We consider the Liouville theory by varying the linear dilaton coupling constant . It is known that, at two different values of coupling constant , the system exhibits two different small superconformal symmetries with central charge and , respectively. In the context of string theory these two theories are considered to describe the Coulomb and Higgs branches of the theory and are expected to be dual to each other. We study the Mathieu and umbral moonshine phenomena in these two theories and discuss their dual descriptions. We mainly consider the case of -type modular invariants.
In this paper we study the supersymmetric Liouville theory in order to have a deeper understanding of the Mathieu and umbral moonshine phenomena discovered recently [1–3]. It is known that, when one perturbs the large theory by varying the strength of the linear dilaton term , there are two special values of where the theory possesses small supersymmetry with the central charges and . These two theories have different symmetries, i.e., different components of of large Liouville theory. In the context of string theory, these describe the Higgs and Coulomb branches of the theory and are expected to have descriptions dual to each other.
While the values of the central charges () appear to differ greatly at large , it is known that the value of the effective central charge does not vary much in Liouville theory as dilaton coupling is varied. Thus we have dual pairs of conformal field theories (CFTs) (cases 1 and 2) with comparable degrees of freedom. We introduce various assumptions on the elliptic genera of the case 1,2 theories and derive algebraic identities and inequalities. We can determine the shadows and elliptic genera of the case 1,2 theories that exhibit nice dual descriptions. We also discuss the relation of Ref. [4] to case 1 theories.
We work mainly with -type modular invariants in this paper; there are some subtleties in the case of -type modular invariants of umbral moonshine, which will be discussed in a future publication.
Preliminaries: Free-field realizations of <inline-formula><tex-math notation="LaTeX" id="ImEquation24"><![CDATA[${\cal N}=4$]]></tex-math></inline-formula> superconformal systems
In this preliminary section we summarize the basic properties of the large/small superconformal algebras and their free-field realizations.
Large <inline-formula><tex-math notation="LaTeX" id="ImEquation26"><![CDATA[${\cal N}=4$]]></tex-math></inline-formula> superconformal algebra
The large superconformal algebra (SCA), often denoted by “”, is defined as the extension of Virasoro algebra [5] (see also Ref. [6] for a good summary). We have a stress--energy tensor , four supercurrents (), two currents , () whose levels are , , one current , and four Majorana fermions (). The unitarity requires that , should be positive integers, and the central charge is given as
c=6k+k−k++k−.
We set , which parameterizes the “mixing” of two currents. The nontrivial part of the large algebra is written as follows:1
where we have introduced the matrices
αab±,i≡12(±δiaδb0∓δibδa0+ϵiab)
(the third term only contributes if ). They obey the commutation relations:
[α±,i,α±,j]=−ϵijkα±,k,[α±,i,α∓,j]=0,{α±,i,α±,j}=−12δij.
Free-field realization of large <inline-formula><tex-math notation="LaTeX" id="ImEquation51"><![CDATA[${\cal N}=4$]]></tex-math></inline-formula> SCA
Let us consider a conformal system composed of a free boson , four Majorana fermions (), and an current . and are normalized as , , and the current algebra is written in our convention as
ji(z)jj(w)∼k2δij(z−w)2+iϵijkz−wjk(w).
We can combine the fermionic components to the currents, and obtain the level “total currents”:
Ji(z):=ji(z)−i2ϵijkψj(z)ψk(z).
We set
T:=−12(∂ϕ)2−12ψa∂ψa+1Njiji,Ga:=iψa∂ϕ−22Nαab+,ijiψb−162Niϵabcdψbψcψd,A+,i:=−i2αab+,iψaψb+ji,A−,i:=−i2αab−,iψaψb,U:=N2i∂ϕ,Qa:=N2ψa.
More explicitly, we can rewrite
G0=iψ0∂ϕ+2N(ψiji−iψ1ψ2ψ3),
which corresponds to the subalgebra ( is identified as the superpartner of ), and also
Gi=iψi∂ϕ−2N(ψ0ji+ϵijkψjjk−i2ϵijkψjψkψ0),A+,i=−i2ψiψ0−i4ϵijkψjψk+ji,A−,i=i2ψiψ0−i4ϵijkψjψk.
