Rankin-Selberg methods for closed string amplitudes

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by  Boris Pioline

Rankin-Selberg methods for closed string amplitudes
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1
0
Boris Pioline
2

l
u Abstract. Afterintegratingoversupermoduliandvertexoperatorpositions,
J scattering amplitudes insuperstringtheory atgenus h≤3arereduced toan

0 integralofaSiegelmodularfunctionofdegreehonafundamental domainof
1 the Siegelupper halfplane. Adirectcomputation isingeneral unwieldy, but
becomesfeasibleiftheintegrandcanbeexpressedasasumoverimagesunder

] asuitablesubgroupoftheSiegelmodulargroup: ifso,theintegrationdomain
h canbeextendedtoasimplerdomainattheexpenseofkeepingasingletermin
t eachorbit–atechnique knownastheRankin-Selbergmethod. Motivatedby
-
p applications to BPS-saturated amplitudes, Angelantonj, Florakis and I have
e appliedthistechniquetoone-loopmodularintegralswheretheintegrandisthe
h productofaSiegel-Narainthetafunctiontimesaweakly,almostholomorphic
[ modular form. I survey our main results, and take some steps in extending

thismethodtogenus greaterthanone.
2
v
5
6
2 1. Introduction
4
. Accordingtothebasicpostulateofsuperstringtheory,scatteringamplitudesof
1
nexternalstatesath-thorderinperturbationtheoryaregivenbycorrelationfunc-
0
tions of n vertex operators,integrated over the moduli space M of n-punctured
4 h,n
1 super-Riemann surfaces Σ of genus h [1, 2]. After integrating over the fermionic
: moduli and the locations of the punctures,1 such amplitudes reduce to an integral
v
over the moduli space of ordinary Riemann surfaces without marked point.
i h,0
X The latter is a quotienMt of the Teichmu¨ller space by the mapping class group
h
T
r Γ . For 1 h 3, is isomorphic (via the period map, Σ Ω Ω +iΩ , away
a h ≤ ≤ Th 7→ ≡ 1 2
from suitable divisors) to the degree h Siegel upper-half plane , while Γ is
h h
identifiedwiththeSiegelmodulargroupSp(h,Z),actingon byHfractionallinear
h
H
transformations. Thus, scattering amplitudes at genus h are ultimately written as
modular integrals
h+1
(1.1) h =R.N. dµhFh(Ω) , dµh = Ω2 − 2 dΩ1dΩ2
A | |
ZFh
where = Γ is a fundamental domain of the action of Γ on , F(Ω) is
h h h h h
F \H H
a function on invariant under Γ , Ω = det[ (Ω)] > 0, dµ is the standard
h h 2 h
H | | ℑ
Contribution to the proceedings of the String-Math conference, June 17-21, 2013, Simons
CenterforGeometryofPhysics,StonyBrook. CERN-PH-TH/2014-007,arXiv:1401.4265v2.
1For h > 2 there is no canonical way of performing these integrations, due in part to the
non-projectivenessofsupermodulispace[3, 4],butdifferentprescriptionsareexpectedtoleadto
thesameintegrandonMh,0,uptototalderivatives.
1
2 BORISPIOLINE
invariant measure on , and R.N. denotes a suitable infrared renormalization
h
H
prescription. For h > 3, the Teichmu¨ller space embeds as a subvariety of codi-
h
T
mensionone or greaterin the Siegelupper-half plane , anddµ is replacedby a
h h
H
suitablemeasurewithsupporton –a complicationwhichweshallnotconfront.
h
F
In general,integralsof the type (1.1) are untractable (except, perhaps, numer-
ically), due to the complicated nature of the modular function F , but also to the
h
unwieldy shape of any fundamental domain: for h = 2, it takes no less than 25
inequalities to define ! [5]Yet, the computation ofsuch integralsis anunavoid-
2
F
able step in extracting any phenomenologicalprediction of superstring theory, and
ininvestigatingsomeofits structuralpropertiessuchasinvarianceunderdualities.
