Supersymmetry Constraints and String Theory on K3
JHE
Supersymmetry Constraints and String Theory on K3
YingHsuan Lin 0 2
ShuHeng Shao 0 2
Yifan Wang 0 1
Xi Yin 0 2
0 Cambridge , MA 02138 U.S.A
1 Center for Theoretical Physics, Massachusetts Institute of Technology
2 Je erson Physical Laboratory, Harvard University
We study supervertices in six dimensional (2; 0) supergravity theories, and derive supersymmetry nonrenormalization conditions on the 4 and 6derivative fourpoint couplings of tensor multiplets. As an application, we obtain exact nonperturbative results of such e ective couplings in type IIB string theory compacti ed on K3 surface, extending previous work on type II/heterotic duality. The weak coupling limit thereof, in particular, gives certain integrated fourpoint functions of halfBPS operators in the nonlinear sigma model on K3 surface, that depend nontrivially on the moduli, and capture worldsheet instanton contributions.
Superstrings and Heterotic Strings; Scattering Amplitudes; Supergravity

HJEP12(05)4
Models
2.1
2.2
2.3
4.1
4.2
5.1
5.2
1 Introduction 2
Supervertices in 6d (2; 0) supergravity
6d (2; 0) Superspinorhelicity formalism
Supervertices for tensor multiplets
Supervertices for supergravity and tensor multiplets
3
4
Di erential constraints on f (4) and f (6) couplings An example of f (4) and f (6) from type II/heterotic duality
Type II/heterotic duality Heterotic string amplitudes and the di erential constraints 4.2.1 4.2.2
Oneloop fourpoint amplitude
Twoloop fourpoint amplitude
5 Implications of f (4) and f (6) for the K3 CFT
Reduction to the K3 CFT moduli space
A1 ALE limit
6
Discussions
A Explicit check of the di erential constraints
A.1 Fourderivative coupling f (4)
A.2 Sixderivative coupling f (6)
B Relation to 5d MSYM amplitudes
B.1 Fourderivative coupling f (4)
B.2 Sixderivative coupling f (6)
We will focus on the 4 and 6derivative couplings of tensor multiplets, of the
schematic form
fa(4bc)d( )HaHbHcHd
and fa(6b;)cd( )D2(HaHb)HcHd;
where stands for the massless scalar moduli elds, that parameterize the moduli space [11]
M = O( 21;5)nSO(21; 5)=(SO(21)
SO(5));
and we have omitted the contraction of the Lorentz indices on the selfdual tensor elds
Ha in the tensor multiplets (not to be confused with the antiselfdual tensor elds in the
supergravity multiplet), a = 1;
fa(6b;)cd( ). These equations are of the schematic form
By consideration of the factorization of sixpoint superamplitudes through graviton
and tensor poles, we derive second order di erential equations that constrain fa(4bc)d( ) and
fi(j4k)` ! gIVIBK`s43 Aijk`(');
oneloop and twoloop heterotic string amplitudes, with the results
fa(4bc)d =
fa(6b;)cd =
IK JL + IL JK
Z
d
2
(yj ; )
( )
;
Z
Y d
2 IJ
(det Im ) 2 10( )
(yj ; ) is the theta function of the
(yj ; ) is an
analogous genus two theta function. The precise expressions of these theta functions will
be given later. The above two expressions depend on the embedding of the lattice
into
R21;5 through the theta functions, and the space of inequivalent embeddings is the same
as the moduli space M (1.1) of the 6d (2; 0) supergravity.
cusp form of SL(2; Z), and
10( ) is the weight 10 Igusa cusp form of Sp(4; Z). The result
for the 4derivative term f (4) has previously been obtained in [12].
( ) = 24( ) is the weight 12
We will verify, through rather lengthy calculations, that (1.3) indeed obey second order
di erential equations of the form (1.2), and x the precise numerical coe cients in these
equations.
While the expressions (1.3) for the coupling coe cients f (4) and f (6) are fully
nonperturbative in type IIB string theory, the results are nontrivial even at string treelevel.
For instance, in the limit of weak IIB string coupling gIIB, f (4) reduces to
(1.1)
(1.2)
(1.3)
(1.4)
HJEP12(05)4
Z d2z
2
; 20), the moduli of the 2d (4; 4) CFT given by
the supersymmetric nonlinear sigma model on K3 (we will refer to this as the K3 CFT).
From the point of view of the worldsheet CFT, we can express Aijk`(') as an integrated
fourpoint function of marginal BPS operators of the K3 CFT, through the expansion
j j
iRR(z) are the weight ( 14 ; 14 ) RR sector superconformal primaries
in the Rsymmetry singlet, related to the NSNS sector weight ( 12 ; 21 ), exactly marginal,
superconformal primaries by spectral ow. The zintegral is de ned using Gamma function
regularization, or equivalently, analytic continuation in s and t from the domain where the
integral converges. While Aijk` gives the treelevel contribution to f (4), Bij;k` captures the
treelevel contribution to f (6).
Note that, in contrast to the Riemannian curvature of the Zamolodchikov metric [13],
which is contained in a contact term of the fourpoint function [14], Aijk` and Bij;k` are
determined by the nonlocal part of the fourpoint function and do not involve the contact
term. Unlike the Zamolodchikov metric which has constant curvature on the moduli space
of K3 (with the exception of orbifold type singularities), Aijk` and Bij;k` are nontrivial
functions of the moduli. In particular, the latter coe cients blow up at the points of the
moduli space where the CFT becomes singular, corresponding to the K3 surface developing
an ADE type singularity, with zero B eld ux through the exceptional divisors.
We can give a simple formula for Aijk` in the case of A1 ALE target space, which may
be viewed as a certain large volume limit of the K3. In this case, the indices i; j; k; ` only
take a single value (denoted by 1), corresponding to a single multiplet that parameterizes
the 4dimensional moduli space
MA1 = R
3
Z2
S1
:
MA1 has two orbifold xed points by the Z2 quotient, one of which corresponds to the
C2=Z2 free orbifold CFT, whereas the other corresponds to a singular CFT, singular in
the sense of a continuous spectrum, that is described by the N = 4 A1 cigar CFT [15{17].
While the Zamolodchikov metric does not exhibit any distinct feature between these two
points on the moduli space, the integrated fourpoint function A1111 does. The latter is a
harmonic function on MA1 , is nite at the free orbifold point, but blows up at the A1 cigar
point. When the A1 singularity is resolved, in the limit of large area of the exceptional
divisor, we nd that A1111 receives a oneloop contribution in 0, plus worldsheet instanton
contributions (5.21).
The paper is organized as follows. In section 2 we set up the superspinorhelicity
formalism in 6d (2; 0) supergravity and classify the supervertices of low derivative orders.
In section 3, we derive the di erential equation constraints on the fourpoint 4 and
6derivative coupling between the tensor multiplets based on the absence of certain sixpoint
supervertices, with some modelindependent constant coe cients yet to be determined.
{ 3 {
In section 4, using type II/heterotic duality, we obtain the exact nonperturbative 4 and
6derivative couplings in type IIB string theory on K3. We verify that these couplings
indeed satisfy the di erential equations and x the constant coe cients in these equations.
In section 5, we consider the weak coupling limit of the above results, which gives the
integrated fourpoint function of BPS primaries in the K3 CFT, with an explicit dependence
on the moduli space. We also consider the A1 ALE sigma model limit of the K3 CFT and
study the 4derivative couplings in that limit.
Supervertices in 6d (2; 0) supergravity
6d (2; 0) Superspinorhelicity formalism
Following [18{20], we adopt the convention for 6d spinorhelicity variables
pAB = A
B
;
pAB
2
1 ABCDpCD = eA _ eB _
_ _ ;
and de ne Grassmannian variables
I and e_ I , where the lower and upper A; B are
SO(5; 1) chiral and antichiral spinor indices respectively, ( ; _ ) are SU(2)
SU(2)
little group indices, and I = 1; 2 is an auxiliary index which may be identi ed with the spinor
index of an SO(3) subgroup of the SO(5) Rsymmetry group.
Let us represent the 1particle states in the (2; 0) tensor multiplet and the (2; 0)
supergravity multiplet as polynomials in the Grassmannian variables
I and ~ _ I . The
1particle states of the (2; 0) tensor multiplet transform in the following representations of
the SU(2)
SU(2) little group,
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
These 1particle states can be represented collectively as a polynomial
up to degree 4 in , but with no ~. In particular, the monomial
to the selfdual two form (3; 1) and the monomials 1;
I J
I J
IJ corresponds
; 4 correspond to the 5
scalars (1; 1).
The 1particle states of the (2; 0) supergravity multiplet, on the other hand, transform
in the following representations of the SU(2)
SU(2) little group,
IJ corresponds to the graviton (3; 3) and the
, and 4 correspond to the 5 antiselfdual tensor
elds (1; 3).
{ 4 {
The 16 supercharges are represented on 1particle states as
They obey the supersymmetry algebra
The 10 SO(5) Rsymmetry generators are
qAI = A
I ;
fqAI ; qBJ g = pAB IJ ;
fq; qg = fq; qg = 0:
J
I
when acting on 1particle states.
In an npoint scattering amplitude, we will associate to each particle spinor helicity
variables iA ; ~iA _ and Grassmannian variables i I ; ~i _ I , with i = 1;
; n.
Correspondingly we de ne the supercharges for each particle,
qiAI = iA
iI ;
:
The supercharges acting on the amplitude are represented by sums of the 1particle
representations
QAI =
X qiAI ;
QAI =
X qiAI ;
and so are the Rsymmetry generators
i
i
X( i2)IJ ;
J
I
:
(2.6)
(2.7)
(2.8)
(2.9)
(2.10)
(2.11)
(2.12)
(2.13)
The solutions to the supersymmetry Ward identities can be expressed in terms of the
superspinorhelicity variables. If such expression is local in these variable, we call it a
supervertex, otherwise it is a superamplitude. Among all the supervertices, the Dterm
type takes the form
where 8(Q) = QA;I QAI , and P is a polynomial in the superspinorhelicity variables
i; ~i; i; ~i associated with the external particles labeled by i = 1; : : : ; n, that is Lorentz
invariant and little group invariant. On the other hand, the Fterm supervertices are of
the form
8(Q)Q8P( i; ~i; i; ~i);
8(Q)F ( i; ~i; i; ~i);
where F is a Lorentz invariant and little group invariant polynomial in the
superspinorhelicity variables that cannot be written in the Dterm form [9, 21{24]. From momentum
counting, we expect Dterm supervertices in general to come at or above 8derivative order.
In the following subsections, we will focus on three and fourpoint supervertices in
the (2; 0) supergravity. We will start with supervertices involving tensor multiplets only,
whose classi cation coincides with that of the (2; 0) SCFT on the tensor branch.
We
will then introduce couplings to the supergravity multiplet and classify the supervertices
thereof. In particular, we will discover that the fourpoint Dterm supervertices involving
supergravitons do not appear until at 12derivative order.
{ 5 {
Among the fourpoint supervertices that only involve the (2; 0) tensor multiplets, the
leading Fterm ones arise at 4 and 6derivative orders and take the form
where QAI = Pi4=1 qiAI and 8(Q) = QA;I QAI . The coe cients f (4); f (6) are constant in
s; t; u but functions of the moduli. Their dependence on the moduli is the main object of
the current paper. The subscripts a; b; c; d label the 21 tensor multiplets. They contain
the H4 and D2H4 couplings, respectively, where H denotes the selfdual three form
eld
strength in the 21 tensor multiplets.
