D6R4 curvature corrections, modular graph functions and Poincaré series
HJE
D6R4 curvature corrections, modular graph functions
Olof Ahl?en 0 1
Axel Kleinschmidt 0 1
0 Am Mu ?hlenberg 1 , DE14476 Potsdam , Germany
1 ULBCampus Plaine CP231 , BE1050 Brussels , Belgium
In this note we study the Uduality invariant coefficient functions of higher curvature corrections to the fourgraviton scattering amplitude in type IIB string theory compactified on a torus. The main focus is on the D6R4 term that is known to satisfy an inhomogeneous Laplace equation. We exhibit a novel method for solving this equation in terms of a Poincar?e series ansatz and recover known results in D = 10 dimensions and find new results in D < 10 dimensions. We also apply the method to modular graph functions as they arise from closed superstring oneloop amplitudes.
MTheory; String Duality; Superstrings and Heterotic Strings

1 Introduction
2 Poincar?e series ansatz
3
4
2.1
2.2
3.1
3.2
A brief reminder of Eisenstein series
R4 and D4R4 curvature corrections
2.3 Inhomogeneous Laplace equation and ansatz
D6R4 solutions in various dimensions
??expansion [1], where ?? = ?s2 is related to the inverse string tension. Via this process the
contribution of massive string states to the scattering of massless particles can be
systematically evaluated. Using further the constraints implied by Uduality and supersymmetry
one can sometimes even determine exactly the perturbative and nonperturbative
contri?
butions to the string scattering amplitude at a fixed order in ? . This strategy was first
employed by Green and Gutperle [2] and further developed in many subsequent papers,
see for example [3?30].
The structure of the (analytic contribution to the) fourgraviton amplitude can be
written in terms of an effective lowenergy theory in D spacetime dimensions (in Einstein
frame) as
L
(D)
1
? (??)3
?
p,q=0
R +
X (??)2p+3qE((pD,q))(g)D4p+6qR4 + . . . ,
(1.1)
? 1 ?
where we have suppressed an overall dimensionful factor. There is a double summation
over two integers due to the two independent symmetric invariants1
that can be constructed for the scattering of four massless particles and that can appear in
the scattering amplitude. The term D4p+6qR4 is shorthand for very specific contractions
of (4p + 6q) covariant derivatives of four Riemann tensors. The main interest lies in the
functions E((pD,q))(g) of the moduli g ? E11?D(R)/K(E11?D), where E11?D is the (split real)
CremmerJulia hidden symmetry group of maximal ungauged supergravity in D spacetime
dimensions [31] and K(E11?D) its maximal compact subgroup. (The Dynkin diagram of
E11?D is given below in figure 1.)
The functions have to be invariant under the discrete Uduality E11?D(Z) [32] and
supersymmetry implies that they also have to satisfy (tensorial) differential equations [6,
10, 23, 27]. The most prominent of these are the second order Poissontype equations for
the first three terms in the expansion [6, 10, 25]
the two functions E((0D,0)) and E(
1,0
) that correspond to the 21 BPS R4 and 41 BPS D4R4
curvature corrections, respectively. Their solutions have been studied in great detail and are
given by (linear combinations of) Eisenstein series, see for instance [14, 15, 17, 25, 35] for
summaries or section 2.2 below. By contrast, equation (1.3c) for E((0D,1)) (that corresponds to
the 81 BPS correction D6R4) is of a qualitatively different nature in that it always contains
a nonlinear source term given by ?(E((0D,0)))2. The equation is therefore an inhomogeneous
Laplace equation or Poisson equation. A similar structure is expected for curvature corrections with even more derivatives [36]. We also note that very similar inhomogeneous
SL(2, Z) invariant equations arise in the study of socalled modular graph functions [37?40]
and we shall also study these cases.
The structure of the inhomogeneous Laplace equation (1.3c) indicates that it will not
be solved by an automorphic function of the standard type as these functions are required
to be finite under all E11?Dinvariant differential operators, see for instance [35]. The
and u = ? ?4 (k1 + k3)2 and satisfy s + t + u = 0 onshell.
1The dimensionless Mandelstam invariants are here defined as s = ? ?4 (k1 + k2)2, t = ? ?4 (k1 + k4)2
?
?
?
? 2 ?
(D)
generalised class of E11?D(Z)invariant functions on E11?D to which the solution E(
0,1
)
belongs has not been identified abstractly. In this article, we will not attempt to define
this class fully but rather present a method for solving equations of the type (1.3c) using
a Poincar?e series ansatz.
Our method is inspired by the recent explicit solution of (1.3c) that was presented
by Green, Miller and Vanhove in [41] for the case of D = 10 type IIB string theory. In
this case one has the CremmerJulia group SL(2, R) and Uduality group SL(2, Z) and
they performed a Fourier expansion of the equation, using SL(2, Z)invariance and the
known source function E((01,00)) that appears on the righthand side of (1.3c). The solution
to the homogeneous equation is fixed by consistency with string perturbation theory as
a boundary condition. In an appendix of [41], the authors rewrite their solution as a
Poincar?e series. The starting point of our approach is that the ?seed? of the Poincar?e series
solves a much simpler Laplace equation than (1.3c). This simpler equation is, however, still
inhomogeneous. We present solutions to this equation. The hard part is now to impose
the correct boundary conditions and to extract the Fourier expansion. Our method is
explained in detail in section 2.
The correction term D6R4 has attracted a fair amount of attention recently [25, 27,
42], in particular in connection to the KawazumiZhang invariant of genus two Riemann
surfaces [43?46]. In the supergravity limit, the corresponding expressions resemble twoloop
field theory amplitudes [25]. This agrees nicely with independent expressions for (parts of)
E((0D,1)) that were recently found using a twoloop calculation in exceptional field theory with
manifest E11?D invariance [29]. Other approaches to solving the inhomogeneous differential
equation include the original approach followed in [10, 41] based on Fourier decomposing
the equation, but one can also try to tackle the equation by using spectral methods [41, 47]
and there might also be a connection to automorphic distributions [41]. Our present work
is complementary to these results.
This article is structured as follows. In section 2, we present the new method for
solving the inhomogeneous Laplace equation (1.3c) based on a Poincar?e series ansatz. The
method is used in section 3 to construct the function E((0D,1)) in various dimensions while we
consider an example from modular graph functions in section 4. Section 5 discusses open
problems arising from our analysis.
2
Poincar?e series ansatz
After first reviewing briefly Eisenstein series and their properties, we expose our ansatz
and how it can be used to simplify the inhomogeneous Laplace equation.
2.1
A brief reminder of Eisenstein series
The homogeneous Laplace equations in (1.3) for E((0D,0)) and E((1D,0)) can be solved in terms
of (combinations of) Eisenstein series E(?, g) on the symmetric space E11?D/K(E11?D).
Such Eisenstein series are invariant under E11?D(Z), can be parametrised by a (complex)
? 3 ?
weight ? of the Lie algebra and satisfy the Laplace equation
where ? is the Weyl vector of the Lie algebra and the normsquares are calculated using
the Killing metric, normalised to 2 on real roots. Eisenstein series can be written as2
? ? 2
Iwasawa decomposition g = nak of a group element g ? E11?D, where n is in the maximal
(upper) unipotent and k ? K(E11?D). As the kcomponent of g does not enter in the
above definition, the Eisenstein series (2.2) descends to a function on the symmetric space
E11?D/K(E11?D). In mathematical terms, it is spherical. Invariance of E(?, g) under
E11?D(Z) is manifest in (2.2) as it is an orbit sum; the quotient by B(Z) = N (Z)A(Z) is
necessary as one has H(?g) = H(g) for ? ? B(Z).