The daunting task of integrating (1.1), however, can be considerably simplified if
the integrand can be written as a sum over images (or Poincar´e series) of a given
function f (Ω),
h
(1.2) F (Ω)= f (Ω)
h h γ
|
γ∈ΓXh,∞\Γh
wheref (Ω)=f (γ Ω)andf (Ω)isinvariantunderasubgroupΓ Γ . Ifthe
h γ h h h,∞ h
| · ⊂
sumisabsolutelyconvergent,exchangingitwiththeintegralextendstheintegration
domaintoalargerfundamentaldomain =Γ ,whilerestrictingthesum
h,∞ ∞,h h
F \H
to a single coset,
(1.3) =R.N. dµ f (Ω) .
h h h
A
ZF∞,h
This‘unfoldingtrick’,attheheartoftheRankin-Selbergmethodinnumbertheory,
is expedient if both and f are simpler than and F .
h,∞ h h h
F F
Aspecialclassofamplitudeswherethismethodisadvantageousariseswhenthe
integrand F (Ω) factorizes as Φ(Ω) Γ , where Φ(Ω) is a (non-holomorphic,
h d+k,d,h
×
in general) Siegel modular form of weight k/2 and Γ is the Siegel-Narain
d+k,d,h
−
theta series
(1.4) Γd+k,d,h(G,B,Y;Ω)= Ω2 d/2 e−πΩ2,αβM2(pα,pβ)+2πiΩ1,αβhpα,pβi
| |
pα∈Λ
X
where the sum runs over h-tuples of vectors pα, α = 1...h in an even self-dual
latticeofsignature(d+k,d)(hencek 8Z)withquadraticform , 2Zequipped
∈ h· ·i∈
with positive definite quadratic form 2(, ). The latter is parametrized by the
M · ·
Grassmannian
(1.5) G =[SO(d+k) SO(d)] SO(d+k,d) ,
d+k,d
× \
whichcanbe coordinatizedby a realpositive definite symmetric matrix G ,a real
ij
antisymmetric matrix B and a real rectangular matrix Y (i,j = 1...d,a =
ij i,a
1...k). This type of theta series arises in any string vacua which involve a d-
dimensional torus with constant metric G and Kalb-Ramond field B , equipped
ij ij
withaU(1)kflatconnectionwithholonomiesY . Importantly,Γ isinvariant
i,a d+k,d,h
under Γ O(d+k,d,Z), where the last factor is the automorphism group of the
h
×
lattice Λ (also known as T-duality group), acting by right-multiplication on the
coset (1.5).
In such cases, the integral
(1.6) =R.N. dµ Γ (G,B,Y;Ω)Φ(Ω) .
h h d+k,d,h
A
ZFh
RANKIN-SELBERG METHODS FOR CLOSED STRING AMPLITUDES 3
can be computed by expressing Γ as a sum of Poincar´eseries under Γ , and
d+k,d,h h
applyingthe unfolding tricktoeachoneofthem. This‘lattice unfoldingtechnique’
has been the method of choice for one-loop amplitudes in the physics literature
[6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17],andhasbeenveryusefulinextracting
asymptoticexpansionsat particularboundarycomponents ofG . However,its
d+k,d
main drawback is that the various terms in the Poincar´e series decomposition are
notinvariantunder SO(d+k,d,Z), eventhoughthe sumis. As a result,the result
of the unfolding trick is not manifestly invariant under SO(d+k,d,Z).
Another option, advocatedin [18]2 and further developed in [21, 22] (see [23]
for a complementary survey), is to represent the other factor Φ(Ω) in the inte-
grand (or the full integrand, in the absence of any Siegel-Narain theta series) as
a Poincar´e series, and use it to unfold the integration domain. This technique is
of course limited by our ability to find absolutely convergent Poincar´e series rep-
resentations for (in general, non-holomorphic) Siegel modular forms. For genus
one, it turns out that any almost, weakly holomorphic modular form of negative
weight under Γ =SL(2,Z) (or congruence subgroups thereof) can be represented
1
as a linear combination of certain absolutely convergent Poincar´e series, first in-
troduced by Niebur [24] and Hejhal [25] and revived in recent mathematical work
[26, 27, 28, 29]. Almost, weakly holomorphic integrands Φ(Ω) are non-generic,
but do occur for certain classes of ’BPS-saturated’ amplitudes, which play a cen-
tral role for determining threshold corrections to gauge couplings and for testing
non-perturbative dualities (see e.g. [30] for a review). As we shall see, the unfold-
ing trick produces a sum over lattice vectors with fixed integer norm, manifestly
invariantunder T-duality. Physically,itcanbe interpretedasa sumoffield-theory
type amplitudes,with BPSstatesrunning inthe loop. Inparticular,itexposesthe
singularitiesoftheamplitude,originatingfromBPSstatesbecomingmassless. The
pricetopayisthatthebehaviorattheboundarycomponentsisobscured,although
itcanberecoveredinsomecases,showingagreementwith–anduncoveringhidden
structure in – the result of the usual lattice unfolding technique.
