Holographic Mellin Amplitudes in Various Dimensions Xinan Zhou C. - - PowerPoint PPT Presentation
Holographic Mellin Amplitudes in Various Dimensions Xinan Zhou C. N. Yang Institute for Theoretical Physics Stony Brook University Based on PRL 118, (2017) 091602, JHEP 1804 (2018) 014, arXiv:1712. 02788 (with L. Rastelli), and
Xinan Zhou YITP , Stony Brook University
Workshop on higher-point correlation functions and integrable AdS/CFT Trinity College Dublin April 18, 2018
Xinan Zhou
Stony Brook University
Based on PRL 118, (2017) 091602, JHEP 1804 (2018) 014, arXiv:1712. 02788 (with L. Rastelli), and arXiv:1712.02800, arXiv:1804.02397
Complicated ways to write zeros! Amplitudes can be “bootstrapped”:
Xinan Zhou YITP , Stony Brook University
n-point process 3 4 5 6 7 …
# of diagrams (cyclic ordered)
1 3 10 38 154 …
Maximally Helicity Violating (MHV) Parke-Taylor Formula:
An[1+ . . . i− . . . j− . . . n+] = hiji4 h12ih23i . . . hn1i
Xinan Zhou YITP , Stony Brook University
Holographic correlators = on-shell scattering amplitudes in a maximally symmetric spacetime We expect nice properties too:
hidden simplicity. AdS5 × S5 ↔ N = 4 Scattering in AdS
Xinan Zhou YITP , Stony Brook University
Infinitely many “particles”: Kaluza-Klein modes from the n-sphere. 1/2 BPS operators: k-fold symmetric-traceless representation of SO(n+1). Dual to scalar fields with . The analogue of S-matrix:
∆ = ✏ k ✏ = d 2 − 1
k
Scattering in AdS
Xinan Zhou YITP , Stony Brook University
hOk1(x1) . . . Ok4(x4)i = X +
The standard algorithm: Witten diagram expansion. n=2,3 is boring, the form is determined by symmetry. Starting from n=4, the dependence on coordinates becomes non-trivial. At sub-leading order in 1/N, External legs: bulk-to-boundary propagators Internal legs: bulk-to-bulk propagators Vertices: expand the effective Lagrangian. Integrate over the AdS.
G∆i
B∂(xi, Z)
G∆,`
BB(Z, W)
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D∆1∆2∆3∆4 = = Z
AdSd+1
dZ G∆1
B∂(x1, Z)G∆2 B∂(x2, Z)G∆3 B∂(x3, Z)G∆4 B∂(x4, Z)
Contact diagrams: is the scalar one-loop box diagram in four dimensions.
D1111
Exchange diagrams: when the quantum numbers satisfy special relations, an exchange diagram can be expressed as a finite sum of contact Witten diagrams [D’Hoker Freedman Rastelli] e.g., condition in the s-channel
∆ − ` = ∆1 + ∆2 − 2m ∆ − ` = ∆3 + ∆4 − 2m0
for positive integers and .
m m0
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hOk1(x1) . . . Ok4(x4)i = X +
A total nightmare to implement!
increased external weights.
Florov for AdS5 quartic vertices).
Only a handful of explicit 1/2 BPS correlators over the last 20 years:
(higher k conjecture) [Dolan Nirchl Osborn], near-extremal: (n+k, n-k, k+2, k+2) [Berdichevsky Naaijkens, Uruchurtu].
What is the organizational principle? Where is the hidden simplicity??
Xinan Zhou YITP , Stony Brook University
Mellin representation of conformal correlators [Mack, Penedones] The integration variables are constrained by
Gconn(xi) = Z [dδij] Y
i<j
(x2
ij)−δijM[δij]
δij = δji , δii = −∆i ,
n
X
j=1
δij = 0
is called the reduced Mellin amplitude. Consider OPE
M[δij]
Then should have simple poles at
M[δij] Oi(xi)Oj(xj) = X
k
c k
ij
⇣ (x2
ij)−
∆i+∆j −∆k 2
Ok(xk) + descendants ⌘ δij = ∆i + ∆j − (∆k + 2n) 2 , n ≥ 0 ⌧k = ∆k − `k
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Let us take n=4, there are two independent variables (“Mandelstam variables”) The constraints are solved by introducing auxiliary “momentum” variables They satisfies “momentum conservation” and “on-shell” condition
δij = pipj X
i
pi = 0 , p2
i = −∆i
s = ∆1 + ∆2 − 2δ12 , t = ∆1 + ∆4 − 2δ14 , u = ∆1 + ∆3 − 2δ13
with the constraint .
s + t + u = ∆1 + ∆2 + ∆3 + ∆4
In the s-channel OPE limit (1,2 come close), a primary operator leads to poles at
s = ⌧O + 2m , m ≥ 0 , ⌧O = ∆O − `
Similar statements for the t and u channels.