8(Q)Q8P( i; ~i; i). For
There are also fourpoint Dterm supervertices of the form
this expression to be nonvanishing, we need P 8to contain at least eight 's. On the other
hand, by exchanging the order of 8(Q) and Q , we see that we cannot have more than
eight 's in P because there are in total 4
4 's from the four 1particle states. Hence
the lowest derivative order Dterm supervertices for tensor multiplets arise at 8derivative
(2.14)
(2.15)
(2.16)
(2.17)
order
Pi<j i j
4 4
.
though we could act Q on other little group singlets made out of eight i's, like for
instance ( 12)IJ ( 22)IJ ( 32)KL( 42)KL, such expressions always turn out to be proportional to
Next, we will show that threepoint supervertices of tensor multiplets are absent. In
general it is more intricate to write down the threepoint supervertices due to the kinematic
constraints,1 and we will work in a frame where the three momenta p1; p2; p3 lie in a null
plane spanned by e0 + e1 (the 0 direction) and e2 + ie3 (the 1 direction). The null plane
is equivalently speci ed by the linear operator,
such that the spinor helicity variables associated with the momenta satisfy
Nb = p1mp2n mn
(p+)2 0 1 ;
NbAB
iB
= 0;
Nb AB eiB _ = 0:
We write both the lower (chiral) and upper (antichiral) SO(5; 1) spinor index A as (
)
which represent spins on the 01 and 23 planes, while the spin in the 45 plane is xed by
the 01 and 23 spins due to the chirality condition. For instance, we write iA
and eiA _ as ei _ . By de nition, is0s1 (or eis_0s1 ) has charge s20 and s21 under the SO(1; 1)01
boost and SO(2)23 rotation in the 01 and 23 planes, respectively. Then by the chirality
as i ,
1As will be shown in this section, for any choice of the three momenta, two QAI 's and two QAI 's vanish.
While this implies that 8(Q) = 0 (and hence the naive construction of the supervertices as in the
fourpoint and higherpoint cases does not apply), the supersymmetry Ward identities associated with the two
vanishing QAI 's also become trivial, which means that the full factor of 8(Q) is not needed in a supervertex.
{ 6 {
in the three orthogonal planes. Note that in the last two rows, we choose a frame where p1 is
parallel to e0 + e1, and p2 is parallel to e2 + ie3.
condition, is0s1 has charge ( 1) s0 +2s1
2
and ei
s0s1 has charge
\tiny group" that rotates the 45 plane. The momentum p+ has charge 12 under both the
SO(1; 1)01 and SO(2)23, and is not charged under the SO(2)45. For clarity, these charges
under the SO(2)45
are summarized in table 1.
The constraint (2.17) implies that i
= ei _
Q
I
and QI
vanish identically. The expression
= 0. Consequently the supercharges
group charge
following form
is thus annihilated by all 16 supercharges QAI and QAI . Since (2.18) has SO(2)45 tiny
1, a general threepoint supervertex for the tensor multiplets must take the
Y
I=1;2
I
Q+ Q +Q++
I
I
0
Q+ Q +Q++
I
I A fabc( i; i);
Symbol
++ / e++
+
/ e+
+ / e +
/ e
p+
p
p
+
1
+
2
1
2
1
2
1
2
1
2
1
2
1
0
SO(1; 1)01
12 /
12 /
where fabc must be annihilated by Q up to terms proportional to Q, invariant with respect
to the little groups, and have charge +1 under tiny group. By consideration of CPT
conjugation,2 fabc cannot depend on i (otherwise the CPT conjugate expression would
2For an npoint (n
4) supervertex or superamplitude
the CPT conjugate is
For a threepoint supervertex
the CPT conjugate is
V = 8(Q)F( i; i; ei; ei);
i :
n
I=1;2
V =
Y QI+ QI +QI++F( i; i; ei; ei);
V =
I=1;2
i :
n
{ 7 {
involve fewer than 6 's and cannot be proportional to Q6). Little group invariance then
forces it to be a function of the momenta only. In particular, since all three momenta are
SO(2)45 tiny group invariant, fabc would have to be tiny group invariant by itself which
then forces it to vanish.
Supervertices for supergravity and tensor multiplets
We will now incorporate the coupling to the supergravity multiplet. Below to ease the
notation, we will de ne
qeAI
eA _ e _ I ;
e
(q2)AB
qeAI qeBJ IJ :
(2.24)
which is at 12derivative order and contains the D4R4 coupling.3
tensor multiplet and two supergravity multiplets as external states
We also have a fourpoint Fterm supervertex at 8derivative order that involves one
e
e
8(Q)(q12)AB(q22)CDp3AC p4BD;
which contains the D2(R2H2) coupling.4
We can also obtain a 10derivative Fterm by
multiplying the 8derivative one (2.27) by s12, which contains the D4(R2H2) coupling. The
lowest derivative order Dterm is
8(Q)Q8 34 44(qe12)AB(q22)CDp3AC p4BD;
e
which is at 12derivative order and contains the D6(R2H2) coupling.
The fact that D term fourpoint supervertices involving the supergravity multiplet only
start appearing at 12derivative order is a special feature of (2; 0) supergravity, in contrast
to the naive momentum counting that may suggest they occur at 8derivative order (as in
the case of maximally supersymmetric gauge theories, with sixteen supersymmetries).
3This is also the only Dterm supervertex of supergravity multiplet at the 12derivative order. The e's
anticommute with the supercharges. Their only role is to form supergraviton states and they must be
contracted with the e's to form little group singlets.
4The 6derivative order supervertex that contains the R2H2 coupling appears to be absent.
{ 8 {
(2.25)
(2.26)
(2.27)
(2.28)
0
1
Y
I=1;2
0
Threepoint supervertices. Let us now discuss the threepoint supervertices between
the (2; 0) supergravity multiplet and the tensor multiplets. Below we will explicitly
construct the 2derivative supervertices and also argue for the absence of threepoint
supervertices at 4derivative order and beyond.
At 2derivative order, the 3supergraviton supervertex is given by
I
Q+ Q +Q++
I
I A (qe12)AA0 (q22)BB0 (q32)CC0 PABCA0B0C0 ( i; i):
e e
(2.30)
I
The power of p+ is xed by the SO(1; 1)01 and SO(2)23 invariance, and this expression
is also invariant under the SO(2)45 tiny group, thereby consistent with the full SO(1; 5)
Lorentz symmetry. More generally, a cubic supervertex of the supergravity multiplet must be of the form
PABCA0B0C0 must be annihilated by Q up to terms proportional to Q, invariant with
respect to the little groups, and must have charge 2 under tiny group scaling. As we have
argued for the 3tensor supervertices in the previous subsection, by applying CPT
conjugation and little group invariance, we conclude PABCA0B0C0 is a tiny group invariant that
only depends the momenta. The tiny group invariance of the full amplitude then forces
(AA0; BB0; CC0) to have a total of 4
's and 8 +'s, and then SO(1; 1)01 and SO(2)23
invariance forces PABCA0B0C0 to scale like (p+) 4, and we are back to the twoderivative
cubic supervertex (2.29). This rules out any higher derivative cubic supervertices of the
supergravity multiplet.
Now let us consider the threepoint supervertex for one supergravity and two tensor
multiplets. We can further choose the lightcone coordinates to be aligned with the momenta
of the rst and second particle, by demanding that p1 = p1+(e0 + e1), and p2 = p2+(e2 + ie3).
This amounts to the restriction
At twoderivative order, the gravitytensortensor supervertex is
1
+ = 0;
+
2
= 0:
1
0
1
Y
I=1;2
I
Q+ Q +Q++
I
I A (qe12)(+ ;+ );
where p1 labels the momentum of the supergraviton. At 4derivative order and beyond,
there do not appear to be threepoint supervertices for the gravitytensortensor coupling,
using the same argument as above. Similarly one can argue that no gravitygravitytensor
supervertex exists.5
5It appears that one can write down a 2derivative order supervertex
1
(p1+)2p2+ I=1;2
Y QI+ QI +QI++ (q12)(+ ;+ )(q22)(++; +) + (1 $ 2):
e e
!
{ 9 {
(2.31)
(2.32)
(2.33)
Supervertices
ggg
gtt
ggt
ttt
gggg
ggtt
tttt
2 only
2 only absent absent Dterms: 12+
Dterms: 12+
Dterms: 8+
Fterms: 8 and possibly 12+
Fterms: 8, 10, and possibly 12+
Fterms: 4, 6, and possibly 8+
and tensor supermultiplets which include R and H, respectively. The derivative order includes the
derivatives implicit in the elds. For example, D2(R2H2) is regarded as an 8derivative supervertex
(2 + 2
2 + 2
1).
It is claimed in [25] that in type IIB string theory on K3, there is a CPodd RH2
e ective coupling that arises at oneloop order, where here H refers to a mixture of the
selfdual twoform in a tensor multiplet and the antiselfdual twoform in the multiplet
that also contains the dilaton . This would seem to correspond to a 4derivative cubic
supervertex. A more careful inspection of the 6d IIB cubic vertex of [25] shows that it in
fact vanishes identically [26], which is consistent with our nding based on the super spinor
helicity formalism.
The classi cation of threepoint and fourpoint supervertices given in this section is
summarized in table 2. In particular, the three and fourpoint supervertices are all
invariant under the SO(5) Rsymmetry (2.8). In other words, our classi cation implies that
SO(5) breaking supervertices in (2; 0) supergravity can only start appearing at vepoint
and higher. The simplest examples of such supervertices are 8(Q) at npoint with n > 4,
which transform in the [n
4; 0] representations of the SO(5) Rsymmetry [27].
3
Di erential constraints on f (4) and f (6) couplings
In this section, we shall deduce the general structures of the di erential constraints on f (4)
and f (6) couplings due to supersymmetry, using superamplitude techniques [9, 10, 23].
The construction of the f (4) and f (6) supervertices in (2; 0) supergravity gives the
onshell supersymmetric completion of the H4 and D2H4 couplings. In particular, given
their relatively low derivative orders, such supervertices must be of Fterm type which are
rather scarce and have been classi ed and explicitly constructed in the previous section.
However, after restoring the full SO(1; 5) Lorentz invariance, the resulting expression cannot be a local
supervertex. This can be seen by noting that the expression is SO(5) Rsymmetry invariant, and there simply
does not exist any 2derivative threepoint coupling that involves two
elds from the gravity multiplet and
multiplet states while the dotted lines stand for the supergravity multiplet states. The black circles
represent the 4derivative fourtensormultiplet supervertex, and the trivalent vertices represent the
2derivative supervertex involving one gravity and two tensor multiplets.
As we shall see below, the absence of certain higher point supervertices of these derivative
orders will lead to di erential constraints on the moduli dependence of the aforementioned
couplings in the quantum e ective action of (2; 0) supergravity.
For example, we can expand the supersymmetric f (4) coupling, in terms of the moduli
elds, and obtain higherpoint vertices. In particular, the resulting sixpoint '2H4 coupling
in the singlet representation of SO(5) Rsymmetry can be related to a symmetric double soft
limit of the corresponding sixpoint superamplitude (at 4derivative order) [10, 27]. The
absence of SO(5) Rsymmetry invariant sixpoint supervertices at 4derivative order [27]
means that this sixpoint '2H4 coupling from expanding f (4) cannot possibly have a local
supersymmetric completion. Rather, it must be related to polar pieces of the
superamplitude via supersymmetry; in other words, it is xed by the residues in all factorization
channels. The '2H4 superamplitude can only factorize through the 4derivative
supervertex for tensor multiplets and 2derivative cubic supervertices for two tensor and one
graviton multiplets (see gure 1), giving rise to
r(e rf)fa(4bc)d = U fa(4bc)d ef + V f((e4()abc d)f) + W fe(f4)(ab cd):
Here a natural SO(21; 5) homogeneous vector bundle W over M arises as the quotient
V
V
where R21;5 transforms as a vector under SO(21) SO(5). We de ne the covariant derivative
rai, where a = 1; : : : ; 21 and i = 1; : : : ; 5, by the SO(21; 5) invariant connection on W that
gives rise to the symmetric space structure of the scalar manifold M. Further imposing
invariance under the SO(5) Rsymmetry means we can focus on the SO(21) subbundle
W. The coupling fa(4bc)d becomes a section of the symmetric product vector bundle
, on which the second order di erential operator r(e rf) acts naturally.