We can also think of the summand as arising from a B(Z)invariant character
?? : B(Z)\B ? C?,
??(na) = eh?+?H(na)i
on the Borel subgroup B that has been extended trivially to all of E11?D = BK(E11?D).
The character property means that
(2.1)
(2.2)
(2.3)
(2.4)
??(bb?) = ??(b)??(b?)
for any b, b? ? B.
2.2
R4 and D4R4 curvature corrections
The functions E((0D,0)) and E((1D,0)) that appear for the R4 and D4R4 curvature corrections,
respectively, satisfy the homogeneous Laplace equations of the first two lines in (1.3) unless
there is a Kronecker source term. In the generic cases without Kronecker source term, the
solutions are given by Eisenstein series of E11?D with weights given as in table 1. In
the table, the weight ?1 refers to the fundamental weight associated with node 1 of the
Dynkin diagram of E11?D that is shown in figure 1. Table 1 lists the relevant weights for
the Eisenstein series.3 In the last column, we also show which representations of E11?D
the automorphic functions belong to. The above answers have passed many consistency
checks [14, 15, 17] and were also found by direct exceptional field theory calculations [29].
The quasicharacters ?? for the particular weight ? = 2s?1 ? ? actually have a larger
invariance than B(Z): also the whole Tduality group SO(10 ? D, 10 ? D; Z) (associated
with the nodes 2, . . . , 11 ? D in the diagram) leaves ?? invariant. Combined with B(Z)
one obtains an invariance under the maximal parabolic subgroup P1(Z) and the Eisenstein
series can also be called a maximal parabolic Eisenstein series [14].
2This expression is absolutely convergent for large enough real parts of ? (with respect to all simple
roots) and can be analytically continued almost everywhere by functional equations [48].
3For D < 9, the Eisenstein series for these weights have to be obtained by analytic continuation from a
functionally related convergent expression of the form (2.2).
? 4 ?
function
E((0D,0)) = 2?(3)E(?, g)
E((1D,0)) = ?(5)E(?, g)
curvature term
R4
D4R4
weight ?
We reproduce the inhomogeneous equation (1.3c) for the D6R4 correction for convenience:
? ?
6(14 ? D)(D ? 6)
D ? 2
the equation is of the form
(? ? ? ) ?(g) = ?(E(?, g))2,
where E(?, g) is an Eisenstein series that is given by a coset sum as in (2.2). This equation
has to be supplemented by appropriate boundary conditions near cusps of G/K. In string
theory language this includes for example consistency with string perturbation theory [41].
More generally, we could also allow for equations involving a polynomial in Eisenstein series
on the righthand side and this happens for higher derivative terms [36] or modular graph
functions [37, 40].
The strategy in [41] for solving the equation (2.6) was to (double) Fourier expand
both sides of the equation and then obtain simpler equations for the Fourier coefficients. A
standard Fourier expansion of the righthand side would require computing the convolution
of the Fourier expansions of the product of two Eisenstein series, something that has
not been carried out in full. The double Fourier expansion consisted in not doing the
convolution sum but rather looking at each summand individually. In appendix A of [41],
the authors rewrite the doubly Fourier expanded solution in terms of a Poincar?e series. This
has served as inspiration for our strategy to directly work with a Poincar?e series ansatz.
Our strategy consists in making the ansatz that ?(g) can be written as a coset sum
?(g) =
X
that has the property that
? : G ? C?
?(bZgk) = ?(g)
for all bZ ? B(Z) and k ? K(E11?D). In other words, the seed ? can be thought of as a
function on the Borel subgroup B that is trivial on the discrete subgroup B(Z) and extended
trivially to all of G by right Kinvariance. Importantly, we do not assume, however, that
? is a character on B. Therefore, the Poincar?e sum ?(g) in (2.7) is in general not an
Eisenstein series.4 Moreover, the ansatz entails that we would like the Poincar?e sum to be
absolutely convergent. As we shall see in the examples below, this is not always obvious
to achieve.
With the ansatz (2.7) and the expression (2.2) one can find solutions of the
inhomogeneous Laplace equation (2.6) if one has a solution of the equation
(? ? ? ) ?(g) = ???(g)E(?, g).
In writing this equation, we have ?folded? the coset sum in one of the Eisenstein factors
on the righthand side. On the lefthand side one also uses the fact that the differential
operator is invariant under the group E11?D.
Since both ?(g) and ??(g) are leftinvariant under the discrete Borel group B(Z), this
group can be further used to Fourier expand equation (2.10). Let U ? B be a unipotent
subgroup of the Borel group B. For example, we could choose U = N , the maximal
unitary character on U , i.e. for abelian U it is of the form5
unipotent subgroup generated by all positive root spaces, but this is not necessary and
other choices might be more convenient in some cases. Let ? : U (Z)\U ? U (
1
) be a
?(u) = ?
exp
dim U
X
k=1
xkEk
!!
= exp 2?i X
dim U
k=1
!
mkxk ,
where Ek are the suitably normalised nilpotent generators of U and mk ? Z are the mode
numbers of the character ? that we will also refer to as instanton numbers or instanton
charges.
defined by
? 6 ?
The Fourier coefficient of any function f (g) that is leftinvariant under U (Z) is then
Z
U(Z)\U
f?(g) =
f (ug)?(u)du,
4The differential equation (2.6) for ? also implies that ? is not an automorphic function in the standard
sense [35]. Standard automorphic functions are what is called Z(g)finite, where Z(g) is the center of
the universal enveloping algebra of the Lie algebra g of E11?D. This center is generated by the Casimir
operators that translate into invariant differential operators. Because of the source (E(?, g))2 in (2.6) the
system is not finite under the action of the Laplace operator.
5For nonabelian U , one only has to include the generators associated with the abelianisation [U, U ]\U .
The Fourier expansion will then be incomplete and has to be refined using also nonabelian Fourier
coefficients [35, 49, 50].
(2.8)
(2.9)
(2.10)
(2.11)
(2.12)
where the integration is over a single period and the Haar measure is normalised such that
U (Z)\U has unit volume. Fourier coefficients satisfy f?(ug) = ?(u)f?(g) for any u ? U .
For abelian U one has the Fourier expansion of the function f given by
The sum is over all unitary characters, i.e. over all choices mk ? Z. The trivial case ? = 1
represents the constant term of f with respect to U ; in physics applications this comprises
typically all perturbative contributions in some modulus.
Taking Fourier coefficients of equation (2.10) one then obtains
(? ? ? ) ??(g) = ? (??E)? (g).
For the product righthand side one needs to take the convolution of the Fourier expansions
of the factors. Due to the Poincar?e series ansatz (2.7) the convolution has become trivial
since the character ?? only has a zero mode in its Fourier expansion. Therefore one obtains
(? ? ? ) ??(g) = ???(g)E?(?, g).
If the Fourier expansion of E(?, g) is known (as it is for the R4 cases in string theory), one
can explicitly write down this equation and try solve it for each mode ?, separately. This
is the strategy we will follow in the following sections.
Besides convergence, one tricky point is the fate of the boundary conditions when
making the Poincar?e series ansatz. The inhomogeneous equation (2.15) will allow also
require solutions to the homogeneous equation and the particular combination that arises
for the wanted solution ? is fixed by boundary conditions.
