Eisenstein series for higher-rank groups and string theory amplitudes
Scattering amplitudes of superstring theory are strongly constrained by the requirement that they be invariant under dualities generated by discrete subgroups, , of simply-laced Lie groups in the series (). In particular, expanding the four-supergraviton amplitude at low energy gives a series of higher derivative corrections to Einstein’s theory, with coefficients that are automorphic functions with a rich dependence on the moduli. Boundary conditions supplied by string and supergravity perturbation theory, together with a chain of relations between successive groups in the series, constrain the constant terms of these coefficients in three distinct parabolic subgroups. Using this information we are able to determine the expressions for the first two higher derivative interactions (which are BPS-protected) in terms of specific Eisenstein series. Further, we determine key features of the coefficient of the third term in the low energy expansion of the four-supergraviton amplitude (which is also BPS-protected) in the case. This is an automorphic function that satisfies an inhomogeneous Laplace equation and has constant terms in certain parabolic subgroups that contain information about all the preceding terms.
Superstring theory is highly constrained by dualities combined with supersymmetry. These constraints are particularly strong in theories with maximal supersymmetry, which can be obtained by compactification of ten-dimensional type II closed-string theories on a -torus, , from dimensions to , in which case the theory is invariant under discrete subgroups of the real split forms of the Lie groups defined in [1, 2]111The symbol will always refer to the real split forms of these groups, which are often denoted elsewhere by or ..
One fruitful direction for investigating the nature of these constraints has been the study of terms in the low energy expansion of string theory amplitudes that generalise the amplitudes of classical supergravity. For example, the four-supergraviton amplitude for either of the compactified type II string theories may be decomposed into the sum of the classical supergravity tree-level contribution, an analytic part and a nonanalytic part,
where the Mandelstam invariants , , ( are quadratic in the momenta of the scattering particles222The Mandelstam invariants are , where () is the null momentum of particle .. The classical supergravity contribution that follows from the Einstein–Hilbert action can be written as
while the analytic part has a low energy expansion in powers of , , , of the form
where , , and is the -dimensional Planck length. The term “supergraviton” refers to the 256 massless physical states of the maximal supergravity multiplet, which have superhelicities that enter in the generalised curvature tensor, . The four powers of this tensor in the kinematic factor are contracted by a rank-sixteen tensor, which is defined in . Since , and are quadratic in momenta, a term of the form contributes a term in an effective action of the form (where the derivatives are contracted into each other in a standard manner). So the infinite series of higher momentum terms translates into a series of higher derivative local interactions in an effective action that generalises the Einstein–Hilbert action. The nonanalytic contribution, , contains threshold singularities in which depend on the dimension, . Although there is generally no unambiguous way of disentangling these from the analytic part, this issue does not affect the terms of low order that are the concern of this paper. Nevertheless, even in the simplest cases the known threshold structure provides strong constraints on the behaviour of the coefficients in (1.3) near the cusp at which the -torus decompactifies to the ()-torus , which will be of importance later.
The duality symmetry of string theory implies that the -dimensional amplitude should be invariant under the action of the duality group, . As a consequence, the coefficient functions, , in (1.3) must be automorphic functions of the symmetric space moduli, , that parameterise the coset space appropriate to compactification on , where is the maximal compact subgroup of the duality group . The list of the various duality groups is given in table 1. These moduli-dependent coefficient functions contain a wealth of information relating perturbative and non-perturbative string theory effects.
Although the structure of generic coefficients appears to be highly intractable, the first three terms, for which , are expected to display simplifying features as a consequence of maximal supersymmetry. These three interactions preserve a fraction of the complete 32-component supersymmetry, and should therefore be described as “F-terms”, or fractional BPS interactions. To be explicit, the interaction is 1/2-BPS, is 1/4-BPS and is 1/8-BPS. A BPS condition on an interaction generally implies that it is protected from receiving perturbative contributions beyond a certain order. In other words, such functions should have a finite number of power-behaved terms when expanded around their cusps. They should also have a calculable spectrum of instanton, or non-zero mode, contributions. However, it is notoriously difficult to determine the extent of the constraints imposed on systems with maximal supersymmetry due to the absence of a covariant off-shell formulation. The next term in the expansion, , is expected to be non-BPS , and therefore not protected by supersymmetry, in which case its coefficient is likely to have an infinite number of perturbative terms (power-behaved components in its expansion around any cusp).
