Threshold corrections, generalised prepotentials and Eichler integrals

0.52 MB·English

by  Carlo Angelantonj

CERN-PH-TH/2015-011
Threshold corrections,
5 generalised prepotentials
1
0
and Eichler integrals
2

n
u
J

6
2
CarloAngelantonj1,IoannisFlorakis2andBorisPioline2,3

]
h
t
- 1 DipartimentodiFisica,UniversitàdiTorino,andINFNSezionediTorino
p
ViaP.Giuria1,10125Torino,Italy
e
h 2 CERNDepPH-TH,1211Geneva23,Switzerland
[
3 LaboratoiredePhysiqueThéoriqueetHautesEnergies,CNRSUMR7589,

2 UniversitéPierreetMarieCurie-Paris6,4placeJussieu,75252Pariscedex05,
v France
7
0
0 ABSTRACT
0
0 Wecontinueourstudyofone-loopintegralsassociatedtoBPS-saturatedamplitudesinN =
.
2 2heteroticvacua. Wecomputetheirlarge-volumebehaviour,andexpressthemasFourierse-
0
riesinthecomplexifiedvolume,withFouriercoefficientsgivenintermsofNiebur-Poincaré
5
1 seriesinthecomplexstructuremodulus. TheclosureofNiebur-Poincaréseriesundermod-
: ular derivatives implies that such integrals derive from holomorphic prepotentials f , gen-
v n
i eralisingthefamiliarprepotentialof N = 2supergravity. Theseholomorphicprepotentials
X
transformanomalouslyunderT-duality, inawaycharacteristicofEichlerintegrals. Weuse
r
a this observation to compute their quantum monodromies under the duality group. We ex-
tend the analysis to modular integrals with respect to Hecke congruence subgroups, which
naturally arise in compactifications on non-factorisable tori and freely-acting orbifolds. In
this case, we derive new explicit results including closed-form expressions for integrals in-
volvingtheΓ (N)Hauptmodul,afullcharacterisationofholomorphicprepotentialsinclud-
0
ingtheirquantummonodromies,aswellasconcreteformulæforholomorphicYukawacou-
plings.
E-mail: [email protected]
[email protected]
[email protected]
Contents
1 Introduction 1
2 ModularintegralsforSL(2;Z) 5
2.1 Niebur-PoincaréseriesforSL(2;Z) . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Evaluationofthemodularintegral . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Thespecialcasesofintegrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3 GeneralisedprepotentialsforSL(2;Z) 16
3.1 Generalisedharmonicprepotentials . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 GeneralisedholomorphicprepotentialsandEichlerintegrals . . . . . . . . . . 18
3.3 Modularpropertiesofgeneralisedholomorphicprepotentials . . . . . . . . . 20
3.4 GeneralisedYukawacouplings . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4 Modularintegralsfor Γ (N) 25
0
4.1 SymmetriesoftheshiftedNarainlattice . . . . . . . . . . . . . . . . . . . . . . 27
4.2 Niebur-Poincaréseriesfor Γ (N) . . . . . . . . . . . . . . . . . . . . . . . . . . 28
0
4.3 Propertiesofthe Γ (N)modularintegrals . . . . . . . . . . . . . . . . . . . . . 31
0
4.4 EvaluationofthemodularintegralI (s;a) . . . . . . . . . . . . . . . . . . . . 32
N
4.5 Generalisedholomorphicprepotentialsfor Γ (N) . . . . . . . . . . . . . . . . 36
0
4.6 GeneralisedYukawacouplingsfor Γ (N) . . . . . . . . . . . . . . . . . . . . . 38
0
A Kloosterman-Selbergzetafunction 40
B ModularpropertiesoftheEichlerintegral 41
1 Introduction
The leading radiative corrections to scattering processes in closed string theories compacti-
fiedonad-dimensionaltorusarecomputedbymodularintegralsoftheform
(cid:90)
I = dµΓ Φ, (1.1)
d+k,d
F
where Γ is the partition function of the Narain lattice with signature (d+k,d), Φ is
d+k,d
an appropriate modular form of negative weight (−k,0) under the modular group Γ =
2
SL(2;Z) oftheworldsheettorus, and F isafundamentaldomainfortheactionof Γ onthe
H
Poincaréupperhalfplane . Inordertoextractquantitativepredictionsfromstringtheory,
itisthereforeessentialtodevelopefficienttoolstocomputesuchmodularintegrals.