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Further define the Mellin amplitude [Mack]
M M(δij) = M(δij) Y
i<j
Γ(δij)
Two benefits:
conformal blocks.
M
For example, has poles that correspond to (recall ) with twist .
O∆1∂`⇤nO∆2 τ = ∆1 + ∆2 + 2n + O(1/N 2) Γ(δ12)
At large N, the Mellin amplitude is meromorphic, with simple poles corresponding to single-trace operators.
δ12 = ∆1 + ∆2 − s 2
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Contact diagrams: for a vertex with 2k derivatives, the Mellin amplitude is a degree-k polynomial [Penedones] Exchange diagrams (s-channel): simple analytic structure Same pole and same residue as a conformal block with the same quantum number! A remarkable simplification: when , the infinite series truncate into a finite sum. Equivalent to the previous truncation in position space. It has a very natural explanation in Mellin space.
τ + 2m0 = 2∆ M(s, t) =
∞
X
m=0
Q`,m(t) s − τ − 2m + P`−1(s, t)
Xinan Zhou YITP , Stony Brook University
G(xi) = 1 x2∆
12 x2∆ 34
Z i∞
−i∞
ds 2 dt 2 U
s 2 V t 2 −∆M(s, t)Γ2[2∆ − s
2 ]Γ2[2∆ − t 2 ]Γ2[2∆ − u 2 ] U = (x12)2(x34)2 (x13)2(x24)2 , V = (x14)2(x23)2 (x13)2(x24)2
The truncation of poles must happen in order to be consistent with the 1/N expansion:
If there is a small anomalous dimension
is produced by a double pole, is produced by a triple pole, etc.
log(U) log(U)2
U
∆−` 2 gcoll
∆,`(V ) + . . .
∆ → ∆ + γ
U τ/2 → U τ/2 + 1 2γU τ/2 log(U) + 1 8γ2U τ/2 log2(U) + . . .
Xinan Zhou YITP , Stony Brook University
On the other hand, using the counting of 4d N=4 SYM, the tree-level Witten diagrams are of order , and the anomalous dimension is also of
Because is multiplied by , only n=1 is allowed at tree level. In terms of the Mellin representation, this means at most double poles are allowed in the integrand. The truncation must occur: the Gamma functions also contains an infinite series of double poles; if the simple poles in overlaps with these double poles, they have to terminate. O(1/N 2) O(1/N 2)
γn logn(U) G(xi) = 1 x2∆
12 x2∆ 34
Z i∞
−i∞
ds 2 dt 2 U
s 2 V t 2 −∆M(s, t)Γ2[2∆ − s
2 ]Γ2[2∆ − t 2 ]Γ2[2∆ − u 2 ] M(s, t)
Flat space limit: the asymptotic Mellin amplitude encodes the flat space scattering amplitude [Penedones, Fitzpatrick Kaplan] If we take all the dimensions fixed (do not scale with R, i.e. ) and consider four-point amplitudes, the prescription further simplifies into
Xinan Zhou YITP , Stony Brook University
T (Sij) = lim
R→∞
1 N Z i∞
−i∞
dαeα α
d−P i ∆i 2
M ✓ δij = −R2Sij 4α , ∆i = Rmi ◆ T (Sij) ∝ lim
β→∞ M(βs, βt)
mi = 0
Conformal cross ratios R-symmetry cross ratios
Xinan Zhou YITP , Stony Brook University
Some kinematics first:
G(xi, ti) ⌘ hOk(x1, t1)Ok(x2, t2)Ok(x3, t3)Ok(x4, t4)i
Exploit the conformal and R-symmetry covariance: where , , .
G(xi, ti) ≡ ✓ t12t34 x2✏
12x2✏ 34
◆k G(U, V ; σ, τ) U = (x12)2(x34)2 (x13)2(x24)2 , V = (x14)2(x23)2 (x13)2(x24)2 σ = (t13)(t24) (t12)(t34) , τ = (t14)(t23) (t12)(t34) . ✏ = d/2 − 1 Ok(x, t) ≡ OI1...Ik
k
(x) tI1 . . . tIk , tItI = 0
(null vectors)
xij ≡ xi − xj tij ≡ ti · tj
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Now we further imposes the constraints from the fermionic generators. We get the superconformal Ward identity [Dolan Gallot Sokatchev]: where we have used a change of variables: R-symmetry group is locally isomorphic to SO(n): 3d OSp(4|N) with even N, 4d (P)SU(2,2|N) with N=2,4, 5d F(4) and 6d OSp(8*|2N) with N=1,2. In the following we first look at the maximally superconformal cases. Maximally supersymmetric theory is very constrained! Non-maximally supersymmetric cases will be discussed in Part II.
