#### M-theory beyond the supergravity approximation

HJE
M-theory beyond the supergravity approximation
Paul Heslop 0 1
Arthur E. Lipstein 0 1
Durham 0 1
0 We also find solutions
1 Department of Mathematical Sciences, Durham University
We analyze the four-point function of stress-tensor multiplets for the 6d quantum field theory with OSp(8∗|4) symmetry which is conjectured to be dual to M-theory on AdS7 × S4, and deduce the leading correction to the tree-level supergravity prediction by obtaining a solution of the crossing equations in the large-N limit with the superconformal partial wave expansion truncated to operators with zero spin. This correction corresponds to the M-theoretic analogue of R4 corrections in string theory. corresponding to higher-spin truncations, but they are subleading compared to the 1-loop supergravity prediction, which has yet to be calculated.
Conformal Field Theory; M-Theory; Supersymmetric Gauge Theory; 1/N
Expansion
1 Introduction
2
3
4
3.1
3.2
4.1
4.2
4.3
5
Conclusion
1
Introduction
Superconformal partial wave expansion
Tree-level supergravity prediction
Corrections to tree-level supergravity
Free theory contribution
Dynamical contribution
General considerations Spin-zero truncation Higher-spin truncations
N = 6 supersymmetry [4]. On the other hand, the M5-brane worldvolume theory is
expected to be a 6d superconformal quantum field theory with OSp(8∗|4) symmetry, and
we will subsequently refer to it as the 6d (2, 0) theory. For the case of a single M5-brane,
it can be formulated in terms of an abelian (2, 0) tensor multiplet, which consists of a
self-dual two-form gauge field, five scalars, and eight fermions [
5–7
], but it is unclear how
to generalize this construction to describe multiple M5-branes. A crucial hint is provided
by the AdS/CFT conjecture which states that the worldvolume theory for a stack of N
M5-branes is dual to M-theory on AdS7 × S4 with N units of flux through the 4-sphere,
which reduces to 11d supergravity on this background in the limit N → ∞ [
8
].
The goal of this paper is to initiate an approach for extending this description of
multiple M5-branes away from N → ∞, or equivalently beyond the supergravity approximation.
Since it is unclear how to formulate the 6d (2, 0) theory, our strategy will be to use
superconformal and crossing symmetry to deduce the structure of the four-point correlator of
stress-tensor multiplets in this theory. Note that the stress tensor belongs to a 1/2-BPS
multiplet whose superconformal primary, TIJ , is a dimension-4 scalar in the two-index
– 1 –
symmetric traceless representation (14) of the R-symmetry group SO(5) ∼ Sp(4), whose
supergravity dual is a scalar in the graviton multiplet with AdS mass m2 = −8 (in units
of the inverse AdS radius). This is closely related to a broader strategy known as the
conformal bootstrap program, which was pioneered long ago in [
9–11
] and revived more
recently following [12]. More concretely, the idea of the conformal bootstrap is to use the
operator product expansion (OPE) to decompose four-point correlators of primary
operators into a sum over intermediate operators labelled by their spin and scaling dimension.
The contribution of an intermediate primary operator and its descendants can be encoded
in a function of two conformal cross-ratios known as a conformal partial wave, and the
coefficients of the OPE expansion encode the three-point functions of primary operators.
These methods were adapted to N = 4 super-Yang-Mills (SYM) in [27] and in this
paper we adapt them to the 6d (2, 0) theory. In particular, we construct analytical solutions
to crossing equations truncated to spin-0,2, and 4 and find the same number of solutions as
in 2d and 4d. Although we cannot determine the numerical coefficients of these solutions,
we can apply the arguments of [
25
] to deduce how these solutions scale in the
large
N limit.
We find that the spin-0 solution scales like N −5, whereas all the higher spin
solutions are suppressed by at least N −19/3. The tree-level supergravity prediction for
the 4-point function scales like N −3 and so the one-loop supergravity prediction (which
has yet to be computed) should scale like N −6, this implies that the spin-0 solution is
the leading correction to the supergravity prediction for the four-point function of
stress1For a CFT dual to string theory, the number of degrees of freedom scales like N 2 in the large-N limit,
but for the 6d (2, 0) theory it should scale like N 3 [26]. Hence, the 6d crossing equations should be expanded
in 1/N 3.