(3.1)
(3.2)
tensor multiplet states while the dotted lines stand for the supergravity multiplet states. The black
and white circles represent the 4 and 6derivative fourtensormultiplet supervertices, respectively,
and the trivalent vertices represent the 2derivative supervertex involving one gravity and two tensor
multiplets.
For the f (6) coupling, recall that it is de ned as the coe cient in the superamplitude
8(Q)(fa(6b;)cds + fa(6c;)bdt + fa(6d);bcu):
Due to the relation s + t + u = 0, there is an ambiguity in the de nition of fa(6b;)cd, where we
can shift f (6) by a term that is totally symmetric in three of the four indices. We x this
ambiguity by demanding that fa(6(b);cd) = 0, which makes f (6) a section of the V
vector
bundle. The corresponding D2('2H4) superamplitude can also factorize through two f (4)
supervertices (see gure 2), and we end up with the following di erential constraint
2r(e rf)fab;cd = u1fab;cd ef + u2(fe(f6;)ab cd + fe(f6;)cd ab) + u02fe(f6;)(c(a b)d)
(6) (6)
+ u3(fe(a6;)fb cd + fe(c6;)fd ab) + u03fe((6c);f(a b)d)
2
+ u4(fe((6c);ab fd) + fe((6a);cd fb)) + u5(fe((6b);a)(c d)f + fe((6d);c)(a b)f )
+
v1 (fg(a4)b(cfd()4e)fg +fg(c4d)(afb()4e)fg)+v2fegabfc(d4f)g +v3fe(g4a)(cfd()4b)fg +(e $ f ) :
(4)
(3.3)
(3.4)
As we shall argue in the next section, the constant coe cients in (3.1) and (3.4) can be xed
using results from the type II/heterotic duality and heterotic string perturbation theory.
4
An example of f (4) and f (6) from type II/heterotic duality
In section 3, we wrote down the di erential constraints (3.1) and (3.4) on the 4 and
6derivative fourpoint couplings f (4) and f (6) between the 21 tensor multiplets in 6d (2; 0)
supergravity, with undetermined modelindependent constant coe cients. To determine
these coe cients, we can consider the speci c example of fourpoint scattering amplitudes
in type IIB string theory on K3. In this section, we will relate the exact nonperturbative
4and 6derivative couplings in type IIB on K3 to a certain limit of the one and twoloop
amplitudes in the T 5 compacti ed heterotic string theory, via a chain of string dualities. With
explicit expressions for the heterotic amplitudes, we verify the di erential constraints (see
appendix A for the detailed computations), and thereby determine the modelindependent
constant coe cients.
4.1
gA
T 4
Mh
We consider type IIB string theory on K3
rB. The 6d limit of interest corresponds to keeping gB
SB1 with string coupling gB, and circle radius
O(1) while sending rB ! 1. We
shall work in units with type II string tension 0 = 1. By Tduality, we can equivalently
look at type IIA string theory on K3
SA1 with string coupling gA = gB=rB
rA and
circle size rA = 1=rB. In terms of type IIA parameters, the 6d limit corresponds to
rA ! 0. Now we use type IIA/heterotic duality to pass to heterotic string theory on
S1 where the size of both T 4 and S1 are of order rA in type II string units. Since the
heterotic string is dual to a wrapped NS5 brane on K3, their tensions satisfy the relation
HJEP12(05)4
1=`h
1=gA
1=rA. The heterotic string coupling, on the other hand, can be xed
by matching the 6d (or 5d) supergravity e ective couplings
to be gh
1=rA
Mh.6 Hence in the limit where the circle of K3
SB1 in the type IIB
picture decompacti es rB ! 1, we have
1
g
2
A
Mh8rA4
g
2
h
gh
Mh ! 1
(4.1)
(4.2)
in the dual T 5 compacti ed heterotic string theory.
Under the duality, the 21 tensor multiplets of (2; 0) supergravity on SB1 are related
to the 21 abelian vector multiplets of heterotic string on T 5. In particular the e ective
action of the tensor multiplets in the (2; 0) supergravity is captured by that of the vector
multiplets in heterotic string. Let us now focus on the fourpoint amplitude of abelian
vector multiplets in heterotic string on T 5. As we shall see in the next subsection, apart
from the treelevel contribution at 2derivative order due to supergraviton exchange, the
fourpoint amplitudes at 4derivative and 6derivative orders receive contributions up to
oneloop and twoloop, respectively. Furthermore we will argue that, in the limit of interest
gh
Mh ! 1, these couplings in the e ective action are free from contributions at higher
loop orders.
can be written as
Relative to the treelevel contribution to the 2derivative amplitude f (2), the
4derivative f (4) coupling in the 6d (2; 0) supergravity from type IIB on K3, which contains H4
f (4)
f (2)
lim
`
2
h
;
(4.3)
where ngh2n is the nloop contribution. By using type I/heterotic duality and con rmed by
twoloop computation in [28{31], it has been argued that the F 4 coupling in heterotic string
is does not receive contributions beyond oneloop, namely
n = 0 for n
2. Therefore, we
expect that f (4) is completely captured by the oneloop contribution 1gh2`2h.7 Indeed, we
6One can also derive this using the equivalence between the IIA string and the wrapped heterotic NS5
brane on T 4.
7Note that the treelevel contribution
0`2h to the 4derivative coupling vanishes in the limit of interest
gh
1=`h ! 1. Similarly, the treelevel and oneloop contributions 0`4h and 1`4hgh2 to the 6derivative
coupling vanish in that limit.
will see in section 4.2.1 that the heterotic oneloop amplitude (4.6) satis es the di erential
constraint for the 4derivative coupling (3.1) in 6d (2; 0) supergravity.
Likewise, the 6derivative f (6) coupling which contains D2H4 can be written as
lim
We will see in section 4.2.2 that the twoloop contribution (4.31) corresponding to the
2gh4`4h term alone satis es the di erential constraint (3.4) for the 6derivative coupling in
6d (2; 0) supergravity. This strongly suggests that the D2F 4 does not receive higher than
twoloop contributions in the T 5 compacti ed heterotic string theory, though we are not
aware of a clear argument.8
HJEP12(05)4
Heterotic string amplitudes and the di erential constraints
In this subsection, we compute the fourpoint amplitude of scalars in the abelian vector
multiplets in
ve dimensions, of heterotic string on T 5, or more precisely, heterotic string
compacti ed on the Narain lattice
21;5. As explained in the previous subsection, the
cove dimensions at genus one and genus
two in heterotic string capture exactly the six dimensional e ective couplings f (4) and f (6)
of type IIB string theory on K3, expanded in the string coupling constant including the
instanton corrections. These results can be extracted by slightly modifying the 10d
heterotic string amplitudes, computed by D'Hoker and Phong (see for instance (6.5) of [29]
and (1.22) of [31]). Furthermore, we x the constant coe cients in the di erential
constraints (3.1) and (3.4) by explicitly varying the heterotic amplitudes with respect to the
moduli elds.
4.2.1
Oneloop fourpoint amplitude
The scalar factor in the 4gauge boson amplitude at oneloop takes the form9
A1 =
Z
d
2
2
( )
i=1
2
( ; ) Y4 d2zi e 12 Pi<j sijG(zi;zj)
* 4
Y jai (zi)
i=1
+
:
(4.5)
Here
denotes the even unimodular lattice
21;5,
( ) =
form of SL(2; Z), and
is the theta function of the lattice
( )24 is the weight 12 cusp
with modular weight ( 221 ; 52 ).
ja stand for the current operators associated to the 5d Cartan gauge elds in the Narain
lattice CFT and G(zi; zj ) is the scalar Green function on the torus. The zi integrals are
performed over the torus and the
over F1 which is the fundamental domain of SL(2; Z)
2
on H . Note that the integrand has total modular weight (2; 2), and hence the integral
8The consistency check with the di erential constraints still allows for the possibility of shifting the
4and 6derivative coupling f (4) and f (6) by eigenfunctions of the covariant Hessian. However, we believe that
f (4) and f (6) are given exactly by the low energy limit of the heterotic one and twoloop contributions.
The above possibility can in principle be ruled out by studying the limit to 6d (2; 0) SCFT, but we will not
demonstrate it here.
9The summation over spin structures has been e ectively carried out already in this expression.
is independent of the choice of the fundamental domain of SL(2; Z). To extract the F 4
coe cient, we can simply set sij to zero in the above scalar factor (4.5), and write
fa(41a)2a3a4
( A1jF 4 )a1a2a3a4 =
Z
d
2
2
4
=
5
2
2
( ; ) Y4 d2zi * 4
( )
i=1
Z d
2
2
5
2
2
2
Y jai (zi)
i=1
( )
;
+
where y is a vector in R21, that lies in the positive subspace of the R21;5 in which the
is embedded. We have rewritten the fourpoint function of the currents as a
fourth derivative on the theta function (see, for example, [32]). The theta function
is
de ned as
variations of the embedding on the lattice
As we have argued previously, the f (4) coupling satis es a di erential constraint (3.1)
on M due to supersymmetry. Given the explicit expression for f (4) (4.6) from the type
II/heterotic duality, we can now proceed to x the constant coe cients in (3.1). The above
expression for the 4derivative term f (4) has previously been determined in [12].
In the following we will explicitly parametrize the coset moduli space M and write
down the covariant Hessian r(e
rf). We will work in a trivialization of the SO(21) vector
bundle V
. In particular, we shall identify the coordinates on the base manifold M with
Let eI be a set of lattice basis vectors, I = 1;
; 26, with the pairing eI
eJ =
IJ
given by the even unimodular quadratic form of
21;5. Let P+ and P
be the linear
projection operator onto the positive and negative subspace, R21 and R5, respectively, of
R21;5. We can write P+eI as (eILa)a=1; ;21, and P eI as (eIRi)i=1; ;5.
are then y
We expand the lattice vectors into components y = yeI eI . The left components of y
a = yeI eILa. The requirement that y lies in the positive subspace means that
y y = yaya stays invariant, y itself must vary, and so does ` y = `I eILaya.
yeI eIRi = 0. This constraint implies that, under a variation of the lattice embedding, while
From
21
X eILaeJLa
a=1
5
i=1
X eIRieJRi =
IJ ;
eILa up to an SO(5) rotation. eIRi by itself is subject to the constraint
we see that (eILa; eIRi) is the inverse matrix of IJ (eJLa; eJRi). Note that eIRi are speci ed by
This constraint leaves 26 5
15 independent components of eIRi. The SO(5) rotation of
the negative subspace further removes 10 degrees of freedom from eIRi, leaving the 21
5 =
IJ eIRieJRj =
ij :
(4.6)
(4.7)
(4.8)
(4.9)
105 moduli of the lattice embedding which give rise to a parametrization of the scalar
manifold M.10
Now consider variation of the lattice embedding,
eIa ! eILa + eILa;
L
eIi ! eIRi + eIi:
R R
subject to the constraints
IJ eIL(a eJb) = O(( e)2);
L
IJ
eILaeJRi + eILa eJi = O(( e)2);
R
IJ eIR(i eJj) = O(( e)2);
R
L L
e(Ia eJ)a
e(Ii eJ)i = O(( e)2):
R R
Let f (eIRi) be a scalar function on the moduli space M of the embedding of 21;5. We can
expand
f (eIRi + eIRi) = f (e) + f Ii(e) eIRi + f IJij (e) eIRi eJRj + O(( e)3):
f Ii and f IJij are subject to shift ambiguities
f Ii
! f Ii +
f IJij
! f IJij +
IJ eJRj gij ;
2
1 IJ gij +
IK eRKkhJKijk +
JK eRKkhIKijk
for arbitrary symmetric gij and hIJijk, due to the constraints on eIRi. We can x these
ambiguities by demanding
eIRj f Ii = 0;
eIRkf IJij = 0:
This can be achieved, for instance, by shifting f Ii with
IJ eJRj gij , for some gij . We can
then de ne
feai = eILaf Ii;
feabij = eILaeJLbf IJij +
1
2 abeIR(if Ij):
Note that these are invariant under the shift (4.15) and hence give rise to wellde ned
di erential operators on the moduli space M.
and we can therefore write the covariant Hessian of fa(4bc)d (4.6) as
This construction can be straightforwardly generalized to nonscalar functions on M,
In appendix A.1, we explicitly compute f~a(4bc)d;efii and nd
10Consider the symmetric matrix MIJ de ned by
We have
MIJ = eILaeJLa + eIRieJRi = 2eILaeJLa
IJ = IJ + 2eIRieJRi:
MIK KLMLJ = eILa abeJLb + eIRi( ij)eJRj = IJ :
The symmetric matrix M , subject to the constraint M
M = , can be used to parameterize the coset
SO(21; 5)=(SO(21)
SO(5)).
