00:02:14,260
And we know that the physics
change with the energy scale

37
00:02:14,260 --> 00:02:16,150
from the field
theory point of view.

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00:02:16,150 --> 00:02:18,340
It's called the
normalization group flow,

39
00:02:18,340 --> 00:02:22,330
how physics evolves when you
change the energy or length

40
00:02:22,330 --> 00:02:23,320
scale.

41
00:02:23,320 --> 00:02:26,430
In the field theory, it's
called normalization group flow.

42
00:02:26,430 --> 00:02:31,610
So you can also view that the
evolution in the gravity side,

43
00:02:31,610 --> 00:02:35,320
say from the boundary
to the interior,

44
00:02:35,320 --> 00:02:38,670
and that this flow
in the z direction

45
00:02:38,670 --> 00:02:44,550
can be considered,
again, to geometrize

46
00:02:44,550 --> 00:02:46,753
the normalization group
flow of the field theory.

47
00:02:49,310 --> 00:02:50,851
So any questions regarding this?

48
00:02:58,831 --> 00:02:59,330
Good.

49
00:02:59,330 --> 00:03:03,725
So now let's talk about some
further aspects of the duality.

50
00:03:07,200 --> 00:03:09,460
So the duality is
that once you realize

51
00:03:09,460 --> 00:03:14,400
there's such relation, since
the two sides are completely

52
00:03:14,400 --> 00:03:19,670
different objects, so the
game is that you really

53
00:03:19,670 --> 00:03:21,196
have to do lots of guess work.

54
00:03:24,639 --> 00:03:26,430
Essentially, you have
two sides if you want

55
00:03:26,430 --> 00:03:28,960
to relate things on two sides.

56
00:03:28,960 --> 00:03:33,710
And you have to do guess work.

57
00:03:33,710 --> 00:03:36,920
How does this quantity translate
into that quantity, et cetera?

58
00:03:36,920 --> 00:03:39,410
And then check the consistency.

59
00:03:39,410 --> 00:03:41,940
Just like you don't
know two languages,

60
00:03:41,940 --> 00:03:45,180
and then you have to guess
between the two languages

61
00:03:45,180 --> 00:03:47,210
and then build up
the dictionary.

62
00:03:47,210 --> 00:03:49,300
And we will be doing the same.

63
00:03:49,300 --> 00:03:51,968
We will be doing the same.

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00:03:51,968 --> 00:03:52,468
Good.

65
00:04:06,020 --> 00:04:10,540
So this is more like a review
of what we already sad.

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00:04:17,610 --> 00:04:18,880
So let me be more explicit.

67
00:04:18,880 --> 00:04:20,671
We have N equals 4
super Yang-Mills theory.

68
00:04:26,170 --> 00:04:28,100
We have N equals 4
super Yang-Mills theory.

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00:04:28,100 --> 00:04:38,930
And then here, you have type
IIB string in Ads5 times ds5.

70
00:04:38,930 --> 00:04:41,825
So let me just write down
Ads5 just for simplicity.

71
00:04:46,640 --> 00:04:54,024
So here on this side, there
is a conformal symmetry

72
00:04:54,024 --> 00:04:56,120
which we explained
before because this

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00:04:56,120 --> 00:04:58,640
is a four dimensional theory.

74
00:04:58,640 --> 00:05:02,660
So the conformal symmetry of a
d dimensional series SO d, 2.

75
00:05:06,430 --> 00:05:08,570
And on this side,
there's precisely

76
00:05:08,570 --> 00:05:15,940
the same group,
which is isometry

77
00:05:15,940 --> 00:05:21,265
of Ads5, which is
precisely also SO 4,

78
00:05:21,265 --> 00:05:26,440
2, which we reviewed before.

79
00:05:26,440 --> 00:05:29,590
And you can write down the
transformation on both sides.

80
00:05:29,590 --> 00:05:34,330
In your pset, you have
checked some of them,

81
00:05:34,330 --> 00:05:37,230
that they actually one to
one correspond to each other.

82
00:05:37,230 --> 00:05:40,440
For example, this special
conformal transformation.

83
00:05:40,440 --> 00:05:44,730
And the others, like translation
or rotation, et cetera,

84
00:05:44,730 --> 00:05:48,380
it's clear.

85
00:05:48,380 --> 00:05:50,350
AUDIENCE: But being
the right hand side,

86
00:05:50,350 --> 00:05:56,486
we still have some SO 5, S5.

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00:05:56,486 --> 00:05:57,860
HONG LIU: I will
talk about that.

88
00:06:01,540 --> 00:06:04,370
So this is more like
a space time symmetry.

89
00:06:04,370 --> 00:06:09,592
So this is a space
time symmetry.

90
00:06:09,592 --> 00:06:13,930
From field theory point of view,
this is a space time symmetry.

91
00:06:13,930 --> 00:06:15,770
So in N equals 4 super
Yang-Mills theory,

92
00:06:15,770 --> 00:06:21,010
there's also global symmetry
we discussed before.

93
00:06:21,010 --> 00:06:23,850
This is also global symmetry.

94
00:06:23,850 --> 00:06:26,580
But this is global symmetry
associated with space time.

95
00:06:26,580 --> 00:06:31,230
And there's also global
internal symmetry,

96
00:06:31,230 --> 00:06:32,670
SO6 internal symmetry.

97
00:06:32,670 --> 00:06:34,420
In N equals 4 super
Yang-Mills theory,

98
00:06:34,420 --> 00:06:37,170
we discussed last time
there are six scalar fields.

99
00:06:37,170 --> 00:06:39,260
You can rotate them each other.

100
00:06:39,260 --> 00:06:42,950
And this can be considered
as coming from the D3 brane.

101
00:06:42,950 --> 00:06:46,930
You can rotate six
transverse directions

102
00:06:46,930 --> 00:06:54,300
the symmetry in rotating the
six transverse directions.

103
00:06:54,300 --> 00:06:57,640
And on this side, there's
exactly the same symmetry.

104
00:06:57,640 --> 00:07:01,340
So now this is isometry of S5.

105
00:07:04,440 --> 00:07:08,660
So the isometry of S5
precisely gives you SO6.

106
00:07:08,660 --> 00:07:11,460
So you have exactly
the same SO6.

107
00:07:16,700 --> 00:07:19,260
And we will not be explicit.

108
00:07:19,260 --> 00:07:22,081
You can also actually map the
supersymmetry between them.

109
00:07:26,320 --> 00:07:30,350
You can also map the
supersymmetry between them.

110
00:07:30,350 --> 00:07:32,150
So N equals 4 supersymmetry.

111
00:07:32,150 --> 00:07:36,990
So there's a 4
supersymmetry, which

112
00:07:36,990 --> 00:07:40,020
just comes from N equal to 4.

113
00:07:40,020 --> 00:07:42,520
And because we have
conformal symmetry,

114
00:07:42,520 --> 00:07:45,800
then the conformal
symmetry does not actually

115
00:07:45,800 --> 00:07:48,190
commute with this
4 supersymmetry,

116
00:07:48,190 --> 00:07:51,790
then generate another
4 supersymmetry.

117
00:07:51,790 --> 00:07:56,179
And anyway, so essentially,
you have eight supercharge

118
00:07:56,179 --> 00:07:56,970
of the [INAUDIBLE].

119
00:07:59,500 --> 00:08:02,935
So this all together is
32 real superchargers.

120
00:08:10,710 --> 00:08:12,780
So when we say N
equal to 4, so this

121
00:08:12,780 --> 00:08:15,990
is N equal to 4 in terms of
four dimensional [INAUDIBLE].

122
00:08:15,990 --> 00:08:26,000
So your supersymmetry has four
[INAUDIBLE] as the supercharge.

123
00:08:26,000 --> 00:08:28,560
But the conformal symmetry
generates another 4.

124
00:08:28,560 --> 00:08:31,570
So all together, you
have eight [INAUDIBLE],

125
00:08:31,570 --> 00:08:35,720
four dimensional [INAUDIBLE],
as the supercharge.

126
00:08:35,720 --> 00:08:37,653
For each [INAUDIBLE]
in four dimensions,

127
00:08:37,653 --> 00:08:38,903
there are how many components?

128
00:08:42,780 --> 00:08:44,834
How many components are
there for [INAUDIBLE]

129
00:08:44,834 --> 00:08:45,625
in four dimensions?

130
00:08:45,625 --> 00:08:46,500
AUDIENCE: Four.

131
00:08:46,500 --> 00:08:47,150
HONG LIU: Four.

132
00:08:47,150 --> 00:08:50,290
Because the two components,
but each component is complex.

133
00:08:50,290 --> 00:08:52,550
And so there are
four real components.

134
00:08:52,550 --> 00:08:55,990
So all together, you
have 32 real charges.

135
00:08:55,990 --> 00:09:00,560
And similarly, you find--
I will not do this side.

136
00:09:00,560 --> 00:09:05,955
You find the same
amount of supersymmetry.

137
00:09:11,710 --> 00:09:15,730
But in this case, for
example, the low energy limit.

138
00:09:15,730 --> 00:09:18,630
Let's just talk about the low
energy limit of this theory.

139
00:09:18,630 --> 00:09:20,760
So the low energy limit,
as we said before,

140
00:09:20,760 --> 00:09:23,420
just has to be super gravity.

141
00:09:23,420 --> 00:09:26,210
So it has to be super gravity.

142
00:09:26,210 --> 00:09:29,990
Then you find you actually
have exactly the same amount

143
00:09:29,990 --> 00:09:32,480
of supersymmetry
in this geometry.

144
00:09:36,940 --> 00:09:42,170
But the interesting thing
is that by definition,

145
00:09:42,170 --> 00:09:45,700
the supersymmetry on the
gravity side is actually local.

146
00:09:45,700 --> 00:09:47,200
So you actually
have the same amount

147
00:09:47,200 --> 00:09:48,490
of local supersymmetries.

148
00:10:07,450 --> 00:10:09,650
So if you look at
this correspondence

149
00:10:09,650 --> 00:10:19,052
between each other, then
you actually see a pattern.

150
00:10:19,052 --> 00:10:20,385
So now let me make some remarks.

151
00:10:26,170 --> 00:10:29,570
So if you look at the
structure, this math in here,

152
00:10:29,570 --> 00:10:30,387
you see a pattern.

153
00:10:34,050 --> 00:10:35,940
On this side, all
these symmetries

154
00:10:35,940 --> 00:10:37,101
are global symmetries.

155
00:10:39,870 --> 00:10:41,510
But on this side,
all these symmetries

156
00:10:41,510 --> 00:10:44,880
are local symmetries.

157
00:10:44,880 --> 00:10:47,440
So this, I just said
that in super gravity,

158
00:10:47,440 --> 00:10:50,540
the supersymmetry is local.

159
00:10:50,540 --> 00:10:59,080
And the space time isometry
is just subset of a space time

160
00:10:59,080 --> 00:11:01,110
coordinate transformation.

161
00:11:01,110 --> 00:11:05,760
And space time
coordinate transformation

162
00:11:05,760 --> 00:11:08,810
are local symmetries.

163
00:11:08,810 --> 00:11:19,440
So the isometry is a
subgroup of diffeomorphism.

164
00:11:24,340 --> 00:11:27,410
So diffeomorphism just means
the coordinate transformations.

165
00:11:30,194 --> 00:11:31,485
And these are local symmetries.

166
00:11:48,580 --> 00:11:52,580
Now we find the mapping is
on the field theory side,

167
00:11:52,580 --> 00:12:00,820
the global symmetries is
mapped on the gravity side

168
00:12:00,820 --> 00:12:02,065
into the local symmetries.

169
00:12:09,810 --> 00:12:12,040
So for each global symmetry
in the field theory side,

170
00:12:12,040 --> 00:12:14,498
there's a corresponding local
symmetry on the gravity side.

171
00:12:21,200 --> 00:12:25,500
But you may immediately ask the
question, on the gravity side,

172
00:12:25,500 --> 00:12:26,940
if you talk about
diffeomorphism,

173
00:12:26,940 --> 00:12:30,050
then this is a huge
group-- certainly much,

174
00:12:30,050 --> 00:12:33,330
much larger than what we are
talking about the isometry

175
00:12:33,330 --> 00:12:34,160
here.

176
00:12:34,160 --> 00:12:35,910
So why we are only
talking about isometry?

