Classical conformal blocks
via AdS/CFT correspondence
Konstantin Alkalaev
I.E.Tamm Theory Department, P.N.Lebedev Physical Institute
K.Alkalaev, V.Belavin, arXiv:1504.05943
Moscow 2015
Konstantin Alkalaev
Classical conformal blocks via AdS/CFT correspondence
Outline
Classical conformal blocks as geodesic lengths
Fitzpatrick, Kaplan, Walters’ 2014
Asplund, Bernamoti, Galli, Hartman’ 2014
Hijano, Kraus, Snively’ 2015
AGT combinatorial realization (instead of Zamolodchikov’s recursion)
Alday, Gaiotto, Tachikawa’ 2010
Alba, Fateev, Litvinov, Tarnopolsky’ 2010
The general consideration of geodesic motions in the bulk
Five-point configurations: explicit results
Conclusions and outlooks
Konstantin Alkalaev
Classical conformal blocks via AdS/CFT correspondence
construction of the correlation functions. They are holomorphic functions of the coordinates zi of
conveniently defined
using the dual pant decomposition
n-pointfieldsclassical
conformal
blockdiagram. In Fig. 1 the dual diagram for
In any
n-point block is given (the meaning of two bold lines is explained later). It represents the contribution
from the states in the Virasoro representations with highest weights r 1 , ..., r n´3 (internal lines) to
the 2
correlation
function offunction
the fields with
conformal
We conformal
denote the
CFT
a correlation
of V
can be decomposed
into
i (external lines).
∆i (zi ) dimensions
corresponding n-point conformal block depending on the parameters of the external and internal
conformal dimensions (as well as on the central charge)
˜ 1 , ..., ∆
˜ n−3 ; c)
F (z1 , ..., zn |∆1 , ..., ∆n ; ∆
Fpz1 , ..., zn | 1 , ..., n ; r 1 , ..., r n´3 ; cq .
(1.1)
blocks
antiholomorphicdepicted
block is defined
which areThe
conveniently
as
Pant
similarly, replacing holomorphic coordinates zk by antiholomorphic z̄k . The construction of the correlation functions involves the summation of the products
of the holomorphic and antiholomorphic blocks over all possible intermediate channels r i weighted
decomposition
with the structure constants of the operator algebra.
z2 ,
z1 ,
¨¨¨¨¨¨
2
zn´2 ,
n´2
zn´1 ,
n´1
zn ,
1
r1
¨¨¨¨¨¨
r n´2
n
r n´3
Figure 1: The n-point conformal block. In the case of the heavy-light conformal block two bold lines
There exist
many
evidences
that
the semiclassical
limit fields
c →depicted
∞ therespectively
conformal
on the
right are
heavy fields,
whileinexternal
fields and intermediate
by blocks
must exponentiate
solid lines and as
wavy lines are light. Using the projective invariance one can fix the coordinates of
three fields as z “ 0, z
“ 1, z “ 8.
1
n´1
n
˜ 1 , ..., ∆
˜ n−3 ; c) ∼ exp cf (z1 , ..., zn |1 , ..., n ; ˜1 , ..., ˜n−3 )
lim F (z1 , ..., zn |∆1 , ..., ∆n ; ∆
c→∞
There exist many evidences ( see, e.g., [11, 12]) that in the classical limit the conformal blocks
must exponentiate as
(
˜k
r , ..., r n´3 ; cq „ exp cf pz1 , ..., zn |✏1 , ..., ✏n ; r
k Fpz1 , ..., zn | 1 ,∆
✏1 , ...,
r
✏n´3 q lim
(1.2)
n ; 1called
and ˜k = ...,
are
classical dimensions and
f (z|,
˜,) is the
k = ∆cÑ8
c
c
where
r
conformalwhere
block
representing
our main interest.
✏ “
and ✏r “
are called classical dimensions and f pz , ..., z
k
k
c
k
k
c
1
✏1 , ..., r
✏n´3 q
n |✏1 , ..., ✏n ; r
classical
is the
classical conformal block representing our main interest.
There are different possible
classicalAlkalaev
limits of the conformal
blocks dependent
behaviour
Konstantin
Classical conformal
blocks on
viathe
AdS/CFT
correspondence
Heavy-light conformal blocks
Different classical limits of the conformal blocks depend on the behavior of the classical
dimensions i and ˜i .
If , ˜ remain finite in the semiclassical limit, the corresponding field is called heavy.
If , ˜ are vanishing in the semiclassical limit, the corresponding field is called light.
All fields are light — global sl(2) conformal block.
All fields are heavy — proper classical block.