However, these boundary
conditions are originally stated for the Poincar?e sum ? in (2.7). We do not currently know
how to generally translate these conditions directly into conditions for ? and so in principle
one has to perform the Poincar?e sum to select the right solution. In the cases below we
will basically follow this logic for fixing the homogeneous solution.
3
D6R4 solutions in various dimensions
in this section we implement the strategy just outlined to determine the D6R4 coefficient
function in D = 10 and D = 7 spacetime dimensions.
3.1
Type IIB in D = 10
In the case of type IIB in D = 10, the differential equation on the Poincar?e upper half
plane H = {z = x + iy ? C y > 0} ?= SL(2, R)/SO(2) is [41]
The R4 function E(0,0) on H is given by
(? ? 12) E(
0,1
) = ? E(0,0) .
2
E(0,0)(z) = 2?(3)E3/2(z),
? 7 ?
(2.13)
(2.14)
(2.15)
(3.1)
(3.2)
E(
0,1
)(z) =
X
??B(Z)\SL(2,Z)
?(?z)
(? ? 12) ?(z) = ?4?(3)2y3/2E3/2(z).
the inhomogeneous Laplace equation (3.1) leads to the ?folded? equation
This equation corresponds to (2.10) in the general discussion and we can further analyse
it by Fourier expanding both sides. The abelian unipotent invariance group here is given
by U (Z) = N (Z) =
1 k
1
k ? Z . Writing the Fourier expansion of ?(z) as
(3.3)
(3.4)
(3.5)
(3.6)
(3.7)
(3.8a)
(3.8b)
(3.9)
with E3/2(z) being the s = 23 case of the nonholomorphic Eisenstein series
Es(z) =
X
??B(Z)\SL(2,Z)
[Im(?z)]s = ys +
?(2(1 ? s)) y1?s + X
?(2s)
n6=0
Fn,s(y)e2?inx
with the nonzero Fourier coefficients
Fn,s(y) =
2
?(2s)
y1/2ns?1/2?1?2s(n)Ks?1/2(2?ny)
and ?(s) = ??s/2?(s/2)?(s). An element ? =
?z = aczz++db .
Kt(y) in the above is a modified Bessel function of the second kind and
induced from the character ?s(z) = ys. The Laplace operator on H is ? = y
?t(k) = Pdk dt is the (positive) divisor sum for k ? Z>0. The Eisenstein series Es(z) is
2 ?x2 + ?y2 .
The physical interpretation of the modulus z ? H here is such that its real part x
corresponds to the RR axion of type IIB string theory while its imaginary part y is the
inverse string coupling gs?1. The Fourier expansion of E3/2(z) given in (3.3) then contains
two zero mode terms, y3/2 = gs?3/2 and y?1/2 = gs1/2. These correspond to the perturbative
treelevel and oneloop contributions to the fourgraviton scattering amplitude expressed
in Einstein frame; the fact that there are no further zero modes is a strong perturbative
a b
c d
? SL(2) acts on z ? H by z 7?
nonrenormalisation theorem [2].
Making the Poincar?e series ansatz (2.7) for E(
0,1
)(z), viz.
leads to the following two equations for the zero and nonzero Fourier modes
(y2?y2 ? 12)c0(y) = ?4?(3)2y3 ? 3
4 ?2?(3)y,
(y2?y2 ? 4?2n2y2 ? 12)cn(y) = ?16??(3)y2n??2(n)K1(2?ny),
where we have used the explicit form of the Fourier coefficients for E3/2(z) given in (3.3).
These equations correspond to (2.15) in the general discussion.
The general solution to equation (3.8a) for the zero mode c0(y) is simple to obtain:
2
3
The solution of equation (3.8b) for the nonzero modes cn(y) is more complicated.
The homogeneous equation can be recast in Bessel form and has only one solution that
falls off at the weak coupling cusp y = gs?1
?
?. For mode number n it is given by
y1/2K7/2(2?ny) and has in fact a finite asymptotic expansion around weak coupling. A
particular solution of (3.8b) can be extracted from the analysis [41]6 and combined with
the homogeneous solution we obtain
+ ?ny1/2K7/2(2?ny),
ny +
10
?2yn
K0(2?ny) +
6
?
+
10
?3y2n2
Here, ? and ? are a priori undetermined integration constants that have to be fixed by
boundary conditions. We note that performing the Poincar?e sum of any term of the form ys
produces the Eisenstein series Es(z) with perturbative terms ys and y1?s, cf. (3.3). Having
either ? or ? nonzero will therefore necessarily lead to the two zero mode terms y?3 and
y4 in the summed E(
0,1
). While the term y?3 = gs3 corresponds to a threeloop contribution
(after changing to string frame), the term y
4 = gs?4 would be a contribution of loop
order ?1/2, something that is incompatible with string perturbation theory. Since both
homogeneous solutions parametrised by ? and ? would lead to this inconsistent behaviour,
we are led to set
? = ? = 0
?
where ?n is the integration constant associated with the homogeneous solution. Again,
this integration constant has to be fixed from asymptotic considerations. In this case, we
consider the strong coupling region y ? 0 where these instantonic contributions dominate.
By Sduality this region is also related to the perturbative regime y ? ?. The absence of
any singular terms in the limit y ? 0 determines the value of ?n to be
?n = ?
128?(3)??2(n) ,
3?p n
 
leading to the final expression after some rearrangements
cn(y) = 8?(3)??2(n)y
1 +
K0(2?ny) +
40
16
(2?ny)2
? 3?(ny)1/2 K7/2(2?ny) .
12
2?ny
+
80
(2?ny)3
K1(2?ny)
(3.13)
This way cn(y) is of order y as y ? 0 and does not contain any singular terms.7 The
resulting expression agrees precisely with the result found in [41] but now obtained directly
from a Poincar?e series ansatz.
6In appendix A, we present an algebraic formalism that also leads to this solution.
7This condition appears slightly stronger than the requirement O(y?2) as y ? 0 for the summed solution
E(
0,1
)(z) that was found by Sduality in Lemma 2.9 in [41].
(3.11)
(3.12)
We note that the seed function ?(z) = c0(y) + Pn6=0 cn(y)e2?inx depends nontrivially
on both Borel coordinates x and y and in particular is not a character on B. This leads
to complications when carrying out the Poincar?e sum. In fact, the full expression is not
known. Let us make a few comments about convergence of the Poincar?e sum over the seed
we just determined. If the sum were absolutely convergent one could perform the sum
over all the terms separately. This cannot be true as it is well known that the Poincar?e
sum for the linear term in y in c0(y) does not converge: it represents the limiting value
of the nonholomorphic Eisenstein series for SL(2). It is therefore desirable to introduce a
regularised version of the solution depending on a parameter with a limit that is related to
the above solution.8 In appendix B, we present a regularised version of the equation and
the solution where this problem does not arise.
The physical content of the solution is obtained by performing a Fourier expansion of
E(
0,1
)(z) = P
? ?(?z) with respect to the periodic real part x that represents the
RamondRamond axion in type IIB theory. The zero mode piece in E(
0,1
)(z) corresponds to the
perturbative terms in gs, together with nonperturbative contributions of vanishing net
instanton charge. These should be interpreted as instantonantiinstanton contributions
and they are exponentially suppressed by e?4?ny [10]. We have not managed to derive the
Fourier expansion from the Poincar?e sum form and here merely quote the result for the
perturbative zero modes obtained in [10, 41]9
The perturbative terms thus obtained correspond to contributions from treelevel up to
threeloops. The numerical values obtained by using SL(2, Z) invariance and the differential
equation have been confirmed by direct string theory calculations [10, 43, 51].