In the traditional lattice unfolding method [1–3], one decomposes the lattice sum Γ
d+k,d
intoorbitsunderthemodulargroup,andreduceseachorbittoasingletermwhileunfolding
the integration domain. Although the resulting expression is very useful in extracting the
asymptotic expansion of I in the limit where the volume of the torus Td becomes large, it
1
dependsonachoiceofchamberintheNarainmodulispace,andT-dualityinvarianceisno
longermanifest.
In a series of recent works [5–7], we proposed a new method, which assumes that Φ
canbewrittenasanabsolutelyconvergentPoincaréseriesunderthemodulargroup. Upon
unfoldingtheintegrationdomainagainstthesumoverimages,andevaluatingtheintegral,
the result is expressed as a sum over charges satisfying a BPS-type condition. This can be
recognisedasaparticulartypeofPoincaréseriesfortheautomorphismgroupO(d+k,d;Z)
oftheNarainlattice,whichisthusmanifestlyinvariantunderT-duality. Theanalyticstruc-
tureoftheintegralnearpointsofenhancedgaugesymmetryisalsomanifest,whilethelarge
volume asymptotics is no longer apparent. In fact, the method can be applied even when
theintegranddoesnotincludetheNarainlattice,e.g. inten-dimensionalstringvacua.
If Φ isagenericnon-holomorphicmodularform,thereisnogeneralknownwaytorep-
resent it as a Poincaré series. For BPS-saturated amplitudes, however, the modular form Φ
turns out to be weakly almost holomorphic, i.e. a polynomial in the ring C[Eˆ ,E ,E ,1/∆].
2 4 6
In that case, it can actually be decomposed as a linear combination of so-called Niebur-
Poincaré series F(s,κ,w;z) [6,8]. These Poincaré series, introduced in the mathematics lit-
H
erature in [9,10], have the virtues of being eigenmodes of the Laplacian on with a pole
of order1 κ at the cusp z = ∞ and of being closed under modular derivatives and Hecke
operators. Inaddition,theyreducetoharmonicMaassformswhens = 1− w,ortomodular
2
derivativesthereofwhen s = 1− w +n with n apositiveinteger. Forsuitablelinearcombi-
2
nations, the non-analytic parts of F(s,κ,w;z) (encoded in the ‘shadow’) cancel, producing
thedesiredweaklyalmostholomorphicform Φ. Theintegral(1.1)isthusreducedtoasum
ofintegralsoftheform
(cid:90)
I(s,κ) = dµΓ F(s,κ,−k/2), (1.2)
d+k,d
F
each of which can be computed by unfolding the fundamental domain against the Niebur-
Poincaréseries[6].
In this work, we compute the large volume expansion of the modular integral (1.2),
thus providing an alternative way to extract the asymptotics of (1.1) within our unfolding
method. The advantage of decomposing Φ into a sum of Niebur-Poincaré series is that the
integral (1.1) is then decomposed into a sum of terms which are separately eigenmodes of
theLaplacianontheNarainmodulispace
G = O(d+k,d;Z)\SO(d+k,d)/SO(d+k)×SO(d), (1.3)
d+k,d
andwhichbehaveinasimplewayundercovariantderivativeswithrespecttothemoduli.
For simplicity, we restrict our attention to integrals involving lattices of signature (2,2),
where the moduli space (1.3) is parametrised by the standard variables (T = T +iT ,U =
1 2
1Whiletheintegrandsappearinginheteroticstringtheoryonlyhaveafirstorderpoleatz=∞,itisusefulto
allowpolesofarbitraryorderκ≥1assuchpolestypicallyoccurinmodularintegralsforcongruentsubgroups,
relevantfororbifoldcompactifications. Moreover, Niebur-Poincaréseries F(s,κ(cid:48),w) withκ(cid:48) > 1occurinthe
Fourierexpansionofmodularintegralsoftheform(1.2),evenifκ=1.