– 2 –
tensor correlators and that higher-spin solutions are subleading compared to the finite
part of the 1-loop supergravity prediction. The analogous spin-0 solution to the crossing
equations in N = 4 SYM, obtained in [27], corresponds schematically to an R4 correction
to 10d supergravity in AdS5 × S5 (where R is the Riemann tensor), and can therefore
be derived using perturbative string theory (see [28–30] for the derivation of such terms
in flat background). On the other hand, the spin-0 solution we obtain corresponds to an
M-theoretic correction to 11d supergravity in AdS7 × S4 and there is currently no other
method to derive this from first principles, although such corrections have been deduced
in flat background using arguments based on dimensional reduction [31–33].
The structure of this paper is as follows. In section 2 we review the CPW expansion
for the 6d (2, 0) theory following [23], and in section 3 we apply it to the supergravity
prediction for the 4-point function of stress-tensor multiplets to deduce the anomalous
dimensions and OPE coefficients. In section 4, we obtain corrections to the supergravity
prediction for the 4-point function by solving the crossing equations truncated to spin-0,2,
and 4. Finally, we present our conclusions in section 5.
2
Superconformal partial wave expansion
Superconformal symmetry in six dimensions has been studied in a number of
papers [
5, 13–15, 18–20, 22, 23
]. It constrains the four-point function of stress-tensor
multiplets in the 6d (2, 0) theory in terms of a single function, or “prepotential”, of two variables
F (x1, x2) as follows
λ4 (g13g24)2 hT1T2T3T4i = D (SF (x1, x2)) + S12F (x1, x1) + S22F (x2, x2) .
(~xi − ~xj )2)
Here we absorb the two SO(5) indices of TIJ at space-time point ~xi using 5d coordinates,
YiI , thus defining Ti := TIJ (~xi)YiI YiJ . Then we have defined the following (with ~xi2j :=
u = x1x2 =
y1y2 =
gij =
~~xx212123~~xx232244 ,
Y1.Y2Y3.Y4
Y1.Y3Y2.Y4
,
Yi.Yj
~xi4j ,
v = (1 − x1) (1 − x2) =
(1 − y1) (1 − y2) =
~~xx212134~~xx222243 ,
Y1.Y4Y2.Y3
Y1.Y3Y2.Y4
and finally D = − (∂1 − ∂2 + λ∂1∂2) λ, Si = (xi − y1) (xi − y2), and S = S1S2. For more
details see [23].
crossing symmetry constraints:
In addition to superconformal symmetry, the prepotential F must satisfy the following
F (u, v) = F (v, u), F (u/v, 1/v) = v2F (u, v).
Note that exchanging u and v corresponds to taking (x1, x2) → (1 − x1, 1 − x2), and taking
(u, v) → (u/v, 1/v) corresponds to taking (x1, x2) →
x1x−11 , x2−1 . It is convenient to
x2
– 3 –
(2.1)
(2.2)
(2.3)
B0
Bl+2
A00
Al+2 0
Al+2 1
An+l n≥2
T
none
[T T ]55
(half BPS)
[T ∂lT ]14 (protected)
[T ∂lT ]10 (protected)
[T ∂l n−2T ]1 (unprotected)
(2.4)
(2.5)
(2.6)
(2.7)
where the functions g and G can be expanded in terms of CPWs as follows:
∞
m=0
g(x) = X Bmgm(x), gm(x) = xm+12F1 (m + 2, m + 1, 2m + 4, x)
∞
∞
where the blocks are defined via an expansion as
Here the t functions are Jack polynomials
Gmn(x1, x2) = X
X cm+4,n+4(a, b) tm+a,n+b (x1, x2) .
xm+1 − x2m+1,
1
(− (m+3)(n+1) (m−n+1) x1m+2xn−1 − (m−n+3) x1m+1x2n − x1↔x2 , n 6= 0
1 2
tm,n (x1, x2) =
cm,n+2(a, b) =
and the c’s are the coefficients of the blocks expanded in t’s and are given explicitly as
m−n−1
µ −1
1−
2a
(m−n−1) (µ +1)
−
2ab
(ma)
2
(nb)2
(m−n−1) (µ +1) (m+n) a! (2m)a b! (2n)b
where µ = m+a−n−b and xn = Γ (x+n) /Γ(x). The indices m, n label operators with
leading scaling dimension m + n and spin m − n.
