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JHEP06(2012)070
Published for SISSA bySpringer
Received: March 16, 2012
Accepted: May 24, 2012
Published: June 12, 2012
Oneloop BPS amplitudes as BPSstate sums
Carlo Angelantonj,a,bIoannis Florakisc,dand Boris Piolineb,e
aDipartimento di Fisica, Universit` a di Torino, and INFN Sezione di Torino,
Via P. Giuria 1, 10125 Torino, Italy
bDep PHTH, CERN,
1211 Geneva 23, Switzerland
cArnold Sommerfeld Center for Theoretical Physics, Fakult¨ at f¨ ur Physik,
LudwigMaximiliansUniversit¨ at M¨ unchen,
Theresienstr. 37, 80333 M¨ unchen, Germany
dMaxPlanckInstitut f¨ ur Physik, WernerHeisenbergInstitut,
80805 M¨ unchen, Germany
eLaboratoire de Physique Th´ eorique et Hautes Energies, CNRS UMR 7589,
Universit´ e Pierre et Marie Curie – Paris 6,
4 place Jussieu, 75252 Paris cedex 05, France
Email: carlo.angelantonj@unito.it, florakis@mppmu.mpg.de,
boris.pioline@cern.ch
Abstract: Recently, we introduced a new procedure for computing a class of oneloop
BPSsaturated amplitudes in String Theory, which expresses them as a sum of oneloop
contributions of all perturbative BPS states in a manifestly Tduality invariant fashion.
In this paper, we extend this procedure to all BPSsaturated amplitudes of the form
?
convergent Poincar´ e series, against which the fundamental domain F can be unfolded. The
resulting BPSstate sum neatly exhibits the singularities of the amplitude at points of gauge
symmetry enhancement, in a chamberindependent fashion. We illustrate our method with
concrete examples of interest in heterotic string compactifications.
FΓd+k,dΦ, with Φ being a weak (almost) holomorphic modular form of weight −k/2. We
use the fact that any such Φ can be expressed as a linear combination of certain absolutely
Keywords: Superstrings and Heterotic Strings, String Duality
ArXiv ePrint: 1203.0566
Open Access
doi:10.1007/JHEP06(2012)070
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JHEP06(2012)070
Contents
1 Introduction1
2NieburPoincar´ e series and almost holomorphic modular forms
2.1Various Poincar´ e series
2.2Fourier expansion of the NieburPoincar´ e series
2.3 Harmonic Maass forms from NieburPoincar´ e series
2.4 Weak holomorphic modular forms from NieburPoincar´ e series
2.5 Weak almost holomorphic modular forms from NieburPoincar´ e series
2.6Summary
4
4
7
8
11
13
14
3 A new road to oneloop modular integrals
3.1 NieburPoincar´ e series in a nutshell
3.2Oneloop BPS amplitudes as BPSstate sums
3.3 Oneloop BPS amplitudes with momentum insertions
3.4 BPSstate sum for integer s
3.5Singularities at points of gauge symmetry enhancement
16
16
17
22
22
25
4 Some examples from string threshold computations
4.1 A gravitational coupling in maximally supersymmetric heterotic vacua
4.2 Gaugethresholds in N = 2 heterotic vacua with/without Wilson lines
4.3K¨ ahler metric corrections in N = 2 heterotic vacua
4.4 An example from noncompact heterotic vacua
26
26
27
28
29
A Notations and useful identities
A.1 Operators acting on modular forms
A.2 Whittaker and hypergeometric functions
A.3 KloostermanSelberg zeta function
29
30
31
34
B SelbergPoincar´ e series vs. NieburPoincar´ e series 35
1Introduction
Scattering amplitudes in closed string theory involve, at hth order in perturbation theory,
an integral over the moduli space of conformal structures on genus h closed Riemann
surfaces. The torus amplitude (corresponding to h = 1) is particularly relevant, as it
encodes the perturbative spectrum of excitations. Moreover, for special choices of vacua
and of external states, corresponding to a special class of Fterm interactions in the low
energy effective action, the torus contribution exhausts the perturbative series, and thus can
serve as a basis for quantitative tests of string dualities (see e.g. [1] and references therein).
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JHEP06(2012)070
The moduli space of conformal metrics on the torus is the Poincar´ e upper half plane H,
parameterised by the complex structure parameter τ = τ1+ iτ2, modulo the action of the
modular group SL(2,Z). After performing the path integral over the worldsheet fields and
over the location of the vertexoperator insertions, the relevant amplitude is then expressed
as a modular integral
?
where F = {τ ∈ H −1
τ−2
2
dτ1dτ2 is the SL(2,Z)invariant integration measure, and A is a modularinvariant
function whose precise expression depends on the problem at hand. With this choice of
integration domain, the imaginary part τ2can be identified with Schwinger’s proper time,
while the real part τ1 is the Lagrange multiplier imposing the levelmatching condition.
Part of the difficulty in evaluating integrals of the form (1.1) is the unwieldy shape of F,
which intertwines the integrals over τ1and τ2.
Depending on the function A(τ1,τ2) methods have been devised to overcome this
problem. If A is a weak almost holomorphic function1of τ (or, alternatively, an anti
holomorphic function), the surface integral over F can be reduced by Stokes’ theorem to a
lineintegral over its boundary ∂F that can be explicitly computed [2]. On the contrary, if
A is a genuine nonholomorphic function, as is the case for the oneloop partition function
of closedoriented strings, no useful method is known to evaluate the integral, but one can
use the RankinSelbergZagier transform [3] to connect the integral to the graded sum of
physical degrees of freedom [4–7]. A frequently encountered intermediate case is that of
modular integrals of the form
?
where
Γd+k,d(G,B,Y ;τ1,τ2) ≡ τd/2
F
dµA(τ1,τ2), (1.1)
2≤ τ1<1
2,τ ≥ 1} is the standard fundamental domain, dµ =
F
dµΓd+k,d(G,B,Y ;τ1,τ2)Φ(τ), (1.2)
2
?
pL,pR
q
1
4p2
L¯ q
1
4p2
R
(1.3)
is the partition function of the Narain lattice of Lorentzian signature (d + k,d), G, B, Y
parameterise the Narain moduli space SO(d+k,d)/SO(d+k)×SO(d), and Φ(τ) is a weak
almost holomorphic modular form of negative weight w = −k/2, which we shall refer to as
the elliptic genus. Such integrals occur in particular in oneloop corrections to certain BPS
saturated couplings in the low energy effective action of heterotic or type II superstrings.
The traditional approach in the physics literature for computing modular integrals
of the form (1.2) has been the orbit method, which proceeds by unfolding the integration
domain F against the lattice partition function Γd+k,d[8–19]. While this procedure yields
an infinite series expansion which is useful in certain limits in Narain moduli space, it does
1By weak almost holomorphic we mean an element in the graded polynomial ring generated by the
holomorphic Eisenstein series E4 and E6, the almost holomorphic Eisenstein seriesˆE2 and the inverse
of the discriminant 1/∆. Our notations for Eisenstein series and other modular forms are collected in
appendix A.1. The adverb weak refers to the fact that the only singularity is, at most, a finite order pole
at the cusp q = 0.
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not make manifest the invariance under the Tduality group O(d + k,d,Z) of the Narain
lattice, nor does it clearly display the singularities of the amplitude at points of gauge
symmetry enhancement.
In [20] we proposed a new method for dealing with modular integrals of the form (1.2),
which relies on representing the elliptic genus Φ as a Poincar´ e series, and on unfolding
the integration domain against it rather than against the lattice partition function. The
advantage of this method is that Tduality remains manifest at all steps, and the result is
valid in all chambers in Narain moduli space, unlike the conventional approach.2Moreover,
the amplitude is expressed as a sum over all BPS states in the spectrum, thus generalising
the constrained Eisenstein series constructed in [21].3
amplitude at points of enhanced gauge symmetry can be immediately readoff from the
contributions of those BPS states which become massless.
The main difficulty in implementing this strategy is due to the fact that the standard
Poincar´ e series representation of a weak holomorphic modular form of weight w ≤ 0 [28–30]
is only conditionally convergent, and therefore unsuited for unfolding. In [20] we attempted
to circumvent this problem by considering a class of nonholomorphic Poincar´ e series
E(s,κ,w) that provide a natural regularisation of the modular forms of interest by inserting
a Kroneckertype convergence factor τs−w/2
2
in the standard sum over images. Therefore,
the resulting Poincar´ e series, originally studied in [31], converges absolutely for ?(s) > 1,
and the modular integral?
w
2, where E(s,κ,w) becomes formally a holomorphic function of τ. This procedure would
then allow to compute the modular integral (1.2) for any Φ which can be expressed as a
linear combination of such E(w
2,κ,w)’s, at least in principle. However, this strategy turned
out to be quite difficult in practice, since this analytic continuation depends on the notori
ously subtle analytic properties of the KloostermanSelberg zeta function which appears in
the Fourier expansion of E(s,κ,w). That is the reason why the analysis [20] was restricted
to the case of zero modular weight, where the analytic continuation is fully under control.
In the present work, we overcome these difficulties by employing a different class of
nonanalytic Poincar´ e series introduced in the mathematics literature by Niebur [32] and
Hejhal [33] and studied more recently by Bruinier, Ono and Bringmann [34–37]. Similarly
to the SelbergPoincar´ e series E(s,κ,w), the NieburPoincar´ e series F(s,κ,w) converges
absolutely for ?(s) > 1, and formally becomes holomorphic in τ at the point s =
However, the NieburPoincar´ e series can be specialised to the other interesting value s =
1−w
Although at this value F(s,κ,w) belongs to the more general class of weak harmonic Maass
forms,4that are typically nonholomorphic functions of τ, it has the important property
Finally, the singularities of the
FΓd,dE(s,κ,w) can be computed by unfolding F against it, at
least for large s. The result should then be analytically continued to the desired value s =
w
2.
2, which lies inside the domain of absolute convergence when the weight w is negative.
2See for instance [18] for a detailed discussion on chamber dependence of the traditional unfolding
method.
3BPS states sums have appeared in earlier works [22–26]. In our approach these BPS sums follow directly
from unfolding the fundamental domain against the elliptic genus, without any further assumption.
4A harmonic Maass form is an eigenmode of the weightw Laplacian on H with the same eigenvalue as
weak holomorphic modular forms. The positive frequency part of a weak harmonic Maass form is sometimes
known as a Mock modular form. See section 2.3 for a more precise definition of weak harmonic Maass forms.
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that any linear combination of F(1 −w
principal part of a given weak holomorphic modular form Φ, is actually a weak holomorphic
modular form, and equals Φ itself. Therefore, given any weak holomorphic modular form
Φ, the integral (1.2) can be computed by decomposing Φ into a sum of NieburPoincar´ e
series, and by unfolding each of them against the integration domain. Moreover, the same
strategy works also for weak almost holomorphic modular forms (i.e. involving powers of
ˆE2), where now one has to specialise the NieburPoincar´ e series to the values s = 1−w
with n a nonnegative integer.
The outline of this work is as follows. In section 2, we introduce the NieburPoincar´ e
series F(s,κ,w), discuss their main properties, present their Fourier coefficients and identify
their limiting values at s = 1 −w
result that any weak almost holomorphic modular form can be represented as a linear
combination of them. In section 3 we evaluate the modular integral?
we use this result to compute a sample of modular integrals of physical interest of the
form (1.2). In appendix A, we define our notation for modular forms, we collect various
definitions and properties of Whittaker and hypergeometric functions, and we introduce the
Kloosterman sums and the KloostermanSelberg zeta function. Finally, in appendix B we
briefly discuss the relation between the Selberg and NieburPoincar´ e series, and between
the “shifted constrained” Epstein zeta series and the above BPSstate sums. The reader
interested only in physics applications may skip section 2 and proceed directly to section 3,
which begins with an executive summary of the main properties of F(s,κ,w).
2,κ,w), whose coefficients are determined by the
2+n,
2+ n. We conclude the section by showing the important
FΓd+k,dF?s,κ,−k
2
?
in terms of certain BPSstate sums and discuss their singularity structure. In section 4,
Note added.
of ref. [34] where similar computations have been performed for general even lattices of
signature (d + k,d) with d = 0,1,2, in particular reproducing Borcherds’ automorphic
products for d = 2 [38]. Unlike [34], we restrict the analysis to even selfdual lattices
(with k = 0 mod 8), which suffices for our physics applications, but we allow for almost
holomorphic modular forms and arbitrary dimension d.