177
00:12:41,334 --> 00:12:43,750
Why we don't talk about other
parts of the diffeomorphism,

178
00:12:43,750 --> 00:12:44,791
only talk about isometry?

179
00:12:47,579 --> 00:12:49,120
So what's special
about the isometry?

180
00:12:52,910 --> 00:12:56,480
So the second
remark, why isometry?

181
00:13:02,430 --> 00:13:03,180
Why only isometry?

182
00:13:05,930 --> 00:13:07,230
Why only look at the isometry?

183
00:13:07,230 --> 00:13:09,250
But let me just save time.

184
00:13:14,430 --> 00:13:23,970
So the isometry is important
for the following reason.

185
00:13:23,970 --> 00:13:29,180
Because even though this
is a subgroup-- so as

186
00:13:29,180 --> 00:13:32,480
I mentioned last time, when
we talk about quantum gravity,

187
00:13:32,480 --> 00:13:36,240
when we talk about string theory
and Anti-de Sitter times S5

188
00:13:36,240 --> 00:13:41,140
in terms of AdS5 times S5,
you should keep in mind

189
00:13:41,140 --> 00:13:46,091
that this AdS5 times S5 refers
to the asymptotic geometry.

190
00:13:46,091 --> 00:13:47,715
Because as a quantum
theory, the theory

191
00:13:47,715 --> 00:13:49,020
can fluctuate in the middle.

192
00:13:52,040 --> 00:13:54,540
And the isometry can be
considered as a subgroup.

193
00:13:59,300 --> 00:14:03,990
But the level is even though
the space time fluctuates,

194
00:14:03,990 --> 00:14:07,940
but the AdS5 times S5 specifies
the asymptotic geometry

195
00:14:07,940 --> 00:14:09,080
of the space time.

196
00:14:13,820 --> 00:14:19,180
And the isometry is
precisely the subgroup

197
00:14:19,180 --> 00:14:34,290
which leaves the asymptotic
form of the metric invariant.

198
00:14:46,260 --> 00:14:50,670
So these are not
ordinary diffeomorphisms.

199
00:14:50,670 --> 00:14:53,630
So the ordinary
diffeomorphisms--

200
00:14:53,630 --> 00:14:55,260
so ordinary gauge
transformations,

201
00:14:55,260 --> 00:14:57,520
let me just say more generally.

202
00:14:57,520 --> 00:15:02,080
By gauge transformations,
also means the diffeomorphism

203
00:15:02,080 --> 00:15:05,480
or just general gauge
transformations.

204
00:15:05,480 --> 00:15:09,450
So when we talk about
gauge transformations, say

205
00:15:09,450 --> 00:15:12,630
for example, in QED
or in QCD, et cetera,

206
00:15:12,630 --> 00:15:20,430
we always assume-- so the
classical gauge transformation,

207
00:15:20,430 --> 00:15:23,920
we can see that we always
take them to be fall off

208
00:15:23,920 --> 00:15:25,310
sufficiently fast at infinity.

209
00:15:44,420 --> 00:15:46,520
When we talk about
gauge transformations,

210
00:15:46,520 --> 00:15:51,090
when we talk about the
typical diffeomorphisms,

211
00:15:51,090 --> 00:15:53,800
these are the kind of
transformations we assume

212
00:15:53,800 --> 00:15:56,110
goes to 1, say, at infinity.

213
00:15:59,850 --> 00:16:03,370
And this isometry, which
leaves the asymptotic

214
00:16:03,370 --> 00:16:07,147
over the metric invariant,
so these are essentially

215
00:16:07,147 --> 00:16:08,480
the large gauge transformations.

216
00:16:20,760 --> 00:16:22,430
So-called large
transformations is

217
00:16:22,430 --> 00:16:26,880
that they don't go to
the Identity at infinity.

218
00:16:26,880 --> 00:16:31,770
And of course, this is precisely
the large gauge transformations

219
00:16:31,770 --> 00:16:34,580
which leaves the
asymptotic invariant.

220
00:16:37,420 --> 00:16:42,310
So in a sense, these large
gauge transformations

221
00:16:42,310 --> 00:16:55,800
can be considered as the global
part of the diffeomorphism.

222
00:17:17,650 --> 00:17:18,894
So any questions on this?

223
00:17:22,310 --> 00:17:23,810
Yes?

224
00:17:23,810 --> 00:17:27,807
AUDIENCE: What does it mean
that those isometries are local?

225
00:17:27,807 --> 00:17:30,292
What do you do with
isometry [INAUDIBLE]?

226
00:17:33,274 --> 00:17:36,270
What do you mean when
you say that isometries--

227
00:17:36,270 --> 00:17:38,748
HONG LIU: Yeah.

228
00:17:38,748 --> 00:17:40,456
These are just coordinate
transformation.

229
00:17:40,456 --> 00:17:43,090
A coordinate transformation is
always defined point by point,

230
00:17:43,090 --> 00:17:45,830
right?

231
00:17:45,830 --> 00:17:49,798
Just these are specific
coordinate transformations.

232
00:17:49,798 --> 00:17:53,130
AUDIENCE: And what happens to
gauge symmetries [INAUDIBLE]?

233
00:17:56,774 --> 00:17:57,440
HONG LIU: Sorry?

234
00:17:57,440 --> 00:17:59,440
AUDIENCE: What happens
to gauge transformations

235
00:17:59,440 --> 00:18:01,789
in the Yang-Mills part?

236
00:18:01,789 --> 00:18:02,330
HONG LIU: No.

237
00:18:02,330 --> 00:18:04,330
The gauge transformation
in the Yang-Mills part,

238
00:18:04,330 --> 00:18:06,690
you don't see it.

239
00:18:06,690 --> 00:18:08,400
Gauge freedom is just
redundant freedom.

240
00:18:08,400 --> 00:18:11,870
You never see it
on the other side.

241
00:18:11,870 --> 00:18:16,295
AUDIENCE: Does this list
exhaust all symmetries

242
00:18:16,295 --> 00:18:17,570
of the Yang-Mills particle?

243
00:18:17,570 --> 00:18:19,984
HONG LIU: Yeah.

244
00:18:19,984 --> 00:18:23,584
Exhausts all global symmetries
in the Yang-Mills theory part.

245
00:18:23,584 --> 00:18:25,876
AUDIENCE: Not local?

246
00:18:25,876 --> 00:18:27,250
HONG LIU: All the
global theories

247
00:18:27,250 --> 00:18:30,345
on the Yang-Mills side,
and not local symmetries.

248
00:18:30,345 --> 00:18:33,990
AUDIENCE: [INAUDIBLE]
discuss them, [INAUDIBLE].

249
00:18:33,990 --> 00:18:36,670
HONG LIU: There's a
u(n) gauge group there,

250
00:18:36,670 --> 00:18:38,890
but they don't have
correspondence in the gravity

251
00:18:38,890 --> 00:18:41,080
side.

252
00:18:41,080 --> 00:18:43,840
The reason we don't consider
the gauge symmetry, because they

253
00:18:43,840 --> 00:18:46,060
correspond to
redundancies and they

254
00:18:46,060 --> 00:18:48,869
don't have to be present
on the other side.

255
00:18:48,869 --> 00:18:51,410
You only need to do the physical
[INAUDIBLE] to be identical.

256
00:18:51,410 --> 00:18:56,960
And this also is the reason
I make this remark here.

257
00:18:56,960 --> 00:18:59,860
Even though all these
global symmetries,

258
00:18:59,860 --> 00:19:01,710
they find the corresponding
local symmetries

259
00:19:01,710 --> 00:19:04,210
on the gravity side.

260
00:19:04,210 --> 00:19:10,160
But precisely due
to this remark,

261
00:19:10,160 --> 00:19:14,160
those things which really
map to the global symmetries

262
00:19:14,160 --> 00:19:15,850
in the field theory
can be considered

263
00:19:15,850 --> 00:19:20,759
as a global part of the local
symmetries on the gravity side.

264
00:19:20,759 --> 00:19:22,550
So even on the gravity
side, in some sense,

265
00:19:22,550 --> 00:19:24,750
you should think of them
as global symmetries.

266
00:19:24,750 --> 00:19:30,622
And they are large
gauge transformations.

267
00:19:30,622 --> 00:19:32,330
Because the ordinary
gauge transformation

268
00:19:32,330 --> 00:19:36,290
is just corresponding
to redundancy

269
00:19:36,290 --> 00:19:37,490
of degrees of freedom.

270
00:19:37,490 --> 00:19:40,265
And that should not be
reflected on the other side.

271
00:19:40,265 --> 00:19:42,640
AUDIENCE: What do you mean by
large gauge transformation?

272
00:19:42,640 --> 00:19:44,140
HONG LIU: Large
gauge transformation

273
00:19:44,140 --> 00:19:47,800
means the gauge transformations
which don't vanish at infinity.

274
00:19:47,800 --> 00:19:49,980
So the ordinary
gauge transformation,

275
00:19:49,980 --> 00:19:55,400
just like in QED, in quantum
electrodynamics, when we talk

276
00:19:55,400 --> 00:19:57,600
about gauge transformation.

277
00:19:57,600 --> 00:20:00,490
The gauge transformations
we consider all the gauge

278
00:20:00,490 --> 00:20:02,842
transformations which
go to zero at infinity.

279
00:20:02,842 --> 00:20:04,800
And if you consider those
gauge transformations

280
00:20:04,800 --> 00:20:06,860
which don't go to
zero at infinity,

281
00:20:06,860 --> 00:20:09,210
and that's what we call
large gauge transformations.

282
00:20:09,210 --> 00:20:11,460
And those large gauge
transformations essentially

283
00:20:11,460 --> 00:20:16,510
is like the global part of
the u(1) gauge symmetry,

284
00:20:16,510 --> 00:20:18,190
in some sense can
be considered as.

285
00:20:21,130 --> 00:20:25,980
AUDIENCE: But that is the gauge
symmetry being gravity side.

286
00:20:25,980 --> 00:20:27,010
HONG LIU: That's right.

287
00:20:27,010 --> 00:20:33,380
So this is the part of the gauge
symmetry on the gravity side.

288
00:20:33,380 --> 00:20:36,510
But the part corresponding
to the boundary

289
00:20:36,510 --> 00:20:38,245
is the part associated
with the infinity.

290
00:20:44,190 --> 00:20:47,270
AUDIENCE: Here in
isometry of AdS5,

291
00:20:47,270 --> 00:20:49,580
we were showing,
like in p-set, it's

292
00:20:49,580 --> 00:20:51,980
point to point local symmetry.

293
00:20:51,980 --> 00:20:55,260
It's like just a
space time point.

294
00:20:55,260 --> 00:21:01,360
But how about this conform
in this Yang-Mills theory?

295
00:21:01,360 --> 00:21:06,430
What is the symmetry
of the object?

296
00:21:06,430 --> 00:21:08,660
What kind of object does
this symmetry correspond

297
00:21:08,660 --> 00:21:09,884
to because it seems--

298
00:21:09,884 --> 00:21:11,300
HONG LIU: It's
conformal symmetry.

299
00:21:16,010 --> 00:21:19,635
AUDIENCE: Conformal symmetry
of the component of the field?

300
00:21:19,635 --> 00:21:20,510
HONG LIU: No, no, no.

301
00:21:20,510 --> 00:21:23,745
It's the conformal symmetry
of the Minkowski space.

302
00:21:26,360 --> 00:21:29,277
So on this side, you
have isometry of AdS.

303
00:21:29,277 --> 00:21:31,193
On the other side, you
have conformal symmetry

304
00:21:31,193 --> 00:21:33,052
of the Minkowski space.

305
00:21:33,052 --> 00:21:35,510
So you're asking what are you
corresponding to on the field

306
00:21:35,510 --> 00:21:36,710
theory side, right?

307
00:21:36,710 --> 00:21:39,550
So that's mapped to
the conformal symmetry

308
00:21:39,550 --> 00:21:41,547
of the Minkowski space.

309
00:21:46,250 --> 00:21:50,500
AUDIENCE: So the
isometry is actually

310
00:21:50,500 --> 00:21:52,004
large gauge transformations.

311
00:21:52,004 --> 00:21:52,670
HONG LIU: Right.

312
00:21:57,007 --> 00:21:58,590
So the story actually
is more general.

313
00:22:03,367 --> 00:22:05,075
So the story actually
works more general.