They generate the large algebra with parameters
c=6(N−1)N(≡6k+k−k++k−),k+=N−1,k−=1,γ=1N.
In fact, the total central charge is calculated as
c=1+4×12+3kk+2=6(k+1)k+2≡6(N−1)N.
As is familiar, the “zero-mode subalgebra” of the large is the super-Lie algebra with generated by , , , , , (for the Neveu--Schwarz (NS) sector).
Deformation by linear dilaton
We next consider a deformation of (2.7) by turning on the linear dilaton (background charge) along . We shall deform it as
T(z)→T˜(z):=T(z)−Q2∂2ϕ,Ga(z)→G˜a(z):=Ga(z)+Qi∂ψa(z).
It keeps the superconformal symmetry generated by , . In other words, behaves as spin-3/2 primary fields with respect to the deformed stress tensor :
T~(z)G~a(w)∼32(z−w)2G~a(w)+1z−w∂G~a(w).
The central charge is shifted as
c~=c+3Q2≡6(1−1N)+3Q2.
In the end, we obtain the modified large SCA generated by as follows:
where is defined as
γ~:=γ−Q2(k+2)≡1N−Q2N.
Here we point out a fact that will be crucial in our arguments: even though the central charge (2.15) depends on the background charge , the effective central charge [7], which counts the net degrees of freedom, is unchanged under the deformation by linear dilaton. In the relevant system, the effective central charge should be
ceff=c≡6(1−1N),
irrespective of the value of .
To conclude this section, we note that the whole SCA (2.16) reduces to the small SCA, if we choose particular values of . In fact, inspecting the operator product expansion (OPE) of given in (2.16), we find that spin-1/2 currents decouple when we set or . We thus obtain the next two “small points”:
Case 1. : small SCA of level ]
This case is just the familiar Callan--Harvey--Strominger (CHS) superconformal system [8]. We have and . Then, , , are decoupled, and generate the small SCA of level 1.
Case 2. : small SCA of level ]
In this case, we have and . Then, , , are decoupled, and generate the small SCA with level .
These types of reductions from the large to the small with level or have already been discussed in Ref. [5], and also potentially utilized in Refs. [9,10] in order to construct the Feigin--Fuchs representation of SCFT. In the context of string theory on the NS5--NS1 background, case 1 is identified with the world-sheet CFT for the “short string” sector (or that describing the “Coulomb branch tube”), while case 2 corresponds to the “long string” sector (or the “Higgs branch tube”) [11]. Therefore, they are expected to be dual to each other from the viewpoints of duality. In our discussions later, this fact would suggest the existence of two different descriptions of the umbral moonshine [2,3] based on cases 1 and 2. Indeed, these two theories have equal effective central charges, as has already been mentioned, which implies essentially the same asymptotic growth of massive (non-Bogomol’nyi--Prasad--Sommerfield (BPS)) excitations characterizing the moonshine phenomena.
Elliptic genera of the <inline-formula><tex-math notation="LaTeX" id="ImEquation123"><![CDATA[${\cal N}=4$]]></tex-math></inline-formula> Liouville modelOutline sketch
Our main purpose is to evaluate the elliptic genera of the Liouville theory with suitable Liouville potentials. However, it seems hard to directly carry out this computation because of the complexity of the Liouville potentials.
We shall thus take another route: we regard the relevant superconformal system as the -orbifold of
SU(2)N/U(1)⊗SL(2)N/U(1)≅SU(2)N/U(1)⊗[N=2 Liouville]Q=2N,
for “case 1” () [12], and then try to deform the system into “case 2” (). The following statements are crucial in our evaluation of the elliptic genus of case 2.
(i) Cases 1 and 2 correspond to theories with different central charges. Nevertheless, these two theories should possess equal effective central charges (2.18):
ceff=6(1−1N),
as we have already emphasized. characterizes the asymptotic growth of the degeneracy of states due to the Cardy formula. In terms of the elliptic genus, since is -invariant, (2.18) implies that the IR behavior of becomes2limτ2→+∞e−2πτ2ceff24|Z(NS)(τ)|≡limτ2→+∞e−2πτ2(14−14N)|Z(NS)(τ)|<∞.