Although the coefficients, , have not been determined in generality, a significant amount of information has accumulated for the first three terms for the cases with (duality groups with ) [6, 7, 8, 9, 10, 11, 12], and there are various conjectures concerning the coefficient of the interaction, , for higher-rank groups [13, 4, 14, 15, 16]. In addition, there are partial results for in dimensions with duality group .
The structure of the coefficients is highly constrained by a combination of string theory and M-theory input, which provides asymptotic information at various cusps in the space of moduli, together with an analysis of the constraints imposed by supersymmetry . Extending this to the exceptional groups, relevant to the theory in dimensions, requires more sophisticated techniques, which we will develop in this paper. In particular, the coefficients and satisfy the Laplace eigenvalue equations ,
where is the Laplace operator on the symmetric space . The Kronecker contributions on the right-hand-side of these equations arise from anomalous behaviour, indicating the presence of polar terms for specific values of for which the eigenvalues in (1.4) and (1.5) vanish. The coefficient satisfies the inhomogeneous Laplace eigenvalue equation
which involves a source term on the right-hand side that is quadratic in the coefficient . The origin of (1.4)-(1.6) and, in particular, the values of the eigenvalues on the left-hand sides of these equations was discussed in appendix H of .
Automorphic functions of moderate growth (which we assume ours are because of physical constraints) are nearly determined by imposing boundary conditions that specify the behaviour of their constant terms in various maximal parabolic subgroups that arise at boundaries of moduli space; the only possibility ambiguity is an additive cusp form. The constant terms are zero Fourier modes with respect to integration over the unipotent radical, , in the Langlands decomposition of a parabolic subgroup, , where is its Levi factor. The expressions for the constant terms corresponding to three particular maximal parabolic subgroups were derived explicitly in  for the with and for will be derived in this paper.
The Levi component has the form , where is a rank subgroup that corresponds to deleting nodes , or from the Dynkin in figure 1, and is given in table 2. Such constant terms contain a finite sum of components, of the form , where are rational numbers, is the parameter for the factor defined in 2.4, and the coefficients are automorphic functions for the subgroup . However, for the three subgroups appropriate to the string theory and supergravity calculations, the coefficients are expected to be maximal parabolic Eisenstein series. The behaviour at these boundaries was discussed in detail in [4, 5], and is summarised as follows.
The subgroup obtained by removing the root associated with the last node of the Dynkin diagram. This is the “decompactification limit” in , in which the radius of one compact dimension, , becomes infinite, where is the -dimensional Planck length. In this case the parabolic subgroup has a Levi factor, , of the form and the constant term for any of the coefficient functions is a sum of a finite number of terms of the form (suppressing some factors of ), where is an automorphic function for the group . This leads to a chain of relations from which it is possible to deduce all of the Eisenstein series from the case,
The subgroup obtained by removing the root associated with the node 1. This is the “string perturbation theory limit” in , in which the amplitude is expanded for small string coupling, . In this case the parabolic subgroup has a Levi factor of the form and the constant term is a sum of a finite number of terms of the form . These correspond to terms in perturbative string theory, which have values that can be obtained by explicit integration over string world-sheets embedded in , where is -dimensional Minkowski space.
The subgroup obtained by removing the root associated to the node 2. This is the limit in which the volume of the M-theory torus, , becomes large. In this case the parabolic subgroup has a Levi factor of the form and the constant term is a sum of a finite number of terms of the form . In this limit the semi-classical approximation to eleven-dimensional supergravity is a good approximation and the values of the constant terms can be determined by evaluating one and two-loop Feynman diagrams embedded in .