2
U +iU ) ∈ H ×H . They correspond to the Kähler modulus and complex structure of
1 2 T U
a two-torus, and are identified under separate actions of the modular group SL(2;Z) on T
and U, and under the involution σ : T ↔ U. This case is relevant when the internal space
includesaT2factor,e.g. inthecontextofheteroticstringscompactifiedonT2(with16super-
charges),onK3×T2 (with8supercharges),oron N = 1orbifoldswith N = 2sub-sectors,
uponrestrictiontobackgroundswithtrivialordiscreteWilsonlines2. Weconsiderboththe
limit where the volume T is scaled to infinity keeping the complex structure modulus U
2
fixed,andthelimitwherebothT andU arescaledtoinfinity(inwhichcasetheasymptotic
2 2
expansiondependsonachoiceofchamber,T > U vs. T < U ). Inthelattercase,theresult
2 2 2 2
is expressed as an infinite sum of a special type of single-valued polylogarithms, known in
themathematicsliteratureasBloch-Wigner-Ramakrishnanpolylogarithms[11],whichcoin-
cide with the real part of the ‘combined polylogarithms’ introduced in [13]. In the former
case, T → ∞withU fixed,weshowthattheFouriercoefficientsoftheintegralI(s,κ)with
2
respecttoT aregivenbythesameNiebur-PoincaréseriesF(s,κ,0)presentintheintegrand,
1
nowasfunctionsoftheU modulus. Thisisamanifestationofaphenomenonfirstobserved
in[14]which,asweshallsee,canbetracedtotheaveragevaluepropertyofeigenmodesof
theLaplacianonthePoincaréupperhalf-plane. Animportantconsequenceofthisobserva-
tionisthat,usingtheclosureofNiebur-Poincaréseriesundermodularderivatives,onemay
expresstheintegral(1.2)asaniteratedderivativeofa‘harmonicprepotential’h [13]
n
(cid:20)(−D D )n (cid:21)
I(n+1,κ) = 4Re T U h (T,U) −24σ (κ)δ logT U . (1.4)
n 1 n,0 2 2
n!
Theprepotentialh isbyconstructionharmonicin(T,U)andtransformscovariantlyunder
n
SL(2;Z) ×SL(2;Z) ,whilethelasttermoriginatesfromtheinfrareddivergenceoftheinte-
T U
gralwhenn = 0. Crucially,itisalsopossibletoconstructaholomorphicprepotential f (T,U)
n
which satisfies the same relation (1.4). This holomorphic prepotential is well-defined up to
the addition of polynomials of degree 2n in T and U with real coefficients, which lie in the
kerneloftheoperatorRe(D D )n. Asaresult,itneednottransformhomogeneouslyunder
T U
T-duality, but may pick up an additional degree 2n polynomial. This type of anomalous
behaviour under modular transformations is characteristic of Eichler integrals. Indeed, we
tracethemodularanomalyof f (T,U) tothepresenceofatermproportionaltotheEichler
n
integraloftheholomorphicEisensteinseriesE2n+2intheFourierexpansion. Weusethisfact
tofullycharacterisetheT-dualitytransformationsofthegeneralisedprepotential f .
n
From a physics viewpoint, the prepotentials f for n > 1 can be viewed as generali-
n
sations of the usual holomorphic prepotential F that governs N = 2 supergravity in four
dimensions – or more accurately, of the one-loop contribution to it [4,15–17]. In fact, it
was already recognised in [15] that the quadratic terms appearing in the behaviour of the
one-loop prepotential f under monodromies around singularities in moduli space can be
1
obtained from contour integrals of suitable modular forms, which are Eichler integrals in
essence. Our results enable us to recover the monodromy matrices computed in [15] in a
streamlinedfashion,andtoextendtheanalysisto n > 1. Thecase n = 2wasstudiedinthe
2Wedefertheanalysisofasymmetriclatticeswithnon-trivialWilsonlinestoaforthcomingpublication.
3
context of F4 couplings in heterotic strings compactified on T2 [18,19], where a relation of
thegeneralisedprepotentials f toperiodsonsymmetricproductsofnK3’swassuggested.
n
However,theanalogueofspecialgeometryforn > 1isnotunderstoodatpresent. Asastep
inthisdirection,weconstructaclassof‘generalisedYukawacouplings’,obtainedfrom f by
n
iterated derivatives with respect to the T and U moduli, which compute modular integrals
withinsertionsofmomentaandarefreeofmodularanomalies.