(4.12)
(4.13)
(4.14)
(4.15)
(4.16)
(4.17)
(4.18)
(4.19)
(4.10)
(4.11)
!I (z) are a basis of holomorphic oneform on the genus two Riemann surface normalized
such that
!J = IJ ;
!J = IJ ;
where the cycles I and J have intersection numbers ( I ; J ) = IJ , ( I ; J ) = ( I ; J ) =
genus two Riemann surface. G(zi; zj) is the Green's function on the genus two Riemann
surface. Again, the contribution to D2F 4 coe cient is simply extracted as
t
u Z
3
Q
To proceed, we need to compute the fourpoint correlation function of
TIa =
d2zja(z)!I (z) =
ja(z)dz;
on the genus two Riemann surface . This allows us to express the correlators of TIa in
terms of the theta function (see for instance [32]),
which allows us to x the constant coe cients in (3.1) to be
3
2
;
U =
V =
2;
The scalar factor in the twoloop heterotic amplitude takes the form [31]
A2 =
Z
Q
where YS is given by
(4.20)
(4.22)
(4.23)
(4.24)
(4.25)
(4.27)
I
I
5
Z
I
I
( ; )
Z
I
I
where
( ; )
=
X
(yj ; );
(4.26)
Thus, we can simplify the result to
Next, we would like to verify that the coe cient functions f (6) extracted from A2 obeys
the di erential constraint (3.4) on the moduli space SO(5; 21)=(SO(5)
SO(21)), and also
x the precise coe cients thereof.
In principle, it should be possible to show that
f (6) is f (6) plus the integral of a total
derivative on the moduli space F2 of the genus two Riemann surface
, which reduces
to a boundary contribution where
is pinched into two genus one surfaces. However,
this calculation is somewhat messy so instead we will x the coe cients of for (f (4))2 by
comparison to similar di erential constraints on the tensor branch of the 6d (2; 0) SCFT.
(yj ; ):
(4.28)
We can write A2jD2F 4 as
HJEP12(05)4
( A2jD2F 4 )a1a2a3a4 = fa(61a)2;a3a4 s12 + fa(61a)3;a2a4 s13 + fa(61a)4;a2a3 s14 ;
However, the de nition (4.29) of fa(61a)2;a3a4 is ambiguous because s12 + s13 + s14 = 0. We
fa(61()a2;a3a4) = 0:
( A1A3 A2A4 + A1A4 A2A3 )
Z
Q
I J d
(yj ; )
y=0
:
(4.29)
(4.30)
(4.31)
(4.32)
(4.33)
(4.34)
(4.35)
x this ambiguity by imposing
Explicitly, fa(61a)2a3a4 is given by,
fa(61a)2;a3a4 enjoys the symmetry
The condition (4.30) gives rise to constraints between the coe cients in (3.4),
fa(61a)2;a3a4 = fa(62a)1;a3a4 = fa(61a)2;a4a3 = fa(63a)4;a1a2 :
u0
2
u2 +
2 = 0; u3 +
3 = 0:
u0
2
We therefore end up with 5 coe cients u1; u2; u3; u4; u5 to determine for the terms
proportional to f (6) on the r.h.s. of the di erential equation (3.4). We determine the ui's by
explicit computation of the covariant Hessian in appendix A.2 and nd
u1 =
On the other hand, to determine the 2 independent coe cients v1; v2 for the (f (4))2
terms in (3.4), we shall take advantage of the following di erential constraint on the
6derivative fourpoint term in the tensor branch e ective action of (2; 0) SCFT [35]
Fg(a4b)(cFd)efg +Fg(c4d)(aFb)efg +w2Fe(g4a)bFc(d4f)g +w3Fe(g4a)(cFd)bfg +(e $ f ) :
(4) (4) (4)
Here F (6) and F (4) are 6derivative and 4derivative fourpoint couplings of tensor
multiplets in the (2; 0) SCFT, which are related to the supergravity couplings f (6) and f (4) by
taking the large volume limit of K3 and zooming in on an ADE singularity that gives rise
to the particular 6d SCFT.11 The 4 and 6derivative terms on the tensor branch of the 6d
(2; 0) SCFT can be in turn computed by the one and twoloop amplitudes in 5d maximal
SYM on its Coulomb branch as discussed in [27]. Explicitly, we have (see appendix B.1
and B.2 for details)
A1jF 4 ! 210 123 F (4);
A2jD2F 4 ! 215 9F (6):
In [35], the coe cients in (4.35) are xed to be12
Hence we x the rest of the constants in (3.4) to be
In summary, the 4 and 6derivative couplings satisfy the following di erential
(4.36)
(4.37)
(4.38)
(4.39)
HJEP12(05)4
+ fe((6c);ab fd + fe((6a);cd fb)
+ (e $ f ) :
3
1
Implications of f (4) and f (6) for the K3 CFT
As alluded to in the introduction, since spacetime supersymmetry imposes di erential
constraints on the fourpoint string perturbative amplitudes which involve, in particular,
integrated correlation functions of exactly marginal operators in the internal K3 CFT, we
will be able to derive nontrivial consequences for the K3 CFT itself. As an illustration, we
will see how the resulting moduli dependence of the f (4) and f (6) couplings at treelevel can
pinpoint the singular points on the moduli space of the K3 CFT which the Zamolodchikov
metric does not detect (since the moduli space is a symmetric space).
Below we will rst explain how to extract the data relevant for K3 CFT from string
treelevel amplitudes and general features thereof. We will then demonstrate their implications,
11The contribution due to supergraviton exchange on the r.h.s. of (3.4) is absent in (4.35) due to this
decoupling limit. A similar reduction of the genus one and two amplitudes in the type II string theory to
supergravity amplitudes was considered in [34].
12The factor 2
in the numerator comes from the relative normalization between the lattice vectors and
the 5d scalars. In particular, the mass square of the W boson is m2 = 2 `2R.
in a particular slice of the K3 CFT moduli space where the K3 CFT is approximated by
the supersymmetric nonlinear sigma model on A1 ALE space.
In principle, we expect to arrive at the same set of constraints from the K3 CFT
worldsheet Ward identities with spin elds associated to the Ramond sector ground states
which appear in the spacetime supercharge. The same set of constraints is expected to
hold for all c = 6 (4; 4) SCFTs.13 We will leave this generalization to future work.
Reduction to the K3 CFT moduli space
The string theory amplitudes we have obtained in the previous section are exact results
which can be regarded as sections of certain SO(21) vector bundles over the full moduli
elds, and the rest 80 NSNS scalars describe the moduli space
SO(5). Among the 105 moduli, one comes from the IIB
Globally the K3 CFT has moduli space [36, 37]
MK3 = O( 20;4)nSO(20; 4)=SO(20)
SO(4);
parametrized by the scalars inside 20 of the 21 tensor multiplet, 'i
from the 6d perspective. From the worldsheet CFT point of view, 'i
with i = 1; 2 : : : ; 20,
are associated with
the BPS superconformal primaries that are doublets of the two SU(2) current algebras.
We will restrict the full string fourpoint amplitude obtained from the type II/heterotic
duality to these 20 tensor multiplets, and expand in the limit of small gIIB.14 In this limit,
the theta function of the
21;5 lattice can be approximated by the product of the theta
function of the 20;4 lattice and that of the 1;1 lattice whose integral basis has the following
embedding in R1;1,
u = (r0; r0);
v =
with r0 ! 1 in the limit. Since at genus one
we have in this limit
1
2r0
;
1;1 ( )
1
2r0
;
r0 and at genus two
1;1 ( )
(5.1)
(5.2)
(5.3)
r02,
(5.4)
(5.5)
full
AH4
r0ArHed4('i);
AfDu2llH4
r02ArDed2H4 ('i):
Now on the other hand, working with the canonically normalized elds (Einstein frame)
which involves rescaling the string frame metric and B elds by
G
! M6 2G ; B
! M6 2B
where M6 = (VK3=gI2IB`s8)1=4 is the 6d Planck scale, we know that the fourpoint coupling
f (4) must scale as M62 and f (6) as M64. From this we conclude that r0
M62.
13The condition c = 6 is used in writing down the spacetime supercharges (in combination with the
spacetime part of the worldsheet CFT) hence making connection to the spacetime supersymmetry constraints
on the integrated CFT correlation functions.
14Note that by doing so we break the SO(5) R symmetry to SU(2) SU(2).
The di erential constraint on A
full in the perturbation expansion implies a similar
red. Focusing on the scalar component of the superamplitude, we have the
constraint on A
following derivative expansion
k '
` )
s
t
u
= s2
ij k` + ik j` + i` jk + Aijk` + Bij;k`s + Bik;j`t + Bi`;jku + O(s2) :
where the rst two terms come from the supergraviton exchange, while Aijkl and Bij;kl
are obtained from the treelevel limit of the f (4) and f (6) couplings respectively. The
coe cients Aijk` and Bij;k` for the N = 4 AK 1 cigar CFT, which is the ZK orbifold of
the supersymmetric SU(2)K =U(1)
SL(2)K =U(1) coset CFT, are studied in [38].
On the other hand, Ared can be evaluated directly from IIB treelevel perturbation
theory. The K3 CFT admits a small (4; 4) superconformal algebra, that contains left and
right moving SU(2) Rcurrent algebra at level k = 1 [39]. Focusing on the left moving
part, the superVirasoro primaries are labeled by its weight h and SU(2) spin `. The BPS
superVirasoro primaries in the (NS,NS) sector consist of the identity operator (h = ` =
h = ` = 0), and 20 others labeled by Oi
with h = ` = h = ` = 1=2 which correspond to
the 20 (1; 1) harmonic forms in the K3 sigma model.15 The BPS primaries Oi
exactly marginal primaries of the K3 CFT, corresponding to the moduli elds 'i . Under
spectral ow, the identity operator is mapped to a unique h = h = 1=4, ` = ` = 1=2 ground
are the
state O0
in the (R,R) sector, whereas the weight1=2 BPS superVirasoro primaries give
rise to h = h = 1=4, ` = ` = 0 (R,R) sector ground states labeled by i
operators for the 6d massless elds all involve these 21 BPS superVirasoro primaries and
RR [39]. The vertex
their spectral owed partners.
More explicitly, the vertex operators in the NSNS sector are
Here eik X comes from the R1;5 part of the worldsheet CFT. The associated 1particle
states transform under the SU(2)
SU(2) little group as
e
e
eik X
eik X 1;
Oi ;
i = 1;
; 20;
The 80 scalars in the second line of (5.7) are denoted by 'i .
On the other hand, the vertex operators in the RR sector are
e
e
=2 =2S _ S _ eik X
=2 =2S S eik X
O0 ;
iRR;
i = 1;
; 20:
15Here Oi
are BPS superconformal primaries of the N = (4; 4) superconformal algebra. With respect
to an N = (2; 2) superconformal subalgebra, Oi
Oi + is an antichiral primary on the left and a chiral primary on the right.