After having obtained most of the results in this paper, we became aware
2 NieburPoincar´ e series and almost holomorphic modular forms
In this section, we introduce the NieburPoincar´ e series F(s,κ,w), a modular invariant
regularisation of the na¨ ıve Poincar´ e series of negative weight. We present its Fourier ex
pansion for general values of s, and analyse its limit as s → 1 −w
nonnegative integer. We explain how to represent any weak almost holomorphic modular
form of negative weight as a suitable linear combinations of such Poincar´ e series.
2+ n where n is any
2.1 Various Poincar´ e series
In order to motivate the construction of the NieburPoincar´ e series, let us start with a brief
overview of Poincar´ e series in general. Let w be an even integer5and f a function on the
5In this paper we shall restrict to the case of even weight w in order to avoid complications with non
trivial multiplier systems, though the construction can be generalised to halfinteger weights.
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Poincar´ e upper half plane H. The action of an element γ =
given by the Petersson slash operator
?a b
c d
?
∈ Γ = SL(2,Z) on f is
(fwγ)(τ) = (cτ + d)−wf(γ · τ),
?1 ?
P(f,w;τ) ≡ P(f,w) =1
γ · τ =aτ + b
cτ + d.
(2.1)
If f is invariant under Γ∞=
0 1
?
⊂ Γ, the Poincar´ e series of seed f and weight w
?
2
γ∈Γ∞\Γ
fwγ (2.2)
defines an automorphic form of weight w on H, which is absolutely convergent provided
f(τ) ? τ1−w
usual holomorphic Poincar´ e series
?
2
2
as τ2→ 0. As an example, the choice f(τ) = q−κwith w > 2 leads to the
P(κ,w) =1
2
(c,d)=1
(cτ + d)−we−2πiκaτ+b
cτ+d, (2.3)
where the pair (a,b) is determined modulo (c,d) by the condition ad − bc = 1. Depending
on the value of κ, eq. (2.3) describes different types of modular forms. For κ = 0, P(κ,w)
is actually an Eisenstein series, while for κ ≤ −1 it is a cusp form, and must therefore
vanish if 2 < w < 12, an observation that will be important later. For κ > 0, eq. (2.3)
represents instead a weak holomorphic modular form with a pole of order κ at q = 0,
P(κ,w) = q−κ+ O(q).
For w ≤ 2, the Poincar´ e series (2.3) is divergent and thus needs to be regularised. One
possible regularisation scheme, introduced in the mathematical literature in [28, 29] and
discussed in the physics literature in [30], is to consider the convergent sum
P(κ,w) =1
2
lim
K→∞
?
c≤K
?
d<K;(c,d)=1
(cτ + d)−we2πiκaτ+b
cτ+dR
?
2πiκ
c(cτ + d)
?
,(2.4)
where R is a specific regulating factor such that R(x) ∼ x1−w/Γ(2 − w) as x → 0 and
approaches 1 as x → ∞. While this regularisation preserves holomorphicity, it does not
necessarily produce a modular form,6except for small w where the modular anomaly can
be shown to vanish. Moreover, the convergence of (2.4) is conditional, which makes it
unsuitable for the unfolding procedure.
Another option, introduced by Selberg [31] and considered in our previous work [20],
is to jettison holomorphicity and introduce a convergence factor ` a la Kronecker, thus
considering the Poincar´ eseries
E(s,κ,w) ≡1
2
?
(c,d)=1
τs−w
2
2
cτ + d2s−w(cτ + d)−we−2πiκaτ+b
cτ+d
(2.5)
6The holomorphic Poincar´ e series (2.4) is in general an Eichler integral, i.e. a function F(τ) which
satisfies F(τ) − (Fwγ)(τ) = rγ(τ) where rγ is a polynomial of degree −w in τ, whose coefficients depend
on a,b,c,d. We shall comment in section 2.3 on the modular completion of P(κ,w).
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associated to the seed f(τ) = τs−w
series. The series (2.5) converges absolutely for ?(s) > 1 and becomes formally holomorphic
at s =
2. However, for w ≤ 2 this value lies outside the convergence domain, and the
analytic continuation to s =
2depends on the analytic properties of the Kloosterman
Selberg Zeta function, defined in appendix B, which are notoriously subtle. In particular
this analytic continuation generally leads to holomorphic anomalies. For this reason, in [20]
we restricted the analysis to the case w = 0, where the analytic continuation is under
control. Another drawback of the SelbergPoincar´ e series (2.5) is that it fails to be an
eigenmode of the Laplacian on H, rather it satisfies [39]
?∆w+1
where ∆wis the weightw hyperbolic Laplacian defined in (A.1). Since E(s+1,κ,w) may
in general have a pole at s =w
2, the analytic continuation of E(s,κ,w) to this value is not
even guaranteed to be harmonic.
To circumvent these problems, following [32–34] we introduce a different regularisa
tion of the Poincar´ e series (2.3) for negative weight, which is both modular invariant and
annihilated by the operator on the l.h.s. of (2.6). Namely, we choose the seed in (2.2) to
be f(τ) = Ms,w(−κτ2)e−2πiκτ1where
Ms,w(y) = 4πy−w
2
2
q−κ. We shall refer to (2.5) as the SelbergPoincar´ e
w
w
2s(1 − s) +1
8w(w + 2)?E(s,κ,w) = 2πκ(s −w
2)E(s + 1,κ,w), (2.6)
2 M w
2sgn(y),s−1
2(4πy) (2.7)
is expressed in terms of the Whittaker function7Mλ,µ(z). We thus define the Niebur
Poincar´ e series
?
=1
2
(c,d)=1
F(s,κ,w) =1
2
γ∈Γ∞\Γ
?
Ms,w(−κτ2)e−2πiκτ1wγ (2.8)
(cτ + d)−wMs,w
?
−
κτ2
cτ + d2
?
exp
?
−2iπκ
?a
c−
cτ1+ d
ccτ + d2
??
.
Since Ms,w(y) ∼ y?(s)−w
dently of w and κ. Moreover, for κ > 0, the case of main interest in this work, the seed
behaves as
Ms,w(−κτ2)e−2πiκτ1∼
Γ(s +w
2 as y → 0, eq. (2.8) converges absolutely for ?(s) > 1, indepen
Γ(2s)
2)q−κ
asτ2→ ∞,(2.9)
so that F(s,κ,w) can indeed be viewed as a regulated version of the na¨ ıve Poincar´ e series
P(q−κ,w), up to an overall normalisation. By construction it is an eigenmode of the
weightw Laplacian on H,
?∆w+1
for all values of s,κ,w. We shall denote by H(s,w) = H(1−s,w) the space of realanalytic
solutions to (2.10) which transform with modular weight w under Γ.
2s(1 − s) +1
8w(w + 2)?F(s,κ,w) = 0, (2.10)
7For a definition of Whittaker functions and some of their properties see appendix A.2.
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The raising and lowering operators Dw,¯Dwdefined in (A.2), map H(s,w) into H(s,w±
2), and have a simple action on the NieburPoincar´ e series
Dw· F(s,κ,w) = 2κ?s +w
¯Dw· F(s,κ,w) =
2
?F(s,κ,w + 2),
1
8κ
?s −w
2
?F(s,κ,w − 2).
(2.11)
Furthermore, under the action of the Hecke operator (A.7) F(s,κ,w) transforms as
?
Tκ? · F(s,κ,w) =
d(κ,κ?)
d1−wF(s,κκ?/d2,w) . (2.12)
In particular, setting κ = 1, the series F(s,κ?,w) is obtained by acting with Tκ? on
F(s,1,w).
While the Poincar´ e series (2.8) converges absolutely only for ?(s) > 1, it is known
to have a meromorphic continuation to the complex splane, holomorphic in the region
?(s) >1
‘completed’ series
Γ(1 − 2s)
Γ?1 − s +w
is known to be invariant under s ?→ 1 − s, up to an additive contribution proportional
to the nonholomorphic Eisenstein series E?(s,w) [32, 40]. In this work however we shall
only consider F(s,κ,w) in its domain of convergence ?(s) > 1, except for w = 0 where we
allow s = 1.
2[32, 40], but with poles on the lines s ∈1
2+ iR and s ∈1
4+ iR. Moreover, the
F?(s,κ,w) =
2sgn(κ)?F(s,κ,w) (2.13)
2.2 Fourier expansion of the NieburPoincar´ e series
The Fourier expansion of F(s,κ,w) can be obtained following the standard procedure of
extracting the contribution from c = 0,d = 1, setting d = d?+ mc in the remaining sum,
and Poisson resumming over m. The result is [34, 36]
F(s,κ,w) = Ms,w(−κτ2)e−2πiκτ1+
?
m∈Z
˜ Fm(s,κ,w)e2πimτ1,(2.14)
where, for zero frequency
˜ F0(s,κ,w) =22−wi−wπ1+s−w
2 κs−w
?Γ?s +w
2 Γ(2s − 1)σ1−2s(κ)
2
Γ?s −w
2
?ζ(2s)
τ1−s−w
2
2
, (2.15)
while for nonvanishing integer frequencies8
˜ Fm(s,κ,w) =
4π κi−wΓ(2s)
Γ?s +w
?
2sgn(m)?
dkdsis the divisor function and Zs(m,−κ) is the
???m
κ
???
w
2Zs(m,−κ)Ws,w(mτ2). (2.16)
In these expressions, σs(k) =
KloostermanSelberg zeta function (A.37), a numbertheoretical function which plays a
8Note that˜ F−κ<0 does not include the contribution from the first term in (2.14).
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JHEP06(2012)070
central rˆ ole in the theory of Poincar´ e series. The function Ws,wis expressed in terms of
the Whittaker Wfunction as
Ws,w(y) = 4πy−w
2 W w
2sgn(y),s−1
2(4πy) ,(2.17)
and is determined uniquely by the requirement that Ws,w(nτ2)e2πimτ1be annihilated by
the Laplace operator on the l.h.s. of (2.10), and be exponentially suppressed as τ2→ ∞.
Using the properties (A.35) and (A.36), it is straightforward to check that all Fourier
modes transform according to (2.11) under the raising and lowering operators Dw,¯Dw.
Moreover, using the action (A.9) of the Hecke operators on the Fourier coefficients, and
the Selberg identity (A.39) satisfied by the Kloosterman sums, one can show that
Tκ· F(s,1,w) = F(s,κ,w). (2.18)
Eq. (2.12) follows then from this equation and from the Hecke algebra (A.8).
2.3 Harmonic Maass forms from NieburPoincar´ e series
Let us focus on the NieburPoincar´ e series F(s,κ,w) at the point s = 1 −w
this value, we recall that any weak holomorphic modular form is an eigenmode of ∆wwith
eigenvalue −w
However, weak holomorphic modular forms are not the only eigenmodes of ∆wwith this
eigenvalue. In fact, the space H(1 −w
forms of weight w, of which weak holomorphic modular forms are only a proper subspace.
The Fourier expansion of a general weak harmonic Maass form Φ of weight w is given
by [41]
2. To motivate
2, and therefore belongs to H(s,w) for s = 1 −w
2(or equivalently, s =w
2).
2,w) is known as the space of weak harmonic Maass
Φ =
−1
?
m=−∞
(−m)w−1¯b−mΓ(1 − w,−4πmτ2)qm+
¯b0(4πτ2)1−w
w − 1
+
∞
?
m=−κ
amqm, (2.19)
where Γ(s,x) is the incomplete Gamma function and am,bmare coefficients constrained
by modular invariance. As a result, a generic weak harmonic Maass form has an infi
nite number of negative frequency components, which are nonholomorphic functions of
τ. A harmonic Maass form splits into the sum Φ = Φa+ Φb of a holomorphic part
Φa =
part Φb. The nonholomorphic and holomorphic parts can be extracted using the lowering
operator¯Dwand the iterated raising operator D1−w
?∞
m=−κamqm, sometimes called a Mock modular form, and a nonholomorphic
w
. Indeed,
• the operator¯Dwannihilates the holomorphic part, and produces, up to powers of τ2,
the complex conjugate of a holomorphic modular form Ψ of weight 2 − w,
¯Dw· Φ =¯Dw· Φb= −21−2w(πτ2)2−wΨ ,Ψ(τ) =
∞
?
m=0
bmqm, (2.20)
sometimes known as the shadow.
