Contents

AEI-2012-035


Eisenstein series for infinite-dimensional

[4mm] U-duality groups
Philipp Fleig, Axel Kleinschmidt

Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut)

Am Mühlenberg 1, DE-14476 Potsdam, Germany Freie Universität Berlin, Institut für Theoretische Physik,

Arnimallee 14, 14195 Berlin, Germany Université de Nice-Sophia Antipolis

Parc Valrose, FR-06108 Nice Cedex 2, France International Solvay Institutes

ULB-Campus Plaine CP231, BE-1050 Brussels, Belgium


We consider Eisenstein series appearing as coefficients of curvature corrections in the low-energy expansion of type II string theory four-graviton scattering amplitudes. We define these Eisenstein series over all groups in the series of string duality groups, and in particular for the infinite-dimensional Kac–Moody groups , and . We show that, remarkably, the so-called constant term of Kac–Moody-Eisenstein series contains only a finite number of terms for particular choices of a parameter appearing in the definition of the series. This resonates with the idea that the constant term of the Eisenstein series encodes perturbative string corrections in BPS-protected sectors allowing only a finite number of corrections. We underpin our findings with an extensive discussion of physical degeneration limits in space-time dimensions.

1 Introduction

Over the last 15 years, a lot of work has been devoted to understanding dualities in string theory. Dualities are discrete symmetries under which string theory is invariant. For toroidal compactifications of type IIB string theory from ten down to space-time dimensions on a -dimensional torus these duality symmetries are thought to be contained in the continuous symmetries of the (maximal) low energy supergravities. These are given by the split real groups for [1, 2, 3, 4] and summarised in the first column of Table 1. Following [5, 6], we have joined conjectural rows for to the table.The (conjectured) duality symmetries [7, 8] are listed in the last column; they are the corresponding Chevalley groups. The Dynkin diagram corresponding to the various groups is given in Figure 1.

Table 1: List of the split real forms of the duality groups one obtains when compactifying type IIB string theory on a -torus to space-time dimensions. We also list the corresponding maximal compact subgroups and the last column contains the discrete versions, which appear in string theory. The last two rows are conjectural as are the corresponding discrete groups for . Note that and as appearing here are thought of as symmetries of the toroidally compactified theory; in contrast to the farther-reaching conjectures of [9] and [10].

Figure 1: Dynkin diagram for .

One way in which the invariance of type IIB string theory under the groups shown in Table 1 becomes manifest is in string scattering amplitudes. The scattering amplitudes depend on the moduli of the compactified theory given by the coset . Here is the maximal compact subgroup of and a complete list is also given in Table 1. The scattering amplitude is then invariant under the discrete group ; it transforms as an automorphic function under it.111It has also been suggested that so-called transforming automorphic forms that are not invariant but covariant under duality play a rôle for correction terms [11, 12, 13, 14, 15]. Starting with the work of [16] it has been shown in [17, 18, 11, 19, 20, 21, 22, 23, 24, 25, 26] that when considering four-graviton scattering, one can make precise statements about the form of the three lowest orders in the low-energy expansion of the four-graviton scattering amplitude. We will now, following loosely [27, 28, 29], briefly present some of the background to our work.

The expansion at low energies of the four-graviton scattering amplitude in space-time dimensions is a function of the Mandelstam variables , and . It can be written as a sum , with the first term being an analytic function of the Mandelstam variables and the second term being non-analytic in these variables [18]. In the present work we will mainly focus on the analytic part . The non-analytic contribution also plays a rôle in the analysis and we will provide some more comments on this term later on. The analytic part in the expansion takes the form222The first term on the right-hand-side of this expansion is the classical supergravity tree-level term, determined by the Einstein-Hilbert action. The function .