Heavy-light classical blocks can be considered as an interpolation between these
two extreme regimes.
Heavy-light blocks (Fitzpatrick, Kaplan, Walters’ 2014)
The classical conformal dimensions of two fields n−1 = n are heavy.
It is instructive to introduce a scale factor δ that we call a lightness parameter.
Schematically, provided that all except two dimensions are rescaled as → δ and ˜ → δ˜
there appear a series expansion
f (z|, ˜) = fδ (z|, ˜) δ + fδ2 (z|, ˜) δ 2 + ... .
The leading contribution fδ (z) yields the heavy-light conformal block, while taking into
account sub-leading contributions approximate the proper conformal block on the left
hand side.
Konstantin Alkalaev
Classical conformal blocks via AdS/CFT correspondence
The AdS/CFT correspondence
The heavy operators with equal conformal dimensions n = n−1 ≡ h produce an
asymptotically AdS3 geometry identified either with an angular deficit or BTZ black hole
geometry parameterized by
p
w
,✏
α = 1 − 4h
The metric reads
n 2
ds 2 =
Here
.
.
.
.
.
.
.
1
α2 − dt 2 + sin2 ρdφ2 + 2 dρ2
2
cos ρ
α
α2 < 0 for an angular deficit
n 2
.
.
.
w1 , ✏1
α2 > 0 for the BTZ black hole
w2 , ✏2
The light fields are realized via particular graph of worldlines of n − 3 classical point
Figure: Multi-particle graph embedded into a constant time slice of a
probes propagating in the background geometry formed
bydefect
thegeometry.
two Solid
boundary
fields.
conical
lines representheavy
external particles,
wavy lines
represent intermediate particles. The original heavy fields produce the
Points wi are boundary attachments of the light operators.
The lightness parameter δ
background geometry with the singularity placed in the center representing
a cubic vertex of two heavy fields and a light intermediate field.
measures a backreaction of the background on a probe.
Konstantin Alkalaev
Classical conformal blocks via AdS/CFT correspondence
The identification
Sclbulk = z γ fδ (z|, ˜) ,
Sclbulk =
n−2
X
i=1
i Li +
n−3
X
˜i L̃i ,
i=1
and Li and L̃i are lengths of different geodesic segments on a fixed time slice.
Konstantin Alkalaev
Classical conformal blocks via AdS/CFT correspondence
AGT representation
Using SL(2) invariance we fix three points z1 = 0, zn−1 = 1, zn = ∞, and replace
zi+1 = qi qi+1 . . . qn−3
for
1≤i ≤n−3
The conformal block is given by the following series expansion
˜ c) = 1 +
F (q|∆, ∆,
X
k
n−3
˜ c)
q1k1 q2k2 . . . qn−3
Fk (∆, ∆,
k1 ,...kn−3
Using the standard Liouville parametrization,
∆i =
Q2
− Pi2 ,
4
2
˜ j = Q − P̃ 2 ,
∆
j
4
c = 1 + 6Q 2 ,
Q =b+
1
,
b
the AGT representation of the n-point conformal block is given as
˜ c) =
F (q|∆, ∆,
n−3
Y n−3
Y
Q
Q
˜ c),
(1 − qr . . . qs )2(Pr +1 − 2 )(Ps+2 + 2 ) Z(q|∆, ∆,
r =1 s=r
where
˜ c) = 1 +
Z(q|∆, ∆,
X
k
n−3
˜ c)
Zk1 ,...kn−3 (∆, ∆,
q1k1 q2k2 . . . qn−3
k1 ,...kn−3
Konstantin Alkalaev
Classical conformal blocks via AdS/CFT correspondence
The diagrammatic coefficients
The Nekrasov functions
Zk1 ,...kn−3 =
X Z (P2 |P1 , ∅; P̃1 , ~λ1 )Z (P3 |P̃1 , ~λ1 ; P̃2 , ~λ2 ) · · · Z (Pn−1 |P̃n−3 , ~λn−3 ; Pn , ∅)
Z ( Q |P̃1 , ~λ1 ; P̃1 , ~λ1 ) · · · Z ( Q |P̃n−3 , ~λn−3 ; P̃n−3 , ~λn−3 )
~
λ1 ,...,~
λn−3
2
2
(1)
(2)
Here, the sum goes over (n − 3) pairs of Young tableaux ~λj = (λj , λj ) with the total
(1)
(2)
number of cells |~λj | ≡ |λ | + |λ | = kj . The explicit form of functions Z reads
j
j
Z (P |P , µ
~ ; P, ~λ) =
00
0
2
Y
Y i,j=1 s∈λi
Q
P 00 − Eλi ,µj (−1)j P 0 − (−1)i P s +
2
×
×
Y Q
P 00 + Eµj ,λi (−1)i P − (−1)j P 0 t −
2
t∈µ
j
where
Eλ,µ x s = x − b lµ (s) + b −1 (aλ (s) + 1)
For a cell s = (m, n) such that m and n label a respective row and a column, the
arm-length function aλ (s) = (λ)m − n and the leg-length function lλ (s) = (λ)T
n − m,
where (λ)m is the length of m-th row of the Young tableau λ, and (λ)T
n the height of the
n-th column, where T stands for a matrix transposition.