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JHEP06(2012)070
• the iterated raising operator D1−w
transform [42], annihilates the nonholomorphic part, and produces a weak holomor
phic modular form Ξ of weight 2 − w,
w
, also known in the physics literature as the Farey
D1−w
w
· Φ = D1−w
w
· Φa= Ξ ,Ξ ≡
∞
?
m=−κ
(−2m)1−wamqm, (2.21)
that we shall call the ghost. The ghost encodes the holomorphic part of the harmonic
Maass form (modulo an additive constant).9
Returning to the NieburPoincar´ e series, we see that by construction the series
F(s,κ,w) at the special point s = 1 −w
gence domain — is a weak harmonic Maass form of weight w. Indeed, using (A.30) we find
that its Fourier expansion (2.14) reduces to
2— which, for w < 0,belongs to the conver
F(1 −w
2,κ,w) = M1−w
2,w(−κτ2)e−2πiκτ1+
?
m∈Z
˜ Fm(1 −w
2,κ,w)e2iπmτ1, (2.22)
where the seed simplifies to a finite sum
M1−w
2,w(−κτ2)e−2πiκτ1= Γ(2 − w)
?
q−κ− ¯ qκ
−w
?
?=0
(4πκτ2)?
?!
?
= [Γ(2 − w) + (1 − w)Γ(1 − w;4πκτ2)]q−κ,
(2.23)
and the remaining Fourier coefficients reduce to
˜ Fm>0(1 −w
2,κ,w) = 4π i−wκΓ(2 − w)
?m
κ
?w
?w
2Z1−w
2(m,−κ)e−2πmτ2,
˜ Fm<0(1 −w
2,κ,w) = 4π i−wκ(1 − w)
4π2κ
(2πi)w
?−m
κ
2
Z1−w
2(m,−κ)Γ(1 − w,−4πmτ2)e−2πmτ2,
˜ Fm=0(1 −w
2,κ,w) =
σw−1(κ)
ζ(2 − w).(2.24)
One thus recognises an expansion of the form (2.19) with coefficients
a−κ= Γ(2 − w),
a−κ<m<0= 0,
a0=
4π2κ
(2πi)w
σw−1(κ)
ζ(2 − w),
am>0= 4π i−wκΓ(2 − w)
b0= 0,
bm>0= (1 − w)κ1−wδm,κ+ 4π iw(1 − w)(mκ)1−w
?m
κ
?w
2Z1−w
2(m,−κ),
2 Z1−w
2(m,κ).
(2.25)
9Notice that the ghost is only defined for integer weight w, unlike the shadow, which extends to the case
of halfinteger weight Mock theta series.
– 9 –
Page 11
JHEP06(2012)070
In particular, b0= 0, so that the shadow of F?1 −w
¯Dw· F(1 −w
2,κ,w?is a cusp form of weight 2−w,
proportional to the holomorphic Poincar´ e series P(−κ,2−w). Indeed, using (2.11) we find
2,κ,w) =1 − w
8κ
=1 − w
8κ
where in the second line we have recognised the Fourier expansion of the standard holo
morphic Poincar´ e series of weight greater than 2. Similarly, using (2.11) the ghost of
F(1 −w
D1−w
w
· F(1 −w
and corresponds to the NieburPoincar´ e series F
the convergence domain. Moreover, the Fourier expansion of the latter reproduces that of
the Poincar´ e series P(κ,w) of positive weight
?m
As an aside, we note that the holomorphic part Fa
series F(s,κ,w) at s = 1−w
w)P(κ,w) defined by holomorphic regularisation as in (2.4) and worked out in [28, 29].
Therefore, the nonholomorphic part Fb
provides the modular completion of the Eichler integral P(κ,w) — a clear advantage of
modularinvariant regularisation over holomorphic regularisation.
To make this discussion less abstract, we shall now exhibit the harmonic Maass form
F?1 −w
coefficients numerically, we find:
F(1 −w
(4π κτ2)2−wP(−κ,2 − w),
2,κ,w − 2)
(2.26)
2,κ,w) is
2,κ,w) = (2κ)1−wΓ(2 − w)F(1 −w
2,κ,2 − w),
, with w?= 2 − w > 2 within
(2.27)
?
w?
2,κ,w??
F(w?
2,κ,w?) = q−κ+ 2π i−w??
m>0
κ
?w?−1
2
?
c>0
S(m,−κ;c)
c
?1 −w
Iw?−1
?4π
c
√κm
?
qm. (2.28)
2,κ,w?
of the NieburPoincar´ e
2reproduces the Fourier expansion of the Poincar´ e series Γ(2−
?1 −w
2,κ,w?
of the same NieburPoincar´ e series
2,κ,w?, its shadow and its ghost for the two cases w = −10 and w = −14 (lower
values of w will be discussed in the next subsection) and κ = 1. Evaluating the Fourier
• For w = −10,
F(6,1,−10)=Fb(6,1,−10)+11!
?
q−1−65520
691
− 1842.89q − 23274.08q2+ ...
?
(2.29)
where Fb(6,1,−10) is the nonholomorphic component. The shadow of (2.29) reads
F(6,1,−12) = (4πτ2)12P(−1,12),
where the modular discriminant ∆ generates the space of cusp forms of weight 12.
The ghost, obtained by acting with D11on the holomorphic part, can be written as
P(−1,12) = β12∆,(2.30)
F(6,1,12) =83E3
4E2
72∆
6− 11E4
6
+ α12(E3
4− E2
∆
6)2
= q−1+ 1842.89q + 47665306.53q2+ ... ,
(2.31)
where α12= 0.201029508104...,β12= 2.840287517... are irrational numbers. The
coefficients 1842.89, 23274.08, 47665306.53 are twodigit approximations of the exact
values 324(9216α12−1847), (60617−69984α12)/2, 1024(60617−69984α12), respec
tively. This example was discussed in detail in [37].
– 10 –
Page 12
JHEP06(2012)070
• Similarly, for w = −14,
F(8,1,−14) = Fb(8,1,−14) + 15!
?
q−1−16320
3617− 45.67q − 366.47q2+ ...
?
(2.32)
where Fb(8,1,−14) is the nonholomorphic component. The shadow of (2.32) reads
F(8,1,−16) = (4πτ2)16P(−1,16),
where E4∆ generate the space of cusp forms of weight 16. The ghost, obtained by
acting with D15on the holomorphic part, can be written as
P(−1,16) = β16E4∆ (2.33)
F(8,1,16) =73E4
4E2
6− E4E4
72∆
6
+ α16E7
4− 2E4
4E2
∆
6+ E4E4
6
= q−1+ 45.67q + 12008361.57q2+ ... ,
(2.34)
where α16= 0.137975847804... and β16= 1.3061364711... are irrational numbers.
The coefficients 45.67, 366.47 in eq. (2.32) are twodigit approximations of the exact
values 36(82944α16− 11443), (314928α16− 37589)/16, respectively.
These two examples illustrate the fact that Fourier coefficients of harmonic Maass
forms are in general irrational numbers.
2.4 Weak holomorphic modular forms from NieburPoincar´ e series
We now come to our main goal, i.e. to find an absolutely convergent Poincar´ e series repre
sentation of any weak holomorphic modular form Φwof weight w ≤ 0 and κorder pole at
the cusp, with given principal part
?
As we shall see, any such Φw can be expressed as a linear combination of the Niebur
Poincar´ e series F(s,κ,w).
We have noted in the previous subsection that the eigenvalue of a weak holomorphic
modular form under the hyperbolic Laplacian ∆w coincides with the eigenvalue of the
NieburPoincar´ e series whenever s = 1 −w
a weak harmonic Maass form, in general not holomorphic. Exceptions to this statement
occur at the special values w ∈ {−2,−4,−6,−8,−12}, where the space of holomorphic
cusp forms of weight 2 − w is empty, and F?1 −w
expansions. For κ = 1 the exact identification is reported in table 1, while for κ > 1, the
proper identification of F?1 −w
For w ≤ 0 outside the list above, the space of cusp forms of weight 2−w is not empty,
and F(1 −w
Nevertheless, it can be shown [34] that the linear combination
?
Φ−
w(τ) =
−κ≤m<0
amqm. (2.35)
2. At this value, however, F?1 −w
2,κ,w?
is
2,κ,w?can be recognised as an element
of the ring of weak holomorphic modular forms by matching the principal part of their
2,κ,w?can be obtained by acting on F?1 −w
2,1,w?with
the Hecke operator Tκ, as given by eq. (2.18).
2,κ,w) is indeed a genuine harmonic Maass form, with nonvanishing shadow.
G(s,w) ≡
1
Γ(2 − w)
−κ≤m<0
amF(s,m,w),(2.36)
– 11 –
Page 13
JHEP06(2012)070
w
F?1 −w
3!E4E6∆−1
5!E2
7!E6∆−1
9!E4∆−1
13!∆−1
2,1,w?
j + 24
F?1 −w
2,1,2 − w?
0E2
4E6∆−1
E4(j − 240)
E6(j + 204)
E2
4(j − 480)
E4E6(j + 264)
E2
4E6(j + 24)
−2
−4
−6
−8
−12
4∆−1
Table 1. Weak holomorphic modular forms obtained as the limit s → 1 −w
values w ∈ {0,−2,−4,−6,−8,−12}. For w negative and outside this range, the limit yields a weak
harmonic Maass form. The second line shows the ghost, which is a weak holomorphic modular form
of weight 2 − w with vanishing constant term (aside from the case w = 0).
2of F(s,1,w), for the
with coefficients amdetermined by the principal part
Φ−
w=
?
−κ≤m<0
amq−m
(2.37)
of any weak holomorphic form Φw of negative weight w, reduces to a weak holomorphic
modular form for s = 1 −w
harmonic Maass forms F?1 −w
1
Γ(2 − w)
−κ≤m<0
or equivalently, using (2.23), as an absolutely convergent Poincar´ e sum
?
−κ≤m<0
where the subtraction in the bracket ensures that the seed is O(τ1−w
that, unlike the holomorphic regularisation in (2.4), the expression (2.39) is manifestly
modular covariant and absolutely convergent.
To illustrate the power of eq. (2.38), let us reconsider the two examples of the previous
subsection, now allowing for κ = 1,2.
2, namely Φwitself. Said differently, the shadows of the weak
2,κ,w?cancel in the linear combination (2.36). As a result,
?
any Φwcan be represented as the linear combination
Φw=amF?1 −w
2,m,w?, (2.38)
Φw=1
2
?
γ∈Γ∞\Γ
Φ−
w−
?
−w
?
?=0
am¯ qm(4πκτ2)?
?!
??????w
2
) as τ2→ 0. We stress
γ ,(2.39)
2
• For w = −10, F(6,2,−10) and F(6,1,−10) are separately weak harmonic Maass
forms with irrational coefficients, but the linear combination
F(6,2,−10)+24F(6,1,−10) = 11!E2
produces (up to an overall normalisation) the unique weak holomorphic form10of
weight −10 with a double pole at q = 0;
10Compare the simplicity of our expression to the corresponding equation in Sec 4.1 of [37].
4E6
∆2
= 11!(q−2+24q−1−196560+...) (2.40)
– 12 –
Page 14
JHEP06(2012)070
• Similarly, for w = −14, F(8,2,−14) and F(8,1,−14) are separately weak harmonic
Maass forms with irrational coefficients, but the linear combination
F(8,2,−14)−216F(8,1,−14) = 15!E4E6
∆2 = 15!(q−2−216q−1−146880+...) (2.41)
produces (up to an overall normalisation) the unique weak holomorphic form of weight
−14 with a double pole at q = 0.
Similar relations occur for higher negative weight w < −14 and higher order κ of the
pole at q = 0.
2.5Weak almost holomorphic modular forms from NieburPoincar´ e series
For physics applications it is important to extend our previous analysis to the case of
weak almost holomorphic modular forms, i.e. elements of the ring generated by the almost
holomorphic Eisenstein seriesˆE2and the ordinary weak holomorphic modular forms, or
equivalently, by the modular derivatives DnΦ of ordinary weak holomorphic modular forms.