(1.1)

Here, is a dimensionless combination of the Mandelstam variables, with being the Planck length in space-time dimensions, and denotes a contraction of four Riemann tensors with a standard 16-index tensor.333The 16-index tensor is the tensor, which can be found for example in [30]. The are automorphic functions of the moduli . The superscript indicates that is an automorphic function under the duality group in space-time dimensions, i.e. . The orders , with positive integers and , have been studied extensively in the literature and a considerable amount of evidence for their precise form has been accumulated in . When translated into the effective action, a term of the form in the scattering amplitude corresponds to a term which is of the form .

In the present work, we will mainly be concerned with the two lowest orders of string theory corrections in the effective action, namely and , where for simplicity of notation we have dropped the explicit moduli dependence. It has been found that for the low-energy expansion of four-graviton scattering in type IIB superstring theory in , and are given by Eisenstein series, multiplied by a suitable normalisation factor444We do not consider the cases where the functions are given by sums of Eisenstein series. [27, 29]

(1.2)

where is the Riemann-Zeta function. These Eisenstein series are of the general form , where is the duality in group in space-time dimensions under which the Eisenstein series are invariant. The label indicates a particular chosen simple root of (see Fig. 1) and is a generically complex parameter, which enters in the definition of the series. The notation used here will be explained in more detail later. It is interesting to also consider the Fourier expansion of such Eisenstein series, since it allows one to give a physical interpretation of the different terms. In such an expansion one will find two types of terms, which differ in their mathematical structure and physical interpretations. The first type is generally referred to as the constant term [31]. The physical interpretation ascribed to this type is that each of its terms encodes a perturbative (string or M-theory) correction of a certain loop order in the scattering process. For finite-dimensional groups, the number of constant terms is bounded by the order of the corresponding Weyl group and therefore finite. This corresponds to a finite number of perturbative corrections (irrespective of supersymmetry). For the actual series (1.2) occurring in string theory, there are many additional cancellations and the number of constant terms is further reduced drastically; this can be seen as a consequence of the large supersymmetry and its associated non-renormalisation properties [16]. The second type of term which appears in the expansion of the Eisenstein series is generally associated with non-perturbative effects, or more precisely instanton corrections, see for example [16, 18, 19, 23, 26]. This type of term is often called (abelian or non-abelian) Fourier coefficients.

As proven in [22] for , the coefficient functions , and of the lowest three orders of curvature corrections each satisfy a Laplace eigenvalue equation defined by the invariant Laplace operator on the moduli space in dimensions. In the first two cases this Laplace eigenvalue equation is homogeneous (with source terms only in dimensions when there is a known divergence). For the third case , the coefficient of the correction, the equation is always inhomogeneous, where the inhomogeneous term is given by . We will give explicit expressions for these Laplace equations later on, supplemented by some further discussion. Duality invariance and the eigenvalue of the Laplace operator in for various terms has also played a rôle in recent discussions of the finiteness (or not) of supergravity, see e.g. [32, 33, 34, 35, 36] and references therein.

The functions (1.2) are subject to a number of strong consistency requirements [27, 28] that arise from the interplay of string theory in various dimensions. The consistency conditions are typically phrased in terms of three limits, corresponding to different combinations of the torus radii555in appropriate units of Planck or string length in the relevant dimensions and the string coupling becoming large. The three standard limits correspond to (i) decompactification from to dimensions, (ii) string perturbation theory and (iii) the M-theory limit. In terms of the diagram this means singling out the nodes , or , respectively, in the three cases. In or above (i.e., up to and including ), the functions in (1.2) have been successfully subjected to the consistency requirements [29, 28, 37]. There are also direct checks of their correctness for some dimensions and parts of their expansions (see [16, 18, 26] and references therein) and general considerations on perturbative expansions for functions constructed from lattice sums (not necessarily satisfying a Laplace equation) [38]. We will provide a heuristic derivation of the parameters entering (1.2) below.