From a mathematics viewpoint, the prepotentials f can also be viewed as interesting
n
generalisations of Borcherds automorphic products [20] – or rather, logarithms thereof. In-
deed,inthespecialcasewhere F(1+n,1,−2n) isaweaklyholomorphicmodularform, f
n
canbewrittenschematicallyasapolylogarithmsum
fn(T,U) = 212n ∑ Fn(NM)Li2n+1(qTMqUN)+P2n(T,U), (1.5)
N,M
where F (M) are the Fourier coefficients of F(1+n,1,−2n) and P is the polynomial am-
n 2n
biguity mentioned above. For n = 0, Li (z) = −log(1−z), so f is recognised as the
1 0
logarithmofaBorcherdsautomorphicproduct. Forn > 0, f isnolongerthelogarithmofa
n
product,butitprovidesaninterestingexampleofanholomorphic‘Mock’modularformon
(H ×H )/σwithknownsingularbehaviouratalldivisors.
T U
Returningtophysics,theintegrals(1.1)donotexhausttheclassofone-loopamplitudes
ofinterestinstringtheory. Infact,incompactificationsonnon-factorisabletori[21,22]oron
freely-acting orbifolds [23–27] the integrand is typically expressed as a sum of various sec-
tors(h,g)involvingshiftedNarainlatticesΓ [h]coupledtomodularformsΦ[h]. Although
2,2 g g
thefullintegrandis,byconstruction,invariantunderthefullmodulargroupSL(2;Z),each
term is only invariant under a suitable congruence subgroup. These setups have not been
extensively analysed so far in the literature and call for a systematic investigation. Upon
partialunfolding[7],onemaycomputetheintegralbyfocusingonasinglesectornowinte-
gratedoverthefundamentaldomainofthecongruencesubgroup. Inthecaseoffreely-acting
Z orbifolds of N = 2 compactifications, the relevant subgroup is the Hecke congruence
N
subgroup Γ (N). In parallel with the SL(2;Z) case, one may decompose Φ[h] as a linear
0 g
combination of the Γ (N) Niebur-Poincaré series F (1+n,κ,0) attached to the cusp a [7].
0 a
One may then evaluate the modular integral by unfolding the fundamental domain F of
N
Γ (N) against the Poincaré series. This procedure was suggested in [7] and applied to the
0
study of radiative corrections to gauge couplings in classically stable non-supersymmetric
heterotic vacua in [26]. The result is a representation of the modular integral as a Poincaré
seriesundertheT-dualitygroup,ingeneralacongruencesubgroupofO(2,2;Z).
In this work, we continue this analysis and extract the large-volume expansion of one-
loop modular integrals for Γ (N) with N prime. We obtain expressions which extend the
0
results for SL(2;Z) and are nicely covariant with respect to the choice of cusp a. The ad-
vantageofourapproach,basedonthespecialpropertiesofF (1+n,κ,0),isthatweobtain
a
a T-Fourier series expansion with U-dependent coefficients which are manifestly invariant
under Γ (N) , and which actually are given in terms of the same class of Niebur-Poincaré
0 U
series. Thispropertyisinstrumentalinallowingustoderivenewexplicitresults. Namely,in
4
then = 0casewere-sumtheintegralintermsoftheΓ (N)Hauptmodulbygeneralisingthe
0
celebratedSL(2;Z)Borcherds’productformulatothecaseofHeckecongruencesubgroups.
For n ≥ 1 we identify generalised holomorphic prepotentials and fully characterise their
transformationpropertiesunderT-duality. Finally,weexpressthegeneralisedholomorphic
Yukawacouplingsintermsofthestandardgeneratorsoftheringof Γ (N)modularforms.
0
Thepaperisorganisedasfollows: inSection2,afterrecallingbasicpropertiesofNiebur-
Poincaré and Eisenstein series, we compute the large volume asymptotics of the modular
integral (1.2) or, equivalently, the Fourier expansion with respect to T , starting from its
1
T-duality invariant representation. We express the results in terms of single-valued poly-
logarithms, and make contact with the literature. In Section 3 we integrate this result in
termsofgeneralisedharmonicandholomorphicprepotentials,explaintheirrelationtoEich-
lerintegrals,anddeterminetheiranomaloustransformationpropertiesunderT-duality. We
constructgeneralisedYukawacouplingsbytakingsuitablemodularderivativesofthegen-
eralisedprepotentials,andconnectthemtomodularintegralswithlattice-momentuminser-
tions. Section4extendsthepreviousanalysistomodularintegralsinvolvingshiftedNarain
lattices. Weobtainnewexplicitresultsincluding‘cusp-covariant’Fourierseriesexpansions,
closed expressions for special classes of integrals involving the Γ (N) Hauptmodul, a full
0
characterisation of the generalised holomorphic prepotentials, as well as concrete formulæ
for the generalised holomorphic Yukawa couplings. Appendix A collects our conventions
for the Kloosterman sums and zeta functions, and Appendix B provides a detailed deriva-
tionofthemodularpropertiesoftherelevantEichlerintegrals.