To this end, it is important to note that for any integer n ≥ 0, it follows from (2.11)
that the NieburPoincar´ e series F(s,κ,w) evaluated at the point s = 1 −w
expressed as
2+ n can be
F?1 −w
2+ n,κ,w?=
1
(2κ)nn!DnF?1 −w
2+ n,κ,w − 2n?, (2.42)
where Dnis the iterated modular derivative (A.6). The NieburPoincar´ e series F(s?,κ,w?)
appearing on the r.h.s. satisfies s?= 1 −w?
As a result, provided that the coefficients am in the linear combination (2.36) are
chosen such that
Φ−
w−2n≡
−κ≤m<0
2, and thus is a harmonic Maass form.
?
am
(2m)nn!qm
(2.43)
is the principal part of a weak holomorphic modular form Φw−2nof weight w − 2n, then
the linear combination G(s,w) in (2.36) evaluated at the point s = 1 −w
an almost holomorphic modular form of weight w,
2+ n reproduces
G?1 −w
2+ n,w?=
1
Γ(2 − w)
?
−κ≤m<0
amF?1 −w
n≥0H?1 −w
2+ n,m,w?= DnΦw−2n.
2+ n,w?as the space of “weak almost
(2.44)
More generally, we refer to the space?
The general Fourier expansion of such forms can be obtained by taking the limit s =
1−w
the series F(s,κ,w) at the point s = 1−w
at the point s?= −w?
harmonic Maass forms”, of which almost holomorphic modular forms are only a subspace.
2+n in eqs. (2.14) and (2.16), and by using the identities (A.33) and (A.34). Similarly,
2+n for n ≤ −2 may be obtained from F(s?,κ,w?)
2by using the lowering operator¯Dw.
– 13 –
Page 15
JHEP06(2012)070

6
•
?
•
•
•
•
•
2
•
•
•
•
•
4
•
•
•
•
•
•
•
•
•
•
•
•
•
•
w
s
−2
−4
1
j + 24
2
3
D
¯D
E2
4E6/∆
weak almost harmonic
s =w
2: weak hol. (ghost)
weak almost hol.
s = −w
2: τ2−w
2
×antihol. (shadow)
s = 1 −w
2: weak harmonic
Figure 1. Phase diagram for the NieburPoincar´ e series F(s,κ,w) for integer values of?w
Maass form, see table 2.
2,s?with
s ≥ 1. For low negative values of w, F(s,κ,w) reduces to an ordinary weak almost holomorphic
2.6 Summary
To summarise this discussion, it is useful to consider the plane of the variables?w
values of s, it is generally a weak almost harmonic Maass form, and on the line s = 1 −w
(and w < 0), F(s,κ,w) becomes a weak harmonic Maass form. On the line s = −w
obtained from the former by acting with the lowering operator D, F(s,κ,w) reduces, up
to an overall multiplicative factor τ−w
2
, to the complex conjugate of a cusp form of weight
2−w, known as the shadow of the harmonic Maass form F(s,κ,w+2). On the line s =w
F(s,κ,w) is instead a weak holomorphic modular form. It is connected to its expression on
the line s = 1−w
it as the ‘ghost’ of the harmonic Maass form F(s,κ,2 − w). In the quadrant w > 2,s > 1,
F(s,κ,w) is more generally a weak almost holomorphic modular form. For low negative val
ues of w and s integer, F(s,κ,w) is in fact always a weak almost holomorphic modular form,
as displayed in table 2. Genuine harmonic Maass forms start appearing at s = 6 and s ≥ 8.
2,s?as in
figure 1. The NieburPoincar´ e series F(s,κ,w) converges absolutely for s > 1. For integer
2
2,
2,
2by the action of the iterated raising operator D1−w, and thus we refer to
– 14 –
Page 16
JHEP06(2012)070
s\w −10 −8
−6
−4
−2
0
2
4
6
8
10
5
0
9!E4
∆
9!
2DE4
∆
9!
8D2 E4
∆
9!
233!D3 E4
∆
9!
244!D4 E4
∆
9!
255!D5 E4
∆
9!
266!D6 E4
∆
9!
277!D7 E4
∆
9!
278!D8 E4
∆
E4E6(j + 264)
4
0
0
7!E6
∆
7!
2DE6
∆
7!
8D2 E6
∆
7!
233!D3 E6
∆
7!
244!D4 E6
∆
7!
255!D5 E6
∆
7!
266!D6 E6
∆
E2
4(j − 480)
7!
288!D8 E6
∆
3
0
0
0
5!E2
4
∆
5!
2DE2
4
∆
5!
8D2E2
4
∆
5!
233!D3E2
4
∆
5!
244!D4E2
4
∆
E6(j + 504)
5!
266!D6E2
4
∆
5!
277!D7E2
4
∆
2
0
0
0
0
3!E4E6
∆
3DE4E6
∆
34D2 E4E6
∆
E4(j − 240)
3!
244!D4 E4E6
∆
3!
255!D5 E4E6
∆
3!
266!D6 E4E6
∆
1
0
0
0
0
0
j + 24
E2
4E6
∆
1
222!D2j
1
233!D3j
1
244!D4j
1
255!D5j
Table 2. NieburPoincar´ e series F(s,1,w) at the special values s = 1 −w
2+ n with n integer, for low negative values of w.
– 15 –
Page 17
JHEP06(2012)070
3A new road to oneloop modular integrals
We are interested in the evaluation of oneloop modular integrals of the form (1.2), while
keeping manifest at all steps the automorphisms of the Narain lattice, i.e. Tduality. Such
integrals encode, for instance, threshold corrections to the running of gauge and gravi
tational couplings. The function Φ, related to the elliptic genus and dependent on the
vacuum under consideration, is in general a weak almost holomorphic modular form of
nonpositive weight. For example, in N = 4 compactifications of the SO(32) heterotic
string (with vanishing Wilson lines) one finds a linear combination of zeroweight weak
almost holomorphic modular forms [2]
Φ(τ) =t8trF4+
1
27325
?ˆE2E4E6
?
E3
∆t8trR4+
ˆE2
4
1
2932
?
ˆE2E4E6
∆
ˆE2
2E2
∆
4
t8(trR2)2
+
1
2832
∆
−
2E2
∆
4
t8trF2trR2
+
1
2932
E3
∆+
4
ˆE2
2E2
∆
4
− 2
− 2732
?
t8(trF2)2,
(3.1)
where t8is the familiar tensor appearing in fourpoint amplitudes of the heterotic string,
and F and R are the gauge field strength and curvature twoform. A similar expression
arises for gauge and gravitational couplings in the E8× E8heterotic string.
While the traditional procedure for evaluating integrals of the form (1.2) has been
to unfold the integration domain F against the lattice partition function Γd+k,d, in [20]
we instead proposed to represent Φ as a Poincar´ e series of the form (2.2), which is then
amenable to the unfolding procedure. The advantage of this approach is that Tduality is
kept manifest at all steps and the final result is expressed as a sum over BPS states which
is manifestly invariant under O(d + k,d;Z). Moreover, singularities associated to states
becoming massless at special points in the Narain moduli space are easily read off from
this representation.
3.1NieburPoincar´ e series in a nutshell
In order to implement this strategy, it is essential to represent Φ as an absolutely convergent
Poincar´ e series, so that the unfolding of the fundamental domain is justified. Fortunately,
as discussed in detail in section 2 and summarised in the following, any weak almost
holomorphic modular form Φwof weight w ≤ 0 can be written as a linear combination of
NieburPoincar´ e series, defined as
?
=1
2
(c,d)=1
F(s,κ,w) =1
2
γ∈Γ∞\Γ
?
Ms,w(−κτ2)e−2πiκτ1wγ(3.2)
(cτ + d)−wMs,w
?
−κτ2
cτ + d2
?
exp
?
−2iπκ
?a
c−
cτ1+ d
ccτ + d2
??
.
Here Ms,wis related to the Whittaker Mfunction via
Ms,w(−y) = (4πy)−w/2M−w
2,s−1
2(4πy),(3.3)
– 16 –
Page 18
JHEP06(2012)070
and s is a complex parameter, the real part of which must be larger than 1 for absolute
convergence. The choice of the Whittaker function in (3.2) is dictated by the requirement
that F(s,κ,w) be an eigenmode of the hyperbolic Laplacian ∆w(see eq. (2.10)), and behave
as q−κat the cusp q ≡ e2πiτ= 0 (see eq. (2.9)), thus reproducing, for κ = 1, the simple pole
associated to the unphysical tachyon of the heterotic string. The set of NieburPoincar´ e
series F(s,κ,w) is closed under the action of the derivative operators Dwand¯Dwdefined
in (A.2), which, according to (2.11), act by raising or lowering the weight w by two units
while keeping s fixed.
At the special point s = 1 −w
convergence, the NieburPoincar´ e series F(s,κ,w) becomes a weak harmonic Maass form.11
In particular, unless w takes one of the special values listed in table 1, it is in general not
holomorphic. Although the values listed in the table essentially exhaust all the cases of
interest in string theory, it is a remarkable fact that linear combinations of NieburPoincar´ e
series, with coefficients determined by the principal part of a weak holomorphic modular
form Φw, are in fact weakly holomorphic, and reproduce Φwitself [34]:
2, which for w < 0 lies within the domain of absolute
Φ−
w=
?
−κ≤m<0
amq−m
⇒
Φw=
1
Γ(2 − w)
?
−κ≤m<0
amF(1 −w
2,m,w) .(3.4)
Moreover, upon using (2.11) one can also relate weak almost holomorphic modular forms
involving (up to) n powers ofˆE2— or equivalently, obtained by acting up to n times with
the derivative operator Dwon a weak holomorphic modular form — to linear combinations
of F(s,κ,w) evaluated at the special points s = 1 −w
In the cases relevant to heterotic string threshold corrections, the elliptic genus Φwhas
a simple pole at q = 0, corresponding to the unphysical tachyon, and therefore the expan
sion (3.4) includes only one term, with κ = m = 1 (modulo an additive constant in the case
w = 0). Moreover, the weight w is related to the signature (d + k,d) of the Narain lattice
by w = −k/2. Since string theory restricts the Narain lattice to be even and selfdual, so
that Γd+k,dis covariant under the full modular group Γ = SL(2,Z), the possible values of
w are w = 0 (corresponding to the point of unbroken E8×E8or SO(32) gauge symmetry),
w = −4 (corresponding to the point of unbroken E8symmetry, with arbitrary Wilson lines
for the other E8factor), or w = −8 (corresponding to generic values of the Wilson lines in
E8× E8or SO(32)). The complete list of weak almost holomorphic modular forms with
a simple pole at q = 0 and modular weights w = 0,−2,−4,−6,−8,−10, together with
their expressions as linear combinations of NieburPoincar´ e series, can be found in table 3.
Although stringtheory applications only require κ = 1, our methods apply equally well for
arbitrary positive integer values of κ, which we therefore keep general until section 3.5.
2+ n?, with 0 ≤ n?≤ n.
3.2Oneloop BPS amplitudes as BPSstate sums
Since any weak almost holomorphic modular form of negative weight can be represented
as a linear combination of NieburPoincar´ e series, for the purpose of computing integrals
11For a definition of weak harmonic Maass forms see section 2.3.