It is natural to ask the question whether these results also extend to the case of the infinite-dimensional symmetry groups , and and their associate duality symmetries , and .666By abuse of notation we will refer to the discrete duality groups as finite-dimensional duality groups for and as infinite-dimensional duality groups for . This sloppiness of terminology helps to make many statements more readable and we will similarly sometimes omit the ‘’ in when it is implicit from the context. The present work is an attempt to answer this question and provide insights into the technical details of how one can extend the analysis to the infinite-dimensional cases. Issues regarding the physical definition of the charges of states and space-time dependence of the moduli fields in low space-time dimensions will not be addressed. This can be (partly) justified by regarding the Eisenstein series for () as unifying objects that give rise to the more physical series for in special limits like the ones to be discussed in section 5.

A central role in our analysis is played by the precise structure of the constant term of the Eisenstein series shown in (1.2), when and , i.e. for the invariance groups and . We will also study the constant terms of the Eisenstein series.777The remaining parts of the Fourier expansion are not addressed in our work. Although we assume that there will be a connection to minimal and next-to-minimal representations as in [39, 28, 37] we do not explore this here. A new feature that should arise in for are instanton corrections of objects that are more non-perturbative than NS branes, i.e., have a string frame ‘tension’ scaling with with  [40, 41]. Half-BPS states are expected to fill out infinite duality multiplets [40, 42, 43]. In particular, as will be explained in more detail later, it is not a priori clear that the constant terms of these series will be made up of a finite number of terms since now the Weyl groups are of infinite order. However, due to the physical interpretation of the constant term as encoding a finite number of perturbative corrections, it is crucial for consistency that the constant terms of the Eisenstein series invariant under infinite-dimensional groups, also only consist of a finite number of terms. Using a technical argument we will show that for special choices of the parameter appearing in the definition of the Eisenstein series, this is indeed the case. This requirement leaves only a small subset of values of out of an initially infinite range. These include the ones that appear in the coefficients of the curvature corrections. In this sense restrictive nature of supersymmetry and the infinite-dimensional duality groups is revealed.

The plan of the article is as follows. In section 2 of the paper we will recapitulate the definition of an Eisenstein series over finite-dimensional groups and introduce some of the concepts required to do so. We will then go on to extend this definition to the case of the infinite-dimensional groups and in particular to the affine groups based on work by Garland. Section 3 will be concerned with the structure of the constant term which appears in a Fourier-like expansion of the Eisenstein series. In section 4 we will give an expression for the constant term of Eisenstein series over affine groups and the results of section 3 will be extended to the infinite-dimensional case. We will show that it is possible that the constant term of an Eisenstein series invariant under an affine group contains only finitely many terms for special values of . Section 5 contains many of the explicit computations which were carried out, others have been relegated to appendices. In particular, we provide constant terms of the -and -Eisenstein series in three different (maximal) parabolic subgroups. These correspond to three physical degeneration limits that we discuss carefully since the case affords physical and technical novelties. The results of this paper were announced in [44].

2 Definition of Eisenstein series

Before we begin, let us fix some general notation used in this paper. We denote the Lie algebra of a group by and the set of roots of the algebra is denoted by . A basis of is given by the choice of a set of simple roots and the number of elements in is equal to the rank of . An element of is generally denoted by , where . We also denote the set of positive and negative roots by and , respectively. The Cartan subalgebra of is denoted by .

In the present article we shall mainly be concerned with the groups of the series of the Cartan classification with , and their infinite-dimensional Kac–Moody extensions for , as given by the Dynkin diagram in Fig. 1. More precisely, we are interested in the split real form of these groups, commonly denoted by . For simplicity of notation, we however denote the split real form simply by . As our discussion is aimed at the series, we will state our results for simple and simply-laced algebras.

As already explained in the introduction, the duality groups appearing in reductions of type II string theory are discrete versions of the groups, which we will denote by and take to be the associated Chevalley groups [45, 46]. These can be thought of as being generated by the integer exponentials of the (real root) generators of in the Chevalley basis.