Detailed knowledge of these boundary conditions is nearly sufficient to determine the solutions to equations (1.4)-(1.6). With these boundary conditions, we show that the solutions of (1.4) and (1.5) are sums of Eisenstein series defined with respect to specific parabolic subgroups of the group – up to the possible additive ambiguity of cusp forms.333Cusp forms often arise as “error terms” in arithmetic expansions, dating back to the classical function , the number of ways an integer can be written as the sum of 4 squares. The generating function , and hence is a modular form of weight 2 for the congruence subgroup . There are no cusp forms of that weight for this group, and so the generating function is exactly an Eisenstein series – resulting in striking identities, such as for prime. However, as one looks at sums of more squares and the weight increases, cusp forms inevitably creep in and complicate the formulas. Our situation is similar: the existence of cusp forms would add a surprising touch of complexity to the Fourier coefficients and asymptotics of the solutions to (1.4)-(1.6). This was demonstrated in detail in  for , i.e, , and is generalized here to . Such cusp forms seem unlikely on purely mathematic grounds, because they have small Laplace eigenvalues. For example, the eigenvalues in (1.4)-(1.6) have the wrong sign to be part of the cuspidal spectrum unless is small enough. Even when the sign is correct, the papers [19, 20] give lower bounds on the cuspidal Laplace spectrum on , for any , which rule out such eigenvalues on this quotient. It seems plausible Langlands functorial lifting from the groups to could (at least conjecturally) reduce our cases of interest here to the results of [19, 20]. Such a link would however require serious technical sophistication, and is beyond the scope of this paper.
In this paper we will extend this analysis to the remaining cases , relating to , and . This involves a detailed analysis of constant terms of Eisenstein series for these groups, which will be the subject of section 2. The general analysis leads to very large numbers of power-behaved components in the constant terms. However, for the very special Eisenstein series of relevance to the string theory considerations there are immense simplifications and the relevant constant terms take the simple form expected according to items (i), (ii) and (iii). The application of these results into string theory language will be the subject of section 3. There, it will be seen that there is precise agreement between the values of the constant terms and the expectations based on string theory.
The complete expressions for the constant terms of relevance are contained in a number of tables in the appendix.
The solutions of (1.6) are more general automorphic functions, . Their constant terms contain exponentially suppressed terms as well as terms that are powers of and were analysed for in [9, 12, 4], and for the case in appendix A of . The relevant constant terms for these coefficients in the case will be determined in section 4. As we will show, the power-behaved components of the constant term in the decompactification limit (i) for the case contain within them all three of the coefficients, , and .
2. Eisenstein series, parabolic subgroups and their constant terms
This section contains an introduction to Langlands Eisenstein series on higher rank groups  and a description of some of his main results, followed by a computation of their constant terms in maximal parabolics. The discussion is geared towards the relevant setting of this paper, though we also make an effort to explain more general phenomena that may later be useful for string theorists. In particular we mainly curtail the discussion to two particular types of Eisenstein series: minimal parabolic Eisenstein series, and maximal parabolic Eisenstein series induced from the constant function (which we shall see are specializations of the former).
We follow Langlands’ Euler Products manuscript  in restricting to split Chevalley groups, as these are the only ones which arise in our investigations. In fact, we only need to study simply laced ones, i.e., either G equals , , or a split form of , , or . Let denote a fixed minimal parabolic “Borel” subgroup of . We decompose , where is its Levi component and its unipotent radical. The Cartan subgroup of shall be denoted by .
2.1. Eisenstein series in classical terminology
Researchers in automorphic forms typically define Eisenstein series in terms of adele groups because of the computational benefits this framework affords. However, this is not necessary to state the definitions. In the present work it is important to understand the connection between Eisenstein series and other lattice constructions common in string theory. Hence we felt it appropriate to define the series in concrete terms, which we shall do in this subsection before recasting the definitions adelically in the next one.
Let us now consider the real points of , and let denote the roots of relative to the Cartan . For each root , let denote the Chevalley basis vector in the Lie algebra of that represents it, and the one-parameter unipotent subgroup it generates. Furthermore we may form the Cartan Lie element , the Lie algebra of . If denotes the positive simple roots, then spans . Thus we may identify elements of the connected component of with the exponentials , each ranging over . The Iwasawa decomposition of states that its elements each have unique decompositions , with , , and , a maximal compact subgroup of . Thus there is a well-defined map such that .
The roots are by definition linear functionals on , and every linear functional is a linear combination of elements of with complex coefficients. In what follows it is helpful to normalize definitions using the linear functional , defined to be half the sum of all positive roots. We denote the pairing between and by . When and are tacitly identified, this corresponds to the usual inner product for the root system. The Weyl group acts both on and dually on in a way which preserves , and can be explicitly identified through any realization of the root system.
The function is visibly unchanged if is multiplied on the left by an element of , and in particular any element of , where is defined as in , or equivalently as the stabilizer in of the lattice spanned by the Chevalley basis . It is likewise invariant under (because this finite group is contained in ), and hence under as well. Eisenstein series are formed by averaging such objects over cosets of a group modulo a subgroup it is invariant under:
The minimal parabolic Eisenstein series for is the coset sum
This sum is absolutely convergent when the real part of has sufficiently large inner products with all .