Konstantin Alkalaev
Classical conformal blocks via AdS/CFT correspondence
The five-point classical conformal block
Q
Q
Q
Q
Q
Q
F (q1 , q2 ) = (1−q1 )2(P2 − 2 )(P3 + 2 ) (1−q1 q2 )2(P2 − 2 )(P4 + 2 ) (1−q2 )2(P3 − 2 )(P4 + 2 ) Z(q1 , q2 ),
where
Z(q1 , q2 ) = 1 +
q1k1 q2k2 Zk1 ,k2 ,
X
k1 ,k2
and
|~
λ1,2 |=k1,2
Zk1 ,k2 =
X
~
λ1 ,~
λ2
Z (P2 |P1 , ∅; P̃1 , ~λ1 )Z (P3 |P̃1 , ~λ1 ; P̃2 , ~λ2 )Z (P4 |P̃2 , ~λ2 ; P5 , ∅)
,
Z ( Q |P̃1 , ~λ1 ; P̃1 , ~λ1 )Z ( Q |P̃2 , ~λ2 ; P̃2 , ~λ2 )
2
2
where on the lower levels the pairs of Young tableaux ~λ = (λ(1) , λ(2) ) with the total
number of cells l = |~λ| are
l =0:
l =1:
{(∅, ∅)}
{(∅, ), (
l =2:
{(∅,
l =3:
{(∅,
, ∅)}
), (∅,
), (
), (∅,
(
,
), (
), (
,
), (∅,
,
), (
, ∅), (
), (
,
, ∅), (
, ∅)}
), (
,
, ∅), (
),
, ∅)} .
In what follows 5-pt conformal blocks are with dimensions P4 = P5 , P1 = P2 , P̃1 = P̃2 .
Konstantin Alkalaev
Classical conformal blocks via AdS/CFT correspondence
We find
Z(q1 , q2 |t) = 1 + (1 + b 2 − 2bP3 )(1 + b 2 + 2bP3 )(q1 + q2 )(8b 2 )−1 t + O(t 2 ) ,
where t m terms take into account contributions q1m1 q2m2 with m = m1 + m2 .
The limit c → 0 can be equivalently understood as b → 0. The classical conformal block
F (q1 , q2 ) = e
−
f (q1 ,q2 )
b2
or
f (q1 , q2 ) = − lim b 2 ln F (q1 , q2 )
b→0
Fields with P4 = P5 are heavy. Recall that the lightness parameter expansion is given
f (q1 , q2 ) = fδ (q1 , q2 )δ + fδ2 (q1 , q2 )δ 2 + ...
Now, 3 (or P3 ) is the new deformation parameter
(0)
(1)
(2)
fδ (q1 , q2 ) = fδ (q1 , q2 ) + 3 fδ (q1 , q2 ) + 23 fδ (q1 , q2 ) + ... .
(0)
Here, the leading term fδ (q1 , q2 ) is identified with the 4-pt classical conformal block,
while the sub-leading terms perturbatively reconstruct the 5-pt classical conformal block
h 2 sinh[ α ln[1−q1 q2 ] ] i
h 4 tanh[ α ln[1−q1 q2 ] ] i
(0)
2
4
fδ (q1 , q2 ) = 21 ln −
− ˜1 ln −
+ 1 ln[1 − q1 q2 ]
αq1 q2
αq1 q2
and
(1)
fδ (q1 , q2 ) = ln sinh[
α(ln[1 − q1 q2 ] − 2 ln[1 − q2 ])
] + ln[1 − q2 ]
2αq2
Konstantin Alkalaev
Classical conformal blocks via AdS/CFT correspondence
ends on the direction of the
flow. The minimal
´ ¯2 radial distance between
p
ngularity is therefore given by tan2 ⇢min “ ↵ . Obviously, the maximal
s to the point ⇢max “ ⇡{2 located on the boundary. Changing variables as
troducing notation
The world-line approach
The semiclassical limit c → ∞. The worldline action (m ∼ )
|p |
Zs “λ00↵
(3.8)
,
q
2
2
2
ctly integrated
on-shellgaction
S =toyield thedλ
tt ṫ + gφφ φ̇ + gρρ ρ̇ ,
0
λ
ˇ⌘2
?