314
00:22:05,075 --> 00:22:07,533
This is the statement for the
N equal to 4 super Yang-Mills

315
00:22:07,533 --> 00:22:09,170
theory.

316
00:22:09,170 --> 00:22:10,380
So the story is more general.

317
00:22:22,030 --> 00:22:27,730
So in the case which you
have, say, more general,

318
00:22:27,730 --> 00:22:32,640
say duality between
a CFT in d dimensions

319
00:22:32,640 --> 00:22:43,855
in Minkowski d with the AdS
d plus 1 dimensional gravity.

320
00:22:47,108 --> 00:22:52,270
In any such duality, then
you always have the conformal

321
00:22:52,270 --> 00:23:00,150
symmetry, which is SO(d, 2)
on this here mapped to AdS

322
00:23:00,150 --> 00:23:00,650
isometry.

323
00:23:09,030 --> 00:23:11,720
And any internal
gauge symmetry, say

324
00:23:11,720 --> 00:23:16,080
if you have some u(s)
global symmetry here,

325
00:23:16,080 --> 00:23:25,320
then this will be mapped
into a u(1) gauge symmetry.

326
00:23:32,880 --> 00:23:36,750
Again, in the sense that
this u(s) global symmetry can

327
00:23:36,750 --> 00:23:40,640
be considered as the global part
of this u(1) gauge symmetry,

328
00:23:40,640 --> 00:23:42,443
the part which does
not vanish at infinity.

329
00:23:45,500 --> 00:23:51,450
And it's always the case that
global supersymmetry here

330
00:23:51,450 --> 00:23:57,020
would be corresponding to
the local supersymmetry here.

331
00:23:57,020 --> 00:23:59,870
And also in exactly
the sense that

332
00:23:59,870 --> 00:24:03,500
even for the local
supersymmetric transformation

333
00:24:03,500 --> 00:24:05,370
in the gravity
side, there's also

334
00:24:05,370 --> 00:24:07,586
a part which does not
vanish at infinity.

335
00:24:07,586 --> 00:24:08,960
And that's the
part corresponding

336
00:24:08,960 --> 00:24:12,770
to the global supersymmetry
on the field theory side.

337
00:24:15,510 --> 00:24:16,350
Yes?

338
00:24:16,350 --> 00:24:21,924
AUDIENCE: What are the fermions
supersymmetry on the AdS side?

339
00:24:21,924 --> 00:24:22,590
HONG LIU: Sorry?

340
00:24:22,590 --> 00:24:25,310
AUDIENCE: In the low energy
limit on the AdS part,

341
00:24:25,310 --> 00:24:29,260
what are the fermion
supersymmetry?

342
00:24:29,260 --> 00:24:32,700
HONG LIU: It's the same thing.

343
00:24:32,700 --> 00:24:34,450
What do you mean,
what about fermions?

344
00:24:34,450 --> 00:24:37,042
So there will be some fermions.

345
00:24:37,042 --> 00:24:39,018
AUDIENCE: [INAUDIBLE] fermions?

346
00:24:39,018 --> 00:24:42,480
So there is more than
gravitons on the AdS side

347
00:24:42,480 --> 00:24:43,820
involved in low energy limits?

348
00:24:43,820 --> 00:24:44,490
HONG LIU: Yeah.

349
00:24:44,490 --> 00:24:46,910
There are gravitons, and
there are also some fermions.

350
00:24:46,910 --> 00:24:48,535
There's something
called a [INAUDIBLE].

351
00:24:48,535 --> 00:24:51,420
There are some other
fermions, et cetera.

352
00:24:51,420 --> 00:24:54,569
AUDIENCE: Is it they who
carry the u(1) gauge charge?

353
00:24:54,569 --> 00:24:55,110
HONG LIU: No.

354
00:24:55,110 --> 00:24:56,090
Here is more general.

355
00:24:58,890 --> 00:25:06,430
In this case, indeed--
so this isometry of S5,

356
00:25:06,430 --> 00:25:11,740
once you deduce to AdS5, that
just again becomes a gauge

357
00:25:11,740 --> 00:25:13,860
symmetry, SO(6) gauge symmetry.

358
00:25:13,860 --> 00:25:19,170
And some fermions indeed are
charged on this SO(6) gauge

359
00:25:19,170 --> 00:25:21,440
symmetry.

360
00:25:21,440 --> 00:25:23,540
And here, it's
just more general.

361
00:25:23,540 --> 00:25:27,540
So whatever things, anytime if
you have some global symmetry

362
00:25:27,540 --> 00:25:32,900
here should be mapped to
some gauge symmetry here.

363
00:25:32,900 --> 00:25:37,742
And we will say this a little
bit more a little bit later.

364
00:25:37,742 --> 00:25:39,075
Do you have any other questions?

365
00:25:41,730 --> 00:25:45,720
So now let's move to the
matching of parameters.

366
00:25:45,720 --> 00:25:51,850
Again, this is
more like a review

367
00:25:51,850 --> 00:25:53,070
of what we discussed before.

368
00:25:56,120 --> 00:26:02,780
And again, first N equals to
4 super Yang-Mills theory.

369
00:26:02,780 --> 00:26:06,780
And then this is type
IIB on AdS5 times S5.

370
00:26:10,740 --> 00:26:14,870
So previously we said, from
the relation of the d-brane,

371
00:26:14,870 --> 00:26:19,530
so the G Yang-Mills square
here is related to the 4 pi GS

372
00:26:19,530 --> 00:26:22,070
here, string coupling.

373
00:26:22,070 --> 00:26:25,780
And also, from the
d3-brane solution,

374
00:26:25,780 --> 00:26:29,310
we find is that the
G Yang-Mills square--

375
00:26:29,310 --> 00:26:33,290
we find, for example,
here the curvature radius

376
00:26:33,290 --> 00:26:41,090
R has the following form, 4
pi GS and alpha prime squared.

377
00:26:41,090 --> 00:26:45,810
So the N is the
same N on this side.

378
00:26:45,810 --> 00:26:54,000
So the N is the gauge group N.
I should say the flux N. Anyway,

379
00:26:54,000 --> 00:26:57,140
let me add something.

380
00:26:57,140 --> 00:27:00,230
So the N is the number of
d-branes on this side which

381
00:27:00,230 --> 00:27:02,880
translate into the flux.

382
00:27:02,880 --> 00:27:06,050
So the flux N. And then
here is corresponding

383
00:27:06,050 --> 00:27:06,800
to you have SU(N).

384
00:27:12,040 --> 00:27:14,670
So this R, this
curvature radius is

385
00:27:14,670 --> 00:27:18,770
related to the alpha prime
squared and the gs in this way.

386
00:27:18,770 --> 00:27:25,760
And so on this side,
the dimension parameter

387
00:27:25,760 --> 00:27:28,680
is given by R squared
divided by alpha prime.

388
00:27:28,680 --> 00:27:32,800
So if you look at
this relation, if you

389
00:27:32,800 --> 00:27:35,960
look at alpha prime 4
to alpha prime squared,

390
00:27:35,960 --> 00:27:39,890
then you find this is
just relating the alpha.

391
00:27:39,890 --> 00:27:42,440
From this relation,
4 pi GS is equal just

392
00:27:42,440 --> 00:27:44,260
to G Yang-Mills square.

393
00:27:44,260 --> 00:27:47,622
And the N is N.
This is the relation

394
00:27:47,622 --> 00:27:48,580
between the parameters.

395
00:27:58,512 --> 00:27:59,720
Any questions regarding this?

396
00:28:06,200 --> 00:28:11,950
So on the gravity side, we said
these are the two parameters.

397
00:28:11,950 --> 00:28:14,410
And of course, you
also have this N.

398
00:28:14,410 --> 00:28:17,940
But these are the
two basic parameters.

399
00:28:17,940 --> 00:28:22,610
And we can also,
instead of using GS,

400
00:28:22,610 --> 00:28:26,190
as we said before, you can
also use the Newton constant.

401
00:28:26,190 --> 00:28:28,860
So the 10 dimensional
Newton constant

402
00:28:28,860 --> 00:28:31,510
is length dimension eight.

403
00:28:31,510 --> 00:28:35,950
Then the dimensionless parameter
would be GN divided by R

404
00:28:35,950 --> 00:28:36,875
to the power 8.

405
00:28:39,990 --> 00:28:44,529
And the GN is related to
the GS and alpha prime

406
00:28:44,529 --> 00:28:45,195
by this formula.

407
00:28:49,820 --> 00:28:53,105
So now you can just use
the relation between the GS

408
00:28:53,105 --> 00:28:57,550
to exactly translate this
into Yang-Mills coupling.

409
00:28:57,550 --> 00:29:00,140
You just can use this
relation and then use

410
00:29:00,140 --> 00:29:03,000
that to translate this into
the parameter in the Yang-Mills

411
00:29:03,000 --> 00:29:04,620
theory side.

412
00:29:04,620 --> 00:29:08,340
Then you find, once you
plug all those relations in,

413
00:29:08,340 --> 00:29:11,890
how the GS and alpha prime,
et cetera related to N,

414
00:29:11,890 --> 00:29:15,360
then you find here
actually, G Yang-Mills

415
00:29:15,360 --> 00:29:17,750
have disappeared
in this relation.

416
00:29:17,750 --> 00:29:21,900
What you find is that the
Newton constant essentially just

417
00:29:21,900 --> 00:29:24,730
related to 1 over N squared.

418
00:29:24,730 --> 00:29:29,710
Up to some parameter, just
related to 1 over N squared.

419
00:29:29,710 --> 00:29:33,121
So if you're expanding G Newton,
just like expanding 1 over N

420
00:29:33,121 --> 00:29:33,621
squared.

421
00:29:39,540 --> 00:29:49,880
So as we said before, we often
do dimensional reduction on S5.

422
00:29:58,160 --> 00:30:01,720
Let me get a five
dimensional Newton constant.

423
00:30:01,720 --> 00:30:03,220
So five dimensional
Newton constant

424
00:30:03,220 --> 00:30:06,770
is equal to the 10
dimensional Newton constant.

425
00:30:06,770 --> 00:30:09,370
And the difference
is the volume of S5.

426
00:30:18,340 --> 00:30:20,440
We wrote this down before.

427
00:30:20,440 --> 00:30:29,350
And the S5 is equal to pi
cubed R to the power fifth.

428
00:30:29,350 --> 00:30:32,940
And then from here,
you can just work out.

429
00:30:32,940 --> 00:30:36,578
So G5 has dimension 3.

430
00:30:36,578 --> 00:30:41,930
Then G5 divided by R cubed,
again only related to N given

431
00:30:41,930 --> 00:30:43,540
by pi divided by 2N squared.

432
00:30:53,450 --> 00:30:55,730
So these relations are
often useful in the future.

433
00:31:12,060 --> 00:31:17,275
So now let's look at [INAUDIBLE]
limits of this relation.

434
00:31:21,660 --> 00:31:23,680
So let's first look at
the classical gravity

435
00:31:23,680 --> 00:31:24,950
limit on the gravity side.

436
00:31:37,601 --> 00:31:42,860
As we discussed at the beginning
of this class, by classic,

437
00:31:42,860 --> 00:31:47,870
we always use h bar equal to 1.

438
00:31:47,870 --> 00:31:49,913
So our h bar is
always equal to 1.

439
00:31:54,020 --> 00:31:57,850
But quantum gravity is
captured by this parameter,

440
00:31:57,850 --> 00:32:00,590
h bar times GN.

441
00:32:00,590 --> 00:32:04,250
So even though h bar equal
to 1, in the limit when

442
00:32:04,250 --> 00:32:07,240
GN goes to zero, then
you are in the regime

443
00:32:07,240 --> 00:32:10,330
in which you can ignore
the quantum gravity effect.

444
00:32:10,330 --> 00:32:13,950
So that's what we mean by
the classical gravity limit

445
00:32:13,950 --> 00:32:16,890
is that this should go to
zero, and then alpha prime

446
00:32:16,890 --> 00:32:19,410
should go to zero.

447
00:32:19,410 --> 00:32:22,450
So this means the string
effect is not important.

448
00:32:22,450 --> 00:32:28,120
And this is all
in the unit of R.

449
00:32:28,120 --> 00:32:30,630
And here, when I
write these relations,

450
00:32:30,630 --> 00:32:35,030
I have all set h bar equal to 1.