It is a famous result of Langlands that it meromorphically continues to all of , to an automorphic function on . When this definition recovers , the usual non-holomorphic Eisenstein series for . One can of course trivially modify the definition to apply to subgroups , though this appears to be unnecessary for our investigations.
The power functions and hence itself are always eigenfunctions of the Laplacian:
This formula for the eigenvalue is crucial for identifying the solutions to (1.4)-(1.6), and is completely analogous to the fact that is an eigenfunction of the hyperbolic Laplacian. It is proven by identifying as a multiple of the Casimir on , and using explicit formulas for the latter (see [24, p. 303]). Actually the power functions and hence Eisenstein series are eigenfunctions of not merely , but furthermore of the full ring of invariant differential operators. This is a crucial fact in Langlands’ meromorphic continuation. So far string theoretic arguments have mainly produced information about and not these other operators, so (2.3) naturally determines only . The structure of the constant terms and the integrality constraint discussed in section 3.1 is then used to pin down exactly there.
We have used the term “minimal parabolic” series for these because of the rôle the Borel subgroup plays in their definition. In general, any subgroup that contains is called a standard parabolic subgroup; is called a maximal parabolic if itself is the only subgroup that properly contains it. All parabolic subgroups have the unique decomposition , where is its Levi component and its unipotent radical. The standard parabolics of are in one-to-one correspondence with subsets as follows: includes all for , while contains all for . In particular, each maximal parabolic subgroup is associated to a single, simple root , and we shall sometimes use the notation to emphasize this dependence.
Another family of important Eisenstein series that arise in our string theory calculations are maximal parabolic Eisenstein series. Let us first explain the simplest versions, which are induced from constant functions – they are in fact generalizations of the classical Epstein Zeta functions. These series are formed in a similar way to (2.2), but with special parameters such that is unchanged if is multiplied on the left by an element of , where is a designated standard maximal parabolic subgroup of . This is equivalent to requiring be orthogonal to any simple root other than the one which defines the maximal parabolic , and restricts to lie on a line in . In terms of the dual basis defined by the condition that , these can be parametrized in terms of single complex variable as
The following definition uses this special choice of , but restricts the range of summation owing to the extra invariance of the summand under :
For , the maximal parabolic Eisenstein series induced from the constant function is
Our normalization of is chosen so that it agrees with the usual non-holomorphic Eisenstein series for . These series of course also have meromorphic continuations to , and specialize to be identically equal to 1 at the special point because of the following fact (whose proof we shall describe later):
The minimal parabolic Eisenstein series equals the constant function 1.
To connect these two definitions, it is worthwhile to consider yet another type of maximal parabolic Eisenstein series that features another ingredient: an additional factor in the summand (2.6) which is an automorphic function on the Levi component of , extended to . Such a sum is still well-defined and has similar convergence properties. Definition 2.5 amounts to setting this function equal to 1. Interestingly, the inclusion of this function allows us to view the maximal parabolic Eisenstein series (2.6) as a special case of (2.2). Indeed, recall the decomposition from above, where is its Levi component and its unipotent radical. The intersection is itself the Borel subgroup of the reductive group , and the coset representatives can be uniquely decomposed into as products with , . Hence we may write (2.2) as the double sum
Any element can be uniquely decomposed as , where is a complex linear combination of simple roots not equal to (the root defining ), and is orthogonal to all such simple roots. This decomposition in particular applies to , expressing it as the sum of (which is itself half the sum of the positive roots of ), and (which is a scalar multiple of ). Using this decomposition, we write the exponent as
in the last step we have used the fact that and have the same inner product with , for any generated by the for , in particular . Hence (2.8) can be expressed in the range of absolute convergence as
where is now a minimal parabolic Eisenstein series for the smaller reductive group . Thus minimal parabolic Eisenstein series are themselves special cases of maximal parabolic Eisenstein series – but induced from the function rather than the constant function.
We can now see that (2.6) is a specialization that coincides with (2.2) when has the form (2.4). In this case , and the Eisenstein series on in the previous paragraph specializes to be constant because of Theorem 2.7. Hence under the special assumption (2.4), the inducing function is constant and the two notions coincide.