The coefficients Bm and Amn in (2.5) and (2.6) are (sums of) squares of OPE
coefficients of operators occurring in the T T OPE. The corresponding operators have lowest
weight states with SO(5) reps, dimension and spin given as:
OPE coefficient Operator φrΔe,pl Example SUGRA operator
∞ ∞
pansion can be summed up and written explicitly in terms of hypergeometric functions [
21
].
In the large-N limit, the four-point function can be computed from tree-level Witten
diagrams for supergravity in an AdS7 × S4 background. The contributions can be written
as the sum of a contribution from free theory (including N 0 and N −3) and the remaining
supergravity contribution (at N −3) [20]. The corresponding prepotential terms we denote
F free and F sugra, respectively.
3.1
Free theory contribution
Let us first analyze the free contribution:
F free = 1 +
1
It is not difficult to check that it satisfies (2.3).
Moreover, decomposing it according to (2.4) gives A = 1, g(x) = 1 +
x
N 3
1
1 − x
.
The OPE coefficients obtained from the expansion (2.6) are then
for m − n even and 0 otherwise.
3.2
Dynamical contribution
is given by [20]
The prepotential for the 4-point function derived from connected AdS Feynman diagrams
F sugra = −
1 λ
2
N 3 uv
The D¯ functions arise from tree-level Feynman diagrams in AdS and are defined in terms
of derivatives of 1-loop massive box functions. For more details, see for example [34]. It is
not difficult to check that (3.2) satisfies (2.3) using the following identities:
DΔ1Δ2Δ3Δ4 (u, v) = D¯ Δ3Δ2Δ1Δ4 (v, u)
¯
DΔ1Δ2Δ3Δ4 (u, v) = vΔ4−ΣD¯ Δ2Δ1Δ3Δ4 (u/v, 1/v),
¯
where Σ = 21 (Δ1 + Δ2 + Δ3 + Δ4). For a more complete list of identities, see [34].
Decomposing the supergravity prepotential according to (2.4) gives A = 0 and
1
N 3
g(x) =
2x 2F1(2, 1, 4, x) −
x +
x
1 − x
.
From (2.5) we see that the first term corresponds to the CPW for the stress tensor
supermultiplet, and the second term cancels the free contribution in (3.1). This agrees with
the discussion around (2.8) that the only twist 4 operator remaining in the supergravity
spectrum is the stress-energy multiplet.
– 5 –
(3.1)
(3.2)
Amnγmsungra ∂m + ∂n
2
Gmn(x1, x2) + AsmungraGmn(x1, x2) , (3.4)
HJEP02(18)4
where γmsungra is the anomalous dimension.2 The derivative of the block (2.7) has the form
(∂m + ∂n)Gmn(x1, x2) = log u Gmn(x1, x2) + Gˆ(x1, x2),
(3.5)
where Gˆ(x1, x2) is analytic as u → 0.
On equating (3.3) with (3.4), we see that the coefficients in the CPW expansion of
Glog correspond to Amnγmsungra/2. Using (2.6), one then obtains the following anomalous
dimensions:
(n−2)(n+1) (n−1)6
γmsungra/2 = − N33 1 + 2(m+n+4)(m−n+3) (m−n+1)(m−n+2)(m+n+5)(m+n+6) ,
for spin m − n even and 0 for m − n odd. Note that they scale like n5 in the large-n limit
(with m − n, the spin, fixed).
On equating the “no-log” part of (3.3) with that of (3.4), we determine the N −3
corrections to the OPE coefficients. We verify that they satisfy the relation
Asmungra =
1
2
∂
∂n
+
∂
∂m
Afmreneγmsungra
(3.6)
seen in other contexts [
25, 27, 35
]. Note in particular that the anomalous dimensions vanish
for n = 0, 1, but their derivatives do not. Nevertheless (3.6) still holds! This corresponds
to the fact that the operators with n = 0, 1 have no anomalous dimensions (see (2.8)), but
have normalisations which do depend on N differently to the free theory. This is in contrast
to N = 4 SYM where the OPE coefficients of protected operators are also protected and
therefore given by the free theory result. This difference will be crucial when deriving
corrections to supergravity in the next section.
Note that we find a similar formula to (3.6) for the corrections to the supergravity
result we find in the next section.