451
00:32:35,030 --> 00:32:37,130
So when I say the
classical gravity limit,

452
00:32:37,130 --> 00:32:40,590
which is this limit in
which h bar equal to 1,

453
00:32:40,590 --> 00:32:44,940
but the effect of the Newton
constant goes to zero.

454
00:32:44,940 --> 00:32:49,160
And the reason I emphasize it
is that if you have some matter

455
00:32:49,160 --> 00:32:55,330
field in this geometry, then
those matter fields should

456
00:32:55,330 --> 00:32:58,230
be treated as full quantum.

457
00:32:58,230 --> 00:33:01,712
Just don't treat the
gravity as quantum,

458
00:33:01,712 --> 00:33:04,420
but those matter fields
should be treated as quantum.

459
00:33:04,420 --> 00:33:05,830
So the classical
gravity limit is

460
00:33:05,830 --> 00:33:11,495
the same as QFT, Quantum Field
Theory in curved space time.

461
00:33:17,410 --> 00:33:19,250
So gravity does not fluctuate.

462
00:33:19,250 --> 00:33:22,180
So you have rigid
curved space time.

463
00:33:22,180 --> 00:33:26,840
But your matter field can
fluctuate, h bar equal to 1.

464
00:33:26,840 --> 00:33:30,110
So essentially, we are
dealing with quantum field

465
00:33:30,110 --> 00:33:32,410
theory in curved space time.

466
00:33:32,410 --> 00:33:38,430
So in the type
IIB super gravity,

467
00:33:38,430 --> 00:33:40,920
there are many, many such
kind of matter fields,

468
00:33:40,920 --> 00:33:44,560
and they all should be
treated quantum mechanically.

469
00:33:48,800 --> 00:33:51,940
It's just that you should
consider this small.

470
00:33:51,940 --> 00:33:55,300
So let's consider
what this means.

471
00:33:55,300 --> 00:34:01,140
So GN small as a
dimensionless parameter,

472
00:34:01,140 --> 00:34:06,540
this translates into
field theory side

473
00:34:06,540 --> 00:34:08,409
if we use this relation.

474
00:34:08,409 --> 00:34:11,130
So that means N
goes to infinity.

475
00:34:11,130 --> 00:34:14,475
So this is the large N limit
of the Yang-Mills theory.

476
00:34:14,475 --> 00:34:16,600
This is the large N limit
of the Yang-Mills theory.

477
00:34:19,179 --> 00:34:20,783
And then alpha
prime goes to zero.

478
00:34:25,170 --> 00:34:29,620
From the relation between
the alpha prime and here,

479
00:34:29,620 --> 00:34:33,819
when alpha prime goes to zero,
so this is in the downstairs.

480
00:34:33,819 --> 00:34:35,610
Then that means this
should go to infinity.

481
00:34:38,190 --> 00:34:43,020
So this is the t Hooft
coupling we defined before.

482
00:34:43,020 --> 00:34:45,390
So this means the t Hooft
coupling goes to infinity.

483
00:34:52,909 --> 00:34:54,677
So now we see a
remarkable relation.

484
00:35:04,690 --> 00:35:08,490
So this is what we expect.

485
00:35:08,490 --> 00:35:12,080
If you still remember what we
did before in the large N gauge

486
00:35:12,080 --> 00:35:15,600
theory, in the large N gauge
theory in the large N limit,

487
00:35:15,600 --> 00:35:18,100
the fluctuations become
very small, et cetera.

488
00:35:18,100 --> 00:35:21,160
So this is consistent
that on the gravity side,

489
00:35:21,160 --> 00:35:25,990
the fluctuation in the
geometry is very small.

490
00:35:25,990 --> 00:35:33,890
And now, we see that the
decoupling of the string effect

491
00:35:33,890 --> 00:35:38,530
requires on the field theory
side the strong coupling.

492
00:35:38,530 --> 00:35:42,410
This is also something
roughly we said before.

493
00:35:42,410 --> 00:35:49,710
Remember, when we talked
about large N gauge theory,

494
00:35:49,710 --> 00:35:51,640
large N gauge theory,
you have planar diagram,

495
00:35:51,640 --> 00:35:55,040
non-planar diagram, et cetera.

496
00:35:55,040 --> 00:35:57,930
And at each level, say
at the planar diagram,

497
00:35:57,930 --> 00:36:01,380
you need to sum over infinite
number of [INAUDIBLE] diagrams

498
00:36:01,380 --> 00:36:04,499
which are all planar topology.

499
00:36:04,499 --> 00:36:07,040
And if you look at just those
[INAUDIBLE] diagrams, of course

500
00:36:07,040 --> 00:36:11,009
you don't see a space
time interpretation

501
00:36:11,009 --> 00:36:12,800
because they are just
[INAUDIBLE] diagrams.

502
00:36:12,800 --> 00:36:16,080
You don't see a
continuous surface.

503
00:36:16,080 --> 00:36:20,080
And I already alluded before
that the continuous surface

504
00:36:20,080 --> 00:36:23,089
can emerge if that
diagram becomes

505
00:36:23,089 --> 00:36:24,130
sufficiently complicated.

506
00:36:26,936 --> 00:36:29,310
And the way to make that
diagram sufficiently complicated

507
00:36:29,310 --> 00:36:32,330
is to make it to be
a strong coupling.

508
00:36:32,330 --> 00:36:34,910
The strong coupling,
then the diagram

509
00:36:34,910 --> 00:36:37,670
with many, many
vertices will dominate.

510
00:36:37,670 --> 00:36:40,470
And then the most dominated
diagrams are those diagrams

511
00:36:40,470 --> 00:36:42,000
with not a lot of vertices.

512
00:36:42,000 --> 00:36:45,600
And they essentially are
going to continue to limits.

513
00:36:45,600 --> 00:36:50,070
And so this relation
[INAUDIBLE] realizes

514
00:36:50,070 --> 00:36:52,380
that kind of intuition.

515
00:36:52,380 --> 00:36:56,230
If you don't remember, go
back to your loads regarding

516
00:36:56,230 --> 00:36:57,280
the large N gauge theory.

517
00:37:01,680 --> 00:37:04,140
So this relation
is also remarkable

518
00:37:04,140 --> 00:37:07,350
for the following reason.

519
00:37:07,350 --> 00:37:09,620
Because this limit on the
gravity side is simple,

520
00:37:09,620 --> 00:37:12,540
we can just deal
with quantum field

521
00:37:12,540 --> 00:37:16,610
theory in the curved space
time, which we know how to do.

522
00:37:16,610 --> 00:37:18,640
But on this side, it's
highly non-trivial

523
00:37:18,640 --> 00:37:21,030
because this is an
infinite coupling limit.

524
00:37:21,030 --> 00:37:31,860
So this will tell you that
the strong coupling limit is

525
00:37:31,860 --> 00:37:40,354
described by classical gravity.

526
00:37:52,480 --> 00:37:56,340
So that means that we can
actually use classical gravity

527
00:37:56,340 --> 00:37:59,404
to, in principle, solve problems
which are strongly coupled.

528
00:38:07,840 --> 00:38:10,470
So also, of course, there
are corrections beyond this.

529
00:38:13,710 --> 00:38:24,610
So quantum gravity
corrections on this side,

530
00:38:24,610 --> 00:38:26,760
so this is a classical
gravity limit

531
00:38:26,760 --> 00:38:29,400
if you take those
parameters to go to zero.

532
00:38:29,400 --> 00:38:32,620
But suppose those parameters
are not zero, just small.

533
00:38:32,620 --> 00:38:35,110
Then you can just expand
in those parameters.

534
00:38:35,110 --> 00:38:39,540
For example, can you
expand in GN divided by 8.

535
00:38:39,540 --> 00:38:42,760
And that expansion essentially
gives you quantum gravity

536
00:38:42,760 --> 00:38:45,600
corrections [INAUDIBLE].

537
00:38:45,600 --> 00:38:51,055
And now, from this relation,
the expansion of GN

538
00:38:51,055 --> 00:38:53,290
then is translated.

539
00:38:53,290 --> 00:39:00,470
So this is expansion
in G Newton.

540
00:39:00,470 --> 00:39:02,390
So from that
relation, we see this

541
00:39:02,390 --> 00:39:07,170
has become the expansion 1 over
N squared in the field theory

542
00:39:07,170 --> 00:39:07,670
side.

543
00:39:14,830 --> 00:39:24,096
Just expansion 1 over N squared.

544
00:39:29,850 --> 00:39:33,500
So on this side, you can
also take into account

545
00:39:33,500 --> 00:39:36,780
that the alpha
prime is non-zero.

546
00:39:36,780 --> 00:39:41,030
Then the alpha prime
corrections in the expansion

547
00:39:41,030 --> 00:39:41,735
in alpha prime.

548
00:39:46,550 --> 00:39:50,180
So from this relation,
translates in the expansion 1

549
00:39:50,180 --> 00:39:51,180
over square root lambda.

550
00:39:57,280 --> 00:40:03,321
So in principle, the
corrections beyond this limit

551
00:40:03,321 --> 00:40:07,220
can again be studied
on the gravity side.

552
00:40:07,220 --> 00:40:09,400
And then the 1 over square
root lambda corrections

553
00:40:09,400 --> 00:40:12,510
then corresponding to
the string G corrections.

554
00:40:12,510 --> 00:40:14,802
And the 1 over N squared
corrections corresponding

555
00:40:14,802 --> 00:40:16,260
to the quantum
gravity corrections.

556
00:40:23,680 --> 00:40:24,980
So this is the classical limit.

557
00:40:24,980 --> 00:40:26,780
You also can see
the classical string

558
00:40:26,780 --> 00:40:32,050
limit we considered before,
we discussed before.

559
00:40:32,050 --> 00:40:34,640
In the classical
string limit, still you

560
00:40:34,640 --> 00:40:40,450
can see the N go to infinity,
which corresponds to GN.

561
00:40:40,450 --> 00:40:43,240
R8 goes to 0.

562
00:40:43,240 --> 00:40:46,156
But here, the alpha
prime can be arbitrary.

563
00:40:51,900 --> 00:40:56,980
And here the [INAUDIBLE]
coupling can be arbitrary.

564
00:40:56,980 --> 00:40:58,160
So it can be finite.

565
00:40:58,160 --> 00:41:03,490
So let me just say alpha
prime finite no longer zero.

566
00:41:03,490 --> 00:41:06,950
And then this is just
corresponding to lambda finite,

567
00:41:06,950 --> 00:41:11,140
which is no longer infinite.

568
00:41:11,140 --> 00:41:16,980
So this is just a
standard t Hooft limit,

569
00:41:16,980 --> 00:41:22,370
which we take N go to infinity
but keep lambda to be finite.

570
00:41:22,370 --> 00:41:25,530
Standard t Hooft limit,
which we talked about before

571
00:41:25,530 --> 00:41:28,930
in the large N gauge theory.

572
00:41:28,930 --> 00:41:30,669
And then again, the
corrections in GN

573
00:41:30,669 --> 00:41:32,335
will be corrections
in 1 over N squared.

574
00:41:43,690 --> 00:41:44,840
So any questions?

575
00:41:44,840 --> 00:41:45,410
Yes?

576
00:41:45,410 --> 00:41:50,479
AUDIENCE: Doesn't small alpha
over R squared mean large GS?

577
00:41:50,479 --> 00:41:51,020
HONG LIU: No.

578
00:41:51,020 --> 00:41:52,311
That has nothing to do with GS.

579
00:41:55,261 --> 00:41:56,802
These are two
independent parameters.

580
00:42:00,536 --> 00:42:05,619
AUDIENCE: But expression
parentheses, so there's also N.

581
00:42:05,619 --> 00:42:06,410
HONG LIU: This is--

582
00:42:13,100 --> 00:42:15,100
AUDIENCE: So in the
classical [INAUDIBLE],

583
00:42:15,100 --> 00:42:18,565
it means that since
lambda is finite,

584
00:42:18,565 --> 00:42:22,520
so this means G Yang-Mills is 0.

585
00:42:22,520 --> 00:42:23,250
HONG LIU: Sorry.

586
00:42:23,250 --> 00:42:27,750
AUDIENCE: I mean lambda is
finite, so G Yang-Mills is 0

587
00:42:27,750 --> 00:42:29,310
and it means weak coupling.

588
00:42:29,310 --> 00:42:30,950
HONG LIU: No.

589
00:42:30,950 --> 00:42:38,490
That's what we discussed before
in the large N gauge theory.

