This section emphasises some interesting number theoretic aspects of GESs. In particular, we discuss the occurrence of the L-values associated with holomorphic cusp forms in the Fourier mode expansion of GESs. Surprisingly, due to some intriguing cancellations such L-values are absent from the modular functions \(}}^w_(\tau )\), which are constructed as theta lifts of local Maass functions (2.32) and correspond to MGFs and SMFs that are relevant to superstring scattering amplitudes.

It is useful to consider the structure of the Fourier mode expansion of GESs with respect to \(\textrm\,\tau =\tau _1\). As in (1.1) we define a GES as the modular invariant solution to the inhomogeneous Laplace eigenvalue equation

$$\begin (\Delta _\tau -s(s-1) )\, }}(s;s_1,s_2; \tau ) = E (s_1;\tau ) E(s_2;\tau ) , \end$$

(4.1)

subject to the condition that \( }}(s;s_1,s_2; \tau )\) has moderate growth at the cusp \(\tau _2\rightarrow \infty \) and fixing the boundary condition that the coefficient of the solution of the homogeneous equation \(\tau _2^s\) vanishes, thus implying the growth condition \( }}(s;s_1,s_2; \tau ) = O(\tau _2^)\) as \(\tau \rightarrow i\, \infty \). For \(s\in \mathbb \), which is the case of present interest, \(}}(s;s_1,s_2; \tau ) \) is the unique modular invariant solution to (4.1) subject to the above boundary conditions.

It follows from the Fourier mode decomposition of the non-holomorphic Eisenstein series (A.3), that the GESs \(}}(s;s_1,s_2;\tau )\) can be decomposed in Fourier modes as

$$\begin }}(s;s_1,s_2;\tau ) = \sum _^\infty }}^(s;s_1,s_2;\tau _2) q^n \bar^m, \end$$

(4.2)

where \(q=e^\) and \(} = e^}\). The mode number is given by \(k=n-m\) and modes with \(k>0\) represent the contributions of instantons while modes with \(k<0\) are contributions of anti-instantons. It is useful to separate the terms with \(k=0\) into purely perturbative contributions, coming from the \(m=n=0\) term, and instanton/anti-instanton contributions where \(m=n >0\).

The \(}}^\) term is a Laurent polynomial that is given byFootnote 8

$$\begin&}}^(s;s_1,s_2;\tau _2) \nonumber \\&\quad =\frac \zeta (2s_1) \zeta (2s_2)}(s_1+s_2-s)(s_1+s_2+s-1)} (\pi \tau _2)^ \nonumber \\&\qquad + \frac-2s_2}}\Gamma (s_1-}) \zeta (2s_1-1)\zeta (2s_2)} (\pi \tau _2)^ \nonumber \\&\qquad + \frac-2s_1}\Gamma (s_2-}) \zeta (2s_2-1)\zeta (2s_1)} (\pi \tau _2)^\nonumber \\&\qquad + \frac})\Gamma (s_2-}) \zeta (2s_1-1)\zeta (2s_2-1)} (\pi \tau _2)^\nonumber \\&\qquad + \beta (s;s_1,s_2) \, (\pi \tau _2)^ \,, \end$$

(4.3)

where the first four terms can be obtained by matching powers of \(\tau _2\) on the left-hand and right-hand sides of (4.1) and the last term satisfies the homogeneous equation and its coefficient is given by [12, 26]Footnote 9

$$\begin & \beta (s;s_1,s_2) :=\over \Gamma (s_)\Gamma (s_)}\,\nonumber \\ & \qquad -s_+1)\zeta ^*(s+s_-s_) \zeta ^*(s-s_+s_)\zeta ^*(s+s_+s_-1)\over (1-2s)\,\zeta ^*(2s)},\nonumber \\ \end$$

(4.4)

with \(\zeta ^*(s):=\zeta (s)\Gamma (s/2)/\pi ^\).