(@χ − ✏↵@α)G(, 0; ↵, ↵0)
U = χχ0 , V = (1 − χ)(1 − χ0) , σ = αα0 , τ = (1 − α)(1 − α0) .
Xinan Zhou YITP , Stony Brook University
AdS4 × S7 AdS5 × S5 AdS7 × S4
stress tensor arbitrary KK modes low-lying KK modes next-next-to-extremal
hO2O2O2O2i hOk1Ok2Ok3Ok4i hOn+kOn−kOk+2Ok+2i hOkOkOkOki , k = 2, 3, 4
Position space method Algebraic bootstrap problem Mellin superconformal Ward identity
PRL ’17, arXiv: 1710 ( L. Rastelli, XZ) arXiv: 1710, 1712.02788 ( L. Rastelli, XZ) arXiv: 1712.02800 (XZ)
Xinan Zhou YITP , Stony Brook University
How it works:
all possible exchange Witten diagrams and contact Witten diagrams. The coeffjcients , could be fixed by using the precise efgective Lagrangian, but it’s devilishly complicated. We instead leave the coeffjcients unfixed.
X ∈ {exchanges} I ∈ {contact}
λX cI
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diagrams (D-functions).
box diagram , , and 1, with rational function coeffjcients of and .
Φ(U, V ) log(U) log(V )
D1111
Φ χ χ0 ∂χΦ = − Φ χ − χ0 − log(V ) χ(χ − χ0) + log(U) (−1 + χ)(χ − χ0) ∂χ0Φ = Φ χ − χ0 + log(V ) χ0(χ − χ0) − log(U) (−1 + χ0)(χ − χ0)
Xinan Zhou YITP , Stony Brook University
identity also has such a unique decomposition. This becomes a set of linear equations of the unknown coeffjcients, which turns out to fix all of them up to an overall scaling
(@χ − ✏↵@α)GAnsatz
+Rlog V (χ, χ0; α0) log(V ) + R1(χ, χ0; α0) = 0 RΦ(χ, χ0; α0) = Rlog U(χ, χ0; α0) = Rlog V (χ, χ0; α0) = R1(χ, χ0; α0) = 0
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The position space method improves the traditional method. It circumvents the diffjculties in obtaining precise vertices from the efgective Lagrangian. But it also has a few obvious shortcomings:
be expressed as finitely many D-functions. E.g. AdS4.
go to Mellin space.
Xinan Zhou YITP , Stony Brook University
The idea of the method:
into an identity in Mellin space. The upshot is a difgerence operator capturing nontrivial structures in the Mellin amplitude.
amplitude (Bose symmetry, analytic property, asymptotic behavior) to formulate an algebraic bootstrap problem.
I will explain this method in the context of N=4 SYM. And for simplicity, I will focus on in the following. The method is similar for the general-weight case and I will show the general answer later.
∆i = k
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Solution of SCFWI in position space:
Gconn(U, V ; σ, τ) = Gfree,conn(U, V ; σ, τ) + R(U, V ; σ, τ) H(U, V ; σ, τ)
= τ 1 + (1 − σ − τ) V + (−τ − στ + τ 2) U + (σ2 − σ − στ) UV + σV 2 + στ R = (1 − χα)(1 − χ0α)(1 − χα0)(1 − χ0α0)
Xinan Zhou YITP , Stony Brook University
Solution of SCFWI in position space:
Gconn(U, V ; σ, τ) = Gfree,conn(U, V ; σ, τ) + R(U, V ; σ, τ) H(U, V ; σ, τ)
= τ 1 + (1 − σ − τ) V + (−τ − στ + τ 2) U + (σ2 − σ − στ) UV + σV 2 + στ R = (1 − χα)(1 − χ0α)(1 − χα0)(1 − χ0α0)
Xinan Zhou YITP , Stony Brook University
Solution of SCFWI in position space:
Gconn(U, V ; σ, τ) = Gfree,conn(U, V ; σ, τ) + R(U, V ; σ, τ) H(U, V ; σ, τ)
= τ 1 + (1 − σ − τ) V + (−τ − στ + τ 2) U + (σ2 − σ − στ) UV + σV 2 + στ R = (1 − χα)(1 − χ0α)(1 − χα0)(1 − χ0α0)
Gconn(U, V ; σ, τ) = Z i∞
−i∞
ds 2 dt 2 U
s 2 V t 2 −kM(s, t; σ, τ)Γ2[2k − s
2 ]Γ2[2k − t 2 ]Γ2[2k − u 2 ]