3.2
Toroidal compactification to D = 7
In the case of D = 7, the Uduality group is SL(5, Z) and the moduli space is SL(5)/SO(5).
Equation (1.3c) becomes
? ? 5
42
(7)
E(
0,1
) = ? E(0,0)
(7)
2
,
where the R4function E((07,)0)(g) = 2?(3)E(3?1 ? ?, g) satisfies
We will analyse equation (3.15) over the mirabolic P1, i.e. the maximal parabolic subgroup
associated with node 1. This is the parabolic that is associated with the string perturbation
8In terms of the Eisenstein series Es(z) induced by the character ys one obtains the socalled (first)
Kronecker limit formula [35] for s ? 1.
9The exponentially suppressed terms in the zero and nonzero modes are not known explicitly.
? +
12
5
(7)
E(0,0) = 0 .
(3.14)
(3.15)
(3.16)
theory expansion; the Levi is SL(
4
) ? GL(
1
) ?= SO(3, 3) ? R
duality moduli space of the string theory threetorus and the string coupling. Explicitly,
we parametrise the group element g ? SL(5) as
g = ulk =
where Q is a fourcomponent row vector and e4 is a general element of SL(
4
) and k ? SO(5).
The variable r equals the inverse string coupling10 in D = 7, Q corresponds to the four
axions that BPSinstantons couple to11 and e4 ? SL(
4
) ?= SO(3, 3) is associated with the
moduli space of T 3. The Laplacian ? = ?SL(5) on SL(5)/SO(5) decomposes in these
coordinates as
?SL(5) =
Before solving the inhomogeneous equation (3.15) using a Poincar?e sum, we first consider
the perturbative pieces similar to (3.14) by considering the zero Fourier modes in an
expansion of E(
0,1
) in the decomposition (3.17). A similar analysis can be found in [14].
Making the ansatz for the perturbative terms up to three loops with Fh denoting the
hloop perturbative piece as12
E(
0,1
)
(7,pert.) = r14/5 r2F0 + r0F1 + r?2F2 + r?4F3
= r24/5F0 + r14/5F1 + r4/5F2 + r?6/5F3
leads to the four equations
?SL(
4
) ? 6 F0 = ?4?(3)2 ,
21
9
?SL(
4
) ? 2
F3 = 0 .
?SL(
4
) ? 2
F1 = ?16?(2)?(3)ESL(
4
)(2?1 ? ?, e4) ,
?SL(
4
) ? 10 F2 = ?16?(2)2ESL(
4
)(2?1 ? ?, e4)2 ,
The equations not involving a squared source on the righthand side are solved by
F0 =
F1 =
2
3
4
3
?(3)2 ,
F3 = 4?(6) ESL(
4
)(6?1 ? ?, e4) + ESL(
4
)(6?3 ? ?, e4) .
?(2)?(3)ESL(
4
)(2?1 ? ?, e4) +
?(7)ESL(
4
)(7?2 ? ?, e4) ,
5?
756
(3.17)
(3.18)
(3.19)
(3.20a)
(3.20b)
(3.20c)
(3.20d)
(3.21a)
(3.21b)
(3.21c)
10We parametrise the string coupling in D dimensions gD such that different orders in perturbation theory
differ by gD2; this is different from the convention used in [17].
11In type IIA language, there are three D0instantons (wrapping one of the three cycles of T 3) and one
D2instanton (wrapping the full torus). In type IIB language, there is one (pointlike) D(?1)instanton and
three D1instantons (wrapping two out of three cycles).
12The overall prefactor comes from relating the string scale to the Planck scale in D = 7 spacetime
dimensions.
The constant treelevel contribution follows by expanding the known amplitude which also
rules out any contributions from the kernel of the differential operator. The oneloop piece
is a theta lift of the Narain partition function [13, 52]
where ? is the complex structure of the string oneloop torus and the combination
?(3)E3(? ) + ?(3) can also be obtained using modular graph functions [37]. The second term
in (3.20b) is in the kernel of the differential operator but is needed for obtaining the right
decompactification limit. The threeloop equation is homogeneous and the particular
combination of the homogeneous solutions is fixed by having the right decompactification limit
consistent with (3.14). It is also given by the genusthree theta lift of the constant
function [14]. The twoloop term is as always the hardest as it satisfies a similar inhomogeneous
equation to the original function. It is connected to an integral over the KawazumiZhang
invariant [43, 44] and constrained by having the right decompactification limit.
E(3?1 ? ?, g) =
fN (r, e4)e2?iQN
X
N?Z4
Solution using a Poincar?e sum
Let us assume that E((07,)1) is a Poincar?e series with respect to the same maximal parabolic
P1 as in (2.7), i.e.
E((07,)1)(g) =
X
??P1(Z)\SL(5,Z)
?(? ? g) .
We note that this particular type of parabolic coset sum is an assumption. Our motivation
for this choice is that is adapted to a string perturbation formulation of the solution, i.e.
the seed will be decomposed into terms at fixed order in string perturbation theory plus
SO(3, 3, Z) Tduality invariant functions coupled to instantons. Of course, this ansatz for
the seed gets spread out by the SL(5, Z) orbit sum into a more complicated Uduality
invariant function. We shall find a solution to the differential equation with our parabolic
ansatz; it is likely that other forms using other parabolics sums exist and it would be very
interesting to study their relation and functional equations.
With the assumption of the parabolic Poincar?e series (3.23), the Laplace equation (3.15)
then unfolds into
Since we assume ? to be left P1(Z)invariant, it can be expressed as the Fourier series
? ? 5
42
?(g) = ?4?(3)2r12/5E(3?1 ? ?, g).
?(g) =
X cN (r, e4)e2?iQN .
N?Z4
As the unipotent of a maximal parabolic subgroup of SL(n) is abelian, this expression
captures the whole of ? without need for nonabelian coefficients. Here, N is a
fourcomponent columnvector. The Eisenstein series on the righthand side of (3.24) can also
be written as a Fourier series over the unipotent as
(3.23)
(3.24)
(3.25)
(3.26)
fN (r, e4) =
One has ?SL(
4
)ESL(
4
)(2?1 ? ?, e4) = ? 2
3 ESL(
4
)(2?1 ? ?, e4) and k = gcd(N ). The notation
here is such that ?1 for SL(
4
) denotes the node adjacent to the one used defining the string
perturbation limit for SL(5) and thus is one of the outer notes of SL(
4
). (From the point
of view of SO(3, 3), this is a spinor node.) We now obtain differential equations for the
HJEP05(218)94
Fourier coefficients cN of ? in (3.25).
For the zero mode c0 we have the equation
5 r2?r2 ? 185
8
r?r +?SL(
4
) ? 5
42
c0(r, e4) = ?4?(3)2r24/5 ?8?(2)?(3)r14/5ESL(
4
)(2?1 ??, e4).