In addition to the purely perturbative terms, the zero-mode sector contains an infinite tower of instanton/anti-instanton contributions, (i.e. all \(}}^\) terms with \(n>0\)), which have the form

$$\begin }}^(s;s_1,s_2;\tau _2) =\frac \sigma _(n)\sigma _(n) }\Phi _( 4\pi n \tau _2)\,, \end$$

(4.5)

where \(\sigma _s(n) :=\sum _d^s\) denotes the standard divisor sigma function. When \(s_1,s_2\in \) the expression \(\Phi _( 4\pi n\tau _2)\) is a polynomial in inverse powers of \(\tau _2\) of degree \(s_2+s_1-2\). The first two perturbative orders in the (n, n) sector do not depend on the eigenvalue s and have the form

$$\begin \Phi _( y) = \frac+ \frac+O(y^)\, . \end$$

(4.6)

Higher-order coefficients do depend on s and can be computed from the differential equation (4.1) as performed in [20, 23] or via resurgence analysis methods [31,32,33]. Note that similar instanton/anti-instanton contributions contribute to the general \(k^\) Fourier mode associated with terms in (4.2),

$$\begin }}^(s;s_1,s_2;\tau _2)=&\fracm^ \sigma _(n)\sigma _(m) }\nonumber \\ &\times \Big ( \frac +O(\tau _2^)\Big ) + (n\leftrightarrow m) \,, \end$$

(4.7)

where \(k=n-m\ne 0\) and both m, n are non-vanishing.

Analysis of the (n, 0) and (0, m) terms requires more care since in these sectors we have the freedom of adding solutions of the homogeneous equation to the differential equation (4.1) without spoiling either the boundary condition or the moderate growth condition at the cusp. From the differential equation (4.1) it is easy to see that the solutions of the homogeneous equation in the zero-mode sector are simply \(\tau _2^s\) and \(\tau _2^\). While the coefficient of the \(\tau _2^s\) term has been set to zero thanks to our choice of boundary condition, the coefficient of \(\tau _2^\) must hence take the value (4.4) as a consequence of modular invariance. For the non-zero Fourier mode the story is more interesting since the solution of the homogeneous equation in the \(n^\) Fourier mode sector (with \(n\ne 0\)) is proportional to \(K_}}(2\pi |n| \tau _2)\), which satisfiesFootnote 10

$$\begin (\Delta _\tau -s(s-1)) \Big [ e^\sqrt K_}}(2\pi |n| \tau _2)\Big ] = 0\,. \end$$

(4.8)

The modified Bessel function is exponentially suppressed at the cusp, i.e.

$$\begin \sqrt K_}}(2\pi |n| \tau _2) = e^ \left( \frac+ \frac+ O(\tau _2^)\right) ,\qquad \tau _2\gg 1.\nonumber \\ \end$$

(4.9)

Furthermore we note for future convenience that at the origin \(\tau _2 \rightarrow 0\) this solution of the homogeneous equation behaves as

$$\begin \sqrt K_}}(2\pi |n| \tau _2)= & \frac\Gamma (s -})} }\nonumber \\ & \left( 1 - \frac + O(\tau _2^4)\right) ,\quad \tau _2\rightarrow 0. \end$$

(4.10)

A key point is that a particular solution of the differential equation (4.1) is not necessarily modular invariant. To ensure that we have indeed constructed the unique modular invariant solution to (4.1) (given the boundary condition above) it may be necessary to add to the particular solution a suitable solution to the homogeneous equation

$$\begin \left( \Delta _\tau -s(s-1) \right) H(\tau )=0, \end$$

(4.11)

which is given by the linear combinationFootnote 11

$$\begin H(\tau ) = \sum _^\infty \alpha _n \sqrt K_}}(2\pi n \tau _2) \big ( e^+ e^), \end$$

(4.12)

with particular real coefficients \(\alpha _n\). We now show that this is precisely the case when constructing GESs relevant for MGFs and SMFs, and the homogeneous coefficients \(\alpha _n\) are crucially related to the Fourier coefficients of certain holomorphic cusp forms.