A third, and most radical option, is to represent 1 as a Poincar´e series, and
use it to unfold the integral. Indeed, it is well-known that 1 is a residue of non-
holomorphic Siegel-Eisenstein series, the simplest conceivable example of Poincar´e
series. This trick, which we refer to as the Rankin-Selberg-Zagier method, is very
usefultoevaluateintegralsofthe type (1.6)withΦ=1(hence k =0). Itexpresses
the result, for any genus, as a Langlands-Eisenstein series of SO(d,d,Z), verifying
a conjecture put forward in [31].
We begin our survey of various applications of the Rankin-Selberg method
(or unfolding trick) to closed string amplitudes in 2 by computing one-loop in-
§
tegrals of symmetric lattice partition functions by means of a non-holomorphic
Eisenstein series insertion. In 3 we move on to more general modular integrals of
§
non-symmetriclatticepartitionfunctionsagainstharmonicellipticgenus,whichwe
computebyrepresentingthe latteraslinearcombinationofNiebur-Poincar´eseries.
In the last section 4 we take steps towards extending the Rankin-Selberg-Zagier
§
method to higher genus.
2TheideaofapplyingRankin-Selberg-typemethodstocomputeone-loopmodularintegrals,
albeitinaratherdifferentwayfrom[18],was firstputforwardin[19]. Steps towardsextending
themtohighergenusweretakenin[20].
4 BORISPIOLINE
Acknowledgements: I wish to thank C. Angelantonj and I. Florakis for a very
enjoyablecollaborationontheresultsreportedin 2-3,K.BringmannandD.Zagier
§
for valuable advice during the course of this project, R. Donagi and R. Russo for
commentsonanearlierversionofthis manuscript,andthe organizersofthe String
Math 2013 conference for their kind invitation to speak.
2. One-loop modular integrals with trivial elliptic genus
We startwith the simplestcase ofone-loopmodular integralsofthe form(1.6)
with Φ =1 (hence k = 0). Such integrals were computed by the ‘lattice unfolding
technique’in[6, 7]ford=1,[8]ford=2,[16]ford 3,andaconjecturalrelation
≥
to constrained Epstein series was put forward in [31]. In this section, we shall
calculatethem insteadbyinserting by handanon-holomorphicEisensteinseriesin
the integrand, computing the integral by the unfolding trick and taking a suitable
residue at the end. We use the standard notations τ = τ +iτ , q = e2πiτ for the
1 2
periodΩ andmodulusofanellipticcurve,andwriteΓ=Γ =SL(2,Z), = ,
11 1 1
H H
etc.
2.1. Non-holomorphic Eisenstein series . The non-holomorphic Eisen-
stein series for Γ=SL(2,Z) is defined by the sum over images
τs
E⋆(s;τ)=ζ⋆(2s) τs γ = 1ζ⋆(2s) 2 ,
(2.1) 2| 2 cτ +d2s
γ∈XΓ∞\Γ (cX,d)=1| |
where Γ is the subgroup of upper-triangular matrices in Γ, and
∞
(2.2) ζ⋆(s) π−s/2Γ(s/2)ζ(s)=ζ⋆(1 s)
≡ −
is the completed Riemann zeta function. The sum convergesabsolutely for (s)>
ℜ
1,andhasameromorphiccontinuationtoall s. Thenormalizationin(2.1)ensures
that E⋆(s;τ) invariant under s 1 s, and has only simple poles at s = 0 and
7→ −
s=1. The key point for our purposes is that the residue at s=1 is constant – in
agreement with the fact that E⋆(s;τ) is an eigenmode of the Laplacian ∆ on ,
H
H
with vanishing eigenvalue at s=0 or 1,
(2.3) ∆ 1s(s 1) E⋆(s;τ)=0 , ∆ =2τ2∂ ∂ .
