Spinning Geodesic Witten Diagrams Strings and Fields 2017 @ YITP - - PowerPoint PPT Presentation
Spinning Geodesic Witten Diagrams Strings and Fields 2017 @ YITP Aug. 10 / 2017 arXiv:1702.08818 [hep-th] (JHEP05 (2017) 070) with Heng-Yu Chen and En-Jui Kuo (National Taiwan Univ.) Hideki Kyono (Kyoto University) Introduction CFT
Strings and Fields 2017 @ YITP Aug. 10 / 2017
O1(x1) O2(x2) O3(x3) O4(x4)
O
O1(x1) O2(x2) O3(x3) O4(x4)
O
1/19
O
O O
O1(x1) O2(x2) O3(x3) O4(x4) O1(x1) O2(x2) O3(x3) O4(x4)
CPW WO(xi) in CFTd GWD WO(xi) in AdSd+1
2/19
Hijano, Kraus, Perlmutter and Snively [1508.00501]
3/19
AdSd+1 ⊂ M1,d+1
Rd ⊂ M1,d+1
(X+, X−, Xa) = 1 z2 (1, z2 + y2, ya), X · X = −1
Poincar´ e AdSd+1; {ya, z}, XA ∈ M1,d+1 Euclid Rd; {ya}, P A ∈ M1,d+1
We consider AdSd+1 and Rd (d > 2)
in the embedding sp. M1,d+1
(c : const.)
4/19
c.f. Weinberg [1006.3480]
We focus on a certain representation; symmetric traceless transverse tensor (STT tensor); ・STT tensor in AdSd+1 ・STT tensor in Rd : polarization vectors W A, ZA ∈ M1,d+1 parametrized by a single integer (rank) J
T(X, W) = W A1...W AJTA1,...,AJ(X) , X · W = W · W = 0 F(P, Z) = ZA1...ZAJFA1,...,AJ(P) , P · Z = Z · Z = 0
5/19
TAB,...(X) = TBA,...(X) , T AA,...(X) = 0 , XATA,...(X) = 0
T (AdS)
µ1...µJ(x) = ∂XA1
∂xµ1 . . . ∂XAJ ∂xµJ TA1...AJ(X) F(R)
a1...aJ(y) = ∂P A1
∂ya1 . . . ∂P AJ ∂yaJ FA1...AJ(P)
W∆,J(Pi) ⇠ Z
∂
dP0 Z ∞
−∞
dν K∆12,∆34,J(ν) ν2 + (∆ h)2 ⇥hO1(P1)O2(P2)Oh+iν,J(P0)ih ˜ Oh−iν,J(P0)O3(P3)O4(P4)i
K∆12,∆34,J(ν) = Γ h+iν+J±∆12
2
h−iν+J±∆34
2
where CPWs have the following integral expression; We will see that a GWD reproduces this expression.
O1(P1) O2(P2) O3(P3) O4(P4)
∆, J
O1(P1) O2(P2) O3(P3) O4(P4)
Oh+iν,J(P0) ˜ Oh−iν,J(P0)
(scalar) (scalar) (scalar) (scalar) (STT tensor)
A CPW can be represented as an integral of two 3-point functions. 6/19
Dolan, Osborn [1108.6194] Chen, Kuo, HK[1702.08818] Sleight [1610.01318]
∼ Z
Rd dP0
Z ∞
−∞
dν
Rd
W∆,J(Pi) =
Costa et al [1107.3554], [1109.6321]
hO∆1,l1(P1, Z1)O∆2,l2(P2, Z2)O∆3,l3(P3, Z3)i = X
nij≥0
λnij 2 4 ∆1 ∆2 ∆3 l1 l2 l3 n23 n31 n12 3 5
3-pt. functions of spinning fields produce some tensor structures;
where
Pij ≡ −2Pi · Pj = (yi − yj)2
∆1 ∆2 ∆3 l1 l2 l3 n23 n13 n12 ≡ Vl1−n12−n31
1,23
Vl2−n23−n12
2,31
Vl3−n31−n23
3,12
Hn12
12 Hn13 13 Hn23 23
(P12)
1 2 (τ1+τ2−τ3) (P13) 1 2 (τ1+τ3−τ2) (P23) 1 2 (τ2+τ3−τ1) .