2.1 Borel and parabolic subalgebras

The Borel subalgebra of an algebra is defined as

(2.1)

A (standard) parabolic subalgebra is a subalgebra of that contains . Parabolic subalgebras decompose in general as the direct sum of the so-called Levi subalgebra and the unipotent radical

(2.2)

A convenient construction of parabolic subalgebras is obtained by selecting a subset of the set of simple roots . This induces a corresponding subset of the set of positive roots , where the are those positive roots that are linear combinations of the simple roots in only. The Levi subalgebra and unipotent radical are then defined as

(2.3)

and

(2.4)

respectively and the parabolic subalgebra is given by

(2.5)

There are two types of parabolic subalgebras which are of importance for us. The first is the minimal parabolic case, which is obtained, when and corresponds to the Borel subalgebra . The second is the maximal parabolic case for which , where is a (single) simple root.888Our terminology differs from that used for example in [29] in that there . Using an abbreviated notation we denote maximal parabolic subalgebras by . We will denote the group associated with the subalgebra by . Similar to the decomposition of shown in (2.2) we also have

(2.6)

where and are the groups associated with the subalgebras and .

2.2 Eisenstein series over finite-dimensional groups

Before discussing the definition of Eisenstein series over infinite-dimensional groups, we want to give the definition for the case of a finite-dimensional group . We define the following (Langlands-)Eisenstein series [31]

(2.7)

where is the Chevalley group of and the corresponding discrete version of the Borel subgroup . is a general weight vector of (not necessarily on the weight lattice) and is the Weyl vector, which is defined as half the sum over all positive roots or alternatively as the sum over all fundamental weights which we will denote by (with ). The function is a map from a general group element to the Cartan subalgebra . Using the standard unique Iwasawa decomposition

(2.8)

we write down the action of the map for a specific group element , decomposed according to (2.8) as , as

(2.9)

This then defines the map . The angled brackets in the definition are the standard pairing between the space of weights and the Cartan subalgebra . We refer to the function defined in (2.7) as a minimal parabolic Eisenstein series since it is associated with the minimal parabolic subgroup . The sum (2.7) converges when the real parts of the inner products for all simple roots are sufficiently large and can be analytically continued to the complexified space of weights except for certain hyperplanes [31].

The Eisenstein series (2.7) is made out of a simple ‘plane-wave type’ function where are coordinates of the Cartan subalgebra and the are constants determined by .999We use the incorrect terminology ‘plane wave’ with an application to quantum gravity in mind, where the Eisenstein series should describe wavefunctions [47, 48]. This function is stabilized by the Borel group and turned into an automorphic function by summing over all its (inequivalent) images determined by . The plane-wave function is trivially an eigenfunction of the quadratic Laplace operator and all higher-order invariant differential operators. Since all these operators are invariant under the group (even ), the Eisenstein series of (2.7) is an eigenfunction of all these operators. In particular, its eigenvalue under the -invariant Laplacian (changing the normalisation of [29]) is

(2.10)

This, of course, is the same eigenvalue as that of the quadratic Casimir on a representation with highest weight up to normalisation. The inner product is normalised such that for simple roots .

One can consider a special case of the Eisenstein series defined in (2.7) by imposing the additional condition

(2.11)

for a chosen . This condition implies that will be orthogonal to all simple roots with . The parameter which appears here can in general be any complex number. However, we will see later that in the cases which are relevant for us in the context of superstring graviton scattering, will be purely real and take half-integer values. With a short calculation, see e.g. [29], one can show that the sum in (2.7) now becomes a sum over the coset and that it takes the form

(2.12)

Here, is the maximal parabolic subgroup defined by the node . For obvious reasons, this function is referred to as a maximal parabolic Eisenstein series and is called the order of the Eisenstein series. In some cases, equivalent expressions in terms of (restricted) lattice sums exist [16, 23, 49, 50, 51].