In this subsection, we clarify how to determine the coefficients in the solution of the homogeneous equation (4.12) so as to obtain a modular invariant solution to (4.1). We first discuss how to construct a particular solution to the differential equation (4.1) and how to check whether or not this is modular invariant. Since the discussion is rather different for the case of GESs relevant to MGFs (where \(s_1\), \(s_2\in ^+\)) and the case relevant to SMFs (where \(s_1\), \(s_2\in +\)) we will discuss these two cases separately. However, the basic arguments that determine the solution of the homogeneous equation are the same for both families of GESs. We will present a conjectured closed-form expression for the coefficient of the solution of the homogeneous equation for the GESs relevant for MGFs (4.20) that reproduces all previously known results. Strikingly, we prove that the analytic continuation (4.39) of this conjectural expression to the case of GESs relevant for SMFs also produces the correct solution for the coefficient of the solution of the homogeneous equation, which will be determined thanks to some recent results on convolution identities for divisor sums [22]. We will highlight these important parallels amongst the two families when they arise in the following discussion.

In [9], two related methods for constructing solutions to the differential equation (4.1) were presented for the case

$$\begin & s_1,s_2\in \mathbb , \quad \textrm\quad s_1,s_2\ge 2,\nonumber \\ & \qquad s\in \.\nonumber \\ \end$$

(4.13)

A first approach is via Poincaré series where the key idea is to write one of the non-holomorphic Eisenstein series on the right hand side of (4.1), say \(E(s_1;\tau )\), as a sum over images under the action of \(B(\mathbb )\backslash \textrm(2,\mathbb )\) of the much simpler function \(\tau _2^\) where \(B(\mathbb ):=\ 1 & n \\ 0 & 1 \end}\right) \,\vert \,n\in \mathbb \}\) is the Borel stabiliser of the cusp \(\tau = i \infty \). It is then possible to find a solution to this modified differential equation via standard methods. Upon summing this solution over all the images under \(B(\mathbb )\backslash \textrm(2,\mathbb )\) we retrieve the modular invariant GES solution.

One of the advantages of this approach is that it immediately generates a modular invariant solution to (4.1). However, the drawback is that while it is possible [31, 32] to extract the zero-mode sector \(\mathcal ^\) (4.5) from the Poincaré series, it becomes much harder, if not impossible, to say anything about the instanton or anti-instanton sectors \(\mathcal ^\) and \(\mathcal ^\).

Interestingly, this Poincaré series approach suggests a different way of tackling the problem of obtaining a particular solution to (4.1) for the spectrum of sources and eigenvalues of present interest. In [9] it was realised that passing to the Poincaré seeds for the GESs motivates a certain notion of depth for the functional space under consideration. Building on [6, 7, 34], in [9] it was shown that a possible Poincaré representation of the GESs with spectrum (4.13) can be given in terms of depth-one iterated integrals of holomorphic Eisenstein series, and their complex conjugates, thus suggesting that GESs themselves are given by depth-two iterated integrals of holomorphic Eisenstein series and their complex conjugates plus possibly lower depth terms.

While invariance under a T-transformation \(\tau \rightarrow \tau +1\) is manifest in the space of iterated Eisenstein integrals, invariance under a S-transformation \(\tau \rightarrow -1/\tau \) is not and has to be checked case by case. Given a particular solution constructed in terms of iterated integrals of two holomorphic Eisenstein series, its S-transform can be evaluated and it can be checked whether or not the particular solution is modular invariant. The difference between an iterated integral of holomorphic Eisenstein series and its S-dual is controlled by a family of periods called multiple modular values. While at depth-one the only multiple modular values we find are standard Riemann zeta values, the same is not true at higher depth. In particular, as discussed in [9, 10] multiple modular values at depth-two display (amongst other things) L-values of holomorphic cusp forms. As a consequence, we see that for the spectrum (4.13) the particular solution to (4.1) constructed via depth-two iterated Eisenstein integrals fails to be modular invariant whenever the eigenvalue s is such that \( \dim \mathcal _ \ne 0\) where \(\mathcal _ \) is the vector space of holomorphic cusp forms with weight 2s. In “Appendix C”, we review some important properties of holomorphic cusp forms and their associated L-values.