Hij ≡ 2{(Zi · Pj)(Zj · Pi) − (Zi · Zj)(Pi · Pj)} Vi,jk ≡ (Pk · Pi)(Zi · Pj) − (Pj · Pi)(Zi · Pk) (Pj · Pk)
τi = ∆i − li
l1 − n12 − n31 ≥ 0 l2 − n23 − n12 ≥ 0 l3 − n31 − n23 ≥ 0
and
All possible tensor structures can be represented by H and V.
O∆i,li(Pi, Zi) = Zi,A1...Zi,Ali O
A1,...,Ali ∆i,li
(Pi)
7/19
Ex. ・2-point function (tensor-tensor) ・3-point function (scalar-scalar-tensor)
hO∆,J(P1, Z1)O∆,J(P2, Z2)i = (H12)J (P12)∆+J
hO∆1(P1)O∆2(P2)O∆3,J(P3, Z3)i ⇠ (V3,12)J (P12)
1 2 (∆1+∆2−∆3+J)(P23) 1 2 (∆2+∆3−∆1+J)(P31) 1 2 (∆3+∆1−∆2+J)
8/19
∆1 ∆2 ∆3 l1 l2 l3 n23 n13 n12 = Hn12
12 Dn13 12 Dn23 21 Dm1 11 Dm2 22
˜ τ1 ˜ τ2 ∆3 l3
˜ τ1 ≡ ∆1 + l1 + (n23 − n13) ˜ τ2 ≡ ∆2 + l2 + (n13 − n23)
Dn10,n20,n12
Left
Definition of D-operators; Note that { } is a linear combination of [ ]
D11 =
1 − (Z1 · P2)P A 1
∂P A
2
+
1 − (Z1 · Z2)P A 1
∂ZA
2
, D12 =
1 − (Z1 · P2)P A 1
∂P A
1
+
1
∂ZA
1
, D22 =
2 − (Z2 · P1)P A 2
∂P A
1
+
2 − (Z1 · Z2)P A 2
∂ZA
1
, D21 =
2 − (Z2 · P1)P A 2
∂P A
2
+
2
∂ZA
2
9/19
Costa et al [1107.3554], [1109.6321]
̶> { } is another basis of 3 point tensor structures
O1(P1) O2(P2) O3(P3) O4(P4)
∆, J
O1(P1) O2(P2) O3(P3) O4(P4)
∼
Z
∂
dP0 Z ∞
−∞
dν
Oh+iν,J(P0) ˜ Oh−iν,J(P0)
Z
∂
dP0 Z ∞
−∞
dν W {n10,n20,n12};{n30,n40,n34}
O∆,J
(Pi, Zi) ≡ Dn10,n20,n12
Left
Dn30,n40,n34
Right
WO∆,J(Pi)
CPWs with external spinning fields can be constructed by using DLeft and DRight.