2In general there will be more than one operator with the same naive dimension and so here and
throughout this paper γmn means an averaged anomalous dimension: γmsungra = Pi Amn;iγmn;i .
Pi Amn;i
– 6 –
4.1
General considerations
In this section, we will deduce corrections to the tree-level supergravity prediction for
the 4-point correlator of stress-tensor multiplets in the 6d (2, 0) theory. We will follow
the strategy set out in [
25
], which solved the crossing equations for a 4-point function in a
generic 2d or 4d CFT with a large-N expansion to first nontrivial order in 1/N by truncating
the spin of the CPW expansion, and showed that the number of solutions with spin at most
L is (L + 2)(L + 4)/8. That paper also provided a simple holographic explanation for this
counting by considering a massive scalar field in AdS with local quartic interactions (which
can be thought of as a toy model for the low-energy effective theory of the gravitational
dual), and observing that up to integration by parts and equations of motion, the local bulk
interactions are in one-to-one correspondence with solutions to the crossing equations. In
particular they showed that there are L/2 + 1 independent quartic interactions which can
create or annihilate a state of spin L, with the number of derivatives ranging from 2L to 3L
(note that only even spins are allowed). For example, there is one spin-0 interaction vertex
φ4, and two spin-2 interaction vertices φ2 (∇μ∇ν φ)2 and φ2 (∇μ∇ν ∇ρφ)2 which contain
four and six derivatives, respectively. More generally, they argued that a basis of 4-point
spin-L interaction vertices can mapped to a basis of 4-point S-matrices in flat space whose
elements take the form
(st)L/2uc, c ∈ {0, . . . , L/2},
where s, t, u are Mandelstam variables satisfying s + t + u = 4m2. This corresponds to
interaction vertices of the form
∇μ1...μL/2ν1...νL/2ρ1...ρc φ∇μ1...μL/2 φ∇ρ1...ρc φ∇ν1...νL/2 φ,
where we associate s, t, u with µ, ν, ρ derivatives, respectively. Hence, the total number of
interactions with spin at most L is PlL=/02(L/2 + 1) = (L + 2)(L + 4)/8. The authors of [
25
]
also argued that the large-twist behaviour of the anomalous dimensions is directly related
to the number of derivatives appearing in the bulk interactions. This makes it possible to
deduce how the solutions to the crossing equations should scale in the large-N limit by
analyzing the large-n behaviour of their anomalous dimensions, where n is the twist.
This approach was implemented in N = 4 SYM in [27] where it was shown that in the
large-n limit
γspin−0/γsugra ∼ n6,
(4.1)
which suggests that the first correction to 10d supergravity in the low-energy effective
action has six more derivatives than the supergravity Lagrangian (stringy corrections to
the stress tensor four-point correlator in N = 4 SYM were also obtained in [36]). By
dimensional analysis, this must be suppressed by (lp10d)6 times some function of the string
coupling gs, where lp10d is the 10d Planck length. For example, in tree-level superstring
theory this would correspond to an α′3 correction and at 1-loop it would have the form
G1N0d/α′, where α′ ∼ gs−1/2(lp10d)2 is the square of the string length and G1N0d ∼ (lp10d)8 is
Newton’s constant in 10d. In the supergravity approximation, the former would vanish and
– 7 –
the latter would correspond to a quadratically divergent 1-loop counterterm. To simplify
the discussion, let us fix the value of the string coupling. The contribution to the tree-level
4-point amplitude arising from such an interaction vertex in the low-energy effective action
then goes like G1N0d(lp10d)6 ∼ (lp10d)14. Recalling that lp10d ∼ N −1/4 for IIB string theory
on AdS5 × S5 with N units of flux through the 5-sphere (in units of the AdS radius), we
conclude that the spin-0 solution to the crossing equations is suppressed by N −7/2.
Using similar arguments, we can deduce that higher-spin solutions are suppressed by
an additional factor of at least (lp10d)4 ∼ N −1 compared to the spin-0 solution since they are
dual to interaction vertices with at least four more derivatives. Recalling that the spin-0
contribution scales like N −7/2 (holding gs fixed), we see that higher-spin contributions are
suppressed by at least a factor of N −9/2 and are therefore subleading with respect to the
1-loop supergravity correction (recently found in [37]), which scales like (G1N0d)2 ∼ N −4.