As argued above, if the particular solution is not modular invariant we must add a suitable solution of the homogeneous equation of the form (4.12) to obtain the unique GES modular invariant solution to (4.1). In [9, 10] it was shown that the failure of modularity for the particular depth-two iterated Eisenstein integral solution can be cancelled by a suitable multiple of the non-modular invariant solution of the homogeneous equation

$$\begin H_(\tau )&:=\sum _^\infty \frac \sqrt K_}(2 \pi n \tau _2)\big ( e^+e^\big )\\&= (-1)^s \frac i } \tau _2^\int _\tau ^ (\tau -\tau ')^ (\bar-\tau ')^\,\Delta (\tau ') \,\textrm\tau ' + \mathrm\,,\nonumber \\ \end$$

(4.14)

which can be thought of as a depth-one iterated integral of the Hecke-normalised, holomorphic cusp form \(\Delta (\tau )= \sum _^\infty a_\Delta (n)q^n \in \mathcal _\). Hecke normalisation implies \(a_\Delta (1)=1\) and we refer again to “Appendix C” for important aspects on the theory of holomorphic cusp forms.

It should be stressed that the Eichler integral \(H_ (\tau )\) is not modular invariant on its own. However, the addition of a suitable multiple of \( H_(\tau )\) to the particular solution \(}}_}(s;s_1,s_2;\tau )\) is crucial for obtaining a modular invariant solution to (4.1). As a result the GESs can be expressed as

$$\begin }}(s;s_1,s_2;\tau )= }}_}(s;s_1,s_2;\tau ) + \sum _}}_} \lambda _(s;s_1,s_2) H_(\tau ), \end$$

(4.15)

where the sum is over a basis of Hecke-normalised cusp forms for \(\mathcal _\). In this expression \(}}_}(s;s_1,s_2;\tau )\) denotes the particular solution constructed in [9], which contains only iterated integrals of holomorphic Eisenstein series with depth less than or equal to two and their complex conjugates.

Recalling the formula for the dimension of the vector space of holomorphic cusp forms:

$$\begin \dim \mathcal _ = \left\ \left\lfloor \frac\right\rfloor -1 & 2s\equiv 2\, \textrm\,12,\\ \left\lfloor \frac\right\rfloor \phantom & \hbox , \end\right. \end$$

(4.16)

we see that the first instance where the particular solution fails to coincide with the GESs happens at eigenvalue \(s=6\) for which we have \( \mathcal _ = \textrm\\}\) where \(\Delta _= \sum _^\infty \tau (n) q^n\) is the discriminant modular form whose q-series coefficients are given by the Ramanujan tau-function \(\tau (n)\) (not to be confused with the modular parameter \(\tau \)). For the spectrum (4.13) the eigenvalue \(s=6\) is attained by considering sources with indices \((s_1,s_2)\in \\) and we find that the corresponding GESs are given by [9, 10]

$$\begin \mathcal (6;2,6;\tau )&= }}_}(6;2,6;\tau ) + \frac \frac,13)},11)} H_}(\tau )\,,\nonumber \\ \mathcal (6;3,5;\tau )&= }}_}(6;3,5;\tau ) - \frac \frac,13)},11)} H_}(\tau )\,,\nonumber \\ \mathcal (6;4,4;\tau )&= }}_}(6;4,4;\tau ) + \frac \frac,13)},11)} H_}(\tau )\,. \end$$

(4.17)

Here \(\Lambda (\Delta ,t)\) denotes the standard completed L-function, defined in (C.8), of the cusp form \(\Delta \in \mathcal _\).

While in [9] the coefficients \(\lambda _(s;s_1,s_2)\) have not been determined in closed form for general parameters \(s,s_1,s_2\) in (4.13), they appear to take the form

$$\begin \lambda _(s;s_1,s_2) = \kappa _ \frac, \end$$

(4.18)

where \(w=s_1+s_2\). From the spectrum (4.13) it is easy to see that since \(w=s_1+s_2\ge s+2\) it follows that the ratio of L-values in the coefficients (4.18) always consists of an odd critical L-value in the denominator and an odd non-critical L-value in the numerator where, for a cusp form \(\Delta \in \mathcal _\), the critical L-values are the numbers \(\Lambda (\Delta ,t)\) with \(t \in \\). The general expression for the coefficients \(\kappa _\) is not determined at this stage but they can be evaluated on a case by case basis by evaluating the multiple modular values associated with the particular solution part of (4.15) and they are generically in the number field associated with \(\mathcal _\).Footnote 12 However, we will shortly propose a strongly motivated conjectural expression for the coefficients in (4.18) based on the expression (4.40) for these coefficients in the context of SMFs, which will be discussed in the next sub-section.