590
00:42:38,490 --> 00:42:42,220
The effective
coupling is lambda.

591
00:42:42,220 --> 00:42:50,030
Lambda is your effective
coupling in the large N limit.

592
00:42:50,030 --> 00:42:53,030
AUDIENCE: When you draw
the diagram, [INAUDIBLE].

593
00:42:53,030 --> 00:42:53,950
HONG LIU: Yeah.

594
00:42:53,950 --> 00:42:56,775
So the lambda is your coupling.

595
00:42:56,775 --> 00:43:01,280
The G Yang-Mills indeed,
in the t Hooft limit,

596
00:43:01,280 --> 00:43:05,050
the G Yang-Mills will go to
0 when N goes to infinity.

597
00:43:05,050 --> 00:43:09,680
But G Yang-Mills is
not the right parameter

598
00:43:09,680 --> 00:43:10,564
to look at things.

599
00:43:16,652 --> 00:43:17,485
Any other questions?

600
00:43:21,120 --> 00:43:22,470
OK, good.

601
00:43:22,470 --> 00:43:25,187
So now we can move on further.

602
00:43:39,190 --> 00:43:49,352
We can talk about the matching
of the spectrum on two sides.

603
00:43:54,920 --> 00:43:59,800
So from now on, I will just
restrict to essentially

604
00:43:59,800 --> 00:44:01,760
the semi classical.

605
00:44:01,760 --> 00:44:06,940
Here I should call semi
classical gravity limit

606
00:44:06,940 --> 00:44:10,120
because we still treat the
matter fields essentially

607
00:44:10,120 --> 00:44:10,690
as quantum.

608
00:44:10,690 --> 00:44:15,680
So we call it semi
classical gravity limit.

609
00:44:15,680 --> 00:44:18,100
So from now on,
so we will mostly

610
00:44:18,100 --> 00:44:20,590
just consider the
semi classical gravity

611
00:44:20,590 --> 00:44:23,580
regime in the gravity side.

612
00:44:23,580 --> 00:44:26,240
Because essentially,
we know very little

613
00:44:26,240 --> 00:44:31,090
about the string theory
in this geometry.

614
00:44:34,300 --> 00:44:38,580
And also, I will
often use the phrase

615
00:44:38,580 --> 00:44:41,170
which applies to the
general correspondence,

616
00:44:41,170 --> 00:44:44,750
and not necessarily just N equal
to 4 super Yang-Mills theory

617
00:44:44,750 --> 00:44:48,504
and the type IIB string theory.

618
00:44:48,504 --> 00:44:50,170
I just use the language
assuming there's

619
00:44:50,170 --> 00:44:53,690
a general correspondence between
some conformal field theory

620
00:44:53,690 --> 00:44:55,415
and some AdS gravity theory.

621
00:45:02,570 --> 00:45:06,320
So if the two theories
have to be the same,

622
00:45:06,320 --> 00:45:08,770
two sides have to
be the same, then

623
00:45:08,770 --> 00:45:10,962
you should be able to
map their spectrum.

624
00:45:10,962 --> 00:45:13,170
So you should be able to
map their [INAUDIBLE] space,

625
00:45:13,170 --> 00:45:13,731
for example.

626
00:45:17,310 --> 00:45:22,180
So again, [INAUDIBLE] now
use the general language

627
00:45:22,180 --> 00:45:24,032
of the boundary and
the [INAUDIBLE].

628
00:45:46,130 --> 00:45:50,220
So the boundary theory
is a conformal theory

629
00:45:50,220 --> 00:45:51,648
with this symmetry.

630
00:45:54,550 --> 00:45:58,270
So for the moment, I don't
have [INAUDIBLE] symmetry.

631
00:45:58,270 --> 00:46:08,900
So that means that [INAUDIBLE]
space should be organized

632
00:46:08,900 --> 00:46:21,436
as the representations
of the conformal group,

633
00:46:21,436 --> 00:46:22,310
say of this SO(d, 2).

634
00:46:27,380 --> 00:46:31,800
And similarly here,
because here, the SO(d,

635
00:46:31,800 --> 00:46:37,400
2) is the isometry group which
lives in infinite invariants.

636
00:46:37,400 --> 00:46:41,490
And again, you should be able to
organize your [INAUDIBLE] space

637
00:46:41,490 --> 00:46:45,940
using the representations
of the SO(d, 2).

638
00:46:45,940 --> 00:46:50,020
And those representations, of
course, should be the same.

639
00:46:50,020 --> 00:46:52,150
So if there's one
representation here,

640
00:46:52,150 --> 00:46:54,360
then there must be a
[INAUDIBLE] representation here.

641
00:46:54,360 --> 00:46:57,400
And if there's two [INAUDIBLE]
representations here,

642
00:46:57,400 --> 00:47:00,920
there would be two identical
representations here.

643
00:47:00,920 --> 00:47:03,550
The representations must match.

644
00:47:03,550 --> 00:47:13,690
And on the boundary
side, the local operators

645
00:47:13,690 --> 00:47:16,120
should also transform
on the representations

646
00:47:16,120 --> 00:47:18,660
of the conformal symmetry.

647
00:47:18,660 --> 00:47:29,920
And again, they conform
local operators can also

648
00:47:29,920 --> 00:47:34,980
be mapped to, say, the
field on the gravity side.

649
00:47:34,980 --> 00:47:46,052
So on the gravity side, the
fields should also transform

650
00:47:46,052 --> 00:47:50,510
under this SO(d, 2) isometry.

651
00:47:50,510 --> 00:47:52,590
That means that it
should be a one to one

652
00:47:52,590 --> 00:47:56,290
correspondence between
the local operators here

653
00:47:56,290 --> 00:48:00,800
and the bulk fields
on the gravity side.

654
00:48:00,800 --> 00:48:04,570
There should be a one to one
correspondence on this side.

655
00:48:04,570 --> 00:48:13,540
For example, in the boundary, if
there is some scalar operator,

656
00:48:13,540 --> 00:48:15,850
there must be a
corresponding scalar

657
00:48:15,850 --> 00:48:17,348
field on the gravity side.

658
00:48:21,820 --> 00:48:25,940
Similarly, if there is some
vector operator in the boundary

659
00:48:25,940 --> 00:48:30,430
theory, there must be a
corresponding bulk vector

660
00:48:30,430 --> 00:48:34,190
field in the gravity side.

661
00:48:34,190 --> 00:48:37,730
And similarly, if you have
some symmetric tensor,

662
00:48:37,730 --> 00:48:42,000
then this must also be related
to some symmetric tensor.

663
00:48:42,000 --> 00:48:44,470
So now I will often use the
notation that I've used.

664
00:48:44,470 --> 00:48:47,820
Mu mu refers to the
boundary indices.

665
00:48:47,820 --> 00:48:53,080
And the capital M, capital N
refers to the bulk indices.

666
00:48:53,080 --> 00:48:55,980
Because they are on
different dimensions.

667
00:48:55,980 --> 00:48:58,165
So then they are
not quite the same.

668
00:49:00,709 --> 00:49:01,875
AUDIENCE: I have a question.

669
00:49:01,875 --> 00:49:03,510
HONG LIU: One second.

670
00:49:03,510 --> 00:49:05,440
Let me just finish.

671
00:49:05,440 --> 00:49:08,780
So here, I'm just talking
about the conformal symmetries.

672
00:49:08,780 --> 00:49:11,890
And if the theory has
some other symmetries,

673
00:49:11,890 --> 00:49:18,530
say some global symmetries or
supersymmetries, then again,

674
00:49:18,530 --> 00:49:21,330
all those fields
and the state, they

675
00:49:21,330 --> 00:49:23,760
should transform on
the representations

676
00:49:23,760 --> 00:49:24,820
of those symmetries.

677
00:49:24,820 --> 00:49:27,600
And again, they should
all match together.

678
00:49:27,600 --> 00:49:30,240
They should all match together.

679
00:49:30,240 --> 00:49:31,230
Yes?

680
00:49:31,230 --> 00:49:40,430
AUDIENCE: So we proved that the
super Yang-Mills theory really

681
00:49:40,430 --> 00:49:42,729
lives on the
boundary of the AdS?

682
00:49:42,729 --> 00:49:43,270
HONG LIU: No.

683
00:49:43,270 --> 00:49:44,910
We did not prove that.

684
00:49:44,910 --> 00:49:46,010
This is just a postulate.

685
00:49:50,580 --> 00:49:54,087
First, let me just repeat again.

686
00:49:56,950 --> 00:50:01,350
Yang-Mills theory lives
on Minkowski space.

687
00:50:01,350 --> 00:50:05,160
And it's just an observation
that the Minkowski space

688
00:50:05,160 --> 00:50:08,010
is the boundary of AdS.

689
00:50:08,010 --> 00:50:13,530
And then you say you can imagine
that this is the boundary,

690
00:50:13,530 --> 00:50:16,700
this relation is related to
the bulk and the boundary.

691
00:50:16,700 --> 00:50:18,916
And this is a postulate
based on that fact.

692
00:50:23,055 --> 00:50:23,554
Yes?

693
00:50:26,220 --> 00:50:29,354
AUDIENCE: I thought one of
the motivations for thinking

694
00:50:29,354 --> 00:50:31,020
about the holographic
duality was to try

695
00:50:31,020 --> 00:50:32,860
to escape [INAUDIBLE] theorem.

696
00:50:35,596 --> 00:50:37,910
And all of a sudden,
it strikes me,

697
00:50:37,910 --> 00:50:42,544
so we're trying to get
on the boundary spin

698
00:50:42,544 --> 00:50:43,460
to massless particles.

699
00:50:46,360 --> 00:50:49,015
Then they will also
exist in the bulk.

700
00:50:49,015 --> 00:50:49,681
HONG LIU: Sorry.

701
00:50:51,875 --> 00:50:53,250
In the field theory
side, there's

702
00:50:53,250 --> 00:50:54,644
no massless spin to particles.

703
00:50:57,490 --> 00:50:59,740
They map to some field in
the gravity side in the five

704
00:50:59,740 --> 00:51:00,850
dimensions.

705
00:51:00,850 --> 00:51:03,990
So this is some four
dimensional operator.

706
00:51:03,990 --> 00:51:06,913
Then this maps to some
five dimensional fields.

707
00:51:06,913 --> 00:51:09,680
AUDIENCE: I need to think
about my question better.

708
00:51:13,520 --> 00:51:14,172
HONG LIU: Yes?

709
00:51:14,172 --> 00:51:16,130
AUDIENCE: So if it's a
postulate that-- I mean,

710
00:51:16,130 --> 00:51:20,797
it's not required that the
theory live on the boundary.

711
00:51:20,797 --> 00:51:23,380
It seems that that's just sort
of a convenient way of thinking

712
00:51:23,380 --> 00:51:24,070
about it.

713
00:51:24,070 --> 00:51:26,784
But what implications would
it have if it actually

714
00:51:26,784 --> 00:51:27,700
lived on the boundary?

715
00:51:27,700 --> 00:51:29,360
Would it change anything?

716
00:51:29,360 --> 00:51:31,650
HONG LIU: That's
what I said before.

717
00:51:31,650 --> 00:51:35,890
So this discussion does
not depend on that.

718
00:51:35,890 --> 00:51:39,190
This discussion does
not depend on that.

719
00:51:39,190 --> 00:51:42,410
You can say I don't need to
worry about that whether this

720
00:51:42,410 --> 00:51:44,790
is on the boundary
or bulk, et cetera.

721
00:51:44,790 --> 00:51:46,950
These are just some
different theories.

722
00:51:46,950 --> 00:51:48,229
I want them to be the same.

723
00:51:48,229 --> 00:51:48,895
AUDIENCE: Right.

724
00:51:48,895 --> 00:51:50,200
OK.

725
00:51:50,200 --> 00:51:53,840
HONG LIU: But, as I said last
time, if you believe this bulk

726
00:51:53,840 --> 00:51:58,370
and boundary relation,
then this is powerful

727
00:51:58,370 --> 00:52:00,220
because then, you
can immediately

728
00:52:00,220 --> 00:52:06,710
deduce that the Yang-Mills
theory on the S3 times time

729
00:52:06,710 --> 00:52:10,760
then is related to the gravity
theory in the global AdS.

730
00:52:10,760 --> 00:52:14,090
And that is a
nontrivial prediction

731
00:52:14,090 --> 00:52:18,090
from thinking the boundary
and the bulk relation.