ˇ
⌘
is?convenient
toˇˇ impose
a
S It
“ ln
,
1 ` ⌘ ` 1 ´ s2 ⌘ ˇ⌘1
2
α2 1
− dt 2 + sin2 ρdφ2 + 2 dρ2
cos2 ρ
α
ds 2 =
the normalization
(3.9) condition
00
Z
q
are initial/final radial positions. Parameter
|ẋ| ≡ gµν (x)ẋ µ ẋs νis an
= integration
1 :
S=
“ cot2 ⇢
rticular form of the geodesic segment.
0
00
0
dλ = (λ − λ ) .
λ
Coordinates t and φ are cyclic — a constant time disk (ρ,φ).
Changing variables as η = cot2 ρ and introducing notation
|p |
s = αφ we find the on-shell action
⇢2
⇢1
w
λ
0
√
η
p
S = ln √
1 + η + 1 − s2η
egments. The graph corresponds to the classical conformal block with two
s of equal dimensions (the arc), and one extremely light intermediate field
η00
0
η
Parameter s is an integration constant that defines a particular
form of the geodesic segment.
geodesic segment is the radial line starting (or ending, depending on the
The radial line has s = 0. For ρ1 = arccos sin(αw /2): Lrad = − ln tan αw
4
ngularity point ⇢2 “ 0, see Fig. 4. In this case, the angular momentum
h
i
After some simpleThe
algebra,
findss from
(3.9)αw
the .radial
Srad “
arconehas
= cot
Thelength
length
Larc = ln sin αw
+ ln 2Λ
2
2
hat Srad is finite implying that a particle reaches the singularity within a
The 4-pt block: f ∼ 1̃ Lradof+the2two
arc
1 Lheavy
erprets the falling into the singularity as a cubic vertex
ator represented by a probe. For the further purpose
we findAlkalaev
a length of
Konstantin
Classical conformal blocks via AdS/CFT correspondence
rges at ⇤ Ñ 8 so that it takes an infinite time to reach the boundary. Note
chosen to compute (3.10) corresponds to the zero value of the radial velocity,
e according to formula (3.7). From the graph in Fig. 4 it is clear that the
en by (3.10).
Five-line configuration
The corresponding particle action reads
le configuration
ne graph on Fig. 5 which is the n “ 5 case
S =of1the
L1 general
+ 2 L2graph
+ 3in
L3Fig.
+ 2.
1̃ L1̃ + 2̃ L2̃
Vertex equilibrium equations
r
2
3
w3
2
r
1
1 + p1 + p2 1st vertex ˜1 p̃µ
=0
1 µ
2 µ x=x1
2 + p3 1 +
2nd vertex ˜1 p̃µ
˜2 p̃µ
=0
3 µ x=x2
1
Angular equations
w1
w2
∆φlines
= w2 −w1 ,
r
1 +∆φ
graph. Solid lines 1, 2, 3 represent external particles, wavy
1, r
22represent
The angles are measured clockwise. In practice, we set w1 “ 0.
∆φ1 +∆φ3 +∆φ̃1 = w3 −w1
icle action reads
S“
ÿ
I
✏ I SI ,
I “ 1, 2, 3, r
1, r
2,
Konstantin Alkalaev
(4.1)
Classical conformal blocks via AdS/CFT correspondence
The complete equation system
Using pρ = gρρ ρ̇, pφ = gφφ φ̇ along with the normalization condition, and recalling that
the angular momenta are motion constants we find
p
p
p
1 − s 2 cot2 ρ00 − is 1 + cot2 ρ00
2
2
p
ρ̇ = cos ρ 1 − s cot ρ ,
iα∆φ = ln p
1 − s 2 cot2 ρ0 − is 1 + cot2 ρ0
Equations to be solved:
Vertex eqs
3
q
q
1 − s32 η2 + ˜1 1 − s̃12 η2 = ˜2 ,
1
q
q
q
1 − s12 η1 + 2 1 − s22 η1 = ˜1 1 − s̃12 η1
Angular eqs
e
e iαw3
iαw2
=
q
q
√
√
1 − s12 η1 − is1 1 + η1
1 − s22 η1 − is2 1 + η1
(1 − is1 )(1 − is2 )
q
q
q
√
√
√
1 − s32 η2 − is3 1 + η2
1 − s̃12 η2 − i s̃1 1 + η2
1 − s12 η1 − is1 1 + η1
q
=
√
(1 − is3 )
1 − s̃12 η1 − i s̃1 1 + η1 (1 − is1 )
5-pt case: a complicated higher order algebraic equation
4-pt case: an exact solution (Hijano, Kraus, Snively, 2015)
Konstantin Alkalaev
Classical conformal blocks via AdS/CFT correspondence
are two light external fields with dimensions ✏1 and ✏2 and one intermediate light
⌫“
.
r
✏1
nr
✏1 . The respective graph is depicted on Fig. 6.