2 Modular integrals for SL(2;Z)
Inthissection,weshallfocusonmodularintegrals(1.1)involvingtheproductofasignature
(2,2) Narain lattice Γ times a weakly almost holomorphic modular functions Φ(τ). As
2,2
anticipated, the latter can be uniquely decomposed in terms of Niebur-Poincaré series of
zero weight [6,8]. Thus, we begin by collecting some useful properties of Eisenstein and
Niebur-Poincaréserieswhichareinstrumentalforthesubsequentanalysis.
2.1 Niebur-PoincaréseriesforSL(2;Z)
TheNiebur-Poincaréseries[8–10]3
F(s,κ,w;z) =1 ∑ M (−κy)e−2πiκx| γ
2 s,w w
γ∈Γ∞\Γ
(cid:18) (cid:19) (cid:26) (cid:18) (cid:19)(cid:27)
=1 ∑ (cz+d)−wM − κy exp −2iπκ a − cx+d .
2 s,w |cz+d|2 c c|cz+d|2
(c,d)=1
(2.1)
3WeshallhenceforthomittheexplicitdependenceoftheNiebur-Poincaréseriesonthemodularparameterz
wheneveritisclearfromthecontext.
5
definesaone-parameterfamilyofmodularformsofnon-positiveweightwandorderκpole
in the nome q = e2iπz at the cusp z ≡ x+iy = ∞. Its seed is expressed in terms of the
Whittaker M-function
Ms,w(t) = |4πt|−w2 Mwsgn(t),s−1 (4π|t|) , (2.2)
2 2
whosebehaviourat t → 0guaranteesabsoluteconvergencewhenRe(s) > 1. ThePoincaré
seriescanbeshowntoadmitameromorphiccontinuationtothecomplexsplane. Moreover,
inthedomainofanalyticity,F(s,κ,w)isaneigenmodeoftheweight-whyperbolicLaplacian
∆ ,
z
(cid:2) (cid:3)
∆ + 1 s(1−s)+ 1 w(w+2) F(s,κ,w) = 0, (2.3)
z 2 8
where
(cid:18) (cid:19)
i iw
∆ = 2D¯ D , with D¯ = −iπy2∂ , D = ∂ − . (2.4)
z z z z z¯ z z
π 2y
ItsFourierexpansionreads
F(s,κ,w) = M (−κy)e−2iπκx+ ∑ F˜ (s,κ,w;y)e2iπmx, (2.5)
s,w m
m∈Z
where,inthemodewithfrequencym = −κ,wehaveseparatedthecontributionoftheseed.
Explicitly,
F˜0(s,κ,w;y) = f0(s,κ,w)y1−s−w2 ,
(2.6)
F˜ (s,κ,w;y) = f (s,κ,w)W (my),
m m s,w
with
f (s,κ,w) = 22−wi−wπ1+s−w2 |κ|s−w2 Γ(2s−1)σ1−2s(κ) ,
0 Γ(s− w)Γ(s+ w)ζ(2s)
2 2 (2.7)
4πi−w|κ|Γ(2s) (cid:12)m(cid:12)w
f (s,κ,w) = (cid:12) (cid:12)2 Z(m,−κ;s).
m Γ(s+ w sgn(m)) (cid:12)κ (cid:12)
2
Intheseexpressions,W (t)isrelatedtotheWhittakerW-function
s,w
Ws,w(t) = |4πt|−w2 Wwsgn(t),s−1 (4π|t|) , (2.8)
2 2
σ (n)isthedivisorfunction,whileZ(a,b;s)istheassociatedKloosterman–Selbergzetafunc-
(cid:96)
tiondefinedin(A.4).