H− 2 − H 2 τ τ¯
Moreprecisely,(cid:2)the Laurentexpan(cid:3)sionats=1is givenby the firstKroneckerlimit
formula,
1
(2.4) E⋆(s;τ)= + 1 γ log(4πτ η(τ)4) + (s 1) ,
2(s 1) 2 E − 2| | O −
−
(cid:0) (cid:1)
where η = q1/24 (1 qn) is the Dedekind eta function and γ is Euler’s
n≥1 − E
constant. All these statements are easy consequences of the Fourier series repre-
Q
sentation of E⋆(s;τ), or Chowla-Selberg formula,
E⋆(s;τ)=ζ⋆(2s)τs+ζ⋆(2s 1)τ1−s
2 − 2
(2.5) +2 |N|s−12σ1−2s(N)τ21/2Ks−12(2π|N|τ2)e2πiNτ1,
N6=0
X
where σ (N) = dt is the divisor function and K (z) is the modified Bessel
t d|N t
function of the second kind. Below we shall denote the first line of (2.5), which
P
dominates the behavior at τ , by E⋆(s;τ).
2 →∞ 0
RANKIN-SELBERG METHODS FOR CLOSED STRING AMPLITUDES 5
ForanymodularfunctionF(τ)ofrapiddecay(suchasthemodulussquare ψ 2
| |
of a holomorphic cusp form), we consider the modular integral (also known as the
Rankin-Selberg transform)
(2.6) ⋆(F;s)= dµE⋆(s;τ)F(τ) .
R
ZF
For (s) > 1, the sum over cosets in (2.1) can be exchanged with the integral, so
ℜ
that the integration domain is extended to the strip
(2.7) =Γ Γ= τ >0, 1 <τ 1 ,
S ∞\ { 2 −2 1 ≤ 2}
at the expense of retaining the contribution of the unit coset only,
∞
(2.8) ⋆(F;s)=ζ⋆(2s) dµτsF(τ)=ζ⋆(2s) dτ τs−2F (τ ) .
R 2 2 2 0 2
ZS Z0
The last equality expresses ⋆(F;s) as a Mellin transform of the zero-th Fourier
R
coefficient F (τ )= 1/2 dτ F(τ). At the same time, ⋆(F;s) inherits the mero-
0 2 −1/2 1 R
morphicity and invariance under s 1 s satisfied by E⋆. In the case where
R 7→ −
F = ψ 2, ⋆(F;s) is proportional to the L-series a 2n−s, whose analyticity
n
| | R | |
and functional equation are thereby determined. This is one of the main uses of
P
the Rankin-Selberg method in number theory [32].
More importantly for our purposes, the fact that the residue of E⋆(s;τ) at
s = 1 is constant implies that the residue of ⋆(F;s) at s = 1 is proportional to
R
the modular integral of F,
(2.9) Res ⋆(F;s)= 1 dµF .
s=1R 2
ZF
Unfortunately,thisstatementonlyholdsforfunctionsF ofrapiddecay,whichrules
out the interesting case F = Γ . In the next section, following [33] we discuss
d,d,1
how the unfolding trick can nevertheless be used after proper regularization.
2.2. Rankin-Selberg-Zagiermethod. Letusnowconsideramodularfunc-
tion F with polynomial growth at the cusp,
(2.10) F(τ) ϕ(τ ) , ϕ(τ )= c τα .
∼ 2 2 α 2
α
X
In order to regulate infrared divergences in the integral (2.6), we truncate the
integration domain to = τ > and define the renormalized integral as
T 2
F F ∩{ T}
(2.11) R.N. dµF(τ)= lim dµF(τ) ϕˆ( ) ,
ZF T→∞(cid:20)ZFT − T (cid:21)
where ϕˆ( ) is the anti-derivative of ϕ(τ ),
2
T
α−1
(2.12) ϕˆ( )= c T +c log .
α 1
T α 1 T
α6=1 −
X
On the other hand, we define the Rankin-Selberg transform of F as the Mellin
transform of the zero-th Fourier coefficient, minus its leading behavior,
∞
(2.13) ⋆(F;s)=ζ⋆(2s) dτ τs−2 (F ϕ) .