2.2. Eisenstein series in adelic terminology
We have just given definitions of the Eisenstein series involved in this paper, in concrete classical terms. Adele groups are often used in automorphic forms as a notational simplification that hints to effective ways to group terms together in calculations. In the context of Eisenstein series, they are used to reparametrize the sums over (whose cosets can be intricate to describe). This application – a brilliant insight of Piatetski-Shapiro that was furthered by Langlands – has been crucial in readily obtaining exact formulas (by comparison, Poisson summation is much harder to execute directly). Since this is crucial to our calculations, we have elected to give a description here.
Let us return to (2.2) and its sum over cosets . As we mentioned above, cosets for this quotient can be difficult to directly describe, especially as the group gets more complicated. This is because is a ring, not a field like (where the corresponding quotient is just the maximal compact subgroup ). Strikingly, the field gives the same coset space as : the inclusion map from into induces the bijection of cosets
This is because our assumptions on imply that (see [22, §2]). Similarly, , and therefore our Eisenstein series can in principal be written as sums over these rational quotients. To do this properly one needs to redefine in such as away that it is invariant under on the left. Note that the existing definition does not qualify, because is dense in and hence cannot be trivial on it.
The remedy is to instead consider the each of the groups , where denotes either a prime number or (in the latter situation, we follow the convention that ). Just as the adeles are the restricted product of all with respect to the (i.e., all but a finite number of components of each element lie in ), the adele group is the restricted product of each with respect to the = the stabilizer of the Chevalley lattice, tensored with .
A variant of the Iwasawa decomposition persists in the -adic case as well: , though this is no longer unique since, for example, is nontrivial. Thus, globally, we have that , where is the product of the real group with all ’s. Since is assumed to be split, is a product of copies of the torus , the ideles of . Strong approximation for the ideles equates the quotient with , where is regarded as diagonally444i.e., into each factor simultaneously. embedded into , and is the product of all ’s. Since the first factor is isomorphic to via the logarithm map, this identification extends from a function on to an -invariant, -invariant function from to . Furthermore, it extends to through the global Iwasawa decomposition to a left -invariant function, with likewise thought of as being diagonally embedded into . We can now define the adelic Eisenstein series as
which we stress agrees with (2.2) when the argument , and defines an extension to that is left-invariant under the diagonally embedded . Similarly
2.3. Constant term formulas
In general the Fourier expansions of Eisenstein series are intricate to state, and are not presently known in full detail. However, a simple natural part of them (defined as follows) have very explicit formulas due to Langlands, and are crucial to the analytic properties of these series.
The constant term of an automorphic form in a standard parabolic subgroup is given by the integral
where is the unipotent radical of , and is Haar measure on normalized to give the quotient volume 1. (Since is unimodular, is simultaneously both a left and right Haar measure.)
We conclude this section by stating the constant term formula for the minimal parabolic Eisenstein series in maximal parabolics. When (2.4) holds, the specialization of the formula below gives the constant terms of the maximal parabolic series in Definition 2.5 – this will be very useful for our applications. The formula involves the functions
which arise from intertwining operators. Here is an element of the Weyl group of , , and is the completed Riemann -function . They satisfy the cocycle identity
Let be any maximal parabolic, and let denote the Weyl group of , thought of as a subgroup of . As above, we decompose any as the sum , where is perpendicular to all simple roots aside from , and is a multiple of .
(Langlands’ Constant term formula – see [25, Proposition II.1.7.ii, p.92])
Of course the last Eisenstein series must be thought of as a product of Eisenstein series on the different reductive factors of when the latter is not simple. Moeglin-Waldspurger actually prove a slightly different statement, where the appear directly as intertwining operators which satisfy a composition law identically compatible to (2.17). Using embedded ’s inside it is easy to see that these operators act there as the scalar (2.16) when is a simple Weyl reflection, and therefore on the full Weyl group as well. Moeglin-Waldspurger also deal with adelic Eisenstein series, which here can be equated to their classical variants via the correspondence described in the previous subsection. Also, the adelic constant term integration over there drops to for the adelization of classical Eisenstein series (this is because of strong approximation for ). Strictly speaking, we have also used the fact that the Weyl group of a Chevalley group sits inside .