Xinan Zhou YITP , Stony Brook University
Solution of SCFWI in position space: zero Mellin amplitude
Gconn(U, V ; σ, τ) = Gfree,conn(U, V ; σ, τ) + R(U, V ; σ, τ) H(U, V ; σ, τ)
= τ 1 + (1 − σ − τ) V + (−τ − στ + τ 2) U + (σ2 − σ − στ) UV + σV 2 + στ R = (1 − χα)(1 − χ0α)(1 − χα0)(1 − χ0α0)
Gconn(U, V ; σ, τ) = Z i∞
−i∞
ds 2 dt 2 U
s 2 V t 2 −kM(s, t; σ, τ)Γ2[2k − s
2 ]Γ2[2k − t 2 ]Γ2[2k − u 2 ]
Xinan Zhou YITP , Stony Brook University
Solution of SCFWI in position space: zero Mellin amplitude
Gconn(U, V ; σ, τ) = Gfree,conn(U, V ; σ, τ) + R(U, V ; σ, τ) H(U, V ; σ, τ) ˜ u = u − 4
= τ 1 + (1 − σ − τ) V + (−τ − στ + τ 2) U + (σ2 − σ − στ) UV + σV 2 + στ R = (1 − χα)(1 − χ0α)(1 − χα0)(1 − χ0α0)
Gconn(U, V ; σ, τ) = Z i∞
−i∞
ds 2 dt 2 U
s 2 V t 2 −kM(s, t; σ, τ)Γ2[2k − s
2 ]Γ2[2k − t 2 ]Γ2[2k − u 2 ] H(U, V ; σ, τ) = Z i∞
−i∞
ds 2 dt 2 U
s 2 V t 2 −k f
M(s, t; σ, τ)Γ2[2k − s 2 ]Γ2[2k − t 2 ]Γ2[2k − ˜ u 2 ]
Xinan Zhou YITP , Stony Brook University
R = τ 1 + (1 − σ − τ) V + (−τ − στ + τ 2) U + (σ2 − σ − στ) UV + σV 2 + στ U 2
U mV n Z ds 2 dt 2 U
s 2 V t 2 −kf(s, t) =
Z ds 2 dt 2 U
s 2 V t 2 −kf(s − 2m, t − 2n)
Note the multiplication of monomials has the efgect of shifting arguments,
We hence interpret each monomial as a difgerence operator where each monomial acts as
Xinan Zhou YITP , Stony Brook University
R = τ 1 + (1 − σ − τ) V + (−τ − στ + τ 2) U + (σ2 − σ − στ) UV + σV 2 + στ U 2
U mV n Z ds 2 dt 2 U
s 2 V t 2 −kf(s, t) =
Z ds 2 dt 2 U
s 2 V t 2 −kf(s − 2m, t − 2n)
Note the multiplication of monomials has the efgect of shifting arguments,
b R = τ b 1 + (1 − σ − τ) b V + (−τ − στ + τ 2) b U + (σ2 − σ − στ) d UV + σ c V 2 + στ c U 2
\ U mV n f M(s, t; σ, τ) ⌘ f M(s 2m, t 2n; σ, τ)
× ✓k1 + k2 − s 2 ◆
m
✓k3 + k4 − s 2 ◆
m
✓k2 + k3 − t 2 ◆
n
× ✓k1 + k4 − t 2 ◆
n
✓k1 + k3 − u 2 ◆
2−m−n
✓k2 + k4 − u 2 ◆
2−m−n
We hence interpret each monomial as a difgerence operator where each monomial acts as
Xinan Zhou YITP , Stony Brook University
R = τ 1 + (1 − σ − τ) V + (−τ − στ + τ 2) U + (σ2 − σ − στ) UV + σV 2 + στ U 2
U mV n Z ds 2 dt 2 U
s 2 V t 2 −kf(s, t) =
Z ds 2 dt 2 U
s 2 V t 2 −kf(s − 2m, t − 2n)
Note the multiplication of monomials has the efgect of shifting arguments,
b R = τ b 1 + (1 − σ − τ) b V + (−τ − στ + τ 2) b U + (σ2 − σ − στ) d UV + σ c V 2 + στ c U 2
\ U mV n f M(s, t; σ, τ) ⌘ f M(s 2m, t 2n; σ, τ)
× ✓k1 + k2 − s 2 ◆
m
✓k3 + k4 − s 2 ◆
m
✓k2 + k3 − t 2 ◆
n
× ✓k1 + k4 − t 2 ◆
n
✓k1 + k3 − u 2 ◆
2−m−n
✓k2 + k4 − u 2 ◆
2−m−n
M = b R f M
A structure of the Mellin amplitude:
Xinan Zhou YITP , Stony Brook University
A bootstrap problem
Other consistency conditions:
M = b R f M σkM(u, t; 1/σ, τ/σ) = M(s, t, ; σ, τ) , τ kM(t, s; σ/τ, 1/τ) = M(s, t; σ, τ) σk−2 f M(˜ u, t; 1/σ, τ/σ) = f M(s, t, ; σ, τ) τ k−2 f M(t, s; σ/τ, 1/τ) = f M(s, t; σ, τ)
Xinan Zhou YITP , Stony Brook University
A bootstrap problem
I. superconformal symmetry:
Other consistency conditions:
M = b R f M
Xinan Zhou YITP , Stony Brook University
A bootstrap problem
I. superconformal symmetry:
Other consistency conditions:
simple poles in s, t, u. The positions of the poles are determined by the twists of exchanged single-trace operators, which are restricted by selection rules of the cubic vertices.