732
00:52:18,090 --> 00:52:21,960
Now you can generalize it.

733
00:52:21,960 --> 00:52:23,780
Because if I just
look at this relation,

734
00:52:23,780 --> 00:52:27,630
I have no reason to suspect why.

735
00:52:27,630 --> 00:52:30,460
You can argue from the
symmetry point of view also,

736
00:52:30,460 --> 00:52:33,270
but that language
is more direct.

737
00:52:40,484 --> 00:52:44,820
In principle, after
establishing the relation

738
00:52:44,820 --> 00:52:47,530
between the Yang-Mills theory
in Minkowski space time

739
00:52:47,530 --> 00:52:51,940
and this gravity in
the Poincare patch,

740
00:52:51,940 --> 00:52:56,380
I may also, just based on
the group theory aspect,

741
00:52:56,380 --> 00:53:01,540
to say that should generalize
to the Yang-Mills theory

742
00:53:01,540 --> 00:53:04,920
on the S sphere
and the global AdS.

743
00:53:04,920 --> 00:53:07,620
You may also be able
to do it that way.

744
00:53:07,620 --> 00:53:10,140
But thinking from the
holographic point of view

745
00:53:10,140 --> 00:53:14,580
will give you a direct
way to argue that,

746
00:53:14,580 --> 00:53:17,035
give you an alternative
way to argue that.

747
00:53:20,320 --> 00:53:24,080
Also, in your p-set, you have
checked this holographic bound.

748
00:53:24,080 --> 00:53:27,230
And so that's a
confirmation of this.

749
00:53:27,230 --> 00:53:27,870
Yes?

750
00:53:27,870 --> 00:53:29,630
AUDIENCE: So what does the
massless [INAUDIBLE] field

751
00:53:29,630 --> 00:53:32,255
on the right map to if it's the
same representation of SO(d,

752
00:53:32,255 --> 00:53:33,824
2)?

753
00:53:33,824 --> 00:53:34,490
HONG LIU: Sorry.

754
00:53:34,490 --> 00:53:35,320
Say it again.

755
00:53:35,320 --> 00:53:38,372
AUDIENCE: So in the
representation line,

756
00:53:38,372 --> 00:53:40,830
so if you have the massless
[INAUDIBLE] field on the right,

757
00:53:40,830 --> 00:53:43,185
what does it map to on the left?

758
00:53:43,185 --> 00:53:43,810
HONG LIU: Yeah.

759
00:53:43,810 --> 00:53:45,510
We are going to talk about it.

760
00:53:45,510 --> 00:53:46,837
We are going to talk about it.

761
00:53:57,056 --> 00:53:58,430
So if there are
other symmetries,

762
00:53:58,430 --> 00:54:01,260
then everything should match.

763
00:54:01,260 --> 00:54:04,590
So now, as an
immediate check, you

764
00:54:04,590 --> 00:54:09,180
can now just open
your old papers.

765
00:54:09,180 --> 00:54:12,670
And if you load them--
because in the '80s,

766
00:54:12,670 --> 00:54:16,520
people have worked out
the supergravity spectrum

767
00:54:16,520 --> 00:54:20,110
precisely on this space.

768
00:54:20,110 --> 00:54:25,530
So even though this relation
was discovered in '97, actually

769
00:54:25,530 --> 00:54:28,730
in the '80s, people
already worked a lot

770
00:54:28,730 --> 00:54:32,170
to consider this type to be
supergravity on this space.

771
00:54:32,170 --> 00:54:34,320
Because this is a maximally
supersymmetric space,

772
00:54:34,320 --> 00:54:35,170
et cetera.

773
00:54:35,170 --> 00:54:39,955
Anyway, so people have
already spent lots of trouble

774
00:54:39,955 --> 00:54:42,920
to actually work
out that spectrum.

775
00:54:42,920 --> 00:54:46,010
And now you can just
open their paper.

776
00:54:46,010 --> 00:54:50,110
Then there's a big table
about the different fields

777
00:54:50,110 --> 00:54:52,630
and their representations,
et cetera.

778
00:54:52,630 --> 00:54:55,710
And then you can immediately
see they actually

779
00:54:55,710 --> 00:55:02,550
map to certain representation
of operators on the N equal to 4

780
00:55:02,550 --> 00:55:03,550
super Yang-Mills theory.

781
00:55:03,550 --> 00:55:05,049
Then you can
immediately check them.

782
00:55:07,540 --> 00:55:12,077
Just based on the group
theory, check them.

783
00:55:15,080 --> 00:55:16,980
So I won't go through
those details.

784
00:55:16,980 --> 00:55:28,670
But let me just mention the most
important such kind of mapping

785
00:55:28,670 --> 00:55:30,300
for these two theories.

786
00:55:30,300 --> 00:55:33,930
And actually does
have consequences

787
00:55:33,930 --> 00:55:39,140
for the general story.

788
00:55:42,070 --> 00:55:50,730
So the most important mapping.

789
00:55:50,730 --> 00:55:52,460
So as we said, on the
string theory side,

790
00:55:52,460 --> 00:55:53,668
you always have this dilaton.

791
00:55:56,010 --> 00:55:58,920
So these is some scalar field.

792
00:55:58,920 --> 00:56:02,740
And this dilaton will appear
in the scalar field, in AdS5,

793
00:56:02,740 --> 00:56:05,260
say, for example.

794
00:56:05,260 --> 00:56:06,870
And it turns out
this dilaton field

795
00:56:06,870 --> 00:56:09,470
is mapped to the Lagrangian
of the N equal to 4

796
00:56:09,470 --> 00:56:10,470
super Yang-Mills theory.

797
00:56:19,510 --> 00:56:22,120
So the Lagrangian
is a Lagrangian

798
00:56:22,120 --> 00:56:24,140
of the N equals to 4
super Yang-Mills theory,

799
00:56:24,140 --> 00:56:27,670
say, trace f squared plus
phi squared, et cetera.

800
00:56:27,670 --> 00:56:29,566
So that's a local operator.

801
00:56:29,566 --> 00:56:31,940
And it turns out that operator
is mapped to this dilaton.

802
00:56:35,812 --> 00:56:38,020
And on the N equal to 4
super Yang-Mills theory side,

803
00:56:38,020 --> 00:56:40,870
we have this SO6 gauge symmetry.

804
00:56:40,870 --> 00:56:43,740
Did I just erase them?

805
00:56:43,740 --> 00:56:46,130
We have this SO6 gauge symmetry.

806
00:56:46,130 --> 00:56:53,989
Then we have the SO6
conserved current.

807
00:56:53,989 --> 00:56:55,780
And it turns out this,
on the gravity side,

808
00:56:55,780 --> 00:57:00,310
just maps to sO6 gauge field.

809
00:57:00,310 --> 00:57:02,841
AUDIENCE: You said it was
an SO6 gauge symmetry?

810
00:57:02,841 --> 00:57:03,382
HONG LIU: No.

811
00:57:03,382 --> 00:57:06,790
In the N equal to 4, this
is a global symmetry.

812
00:57:06,790 --> 00:57:12,060
But on this side, so this
appears as isometry on the S5.

813
00:57:12,060 --> 00:57:16,790
But when you dimensionally
reduce on S5, then in AdS5,

814
00:57:16,790 --> 00:57:18,610
then there will be
a pure gauge field

815
00:57:18,610 --> 00:57:23,890
corresponding to each symmetry
generated [INAUDIBLE] S5.

816
00:57:23,890 --> 00:57:28,390
And that naturally maps
to this conserved current

817
00:57:28,390 --> 00:57:31,700
on the field theory side.

818
00:57:31,700 --> 00:57:35,000
And then another universal
operator on the field theory

819
00:57:35,000 --> 00:57:36,787
side is the stress tensor.

820
00:57:36,787 --> 00:57:38,370
So no matter what
theory you consider,

821
00:57:38,370 --> 00:57:39,590
you have stress tensor.

822
00:57:44,010 --> 00:57:47,715
And this is mapped
to-- turns out,

823
00:57:47,715 --> 00:57:48,881
to the metric perturbations.

824
00:57:57,140 --> 00:57:59,051
It's a deviation
from the AdS metric.

825
00:58:04,910 --> 00:58:11,080
Again, so at the representation
level, this is very natural.

826
00:58:11,080 --> 00:58:12,330
So this is a symmetric tensor.

827
00:58:12,330 --> 00:58:13,462
This is a symmetric tensor.

828
00:58:15,970 --> 00:58:18,649
But physically, this
is also natural.

829
00:58:18,649 --> 00:58:19,690
Physically, also natural.

830
00:58:19,690 --> 00:58:25,190
I will elaborate this
a little bit further

831
00:58:25,190 --> 00:58:32,460
from a different perspective
in a few minutes.

832
00:58:37,442 --> 00:58:39,275
So do you have any
questions regarding this?

833
00:58:45,590 --> 00:58:48,810
So now, given this
mapping, any operator

834
00:58:48,810 --> 00:58:51,570
is due to a bulk field.

835
00:58:51,570 --> 00:58:55,360
Then you can ask some
immediate questions.

836
00:58:55,360 --> 00:59:01,510
For example, the quantum
numbers of these operators

837
00:59:01,510 --> 00:59:04,290
will map to the quantum
numbers of the bulk fields.

838
00:59:04,290 --> 00:59:09,160
And that's something I said
you can check their symmetries.

839
00:59:09,160 --> 00:59:12,950
For example, not only
the representations

840
00:59:12,950 --> 00:59:14,450
under this conformal
symmetry should

841
00:59:14,450 --> 00:59:16,690
map under that representation.

842
00:59:16,690 --> 00:59:22,860
And also, the
representation under that

843
00:59:22,860 --> 00:59:24,430
SO6 we also match, et cetera.

844
00:59:28,700 --> 00:59:34,161
So for local operator on
the field theory side,

845
00:59:34,161 --> 00:59:36,160
so once we have this
mapping, we can immediately

846
00:59:36,160 --> 00:59:42,580
ask questions related to
operators on this side,

847
00:59:42,580 --> 00:59:45,250
and try to ask what's the
counterpart on the other side.

848
00:59:45,250 --> 00:59:49,510
And ask the story about the
field on the gravity side,

849
00:59:49,510 --> 00:59:51,320
and ask what's
happened on this side.

850
00:59:51,320 --> 00:59:55,610
We can start developing
the relations.

851
00:59:55,610 --> 00:59:59,120
So now let's start
discussing them.

852
00:59:59,120 --> 01:00:02,270
So the first thing you can
do, so immediate question

853
01:00:02,270 --> 01:00:08,710
you can ask is that
given the operator,

854
01:00:08,710 --> 01:00:14,370
say operator O in
the field theory.

855
01:00:21,912 --> 01:00:23,370
So there are some
natural questions

856
01:00:23,370 --> 01:00:25,250
one can raise about
this operator.

857
01:00:25,250 --> 01:00:28,360
For example, what is correlation
functions, et cetera?

858
01:00:28,360 --> 01:00:30,000
And that we will
discuss a bit later.

859
01:00:30,000 --> 01:00:34,940
But now let me discuss
another natural question.

860
01:00:34,940 --> 01:01:00,270
And the natural thing to do is
to deform your original theory

861
01:01:00,270 --> 01:01:05,040
by adding this operator
to the Lagrangian.

862
01:01:15,600 --> 01:01:17,210
So I have an operator.

863
01:01:17,210 --> 01:01:18,240
I can put the source.

864
01:01:18,240 --> 01:01:19,870
I can put the coefficients.

865
01:01:19,870 --> 01:01:22,090
And these coefficients
may depend or not

866
01:01:22,090 --> 01:01:25,010
depend on the space
time coordinates.

867
01:01:25,010 --> 01:01:28,060
And I can always add such
a term to a Lagrangian.

868
01:01:28,060 --> 01:01:31,140
And then this deforms my theory
away from the original theory.

869
01:01:31,140 --> 01:01:32,990
So this is a
natural thing to do.

870
01:01:32,990 --> 01:01:34,902
And this phi 0 is often
called the source.

871
01:01:42,510 --> 01:01:44,076
So this phi 0 is
called the source.

872
01:01:54,560 --> 01:01:57,970
So immediate question
you can ask--

873
01:01:57,970 --> 01:02:05,430
and when phi 0 is
equal to constant,

874
01:02:05,430 --> 01:02:18,280
then this corresponding to
a change in the coupling

875
01:02:18,280 --> 01:02:20,050
for this operator.