Niebur-Poincaréseriesareclosedundertheactionofthemodularderivatives,
D F(s,κ,w) = 2κ(s+ w)F(s,κ,w+2),
z 2
(2.9)
1
D¯ F(s,κ,w) = (s− w)F(s,κ,w−2),
z 8κ 2
6
andoftheHeckeoperator4
F(s,κ,w) = H(z)F(s,1,w). (2.10)
κ
For the special values s = w and s = 1− w, within the domain of analyticity, F(s,κ,w)
2 2
becomes a harmonic Maaß form. For negative weight, the case of interest for most appli-
cations to string theory [6], the Niebur-Poincaré series is absolutely convergent at the latter
value s = 1− w. Moreover, sincethefirstcuspformforSL(2;Z) appearsatweighttwelve,
2
for w ∈ {0,−2,−4,−6,−8,−12} the shadow of F(1− w,κ,w) vanishes and the Niebur-
2
Poincaré series is actually holomorphic, hence valued in the polynomial ring generated by
E , E and ∆−1, where E are the holomorphic Eisenstein series and ∆is the modular dis-
4 6 2n
criminant. OtherNiebur-PoincaréseriesofinterestareF(1− w +n,κ,w),withnapositive
2
integer, which can be reached from the harmonic Maaß form F(1− w +n,κ,w−2n) by
2
acting with n modular derivatives. In fact, one can prove [6–8] that any weak almost holo-
morphicmodularformofnegativeweightinthepolynomialringgeneratedbyEˆ ,E ,E and
2 4 6
∆−1,with Eˆ beingthealmostholomorphicEisensteinseriesofweighttwo,canbeuniquely
2
decomposedintoalinearcombinationofNiebur-Poincaréseries,
n
Φ(z) = ∑ ∑ d (m)F (cid:0)1− w +(cid:96),m,w(cid:1) . (2.11)
(cid:96) 2
0<m≤κ (cid:96)=0
Thecoefficientsd (m)aredeterminedbytheprincipalpartoftheq-Laurentexpansionof Φ
(cid:96)
itself[7]. TheintegerncountsthemaximalpowerofEˆ intheringdecompositionofΦ,and
2
is known as the depth. An important fact is that for each value of (cid:96), the shadows cancel in
thelinearcombination(2.11).
Itwillalsobeusefultointroducethenon-holomorphicweight-wEisensteinseries
E(s,w;z) = 21 ∑ ys−w2|wγ
γ∈Γ∞\Γ
= 1 ∑ ys−w2 (2.12)
2 (cz+d)w|cz+d|2s−w
(c,d)=1
whichisabsolutelyconvergentforRe(s) > 1,andadmitstheFourierseriesexpansion
41−si−wπΓ(2s−1)ζ(2s−1)
E(s,w;z) =ys−w2 + y1−s−w2
Γ(s+ w)Γ(s− w)ζ(2s)
2 2
(2.13)
+ ∑ Γ(si+−ww(4sπgn|m(m|))w2)πζs(2s) |m|−sσ2s−1(|m|)Ws,w(my)e2iπmx.
m(cid:54)=0 2
4TheHeckeoperatorH(z)actsontheFouriermodesofaweight-wmodularformΦ=∑ Φ(m,y)e2iπmxas
n m
H(z)Φ(m,y)=n1−w ∑ dw−1Φ(nm/d2,d2y/n).
n
d|(m,n)
ItsatisfiesthecommutativealgebraHκ(z)Hλ(z) =∑d|(κ,λ)d1−wHκ(zλ)/d2.
7
TheEisensteinseriesE(s,w)canbeobtainedformallyasthelimitκ → 0of|4πκ|w2−sF(s,κ,w),
and thus shares several of the properties of the Niebur-Poincaré series. Namely, it is an
eigenmodeoftheweight-whyperbolicLaplacian
(cid:2) (cid:3)
∆ + 1s(1−s)+ 1w(w+2) E(s,w;z) = 0, (2.14)
w 2 8
isclosedundertheactionofthemodularderivatives
1
D E(s,w;z) = (s+ w)E(s,w+2;z),
z 2π 2
(2.15)
π
D¯ E(s,w;z) = (s− w)E(s,w−2;z),
z 2 2
andbecomesharmonicinzwhens = w ors = 1− w. Forw > 2,E(w,w)coincideswiththe
2 2 2
holomorphicEisensteinseries E .