Starting with the homogeneous equation, one can make a separated ansatz of the form
c(0h)(r, e4) = r?F?(e4). This leads to
c(Np) = 32?2?(3)?2(k)r24/5 K0 2?re?1N 2 +12
is a particular solution of this equation. The proof relies on writing out the SL(
4
) Laplacian
and using properties of the Bessel functions in a way similar to demonstrating (3.11). As
In order for this to produce terms consistent with string perturbation theory only the values
? ? { 5
24 , 154 , 4 , ? 5 } are allowed, leading to the eigenvalues {6, 221 , 10, 29 } as in (3.20). The
6
5
remaining equation for F?(e4) is then solved by appropriate SL(
4
) Eisenstein series.
Looking for a particular solution of the form ?1r24/5 + ?2r14/5ESL(
4
)(2?1 ? ?, e4) we
are led to the particular solution
2
3
2
3
c(0p)(r, e4) = ?(3)2r24/5 + ?(2)?(3)r14/5E(2?1 ? ?, e4).
The full solution for the zero mode is now c0 = c(0h)+c(0p). Comparison with the IIB case
suggests that one should choose the homogeneous solutions c(0h) = 155?12 ?(7)r14/5ESL(
4
)(7?2 ?
?, e4) but without computing the Fourier expansion of the Poincar?e sum (3.23) and
comparing with (3.20) we cannot fix the homogeneous term definitively.
For the nonzero modes cN (r, e4), we have the equation
5 r2?r2 ? 185
8
r?r ?4?2r2e4?1N 2 +?SL(
4
) ? 5
42
We show in appendix A that
cN = ?16??(3)?2(k)r19/5 K1(2?re4?1N ) .
also explained in the appendix, there are solutions to the homogeneous equation given by
? 15?? (2?re4
?1N )5/2
If we impose the same constraint as for SL(2), namely that the strong coupling behaviour
r ? 0 is regular, this rules out the second solution and selects the combination
cN (r, e4) = 32?2?(3)?2(k)r24/5 K0 2?re?1N  + 12 K1 2?re?1N  + 40 K2 2?re?1N 
2 3 4
Eisenstein series associated with the minimal series on node 1. Looking for such an Eisenstein series leads to
irrational powers of r. There is, however, a wellknown homogeneous solution given by ESL(5)(7?2 ? ?) [25].
?
y
5
(3.33)
(3.34)
d? ,
(4.1)
(4.2)
Modular graph functions
Another family of automorphic functions satisfying inhomogeneous Laplace equation is
provided by modular graph functions [37]. These are functions that are invariant
under SL(2, Z) and have an explicit lattice sum description. Moreover, they satisfy
typically inhomogeneous Laplace equations. We note that for any Poincar?e sum of the form
?(z) = P??B(Z)\SL(2,Z) ?(?z), where z = x + iy and the periodic seed has the expansion
?(x + iy) = Pn?Z cn(y)e2?inx, the Fourier modes of ?(z) = Pn?Z fn(y)e2?inx are given
involving the Kloosterman sums
exp ?2?i?n?2?im d2(?2 +y2)
cm
y
d2(?2 +y2)
S(m, n; d) =
X
q?(Z/dZ)?
e2?i(qm+q?1n)/d .
This can be shown by writing out explicitly the coset sum. We shall try to apply this
formalism to rederive some results on modular graph functions.
As an example we consider the function C3,1,1(z) in the notation of [37]. It can be
defined explicitly from a multiple lattice sum as
C3,1,1(z) =
X
(mi,ni)6=(0,0)
(m1+m2,n1+n2)6=(0,0)
(m1,n1),(m2,n2)?Z2 ?5m1z + n16m2z + n22(m1 + m2)z + (n1 + n2)2
(? ? 6) ?(z) =
172?5
10
where we have explicitly written out the normalising factors and y? represents the regulator
for the constant. In writing the equation we have chosen to ?fold? E3. The boundary
condition for solving this equation is that the solution f (z) should not have a term growing
as y3 when approaching y ? ?.
Reducing the equation to Fourier modes ?(z) = Pn?Z cn(y)e2?inx leads to
A particular solution to these equations is given by
y2?y2 ? 6 c0(y) =
= ? 945 ??3(n)y2(1 + 2?ny)e?2?ny .
As shown in [37, eq. (3.19)] it satisfies the equation
(? ? 6) C3,1,1(z) =
,
8?2
c0(y) =
945
?(5)
(The homogeneous solutions correspond to the various Fourier modes of E2(z), but we will
not require them here.)
Using the general formula for Fourier expansions of Poincar?e sums one can now
construct the zero mode of the C3,1,1(z) = Pn?Z fn(y)e2?inx by computing
f0(y) = c0(y) + X
X S(m, 0; d)
Z
s = 2??s?(2s)Es is the nonholomorphic Eisenstein series Es in a different
We shall now forget that we have an explicit solution for C3,1,1(z) and try to construct
one by solving the Laplace equation (4.4) using a Poincar?e ansatz. In order to treat the
finite constant on the righthand side, we replace it by an Eisenstein series E? and send
? ? 0 at the end, using analytic continuation. Writing C3,1,1(z) = P??B(Z)\SL(2,Z) ?(?z)
in Poincar?e form, we have to solve the equation
This expression can be evaluated for the present case as follows.
We first rescale the
integration variable and restrict to the terms with m 6= 0 in the sum. They are
y X
f0(y) =
+
?
2?5
y5 +
The ksummation terminates due to the zeroes of the Riemann zeta function at negative
even integers. The case k = 2 requires taking a limit.
To complete the calculation of the zero mode f0(y) we also have to take into account
the contribution from c0(y) and the terms with m = 0 in the general expression. These are
just the usual constant terms for Eisenstein series [35]. This leads in total to
?(5)
1312??(94) y?4
10(?(? ? 1) ? 6) ?(2?)
?(2? ? 1) y1??
As is evident from this expression the sum over m is divergent and so the two remaining
summations cannot be interchanged strictly. One can partly make sense of the sum over
m by analytically continuing the Ramanujan identity
X ?a(m)?b(m)m?s =
m>0
?(s)?(s ? a)?(s ? b)?(s ? a ? b)
?(2s ? a ? b)
from the convergent region with large real part of s to s = ?k. This leads to
?
3
945
= ? 21?
2?(3)2
k!
k?0
y?1 +
(4.9) =
y?1 X k + 2 ?(?k)?(3 ? k)?(k + 3)?(k + 6)
(??y?1)k
?(4 + 2k)?(6)
7?(7) y?2
16?2
?
2?3
?(3)?(5) y?3 +
1312??(94) y?4 .
(4.9)
(4.10)
(4.11)
= ???3({4?z+22kk()m)
X k + 2 ??3(m)??3?2k(m)mk
?(4 + 2k)
(??y?1)k .
X S(m, 0; d)d?4?2k mk??3(m)(?2?y?1)k Z
(1 + t2)?k?2(1 + it)kdt

R
=2?k?{2z(k+2)?
}
This agrees with the Laurent polynomial stated in [37, eq. (6.2)] except for the constant
term in y0.14 The reason for this appears to be that the constant term of E2E3 has a
14We note that this Laurent polynomial can be deduced by considering the zero mode of equation (4.4)
and applying the RankinSelberg method to fix the solution to the homogeneous solution.
contribution to y0 that is missed by the above construction and this seems to be a general
feature that requires additional study. A further point that requires investigation is that
our method of treating the divergent sum over m seems to have lost all the exponentially
suppressed terms of the form O(e?y) in the zero mode while one would expect them from
the known function C3,1,1(z). However, we note that the method produces also the right
combination of solutions to the homogeneous equation found in [37].