R 2 2 0−
Z0
6 BORISPIOLINE
The renormalized integral (2.11) is then related to the Rankin-Selberg transform
(2.13) via a generalizationof (2.9) [33]
(2.14)
R.N. dµF(τ)=2Res [ ⋆(F;s)+ζ⋆(2s)h (s)+ζ⋆(2s 1)h (1 s)] ϕˆ( ),
s=1 T T
R − − − T
ZF
where h (s) is the meromorphic function of s defined by
T
T α+s−1
(2.15) h (s)= dτ ϕ(τ )τs−2 = c T .
T 2 2 2 α α+s 1
Z0 α −
X
Notethattheright-handsideof(2.14)isbyconstructionindependentoftheinfrared
cut-off . The derivation of (2.14) is based on the generalized unfolding trick for
T
modular integrals on the truncated fundamental domain,
(2.16) dµF f = dµF f dµF f ,
γ γ
| − |
ZFT γ∈XΓ∞\Γ ZST ZF−FT γ∈XγΓ6=∞1\Γ
where is the truncated strip τ < , 1/2 < τ < 1/2 . Applying this ob-
T 2 1
S { T − }
servation to the regulated integral R⋆(F;s) dµFE⋆(s;τ) and reorganizing
T ≡ FT
terms gives [33, Eq. (27)]
R
⋆(F;s)=R⋆(F;s)+ dµ (F E⋆(s;τ) ϕE⋆(s;τ ))
(2.17) R T ZF−FT − 0 2
ζ⋆(2s)h (s) ζ⋆(2s 1)h (1 s) ,
T T
− − − −
fromwhich(2.14)follows. Anotherconsequenceof (2.17)isthattheRankin-Selberg
transform ⋆(F;s)hasameromorphiccontinuationins,invariantunders 1 s,
R 7→ −
and analytic away from s = 0,1,α ,1 α . A particularly pleasant feature of the
i i
−
renormalization prescription (2.11) is that the Rankin-Selberg transform ⋆(F;s)
R
coincides with the renormalized integral
(2.18) ⋆(F;s)=R.N. dµF(τ)E⋆(s;τ) .
R
ZF
Moreover,ifF isconstant, ⋆(F;s)vanishesandthereforeR.N. dµE⋆(s;τ)=0.
R F
R
2.3. Constrained Epstein series. The Rankin-Selberg-Zagier method dis-
cussedintheprevioussubsectionappliesimmediatelytomodularintegralsofSiegel-
Narain theta series for even self-dual lattices of signature (d,d),
(2.19) Γ (G,B;τ)=τd/2 e−πτ2M2(mi,ni)+2πiτ1mini ,
d,d 2
(mi,Xni)∈Z2d
where 2 is the positive definite quadratic form
M
(2.20) 2(m ,ni)=(m +B nk)Gij(m +B nl)+niG nj
i i ik j jl ij
M
and(G ,B )=G , B )parametrizetheGrassmannianG . Eq. (2.19)defines
ij ij ji ji d,d
−
a modular function F on of polynomial growth characterized by
H
d d
(2.21) ϕ(τ2)=τ2d2 , hT(s)= sT+s+d2−11, ϕˆ(τ2)=τlo22g−τ1/(d2 −1) iiff dd=6=22 .
2 −  2

RANKIN-SELBERG METHODS FOR CLOSED STRING AMPLITUDES 7
Its Rankin-Selberg transform is
∞ d
⋆(Γ ;s)=ζ⋆(2s) dτ τs+2−2 e−πτ2M2(mi,ni)
R d,d 2 2
(2.22) Z0 (mi,ni)X∈Z2d\(0,0)
mini=0
= SO(d,d),⋆(s+ d 1;G,B)
EV 2 −
where SO(d,d),⋆(s) is the completed, constrained Epstein series defined by [31]
EV
SO(d,d),⋆(s)=ζ⋆(2s)ζ⋆(2s+2 d) SO(d,d)(s)
EV − EV
1
(2.23) SO(d,d)(s;G,B)= [ 2(m ,ni)]−s ,
EV ζ(2s) M i
(mi,ni)X∈Z2d\(0,0)
mini=0
the sum being absolutely convergent for (s) > d. The results of 2.2 show that
ℜ §
SO(d,d),⋆(s)admitsameromorphiccontinuationins,invariantunders d 1 s,
EV 7→ − −
with simple poles at s = 0,d 1,d,d 1 (or double poles at s = 0 and s = 1 if
2 − 2 −
d=2). For d=2, the residues at s= d or s= d 1 produce the modular integral
6 2 2−
of interest:
(2.24)
R.N. dµΓ =2Res SO(d,d),⋆(s;G,B)=ζ⋆(d 2) SO(d,d) d 1;G,B ,
d,d s=dEV − EV 2 −
ZF 2
(cid:0) (cid:1)
rigorously proving a conjecture in [31]. Physically, the integral (2.24) computes
(among other things) the one-loop contribution to 4 couplings in type II strings
R
compactified on a torus Td [34, 16]. The right-hand side is interpreted as a sum
of one-loop contributions from particles of momentum m and winding ni along
i
the torus, with mass 2(m ,ni), satisfying the BPS constraint m ni = 0. It is
i i
manifestly invariant uMnder the T-duality group SO(d,d,Z), under which m ,ni
i
transform in the vector (defining) representation. For s 1 the s-dependent
→
generalization(2.22)canbethoughtofasthedimensionallyregularized3amplitude,
i.e. the amplitude in D = 10 (d+2s 2) non-compact dimensions. The case
− −
s=2alsocomputesD4 4 couplingsatone-loopintypeIIstringtheoryonTd[38].