Related to the vanishing of extremal cubic coupling in supergravity, also explained by the aforementioned truncation of poles in Mellin space. The residue at each simple pole is a polynomial in the other independent Mandelstam variable.
M = b R f M s0 , t0 , u0 = 2 , 4 , . . . , 2k − 2 τ < ∆1 + ∆2 = 2k
Xinan Zhou YITP , Stony Brook University
A bootstrap problem
I. superconformal symmetry:
Other consistency conditions:
M = b R f M
Xinan Zhou YITP , Stony Brook University
A bootstrap problem
I. superconformal symmetry:
Other consistency conditions:
variables is linear. Required by the good flat space limit. A spin-J exchange diagram has growth J-1, a contact diagram with 2k derivatives has growth rate k. We have particles with highest spin two. And efgectively there are at most two derivatives in the quartic vertices [Arutyunov
Frolov, Klabbers, Savin].
The correct behavior of flat space scalar amplitude in IIB supergravity.
M = b R f M M(βs, βt) ∼ β1 , β → ∞
Xinan Zhou YITP , Stony Brook University
A bootstrap problem
I. superconformal symmetry:
Other consistency conditions:
variables is linear. Required by the good flat space limit. A spin-J exchange diagram has growth J-1, a contact diagram with 2k derivatives has growth rate k. We have particles with highest spin two. And efgectively there are at most two derivatives in the quartic vertices [Arutyunov
Frolov, Klabbers, Savin].
The correct behavior of flat space scalar amplitude in IIB supergravity. Defines a very constrained bootstrap problem! E.g. we can prove that is a rational function.
M = b R f M M(βs, βt) ∼ β1 , β → ∞ f M
Solution
Some experimentation leads to where and . The coeffjcient is uniquely solved up to a constant Reproduces all the position space results: k=2,3,4,5. So far we can only prove the uniqueness of the ansatz for k=2. But we believe the formula holds in general.
I + J + K = k − 2, 0 ≤ I, J, K ≤ k − 2
˜ u = u − 4 aIJK f M(s, t, ˜ u; σ, τ) = X aIJKσIτ J (s − sM + 2K)(t − tM + 2J)(˜ u − uM + 2I) aIJK = Ckkkk ✓ k − 2 I, J, K ◆2 sM = tM = uM = k − 2
Xinan Zhou YITP , Stony Brook University
General solution for arbitrary weights
The solution for arbitrary weights: where and we assumed and is uniquely determined by the bootstrap conditions. can be fixed by taking a lightlike limit in position space. [Aprile,
Drummond, Heslop, Paul]
˜ u = u − 4 aIJK k1 ≥ k2 ≥ k3 ≥ k4
I + J + K = L − 2, 0 ≤ I, J, K ≤ L − 2,
f M(s, t, ˜ u; σ, τ) = X aIJKσIτ J (s − sM + 2K)(t − tM + 2J)(˜ u − uM + 2I)
L = min ⇢ k4, k2 + k3 + k4 − k1 2
tM = min{k1 + k4, k2 + k3} − 2 uM = min{k1 + k3, k2 + k4} − 2
aIJK = Ck1k2k3k4 L−2
I,J,K
2
)I(1 + |k1+k4−k2−k3|
2
)J(1 + |k1+k2−k3−k4|
2
)K
Ck1k2k3k4
Xinan Zhou YITP , Stony Brook University
Xinan Zhou YITP , Stony Brook University
algebraic problem. Avoid the detour into position space. Never computes a single Witten diagram.
simpler auxiliary amplitude. Interesting structures.
3d involves non-local difgerential operator, hard to interpret in Mellin space.
Xinan Zhou YITP , Stony Brook University
Xinan Zhou YITP , Stony Brook University
The solution to the superconformal Ward identity in difgerent spacetime dimensions looks very difgerent. For example, non-local difgerential operators appear in the solution for odd dimensions (but not in, e.g., 4d). However the identity itself takes a universal form Translate the identity into Mellin space! Then we obtain the superconformal Ward identity for Mellin amplitudes.