++ is a chiral primary both on the left and the right, whereas
4h=h= 14
20
1h=h= 14
6d multiplet supergravity + tensor 20 tensors
K3. The 1 and 4 denote the trivial and ` = ` = 12 representations of the
worldsheet SU(2) Rsymmetry, and the arrows represent the spectral ow.
The chiralities of the spin
elds are dictated by the IIB GSO projection in RR sector,
which depends on the SU(2) Rcharge of the vertex operator.16 The associated 1particle
SU(2) little group. (5.8) and (5.11) together give the 1particles states
in the (2,0) supergravity multiplet and the 21 tensor multiplets. See table 3 for summary.
The fourscalar amplitude of 'i
in treelevel string theory is given by
k '
` )
=
Z d2z D
2
G 1 G 12 Oi
2
++eik1 X (z) G 1 G 12 Oj
2
++eik2 X (0) Ok eik3 X (1) O` eik4 X (1)
account in the above.
correlation function vanishes identically
2
2 i
where G(z) is the N = 1 superVirasoro current, which acts on both Oi and eik X .17 We
have put two vertex operators in the ( 1; 1) picture and the other two in the (0; 0)
picture to add up to the total picture number ( 2; 2) for the treelevel string scattering
amplitude. The correlator of the superconformal ghosts have already been taken into
By deforming the contour of G 1 = H dw G (w), it is easy to see that the following
D
G 1 G 12 Oi++(z) G 1 G 12 Oj++(0) Ok (1) O` (1)
2 2
E
=
G 1 G 12 Oi++(z) G 1 G 12 Oj++(0) Ok (1) O` (1) :
2 2
16The IIB GSO projection in the RR sector is [17, 38]
FL + 2JL3
1
2 2 2Z; FR + 2JR3
1
2 2 2Z
where FL;R are the left and right worldsheet fermion numbers in R1;5, and JL3;R denote the left and right
SU(2) Cartan Rcharges of the internal K3 CFT.
17The sigma model on R1;5
current G(z) is the sum of N = 2 superVirasoro currents G+(z) + G (z), and G
of the N = 4 superVirasoro currents. The U(1) charge of the N = 2 algebra coincides with the J3 charge
are each a combination
K3 has N
= 2 worldsheet supersymmetry. The N = 1 superVirasoro
of the N = 4.
(5.11)
E
(5.12)
(5.13)
(5.10)
Therefore in (5.12) we can take G 1 ; G 1 to act on eik X only, which gives
2
2
k '
` ) = s2
Z d2z
2
j j
z s 2 1
j
zj t DOi++(z)Oj++(0)Ok (1)O` (1)E :
Thus, comparing with (5.6), we obtain the relation
Z d2z
2
j j
z s 2 1
j
zj t DOi++(z)Oj++(0)Ok (1)O` (1)
E
s
t
u
=
partners iRR,
all channels,
Z d2z
2
From the CFT perspective, the polar terms in t and u are simply due to the appearance
of the identity operator in the OPE of O
++ with O
, while Aijk` and Bijk` capture
information about all intermediate primaries in the conformal block decomposition of the
fourpoint function of the marginal operators. It is then natural to expect this relation
to hold for exactly marginal operators in any c = 6 (4; 4) SCFT. Furthermore, we expect
Aijk` and Bijk` to obey the same kind of di erential equations as f (4) and f (6), for any
Using the relation between the correlation function of Oi
and their spectral owed
j j
ij k` + ik j` + i` jk + Aijk` + Bij;k`s + Bik;j`t + Bi`;jku + O(s2; t2; u2):
(5.14)
(5.15)
(5.17)
(5.18)
DOi++(z)Oj++(0)Ok (1)O` (1)E =
z
j j
j
1
z
j
iRR(z) jRR(0) kRR(1) `RR(1) ;
(5.16)
we can put (5.15) into an equivalent form, where the crossing symmetries are manifest in
To illustrate the power of the relation (5.15), we consider the A1 ALE limit where we
zoom in on and resolve an A1 singularity. In other words, we focus on a slice near the
boundary of the full moduli space MK3, where the K3 CFT is reduced to a sigma model
on A1 ALE space, which is related to the sigma model on C2=Z2 by exactly marginal
deformations [37, 40, 41].
The slice of interest is parametrized by the normalizable exactly marginal deformations
of the orbifold CFT C2=Z2, which is simply the moduli space of the A1 SCFT18
MA1 =
R
3
Z2
S1
;
18In [
42
], the moduli space of the nonlinear sigma model on a general hyperkahler manifold is
discussed. For the A1 ALE space sigma model, the moduli space metric is at because we have scaled the
Zamolodchikov metric by an in nite volume factor of the target space.
3 corresponds to the Kahler and complex structure deformations associated with
the exceptional divisor of the C2=Z2, and the S1 is parameterized by the integral of the B
eld on the exceptional divisor. This Z2 can be understood from the fact that the SO(3)
rotation of the asymptotic geometry of the circle bration of the EguchiHanson geometry
that exchanges the two points of degenerate
ber e ectively also ips the orientation of
the P1 hence re ects the B eld ux. The two orbifold singularities on the moduli space
corresponds to the free orbifold point and the singular CFT point where a linear dilaton
throat develops. The distinction between these two points on the moduli space is not
detected by the Zamolodchikov metric, but should be detected by f (4) restricted to the
single tensor multiplet corresponding to this exceptional divisor (or rather A1111).19
Since the overall volume of the CFT target space is in nite, A1111 is a harmonic function
on the moduli space.20 Near the singular CFT point, A1111 goes like 1=j'~j2, where '~ is a
local Euclidean coordinate on the moduli space, as in the case of the A1 DSLST at
treelevel (either (2; 0) or (1; 1)) [38, 43]. At the free orbifold point, on the other hand, the
fourpoint function of marginal operators are perfectly nonsingular, and A1111 should be
nite. This together with the harmonicity and Rsymmetry determines A1111 to be (up to
an overall coe cient)
;
where R is the radius of the S1 of the moduli space.21 It is easy to identify from (5.19) that,
'~ = (0; 0; 0; 0) is the singular CFT point, and '~ = (0; 0; 0; R) is the free orbifold point,
since A1111 is nonsingular at the latter point, and the Z2 symmetry is clearly preserved.
Let us de ne r2 = Pi3=1 'i2, '4 =
R + y. Then near the free orbifold point, r; y are
small, we have
A1111 =
1
4R2 +
3y2
r
2
48R4
+ O(r4; y4; r2y2):
One should be able to con rm this using conformal perturbation theory [
44
].
In the large ' regime, where the CFT is described by a nonlinear sigma model on
T CP1, performing Poisson summation on (5.19), we can write A1111 as the expansion
A1111 =
1
2Rr
1
n=1
1 + X( )ne
n(r+iy)
R
+ X( )ne
n(r iy) #
R
:
1
n=1
Since r scales like the area of the CP1, the leading 1=r contribution should come from
oneloop order in 0 perturbation theory. The e nr=R corrections, on the other hand, are
expected to come from worldsheet instanton e ects. Moreover, the phase e iny=R
indicates that there are contributions from both holomorphic and antiholomorphic worldsheet
19Note that f (6) vanishes in this case because there is only one tensor multiplet involved.
20The contribution from supergraviton exchange on the r.h.s. of (3.1) is suppressed in this limit.
21The S
1 parameterized by the B eld
ux through the exceptional divisor P1 is of constant size along
the R3. This is because the marginal primary operator associated with the normalizable harmonic 2form
on the ALE space with unit integral on the P1 also has a normalized twopoint function.
(5.19)
(5.20)
(5.21)
instantons. In other words, our exact result based on supersymmetry constraints gives
the striking prediction that in 0 perturbation theory, A1111 which is related to the
fourpoint function of exactly marginal operators of the A1 SCFT, receives only oneloop plus
worldsheet instanton contributions.
It would be interesting to understand if a similar worldsheet instanton expansion
applies for the K3 CFT at
string [45{48]. In particular, the N = 4 topological string amplitudes are written as
integrals over the fundamental domain F1 and also satisfy certain di erential equations on the
moduli space [48].
6
The main result of this paper is the exact nonperturbative coupling of tensor multiplets
at 4 and 6derivative orders in type IIB string theory compacti ed on K3, fa(4bc)d( ) and
fa(6b;)cd( ), and the di erential equations they obey on the 105dimensional moduli space. In
the weak coupling limit (treelevel string theory), as described in section 5, they reduce to
(up to a factor involving the IIB string coupling) the functions Aijk`(') and Bij;k`(') on
the 80dimensional moduli space of the K3 CFT. Aijk` and Bij;k` are integrated fourpoint
functions of 12 BPS operators in the K3 CFT on the sphere. Unlike the Zamolodchikov
metric or its curvature [14], Aijk` and Bij;k` do not receive contribution from contact
terms, and depend nontrivially on the moduli. In particular, these functions diverge at the
points in the moduli space where the CFT develops a continuous spectrum (corresponding
to ADE type singularities on the K3 surface, with no B eld through the exceptional
divisors [37]). This allows us to pinpoint the location on the moduli space using CFT data
alone (as opposed to, say, BPS spectrum of string theory), and makes it possible to study
the K3 CFT through the superconformal bootstrap [49{52] (e.g. constraining the nonBPS
spectrum of the CFT) at any given point on its moduli space. This is currently under
investigation [53].
In the full type IIB string theory on K3, at the ADE points on the moduli space, there
are new strongly interacting massless degrees of freedom, characterized by the 6d (2; 0)
superconformal theory at low energies. Near these points, the components of fa(4bc)d( ) and
fa(6b;)cd( ) associated with the moduli that resolve the singularities are precisely the H4 and
D2H4 couplings on the tensor branch of the (2; 0) SCFT, studied in [27, 54]. Note that
this is di erent from the ALE space limit discussed in section 5.2, which was restricted to
the weak string coupling regime.
As pointed out in section 2, there are Fterm supervertices involving the supergraviton
in 6d (2; 0) supergravity theories as well, including one that corresponds to a coupling
of the schematic form fR( )R4 +
. It appears that a sixpoint supervertex involving 4
supergravitons and 2 tensor multiplets in the SO(5)R singlet does not exist at this derivative
order (namely 8), and so by the same reasoning as section 3, we expect that fR( ) obeys
a second order di erential equation with respect to the moduli, whose form is determined
by the factorization structure of the sixpoint superamplitude of 4 supergravitons and 2
tensor multiplets. One complication here is the potential mixing of the coe cients of R4,
D2(R2H2), and D4H4, in the di erential constraining equations. In particular, D4H4 is
a Dterm, and by itself is not subject to such constraining equations. We leave a detailed
analysis of the supersymmetry constraints on the higher derivative supergraviton couplings
in (2; 0) supergravity to future work.
One can similarly classify the supervertices in the 6d (1,1) supergravity theory and
derive di erential constraints for the higher derivative couplings. In this case however, the
string coupling lies in the 6d supergraviton multiplet rather than the vector multiplets,
and its dependence is not controlled by the same type of di erential equations considered
in this paper.
Finally, one may wonder whether our exact results for integrated correlators in the
HJEP12(05)4
K3 CFT can be extended to 2d (4; 4) SCFTs with c = 6k for k > 1, such as the D1D5
CFT [55, 56]. While this is conceivable, the arguments used in this paper are based on
the spacetime supersymmetry of the string theory and cannot be applied directly to the
k > 1 case. In the CFT language, our constraints can be recast as Ward identities involving
insertions of spin elds, and we have implicitly used the property that the spin elds of the
c = 6 (4; 4) SCFT transform in a doublet of the SU(2)R symmetry. It would be interesting
to understand whether there are analogous Ward identities in the c = 6k (4; 4) SCFTs,
where the spin elds carry SU(2)R spin j = k2 .22
Acknowledgments
We would like to thank Clay Cordova, Thomas Dumitrescu, Hirosi Ooguri, David
SimmonsDu n, Cumrun Vafa for discussions. We would like to thank the Quantum Gravity
Foundations program at Kavli Institute for Theoretical Physics, the workshop \From Scattering
Amplitudes to the Conformal Bootstrap" at Aspen Center for Physics, and the Simons
Summer Workshop in Mathematics and Physics 2015 for hospitality during the course of
this work. SHS is supported by a Kao Fellowship at Harvard University. YW is supported
in part by the U.S. Department of Energy under grant Contract Number DESC00012567.