R
Mathematically, (2.23) is recognized as the degenerate Langlands-Eisenstein series
of SO(d,d) with infinitesimal character ρ 2sλ (where ρ is the Weyl vector and
1
−
λ theweightofthevectorrepresentation)[39, 40]. TheresidueofthisLanglands-
1
Eisensteinseriesats= d yieldsthe minimalthetaseriesassociatedtothe minimal
2
representationof SO(d,d) [41].
Ford=2,theGrassmannianG reducestoaproductoftwoupperhalfplanes
2,2
,whereT andU aretheK¨ahlermodulusandcomplexstructuremodulus,
T U
Hresp×ecHtively, while the T-duality group decomposes into SL(2,Z) SL(2,Z) ⋉
T U
×
σ ,whereσ isaninvolutionexchangingT andU. TheBPSconstraintm n1+
T,U T,U 1
m n2 =0canbesolvedexplicitly,allowingtorewritetheconstrainedEpsteinseries
2
as a product of two non-holomorphic Eisenstein series [18],
(2.25) SO(2,2),⋆(s;T,U)=2E⋆(s;T)E⋆(s;U) .
EV
3Otherversionsofdimensionalregularizationinstringtheorywerediscussedin[35,36,37].
8 BORISPIOLINE
Extracting the residue at s=0 or 1 by means of the Kroneckerlimit formula (2.4)
leads to
(2.26) Γ (T,U;τ)dµ= log T U η(T)η(U)4 +cte ,
2,2 2 2
− | |
ZF
whichagreeswith[8],uptoarenormalizat(cid:0)ionscheme-depend(cid:1)entadditiveconstant.
The reader familiar with [8] may appreciate the elegance of the Rankin-Selberg-
Zagier method compared with the lattice unfolding method.
3. One-loop modular integrals with harmonic elliptic genus
While the integrand Φ in type II one-loop string amplitudes is of polynomial
growth at the cusp τ i , this is not the case for heterotic strings, due to the
→ ∞
tachyoninthespectrumbeforeimposingtheGSOprojection. Instead,Φisamod-
ular form of negative modular weight w = k/2 with a first order pole at the
−
cusp. The Rankin-Selberg-Zagier method described in 2.2 is not directly appli-
§
cable, however the unfolding trick could still be used provided Φ had a uniformly
convergentPoincar´eseries representation
(3.1) Φ= f γ
w
|
γ∈XΓ∞\Γ
with suitable Γ -invariant seed f(τ). f should grow as 1/qκ at τ if (3.1)
∞ 2
→ ∞
is to represent a modular form with a κ-th order pole at the cusp, but uniform
convergencerequires f(τ) τ1−w2 as τ 0. The naive choice f(τ)=1/qκ is fine
≪ 2 2 →
for weight w>2 but fails for w 2.