Since the counting of bulk interaction vertices described in [
25
] is not tied to any
particular dimension of AdS, this suggests that the solution counting for the crossing equations
in 2d and 4d should also hold in 6d, which is consistent with our findings below. In
particular, we will find that (4.1) is once again satisfied, indicating that the leading correction
to 11d supergravity contains six more derivatives than the supergravity Lagrangian and
must be suppressed by (lp11d)6. Hence, the contribution to the 4-point amplitude arising
from this interaction vertex goes like G1N1d(lp11d)6 ∼ (lp11d)15 ∼ N −5, where we noted that
for M-theory in AdS7 × S4 with N units of flux through the 4-sphere, lp11d ∼ N −1/3 in units
of the AdS radius. Hence, the spin-0 solution to the crossing equations must be suppressed
by a factor of N −5. Note that for the 6d (2, 0) theory, the only tunable parameter is N .
Using similar arguments one can deduce that if the anomalous dimension for a solution to
the 6d crossing equation scales like nα, then it must be suppressed by a factor of
G1N1d(lp11d)α−5 ∼ N −(3+(α−5)/3).
(4.2)
From this equation, we see that solutions with spin greater than zero are subleading with
respect to the 1-loop supergravity prediction which scales like (G1N1d)2 ∼ N −6 (note that the
1-loop supergravity amplitude contains a cubic divergence [38], so when the corresponding
counterterm is lifted to M-theory it is expected to scale like G1N1d/(lp11d)3 ∼ N −5).
In the following subsections, we will present explicit solutions to the crossing
equations for spin L = 0, 2, 4, confirming the claims above. In the language of section 2, we
will look for solutions to the crossing equations for which the coefficients of the CPW
expansion are zero for m − n > L. Solutions corresponding to a spin-L truncation will be
labelled with subscripts running from 2L to 3L in increments of 2 (corresponding to the
number of derivatives in the corresponding bulk interaction vertices, according to the toy
model of [
25
]).
To obtain the solutions we use the following ansatz (similar to the procedure used for
N = 4 SYM in [27]) for the prepotential:
F (u, v) = λ2uavbD¯ p1p2p3p4 + crossing ,
(4.3)
– 8 –
where “+ crossing” means we sum all 6 terms obtained by permuting the external points.
We split this into a piece proportional to log u and a piece analytic as u → 0
F (x1, x2) = Fno−log (u, v) + log u Flog (u, v)
and we impose the following small u behaviour:
Flog(u, v) = O(u)
Fno−log(u, v) = O(u0) .
Note that as u → 0 the blocks (2.7) are of the form Gmn = O(un−1) for n > 0. Thus the first
constraint in (4.5) arises from insisting that the lowest twist operators which can develop
anomalous dimensions (controlled by the log u piece) are the two-particle supergravity
states T ∂lT in the singlet rep (see (2.8)) with corresponding blocks Gmn with n ≥ 2 of
O(u). However as seen already in the supergravity approximation, the OPE coefficients
(controlled by the non-log piece) of protected operators (with n = 0, 1) can be N -dependent
even though they have no anomalous dimension. These are operators of the form T ∂lT
but in non-singlet reps and so correspondingly the non-log part of F is O(u0).
The small u behaviour of the D¯ functions themselves is
(4.4)
(4.5)
(4.6)
(4.7)
(4.8)
(4.9)
¯
Dp1p2p3p4 |log =
¯
Dp1p2p3p4 |no−log =
( O(up) p ≥ 0
O(u0) p < 0
( O(1/up) p ≥ 0
p := 12 (−p1 − p2 + p3 + p4).
Using this we find that the constraint (4.5) (including all crossing channels) yields six
inequalities on the 6 parameters a, b, pi of (4.3):
n2a ≥ max (0, p1 + p2 − p3 − p4) , 2b ≥ max (0, −p1 + p2 + p3 − p4) , −2a − 2b
+ p1 + p2 + p3 − p4 − 8 ≥ max (0, p1 − p2 + p3 − p4) , max (0, −p1 − p2 + p3 + p4)
+ 2a ≥ 2, max (0, p1 − p2 − p3 + p4) + 2b ≥ 2, max (0, −p1 + p2 − p3 + p4)
− 2a − 2b + p1 + p2 + p3 − p4 − 8 ≥ 2 .
o
The sum of these six inequalities yields a simple constraint on the sum of the indices of
the D¯ function
p1 + p2 + p3 + p4 ≥ 22 .