XY is supported by a Sloan Fellowship and a Simons Investigator Award from the Simons
Foundation.
A
A.1
Explicit check of the di erential constraints
Fourderivative coupling f (4)
In this appendix we will explicitly show that the 4derivative term coe cient fa(4bc)d between
the 21 tensor multiplets satis es the following di erential equation and determine the
22In the case of the c = 12 (4; 4) SCFT, say described by the nonlinear sigma model on a hyperKahler
4fold, one may compactify type IIB string theory to 2d, which generally leads to a (6; 0) supergravity
theory in two dimension [57, 58], and examine the 4derivative Fterm coupling of moduli
elds in this
theory. However, we are not able to derive di erential constraining equations on these couplings based on
soft limits of superamplitudes, due to the existence of local supervertices for the relevant sixpoint couplings
at the same derivative order, in contrast to the 6d (2; 0) supergravity theory.
coe cients U; V; W ,
re rf fa(4bc)d = U fa(4bc)d ef + V f((e4()abc d)f) + W fe(f4)(ab cd) :
Let us rst decompose the 4derivative coe cient fa(4bc)d into the
fa(4bc)d = Aabcd + (abBcd) + (ab cd)C
where Aabcd and Bcd are symmetric and traceless. The covariant Hessian r(a
rb) of these
tensors can be expressed, through a set of relations similar to (4.14) and (4.16), in the form
The di erential constraints (A.1) can be expressed as
X Aeabcd;efii;
5
i=1
X Beab;cdii;
5
X Ceabii:
i=1
5
i=1
e;i
c;i
X Aeabcd;eeii = aAabcd;
X Beab;ccii = bBab;
X Ceaaii = cC;
a;i
1 ef X Aeabcd;ggii = u (e(aAf)cde) + v e(a fbBcd)
traces;
X Beab;eeii = xAabcd + y (c(aBd)b) + z c(a b)dC
traces;
`2
`2
F
We will relate the coe cients a; b; c; u; v; x; y; z; w to U; V; W later.
To start with, let us determine the constant in the di erential equation for the scalar
function C. From (4.6) and (A.2), we rst write C as
C = 4!
Z d
2
Z d
2
F
F
1
2
2
1
2
2
( )
( )
X q `2` e 2 2`I eIRieJRi`J
X q `2`
" 2
3
21 2
`I `J eILaeJLa +
+
(`I `J eILaeJLa)2
2#
e 2 2`I eIRieJRi`J ;
where we have used `I `J eILaeJLa = ` ` + `I `J eIRieJRi. After integration by part, we have
16 4 Z d
( )
"
X q `2` (` `)2
`2
11
2
` ` +
33 #
2
2 2 e 2 2`I eIRieJRi`J :
(A.1)
(A.2)
(A.3)
(A.4)
(A.5)
(A.6)
Under the variation eR
! eR + eR, the rst and second order variations of C are given by
CIi eIRi + C
IJij R
R
eIi eJj
Z
2
F
( )
X
`2
` `
q 2
(` `)2 +
2
` `
` `
Z d
2
F
( )
1
2
e 2 2`IeIRieJRi`J n
2
` `
q 2
2
8 22
3
` `
2 ab`I `J eIRieJRj +
4
2
1
2
+
` `
We can now compute the Laplacian of C,
21
X
ra raC =
Ceaaii
X
a;i
Z d
2
F
( )
n
1
2
X
a;i
Ceaaii =
25
2
C:
This xes the constant c in (A.4) to be c = 25=2.
where the hatted indices are taken to be symmetric traceless combinations.
R
eIi eJj
R o
:
(A.7)
` `
` `
2
4
4
1
2
+
2 4
3 161
(` `)2 +
e 2 2`IeIRieJRi`J :
` `
2
2
1
"
Similarly we can write
Aabcd = (2 i)4
Bab =
d
F
F
d
2
1
2
2
( )
1
2
2
( )
X
`2
X
`2
` `
q 2 e
#
` `
q 2 e
` `
#
`I `J eILa^eJL^b;
r(c rd)Bab = B~ab;cdii
`I `J eILa^eJ^b
L
24`M `N eLMceNd + cd
L
+ a^c ^bd
` ` +
`I `J eILa^eJ(d c)^b
L
` ` +
` `
q 2 e
` `
r(a rb)C = Cea^^bii
704 4 Z
2
161
F
` `
q 2 e
"
` `
where we have used ( xing the SO(21) freedom)
eJa = eJRieaLI R
L
eIi +
eLI R
a
R
eIi eJi
eJRj ejRM eLN
a
R R
eMk eNk + : : : :
After a somewhat tedious but straightforward calculation, we obtain all the di erential
equations in (A.4),
The covariant Hessians of Aabcd, Bab, and C can be computed straightforwardly to be
r(e rf)Aabcd = Aeabcd;efii
= 16
4
Z
2
` `
q 2 e
"
2`I `J `M `N eILa^eJL^beMc^eN(e f)d^
L L
+
2
3
2
`I `J eILa^eJ^b c^e d^f
L
` `
q 2 e
2 2`I eIRieJRi`J `I `J `M `N eILa^eJL^beMc^ Nd^
L L
e ;
`2
6
2
1
2
2
( )
4
X
`2
1
2
1
2
#
#
;
18
2
#
6
2
`I `J eILa^eJL^b;
(A.12)
(A.13)
F
4
F
24
ef
2
25
2
2
( )
X
`2
Z
F
1
2
2
( )
2
1
2
2
( )
X
`2
1
2
67
2
17
2
Aabcd;
Bab;
Aeabcd;eeii =
Bab;ccii =
X
e;i
X
i
X
c;i
X
i
X
i
144
25
550
483
Cea^^bii
Bab;
71
25
A
eabcd;e^f^ii
2Aa^^bc^(e f)d^
Ba^^b c^(e f)d^
trace in (ef );
(A.14)
B
eab;c^d^ii =
Aabcd +
Ba^(c d)^b + 2C a^(c d)^b
trace in (cd);
where the hatted indices are taken to be symmetric and traceless. Together with (A.10),
we have thus determined all the coe cients in (A.4),
a =
x =
67
2
144
25
b =
y =
17
2
;
;
71
25
c =
z = 2;
25
2
;
u =
w =
2;
550
483
:
v =
1;
(A.15)
Determination of U; V; W .
With the above 9 coe cients determined, we now arrange
them into the form (A.1) and determine U; V; W . Let us start by inspecting the trace part
in (ef ) of (A.1),
r2fa(4bc)d = (21U + V )fa(4bc)d + W
ef fe(f4)(ab cd):
Noting that ef fe(f4)ab = 265 Bab + 233 abC, we obtain the rst three equations in (A.4),
25
6
23
3
W )Bab;
W )C:
V
2 e(aBbc d)f)
Aefab +
W
Bef :
trace in (ef), (abcd);
29
12
V +
25
9
W
B(e(a b)f)
(e(a b)f)C
trace in (ef), (ab);
(A.16)
(A.17)
(A.18)
(A.19)
Next, the traceless part in (ef ) of (A.1) can be written as
r(e rf)
r(e rf)
1
21 ef r
1
21 ef r
2 Aabcd = V A(e(abc d)f) +
2 Bab =
r(e rf)
1
21 ef r
2 C =
r2Aabcd = (21U + V )Aabcd;
r2Bab = (21U + V +
r2C = (21U + V +
6
25
+
1
161
3
2
25
6
V +
V
2
25
6
25
6
25
9
W
W
575
18
V +
Matching (A.17) and (A.18) with (A.4), we nd the 9 coe cients a; b; c; u; v; x; y; z; w
are indeed determined by U; V; W , which are
U =
V =
2;
Sixderivative coupling f (6)
In this appendix we will show that the 6derivative term between the 21 tensor multiplets
f (6) de ned in (4.29) satis es the following di erential equation,
r(e rf)fa(61a)2;a3a4 = u1fa(61a)2;a3a4 ef + u2 fe(f6;)a1a2 a3a4 + fe(f6;)a3a4 a1a2
+ u4 fe((6a)3;a1a2 fa4) + fe((6a)1;a3a4 fa2)
+ u5 fe((6a)2;a1)(a3 a4)f + fe((6a)4;a3)(a1 a2)f ;
modulo terms of the schematic form (f (4))2. In the following the symmetrization on the
indices (ef ) is always understood if not explicitly written. We have already taken the
condition (4.30) and its consequence (4.33) into account.
In the following we will use an abbreviated notation to simplify the notations, eIa
e~Ii = eIRi, and MAB
Im
From (4.31), we can write fa(61a)2a3a4 as
G H d
6
3
+ 8 3M 1 MAB`AI `BJ (e2)a1a2;a3a4;IJ
( a1a2 a3a4
a1(a3 a4)a2 ) ;
exp i
AB`A
B
2 MAB`AI `BJ e~Iie~Ji
HJEP12(05)4
eILa,
(A.21)
(A.22)
(A.23)
(A.24)
1
1
2
1
2
1
2
1
They can be computed straightforwardly to be
GIKJi;ab = e~IieaK eJb + e~JiebK eIa;
+ e~Iie~Jj eaM ebN ;
EII1iI2I3I4;a1a2a3a4 = e~I1ieIa1 eI2a2 eI3a3 eI4a4 + 3 more;
FIM1Ii2NI3jI4;a1a2a3a4 =
e(aM1 IN1)
e~I1ke~(kM eaN1) eI2a2 eI3a3 eI4a4 ij + 3 more
HIMJ;iaNb j =
e(M N)
a
I
e~Ike~(kM eN) eJb ij +
a
b
e(M N)
J
e~Jke~(kM eN) eIa ij
b
(e2)a1a2;a3a4;IJ :=
a3a4 eIa1 eJa2
a1a2 eIa3 eJa4
with symmetrization on the (IJ ) indices.
2 a2a3 eIa4 eJa1 +
2 a1a4 eIa2 eJa3 +
2 a2a4 eIa1 eJa3 +
2 a1a3 eIa2 eJa4 ;
1
Recall that under the variation e~ ! e~ + e~, eIa transforms as, up to second order,
eIa = e~IieaJ e~Ji +
e(M N)
a
I
e~Ij e~(jM eN)
a
e~Mi e~Ni;
where the M; N indices are raised by
MN . We will de ne the tensor G; H; E; F as
(eIaeJb) = GIKJi;ab e~Ki + HIMJ;iaNb j e~Mi e~Nj ;
(eI1a1 eI2a2 eI3a3 eI4a4 ) = EII1iI2I3I4;a1a2a3a4 e~Ii + FIM1Ii2NI3jI4;a1a2a3a4 e~Mi e~Nj :
1
2
+ e~I1ie~I2j eaM1 eaN2 eI3a3 eI4a4 + 5 more:
(A.25)
f (6)
a1a2;a3a4= f (6) Ii
e~Ii + f (6) MiNj e~Mi e~Nj
a1a2;a3a4
n h
Q
G H d
6
GH
1
X
exp i
AB`
A
`
B
32 4( 2e4`4)a1a2;(a3a4) + 8 3M
1 MAB`AI `BJ (e2)a1a2;a3a4;IJ
+4 2M
1
4 MAB`AI `BJ e~Ii e~Ji
2 MAB`AI `BJ
ij + 8 2MABMCD`AI `BM `CJ `DN e~Mie~Nj
e~Ii e~Jj
+ 32 4
+8 3M
AB CD`AI1 `CI2 `BI3 `DI4hEIi
I1I2I3I4;a1a2(a3a4) e~Ii +F MiNj
I1I2I3I4;a1a2(a3a4) e~Mi e~Nj
1(MEF `EI `F J ) h
e~Ki + HIMJ;iaN1aj2
e~Mi e~Nj
+ 5 more
AB CD`AI1 `CI2 `BI3 `DI4 EMi
I1I2I3I4;a1a2(a3a4)
1 MEF `EI `F J
a3a4 GIMJ;ia1a2+5 more
4 MAB`AK `BN e~Kj
e~Mi e~Nj
i
HJEP12(05)4
o
(A.26)
(A.27)
where we have de ned
X
f
a1a2;a3a4;ef ii =
( 2e4`4)a1a2;a3a4 =
AB CD`AI `BJ `CM `DN eIa1 eMa2 eJa3 eN a4 :
Note that ( 2e4`4)a1a2;(a3a4) = ( 2e4`4)(a1a2);(a3a4) = ( 2e4`4)(a3a4);(a1a2).