w
2.2 Evaluationofthemodularintegral
We can now proceed to the study of the integral (1.1). Since any weak almost holomorphic
modularform Φ canbedecomposedasin(2.11)itissufficienttofocusourattentiononthe
basicregularisedmodularintegral
(cid:90)
I(s,κ;T,U) = R.N. dµΓ (T,U;τ)F(s,κ,0;τ), (2.16)
2,2
F
where Γ (T,U;τ) is the partition function of the even self-dual lattice of signature (2,2),
2,2
andR.N.standsforthemodularinvariantrenormalisationprescriptionof[5,6,28]fortreat-
ing the infra-red divergences arising at the cusp. For s = 1, the integral (2.16) develops a
simplepole,whichmustbesubtracted[6,8]. ByabuseofnotationwedenotebyI(1,κ;T,U)
theresultofthissubtraction. Thepartitionfunction
Γ2,2(T,U;τ) = τ2 ∑ q41|pL|2q¯14|pR|2, (2.17)
m1,m2,n1,n2∈Z
which we refer to as the Narain lattice, is a sum over integers m and ni identified with the
i
Kaluza-Klein momentum and winding numbers along a two-dimensional torus. The left-
movingandright-movingmomentaonthetwo-dimensionallattice
m −Um +Tn1+TUn2
p = 2 √1 ,
R
T U
2 2
(2.18)
m −Um +T¯n1+T¯Un2
p = 2 √1 ,
L
T U
2 2
dependontheKählermodulus T = T +iT andthecomplexstructuremodulusU = U +
1 2 1
iU of the two-torus. Γ (T,U;τ) is modular invariant under SL(2;Z), which requires the
2 2,2
modularweightoftheNiebur-Poincaréseriesin(2.16)tovanish.
Eq. (2.17) corresponds to the so-called Hamiltonian representation of the Narain lat-
8
tice, and exhibits manifest invariance under the duality group O(2,2;Z) ∼ SL(2;Z) ×
T
SL(2;Z) (cid:110)σ , where the Z generator σ exchanges the T and U moduli. A Poisson
U T,U 2 T,U
summationoverthemomentam yieldstheso-calledLagrangianrepresentation
i
 (cid:12) (cid:32) (cid:33)(cid:12)2
Γ2,2(T,U;τ) = T2 ∑ e−2πiTdet(A)exp− πT2 (cid:12)(cid:12)(cid:12)(cid:16)1 U(cid:17) A −τ (cid:12)(cid:12)(cid:12)  , (2.19)
A∈Mat2×2(Z) τ2U2 (cid:12) 1 (cid:12)
which now exhibits manifest invariance under SL(2;Z) ×SL(2;Z) (cid:110)σ . As a conse-
τ U τ,U
quence, Γ (T,U;τ) is invariant under the action of SL(2;Z) ×SL(2;Z) ×SL(2;Z) (cid:110)
2,2 τ T U
σ ,whereσ permutestheτ, T andU moduli. Thistrialityresultsintheidentities
τ,T,U τ,T,U
∆ Γ ≡ ∆ Γ ≡ ∆ Γ , H(τ)Γ ≡ H(T)Γ ≡ H(U)Γ , (2.20)
T 2,2 U 2,2 τ 2,2 n 2,2 n 2,2 n 2,2
where ∆ is the weight-zero hyperbolic Laplacian in the variable z. Together with the dif-
z
ferential equation (2.3), the first equation in (2.20) implies that the modular integral (2.16)
satisfy
(cid:2) (cid:3) (cid:2) (cid:3)
∆ + 1s(1−s) I(s,κ) = ∆ + 1s(1−s) I(s,κ) = 12σ (κ)δ , (2.21)
T 2 U 2 1 s,1
where the source term is due to the subtraction of the simple pole in the integral (2.16) at
s = 1.
Moreover, it suffices to consider the integral I(s) ≡ I(s,1) since eqs. (2.10), (2.20) to-
gether with the self-adjointness of the Hecke operator allows one to express the integral
(2.16)forgeneralpositiveintegerκ as
I(s,κ) = H(T)I(s) = H(U)I(s). (2.22)
κ κ
2.2.1 TheBPSstatesumrepresentation
A novel procedure for evaluating the integral (2.16) has been discussed in [6]. It amounts
tounfoldingtheSL(2;Z)fundamentaldomainF againsttheabsolutelyconvergentNiebur-
Poincaréseries. OnethusarrivesattheSchwinger-likerepresentation
I(s) = ∑ (cid:90) ∞ dτ2 Ms,0(−τ2)e−π2τ2(|pL|2+|pR|2), (2.23)
τ
BPS 0 2
where∑ denotesasumoverintegersm ,ni subjecttothe‘BPSconstraint’5
BPS i
m n1+m n2 = 1. (2.24)
1 2
5BPSstatesexistforanychargessuchthatm n1+m n2 ≤1,so(2.24)picksonlyasubsetoftheallowedBPS
1 2
states associated to the q-pole, which corresponds to the ground state of the non-supersymmetric side of the
heteroticstring.
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