The D¯ functions satisfy a number of identities, which can be used to reduce any D¯ function
into one of the forms D¯ nnnn, D¯ n+1 n+1 nn, D¯ nnn n+2, D¯ nn n+1 n−1 [34]. Inputting these forms
into the above inequalities allows us to enumerate all possible solutions, although there are
still further identities satisfied by the solutions obtained via the above procedure which we
manually impose to reduce the solution space further. We then order the solutions thus
derived by their value of Pi pi.
In the next subsections, we give the first few cases in this list and perform a CPW
expansion of them. Very nicely, we find that the CPW expansions do indeed all truncate
to small spin, the first to spin-0, the next two to spin-2 and the next three to spin-4 etc.
We further find that in all cases the corrections to the OPE coefficients are related to the
anomalous dimensions by the analogous formula to the supergravity case (3.6)
Acmonrr. =
1
2
∂
∂m
+
∂
∂n
Afmreneγmn
corr. .
(4.10)
The solution with smallest value of P pi = 22 is
we find that it indeed corresponds to the spin-0 truncation of the 6d crossing equations.
Indeed, using (2.4) and (2.6), we deduce the following (averaged) anomalous dimensions:
γnsp, nin−0 = −
(n − 1)8 n6
2240(2n + 3)(2n + 5)(2n + 7)
.
Note that they scale like n11 in the large-n limit. Using (4.2), we see that this solution
should therefore be suppressed by a factor of N −5.
4.3
Higher-spin truncations
The next two solutions are
F spin−2 = 2λ2uv D¯ 6776 + D¯ 7676 + D¯ 7766 , F spin−2 = 6λ2uvD¯ 7777,
4 6
where the overall numerical coefficients are once again unfixed. We find that these two
solutions correspond to spin-2 truncations of the 6d crossing equations. The anomalous
dimensions for the first solution F spin−2 are given by
4
Moreover the anomalous dimensions for the second solution F spin−2 are
6
1 spin−2
2 γ(4) n+2,n =
21 γ(s4p)inn−,n2 =
1 spin−2
2 γ(6) n+2,n = −
21 γ(s6p)inn−,n2 =
Note that the anomalous dimensions for F spin−2 and F spin−2 scale like n15 and n17,
respectively. This agrees with the holographic arguments of [
25
], which predict two spin-2
solutions which scale like n4 and n6 times the spin-0 solution since the corresponding bulk
interaction vertices contain four and six additional derivatives, respectively. Plugging these
large-n scalings into (4.2), we then find that F4spin−2 and F spin−2 should be suppressed by
6
factors of N −19/3 and N −7, respectively, and are therefore subleading with respect to 1-loop
supergravity correction which should scale like N −6 but has not been computed yet.
Finally, the next three solutions are
F spin−4 = uv uD¯7887 + uD¯8787 + vD¯8787 + vD¯8877 + D¯7887 + D¯8877
8
F1s0pin−4 = 2uvD¯8888(u + v + 1)
F1s2pin−4 = 2uv u2D¯9988 + v2D¯8998 + D¯9898 .
As the labelling suggests, one finds that all three of these truncate at spin-4 in a CPW
expansion. The averaged anomalous dimensions computed from the first one are
HJEP02(18)4
(n − 1)10n8(n + 3)2 p6(n)
(n − 1)8n6 p12(n)
,
((n − 1)12)2(n + 3)4
((n − 1)10)2(n + 3)2 q6(n)
((n − 1)8)2q12(n)
2554675200(2n − 1)(2n+1)(2n+3)(2n+5)(2n+7)(2n+9)(2n+11)
The averaged anomalous dimensions of the second spin-4 truncated function are
1 spin−4
2 γ(8) n+4,n =
where
1 spin−4
2 γ(10) n+4,n = −
q6(n) = 13n6 + 273n5 + 1543n4 − 693n3 − 12380n2 + 39564n + 22530
The averaged anomalous dimensions of the third spin-4 truncated function are
((n − 1)10)2(n + 3)2 r8(n)
((n − 1)8)2r14(n)
66421555200(2n − 1)(2n+1)(2n+3)(2n+5)(2n+7)(2n+9)(2n+11)
where
(4.22)
(4.23)
,
(4.24)
(4.25)
(4.26)
r8(n) = 21n8 + 588n7 + 6631n6 + 38409n5 + 282976n4 + 2472015n3 + 10181116n2
We see that the anomalous dimensions for F spin−4,F1s0pin−4 and F1s2pin−4 scale like n19,
8
n21 and n23 respectively. This also agrees with the holographic arguments of [
25
], which
predict three spin-4 solutions which scale like n8, n10, n12 times the spin-0 solution since
the corresponding bulk interaction vertices contain 8, 10 and 12 additional derivatives,
F1s2pin−4 should be suppressed by factors of N −23/3, N −25/3, N −9 respectively.