In order to make contact with the expression for \(\lambda _\Delta \) that we will find for SMFs in the next sub-section, it is convenient to re-express (4.18) so that the critical L-value in the denominator is converted to the Petersson norm \(\langle \Delta ,\Delta \rangle \) defined in (C.15). This makes use of a classic result by Rankin [35] that states that the Petersson norm of a Hecke eigenform \(\Delta \) can be written in terms of the product of two critical L-values with opposite parity. For example the Petersson norm of the Ramanujan cusp form \(\Delta _\),  i.e. the unique Hecke normalised element of \(\mathcal _\), can be written as (see (9.1) of [35])

$$\begin ,\Delta _ \rangle }=\frac \Lambda (\Delta _,8) \Lambda (\Delta _,11). \end$$

(4.19)

The general expression for the norms of arbitrary cusp forms follows from Rankin’s theorem stated explicitly in (C.16).

Furthermore, a beautiful result by Manin [36] known as the periods theorem and briefly reviewed in “Appendix C”, proves that the ratio of any two even critical L-values or any two odd critical L-values are rational over the algebraic number field generated by the Fourier coefficients of the cusp forms. Hence by combining (C.16) and Manin’s theorem we can rewrite (4.18) as

$$\begin \lambda _(s;s_1,s_2) = \tilde_ \frac. \end$$

(4.20)

According to Conjecture 1 presented in Sect. 1.2 the expression for the coefficients of the solution of the homogeneous equation, \(\lambda _(s;s_1,s_2)\), in the MGF case (with integer \(s_1\) and \(s_2\)) are the same as those in the SMF case with (half-integer \(s_1\) and \(s_2\)). An explicit expression for the latter will be derived in the next sub-section and is given in (4.39)–(4.40). As a result the coefficient \(\tilde_\) which appears in (4.20) is given by

$$\begin \tilde_ = (-1)^ } \frac\right) }\Gamma \left( \frac\right) \widetilde_} , \end$$

(4.21)

with \(w=s_1+s_2\) and where \(\widetilde_\) is the rational number defined in (3.49). Our choice of normalisation ensures that the overall coefficient \(\tilde_\) in (4.21) is manifestly rational for the spectrum (4.13). Furthermore, for this spectrum \(\lambda _\Delta (s;s_1,s_2)\) always contains the product of an even critical L-value and an odd non-critical L-value. Lastly, while the particular results provided in [9, 10] have the completed L-values multiplied by an overall coefficient which explicitly belongs to the number field associated with \(\mathcal _\), we see that in our expression (4.20) the dependence on the number field is implicit and it is entirely captured by the Petersson norm \(\langle \Delta , \Delta \rangle \) in the denominator, as we see in Rankin’s expression (C.16) for the Petersson norm.

We have checked that our conjectural expression (4.20) does indeed reproduce all particular examples presented in the ancillary file of [9]. If we rewrite the particular examples (4.17) using (4.19) in order to implement the above representation we can check that (4.20) is correct for \(s=6\) and we find

$$\begin \mathcal (6;2,6;\tau )&= }}_}(6;2,6;\tau ) + \frac \frac,10)\Lambda (\Delta _,13)},\Delta _\rangle } H_}(\tau )\,,\nonumber \\ \mathcal (6;3,5;\tau )&= }}_}(6;3,5;\tau ) - \frac \cdot \frac \frac,10)\Lambda (\Delta _,13)},\Delta _\rangle } H_}(\tau )\,,\nonumber \\ \mathcal (6;4,4;\tau )&= }}_}(6;4,4;\tau ) +\frac \cdot \frac \frac,10)\Lambda (\Delta _,13)},\Delta _\rangle } H_}(\tau )\, , \end$$

(4.22)

where we have expressed all the critical L-values in terms of \(\Lambda (\Delta _,10)\) by using Manin’s relations [36]. We will shortly see that the coefficients in front of the solutions of the homogeneous equation appearing in (4.22) are actually deeply connected with the particular combinations of GESs obtained from the theta-lifted local Maass functions discussed in Sect. 3. However, before doing that we will now review how a similar interplay between the particular solution and the solution of the homogeneous equation related to holomorphic cusp forms is also present in the context of GESs associated with SMFs.