5
Discussion
In the present paper, we have outlined a method for solving inhomogeneous automorphic
differential equations of the type that appear in string theory in several places.
method relied on making an ansatz for the solution of a Poincar?e sum form as in (2.7).
The advantage of this method is that the resulting differential equation for the seed ? of
the Poincar?e sum is less involved than the original equation and is solved in a Fourier
expansion. We exemplified this method for four examples, namely the D6R4 correction
for tendimensional type IIB string theory and the D6R4 correction in D = 7 spacetime
dimensions. The first example reproduced a known result from [10, 41] while the second
example gave a new proposal for D = 7. The third example dealt with a modular graph
function and how to reproduce its Laurent polynomial. A last family of examples was a
generalisation of the D6R4 function in D = 10 presented in appendix B.
While the method seems powerful and convenient for producing formal solutions to
the equations, there are a number of important and interesting points that require further
investigation for bringing the method to its full power. Besides the question of convergence
of the Poincar?e sum, these are
1. If the original automorphic differential equation comes equipped with boundary
conditions such as compatibility with perturbation theory, these boundary conditions
must be rephrased for the new differential equation for the seed ?. The direct
translation is not obvious and a general procedure will probably rely on a solution to the
second point below. In the examples in this paper we have given a heuristic set of
boundary conditions based on strong couplings limits at the level of the seed.
2. The boundary conditions and the physical content of the Poincar?e sum
?(g) = P
? ?(?g) are expressed through the Fourier expansion of ?. Even though
? was solved using Fourier expansion, the direct translation of this into the Fourier
expansion of ? is very hard. For the case of SL(2, Z) some intuition can be gleaned by
considering the form of the Fourier expansion of the solution given in general in (4.1).
These expressions, though explicit, seem impossible to evaluate for the seeds ? that
we found for the D6R4 solution15 and also in the case of modular graph functions in
section 4 we had to deal with divergent sums.
15Plugging in our solution (3.10) and (3.13) this agrees with [41, eq. (B.13)]. In both cases, there appears
to be a problem with absolute convergence as the na??ve separate evaluation of the m = 0 term in the zero
mode f0(y) leads to Pd>0 S(0, 0; d)d?2s = ?(2s?1)/?(2s) ? ? for s ? 1 from the linear term in y in c0(y).
This problem is related to the lack of convergence of the Poincar?e series for this term alone that was
mentioned below (3.13) as this term normally produces the second constant term of the Eisenstein series Es(z).
However, we can see from (4.1) that the identity element in the Poincar?e sum always
yields the Fourier mode cn of the seed and this is why we relied on the properties of
this in our heuristic analysis of the boundary conditions. One obtains the same types
of integral if one does the direct Fourier expansion of the solution in the SL(5) case. It
would be very good to develop general methods for this and also for higher rank cases.
3. Automorphic forms solving homogeneous equations can belong to principal series
representations and satisfy interesting functional equations [48]. Is there a similar
theory underlying the solutions to the inhomogeneous equations? There is no obvious
such functional relation for the oneparameter family of solutions in appendix B. It
is also not clear what the representationtheoretic meaning of the solutions is. They
most probably represent vectors (or packets) in the tensor product of principal series
representations.
4. The solution we found for D6R4 in D = 7 spacetime dimensions is very different
from the integral formulas found in [25, 29]. Since a Fourier expansion has not been
achieved in those cases either, it is hard to compare the two results. The solution we
constructed was based on the Laplace equation. For higher rank groups and
curvature corrections one typically has more differential equations to solve and they can
be of higher order in derivatives [27], either of homogeneous or inhomogeneous type.
They represent elements in the center of the universal enveloping algebra and
generated the annihilator ideal for standard automorphic forms. It would be interesting
to investigate these tensorial type equations for Poincar?e sum solutions and they will
constrain the homogeneous solutions further.
Acknowledgments
We would like to thank G. Bossard, S. Friedberg, M.B. Green, S.D. Miller, D. Persson,
B. Pioline, P. Vanhove and D. Zagier for discussions. AK gratefully acknowledges the
warm hospitality of the Max Planck Institute for Mathematics and the Hausdorff Institute
in Bonn during the final stages of this work.
A
Particular solution of the equation for cN for SL(5)
In this appendix, we construct the particular solution for the nonzero Fourier mode
cN (r, e4) that was stated in (3.31). The equation to solve is (with k = gcd(N )):

5 r2?r2 ? 8
8
15
r?r ? 4?2r2e4?1N 2 + ?SL(
4
) ? 5
{z
D
42
}

{z
Ak
}
cN = ?16??(3)?2(k) r19/5 K1(2?re4?1N ) .
The SL(
4
) invariant scalar Laplacian on SL(
4
)/SO(
4
) is given by
?SL(
4
) =
1
2 gikgjl?ij ?kl
1
? 8
gij ?ij 2
+
5
2 gij ?ij
(A.2)
where
?
? ?gij
g = e4e4T
and
?ij
satifies ?ij gkl = ?ki?lj + ?kj?li
as in [7, appendix A] up to an overall normalisation consistent with our conventions.16 We
introduce some helpful notation
u = e4?1N  = pN T g?1N = q
Nigij Nj ,
x = 2?re4?1N  = 2?ru
HJEP05(218)94
and define the functions
f ?? = r?u?Ks ,
s
f ??? = r?u?Ks? ,
s
f ???? = r?u?Ks?? ,
s
so that the prime only affects the Bessel function. All these functions have the Bessel
part evaluated at x = 2?ru, e.g. fs???
? r?u?Ks?(x). Note that the righthand side of
the differential equation (A.1) is of the form Ak f119/5,?1 and so contains a function in the
class (A.5) that we shall use below to construct an ansatz for the solution.
We record some helpful identities
gij ?ij u = ?u ,
gikgjl?ij ?klu = 9u ,
gikgjl ?ij u
?klu
= u2 ,
where repeated indices are summed over. We also record how the differential operators
in (A.1) act on a function fs?? from (A.5). By using the identities (A.6) one can derive the
following relations
r2?r2fs?? = ? (? ? 1) fs?? + 2?xfs??? + x2fs???? ,
r?rfs?? = ?fs?? + xfs??? ,
Acting with D from (A.1) on a term fs?? then gives
Dfs?? = x2fs???? +
5? + 3?
4
xfs??? +
5
8
? (? ? 4) +
3
8
? (? + 4) ? x2 ? 5
42
This expression can be further reduced to a more algebraic equation by using properties
of the Bessel function Ks(x) = K?s(x). The first identity is the modified Bessel equation
x2Ks??(x)+xKs?(x)?(x2 +s2)Ks(x) = 0 or x2fs???? +xfs??? ?(x2 +s2)fs?? = 0
(A.9)
16The derivative ?ij is secretly with respect to the matrix with diagonal elements rescaled by 2 as in [7],
but what we need is its characteristic property when differentiating the symmetric metric gkl.
(A.3)
(A.4a)
(A.4b)
(A.5)
(A.6a)
(A.6b)
(A.6c)
(A.7a)
(A.7b)
(A.7c)
(A.7d)
(A.7e)
that can be used to eliminate the second derivative of the Bessel function without changing
the order s of the Bessel function. Moreover, we have the recursive Bessel relation
xKs?(x) = sKs(x) ? xKs+1(x) or xfs??? = sfs?? ? 2?fs?++11,?+1
that can be used to replace first derivatives of the Bessel function at the cost of changing
the order.