≤
3.1. Selberg-Poincar´e and Niebur-Poincar´e series. A first, natural op-
tion for regulating the sum is to insert a non-holomorphic convergence factor `a
la Hecke-Kronecker, i.e. choose a seed f(τ) = τs−w2 q−κ. The resulting Selberg-
2
Poincar´e series
(3.2) E(s,κ,w;τ) 1 (cτ +d)−wτ2s−w2 e−2πiκcaττ++db
≡ 2 cτ +d2s−w
(cX,d)=1 | |
converges absolutely for (s) > 1, but analytic continuation to the desired value
ℜ
s = w is non-trivial, as it depends on deep properties of Kloosterman sums is
2
tricky, and in general non-holomorphic [42, 43]. Another undesirable feature of
the Selberg-Poincar´e series (3.2) is that it is not an eigenmode of the weight w
Laplacian on , rather4
H
1
(3.3) ∆ + (s w)(1 w s) E(s,κ,w)=2πκ(s w)E(s+1,κ,w) ,
H,w 2 − 2 − 2 − − 2
(cid:20) (cid:21)
sothattheanalyticcontinuationtos= w isnotguaranteedtoyieldaholomorphic
2
result, nor even harmonic [44].
A much more convenient choice, which does not require analytic continuation,
is the Niebur-Poincar´e series5
κτ
(3.4) F(s,κ,w;τ)= 21 (cτ +d)−wMs,w − cτ +2d2 e−2πiκℜ(caττ++db)
(cX,d)=1 (cid:18) | | (cid:19)
4Here∆H,w =2Dw−2D¯w whereDw,D¯w aredefinedin(3.8). Noticethechangeofconven-
tioncomparedto[21].
5TherelationbetweenthePoincar´eseries(3.2)and(3.4)canbefoundin[21,App. B].
RANKIN-SELBERG METHODS FOR CLOSED STRING AMPLITUDES 9
first introduced by Niebur [24] (for weight zero) and Hejhal [25] and revived in
recent mathematical work on Mock modular forms[26, 27, 28, 29]. The seed
f(τ)= s,w( κτ2)e−2πiκτ1, where
M −
(3.5) Ms,w(y)=|4πy|−w2 Mw2sgn(y),s−12 (4π|y|) ,
is proportional to the Whittaker function M (z), is uniquely determined by the
λ,µ
requirements that (s,κ,w;τ) be an eigenmode of the Laplacian on ,
F H
(3.6) ∆ + 1(s w)(1 w s) (s,κ,w;τ)=0 ,
H,w 2 − 2 − 2 − F
and that f(τ) ha(cid:2)s the desired growth at τ2 (cid:3) and τ2 0,
→∞ →
Γ(2s)
(3.7) f(τ)∼τ2→∞ Γ(s+ w)q−κ , f(τ)∼τ2→0 |4πκτ2|s−w2e−2πiκτ1 .
2
The last property ensures that (s,κ,w) converges absolutely for (s) >1, while
F ℜ
a more detailed argument based on the Fourier expansion of (s,κ,w) (which can
F
be found in [29, 21]) shows that (s,κ,w) is holomorphic for (s) > 3 [45, 46].
F ℜ 4
Besides being an eigenmode of ∆ , (s,κ,w) also transforms in a simple way
H,w
under the raising, lowering and Hecke oFperators D,D¯,H defined by
m
iw
(3.8) D = i ∂ , D¯ = iπτ2∂ ,
w π τ − 2τ w − 2 τ¯
(cid:18) 2(cid:19)
aτ +b
(3.9) (H Φ)(τ)= d−wΦ ,
m
· d
a,d>0bmodd (cid:18) (cid:19)
X X
ad=m
namely [21]
D (s,κ,w;τ)=2κ(s+ w) (s,κ,w+2;τ),
w·F 2 F
1
D¯ (s,κ,w;τ)= (s w) (s,κ,w 2;τ).
(3.10) w·F 8κ − 2 F −
H (s,κ,w;τ)= d1−w (s,κm/d2,w;τ) .
m
·F F
d|X(κ,m)
The decisive advantage of Niebur’s Poincar´e series over Selberg’s, however, is
thatthevalues=1 w,degeneratewiththevalues= w undertheLaplacian(3.6),
−2 2
lies in the convergence domain (s)> 1 (except for w = 0, which requires a more
ℜ
careful treatment). Eigenmodes of the Laplacian (3.6) with s = w or equivalently
2
s=1 w areknownas weak harmonic Maass forms (WHMS), and havea Fourier
− 2
expansion near τ =i of the form6
∞
∞ ∞
(3.11) Φ= a qm+ (4πτ2)1−w¯b + mw−1¯b Γ(1 w,4πmτ )q−m
m w−1 0 m − 2
m=−κ m=1
X X
Weak holomorphic modular forms are a special case of WHMS, where the neg-
ative frequency coefficients ¯b vanish. Mock modular forms are defined as the
m
analytic part Φ+ = ∞ a qm of a WHMS. Acting on any WHMS Φ of
m=−κ m
weight w with the lowering operator D¯ produces the complex conjugate of a holo-
P
morphic modular form Ψ = b qm of weight 2 w (the shadow) while
m≥1 m −
the iterated raising operator D1−w produces a weakly holomorphic modular form
P
6Herewerestricttothecasewheretheshadowisregularatτ =i∞,see[46]forthegeneral
expansion.