We now consider the GES solution to (4.1) with spectrumFootnote 13

$$\begin s_1,s_2\in }+\frac,\qquad s\in \. \end$$

(4.23)

An algorithm for constructing a particular solution, \(}}_}(s;s_1,s_2;\tau )\), to (4.1) for the special case \(}}_}(4;},};\tau )\) in [37] (see also [20]), and later generalised in [23] to the spectrum (4.23). Particularly important for the present discussion is the form of the particular solution for \(}}^_}\) and \(}}^_}\) with \(n\ne 0\) as defined in (4.2).

Given the differential equation (4.1) and the Fourier mode expansion for the non-holomorphic Eisenstein series (A.3) we see that \(}}^_}\) with \(n_1+n_2\ne 0\) can be represented as

$$\begin e^}}^_}(s;s_1,s_2;\tau _2) = \hat_(\tau _2), \end$$

(4.24)

where the functions \( \hat_(\tau _2)\) are solutions to the second order differential equations

$$\begin&\left[ \tau _2^2\partial _2 -(2\pi n \tau _2)^2 -s(s-1) \right] ( \hat_(\tau _2) + \hat_(\tau _2))\nonumber \\&\quad =\Big ( \frac}} + \frac)}}\Gamma (s_1)} \zeta (2s_1-1)\tau _2^} \Big ) \nonumber \\&\qquad \times \frac} n^}} \sigma _(n) \, K_} (2 \pi n\tau _2) +(s_1\leftrightarrow s_2) \, , \end$$

(4.25)

and

$$\begin&\left[ \tau _2^2\partial _2 -(2\pi n \tau _2)^2 -s(s-1) \right] ( \hat_(\tau _2) + \hat_(\tau _2))\nonumber \\ &\quad =\Big ( \frac}} + \frac)}}\Gamma (s_1)} \zeta (2s_1-1)\tau _2^} \Big ) \nonumber \\ &\qquad \times \frac} n^}} \sigma _(n) \, K_} (2 \pi n\tau _2) +(s_1\leftrightarrow s_2) \, , \end$$

(4.26)

The terms \(\hat_(\tau _2) \) and \(\hat_(\tau _2)\) in (4.25) originate from contributions to the source term coming from the product of the n-instanton sector of \(E(s_1;\tau )\) and the perturbative expansion of \(E(s_2;\tau )\) (together with the terms with \(s_1\leftrightarrow s_2\)) while \(\hat_(\tau _2)\) in (4.26) comes from contributions from the product of the \(n_1\)-instanton sector of \(E(s_1;\tau )\) with the \(n_2\)-instanton sector of \(E(s_2;\tau )\).

Equation (4.25) can be solved by standard methods, while (4.26) can be brought into a nicer form by writing it as

$$\begin \hat_(\tau _2)=&\frac n_1^} n_2^}\,\nonumber \\ &\times \sigma _(n_1)\sigma _(n_2) G(2\pi n_1,2\pi n_2,\tau _2), \end$$

(4.27)

where we have used \(\sigma _(n) =n^\sigma _(n)\). The auxiliary function \(G(n_1,n_2,y)\) solves:

$$\begin \left[ y^2 \partial _y^2 - ( (n_1+n_2) y)^2 -s(s-1) \right] G(n_1,n_2,y)=y K_}( n_1 y)K_}( n_2 y).\nonumber \\ \end$$

(4.28)

The results in [23] lead to a particular solution of the form:

$$\begin G(n_1,n_2,y) = \sum _^1 \eta _(n_1,n_2,y) K_i(n_1 y) K_j (n_2 y), \end$$

(4.29)

with \( \eta _(n_1,n_2,y)\) a rational function depending on the parameters \(s_1,s_2\) and s. The key question now is whether this particular solution of (4.1) is modular invariant and is therefore a GES. In order to answer this question we recall an important lemma proved in [37]