876
01:02:24,760 --> 01:02:29,660
So if your Lagrangian previously
already included this operator,

877
01:02:29,660 --> 01:02:32,770
and then if you add such a term,
so phi 0 equal to constant,

878
01:02:32,770 --> 01:02:35,990
then you are just changing the
coupling for that operator.

879
01:02:35,990 --> 01:02:38,230
And if this operator
is not there,

880
01:02:38,230 --> 01:02:42,337
then you just add
the new coupling.

881
01:02:42,337 --> 01:02:44,545
But in general, you can make
it space time dependent.

882
01:02:49,570 --> 01:02:51,470
AUDIENCE: Why is it
coupling since there's

883
01:02:51,470 --> 01:02:55,180
no other operators
coupling without if we

884
01:02:55,180 --> 01:02:57,829
say phi 0 is a constant?

885
01:02:57,829 --> 01:02:58,495
HONG LIU: Sorry?

886
01:02:58,495 --> 01:03:00,000
AUDIENCE: If phi
0 is a constant?

887
01:03:00,000 --> 01:03:00,200
HONG LIU: No.

888
01:03:00,200 --> 01:03:01,090
O is the operator.

889
01:03:01,090 --> 01:03:02,358
It's the local operator.

890
01:03:02,358 --> 01:03:06,820
AUDIENCE: Yes, but there
is no other operator

891
01:03:06,820 --> 01:03:09,920
this operator O coupling with.

892
01:03:09,920 --> 01:03:12,350
HONG LIU: No, no, no.

893
01:03:12,350 --> 01:03:14,410
What is the meaning of operator?

894
01:03:14,410 --> 01:03:20,980
The meaning of
operator is that O is

895
01:03:20,980 --> 01:03:24,630
a sum of the product of fields.

896
01:03:24,630 --> 01:03:28,010
And so there's already--
so this O may be phi cubed,

897
01:03:28,010 --> 01:03:29,360
or may be phi 4, et cetera.

898
01:03:34,520 --> 01:03:37,610
So example is O in the
scalar field theory.

899
01:03:42,040 --> 01:03:45,610
If a scalar field theory
has a gauge field,

900
01:03:45,610 --> 01:03:49,000
then I can write O
as trace F squared.

901
01:03:49,000 --> 01:03:52,080
Then this corresponding
to changing the coupling

902
01:03:52,080 --> 01:03:53,780
for trace F squared.

903
01:03:59,010 --> 01:04:01,674
So immediate question is,
what does this operation-- so

904
01:04:01,674 --> 01:04:03,090
in the field theory
point of view,

905
01:04:03,090 --> 01:04:05,489
you can always do
this operation.

906
01:04:05,489 --> 01:04:07,780
So the immediate question,
what does this corresponding

907
01:04:07,780 --> 01:04:08,524
to in the bulk?

908
01:04:13,610 --> 01:04:27,985
What does this operation
correspond to in the bulk?

909
01:04:36,080 --> 01:04:39,800
So now I'll try to
answer this question.

910
01:04:39,800 --> 01:04:45,250
As I said before, in
establishing such dictionary,

911
01:04:45,250 --> 01:04:49,380
you often have to do lots of
guesswork, and then you check.

912
01:04:49,380 --> 01:04:55,870
And the guesswork is based
on some very small clues.

913
01:04:55,870 --> 01:05:00,120
So people who are able
to make good guesses

914
01:05:00,120 --> 01:05:04,790
is that they are able to grasp
important clues sometimes

915
01:05:04,790 --> 01:05:05,885
from very simple facts.

916
01:05:08,750 --> 01:05:11,150
And so that is what's
good physicists do

917
01:05:11,150 --> 01:05:14,230
is that they can see
non-ordinary things

918
01:05:14,230 --> 01:05:16,530
from ordinary things.

919
01:05:16,530 --> 01:05:27,156
So here, I will try to deduce
the answer to this question

920
01:05:27,156 --> 01:05:28,447
by starting from this relation.

921
01:05:35,330 --> 01:05:37,281
So let me start
with that relation.

922
01:05:46,520 --> 01:05:49,520
Related to the GS.

923
01:05:49,520 --> 01:05:52,940
Let's forget about p for pi.

924
01:05:52,940 --> 01:05:56,890
And now, we've talked about
before GS string coupling can

925
01:05:56,890 --> 01:06:01,310
be considered as the expectation
value of the dilaton field.

926
01:06:01,310 --> 01:06:02,858
Let me actually
call it capital Phi.

927
01:06:07,140 --> 01:06:09,694
So we mentioned before when
we talked about string theory

928
01:06:09,694 --> 01:06:12,110
that this string coupling can
be considered as expectation

929
01:06:12,110 --> 01:06:15,278
value of the dilaton field.

930
01:06:19,904 --> 01:06:21,570
Of course in general,
again, the dilaton

931
01:06:21,570 --> 01:06:22,611
can fluctuate, et cetera.

932
01:06:25,680 --> 01:06:27,300
So normally, in
flat space, when we

933
01:06:27,300 --> 01:06:31,950
say the dilaton has expectation
value, what we normally

934
01:06:31,950 --> 01:06:35,760
say is that the value of
the dilaton at infinity

935
01:06:35,760 --> 01:06:40,340
because the boundary by
infinity, that's how you fix.

936
01:06:45,000 --> 01:06:51,160
And similarly, for space time
like AdS with a boundary,

937
01:06:51,160 --> 01:06:55,300
and the expectation
value of this dilaton

938
01:06:55,300 --> 01:06:58,620
can be identified with
the value of the dilaton

939
01:06:58,620 --> 01:07:00,310
at the boundary of AdS.

940
01:07:03,010 --> 01:07:06,345
So here, I write partial AdS
means the boundary of AdS.

941
01:07:10,100 --> 01:07:14,816
So this is the value of AdS,
value of phi at the boundary.

942
01:07:22,650 --> 01:07:24,585
So expectation value
essentially can

943
01:07:24,585 --> 01:07:26,710
be associated with the
boundary value of the field.

944
01:07:29,900 --> 01:07:32,410
AUDIENCE: You mean
if it is a constant?

945
01:07:32,410 --> 01:07:36,490
You mean if expectation
value of the phi is constant?

946
01:07:36,490 --> 01:07:40,778
Or if it's, function-wise,
equal to the boundary limit?

947
01:07:40,778 --> 01:07:42,210
HONG LIU: No, no, no.

948
01:07:42,210 --> 01:07:43,990
Think about the following thing.

949
01:07:43,990 --> 01:07:49,340
As I said, phi in
principle can fluctuate.

950
01:07:49,340 --> 01:07:55,040
And its expectation may also be
able to fluctuate in the space

951
01:07:55,040 --> 01:07:55,970
time.

952
01:07:55,970 --> 01:07:59,670
But the only sensible way to
talk about expectation value

953
01:07:59,670 --> 01:08:01,640
is to talk about its
value at the boundary

954
01:08:01,640 --> 01:08:05,460
because we assume the boundary
conditions don't fluctuate.

955
01:08:05,460 --> 01:08:07,357
It's the same thing
for flat space.

956
01:08:07,357 --> 01:08:09,190
And we always specify
the boundary condition

957
01:08:09,190 --> 01:08:11,030
at the spatial
infinity, for example.

958
01:08:11,030 --> 01:08:14,600
And in the AdS, which is also
space time with a boundary,

959
01:08:14,600 --> 01:08:19,240
than we can associate
the constant parts

960
01:08:19,240 --> 01:08:23,217
of the expectation value as
the value at the boundary.

961
01:08:34,569 --> 01:08:37,230
Now you have to use a little
bit stretch of imagination.

962
01:08:41,020 --> 01:08:43,310
The following fact,
you don't need to.

963
01:08:43,310 --> 01:08:47,420
So now, with this relation, we
have established a connection

964
01:08:47,420 --> 01:08:52,290
between the Yang-Mills theory
coupling with the value of phi

965
01:08:52,290 --> 01:08:54,116
at the boundary of AdS.

966
01:09:01,439 --> 01:09:07,229
And now remember, this
Yang-Mills coupling,

967
01:09:07,229 --> 01:09:10,330
so I will not be careful about
1 over Yang-Mills squared.

968
01:09:10,330 --> 01:09:11,550
This is related.

969
01:09:11,550 --> 01:09:20,832
This is the coupling or source
for the Lagrangian of N equals

970
01:09:20,832 --> 01:09:22,040
to 4 super Yang-Mills theory.

971
01:09:40,750 --> 01:09:47,580
So now we can deduce
here two things.

972
01:09:47,580 --> 01:09:58,170
In particular, if we
deform the Lagrangian,

973
01:09:58,170 --> 01:10:06,160
say, change this coupling,
deform the boundary

974
01:10:06,160 --> 01:10:08,646
by changing the coupling,
which corresponds

975
01:10:08,646 --> 01:10:12,290
to you add some delta G,
say the Lagrangian of N

976
01:10:12,290 --> 01:10:16,650
equal to 4 super Yang-Mills,
deform the boundary

977
01:10:16,650 --> 01:10:18,440
set by changing the coupling.

978
01:10:18,440 --> 01:10:21,650
And this corresponding
to you essentially

979
01:10:21,650 --> 01:10:35,637
change the boundary
value of dilaton

980
01:10:35,637 --> 01:10:36,720
because these are related.

981
01:10:40,790 --> 01:10:45,320
So now, this example gives us
the answer to this question.

982
01:10:51,600 --> 01:10:58,065
This example gives
us this question.

983
01:11:01,540 --> 01:11:12,600
From here, we can deduce first
that the operator corresponding

984
01:11:12,600 --> 01:11:15,620
to the dilaton must be the N
equal to 4 super Yang-Mills

985
01:11:15,620 --> 01:11:21,690
Lagrangian, as you can already
deduce from some other methods,

986
01:11:21,690 --> 01:11:23,670
say using group
theory, et cetera.

987
01:11:27,300 --> 01:11:36,980
And the second is that phi
0 O, in the boundary theory,

988
01:11:36,980 --> 01:11:43,430
adding a phi 0 O in
the boundary theory

989
01:11:43,430 --> 01:11:48,250
must be related-- now I'm
generalizing this story.

990
01:11:48,250 --> 01:11:52,530
The bulk field phi-- so
now I'm no longer not

991
01:11:52,530 --> 01:11:54,290
necessarily using the dilaton.

992
01:11:54,290 --> 01:12:09,170
The bulk field phi due to O
has a boundary value phi 0.

993
01:12:18,600 --> 01:12:20,170
Any questions about this?

994
01:12:25,380 --> 01:12:25,880
Yes?

995
01:12:25,880 --> 01:12:27,416
AUDIENCE: Your
choice of operator O

996
01:12:27,416 --> 01:12:29,416
should be consistent with
all symmetries, right?

997
01:12:29,416 --> 01:12:30,326
HONG LIU: Hm?

998
01:12:30,326 --> 01:12:31,826
AUDIENCE: You're
not pre-choosing O?

999
01:12:31,826 --> 01:12:32,367
HONG LIU: No.

1000
01:12:32,367 --> 01:12:34,035
O can be anything.

1001
01:12:34,035 --> 01:12:36,284
AUDIENCE: Consistent with
original symmetries of the--

1002
01:12:36,284 --> 01:12:37,784
HONG LIU: No, it
doesn't have to be.

1003
01:12:37,784 --> 01:12:40,200
You can deform the
theory without-- you

1004
01:12:40,200 --> 01:12:41,930
can break the symmetry.

1005
01:12:41,930 --> 01:12:44,150
That deformation can
break the symmetry.

1006
01:12:44,150 --> 01:12:45,650
If you want to
preserve the symmetry

1007
01:12:45,650 --> 01:12:49,390
of the original Lagrangian,
then you can choose a certain O.

1008
01:12:49,390 --> 01:12:51,910
But in principle, you
can choose any O. You can

1009
01:12:51,910 --> 01:12:54,510
break the symmetry if you want.

1010
01:12:54,510 --> 01:12:57,206
AUDIENCE: Do you have
to always then break

1011
01:12:57,206 --> 01:12:59,485
some symmetry on the AdS part?

1012
01:12:59,485 --> 01:13:00,110
HONG LIU: Yeah.