The 5-pt case as a deformation of the 4-pt case
2
r
1
r
2
3
1
w3
2
0
w2
r
1
r
1
1
w1
w2
The lines of the resulting five-line configuration are characterized by the deformed angular
e graph.momenta
Solid lines represent external
wavy linesmethod.
representVertex
intermediate
r
Figureparticles,
7: A deformation
1´r
2 ´ 3 originates from the seed vert
2
)
,
I
=
1, 2, 3, 1̃,
2̃ , r
sthe
bI + line
+
O(
3 cI r
3
I =radial
1. The deformation produces
lines
1 and r
2 from the original line r
1
vertex
point
using
line
3.
Solid
lines
correspond
to
the
4-pt
where bI are the angular momenta of the seed three-line configuration and cI case,
are while dotted on
um and corrections.
the angular equations
read
deformation.
Note that
s̃2 = b2̃ = 0 remain intact, and the seed line 1̃ is radial so that
bb = 0. By convention,
b
b23The
is the
seed momentum assigned to line 3. The total action
1̃
same constraints have been used in the boundary computations (2.14).
✏1 ,
✏1 reads
1 ´ s21 ⌘ ` ✏2 1 ´ s22 ⌘ “ r
✏ 1 s1 ´ ✏ 2 s2 “ 0 ,
(5.1)
2
e
i↵w2
2 , w3 ) = S0 (w2
a
? S(wa
?) + 3 S1 (w2 , w3 ) + O(3 ) ,20
p 1 ´ s21 ⌘ ´ is1 1 ` ⌘qp 1 ´ s22 ⌘ ´ is2 1 ` ⌘q
“
, configuration,(5.2)
where S0 = S0 (wp1
isisthe
action of the three-line
while S1 (w2 , w3 ) is a
2 )´
1 qp1 ´ is2 q
correction.
oordinate of the vertex is
Konstantin Alkalaev
Classical conformal blocks via AdS/CFT correspondence
The total length
We set ˜1 = ˜2 , 1 = 2 . Denote
ν = 3 /˜
1 ,
κ = ˜1 /1 ,
θi =
αwi
2
The first order solution to the 5-pt configuration reads
s1 = − cot θ2 +
s3 = − cot(2θ3 − θ2 ) + ν
κ
νκ
−
cot(2θ3 − θ2 ) + O(ν 2 ) ,
2 sin θ2
2
cos(2θ2 − 4θ3 ) − 2 cos(2θ2 − 2θ3 ) − 2 cos 2θ3 + 3
+ O(ν 2 ) ,
4 sin3 (θ2 − 2θ3 )
and
s2 = s1 − νκs3 ,
s̃1 = νs3 ,
s̃2 = 0 .
The final action
S(w2 , w3 ) = −21 ln sin θ2 + ˜1 ln tan
θ2
− 3 ln sin(2θ3 − θ2 ) + O(23 )
2
According to the general prescription the action is related to the conformal block as
fδ (q1 , q2 ) ∼ −S(θ1 , θ2 )
The identification is achieved by the following conformal transformations to the plane
θ2 =
iα
ln(1 − q1 q2 ) ,
2
Konstantin Alkalaev
θ3 =
iα
ln(1 − q2 )
2
Classical conformal blocks via AdS/CFT correspondence
Conclusions & outlooks
Done:
We have proposed the general identification between the pant decomposition with
n legs on the boundary and the corresponding multi-line graph in the bulk.
We have written down the general system of equations describing the dynamics of
probes in the bulk background.
We have performed explicit computations in the n = 5 case establishing the
correspondence in the first order in the conformal dimension of one of fields while
keeping other dimensions arbitrary.
To be done:
The n-point configurations explicitly. On the boundary side we can use the
monodromic approach.
The heavy-light classical blocks with arbitrary number of heavy operators.
The AdS/CFT semiclassical calculations from the first principles.
Konstantin Alkalaev
Classical conformal blocks via AdS/CFT correspondence