The second derivative of f (6) is then given by
X
eIeeJf f (6) IJii +
e~Iif (6) Ii
2
Q
G H d
6
GH
1
X
`1;`22
exp i
AB `
A
`
B i
32
2M
1
4( 2e4`4)a1a2;(a3a4) + 8
3M
1 MAB `AI `BJ (e2)a1a2;a3a4;IJ
+ 32
+ 8
3M
AB CD`AI1 `CI2 `BI3 `DI4 F M iNi
I1I2I3I4;a1a2(a3a4) eM eeN f
1
(MEF `EI `F J )eM eeNf
a3a4 HIMJ;iaN1 ai2
+ 5 more
+
32
4
AB CD`AI1 `CI2 `BI3 `DI4 EM i
I1I2I3I4;a1a2(a3a4)
+ 8
3M
1
(MEF `EI `F J
a3a4 GIMJ;ia1a2
+ 5 more
4 MAB `
AK `BN e~Ki eM (eeNf )
exp
2 MAB `AI `BJ e~Iie~Ji ;
(A.28)
where we have used EIi
I1I2I3I4;a1a2(a3a4)e~Ij = 0 and GIi
I1I2;a1a2 e~Ij = 0.
Let us now study the di erent powers of ` terms in the integrand. Note that since we
can replace `AI `BJ e~Iie~Ji by
21 @M@AB , e~Ii should be treated as ` 1 in the power counting.
Also note that the tensors G; H; E; F contain factors of e~Ii.
First let us note that the `6 terms cancel as in the 4derivative case after integration
by parts. Moving on to the `4 terms, they can be organized to be
X f~(6)
a1a2;a3a4;efii `4 3
G H d
X
exp hi
AB`A `Bi
10( )M 21 `1;`22
64 4 ef ( 2e4`4)a1a2;(a3a4)
a3a4 ( 2e4`4)ef;(a1a2) + a1a2 ( 2e4`4)ef;(a3a4) 2 a2a3 ( 2e4`4)ef;(a1a4)
1
2 a2a4 ( 2e4`4)ef;(a1a3)
2 a1a3 ( 2e4`4)ef;(a2a4)
+16 4
a1e( 2e4`4)fa2;(a3a4) + a2e( 2e4`4)a1f;(a3a4) + a3e( 2e4`4)a1a2;(fa4)
1
1
+ a4e( 2e4`4)a1a2;(a3f)
:
u1 a1a2 a3a4
a1(a3 a4)a2
ef
+ 2u2 ef a1a2 a3a4
+ 2u4 ea3 a4)f a1a2
+ (a1 $ a3; a2 $ a4) = 0:
e(a1 a2)f a3a4
e(a1 a2)(a3 a4)f
Again, the symmetrization on the indices (ef ) is implicitly understood.
ef (a3(a1 a2)a4) + e(a1 a2)(a3 a4)f
This already xes ui's to be
u1 =
In the following we will show that the terms with `2 and `0 in the integrand also
satis es the same di erential equation (A.20) with the same values of ui's. Let us start
with the `0 term in the covariant Hessian (l.h.s. of (A.20)),
X f~(6)
a1a2;a3a4;efii `0 / 3
G H d
6
GH
1
X
`1;`22
exp
exp i
AB`A
Hence we need to show that the righthand side of (A.20) is also zero when replacing
fa(61a)2;a3a4 by its `0 term in the integrand, namely, fa(61a)2;a3a4 ! ( a1a2 a3a4
a1(a3 a4)a2 ).
Indeed, under this replacement the righthand side of (A.20) is zero with ui's given by (A.30)
(A.29)
(A.30)
(A.31)
(A.32)
exp i
AB`A
B
2 MAB`AI `BJ e~Iie~Ji
eJf
nh
4 3h
8 3
ef (e2)a1a2;a3a4;IJ 16 3
a1(a3 a4)a2 eIeeJf
4 3 2 a1e a2f eIa3 eJa4 + 2 a3e a4f eIa1 eJa2
a1e a3f eIa2 eJa4 a1e a4f eIa2 eJa3 a2e a3f eIa1 eJa4 a2e a4f eIa1 eJa3
a3a4 ( a1eeIa2 + a2eeIa1 )
a1a2 ( a3eeIa4 + a4eeIa3 )
1
2 a2a3 ( a1eeIa4 + a4eeIa1 ) + 3 moreio:
Xf~(6)
a1a2;a3a4;efii `2 3
G H d
6
GH
1
X
i
(A.33)
(A.34)
We need to match the second derivative of f (6) given above with the righthand side of (A.20)
at the `2 order in the integrand. For example, the coe cient for ef (e2)a1a2;a3a4;IJ on the
righthand side of (A.20) is 8 3(u1+u2) =
8 3, which agrees with the coe cient the second
derivative f~(6). Similarly one can show that the `2 terms agree on both sides of (A.20).
In conclusion, we have checked that fa(61a)2;a3a4 given in (4.31) satis es the following
2r(e rf)fa(61a)2;a3a4 =
2fa(61a)2;a3a4 ef + fe(f6;)a1a2 a3a4 + fe(f6;)a3a4 a1a2
+ fe((6a)3;a1a2 fa4) + fe((6a)1;a3a4 fa2) + (e $ f ) ;
modulo the (f (4))2 term that is determined in section 4 and appendix B.
B
Relation to 5d MSYM amplitudes
In section 4, we discuss how the numerical coe cients v1; v2; v3 for the (f (4))2 term in (3.4)
can be xed from the 6d (2; 0) SCFT limit, where a similar di erential equation holds [35].
The fourpoint 4 and 6derivative couplings on the tensor branch of the 6d (2; 0) SCFT
can be in turn computed by the one and twoloop amplitudes in 5d maximal SYM on its
Coulomb branch [27]. Therefore, to determine these coe cients, we will x the relative
normalization between the F 4 and D2F 4 couplings in the Coulomb branch e ective action
of 5d maximal SYM and the T 5 compacti ed heterotic string amplitudes in this appendix.
B.1
In this subsection, we would like to x the relative normalization between the F 4 coupling
from oneloop heterotic string amplitude and that from oneloop 5d maximal SYM on its
Coulomb branch by looking at a point of enhanced ADE gauge symmetry in the heterotic
moduli space and a degeneration limit of the genus one Riemann surface (see
gure 3).
A similar reduction of the genus one and two amplitudes in the type II string theory to
supergravity amplitudes was considered in [34].
oneloop amplitude A1SYM in 5d maximal SYM.
Recall that the heterotic oneloop amplitude is
with the theta function
de ned by
A1jF 4 =
Z d
i `2R+2 i` y
= e 2 2 y y X e i ` ` 2 2`2R+2 i` y:
1
(B.2)
(B.4)
Let us inspect the contributions to the integral in the large 2 regime, where
( ) can
be approximated by q = e2 i . Then
2
L
`2R = 2, and we have
is dominated by the contribution from ` ` =
A1jF 4 ! (2 )
`aL1 `aL2 `aL3 `aL4 e 2 2`2R :
In the limit of the moduli space where `R ! 0 for some of the ` ` = 2 lattice vectors, the
dominant contribution takes the form of the oneloop contribution from integrating out
W bosons labeled the root vectors ` in 5d maximal SYM. Here `2R is proportional to the
W boson mass squared, and `aL labels the charge of the W boson with respect to the ath
Cartan generator.
To compare the normalization with the 5d SYM oneloop amplitude, we use the
Schwinger parametrization to write down the contribution from the diagrams involving
light internal W bosons, which are labeled by the root vectors `L,
A1
SYM =
=
X
(`L)2=2
1
26 25
Z
3
Z
dt
t
3
3! `aL1 `aL2 `aL3 `a4
e t(p2+m2)
dt t 21
X
(`L)2=2
`aL1 `aL2 `aL3 `aL4 e tm2
`2
`2
X
` `=2
Identifying m2 = 2 `2R, we x the relative normalization to be
A1jF 4 ! 210 123 A1SYM:
B.2
Sixderivative coupling f (6)
In this subsection, we would like to x the relative normalization between the D2F 4
coupling from twoloop heterotic string amplitude and that from twoloop 5d maximal SYM
on its Coulomb branch by looking at a point of enhanced ADE gauge symmetry in the
heterotic moduli space and a degeneration limit of the genus two Riemann surface (see
gure 4).
Recall that the heterotic twoloop amplitude is
t u
3
+ (2 perms)
# Z
Q
10( ) = e2 i( + + )
e2 i(n +k +` ) :
Y
(n;k;`)>0
with the theta function given by
X
`1;`22
X
`1;`22
e i AB`LA `LB i AB`RA `RB+2 i`A yA+ 2 ((Im ) 1)AByA yB
e
i AB`A `B 2 Im AB`RA `RB+2 i`A yA+ 2 ((Im ) 1)AByA yB :
Each component of Re AB has periodicity 1. The imaginary part of the period matrix can
be written as
Im
t1 + t3
;
with det Im
= t1t2 + t1t3 + t2t3. In the limit of large positive t1; t2; t3, this corresponds
to the genus two Riemann surface degenerating into three long tubes, of length t1; t2; t3
respectively. We can also write
Im AB`A `B = t1(`1)2 + t2(`2)2 + t3(`1 + `2)2;
((Im ) 1)AByA yB = t1y22 + t2y12 + t3(y1
y2)2 :
t1t2 + t1t3 + t2t3
In the limit of large positive t1; t2; t3, the theta function, apart from the term 1 which
vanishes upon taking yderivative, is dominated by the terms involving lattice vectors `
such that `2L + `2R is close to 2, when the lattice embedding is near an ADE point in the
moduli space. The Igusa cusp form
10( ), on the other hand, behaves as
10( ) ! e2 iRe( 11+ 22 12)e 2 (t1+t2+t3);
where we have used the product expression for 10( ),
(B.5)
(B.6)
(B.8)
(B.10)
(B.11)
giving the factor
HJEP12(05)4
exp
2 (t1(`1R)2 + t2(`2R)2 + t3(`1R + `2R)2) + 2 i`A
yA :
We are interested in the limit where (`1R)2, (`2R)2, and (`1R + `2R)2 are small, and correspond
to W boson masses of three propagators in the twoloop diagram. We have (in the rest of
this section we will not distinguish `Ia with (`L)Ia since in the limit of interest (`R)Ia ! 0)
A2jD2F 4 !
4
X
(`1)2=(`2)2=(`1+`2)2=2
IJ KL`Ia1 `aJ2 `aK3 `aL4
3
u
t
Z
dt1dt2dt3
(t1t2 + t1t3 + t2t3) 2
1 e 2 (t1(`1R)2+t2(`2R)2+t3(`1R+`2R)2) +(cyclic perms in 2; 3; 4):
Here (n; k; `) > 0 means that n; k
0, ` 2 Z, and in the case when n = k = 0, the product
is only over ` < 0. In the above expression we parametrize
as
The integration over Re AB then picks out the terms in the theta function with
1 `1 = `2 `2 = (`1 + `2)2 = 2;
(B.12)
(B.13)
(B.14)
(B.15)
Here `Ia is the eigenvalue of the Cartan generator Ta on the W boson labeled by the root
vector `I , on the propagator of length tI , I = 1; 2. On the third propagator of length t3,
the W boson has charge `1a + `2a with respect to Ta.