2.3 Eisenstein series over Kac–Moody groups

The theory of Eisenstein series defined over affine (loop) groups was first developed by Garland and is comprehensively described in [46] (see also [52]). Indeed, the definition of Eisenstein series over affine groups proceeds in much the same way as the one for the finite groups. There are, however, some subtleties which we shall explain in the following.

A hat is used to denote objects of affine type. Starting from a finite-dimensional, simple and -split Lie algebra one constructs the non-twisted affine extensions as

(2.13)

The first summand is the algebra of formal Laurent series over (the loop algebra) and the other summands are the central extension and derivation, respectively. The algebra has a Cartan subalgebra of dimension and its roots decompose into real roots and imaginary roots (see [53]).

The real affine group (in a given representation over ) is defined by taking the closure of exponentials of the real root generators of the non-twisted affine algebra. Due to the structure of the commutation relations where never appears on the right hand side, the group thus generated will not use the derivation generator. has the following Iwasawa decomposition

(2.14)

in an analogous way to (2.8), but now is the exponential of the dimensional abelian algebra only [46]. does not include the group generated by the derivation ; we denote by the group with the derivation added to it.

Similar to the definition of the Eisenstein series over finite-dimensional groups, in the infinite-dimensional case one can define in a meaningful manner

(2.15)

where parameterises the group associated with the derivation generator and is written as in [46]. This definition of the Eisenstein series is derived in [46] and the convergence of the series is proven for for . The domain of definition can be extended by meromorphic continuation. One important special property of the affine case that enters in (2.15) is the definition of the affine Weyl vector : The usual requirement that the Weyl vector have inner product with all affine simple roots does not fix completely; it is only defined up to shifts by the so-called (primitive) null root that has vanishing inner product with all  [53]. We choose the standard convention that is the sum of all the fundamental weights [53], i.e., it acts on the derivation by . Associated with the existence of the null element is also the existence of a particular type of a fairly simple automorphic function given for any , where is the primitive affine null root of the affine root system. This is the automorphic version of the fact that there are infinitely many trivial representations whose characters differ by factors  [53]. We denote these special Eisenstein series by

(2.16)

More generally, we can always multiply any affine Eisenstein series by an arbitrary power of and still obtain an Eisenstein series.

As in the finite-dimensional case, the Eisenstein series (2.15) is an eigenfunction of the full affine Laplacian and has eigenvalue

(2.17)

The Laplacian itself is not unambiguously defined because of the ambiguity in (related to a rescaling of the overall volume of moduli space). We reiterate that we adopt consistently the convention that has no part. An important difference to the finite-dimensional case is that there are no higher order polynomial invariant differential operators that help to determine but only transcendental ones [54]. We have not investigated their action on (2.15).

By imposing the additional condition on the (minimal) Eisenstein series defined above in (2.15) one can again obtain a maximal parabolic Eisenstein series:

(2.18)

Turning to more general Kac–Moody groups, we will assume that the Eisenstein series for with are defined formally exactly as in (2.7). A proof for the validity of this formula (i.e. existence via convergence) is not known to our knowledge but for sufficiently large real parts of one should obtain a convergent bounding integral and then continue meromorphically. The definition of the real group and the Chevalley group proceeds along the same lines as in the affine case [54]. The expression for the Laplace eigenvalue is as before and is unambiguous for with . We do not address the issue of square integrability of Eisenstein series for Kac–Moody groups.

3 Eisenstein series and constant terms

We now turn to the analysis of the constant terms of Eisenstein series of the type (2.7) or (2.15). In this section we restrict mainly to the finite-dimensional duality groups and treat the infinite-dimensional case in the next section.

3.1 Constant term formulæ

The constant terms are those terms that do not depend on those coset space coordinates associated with the unipotent part in (2.8) but only on the Cartan subalgebra coordinates. They are hence obtained by integrating out the unipotent part (using the invariant Haar measure):101010We note the similarity with Weyl character formula.