– 17 –
Page 19
JHEP06(2012)070
w = 0
ˆ E2E4E6
∆
ˆ E2
= F(2,1,0) − 5F(1,1,0) − 144
=1
5F(3,1,0) − 4F(2,1,0) + 13F(1,1,0) + 144
=
2E2
∆
ˆ E3
2E6
∆
ˆ E4
2E4
∆
4
3
175F(4,1,0) −3
1
1225F(5,1,0) −
+29
5F(3,1,0) +33
6
175F(4,1,0) +18
5F(2,1,0) − 17F(1,1,0) − 144
35F(3,1,0) −16
=
5F(2,1,0)
5F(1,1,0) +144
1
1926925F(7,1,0) −
+12
5
ˆ E6
∆=
2
3
2695F(5,1,0) +
7F(1,1,0) −144
w = −2
3F(2,1,−2)
1
20F(3,1,−2) +11
1
350F(4,1,−2) +
1
12936F(5,1,−2) +
6
175F(4,1,0) −3
7F(3,1,0)
5F(2,1,0) −29
7
ˆ E2E2
∆
ˆ E2
2E6
∆
ˆ E3
2E4
∆
ˆ E5
4
=
1
40F(3,1,−2) −1
1
525F(4,1,−2) −
1
11760F(5,1,−2) −
1
19819800F(7,1,−2) −
+1
15F(2,1,−2)
=
30F(2,1,−2)
9
280F(3,1,−2) −
1
525F(4,1,−2) −
=
2
15F(2,1,−2)
1
56F(3,1,−2)
2
∆=
w = −4
ˆ E2E6
∆
ˆ E2
2E4
∆
ˆ E4
=
1
2520F(4,1,−4) −
1
70560F(5,1,−4) −
1
148648500F(7,1,−4) −
1
120F(3,1,−4)
1
2520F(4,1,−4) +
1
129360F(5,1,−4) +
w = −6
241920F(5,1,−6) −
1
792792000F(7,1,−6) −
=
1
280F(3,1,−4)
1
6300F(4,1,−4) −
2
∆=
1
840F(3,1,−4)
ˆ E2E4
∆
ˆ E3
=
11
10080F(4,1,−6)
1
887040F(5,1,−6) +
w = −8
2
∆=
1
50400F(4,1,−6)
ˆ E2
∆=
2
1
2854051200F(7,1,−8) −
1
3991680F(5,1,−8)
w = −10
1
13!F(7,1,−10)
ˆ E2
∆=
Table 3. List of all weak almost holomorphic modular forms of negative weight with a simple pole
at q = 0, as linear combination of NieburPoincar´ e series F(1−w
appear in the first column of table 1).
2+n,1,w) (the holomorphic ones
– 18 –
Page 20
JHEP06(2012)070
of the form (1.2) it suffices to consider the basic integral
Id+k,d(G,B,Y ;s,κ;T ) ≡ Id+k,d(s,κ;T ) =
?
FT
dµΓd+k,d(G,B,Y )F?s,κ,−k
2
?,(3.5)
where the modular weight w = −k/2 of the NieburPoincar´ e series is determined, via
modular invariance, by the signature of the Narain lattice. In order to regulate potential
infrared divergences, associated to massless string states, we have introduced in (3.5) an
infrared cutoff T , which we shall eventually take to infinity.
According to the unfolding procedure, extended in the presence of a hard cutoff T
in [3], the truncated fundamental domain FT can be extended to the truncated strip
?0 < τ2 < T ,−1
trivial ones integrated over the complement F − FT. In equations
?T
−
F−FT
Using the asymptotic behaviours
2≤ τ1 <
1
2
?
at the expense of restricting the sum over images in the
NieburPoincar´ e series to the trivial coset, and subtracting the contribution of the non
Id+k,d(s,κ,T ) =
0
dτ2
τ2
2
?1/2
dµΓd+k,d
−1/2
dτ1Γd+k,dMs,−k
?
2(−κτ2)e−2iπκτ1
?− Ms,−k
?
F?s,κ,−k
2
2(−κτ2)e−2iπκτ1?
.
(3.6)
Ms,−k
2(−κτ2) ∼ τs+k
4
2
,Γd+k,d∼ τ−d+k
2
2
,asτ2→ 0, (3.7)
and
Γd+k,d∼ τ
d
2
2
asτ2→ ∞,(3.8)
together with the Fourier expansion (2.14), one can show that the second integral in (3.6)
converges for ?(s) >
1 +1
4(2d + k). For ?(s) in this range, one may then remove the IR cutoff and extend, in
the first integral, the τ2range to the full R+. Moreover, the τ1integral vanishes unless the
lattice vector satisfies the levelmatching constraint
1
4(2d + k), while the first integral in (3.6) converges for ?(s) >
p2
L− p2
R= 4κ.(3.9)
In heterotic string vacua (with κ = 1) this condition selects the contributions of the half
BPS states in the perturbative spectrum, and thus the first integral in (3.6) can be written
as a BPSstate sum
?
Here we have introduced the shorthand notation
?
to denote the sum over those lattice vectors satisfying the levelmatching condition (3.9),
and corresponding to halfBPS states if κ = 1. By the previous estimates, this sum is
Id+k,d(s,κ) ≡
BPS
?∞
0
dτ2
τ2
2
Ms,−k
2(−κτ2)τd/2
2
e−πτ2(p2
L+p2
R)/2.(3.10)
BPS
≡
?
pL,pR
δ(p2
L− p2
R− 4κ)(3.11)
– 19 –
Page 21
JHEP06(2012)070
absolutely convergent for ?(s) > 1 +1
in this range.
To relate the BPSstate sum to the modular integral of interest, we note that upon
using (3.6) and rearranging terms, eq. (3.10) may be rewritten as
4(2d + k), and thus defines an analytic function of s
Id+k,d(s,κ) =Id+k,d(s,κ,T )
+
?
?
?
F−FT
dµΓd+k,d
?
F?s,κ,−k
d
2
2)
2
?− Ms,−k
Ms,−k
2(−κτ2)e−2iπκτ1− f0(s)τ1−s+k
4
2
?
+
F−FT
dµ(Γd+k,d− τ
?
?
2(−κτ2)e−2iπκτ1+ f0(s)τ1−s+k
4
2
?
+
F−FT
dµτ
d
2
2
Ms,−k
2(−κτ2)e−2iπκτ1+ f0(s)τ1−s+k
4
2
?
, (3.12)
where
f0(s) =(4π)1+k
4πsi
Γ?s +k
k
2 Γ(2s − 1)κs+k
4
4 σ1−2s(κ)
?ζ(2s)
?Γ?s −k
4
(3.13)
is the coefficient of the zerofrequency Fourier mode (2.15), and the r.h.s. of (3.12) is
independent of T . The first three lines in (3.12) are analytic functions of s for ?(s) > 1,
since Id+k,d(s,κ,T ) is integrated over the compact domain FT, while the integrands in the
second and third line are exponentially suppressed as τ2→ ∞, away from the points of
enhanced gauge symmetry. The fourth line, however, evaluates to
?∞
and is therefore analytic in s, except for a simple pole at s =1
that the BPS state sum (3.10) admits a meromorphic continuation to ?(s) > 1, with a sim
ple pole at s =2d+k
4
with residue f0
4
we find that the BPSstate sum (3.10) is actually equal to the renormalised integral
?
f0(s)
T
dτ2τ−1−s+2d+k
2
4
= f0(s)T
2d+k
4
−s
s −2d+k
4
(3.14)
4(2d+k). We thus conclude
?2d+k
?. Moreover, taking the limit T → ∞ in (3.12),
R.N.
?
F
dµΓd+k,dF?s,κ,−k
2
?= lim
= Id+k,d(s,κ)
T →∞
Id+k,d(s,κ,T ) + f0(s)T
2d+k
4
−s
s −2d+k
4
?
(3.15)
for generic values of s ?=2d+k
equal to the constant term in the Laurent expansion of Id+k,d(s,κ) around s =2d+k
?
=ˆId+k,d
where f?
properly subtracted,
?2d + k
4. At the point s =2d+k
4, the renormalised integral is instead
4,
?2d+k
(3.16)
R.N.
F
dµΓd+k,dF(s,κ,−k
2) = lim
T →∞
?Id+k,d
?2d+k
4,κ,T?− f0
?2d+k
4
?logT + f?
0
4
??
?2d+k
4,κ?,
0(s) = df0/ds, and the r.h.s. is defined as the limit of Id+k,d(s,κ) after the pole is
ˆId+k,d
4
,κ
?
≡
lim
s→2d+k
4
?
Id+k,d(s,κ) −f0
?2d+k
4
?
s −2d+k
4
?
.(3.17)
– 20 –
Page 22
JHEP06(2012)070
Eqs. (3.15) and (3.16) relate the renormalised integral to the BPS state sum (3.10), or to its
analytic continuation whenever ?(s) > 1. We note that this renormalisation prescription
amounts to subtracting only the infrared divergent contribution of the massless states,
unlike other schemes used in the literature where the full contribution of the massless
states is subtracted. Of course, any two renormalisation schemes differ by an additive
constant independent of the moduli.
Having discussed the analytic properties of the BPSstate sum (3.10), and its relation
to the regulated integral (3.5), let us now evaluate the integral in (3.10). Using the rela
tion (A.16) between the Whittaker Mfunction and the confluent hypergeometric function
1F1, as well as the identity
?∞
0
dtta−1e−z t1F1(b;c;t) = z−aΓ(a)2F1(a,b;c;z−1),(3.18)
we arrive at our main result
Id+k,d(s,κ) =(4πκ)1−d
2Γ(s +2d+k
?
4
− 1)
×
?
BPS
2F1
s −k
4, s +2d + k
4
− 1; 2s;4κ
p2
L
? ?p2
L
4κ
?1−s−2d+k
4
.
(3.19)
The sum in (3.19) converges absolutely for ?(s) >2d+k
to a meromorphic function on ?(s) > 1 with a simple pole at s =2d+k
κ = 1 the sum in (3.19) can be physically interpreted as a sum of the oneloop contributions
of all physical BPS states satisfying the levelmatching condition (3.9). This expression is
manifestly invariant under Tduality, independent of any choice of chamber, and generalises
the constrained Epstein zeta series considered in [20, 21] to the case of a nontrivial elliptic
genus. We would like to stress that these properties follow directly from our approach, as
opposed to the conventional unfolding method, which depends on a choice of chamber to
ensure convergence.
Moreover, using the fact that the lattice partition function satisfies the differential
equation [21]
?∆SO(d+k,d)− 2∆−k/2+1
we find that the BPS state sum (3.19) is an eigenmode of the Laplacian ∆SO(d+k,d)on the
Narain moduli space
4
and can be analytically continued
4
[34]. Again, for
4d(d + k − 2)?Γd+k,d= 0, (3.20)
?∆SO(d,d+k)+
4, the eigenvalue vanishes but the BPS state sum Id+k,d(s,κ) has a pole. After
subtracting the pole, one finds that the renormalised BPS state sum is an almost harmonic
function on the Narain moduli space, namely its image under the Laplacian is a constant
?2d + k
1
16(2d + k − 4s)(2d + k + 4s − 4)?Id+k,d(s,κ) = 0 .
(3.21)
For s =2d+k
∆SO(d,d+k)ˆId+k,d
4
,κ
?
= (1 − d −k
2)f0
?2d + k
4
?
.(3.22)
– 21 –
Page 23
JHEP06(2012)070
3.3Oneloop BPS amplitudes with momentum insertions
Our method carries over straightforwardly to cases where insertions of leftmoving or right
moving momenta appear in the lattice sum, i.e. to modular integrals of the type
?
F
dµ
?
τ−λ/2
2
?
pL,pR
ρ(pL√τ2,pR√τ2) q
1
4p2
L¯ q
1
4p2
R
?
Φ(τ), (3.23)
considered for example in [26, 38]. The term in the square bracket is a modular form of
weight (λ + d +k
2,0), provided that the function ρ(xL, xR) satisfies
?∂2
and that ρ(xL, xR)e−π (x2
upon choosing ρ = τ2p2
derivative D·Γd+k,d(respectively, D·Γd+k,d) of the usual Narain lattice partition function.
The integrand in (3.23) is then modular invariant provided λ + d +k
As usual, expressing the elliptic genus as a linear combination of Niebur Poincar´ e
series, one is left to consider integrals of the form
?
pL,pR
xL− ∂2
xR− 2π (xL∂xL− xR∂xR− λ − d)?ρ(xL, xR) = 0,
L+x2
(3.24)
R)should decay sufficiently fast at infinity [27].12For example,
L−d+k
2π(respectively ρ = τ2p2
R−d
2π), it is proportional to the modular
2= −w.
F
dµτ−λ/2
2
?
ρ(pL√τ2,pR√τ2) q
1
4p2
L¯ q
1
4p2
RF(s,κ,w),(3.25)
and following similar steps as in the previous subsection, one finds the result
(4πκ)1+λ
2?
BPS
?∞
0
dtts+2d+k
4
−21F1
?
s −2λ+2d+k
4
;2s;t
?
ρ
?
pL
√4πκ
√t,
pR
√4πκ
√t
?
e−tp2
L/4κ.
(3.26)
In most applications, ρ is a polynomial in pa
ing (3.18). As a result, each monomial can be evaluated to
L,pb
R, and the integral can be evaluated us
?
F
dµτδ
2
?
pL,pR
pa1
L···paα
Lpb1
R···pbβ
Rq
1
4p2
L¯ q
1
4p2
RF(s,κ,w)⇒ (4πκ)1−δΓ(s +w
2+ δ−1) (3.27)
×
?