(@χ − ✏↵@α)G(, 0; ↵, ↵0)
Xinan Zhou YITP , Stony Brook University
We want such that we get some difgerence operators. But solving , in terms of U and V leads to nasty square roots due to the asymmetric appearance of and .
(@χ − ✏↵@α)G(, 0; ↵, ↵0)
U mV n Z ds 2 dt 2 U
s 2 V t 2 −kf(s, t) =
Z ds 2 dt 2 U
s 2 V t 2 −kf(s − 2m, t − 2n)
χ χ0 χ χ0
The position space superconformal Ward identity: How do we exploit this in Mellin space?
Xinan Zhou YITP , Stony Brook University
We want such that we get some difgerence operators. But solving , in terms of U and V leads to nasty square roots due to the asymmetric appearance of and .
(@χ − ✏↵@α)G(, 0; ↵, ↵0)
U mV n Z ds 2 dt 2 U
s 2 V t 2 −kf(s, t) =
Z ds 2 dt 2 U
s 2 V t 2 −kf(s − 2m, t − 2n)
χ χ0 χ χ0
The position space superconformal Ward identity: How do we exploit this in Mellin space? Make it symmetric!
Xinan Zhou YITP , Stony Brook University
(@χ − ✏↵@α)G(, 0; ↵, ↵0)
The position space superconformal Ward identity: We write act with the derivatives and set . We bring all the terms to the minimal common denominator. The denominator is where is the degree of the correlator as a polynomial of the R- symmetry variables. The numerator is a polynomial in of degree
χ ∂ ∂χ = U ∂ ∂U + V ∂ ∂V − 1 1 − χV ∂ ∂V α α = 1/χ χL(1 − χ) L χ L + 1 f0(U, V ; α0) + χf1(U, V ; α0) + χ2f2(U, V ; α0) + . . . + χL+1fL+1(U, V ; α0) = 0
Notice there is an ambiguity in namely, exchanging , does not change U and V. From the previous identity we get another copy for free We add up the two identities and note the following combination is always a polynomial of U and V and we can then interpret them as difgerence
Xinan Zhou YITP , Stony Brook University
χ χ0 U = χχ0 , V = (1 − χ)(1 − χ0) , χn + χ0n f0(U, V ; α0) + χf1(U, V ; α0) + χ2f2(U, V ; α0) + . . . + χL+1fL+1(U, V ; α0) = 0 f0(U, V ; α0) + χ0f1(U, V ; α0) + χ02f2(U, V ; α0) + . . . + χ0L+1fL+1(U, V ; α0) = 0
Procedure of imposing superconformal constraints in Mellin space:
and postpone the action of the U, V derivatives.
and act with the derivative and set .
the , as polynomials of U and V.
Xinan Zhou YITP , Stony Brook University
χ ∂ ∂χ = U ∂ ∂U + V ∂ ∂V − 1 1 − χV ∂ ∂V χ α α = 1/χ (1 − χ)−1χ−L χ χ χ0 χ χ0 G(U, V ; σ, τ) = X
L+M+N=L
σMτ NGLMN(U, V )
Procedure of imposing superconformal constraints in Mellin space:
and reinterpret as follows:
Xinan Zhou YITP , Stony Brook University
GLMN(U, V ) = Z i∞
−i∞
ds 2 dt 2 U
s 2 − ✏(k3+k4) 2
+✏L V
t 2 − ✏ min{k1+k4,k2+k3} 2
MLMN(s, t)Γk1k2k3k4
U @ @U ⇒ s 2 − ✏(k3 + k4) 2 + ✏L
V @ @V ⇒ t 2 − ✏ min{k1 + k4, k2 + k3} 2
U mV n ⇒
shift s by -2m and shift t by -2n This procedure works for all the spacetime dimensions.
Xinan Zhou YITP , Stony Brook University
Next-next-to-extremal correlators in AdS7 (6d (2,0) theories at large c) (“extremality”) E=0 and 2 are called extremal and next-to-extremal. E=4 is the first case where the supergravity computation is non-trivial.
k1 = n + k , k2 = n − k , k3 = k4 = k + 2 E = k2 + k3 + k4 − k1 = 4
Now let’s use the Mellin technique to solve the four-point functions. Analytic structure of the four-point next-next-to-extremal Mellin amplitude is simple: it is composed of a singular part and a regular part. The singular part can be written as a sum of three channels.