Let us first apply the modified Bessel equation to (A.8). This yields
Dfs?? =
5? + 3? ? 4 xfs??? +
4
5
8
3
? (? ? 4) + 8 ? (? + 4) + s2 ? 5
Applying then the Bessel relation (A.10) to this leads to
Dfs?? =
5
8
which implements the differential operator D as a completely algebraic operation on the
space of functions {fs??}.
As a check on the rewriting of the differential operator, one can work out the SL(5)
Laplacian (i.e. removing the ?42/5 from D) on the Fourier mode (3.27a) of the R4
correction term which corresponds to ? = 7/5, ? = ?1 and s = 1 to obtain the eigenvalue
?12/5 as needed. The last term with shifted order drops out in this case as needed. We
also note that the following functions are in the kernel of D:
Dfs((4??)3?)/5,? = 0
(A.10)
(A.11)
(A.12)
(A.13)
(A.14)
(A.15)
(A.16)
with s(?) = p(50 ? 12? ? 3?2)/5. This is not necessarily a complete description of the
kernel.
There is one more algebraic relation that we record for special low order
K2(x) =
2
x K1(x) + K0(x) or f ?? =
2
? 1
1 f ??1,??1 + f0?? .
A.2
Particular solution by recursion
In order to find a particular solution to (3.31) we make the ansatz
c(Np) = Ak X Bir?iu?iKsi = Ak X Bifs?ii?i with Bi ? R
i
i
where the number of terms in the sum is to be determined and we take out the overall
numerical factor Ak. From the SL(2) example in (3.13) we expect that a small and finite
Plugging the ansatz (A.15) into the differential equation (A.1), we get upon use
number suffices.
of (A.12)
X Bi
i
3
5
8 ?i (?i ? 4) + 8 ?i (?i + 4) + si2 +
5?i + 3?i ? 4 si ? 5
4
42
fs?ii?i
4
? 2?
5?i + 3?i ? 4 f ?i+1,?i+1
si+1
= f119/5,?1 .
Our strategy will be to solve this recursively such that we always generate the
rightbe generated by acting on f014/5,?2 from the last term in (A.12). Since
hand side from the nonorder preserving term in (A.12). The righthand side f119/5,?1 can
1
? 2?
Df014/5,?2 =
we start the recursion with ? 21? f014/5,?2 to generate the original right hand side. This
produces a new righthand side involving ? 21?2 f014/5,?2 that we now cancel by the same
method. Since
HJEP05(218)94
we can find a linear combination of the two functions that produces exactly the new
righthand side, viz.
terminates.
12
D
? (2?)2 f ?9/15,?3
? (2?)3 f ?4/25,?4
40
= ? 2? 0
Here, it is crucial that in (A.18b) no eigenvalue term is produced and our method
(A.17)
(A.18a)
(A.18b)
(A.19)
(A.20)
(A.21)
upon substituting back the constants and definitions, together with the symmetry
K?s = Ks. In the final rewriting, we have used the identity (A.14) to eliminate the
K2 Bessel function and make the solution more similar to (3.11) that arose in the
tendimensional type IIB case.
Similar to the solution of the homogeneous equation in the type IIB case we can expect
to have at least a homogeneous solution involving K7/2. This can be manufactured using
one of the functions in (A.13) using ? = ? 52 or ? = ? 23 , i.e. the functions
f72/32/10,?5/2
and
f71/72/10,?3/2
(A.22)
are homogeneous solutions to (A.1). They are used in the main text to propose the seed
of the D6R4 threshold function for D = 7 spacetime dimensions.
Thus, altogether we obtain the relation
D
? 2? 0
leading to the particular solution for the Fourier mode of the seed
c(Np)(r, e4) = 32?2?(3)?2(k)r24/5
= 32?2?(3)?2(k)r24/5
1 +
K0(2?re4?1N ) + 12 K1(2?re4?1N ) + 40 K2(2?re4?1N )
Regularisation of inhomogeneous Laplace equations in D = 10
In this appendix, we generalise the solution to the inhomogeneous Laplace equation (3.1)
found in section 3.1.
The generalisation consists in deforming the inhomogeneous
equation to
(? ? (3 ? ?)(4 ? ?)) f?(z) = ?4?(3)?(3 + 2?)E3/2(z)E3/2+?(z)
(B.1)
The value ? = 0 corresponds to the actual D6R4 correction in D = 10. The reason for
this generalisation is that we would like to avoid the problem with the apparent singular
behaviour when carrying out the Poincar?e sum. Note that the generalisation is exact in ?.
This equation can be treated by the same method as in section 3.1.
We fold the
sum on the deformed Eisenstein series on the righthand side to replace it by y3/2+? and
obtain the following equations for the zero mode and nonzero modes of f?(z) = d0(y) +
Pn6=0 dn(y)e2?inx:
y2?y2 ?(3??)(4??) d0(y) = ?4?(3)?(3+2?)y3+? ? 43 ?2?(3+2?)y1+?,
(B.2a)
y2?y2 ?4?2n2y2 ?(3??)(4??) dn(y) = ?16??(3+2?)y2+?n??2(n)K1(2?ny). (B.2b)
The solution to the zero mode equation (B.2a) is given by
d0(y) =
2
3 ? 6?
?
2
9(1 ? 6?)
?(3)?(3 + 2?)y3+? +
?(3 + 2?)y1+?.
(B.3)
in section 3.1.
dn(y) =
Here, we have fixed the solution of the homogeneous equation to zero in the same way as
The homogeneous form of (B.2b) for the nonzero modes has the solution
y1/2K7/2??(2?ny). Combining it with a particular solution we find
+
8?(3+2?)??2(n)y1+?
1?4?2
6??
10?4?
+ ?3n3y3
1?2?+ ?2(ny)2
10?4?
K0(2?ny)
10?4?
K1(2?ny)? ? 72 ?? (?ny)1/2+? K7/2??(2?ny) , (B.4)
where we have fixed the homogeneous solution such that the most singular terms at strong
coupling (y ? 0) are absent. This is in correspondence with the choice for d0(y) at weak
coupling (y ? ?). One can check that the above solution tends to (3.13) when ? ? 0.
The advantage of this solution is that the Poincar?e sum of the zero mode (B.3)
converges for ? > 0.
Open Access.
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Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
Nucl. Phys. B 277 (1986) 1 [INSPIRE].
HJEP05(218)94
instantons, Nucl. Phys. B 508 (1997) 509 [hepth/9707018] [INSPIRE].
[hepth/9804023] [INSPIRE].
[6] M.B. Green and S. Sethi, Supersymmetry constraints on type IIB supergravity,
Phys. Rev. D 59 (1999) 046006 [hepth/9808061] [INSPIRE].
[7] N.A. Obers and B. Pioline, Eisenstein series and string thresholds,
Commun. Math. Phys. 209 (2000) 275 [hepth/9903113] [INSPIRE].
[8] M.B. Green, H.h. Kwon and P. Vanhove, Two loops in elevendimensions,
Phys. Rev. D 61 (2000) 104010 [hepth/9910055] [INSPIRE].
[9] B. Pioline, H. Nicolai, J. Plefka and A. Waldron, R4 couplings, the fundamental membrane
and exceptional theta correspondences, JHEP 03 (2001) 036 [hepth/0102123] [INSPIRE].
[10] M.B. Green and P. Vanhove, Duality and higher derivative terms in Mtheory,
JHEP 01 (2006) 093 [hepth/0510027] [INSPIRE].