BPS
pa1
L···paα
Lpb1
R···pbβ
R2F1
?
s −w
2,s +w
2
+ δ − 1;2s;4κ
p2
L
??p2
L
4κ
?1−s−w
2−δ
,
with δ = (α+β−λ)/2. Clearly, this result is meaningful only when the various monomials
are combined into a solution of (3.24), as required by modular invariance.
3.4BPSstate sum for integer s
For special values of s and w, the hypergeometric function2F1appearing in the BPSstate
sum (3.19) can actually be expressed in terms of elementary functions. For example, for
12We are grateful to J. Manschot for pointing out this reference.
– 22 –
Page 24
JHEP06(2012)070
d = 1 and w = 0,2F1(s,s −1
2,2s;z) = 22s−1(1 +√1 − z)1−2s, and thus
??
√π (16κ)1+nΓ?n +1
pq=κ
I1,1(1 + n,κ) =
√4πκ21+2nΓ?n +1
=1
2
2
??
BPS
??
p2
4κ+
???pR + q R−1??+??pR − q R−1???−1−2n,
L
?
p2
4κ
R
?−1−2n
(3.28)
2
p,q∈Z
with s = 1 + n. For n = 0, this agrees with the expression derived in [20] using the
SelbergPoincar´ e series E(s,κ,w) at s = 0.
More generally, similar simplifications also take place for s = 1 −w
with n a positive integer, which are the special values relevant for representing weak almost
holomorphic modular forms, and are thus of interest for our physical applications. While
it is cumbersome to express2F1directly in terms of elementary functions, it is simpler to
notice that the Whittaker Mfunction appearing in (3.10) reduces to the finite sum (A.33).
As a result, the integral (3.10) reduces to
?∞
=(4πκ)1−d
2+ n = 1 +k
4+ n,
Id+k,d(1+k
4+ n,κ) =
?
BPS
0
dτ2τ
d
2−2+α
2
M1+k
4+n,−k
?Γ?n +d+k
2(−κτ2)e−πτ2(p2
?
?
L+p2
R)/2, (3.29)
2Γ?2(n + 1) +k
22
n!
n
?
m=0
?n
m
?
(−1)m
Γ?n − m +d+k
L/4κ
2
?
×
?
BPS
?p2
L
4κ
?n−m?∞
0
dz z
d
2−m−2
e−z p2
R/4κ− e−z p2
2n+k
?
2
?=0
z?
?!
?
,
where, in going from the first to the second line we have set z = 4πκτ2, and have integrated
by parts n times, and we used the fact that the boundary terms vanish. Although the full
integrand vanishes rapidly enough as z → 0, so that the integral exists, this is not true of
each individual term, unlessd
2− n − 1 > 0. To regulate these unphysical divergences, we
introduce a convergence factor τα
2in the integrand, and evaluate each integral in (3.29) for
large enough α. The desired result is then expressed as the limit
Id+k,d(1+k
4+ n,κ) =(4πκ)1−d
2Γ?2(n + 1) +k
?p2
2n+k/2
?
2
?Γ(n +d+k
2−m−1 + α??p2
?p2
2)
n!
n
?
m=0
?n
m
?m+1−d
2−?−α?
?
(−1)m
Γ?n − m +d+k
2
?
×
?
BPS
L
4κ
?n−m
Γ?d
lim
α→0
?
Γ?d
R
4κ
2−α
−
?=0
2− m − 1 + ? + α?
?!
L
4κ
?1+m−d
.(3.30)
The series (3.30) converges absolutely for n >d
original modular integral. For n ≤d
which nevertheless captures the singularities of the amplitude at points of gauge symme
try enhancement.
2− 1, as a result of the finiteness of the
2− 1, it is a formal (divergent) sum over BPS states,
– 23 –
Page 25
JHEP06(2012)070
For n <d
leading to Id+k,d(s,κ) = I(1)
2− 1, or whenever d is odd, independently of n, the limit α → 0 is trivial,
d+k,d(s,κ) where
I(1)
d+k,d(1 +k
4+ n,κ) =(4πκ)1−d
2Γ(2(n + 1) +k
2)Γ(n +d+k
n!
2)
×
d/2−2
?
?
m=0
?n
2− m − 1??p2
2n+k/2
?
m
?
(−1)m
Γ(n − m +d+k
2)
?
2
BPS
?p2
L
4κ
?n−m
×
Γ?d
R
4κ
?m+1−d
−
?=0
Γ?d
2− m − 1 + ??
?!
?p2
L
4κ
?1+m−d
2−??
.
(3.31)
If d is even and n ≥d
term is still given by (3.31) and the second term is
2−1 one finds Id+k,d(s,κ) = I(1)
d+k,d(s,κ)+I(2)
d+k,d(s,κ) where the first
I(2)
d+k,d(1 +k
4+ n,κ) =(4πκ)1−d
2Γ?2(n + 1) +k
n
?
2n+k/2
?
(−1)m+1−d
Γ(m + 2 −d
1
Γ?m+2−d
k=1k−1is the Nth harmonic number. The combination (3.31) vanishes for
d = 2, the sum over m being void. The results (3.31) and (3.32) allow to write any integral
of the type (1.2) as a formal sum over physical BPS states (which converges absolutely
for n >d
2− 1). In particular, the result is manifestly invariant under the Tduality group
O(d + k,d;Z)
We conclude this subsection with some simple examples for special values of n and k.
For n = 0 the sum over m in (3.29) is void and only few terms contribute to the integral,
corresponding to the various terms in (2.23). When d ?= 2 the limit α → 0 is trivial and
one arrives at the simple expression
?
2
?Γ?n +d+k
n!
(−1)m
Γ?n − m +d+k
Γ?d
?!
?p2
m+1−d/2
?
2
?
?p2
?1+m−d
(3.32)
×
?
BPS
m=d/2−1
?n
m
?
2
?
L
4κ
?n−m
×
−
?=m+2−d/2
2− m − 1 + ??
?m+1−d
?m+1−d
?p2
L
4κ
2−?
+
2
2)
R
4κ
2?
Hm+1−d
??
2− log
?m+1−d
?p2
2−?
R
p2
L
??
−
2
?
?=0
2
?
−p2
4κ
L
Hm+1−d
2−?
,
where HN=?N
Id+k,d(1 +k
4,κ) = (4πκ)1−d
2Γ(2 +k
2)
?
BPS
Γ(d
2− 1)
?p2
R
4κ
?1−d
2
−
k/2
?
?=0
Γ(d
2+ ? − 1)
?!
?p2
L
4κ
?1−d
2−??
.
(3.33)
– 24 –
Page 26
JHEP06(2012)070
When d = 2, the limit α → 0 is subtler and leads to logarithmic contributions. One
obtains, for n = 0, any k,
I2+k,2(1 +k
4,κ) = −Γ(2 +k
2)
?
BPS
log
?p2
R
p2
L
?
+
k/2
?
?=1
1
?
?p2
L
4κ
?−?
, (3.34)
and for k = 0, any n,
I2,2(1+n,κ) =(2n+1)!
n!
?
BPS
?p2
?m
L
4κ
??p2
?n
n
?
?−?
m=0
?n
m
?2??p2
R
p2
L
?m?
Hm− log
?p2
R
p2
L
??
(3.35)
−
m
?
?=0
(−1)?
?
L
4κ
Hm−?− (−1)m
2n
?
?=m+1
Γ(? − m)m!
?!
?p2
L
4κ
?−??
.
As usual, the left and righthanded momenta are defined by
pL,I= (mi+ Ya
pR,i= mi+ Ya
iQa+1
iQa+1
2Ya
2Ya
iYa
iYa
jnj+ (G + B)ijnj, Qa+ Ya
jnj− (G − B)ijnj,
jnj),
(3.36)
with miand nithe KaluzaKlein and winding numbers and Qaare the charge vectors. In
the d = 2 case, and in the absence of Wilson lines, it is often convenient to express them
in terms of the K¨ ahler modulus T and of the complex structure modulus U as
p2
L=
1
T2U2
1
T2U2
??m2− U m1+¯T(n1+ U n2)??2,
p2
R=
??m2− U m1+ T(n1+ U n2)??2.
(3.37)
The relation between these results (for k = 0) and the ‘shifted constrained Epstein
zeta series’ of [20] is discussed in appendix B.
3.5Singularities at points of gauge symmetry enhancement
In addition to keeping Tduality manifest, another advantage of this approach for the
evaluation of oneloop modular integrals is that it allows to easily readoff the singular
ity structure of the amplitudes at point of enhanced gauge symmetry. These points are
characterised by the appearance of extra massless states with pR= 0. Depending on the
dimension of the Narain lattice, as well as on the value of n, the amplitude may diverge
(we refer to this case as real singularity) or one of its derivatives can be discontinuous (we
refer to this case as conical singularity).
For odd dimension d, the modular integral Id+k,d(s,κ) always develops conical singu
larities, as exemplified in the onedimensional case by eq. (3.28). In addition, for d ≥ 3
real singularities appear from terms with m <d
For even dimension real singularities always appear. They are are powerlike in I(1)
whenever d ≥ 4 and logarithmic in I(2)for any even d ≤ 2n + 2. Moreover, conical
singularities do not appear.
2− 1 in (3.30).
– 25 –
Page 27
JHEP06(2012)070
Notice that for d = 2 the singularities cancel in the combination
I2,2(1 + n,1) −(2n + 1)!
n!
ˆI2,2(1,1), (3.38)
which is therefore a continuous function over the Narain moduli space, including at points of
enhanced gauge symmetry. Since, using the results in [10, 18] and the fact that F(1,1,0) =
j + 24,
ˆI2,2(1,1) = −logj(T) − j(U)4− 24 log?T2U2η(T)η(U)4?+ const,
we conclude that all integrals I2,2(1 + n,1) exhibit the same universal singular behaviour
up to an overall normalisation
(3.39)
I2,2(1 + n,1) ∼ −(2n + 1)!
n!
logj(T) − j(U)4. (3.40)
This expression can be generalised as in [18] if Wilson lines are turned on.
4Some examples from string threshold computations
In this section we evaluate a sample of modular integrals that enter in threshold corrections
to gauge and gravitational couplings in heterotic string vacua using the method developed
in the previous section. We express the elliptic genus as a linear combination of Niebur
Poincar´ e series, and we evaluate the modular integral in terms of the BPSstate sums
Id+k,d(s,κ) defined in eqs. (3.19) and (3.30).
4.1 A gravitational coupling in maximally supersymmetric heterotic vacua
Let us start with the example of toroidally compactified SO(32) heterotic string, for which
the elliptic genus takes the form (3.1). Using table 3 and the relation E3
this can be conveniently expressed in terms of the NieburPoincar´ e series as
4∆−1= j + 744,
Φ(τ) =t8trF4+
1
27325[F(1,1,0) + 720] t8trR4
5F(3,1,0) − 4F(2,1,0) + 13F(1,1,0) + 144?t8(trR2)2
?−1
?1
Therefore, using the results in the previous section, the renormalised modular integral (1.2)
can be expressed as the linear combination
?
+
2932
1
2832
1
2932
+
1
2932
1
2832
1
2932
?1
+
5F(3,1,0) + 5F(2,1,0) − 18F(1,1,0) + 288?t8trR2trF2
+
5F(3,1,0) − 6F(2,1,0) + 24F(1,1,0) − 576?t8(trF2)2.
(4.1)
R.N.
F
dµΓd,dΦ =Id,dt8trF4+
1
27325[Id,d(1,1) + 720Id,d] t8trR4
(4.2)
1
?1
?−1
?1
5Id,d(3,1) − 4Id,d(2,1) + 13Id,d(1,1) + 144Id,d
?t8(trR2)2
?t8trR2trF2
?t8(trF2)2,
+
5Id,d(3,1)+5Id,d(2,1)−18Id,d(1,1)+288Id,d
+
5Id,d(3,1) − 6Id,d(2,1) + 24Id,d(1,1) − 576Id,d
– 26 –
Page 28
JHEP06(2012)070
where, as computed in [20, 21],
Id,d≡ R.N.
?
F
dµΓd,d(G,B) =Γ?d
2− 1?
π
d
2−1
Ed
V
?G,B;d
2− 1?, (4.3)
with Ed
any time n =d
as explained in section 3.2.