Mansatz = Ms + Mt + Mu | {z }
singular
+ Mc |{z}
regular
Moreover the singular part contains only finitely many simple poles, and we know precisely the positions of the poles. They are determined by exchanged single-trace operators subject to the cubic coupling selection rules (R-symmetry and twist cut-ofg). In s-channel, there are simple poles at s=4k+4, 4k+6, It has degree-2 in the R-symmetry cross ratios because it’s next-next-to- extremal, and degree-2 in t because the particles have at most spin 2. In t and u-channels, poles are at t=2n, 2n+2, u=2n, 2n+2.
Xinan Zhou YITP , Stony Brook University
Ms(s, t; σ, τ) = X X λ(s,1)
ij;a σiτ jta
s − (4k + 4) + X X λ(s,2)
ij;a σiτ jta
s − (4k + 6)
0 ≤ i, j ≤ 2 0 ≤ i, j ≤ 2 0 ≤ i + j ≤ 2 0 ≤ i + j ≤ 2 0 ≤ a ≤ 2 0 ≤ a ≤ 2
The regular piece is a polynomial, which account for the contact interactions: It is linear in s and t in order to be consistent with the flat space limit: in flat space we have eleven-dimensional supergravity which contains two derivatives. Such an ansatz is the most general ansatz one can write down that is compatible with the qualitative features of the bulk supergravity.
Xinan Zhou YITP , Stony Brook University
Mc(s, t; σ, τ) = X X µij;ab σiτ jsatb
0 ≤ i, j ≤ 2 0 ≤ i + j ≤ 2 0 ≤ a, b ≤ 1 0 ≤ a + b ≤ 1
Xinan Zhou YITP , Stony Brook University
Impose superconformal symmetry using the previous procedure. This is a finite problem. All the coeffjcients are fixed up to an overall normalization. There is a hidden simplicity. In 6d we can also find a difgerence operator to repackage the full Mellin amplitude into an auxiliary Mellin amplitude. All the next-next-to-extremal auxiliary Mellin amplitude contains one term: Very similar to the case in AdS5. f M(s, t) = Cn,k (s − 4k − 6)(s − 4k − 4)(t − 2n − 2)(t − 2n)(˜ u − 2n − 2)(˜ u − 2n)
Stress-tensor multiplet four-point function for
representation of SO(8) (or the rank-2 symmetric-traceless representation of ). It is dual to a scalar in the bulk.
✤ The scalar field itself. ✤ A vector field with and in the representation . ✤A graviton field with and R-symmetry singlet. All the three fields have twist . Recall the truncation condition is . It implies the series doesn’t truncate and we infinitely many poles.
Xinan Zhou YITP , Stony Brook University
Ok=2 35c ∆ = 1 ∆ = 2 28 ∆ = 3 τ = 1 AdS4 × S7 8c
∆1 + ∆2 − τ = 2m
We will proceed with a simpler ansatz (as in position space approach):
The singular parts are the same as those of the conformal blocks.
three channels, and a regular piece which is a polynomial.
Xinan Zhou YITP , Stony Brook University
Ms-exchange = λgMgraviton(s, t) + λv(σ − τ)Mvector(s, t) + λs(4σ + 4τ − 1)Mscalar(s, t)
Mvector =
∞
X
n=0
√π cos[nπ] (1 + 2n)Γ[ 1
2 − n]Γ[1 + n]
2t + s − 4 s − (2n + 1) Mscalar =
∞
X
n=0
√π cos[nπ] n!Γ[ 1
2 − n]
1 s − (2n + 1) Mgraviton =
∞
X
n=0
−3√π cos[nπ]Γ[− 3
2 − n]
4n!Γ[ 1
2 − n]2
4n2 − 8ns + 8n + 4s2 + 8st − 20s + 8t2 − 32t + 35 s − (2n + 1)
Exchange amplitudes [Mack, Fitzpatrick Kaplan]:
Xinan Zhou YITP , Stony Brook University
Total ansatz: The t and u-channel exchange amplitudes are related to the s-channel by crossing symmetry.
M(s, t; σ, τ) = Ms-exchange + Mt-exchange + Mu-exchange + Mcontact
Mc(s, t; σ, τ) = X X µij;ab σiτ jsatb
0 ≤ i, j ≤ 2 0 ≤ i + j ≤ 2 0 ≤ a, b ≤ 1 0 ≤ a + b ≤ 1
The regular piece is the same as before: degree-2 in R-symmetry and linear in s and t.
Xinan Zhou YITP , Stony Brook University
Solution:
The superconformal symmetry fixes all the relative coeffjcients λv = −4λs , λg = λs 3
Mcontact = πλs 2
Xinan Zhou YITP , Stony Brook University
Solution:
The superconformal symmetry fixes all the relative coeffjcients λv = −4λs , λg = λs 3
Mcontact = πλs 2
λs = − 3 √ 2 4π2N 3/2
Xinan Zhou YITP , Stony Brook University
Solution:
The superconformal symmetry fixes all the relative coeffjcients λv = −4λs , λg = λs 3
Mcontact = πλs 2
λs = − 3 √ 2 4π2N 3/2
Anomalous dimension of
: OIJ
2 OIJ 2
:
Numerical estimate
[Agmon, Chester, Pufu]
∆0 = 2 − 1120 π2 1 CT + . . . ≈ 2 − 113.5 1 CT + . . .