Formula (2.19) is consistent with two other important identities about the minimal parabolic Eisenstein series: the functional equation
and Langlands’ constant terms formula in the minimal parabolic
For example, when the inner product of and any simple root is , a point at which vanishes. Any nontrivial Weyl group element flips the sign of at least one simple root, so that vanishes for all but the identity element . This means the constant term (2.21) is identically one, consistent with Theorem 2.7. Moreover, it is not hard to deduce Theorem 2.7 from this calculation. Indeed, the general theory of Eisenstein series describes how the constant terms control the growth of Eisenstein series: since this one is bounded, so is the full series. It is furthermore a Laplace eigenfunction with eigenvalue 0 when , and hence it is constant.
As noted above, because specializes to the maximal parabolic Eisenstein series under (2.4), Theorem 2.18 provides the constant terms of those objects as well. It is possible to prove those formulas more directly, without reference to the minimal parabolic Eisenstein series, for example as is argued for some of the relevant cases earlier in . However, we felt it was important to calculate everything via the minimal parabolic Eisenstein series for two different reasons. The first is that we will rely heavily on special identities relating various via (2.20) that are more apparent as specializations, rather than as identities of sums over completely different coset spaces. The second reason is that the constant term calculation in the proof of [26, Theorem 2.3] does not carry over to all Eisenstein series, because they assert that the intersection of the Levi component of a maximal parabolic with the conjugate of another parabolic is itself a maximal parabolic subgroup of this Levi. However, this is false in general – including for a number of subtle examples we faced in the present work, such as the case of when and the constant term is taken in the maximal parabolic corresponding to (in the numbering of Figure 1). We would like to make clear that this in no way affects the validity of the results in , because the assertion is valid in the cases they study – rather it only affects extensions of their work to different situations. What it means is that non-maximal parabolic Eisenstein series can arise in some of the constant terms we study. Strikingly, these series tend to vanish at the special points of interest.
Finally, in addition to terms vanishing because a simple root is flipped, sometimes Eisenstein series vanish for a more subtle reason: all their constant terms vanish. In such a case, the Eisenstein series is by definition a cusp form, which much itself vanish because Eisenstein series are orthogonal to all cusp forms. It follows by induction that this is the case if and only if (2.21) vanishes, which can happen even at points where the individual are singular because of cancellation between terms. This can be very tedious to computationally check, even with the methods of the paragraph below. This vanishing, however, is ultimately responsible for many of our simple formulas for constant terms at special points.
2.4. Brief description of explicit computations used later in the paper
Theorem 2.18 gives explicit formulas for all constant terms we are after, since it is possible to enumerate the Weyl group, describe its action on , and calculate the exact factors from (2.16). In practice, the Weyl group is so large that it is difficult to have an a priori explanation of what the calculation will give. Langlands noticed already in his example of the rank two group in [21, Appendix 3] that many terms vanish at special points. Indeed, such vanishing is absolutely crucial to the physical conclusions we draw from the case of and its subgroups. Mathematicians have studied similar vanishing in different settings, and giving explanations for it (e.g. [27, 26]).
Unfortunately, it was not initially obvious what configurations of parabolics and maximal parabolic Eisenstein series to investigate, and it became a practical necessity to have a fast way to obtain explicit constant terms for large swaths of examples in order to unravel structures which were important to us from a string-theoretic point of view. Economical mathematical explanations for the phenomena at hand were not as important as speed, especially in light of the complexity of the objects involved. To get around this issue, we first precomputed which Weyl group elements flip a simple root . For such , , which forces the product (2.16) to vanish. These terms, which provide the vast majority, can hence be discarded from the constant term calculations. The remaining terms are stored for later calculations. For example, out of the 696,729,600 Weyl group elements, only 240 are needed when is the 8-th root of , i.e. the one which is added on from . To lessen storage requirements, we worked with a factorization of the Weyl groups in terms of the Weyl group, and a fixed set of coset representatives. The resulting computer calculations were then very efficient, and allowed us to explore the properties of a large number of cases and identify significant patterns. Ultimately the constant term calculations were automated, only rarely taking more than 30 seconds a piece
The constant term is a function on the Levi component of the maximal parabolic , which is the product of a one-dimensional group with all Chevalley groups whose Dynkin diagrams compose the connected components of the Dynkin diagram of , once the node for the simple root is deleted (note that this is not in general the same root that defines the Eisenstein series). This second factor is the group that the Eisenstein series on the right hand side of (2.19) is defined on. The exponential factor multiplying it depends on the one-dimensional piece. To parameterize it uniformly, factor as , where is in this one-dimensional piece and is in the product of smaller Chevalley groups. By construction, acts trivially on the roots spaces spanned by the contained inside , and is uniquely determined by its eigenvalue on , which we parametrize as . Hence the exponential factor in (2.19) is a power of , and the constant term is a polynomial with exponents depending on whose coefficients are lower-rank Eisenstein series.