W {n10,n20,n12};{n30,n40,n34}
O∆,J
(Pi, Zi)
Dn10,n20,n12
Left
Dn30,n40,n34
Right
←
O1(P1) O2(P2) O3(P3) O4(P4) Oh+iν,J(P0)
˜ Oh−iν,J(P0)
Z
∂
dP0 Z ∞
−∞
dν
∆, J
O∆1,l1(P1, Z1) O∆2,l2(P2, Z2) O∆3,l3(P3, Z3) O∆4,l4(P4, Z4)
∆1 ∆2 h + iν l1 l2 J n20 n10 n12 · ∆3 ∆4 h − iν l3 l4 J n40 n30 n34
Using the integral expression;
(P
10/19
Φ1 rA1,...,AJΦ2 T A1,...,AJ
3
bulk-to-boundary propagator;
γ12 : XA(λ) = P A
1 eλ + P A 2 e−λ
(P12)
1 2
X(λ)
O∆1(P1) O∆2(P2)
O∆3,J(P3, Z3)
γ12 3-point GWD with a derivative interaction;
X(λ) O∆1(P1) O∆2(P2) O∆3,J(P3, Z3)
γ12
geodesic connecting 1-2;
Π∆,J(P, X; Z, W) = C∆,J 2{(W · P)(Z · X) − (W · Z)(P · X)}J (−2P · X)∆+J
λ parametrizes the geodesic
This 3-point GWD is proportional to 3-pt function with (0,0,J) spins; 11/19
Costa et al [1404.5625]
= Z
γ12
Π∆1,0(P1, X)(K · r)JΠ∆2,0(P2, X)Π∆3,J(P3, X; Z3, W)
Π∆,J(X, ˜ X; W, ˜ W) = 1 πJ!(h − 1)J Z ∞
−∞
dν Z
∂
dP0 ν2 ν2 + (∆ − h)2 ×Πh+iν,J(X, P0; W, DZ)Πh−iν,J( ˜ X, P0; ˜ W, Z0)
X(λ) ˜ X(λ0)
O∆1(P1) O∆2(P2) O∆3(P3) O∆4(P4)
∆, J
⇠ Z
γ12
Z
γ34
Π∆1,0(P1, X)(K · rX)JΠ∆2,0(P2, X) ⇥ Π∆,J(X, ˜ X; W, ˜ W) Π∆3,0(P3, ˜ X)( ˜ K · ˜ r ˜
X)JΠ∆4,0(P4, ˜
X) the split (integral) representation of bulk-to-bulk propagator; contracted
bulk-to-bulk (spinning) bulk-to-boundary (scalar)
12/19
Costa et al [1404.5625]
O1(P1) O2(P2) O3(P3) O4(P4)
∆, J
O1(P1) O2(P2) O3(P3) O4(P4)
∼
Z
∂
dP0 Z ∞
−∞
dν
Oh+iν,J(P0) ˜ Oh−iν,J(P0)
W∆,J(Pi) = Z dP0 Z ∞
−∞
dν ν2Ch+iν,JCh−iν,Jβ∆12,h+iν+Jβ∆34,h−iν+J ν2 + (∆ − h)2 × 2 4 ∆1 ∆2 h + iν J 3 5 · 2 4 ∆3 ∆4 h − iν J 3 5
X(λ) ˜ X(λ0)
O∆1(P1) O∆2(P2) O∆3(P3) O∆4(P4)
P0
h + iν, J h − iν, J
X(λ) ˜ X(λ0) O∆1(P1) O∆2(P2) O∆3(P3) O∆4(P4) ∆, J
Z
∂
dP0 Z ∞
−∞
dν
A bulk-to-bulk prop. can be split into two bulk-to-boundary prop. Using this relation, we can compute 4-pt. diagrams explicitly; This is proportional to the integral expression of CPW;
W∆,J(Pi) =
(CPW) (GWD) 13/19
This relation holds even for spinning CPWs and GWDs.
Dn10,n20,n12
Left
Dn30,n40,n34
Right
W∆,J(Pi) ∼ Dn10,n20,n12
Left
Dn30,n40,n34
Right
W∆,J(Pi)
Using
X(λ) ˜ X(λ0)
∆, J
∆, J
O∆1,l1(P1, Z1) O∆2,l2(P2, Z2) O∆3,l3(P3, Z3) O∆4,l4(P4, Z4)
Z
∂
dP0 Z ∞
−∞
dν K∆12,∆34,J(ν) ν2 + (∆ − h)2 8 < : ∆1 ∆2 h + iν l1 l2 J n12 n20 n01 9 = ; · 8 < : ∆3 ∆4 h − iν l3 l4 J n34 n40 n03 9 = ;
O∆1,l1(P1, Z1) O∆2,l2(P2, Z2)
O∆3,l3(P3, Z3) O∆4,l4(P4, Z4)
14/19
J n1,n2,n0
l1,l2,l0
(T r) = Yl1−n2−n0
1
Yl2−n0−n1
2
Yl0−n1−n2
3
Hn1
1 Hn2 2 Hn0
T 1(X1, W1)T 2(X2, W2)T 0(X0, W0) |Xr=X(λ)
Y1 = ∂W1 · ∂X0, Y2 = ∂W2 · ∂X0, Y3 = ∂W0 · ∂X1, H1 = ∂W2 · ∂W0, H2 = ∂W0 · ∂W1, H0 = ∂W1 · ∂W2.
where … contraction between two tensor fields; … contraction with a derivative;
Yi
1
...