In the onedimensional case the constrained sums can be easily evaluated, leading to
?
+
2532
+
233
−
233
Vbeing the constrained Epstein zeta function defined in [20, 21]. In this expression,
2− 1 the BPSstate sum Id,d(1 + n,κ) should be replaced byˆId,d(1 + n,κ),
F
dµΓ1,1Φ =π
3(R + R−1)t8trF4+
π
?3R + 16R−1− 24R−3+ 12R−5?t8(trR2)2
π
π
23325
?15R + 16R−1?t8trR4
?R − 2R−1+ 5R−3− 2R−5?t8trR2trF2
π
?R − R−1+ 3R−3− R−5?t8(trF2)2,
(4.4)
for R > 1.
expression R ?→ R−1. Notice that, aside from the threshold correction to t8trF4, all other
terms develop a conical singularity at the selfdual radius R = 1.
The expression for R < 1 can be obtained by replacing in the previous
4.2 Gaugethresholds in N = 2 heterotic vacua with/without Wilson lines
Let us turn now to N = 2 heterotic vacua in the orbifold limit T2×T4/Z2, with a standard
embedding on the gauge sector. At the orbifold point, the gauge group is broken to
E8× E8→ E8× E7× SU(2),(4.5)
and, in the absence of Wilson lines, gauge threshold corrections read
∆E8= −1
12
?
?
F
dµΓ2,2
ˆE2E4E6− E2
∆
ˆE2E4E6− E3
∆
6
,
∆E7= −1
12
F
dµΓ2,2
4
.
(4.6)
From table 3 one can read that
ˆE2E4E6− E2
∆
ˆE2E4E6− E3
∆
6
= F(2,1,0) − 6F(1,1,0) + 864,
4
= F(2,1,0) − 6F(1,1,0) − 864,
(4.7)
and thus
∆E8=
?
?
BPS
?
?
1 +p2
R
4
log
?p2
?p2
R
p2
L
??
??
+ 72 log
?
?
T2U2η(T)η(U)4?
T2U2η(T)η(U)4?
+ const,
∆E7=
BPS
1 +p2
R
4
log
R
p2
L
− 72 log+ const
(4.8)
– 27 –
Page 29
JHEP06(2012)070
Notice that the combination I2(2,1,0)−6I2(1,1,0) is regular at any point in moduli space
(and in any chamber), as expected since the unphysical tachyon is neutral and therefore
does not contribute to the running of the nonAbelian gauge couplings.
Turning on Wilson lines on the E8group factor along the spectator T2, yields
?
Using table 3, one easily finds
∆E7= −1
12
F
dµΓ2,10
ˆE2E6− E2
∆
4
. (4.9)
ˆE2E6− E2
∆
4
=2
7!F(4,1,−4) −2
5!F(3,1,−4),(4.10)
and thus
∆E7= −
1
720
?1
1 +p2
42I10,2(4,1) − I10,2(3,1)
?p2
L
?
8
3p4
=
?
BPS
?
R
4
log
R
p2
?
−
2
p2
L
−
L
−
16
3p6
L
−
64
5p8
L
?
.
(4.11)
In this expression pL,Rdepend also on the Wilson lines, and the constraint in the BPSsum
now reads
p2
R= 4
⇒
where Q is the U(1)charge vector in the Cartan subalgebra of E8.
L− p2
mTn +1
2QTQ = 1, (4.12)
4.3K¨ ahler metric corrections in N = 2 heterotic vacua
Our procedure can also be used to compute loop corrections to K¨ ahler metric and other
terms in the lowenergy effective action. For instance, in N = 2 heterotic vacua at the
orbifold point, the oneloop correction to the K¨ ahler metric for the T modulus reads
???1−loop=
=
72π T2
2
F
where we have used the relation between E4E6∆−1and F(s,κ,w) from table 1. Integrating
by parts, and using the action of the modular derivative on the NieburPoincar´ e series, one
immediately finds
???1−loop= −
=
36T2
2
F
=
36T2
2
KT¯T
i
12π T2
i
2
?
?
F
dµE4E6
∆
∂τΓ2,2
dµF(2,1,−2)∂τΓ2,2,
(4.13)
KT¯T
i
72π T2
1
2
?
dµΓ2,2F(2,1,0)
F
dµΓ2,2D−2F(2,1,−2)
?
I2,2(2,1).
1
(4.14)
Similar results can be obtained for higherderivative couplings in N = 4 vacua.
– 28 –
Page 30
JHEP06(2012)070
4.4An example from noncompact heterotic vacua
In some heterotic constructions on ALE spaces and in the presence of background NS5
branes, gauge threshold corrections include a contribution of the (finite) integral [45]
L =
?
F
dµ(√τ2η ¯ η)3
ˆE2E4
?ˆE2E4− 2E6
∆
?
. (4.15)
Despite its apparent complexity, this integral can be easily computed using our techniques.
In fact, the relation
ϑ?
1(0τ) = 2πη3, (4.16)
and the standard bosonisation formulae, allow one to write
(√τ2η¯ η)3= −1
8π
∂
∂R
?1
R(Γ1,1(2R) − Γ1,1(R))
?
R=1/√2
.(4.17)
Combining this observation with table 3, eq. (3.28), and with the standard result
?
L = −
8π ∂RR
F
×?1
= −
8π ∂RR
FdµΓ1,1(R) =π
3(R + R−1), the integral reduces to
?1
5F(3,1,0) − 6F(2,1,0) + 23F(1,1,0) + 432??
1∂1
?
−?1
= − 20√2.
1∂
?
dµ (Γ1,1(2R) − Γ1,1(R))
R=1/√2
?
1
5I1,1(2R;3,1) − 6I1,1(2R;2,1) + 23I1,1(2R;1,1)
5I1,1(R;3,1) − 6I1,1(R;2,1) + 23I1,1(R;1,1)?+ 144π?R −1
2R−1???
R=1/√2
(4.18)
Acknowledgments
We are grateful to S. Hohenegger, J. Manschot, S. Murthy for interesting comments and
discussions. B. P. is grateful to R. Bruggeman, J. H. Bruinier, W. Pribitkin and´A. T´ oth for
helpful remarks during the Krupp Symposium on Modular Forms, Mock Theta Functions
and Applications in K¨ oln, Feb 27  March 1, 2012. C.A and I.F. would like to thank the TH
Unit at CERN for hospitality while this project was in progress. This work was partially
supported by the European ERC Advanced Grant no. 226455 “Supersymmetry, Quantum
Gravity and Gauge Fields” (SUPERFIELDS) and by the Italian MIURPRIN contract
2009KHZKRX007 “Symmetries of the Universe and of the Fundamental Interactions”.
A Notations and useful identities
In this appendix we collect various definitions, notations and formulae used in the text.
– 29 –
Page 31
JHEP06(2012)070
A.1Operators acting on modular forms
The hyperbolic Laplacian acts on modular forms of weight w via13
∆w= 2τ2
2∂¯ τ
?
∂τ−iw
2τ2
?
. (A.1)
We denote by H(s,w) the eigenspace of ∆wwith eigenvalue1
space of real analytic functions of modular weight w under Γ = SL(2,Z). The raising and
lowering operators Dw,¯Dwdefined by
?
map H(s,w) to H(s,w ± 2),
¯ Dw
←− H(s,w)
2s(s−1)−1
8w(w +2), in the
Dw=i
π
∂τ−iw
2τ2
?
,
¯Dw= −iπ τ2
2∂¯ τ, (A.2)
H(s,w − 2)
Dw
−→ H(s,w + 2), (A.3)
and satisfy the commutation identity
Dw−2·¯Dw−¯Dw+2· Dw=w
4.(A.4)
The operator Dw(and of course,¯Dw) satisfies the Leibniz rule
Dw+w? (fwfw?) = (Dwfw)fw? + fw(Dw? fw?), (A.5)
where fwis a modular form of weight w. We denote by Dr
derivative Dw+2r−2· ... · Dw+2· Dw· fw, a modular form of weight w + 2r. One has
?i
j=0
wfw(or simply Drf) the iterated
Dr
w=
π
?r
r
?
r!
j!(r − j)!
Γ(w + r)
Γ(w + j)(2iτ2)j−r∂j
τ. (A.6)
For w ≤ 0, the operator D1−w
the physics literature as the Farey transform [42].
The Hecke operators Tκare defined by
w
simplifies to (i/π)1−w∂1−w
τ
(Bol’s identity), and is known in
(Tκ· Φ)(τ) =
?
a,d>0,ad=κ
?
b mod d
d−wΦ
?aτ + b
d
?
, (A.7)
and satisfy the commutative algebra
TκTκ? =
?
d(κ,κ?)
d1−wTκκ?/d2 ,(A.8)
If Φ =?
n∈ZΦ(n,τ2)e2πnτ1is a modular form of weight w, then the Fourier coefficients of
Tκ· Φ are
(Tκ· Φ)(n,τ2) = κ1−w
d(n,κ)
?
dw−1Φ?nκ/d2,d2τ2/κ?.
2∆w;BO−w
(A.9)
13Our Laplacian is related to the one used e.g in [36] via ∆w = −1
2.
– 30 –
Page 32
JHEP06(2012)070
The generators of the ring of holomorphic modular forms are the normalised Eisen
stein series
E4= 1 + 240
∞
?
n=0
n3qn
1 − qn
andE6= 1 − 504
∞
?
n=0
n5qn
1 − qn, (A.10)
with modular weight 4 and 6, respectively. The discriminant function is the weight 12
cusp form
∞
?
The generators of the ring of weak holomorphic modular forms are E4,E6and 1/∆. The
modular jinvariant is the unique weak holomorphic modular form of weight zero with
j = 1/q + O(q),
j =E3
∆ = q
n=1
(1 − qn)24=
1
1728(E3
4− E2
6). (A.11)
4
∆− 744 =E2
6
∆+ 984 .(A.12)
The ring of weak almost holomorphic modular forms is obtained by adding to E4,E6,1/∆
the almost holomorphic Eisenstein series
ˆE2= E2−
3
πτ2
= 1 − 24
∞
?
n=0
nqn
1 − qn−
3
πτ2
. (A.13)
Under the raising operator Dwone has
DˆE2=1
6(E4−ˆE2
2), DE4=2
3(E6−ˆE2E4), DE6= E2
4−ˆE2E6, D(1/∆) = 2ˆE2/∆,
(A.14)
where, for simplicity, we have left implicit the specification of the weight in D. Using the
Leibniz rule (A.5), this allows to compute the action of D on any weak almost holomorphic
modular form.
Finally, the operator Tκmaps the weak holomorphic modular form Φ = 1/q +O(q) to
TκΦ = 1/qκ+ O(q).
A.2Whittaker and hypergeometric functions
Whittaker functions and hypergeometric functions, more in general, are central in the
analysis of the NieburPoincar´ e series and the evaluation of oneloop modular integrals.
We summarise here their definitions and some of their main properties.
Whittaker functions are solutions of the secondorder differential equation
?
u??+
−1
4+λ
z+
1
4− µ2
z2
?
u = 0.(A.15)
For 2µ not integer the two independent solutions are given by the Whittaker Mfunctions
Mλ,±µ(z) = e−z/2z±µ+1
21F1
?±µ − λ +1
2;1 ± 2µ;z?,(A.16)
– 31 –
Page 33
JHEP06(2012)070
and are expressed in terms of the confluent hypergeometric function
1F1(a;b;z) =Γ(b)
Γ(a)
∞
?
n=0
Γ(a + n)
Γ(b + n)
zn
n!.(A.17)
When 2µ is an integer, however, the second solution is not defined anymore, and thus it is
useful to introduce a second Whittaker function defined by
?(0+)
where arg(−t) ≤ π, and the contour does not contain the point t = −z and circles the ori
gin counterclockwise. The functions Wλ,µ(z) and W−λ,µ(−z) are then the two independent
solutions of the differential equation (A.15), and have the asymptotic behaviour
Wλ,µ(z)=−1
2πiΓ(λ +1
2− µ)e−z/2zλ
∞
(−t)−λ−1
2+µ
?