respectively. Plugging these large-n scalings into (4.2), we then find that F spin−4, F1s0pin−4,
8
5
Conclusion
We have explored some implications of superconformal and crossing symmetry for 4-point
correlators of stress tensor multiplets in the 6d (2, 0) theory in order to gain new insight
into M-theory beyond the supergravity approximation.
We did so by finding crossing
symmetric functions with single discontinuities which have CPW expansions that truncate
to finite spin. Adapting the holographic arguments developed in [
25
], we deduced how
these solutions scale in the the large-N limit by studying the large-twist behaviour of their
anomalous dimensions and find that while the spin-0 solution scales like N −5, all higher-spin
solutions are suppressed by at least N −19/3. As a result, the spin-0 solution corresponds to
the leading correction to the supergravity prediction for the 4-point correlator (which scales
like N −3) and all higher-spin solutions are subleading compared to the 1-loop supergravity
prediction (which scales like N −6 but has not been computed yet).
Our results provide important hints about the low energy effective action of M-theory
on AdS7 × S4. In particular, noting that the anomalous dimensions of the spin-0 solution
scale like n6 compared to those of the supergravity prediction (where n is the twist), this
suggests that the corresponding terms in the effective action contain six more derivatives
than the supergravity Lagrangian, and are subsequently of the form R4, where R is the
Riemann tensor. M-theoretic corrections to 11d supergravity of this form were previously
deduced in flat background using arguments based on dimensional reduction to string
theory [31–33], so it would be very interesting to lift them to AdS7 ×S4 and check that when
they are added to the Lagrangian for 11d supergravity in this background, the resulting
4-point amplitude agrees with the spin-0 solution we obtained. It would also be interesting
to see if the analogous spin-0 solution for N = 4 SYM obtained in [27] can be derived by
adding R4 terms to IIB supergravity in AdS5 × S5.
As mentioned above, after the spin-0 solution to the crossing equations, the next
correction to the 4-point correlator corresponds to a 1-loop supergravity amplitude in AdS7 × S4.
It would therefore be very interesting to compute this correction. This was recently done for
N = 4 SYM for the stress-tensor multiplet correlator in [37] and a correlator involving the
next higher charge half-BPS operator in [39]. This involved disentangling the contributions
of degenerate operators to the four point function [
39, 40
] by using free and
supergravitycorrected correlation functions of the higher charge half BPS operators (corresponding to
higher Kaluza-Klein mode supergravity states) which were recently obtained in [
41, 42
].
An alternative approach which directly gives the data contained in the correlators has also
recently been developed [40, 43–45].
In order to apply these methods to the (2, 0) theory, we would thus need to first
generalize the tree-level supergravity prediction for the 4-point correlator of stress tensor
multiplets to operators with higher R-charges (the first steps in this direction were recently
taken in [46, 47]) and also have better control of the CPWs for higher charge correlators.
Ultimately, we hope that this approach will provide a systematic way to construct
correlation functions of the 6d (2, 0) theory, or equivalently the low-energy effective action of
M-theory on AdS7 × S4, from first principles.
Acknowledgments
We thank Fernando Alday, Francesco Aprile, Agnese Bissi, James Drummond, Tomasz
Lukowski, and Hynek Paul for useful conversations. AL is supported by the Royal Society as
a Royal Society University Research Fellowship holder, and PH by an STFC Consolidated
Grant ST/P000371/1. Both authors express appreciation for MIAPP hosting the workshop
“Mathematics and Physics of Scattering Amplitudes” where this work was initiated.
Open Access.
This article is distributed under the terms of the Creative Commons
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any medium, provided the original author(s) and source are credited.
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