Let us compare this with the twoloop amplitude at 6derivative order in 5d SYM,
whose contribution from the diagrams involving two light internal W bosons takes the form
A2
SYM =
2
X
(`1L)2=(`2L)2=(`1L+`2L)2=2
t21t22`1a1 `1a2 `2a3 `2a4 + 5 more
Z
dt1dt2dt3
t12t2t3`1a1 `1a2 ( `
1
`2a3 )`2a4
t12t2t3( `
1
a1
`2a1 )`2a2 `1a3 `1a4 + 10 more
Z d5p1d5p2
(2 )10 e
Pi3=1 ti(pi2+mi2) + (cyclic perms in 2; 3; 4);
where the rst and the second lines come from the rst and the second twoloop diagrams
in gure 4, respectively. The term proportional to t21t22, for instance, comes from the
twoloop diagram with two external lines (with Cartan label a1; a2) attached to the propagator
of length t1 and two external lines (with Cartan label a3; a4) attached to the propagator of
length t2. The
`1; `2; `
1
`2 to each internal propagator.
stand for all the other possible assignments of the W boson root vectors
4
2
twoloop amplitudes A2SYM in 5d maximal SYM.
We can identify m21 = 2 (`1R)2, m22 = 2 (`2R)2, m23 = 2 (`1R + `2R)2. The factor in the
bracket, after multiplication by s and summation over permutations, can be organized into
the form (taking into account s + t + u = 0)
s(t1t2 + t1t3 + t2t3)2 `1a1 `1a2 `2a3 `2a4 + `2a1 `2a2 `1a3 `1a4
+ (cyclic perms in 2; 3; 4)
2
3
s(t1t2 + t1t3 + t2t3)2 `1a1 `1a2 `2a3 `2a4 + `2a1 `2a2 `1a3 `1a4
2`(1a1 `2a2)`(a3 `a4)
1 2
+ (cyclic perms in 2; 3; 4)
3 (t1t2 + t1t3 + t2t3)2( IK JL + IL JK )`Ia1 `aJ2 `aK3 `aL4 + (cyclic perms in 2; 3; 4):
Notice that only terms with two `1 and two `2 will survive after summing over the s; t; u
channels. Hence the SYM twoloop amplitude can be put into the form
(B.18)
(B.19)
A2
SYM = 2 11
s
3
Z
( IK JL + IL JK )`Ia1 `aJ2 `aK3 `aL4 + (cyclic perms in 2; 3; 4)
dt1dt2dt3
(t1t2 + t1t3 + t2t3) 2
1 e
Pi timi2 :
This is indeed proportional to (B.15),
A2jD2F 4 ! 215 9 SYM:
A2
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
[1] M.B. Green and M. Gutperle, E ects of D instantons, Nucl. Phys. B 498 (1997) 195
[hepth/9701093] [INSPIRE].
[2] M.B. Green and S. Sethi, Supersymmetry constraints on type IIB supergravity, Phys. Rev. D
59 (1999) 046006 [hepth/9808061] [INSPIRE].
theory and maximal supergravity, JHEP 02 (2007) 099 [hepth/0610299] [INSPIRE].
[4] M.B. Green, J.G. Russo and P. Vanhove, Automorphic properties of low energy string
amplitudes in various dimensions, Phys. Rev. D 81 (2010) 086008 [arXiv:1001.2535]
to the D6R4 interaction, Commun. Num. Theor. Phys. 09 (2015) 307 [arXiv:1404.2192]
JHEP 01 (2000) 029 [hepth/0001083] [INSPIRE].
Theory, JETP Lett. 43 (1986) 730 [INSPIRE].
Lett. B 220 (1989) 153 [INSPIRE].
Phys. B 463 (1996) 55 [hepth/9511164] [INSPIRE].
034 [hepth/9909110] [INSPIRE].
JHEP 04 (2010) 127 [arXiv:0910.2688] [INSPIRE].
(2009) 075 [arXiv:0902.0981] [INSPIRE].
163 [arXiv:1201.2653] [INSPIRE].
[6] G. Bossard and V. Verschinin, Er4R4 type invariants and their gradient expansion, JHEP
03 (2015) 089 [arXiv:1411.3373] [INSPIRE].
[7] G. Bossard and V. Verschinin, The two r6R4 type invariants and their higher order
generalisation, JHEP 07 (2015) 154 [arXiv:1503.04230] [INSPIRE].
[8] G. Bossard and A. Kleinschmidt, Supergravity divergences, supersymmetry and automorphic
forms, JHEP 08 (2015) 102 [arXiv:1506.00657] [INSPIRE].
[9] Y. Wang and X. Yin, Constraining Higher Derivative Supergravity with Scattering
Amplitudes, Phys. Rev. D 92 (2015) 041701 [arXiv:1502.03810] [INSPIRE].
[10] Y. Wang and X. Yin, Supervertices and Nonrenormalization Conditions in Maximal
Supergravity Theories, arXiv:1505.05861 [INSPIRE].
[11] P.S. Aspinwall, K3 surfaces and string duality, in proceedings of Theoretical Advanced Study
Institute in Elementary Particle Physics (TASI 96): Fields, Strings, and Duality, Boulder,
U.S.A., 2{28 Jun 1996, pp. 421{540 [hepth/9611137] [INSPIRE].
[12] E. Kiritsis, N.A. Obers and B. Pioline, Heterotic/typeII triality and instantons on K3,
[13] A.B. Zamolodchikov, Irreversibility of the Flux of the Renormalization Group in a 2D Field
[14] D. Kutasov, Geometry on the Space of Conformal Field Theories and Contact Terms, Phys.
[15] H. Ooguri and C. Vafa, Twodimensional black hole and singularities of CY manifolds, Nucl.
[16] D. Kutasov, Orbifolds and solitons, Phys. Lett. B 383 (1996) 48 [hepth/9512145] [INSPIRE].
[17] A. Giveon and D. Kutasov, Little string theory in a double scaling limit, JHEP 10 (1999)
[18] T. Dennen, Y.t. Huang and W. Siegel, Supertwistor space for 6D maximal super YangMills,
[19] C. Cheung and D. O'Connell, Amplitudes and SpinorHelicity in Six Dimensions, JHEP 07
[20] R.H. Boels and D. O'Connell, Simple superamplitudes in higher dimensions, JHEP 06 (2012)
[21] H. Elvang, D.Z. Freedman and M. Kiermaier, Solution to the Ward Identities for
Superamplitudes, JHEP 10 (2010) 103 [arXiv:0911.3169] [INSPIRE].
supergravity, JHEP 11 (2010) 016 [arXiv:1003.5018] [INSPIRE].
[23] Y.H. Lin, S.H. Shao, Y. Wang and X. Yin, On Higher Derivative Couplings in Theories
with Sixteen Supersymmetries, arXiv:1503.02077 [INSPIRE].
[24] W.M. Chen, Y.t. Huang and C. Wen, Exact coe cients for higher dimensional operators
with sixteen supersymmetries, JHEP 09 (2015) 098 [arXiv:1505.07093] [INSPIRE].
[25] A. Gregori, E. Kiritsis, C. Kounnas, N.A. Obers, P.M. Petropoulos and B. Pioline, R2
(1998) 423 [hepth/9708062] [INSPIRE].
[26] J.T. Liu and R. Minasian, Higherderivative couplings in string theory: dualities and the
B eld, Nucl. Phys. B 874 (2013) 413 [arXiv:1304.3137] [INSPIRE].
[27] C. Cordova, T.T. Dumitrescu and X. Yin, Higher Derivative Terms, Toroidal
Compacti cation and Weyl Anomalies in SixDimensional (2; 0) Theories,
arXiv:1505.03850 [INSPIRE].
B 291 (1987) 41 [INSPIRE].
24 [hepth/9410152] [INSPIRE].
[28] D.J. Gross and J.H. Sloan, The Quartic E ective Action for the Heterotic String, Nucl. Phys.
[29] E. D'Hoker and D.H. Phong, The Box graph in superstring theory, Nucl. Phys. B 440 (1995)
[30] S. Stieberger and T.R. Taylor, NonAbelian BornInfeld action and type Iheterotic duality
(II): Nonrenormalization theorems, Nucl. Phys. B 648 (2003) 3 [hepth/0209064] [INSPIRE].
[31] E. D'Hoker and D.H. Phong, Twoloop superstrings VI: Nonrenormalization theorems and
the 4point function, Nucl. Phys. B 715 (2005) 3 [hepth/0501197] [INSPIRE].
[32] J. Polchinski, String theory. Volume II: Superstring theory and beyond, Cambridge University
[33] G.W. Moore, Modular Forms and Two Loop String Physics, Phys. Lett. B 176 (1986) 369
[34] P. Tourkine, Tropical Amplitudes, arXiv:1309.3551 [INSPIRE].
[35] C. Cordova, T.T. Dumitrescu, Y.H. Lin and X. Yin, work in progress.
[36] N. Seiberg, Observations on the Moduli Space of Superconformal Field Theories, Nucl. Phys.
B 303 (1988) 286 [INSPIRE].
[37] P.S. Aspinwall and D.R. Morrison, String theory on K3 surfaces, hepth/9404151 [INSPIRE].
[38] C.M. Chang, Y.H. Lin, S.H. Shao, Y. Wang and X. Yin, Little String Amplitudes (and the
Unreasonable E ectiveness of 6D SYM), JHEP 12 (2014) 176 [arXiv:1407.7511] [INSPIRE].
[39] T. Eguchi and A. Taormina, Unitary Representations of N = 4 Superconformal Algebra,
Phys. Lett. B 196 (1987) 75 [INSPIRE].
(1985) 678 [INSPIRE].
Int. J. Mod. Phys. A 6 (1991) 1749 [INSPIRE].
[40] L.J. Dixon, J.A. Harvey, C. Vafa and E. Witten, Strings on Orbifolds, Nucl. Phys. B 261
[41] S. Cecotti, N = 2 LandauGinzburg versus CalabiYau models: Nonperturbative aspects,
Phys. 219 (2001) 399 [hepth/0006196] [INSPIRE].
[hepth/9407190] [INSPIRE].
HJEP12(05)4
E ective Action, Nucl. Phys. B 771 (2007) 40 [hepth/0610258] [INSPIRE].
Theory, Phys. Rev. D 83 (2011) 046011 [arXiv:1009.2725] [INSPIRE].
Theories, JHEP 05 (2011) 017 [arXiv:1009.2087] [INSPIRE].
[arXiv:1203.6064] [INSPIRE].
arXiv:1507.05637 [INSPIRE].
Bootstrap of the K3 CFT, arXiv:1511.04065 [INSPIRE].
[arXiv:1204.2002] [INSPIRE].
[hepth/9903224] [INSPIRE].
[42] R. Dijkgraaf , Instanton strings and hyperKahler geometry, Nucl. Phys. B 543 ( 1999 ) 545 [43] O. Aharony , B. Fiol , D. Kutasov and D.A. Sahakyan , Little string theory and heterotic/typeII duality, Nucl . Phys. B 679 ( 2004 ) 3 [ hep th/0310197] [INSPIRE].
[44] O. Lunin and S.D. Mathur , Correlation functions for M N =SN orbifolds , Commun. Math.
[45] N. Berkovits and C. Vafa , N = 4 topological strings , Nucl. Phys. B 433 ( 1995 ) 123 [46] I. Antoniadis , S. Hohenegger and K.S. Narain , N = 4 Topological Amplitudes and String [47] I. Antoniadis , S. Hohenegger , K.S. Narain and E. Sokatchev , Harmonicity in N = 4