(3.1)

where we have already applied Langlands’ constant term formula [31] for evaluating the integral. The constant terms are hence given by a sum over the Weyl group of with individual summands being the numerical factor times a monomial of the Cartan subalgebra coordinates. The numerical factors are given explicitly by

(3.2)

The product runs over all positive roots, which also satisfy the condition that be a negative root for the Weyl group element . The function is the completed Riemann -function and is defined as .111111The Riemann function can be seen to occur by using the -adic approach to automorphic functions [31, 46]. The expansion (3.1) will be referred to as minimal parabolic expansion of the constant terms.

There is another way of arranging the constant terms that corresponds to choosing a maximal parabolic subgroups defined by a node as in (2.6). In order to introduce it, let us remark that the Levi component can be written as the product of two groups, namely

(3.3)

where is the subgroup of which is determined by our choice of a simple root in the Dynkin diagram of . The Dynkin diagram of is given by the diagram which is left once one has deleted the node associated with from the Dynkin diagram of . The one-parameter group can be parameterised by a single variable .

The corresponding arrangement then highlights the dependence on only one of the parameters, namely , corresponding to the single node (say, a decompactifying circle) and maintains the invariance under the remaining group in the decomposition (3.3). In that case the constant terms can be packaged using cosets of the Weyl group . Denoting the Weyl group of by , the constant terms read [55, 56, 57, 29]

(3.4)

Let us explain some of the notation introduced here. For a weight , is a projection operator on the component of proportional to the fundamental weight , and is orthogonal to , i.e., a linear combination of the simple roots of . The Eisenstein series on the right does not depend on the factor in (3.3) since the dependence on the abelian group is explicitly factored out using the projections. The expression (3.4) does not depend solely on the Cartan subalgebra coordinates but also retains dependence on some of the positive step operators that appear in the Eisenstein series defined with respect to the reductive factor . Even though indicated as depending on , the Eisenstein series on the r.h.s. of (3.4) effectively depends only on . This type of expansion is called maximal parabolic expansion of the constant terms of an Eisenstein series.

For finite-dimensional groups the number of terms contained in the constant term in the minimal parabolic expansion (3.1) is generically equal to the finite order of the Weyl group and in the maximal parabolic expansion (3.4) to the number of Weyl group cosets. For special choices of there can be vast cancellations reducing the number of constant terms. These are the values that are relevant in string theory.

For affine or general Kac–Moody groups one would expect generically infinitely many constant terms but again, as we will show, there are special choices for where the number reduces to a small finite number. We will treat these cases in the next section but first describe more properties of the coefficients that control the cancellations.

3.2 Functional relation and properties of

The factors are easily seen to satisfy the multiplicative identity

(3.5)

One also has the following functional relation for minimal Eisenstein series [31]

(3.6)

This relation together with (3.5) is useful in showing that the sum in (3.4) is independent of the Weyl coset representative.

The completed Riemann function entering in (3.2) satisfies the simple functional equation

(3.7)

which is at the heart of the meromorphic continuation of the Riemann -function. Defining the function by

(3.8)

the functional equation (3.7) implies

(3.9)

The only (simple) zero of occurs for ; consequently has a (simple) pole at :

(3.10)

If, for a given , the product contains more than factors, then will vanish. This is exactly what happens for minimal Eisenstein series when is chosen suitably as we will now explain in more detail (see also [56, 28, 29, 37]).

We now restrict to the case of interest, namely , relevant for the maximal parabolic Eisenstein series (2.12). The argument of the -function appearing in is . Now, for a simple root

(3.11)

Therefore, for simple roots . This reduces the number of terms in the constant term considerably, namely one can restrict the sum over the Weyl group to the following subset [29]

(3.12)

If then there will be at least one simple root included in the product (3.2) and consequently vanishes and the corresponding term in sum (3.1) disappears. The zero coming from the simple root cannot be cancelled by contributions from other roots; this can be argued by analytic continuation in  [37].