1013
01:13:02,660 --> 01:13:05,170
Impose this kind of
boundary conditions.

1014
01:13:05,170 --> 01:13:09,260
Then you may break
AdS symmetry too.

1015
01:13:09,260 --> 01:13:11,470
So now we have
answered this question.

1016
01:13:11,470 --> 01:13:15,360
And then of course, this
just provides the answer.

1017
01:13:15,360 --> 01:13:17,190
Then you would check it.

1018
01:13:17,190 --> 01:13:20,180
And we will check it
later using other methods.

1019
01:13:20,180 --> 01:13:23,250
Now I'm saying because I
already know it's true.

1020
01:13:23,250 --> 01:13:26,820
But in real life, what you
will do from this example,

1021
01:13:26,820 --> 01:13:28,920
you will say, ah,
this must be the case.

1022
01:13:28,920 --> 01:13:32,635
Then you will start trying
to find examples to check it.

1023
01:13:32,635 --> 01:13:35,530
Start to find other
ways to check it.

1024
01:13:35,530 --> 01:13:38,160
And we will describe it later.

1025
01:13:38,160 --> 01:13:40,690
So in the end, what
we will describe

1026
01:13:40,690 --> 01:13:41,890
is a self consistent story.

1027
01:13:41,890 --> 01:13:47,120
I will not contradict myself
in my later discussion.

1028
01:13:47,120 --> 01:14:02,195
So if we assume this, I
can also use this to argue.

1029
01:14:13,510 --> 01:14:21,880
I can also use
this identification

1030
01:14:21,880 --> 01:14:23,930
to argue this too.

1031
01:14:23,930 --> 01:14:30,060
So let me call this
star and star star.

1032
01:14:30,060 --> 01:14:40,990
I can use this identification
to make star and star star

1033
01:14:40,990 --> 01:14:50,875
natural for any
duality, not only N

1034
01:14:50,875 --> 01:14:56,555
equals 4 super Yang-Mills theory
and this type IIB gravity.

1035
01:14:59,830 --> 01:15:02,210
It's that any conserved
curve in the boundary theory

1036
01:15:02,210 --> 01:15:06,040
must be equal to some gauge
field in the gravity side,

1037
01:15:06,040 --> 01:15:10,630
and the stress tensor should
always be due to the metric.

1038
01:15:10,630 --> 01:15:15,530
So now I'm going to
use this argument

1039
01:15:15,530 --> 01:15:17,190
to make that a little
bit more natural.

1040
01:15:23,060 --> 01:15:27,080
So let's first consider this.

1041
01:15:27,080 --> 01:15:35,810
So suppose I have a conserved
current, J, J. I have a J mu.

1042
01:15:38,430 --> 01:15:41,550
For simplicity, let me
just take it to be u(1).

1043
01:15:41,550 --> 01:15:44,410
Then I can deform
the boundary theory

1044
01:15:44,410 --> 01:15:50,550
by adding a source for this J mu
by adding a boundary Lagrangian

1045
01:15:50,550 --> 01:15:51,860
term like this.

1046
01:15:51,860 --> 01:15:52,835
And a mu is the source.

1047
01:15:58,890 --> 01:16:03,670
And then according to
this identification,

1048
01:16:03,670 --> 01:16:11,570
the A mu must be A mu-- we
should be able to identify it

1049
01:16:11,570 --> 01:16:15,470
as the boundary value of some
bulk field, A mu evaluated

1050
01:16:15,470 --> 01:16:16,974
at z equal to 0.

1051
01:16:20,790 --> 01:16:22,437
So this is a bulk vector field.

1052
01:16:28,310 --> 01:16:32,050
So if this is true, there
must exist a vector field

1053
01:16:32,050 --> 01:16:34,480
due to this vector field.

1054
01:16:34,480 --> 01:16:38,170
And this must correspond to
the boundary value of that.

1055
01:16:38,170 --> 01:16:42,630
But now we can argue why
this should be a gauge field.

1056
01:16:42,630 --> 01:16:45,430
So now I want to argue
this is a gauge field.

1057
01:16:45,430 --> 01:16:49,660
And to see this is very
easy because since J mu is

1058
01:16:49,660 --> 01:16:53,820
conserved, then this
coupling-- so let

1059
01:16:53,820 --> 01:16:57,500
me call this star star star.

1060
01:16:57,500 --> 01:17:03,590
Let me call it star star star.

1061
01:17:03,590 --> 01:17:10,700
So since this is conserved,
and this star star star

1062
01:17:10,700 --> 01:17:18,210
is invariant under the
following transformation,

1063
01:17:18,210 --> 01:17:24,940
A mu goes to A mu plus partial
mu lambda x for any lambda x.

1064
01:17:35,050 --> 01:17:42,530
So that means the A mu, this
A is the boundary value of A.

1065
01:17:42,530 --> 01:17:46,290
So that means this
capital A, that

1066
01:17:46,290 --> 01:17:55,090
means the dynamics of this A
mu should also be invariant.

1067
01:18:01,740 --> 01:18:04,520
If I change the boundary value
of A by this transformation,

1068
01:18:04,520 --> 01:18:07,210
the dynamics of A mu
should also not change.

1069
01:18:07,210 --> 01:18:13,440
Let me call it AM so the bulk
and general will be different.

1070
01:18:13,440 --> 01:18:17,210
But this is like a
gauge transformation.

1071
01:18:17,210 --> 01:18:19,840
So this is like a
gauge transformation.

1072
01:18:19,840 --> 01:18:22,850
So we deduce that
somehow, this must

1073
01:18:22,850 --> 01:18:24,960
be some subset of the
bulk gauge transformation.

1074
01:18:32,492 --> 01:18:36,770
Now call it star to the power 4.

1075
01:18:36,770 --> 01:18:44,240
Then star to the
power 4 also must

1076
01:18:44,240 --> 01:18:52,885
be a subset of some bulk
gauge transformation.

1077
01:19:03,400 --> 01:19:06,100
So this does not prove it,
but makes it more natural.

1078
01:19:06,100 --> 01:19:07,620
This argument makes
it more natural.

1079
01:19:07,620 --> 01:19:10,910
The fact that the
boundary value of A

1080
01:19:10,910 --> 01:19:12,340
has such kind of gauge symmetry.

1081
01:19:15,140 --> 01:19:18,440
So in some sense,
this must generalize

1082
01:19:18,440 --> 01:19:23,270
to some gauge symmetry
in the full gravity side.

1083
01:19:23,270 --> 01:19:28,049
So similarly, you can do this
for the metric for the stress

1084
01:19:28,049 --> 01:19:28,549
tensor.

1085
01:19:32,165 --> 01:19:34,206
Similarly, you can do this
for the stress tensor.

1086
01:19:39,020 --> 01:19:49,370
So for the stress tensor,
so for the star star, again,

1087
01:19:49,370 --> 01:20:01,380
we can add h mu mu x T mu mu
to the boundary Lagrangian.

1088
01:20:01,380 --> 01:20:05,920
And this is the source
to the stress tensor.

1089
01:20:11,050 --> 01:20:16,152
And now, from your knowledge
of quantum field theory,

1090
01:20:16,152 --> 01:20:23,890
when I add such a term to the
field theory, it's the same.

1091
01:20:23,890 --> 01:20:34,140
So this is equivalent to I
deform the boundary space

1092
01:20:34,140 --> 01:20:45,620
time metric from just eta mu
mu to eta mu mu plus h mu mu.

1093
01:20:48,220 --> 01:20:51,667
So let me call this
quantity g mu mu b.

1094
01:20:58,270 --> 01:21:02,200
So adding such a term
source for the stress tensor

1095
01:21:02,200 --> 01:21:04,140
is like you deform your
space time geometry

1096
01:21:04,140 --> 01:21:06,710
a little bit by this source.

1097
01:21:06,710 --> 01:21:09,435
This is when h is very small.

1098
01:21:09,435 --> 01:21:10,810
When we consider
the information,

1099
01:21:10,810 --> 01:21:12,434
we always consider
the source is small.

1100
01:21:16,430 --> 01:21:22,980
But now, we can argue this
thing, mass corresponding

1101
01:21:22,980 --> 01:21:28,220
to the boundary
value of the metric

1102
01:21:28,220 --> 01:21:31,750
in the gravity side for
the following reason.

1103
01:21:31,750 --> 01:21:34,890
So if you look at
this expression,

1104
01:21:34,890 --> 01:21:36,380
so if you look at
even the pure AdS

1105
01:21:36,380 --> 01:21:38,540
metric before you
do any deformation,

1106
01:21:38,540 --> 01:21:44,820
so this is just due to
the original theory.

1107
01:21:44,820 --> 01:21:55,550
So from the AdS metric, so
the original AdS metric,

1108
01:21:55,550 --> 01:21:56,590
the g mu mu component.

1109
01:22:00,590 --> 01:22:05,320
So let me write this
AdS metric in the form,

1110
01:22:05,320 --> 01:22:08,255
say, R squared, z
squared, separate

1111
01:22:08,255 --> 01:22:10,050
the z squared with the rest.

1112
01:22:14,940 --> 01:22:18,492
So this is essentially the
component of the bulk metric

1113
01:22:18,492 --> 01:22:19,700
along the boundary direction.

1114
01:22:26,430 --> 01:22:34,540
The boundary value of this
metric component evaluates

1115
01:22:34,540 --> 01:22:37,490
z equal to 0 is
equal to R squared

1116
01:22:37,490 --> 01:22:42,370
divided by z squared eta mu mu.

1117
01:22:42,370 --> 01:22:47,100
So that just comes from the
AdS metric, just this g mu mu.

1118
01:22:47,100 --> 01:22:49,960
And when you go the
boundary, just given by this.

1119
01:22:49,960 --> 01:22:50,910
And this is eta mu mu.

1120
01:22:50,910 --> 01:22:53,190
We recognize it just
as the boundary metric.

1121
01:23:01,300 --> 01:23:02,684
So these correspond to deform.

1122
01:23:06,634 --> 01:23:08,050
Since adding this
term corresponds

1123
01:23:08,050 --> 01:23:11,400
to deform the boundary
metric, so we expect,

1124
01:23:11,400 --> 01:23:13,450
when we add this term,
so this corresponding

1125
01:23:13,450 --> 01:23:21,413
to deform the g mu mu
now to the 0 become.

1126
01:23:24,140 --> 01:23:28,582
So we expect this becomes
equal to g mu mu b.

1127
01:23:36,400 --> 01:23:40,400
So this tells you
that the stress tensor

1128
01:23:40,400 --> 01:23:44,710
must be due to the metric
perturbations in the gravity

1129
01:23:44,710 --> 01:23:45,210
side.

1130
01:23:48,070 --> 01:23:50,844
The stress tensor
must be corresponding

1131
01:23:50,844 --> 01:23:52,760
to the metric perturbation
on the gravity side

1132
01:23:52,760 --> 01:23:54,843
because it's corresponding
to perturb the boundary

1133
01:23:54,843 --> 01:23:59,430
conditions of the bulk
metric-- corresponding

1134
01:23:59,430 --> 01:24:02,350
to preserve the boundary
condition for the bulk metric.

1135
01:24:02,350 --> 01:24:07,650
So this is a very
general statement

1136
01:24:07,650 --> 01:24:11,740
valid for any correspondence
between the gravity

1137
01:24:11,740 --> 01:24:14,029
and the field theory.

1138
01:24:14,029 --> 01:24:16,320
And that also tells you that
if you have a theory which

1139
01:24:16,320 --> 01:24:20,390
due into a higher dimensional
theory, and then that theory,

1140
01:24:20,390 --> 01:24:23,230
if the field theory
has a stress tensor,

1141
01:24:23,230 --> 01:24:28,630
then this bulk theory
must have gravity

1142
01:24:28,630 --> 01:24:32,480
because this bulk theory
must have a dynamical metric.

1143
01:24:36,510 --> 01:24:39,200
And if you have a dynamical
metric, then you have gravity.

1144
01:24:39,200 --> 01:24:42,630
So you can say, if
any field theory

1145
01:24:42,630 --> 01:24:47,030
is due to a theory of one higher
dimension, that theory of one

1146
01:24:47,030 --> 01:24:52,220
higher dimension must involve
gravity-- nothing about quantum

1147
01:24:52,220 --> 01:24:55,150
gravity.

1148
01:24:55,150 --> 01:24:57,116
Let's stop here.