[11] A. Basu, The D4R4 term in type IIB string theory on T 2 and Uduality,
Phys. Rev. D 77 (2008) 106003 [arXiv:0708.2950] [INSPIRE].
[12] A. Basu, The D6R4 term in type IIB string theory on T 2 and Uduality,
Phys. Rev. D 77 (2008) 106004 [arXiv:0712.1252] [INSPIRE].
[13] M.B. Green, J.G. Russo and P. Vanhove, Low energy expansion of the fourparticle genusone
amplitude in typeII superstring theory, JHEP 02 (2008) 020 [arXiv:0801.0322] [INSPIRE].
[14] 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]
[15] B. Pioline, R4 couplings and automorphic unipotent representations, JHEP 03 (2010) 116
[16] F. Gubay, N. Lambert and P. West, Constraints on Automorphic Forms of Higher Derivative
Terms from Compactification, JHEP 08 (2010) 028 [arXiv:1002.1068] [INSPIRE].
[17] M.B. Green, S.D. Miller, J.G. Russo and P. Vanhove, Eisenstein series for higherrank
groups and string theory amplitudes, Commun. Num. Theor. Phys. 4 (2010) 551
[arXiv:1004.0163] [INSPIRE].
[18] A. Basu, Supersymmetry constraints on the R4 multiplet in type IIB on T 2,
Class. Quant. Grav. 28 (2011) 225018 [arXiv:1107.3353] [INSPIRE].
JHEP 06 (2012) 054 [arXiv:1204.3043] [INSPIRE].
JHEP 10 (2014) 008 [arXiv:1406.5527] [INSPIRE].
[23] G. Bossard and V. Verschinin, Minimal unitary representations from supersymmetry,
[24] G. Bossard and V. Verschinin, E?4R4 type invariants and their gradient expansion,
JHEP 03 (2015) 089 [arXiv:1411.3373] [INSPIRE].
[arXiv:1502.03377] [INSPIRE].
[26] Y. Wang and X. Yin, Constraining Higher Derivative Supergravity with Scattering
Amplitudes, Phys. Rev. D 92 (2015) 041701 [arXiv:1502.03810] [INSPIRE].
[27] G. Bossard and V. Verschinin, The two ?6R4 type invariants and their higher order
generalisation, JHEP 07 (2015) 154 [arXiv:1503.04230] [INSPIRE].
[28] Y. Wang and X. Yin, Supervertices and Nonrenormalization Conditions in Maximal
Supergravity Theories, arXiv:1505.05861 [INSPIRE].
[29] G. Bossard and A. Kleinschmidt, Loops in exceptional field theory, JHEP 01 (2016) 164
[arXiv:1510.07859] [INSPIRE].
[30] G. Bossard and A. Kleinschmidt, Cancellation of divergences up to three loops in exceptional
field theory, JHEP 03 (2018) 100 [arXiv:1712.02793] [INSPIRE].
[hepth/9410167] [INSPIRE].
JHEP 06 (2010) 075 [arXiv:1002.3805] [INSPIRE].
[33] M.B. Green, J.G. Russo and P. Vanhove, String theory dualities and supergravity divergences,
[34] G. Bossard and A. Kleinschmidt, Supergravity divergences, supersymmetry and automorphic
forms, JHEP 08 (2015) 102 [arXiv:1506.00657] [INSPIRE].
[35] P. Fleig, H.P.A. Gustafsson, A. Kleinschmidt and D. Persson, Eisenstein series and
automorphic representations ? with applications to string theory, Cambridge University
Press, to appear (2018) [arXiv:1511.04265] [INSPIRE].
[36] M.B. Green, J.G. Russo and P. Vanhove, Modular properties of twoloop maximal supergravity
and connections with string theory, JHEP 07 (2008) 126 [arXiv:0807.0389] [INSPIRE].
[37] E. D?Hoker, M.B. Green and P. Vanhove, On the modular structure of the genusone Type II
superstring low energy expansion, JHEP 08 (2015) 041 [arXiv:1502.06698] [INSPIRE].
[38] E. D?Hoker, M.B. Green, O? . Gu?rdogan and P. Vanhove, Modular Graph Functions,
Commun. Num. Theor. Phys. 11 (2017) 165 [arXiv:1512.06779] [INSPIRE].
JHEP 09 (2017) 155 [arXiv:1706.01889] [INSPIRE].
twoloops, JHEP 01 (2015) 031 [arXiv:1405.6226] [INSPIRE].
Class. Quant. Grav. 31 (2014) 245002 [arXiv:1407.0535] [INSPIRE].
superstring effective action, JHEP 12 (2015) 102 [arXiv:1510.02409] [INSPIRE].
Mathematics, Cambridge University Press, Cambridge (1997).
JHEP 10 (2013) 217 [arXiv:1308.6567] [INSPIRE].
[1] D.J. Gross and E. Witten , Superstring Modifications of Einstein's Equations, [2] M.B. Green and M. Gutperle , Effects of D instantons, Nucl. Phys. B 498 ( 1997 ) 195 [3] M.B. Green , M. Gutperle and P. Vanhove , One loop in elevendimensions , Phys. Lett. B 409 ( 1997 ) 177 [ hep th/9706175] [INSPIRE].
[4] E. Kiritsis and B. Pioline , On R4 threshold corrections in IIB string theory and (p, q ) string [5] B. Pioline , A Note on nonperturbative R4 couplings , Phys. Lett. B 431 ( 1998 ) 73 [19] M.B. Green , S.D. Miller and P. Vanhove , Small representations, string instantons and Fourier modes of Eisenstein series , J. Number Theor . 146 ( 2015 ) 187 [arXiv: 1111 .2983] [20] F. Gubay and P. West , Parameters, limits and higher derivative typeII string corrections , [21] P. Fleig and A. Kleinschmidt , Eisenstein series for infinitedimensional Uduality groups , [22] M.R. Garousi , Sduality invariant dilaton couplings at order ??3 , JHEP 10 ( 2013 ) 076 [25] B. Pioline , D6R4 amplitudes in various dimensions , JHEP 04 ( 2015 ) 057 [31] E. Cremmer and B. Julia , The SO(8) Supergravity, Nucl . Phys. B 159 ( 1979 ) 141 [INSPIRE].
[32] C.M. Hull and P.K. Townsend , Unity of superstring dualities, Nucl. Phys. B 438 ( 1995 ) 109 [39] A. Basu , Proving relations between modular graph functions , Class. Quant. Grav . 33 ( 2016 ) 235011 [arXiv: 1606 .07084] [INSPIRE].
[40] A. Kleinschmidt and V. Verschinin , Tetrahedral modular graph functions, [41] M.B. Green , S.D. Miller and P. Vanhove , SL(2, Z ) invariance and Dinstanton contributions to the D6R4 interaction, Commun . Num. Theor. Phys . 09 ( 2015 ) 307 [arXiv: 1404 .2192] [44] E. D'Hoker , M.B. Green , B. Pioline and R. Russo , Matching the D6R4 interaction at [43] E. D'Hoker and M.B. Green , ZhangKawazumi Invariants and Superstring Amplitudes, [47] K. KlingerLogan , Differential equations in automorphic forms , arXiv: 1801 .00838.
[48] R.P. Langlands , On the Functional Equations Satisfied by Eisenstein Series, Lecture Notes in Mathematics , vol. 544 , SpringerVerlag, New York, BerlinHeidelberg ( 1976 ).