∂AT1... T2
A... X(λ)
case
A T2 B T0 C1C2
[J1] ≡ h J0,0,0
1,1,2
i , [J2] ≡ h J1,0,0
1,1,2
i , [J3] ≡ h J0,1,0
1,1,2
i , [J4] ≡ h J1,1,0
1,1,2
i , [J5] ≡ h J0,0,1
1,1,2
i .
five possible interactions
l1,l2,l0
15/19
Sleight and Taronna [1603.00022] Chen, Kuo, HK[1702.08818]
case
[I1] ≡ ∆1 ∆2 ∆0 1 1 2 , [I2] ≡ ∆1 ∆2 ∆0 1 1 2 1 , [I3] ≡ ∆1 ∆2 ∆0 1 1 2 1 , [I4] ≡ ∆1 ∆2 ∆0 1 1 2 1 1 , [I5] ≡ ∆1 ∆2 ∆0 1 1 2 1 .
There are five possible tensor structures (in the box basis);
Tab = 4 (1 + ∆1) β0,∆+2 Q3
r=1 C∆r,lr
× B B B B B @ −
(2 + ∆1)
2(2+∆)(1+∆+∆1) ∆
2(2 + ∆) (2 + ∆1)
2(2+∆)(1+∆+∆1) ∆
−∆ −1 − ∆ − ∆+∆2+2∆1
∆+∆∆1
− (1+∆)(∆+∆1)
∆(1+∆1)
−2 + ∆ −2 −1 − 2
∆ + ∆
− 1+∆
∆ 1 1+∆1 1+∆ ∆+∆∆1 1+∆ ∆+∆∆1 1+∆ ∆+∆∆1
∆1 1 C C C C C A
These two bases relate each other as; where
We set
∆1 = ∆2,
∆0 = ∆
16/19
case
Aab = 1 − 1
4∆(4 + ∆)
− ∆
2
− ∆
2
− 1
2 2−∆ 4
− 1
4(−2 + ∆)∆ ∆ 2
1 − ∆
2 1 2
− ∆
4
− 1
4(−2 + ∆)∆
1 − ∆
2 ∆ 2 1 2
− ∆
4
− 1
4(−2 + ∆)2 1 2(−2 + ∆) 1 2(−2 + ∆)
− 1
2 2−∆ 4
1
{Da} = Aab[Ib]
{D1} ≡ ∆1 ∆2 ∆0 1 1 2 , {D2} ≡ ...
where
Therefore the differential basis and the bulk-interactions relate as; In this way, the bulk interaction which corresponds to a differential basis can be specified. 17/19
= − ˆ N GWD
J
Z ∞
−∞
dν 1 ν2 + (∆ − h)2 Z i∞
−i∞
dsdt (4πi)2 ×ωGWD
ν,J
(t)Pν,J(s, t) Y
i<j
Γ(δij)P −δij
ij
ˆ N GWD
J
≡ 2 πh (∆2)J(∆4)J , ωGWD
ν,J
(t) = Γ h±iν−J−t
2
∆1+∆2−t
2
∆3+∆4−t
2
˜ X(λ0) O∆1(P1) O∆2(P2) O∆3(P3) O∆4(P4) ∆, J
Pν,J(s, t) : the Mack polynomial
there is a “good” basis in the Mellin space?
The spectrum function differ from one of usual Witten diagram. There is no double trace pole.
Spinning case; 18/19 DLeftDRight Z i∞
−i∞
dsdt (4πi)2 ...
c.f. Sen’s talk
19/19 Summary; Future works; ・Geodesic Witten diagrams correspond to conformal partial waves in the CFT side. ・By using derivative operators, the relation between CPWs and GWDs is generalized to spinning cases. ・Parameterizing bulk interactions on geodesics appropriately, we can see which tensor structures correspond to a certain interaction. ・Generalization to other representation? (aside from STT tensor) ・Spinning Mellin representation?
Fitzpatrick et al. [1107.1499], Paulos[1107.1504] Tamaoka [1707.07934], Isono [1706.02835],