(CT = 64 √ 2 3π N 3/2)
∆∗
0 & 2.01 − 109 1
CT
Xinan Zhou YITP , Stony Brook University
The technique for imposing superconformal symmetry in Mellin space is also useful beyond the supergravity limit. Together with input from localization, the leading M- theory correction was also recently computed [Chester, Pufu, Yin]. More CFT data can be extracted from the amplitude [Chester] (table on the right). The OPE coeffjcients match precisely with the localization results and also agree very well with the numerical estimates from the conformal bootstrap.
Xinan Zhou YITP , Stony Brook University
The strategy and techniques also extend to SCFTs with non-maximally superconformal symmetry (eight superchareges in particular). We look at the four-point functions of moment map operators of the flavor current multiplet has dimension , adjoint of SU(2) R-symmetry and the flavor group. Used in the superconformal bootstrap program with eight supercharges:
hOa(x1, t1)Ob(x2, t2)Oc(x3, t3)Od(x4, t4)i Oa
αβ(x)
∆ = 2✏
Xinan Zhou YITP , Stony Brook University
Examples of SCFTs with eight supercharges include: 5d F(4): Seiberg theories Romans F(4) supergravity coupled to matter 6d (1,0): E-string theory N=2 gauged supergravity coupled to matter
⇒ ENf +1 E8 ⇒
The method of imposing superconformal constraints is essentially the same. But the computation involves two new features:
identity weaker.
All together the constraining power is still pretty strong.
Xinan Zhou YITP , Stony Brook University
Ansatz: exchange (flavor current multi.) + exchange (stress tensor multi.) + contact
Xinan Zhou YITP , Stony Brook University
Ansatz: exchange (flavor current multi.) + exchange (stress tensor multi.) + contact Solution: gives the superconformal block of the flavor current multiplet gives the superconformal block of the stress tensor multiplet All the parameters in the contact part are solved in terms of and .
λF λS λS λF
Xinan Zhou YITP , Stony Brook University
An example: Seiberg theory with s-channel exchange: , if , , if , , otherwise. The t-channel and u-channel exchange are related to the s-channel by crossing. The ansatz for the contact part is linear in s and t.
E8 248 ⊗ 248 = 1 ⊕ 3875 ⊕ 27000 | {z }
S
⊕ 248 ⊕ 30380 | {z }
A
MI
s(s, t; α) = λS,gMgraviton(s, t) + λS,vP1(2α − 1)Mvector(s, t) + λS,sMscalar(s, t)
MI
s(s, t; α) = λF,vMvector(s, t) + λF,sP1(2α − 1)Mscalar(s, t)
I = 1 I = 248 (Adj)
MI
s(s, t; α) = 0
Xinan Zhou YITP , Stony Brook University
Superconformal symmetry fixes and
M1
con =
5πλS(−16350 + 1736s + 3465t) 47616 − α5πλS(−31146 + 3458s + 3465t) 23808 +α2 5πλS(−20712 + 3451s) 47616 + 15π 128 (2α2 − 2α − 1)λF , M3875
con
= −5πλS(−18 + 7t) 47616 + α5πλS(−102 + 14s + 7t) 23808 − α2 5πλS(−40 + 7s) 15872 + 3π 128(2α2 − 2α − 1)λF , M27000
con
= −5πλS(−18 + 7t) 47616 + α5πλS(−102 + 14s + 7t) 23808 − α2 5πλS(−40 + 7s) 15872 − π 256(2α2 − 2α − 1)λF ,
M248
con =
5πλS(−150 + 28s + 21t) 47616 − α5πλS(18 + 14s − 7t) 23808 − α2 35πλS(−12 + s + 2t) 47616 +45π 256 (2α − 1)λF ,
M30380
con
= 5πλS(−150 + 28s + 21t) 47616 − α5πλS(18 + 14s − 7t) 23808 − α2 35πλS(−12 + s + 2t) 47616 .
λS,s = λS , λS,v = −4 3λS , λS,g = 1 15λS , λF,s = λF , λF,v = −1 3λF ,
λS = − 2480 √ 2 3π2N 5/2 λF = 240 √ 2 π2N 3/2
and are fixed by central charges:
λF λS
Xinan Zhou YITP , Stony Brook University
A lot for the future:
Xinan Zhou YITP , Stony Brook University