3.3 Solving the condition in

Now we want to give a more manageable description of the set in (3.12). From the definition it follows that, as a set,

(3.13)

where is the Weyl group of the Levi factor , i.e., the Weyl group described by the diagram where the node has been removed; can also be defined as the stabiliser in of the fundamental weight . The quotient (3.13) has to arise since any non-trivial element in maps at least one of the simple roots of to a negative root. Therefore we should remove any element that appears at the right end of a Weyl word. Once this is done the Weyl words appearing in start with on the right and will never map any positive root of to a negative root.

A different and more explicit description of this fact can be given by constructively computing the set by using the Weyl orbit of the fundamental weight . The Weyl words necessary for the orbit are exactly those appearing in .

We illustrate the procedure for the specific example of and , so that the Dynkin labels of the fundamental weight are . The only fundamental Weyl reflection that acts non-trivially on is , yielding the weight . In order to create a new weight we can only act with , yielding . Then one can only act with , giving . At this point we have two possibilities of fundamental Weyl reflections to act with, namely and , giving us and respectively. We continue in this way iteratively until we are left with weights with entries being only or .121212This only happens for finite-dimensional Weyl groups and the final element in the orbit is the negative of a dominant weight. The first few Weyl words generated in this way are summarised in Table 2. In this way one computes efficiently all the elements of from the orbit of .

Weyl words Weights in Orbit
id (dominant weight)
; ;
Table 2: Weyl words and weights in the Weyl orbit of for .

The size of the Weyl orbit of in the finite-dimensional case is given by

(3.14)

For our example and the size of the orbit is . Therefore we have distinct Weyl words in the left column of Table 2.

We now prove formally that each Weyl word that generates an element of satisfies for . This establishes a one-to-one correspondence between elements in the Weyl orbit and . The proof is by induction (on the length of the Weyl word/height of the weight in the orbit).

The identity element is in and corresponds to the weight . Suppose now that a particular Weyl word corresponds to a weight in the orbit . To continue the orbit we need to analyse the Dynkin labels of ; these are given by for . We have to distinguish the three cases when a given is positive, negative or vanishes, and consider in all cases whether we is in .

First suppose that we have (for a fixed )

(3.15)

where is the simple root. By invariance of the product we also have . From this we see that the root is a linear combination of all simple roots other than . i.e., it is a root of the Levi factor . Writing either all are non-negative or non-positive. Applying to yields . But by assumption for all ; the equation can only be true if for some . But this implies immediately that

(3.16)

and we conclude that . Similarly, if then will leave invariant and therefore does not produce a new element of the Weyl orbit.

Secondly, we consider the case of

(3.17)

By invariance again and we conclude that

(3.18)

where . Now suppose for some . This can only happen if since is the only positive root that is mapped to a negative root by and is positive by the induction assumption. But then which cannot happen since has a non-vanishing component along . Therefore for all and therefore . Similarly, when the element has a lower height than and hence is also a new element of the orbit .

Finally, we consider the case

(3.19)

Here, and hence the height is larger than that of and is an element of the orbit that has already computed. But this means that has an equivalent representative in of shorter length that has already been accounted for in . Therefore, the element is in but not a new one in the same way that is not a new element of the Weyl orbit . This completes the proof.

In summary, there is a one-to-one correspondence between the elements of and Weyl words that make up the orbit . This correspondence gives also a very manageable way of constructing the set by starting from the dominant weight and computing its Weyl orbit as a rooted and branched tree of Weyl words of increasing length.131313There is a natural partial order induced on the constant terms from the Weyl orbit; this can be used to display the constant term structure in terms of a Hasse diagram. By the multiplicative identity (3.5), one obtains that when going down the tree one has that if vanishes, the subsequent will also vanish. Therefore one can stop the construction of the tree along a given branch once the factor on a vertex vanishes.141414Again, it cannot happen that the zero of gets balanced by a diverging .