1 +t
z
?λ−1
2+µ
e−tdt,(A.18)
Ws,w(y) ∼ 4πy
w
2(sgn(y)−1)e−2πy
as
y → ∞. (A.19)
The Whittaker Mfunction can then be expressed as the linear combination
Mλ,µ(z) =
Γ(2µ + 1)
Γ?µ − λ +1
2
?eiπλW−λ,µ(eiπz) +
Γ(2µ + 1)
Γ?µ + λ +1
2
?eiπ(λ−µ−1
2)Wλ,µ(z).(A.20)
Using the symmetry of the Wfunctions, Wλ,µ(z) = Wλ,−µ(z), one can invert the previous
relation and write
Wλ,µ(z) =
Γ(−2µ)
Γ?1
2− µ − λ? Mλ,µ(z) +
Γ(2µ)
2+ µ − λ? Mλ,−µ(z).
Γ?1
(A.21)
This implies that the functions Ms,wand Ws,wobey
Γ(1 − 2s)
Γ?1 − s −w
For special values of λ and µ, the Whittaker functions reduce to elementary functions
or to other special functions.This derives from the properties of the hypergeometric
functions, for instance
Ws,w(y) =
2sgn(y)? Ms,w(y) +
Γ(2s − 1)
Γ?s −w
2sgn(y)? M1−s,w(y). (A.22)
1F1(a;a;z) = ez,
1F1(1,a,z) = (a − 1)z1−aezγ(a − 1,z),
1F1(a,a + 1,z) = a(−z)−aγ(a,−z),
1F1(a;2a;z) = ez/2?1
?z
is the incomplete Gamma function, and
4z?1
2−aΓ?a +1
2
?Ia−1
2(z/2),
(A.23)
where
γ(a,z) =
0
e−tta−1dt = Γ(a) − Γ(a,z)(A.24)
Iν(z) =
∞
?
m=0
(z/2)2m+ν
m!Γ(m + ν + 1)= i−νJν(iz)(A.25)
is the modified Bessel function and Jν(z) is the Bessel function of the first kind.
– 32 –
Page 34
JHEP06(2012)070
Given the definitions (A.16), (2.7) and
Kν(z) =π
2
I−ν(z) − Iν(z)
sin πν
,(A.26)
one can then show that (y > 0)
Ms,0(±y) = 22s−1Γ(s +1
Ws,0(±y) = 2y
2,w(−y) = e2πy,
Ww
Ww
2)(4πy)
1
2Is−1
2(2πy),
1
2Ks−1
2(2πy), (A.27)
Mw
2,w(−y) = Γ(1 − w,4πy)e2πy,
2,w(y) = e−2πy, (A.28)
M−w
W−w
W−w
2,w(−y) = (4πy)we−2πy,
2,w(−y) = (4πy)−we−2πy,
2,w(y) = (4πy)−wΓ(1 + w,4πy)e2πy, (A.29)
M1−w
W1−w
W1−w
2,w(−y) = (1 − w)γ(1 − w,4πy)e2πy,
2,w(−y) = Γ(1 − w,4πy)e2πy,
2,w(y) = e−2πy.(A.30)
For integer values of the arguments, the confluent hypergeometric function, and thus
the Whittaker functions, take a particularly simple expression
?
= Γ(a)z1−a?
where
L(k)
n!
are the associated Laguerre polynomials.
As a result, the seed function that enters in the definition of the NieburPoincar´ e series
involves only a finite number of terms when s = 1 −w
1F1(n + 1,a,z) =Γ(a)
n!
dn
dzn
zn+1−a
?
ez−
a−2
?
k=0
zk
k!
??
ezL(1−a)
n
(−z) − L(1−a)
a−2−n(z)
?
,
(A.31)
n(x) =x−kex
dn
dxn
?
xn+ke−x?
(A.32)
2+ n, and reads
M1−w
2+n,w(−y) =(4πy)1−w+ne−2πy(2n + 1 − w)!
dn
d(4πy)n
n!
×
?
(4πy)w−n−1
?
e4πy−
2n−w
?
k=0
(4πy)k
k!
??
=Γ(2n + 2 − w)(4πy)−n
×
?
e2πyL(−1−2n+w)
n
(−4πy) − e−2πyL(−1−2n+w)
n−w
(4πy)
?
.
(A.33)
– 33 –
Page 35
JHEP06(2012)070
Similarly,
W1−w
W1−w
The modular derivatives (A.2) have a natural action on the Whittaker functions,
so that
Dw·?Ms,w(−κτ2)e−2πiκτ1?= 2κ(s +w
1
8κ(s −w
and
2+n,w(y) =(−1)nn!(4πy)−ne−2πyL(−1−2n+w)
2+n,w(−y) =(−1)n−wΓ(n − w + 1)(4πy)−ne2πyL(−1−2n+w)
n
(4πy),
n−w
(−4πy).
(A.34)
2)Ms,w+2(−κτ2)e−2πiκτ1,
2)Ms,w−2(−κτ2)e−2πiκτ1,
¯Dw·?Ms,w(−κτ2)e−2πiκτ1?=
(A.35)
Dw·?Ws,w(nτ2)e2πinτ1?= Ws,w+2(nτ2)e2πinτ1×
¯Dw·?Ws,w(nτ2)e2πinτ1?= Ws,w−2(nτ2)e2πinτ1×
?
?
−2n,
2n?s +w
1
8n
−1
n > 0,
2
??s −w
??s +w
2− 1?,
2− 1?
n < 0,
n < 0,
8n
?s −w
2
n > 0.
(A.36)
A.3KloostermanSelberg zeta function
The KloostermanSelberg zeta function entering in the expression (2.16) of the Fourier
coefficients of the NieburPoincar´ e series is defined as [43]
where Is(x) and Js(x) are the Bessel I and J functions, and S(a,b;c) are the classical
Kloosterman sums for the modular group Γ = SL(2,Z),
?
Here a, b and c are integers, and d−1is the inverse of d mod c. S(a,b;c) is clearly symmetric
under the exchange of a and b. Less evidently, it satisfies the Selberg identity
?
In the special case a ?= 0, b = 0, the Kloosterman sum reduces to the Ramanujan sum
?
with µ(n) the M¨ obius function. For a = b = 0, S(a,b;c) reduces instead to the Euler
totient function φ(c), and one can verify that
?
Zs(a,b) =
1
2?ab
?
c>0
S(a,b;c)
c
×
J2s−1
I2s−1
?
4π
c
√ab
√−ab?
?
ifab > 0,
?4π
c
ifab < 0,
(A.37)
S(a,b;c) =
d∈(Z/cZ)∗
exp
?2πi
c
(ad + bd−1)
?
. (A.38)
S(a,b;c) =
dgcd(a,b,c)
dS(ab/d2,1;c/d).(A.39)
S(a,0;c) = S(0,a;c) =
d∈(Z/cZ)∗
exp
?2πi
c
ad
?
=
?
dgcd(c,a)
dµ(c/d),(A.40)
c>0
S(0,0;c)
c2s
=ζ(2s − 1)
ζ(2s)
,
?
c>0
S(0,±κ;c)
c2s
=σ1−2s(κ)
ζ(2s)
(κ ?= 0), (A.41)
– 34 –
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JHEP06(2012)070
with σx(n) the divisor function. Under complex conjugation, Zs(a,b) transforms as
Zs(a,b) = Z¯ s(−a,−b) .(A.42)
The KloostermanSelberg zeta function defined in (A.37) is related to the zeta function
Z(a,b;s) ≡
?
c>0
S(a,b;c)
c2s
(A.43)
originally considered in [31] and used in [20] via
Zs(a,b) = π (4π2ab)s−1
∞
?
m=0
(−4π2ab)m
m!Γ(2s + m)Z(a,b;s + m).(A.44)
BSelbergPoincar´ e series vs. NieburPoincar´ e series
In this section, we briefly discuss the relation between the NieburPoincar´ e series (2.8)
and the SelbergPoincar´ e series (2.5), considered in our previous work [20] in the special
case w = 0, as well as the relation between the BPSstate sum (3.19) and the “shifted
constrained Epstein zeta series” considered in [20].
Comparing the differential equations (2.10) and (2.6), it is easily seen that a set of
solutions of one can be converted into a set of solutions of the other by considering the
linear combinations [33, 44]
?
E(s,κ,w) =b(s,κ,w,m)F(s + m,κ,w),
F(s,κ,w) =
m≥0
?
a(s,κ,w,m)E(s + m,κ,w),
m≥0
(B.1)
such that the coefficients satisfy the recursion relations
= −4πκ?s + m −w
a(s,κ,w,m + 1)
a(s,κ,w,m)
2
?
(m + 1)(m + 2s),
b(s,κ,w,m + 1)
b(s + 1,κ,w,m)=
4πκ?s −w
2
?
(m + 1)(m + 2s). (B.2)
Comparing also the constant term (2.15) of the NieburPoincar´ e series and the con
stant term
˜E0(s,κ,w) =
∞
?
m=0
22(1−s)π i−w(πκ)mΓ(2s + m − 1)σ1−2s−2m(κ)
m!Γ?w
of the SelbergPoincar´ e series, we find that the coefficients are given by
2+ s + m?Γ?s −w
2
?ζ(2s + 2m)
τ
1−s−m−w
2
2
,(B.3)
a(s,κ,w,m) = (−1)m22s+2m−w(πκ)s−w
b(s,κ,w,m) =2w−2s(πκ)−s+w
2+mΓ(2s)Γ?s + m −w
2 Γ(2s + m − 1)Γ?s + m −w
2
?
m!Γ(2s + m)Γ?s −w
m!Γ(2s + 2m − 1)Γ?s −w
2
?
,
2
?
2
?
.
(B.4)
– 35 –
Page 37
JHEP06(2012)070
In particular, in the limit s →w
becomes holomorphic, one finds E(s,κ,w) = F(w
2where the summand of the SelbergPoincar´ e series (2.5)
2,κ,w) for w ≥ 2, but
E?w
2,κ,w?=F?w
2,κ,w?+
−w
?
2−1
m=1
b?
mRess=w
2+mF(s,κ,w)+
1−w
?
m=−w
2+1
bmF?w
2+ m,κ,w?
(B.5)
for w ≤ 0, where bm ≡ lims→w
ing (B.5), we have assumed that the singularities of F(s,κ,w) on the real saxis can be
read off from the constant term (2.15), namely that F(s,κ,w) is regular at integer values
of s provided s ≥ 0 or s ≤ −w/2, and has simple poles for integer values of s such that
−w
tional to the harmonic Maass form F(1 −w
the present work, but is contaminated by other NieburPoincar´ e series lying outside the
convergence domain ?(s) > 1. For example, for w = 0, we find
2b(s,κ,w,m) and b?m≡ lims→w
2
d
dsb(s,κ,w,m). In writ
2< s < 0. In particular, E(w
2,κ,w) receives a contribution (for m = 1 − w) propor
2,κ,w), which is the main object of interest in
E(0,κ,0) = F(0,κ,0) +1
2F(1,κ,0), (B.6)
consistently with the identifications
E(0,κ,0) = Tκj + 12σ(κ),
F(1,κ,0) = Tκj + 24σ(κ),
F(0,κ,0) =1
2Tκj .(B.7)
Moreover, the relation (B.1) between the NieburPoincar´ e and SelbergPoincar´ e series im
plies a similar relation between the BPSstate sum (3.19) and the “shifted constrained
Epstein zeta series”
Ed
V(G,B,Y ;s,κ) ≡ 2s?
BPS
(p2
L+ p2
R− 4κ)−s=
?
BPS
(p2
R)−s
(B.8)
generalising the constructio in [20] to the case k ?= 0. Namely, using the same techniques
as in our previous paper, one may show that (B.8) arises from the modular integral
lim
T →∞
?
FT
dµΓd+k,d(G,B,Y )E?s,κ,−k
Since the BPSstate sum Id+k,d(s,κ) arises from the limit T → ∞ of the integral (3.15),
from (B.1) we conclude that
2
?=Γ?s +2d+k
4
− 1?
πs+2d+k
4
−1
Ed
V
?G,B,Y ;s +2s+k
4
− 1,κ?.
(B.9)
Ed
V
?G,B,Y ;s +2d+k
for large ?(s).
4
− 1,κ?=
πs+2d+k
Γ?s +2d+k
4
−1
4
− 1?
?
m≥0
b(s,κ,w,m)Id+k,d(s+m,κ), (B.10)
Open Access.
Attribution License which permits any use, distribution and reproduction in any medium,
provided the original author(s) and source are credited.
This article is distributed under the terms of the Creative Commons
– 36 –
Page 38
JHEP06(2012)070
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