10 BORISPIOLINE
Ξ= ∞ m1−wa qm of weight 2 w (the ghost ?) such that Φ+ is an Eichler
m=−κ m −
integralofΞ. In the caseof (1 w,κ,w), the shadow is the usualPoincar´eseries
P F − 2
Ψ P( κ,2 w) = qκ , while the ghost is the Niebur-Poincar´e series
γ,2−w
∝ − − |
Ξ (1 w,κ,2 w). In particular, Ψ is a cusp form of weight 2 w, so must
van∝isFh for−w2= 0, −2, P4, 6, 8, 12. Indeed, for these values, (1− w,κ,w) is
− − − − − F − 2
an ordinary weak holomorphic modular form, e.g.
(3.12) (1,1,0)=J +24 , (2,1, 2)=3!E4E6 , (7,1, 12)=13!/∆ ,...
F F − ∆ F −
whereE ,E aretheusualEisensteinseriesofweight4,6underSL(2,Z),∆=η24is
4 6
themodulardiscriminantandJ = E43 744=1/q+ (q)istheusualHauptmodul.
∆ − O
Incontrast,forw= 10, (1,1, 10)isa genuineWHMS, withirrationalpositive
− F −
frequency Fourier coefficients and non trivial shadow, proportional to ∆ [47].
It is worth noting that for s = 1 w, the seed of the Niebur-Poincar´e series
− 2
simplifies to
−w (4πκτ )ℓ
(3.13) f(τ)=Γ(2 w) q−κ q¯κ 2 ,
− − ℓ! !
ℓ=0
X
whichplainlyshowstheimprovedultravioletbehaviorcomparedtothenaivechoice
f(τ) q−κ.
∼
Using the Niebur-Poincar´eseries (s,κ,w) with s=1 w, we can now repre-
F − 2
sent any weakly holomorphic modular form Φ of weight w 0 as a linear combi-
≤
nation of Niebur-Poincar´e series7
1
(3.14) Φ= a (1 w, m,w;τ)+a′ δ
Γ(2 w) mF − 2 − 0 w,0
− −κ≤m<0
X
where the coefficients are read off from the polar part Φ = a qm +
−κ≤m≤0 m
(1) at the cusp τ i . Indeed, the difference between the left and right-hand
O → ∞ P
sides of (3.14) is a harmonic Maass form of negative weight which is exponentially
suppressed at the cusp (for suitable choice of a′ if w =0), hence vanishes [26]. In
0
particular, while each term in (3.14) may be a WHMF with non-trivial shadow,
the shadow cancels in the linear combination (3.14). More generally, using the
fact that almost, weakly holomorphic modular forms of weight w < 0 are linear
combinations8 of iterated derivatives DnΦ of weakly holomorphic modular
w−2n
forms of weight w 2n, along with (3.10), we can similarly represent any almost,
−
weakly holomorphic modular form of weight w <0 as a linear combination
n
(3.15) Φ= a(p)(m) (1 w +p, m,w;τ) +a′ δ
F − 2 − 0 w,0
p=0−κ≤m<0
X X
wherenisthedepth(i.e. themaximalpowerofEˆ ). Asanexamplerelevantforthe
2
computation of threshold corrections to gauge couplings in heterotic string theory
7Similarly, modular forms under congruence subgroups of SL(2,Z) can be represented as
linear combinations of Niebur-Poincar´e series attached to all cusps, see [22] for the example of
theHeckecongruence subgroupΓ0(N).
8Thisisnotthecaseforalmost,weaklyholomorphicmodularformsofweightw>0,(Eˆ2)nJ
beinga counter-example. Such cases can be treated by considering s-derivatives of F(s,κ,w)at
s=w/2[48].