The analysis in this section can clearly be extended to the case where entering in the definition (2.7) of the minimal parabolic Eisenstein series is not proportional to a single but has support on several fundamental weights. The contributing Weyl words are still in one-to-one correspondence with the orbit of .

The restriction of the sum to the quotient for the constant terms expanded in the minimal parabolic subalgebra has also consequences for the expansion in maximal parabolic algebras as described by formula (3.4). The constant terms in this case are described by double cosets via (see also [56])

(3.20)

These are typically very few in number. The rooted tree mentioned above can be contracted further in this case thanks to the double coset structure.

3.4 The order and ‘guessing’ the right Eisenstein series

From the previous section we know that the constant term is given by a polynomial in the Cartan subalgebra coordinates with a total of at most terms. This is the correct number of constant terms for generic but one can make the observation that for specific choices of the parameter only a small fraction of these terms will survive, with all the other terms being zero. The reason is that for such special choices of , the factor (which of course depends on through ) will vanish. This has the remarkable effect that even for large groups , the number of constant terms is reduced drastically. The inner product enters in via the factor and the properties of the -function imply that will only vanish if for some . But this generically151515There can be exceptions when for . This arose in none of the cases we have considered. It is not fully inconceivable that for infinite-dimensional algebras such exceptions might happen since there is not bounded. only happens if is on the weight lattice and hence ; therefore the weight lattice plays a distinguished role.

As a mathematical exercise there could now be many interesting integral weights to consider, maybe associated with general minimal parabolic Eisenstein series, but string theory suggests which to select. As the Eisenstein series are meant to occur at a fixed order in the and T-duality is an exact symmetry at each order in  [58, 59, 60]. That in particular the tree-level term –that we associate with the identity Weyl element in the expansion– be invariant under T-duality implies that the weight entering in the definition of the minimal parabolic Eisenstein series should be invariant under . In other words, is proportional to (in the numbering of Fig. 1), i.e., we immediately arrive at (for )

(3.21)

for string theory applications of minimal parabolic Eisenstein series. This assumes that the whole function is given by a (single) minimal parabolic Eisenstein series, something that is not true for all and . Choosing a weight determined by is also the only way of getting string perturbation theory right, see (5.5) below. A similar conclusion was reached in [61]. One could use the functional relation (3.6) to replace by any element in its Weyl orbit.

The only remaining question then is to fix the parameter for the various types of higher derivative corrections. This can be done for example as follows. Supposing one knows the Laplace eigenvalue of the Eisenstein series from different considerations (e.g., as in [62]), then one needs to fix such that the quadratic Casimir gives the correct value.161616If one knew all eigenvalues under the full set of higher order invariant differential operators one could determine without making recourse to T-duality invariance. Another comment is that it is not a priori clear that the value of is independent of the dimension. It turns out that this can be achieved for and . For the curvature correction term at order this implies , and for the curvature correction term at order this gives . Alternatively, one can compute this from the leading wall in a cosmological billiard (BKL) analysis, see [63, 64] and section 5.7 below. This would immediately give at order . Finally, can be determined from comparing to known results from string scattering calculations, e.g. [30, 19] and (5.5) below.

By the functional relation (3.6) one can also check which terms lift to ; this requires that there is a Weyl-equivalent such that is integrally proportional to . This happens for but not for , consistent with the fact that there is a curvature correction term in , whereas there is no such term.

That the corresponding Eisenstein series for and , normalised as in (1.2) produce the right constant terms and abelian Fourier coefficients was checked in [28, 29, 37]. In dimensions the coefficient functions and are also known as sums of Eisenstein series [29].

4 Constant terms: infinite-dimensional case

The constant term in the full expansion of the maximal parabolic Eisenstein series over an affine group is given by

(4.1)

The constant term in the expansion with respect to a particular maximal parabolic subgroup is given by171717Note that the Levi factor in this case is a finite-dimensional group.