Example: , , .
Here all but 2,160 out of the 696,729,600 Weyl group elements give zero contribution, and the polynomial just mentioned is equal to , provided the Eisenstein series are suppressed (otherwise the formula is even more unwieldy). At the special point all but the two smallest powers of vanish, giving
as the constant term.
3. String theory amplitudes and their degeneration limits
The results of the previous section allow us to extend the analysis of the structure of the automorphic functions that arise in the expansion of the string theory amplitude  to a more general setting that includes the exceptional duality groups.
It is useful to translate the terms in the low energy expansion of the analytic part of the scattering amplitude, (1.3) into local terms in an effective action, so that the first three terms beyond classical Einstein theory in dimensions are
We will first consider the solutions for the coefficients and , which satisfy the Laplace eigenvalue equations (1.4) and (1.5). The discussion of the automorphic coefficient , which satisfies the more elaborate equation (1.6), will be deferred to section 4.
Note on conventions
The solutions will involve linear combinations of Eisenstein series of the kind described in the last section. In describing the string theory results it will prove convenient to use a normalisation for maximal parabolic Eisentein series that includes a factor of , so we will define
where is the Dynkin label associated with the simple root in the definition (2.4). Furthermore, since the conventional Eisenstein series has a trivial Dynkin label it will be written as . For the non-Epstein series the normalisation differs by a factor of with respect to the one used in [4, 5].
The parameter , defined in section 2.4, associated with the factor in section 2 translates into distinct physical parameters in each of the three degeneration limits that correspond to deleting nodes , and , respectively, of the Dynkin diagram in fig. 1. These are summarised as follows:
The -dimensional string coupling constant is defined by , where is the IIA or IIB string coupling and is the volume of in string units. The Planck scales in different dimensions that enter in (3.1) and (3.2) are related to each other and the string scale, , by
where is the IIA string coupling.
3.1. Solutions for the coefficients and .
for (or ). Substituting the expression (2.4) for in terms of into equation (2.3), it follows that the solutions (3.6) and (3.7) satisfy the Laplace eigenvalue equations (1.4) and (1.5) with and respectively. We will shortly show that these functions also satisfy the requisite boundary conditions in the three limits of interest. An important general comment about these boundary conditions is that in each of the three limits the automorphic coefficients have moderate power-like growth for . Another necessary condition for these being acceptable solutions is that in the limits (i) and limit (ii) they give rise to integer powers of the radius and the string coupling . Strictly speaking our calculations merely show that (3.6) and (3.7) solve all relevant equations, but do not directly rigorously show that they are unique solutions. However, any two solutions differ only by a linear combination of cusp forms and other Eisenstein series. The constant terms of these series can be computed explicitly as well. Though we do not fully investigate this here, the possibility of other solutions seems unlikely because of known non-existence results for cusp forms on [19, 20], the conjectured properties of Langlands’ functorial lifts, and the rationality of the cuspidal eigenvalues the above integrality constraint dictates.
First we will comment on the form of these solutions. From the general expression (2.4) for the weight vector that defines a maximal parabolic Eisenstein series, the vector associated with a maximal parabolic Eisenstein series is . For the series in (3.6) and (3.7) this has the form , where is the weight vector for the simple root labelling the first node of the Dynkin diagram in figure 1. Therefore, the weight vector associated with the series is , while for the series it is . As we know from earlier examples, there are many equivalent ways of expressing the same series as those in (3.6) and (3.7). For the exceptional groups, with , the weight vectors , and are in the same orbit under the action of the Weyl group, . Similarly, and are also in the same Weyl orbit. This means that as a consequence of the functional equation (2.20) satisfied by the minimal parabolic Eisenstein series, the maximal parabolic Eisenstein series satisfy the following relationships, among many others:
The symbol means that the quantities are equal up to a constant of proportionality, which may depend on . In this manner our solutions can be rewritten in many different ways. Such a relationship was pointed out in the case in .
(i) Decompactification from to
This is the limit associated with the parabolic subgroup , for node . Consistency under decompactification in this limit requires (see equations (2.10) and (2.11) of ),