For case 1, we can easily confirm that this condition is satisfied. We will later discuss how we can refine the constraint (3.1) due to the symmetry that is part of the superconformal symmetry in case 2. Indeed, the resultant constraint will play a crucial role in determining the elliptic genus of case 2.
(ii) When evaluating the elliptic genera, the contributing states in these systems are weighted by different currents. Namely, setting
ψ±:=12(ψ3∓iψ0),χ±:=12(ψ1±iψ2),
the current for each case is written as follows:
case 1 :
J(c^=2):=2A−,3=i(ψ3ψ0−ψ1ψ2)=ψ+ψ−+χ+χ−.
case 2 :
J(c^=2(N−1))≡2A+,3:=i(−ψ3ψ0−ψ1ψ2)+2j3=−ψ+ψ−+χ+χ−+2j3.
(iii) Elliptic genera should be invariant under generic marginal deformations, at least for the holomorphic part that is contributed from the BPS states. Moreover, the nonholomorphic part (“holomorphic anomaly”) only includes the continuous spectrum propagating in the asymptotic region where the relevant Liouville potentials are negligible. These facts imply that both the holomorphic and nonholomorphic terms of the elliptic genus of case 2 do not depend on the details of the Liouville potential.
Based on these considerations, we propose that the elliptic genus of case 2 can be uniquely determined in the following way:
We first evaluate the elliptic genus of case 1, i.e., .
Secondly, we deform the holomorphic anomaly term in , taking account of the distinction between the currents of (3.3) and (3.4). The expected nonholomorphic term should be expanded in terms of the massive characters of .
Finally, we determine the holomorphic part of the elliptic genus desired, which is expected to be written in terms of the massless characters of . It will be crucial that the possible ambiguity by adding general holomorphic Jacobi forms can be removed by examining the IR behavior of the NS-sector elliptic genus, which extends the argument of effective central charge given above.
PreliminariesBranching relation for the <inline-formula><tex-math notation="LaTeX" id="ImEquation145"><![CDATA[${\cal N}=2$]]></tex-math></inline-formula> minimal model
As preparation, we start with recalling the coset construction of the minimal model.
The character for the spin--integrable representation is given as
χℓ(k)(τ,z):=1iθ1(τ,z)[Θℓ+1,k+2(τ,z)−Θℓ+1,k+2(τ,−z)]≡2iΘ1,(τ,z)Θℓ+1,k+2[−](τ,z),(ℓ=0,1,…,k)
and the string function is defined by the expansion
χℓ(k)(τ,z)=∑m∈Z2kcmℓ(τ)Θm,k(τ,z).
The branching relation describing the minimal model as the supercoset
SU(2)k×SO(2)1U(1)k+2
is written as3χℓ(k)(τ,w)Θs,2(τ,w−z)=∑m∈Z2(k+2)χmℓ,s(τ,z)Θm,k+2(τ,w−2z/(k+2)),
where the branching function is explicitly written as
χmℓ,s(τ,z)=∑r∈Zkcm−s+4rℓ(τ)Θ2m+(k+2)(−s+4r),2k(k+2)(τ,z/(k+2)).
The characters of the minimal model are written in terms of the branching functions as follows:
ℓ,m(NS)(τ,z)chℓ,m(NS˜)(τ,z)chℓ,m(R)(τ,z)chℓ,m(R˜)(τ,z)====χmℓ,0(τ,z)+χmℓ,2(τ,z),χmℓ,0(τ,z)−χmℓ,2(τ,z),χmℓ,1(τ,z)+χmℓ,3(τ,z),χmℓ,1(τ,z)−χmℓ,3(τ,z).
Now, the parameter in (3.7) is interpreted as the deformation parameter (or ) in the previous section. It is explicitly identified as
w=2γ~z,
and the corresponding current is given as
Jγ~=2γ~(A+,3−A−,3)+2A−,3≡2γ~(j3−ψ+ψ−)+(ψ+ψ−+χ+χ−).
The relevant branching relations are summarized as
Let us introduce some notations. For 0})$]]>, we set
f(N)(τ,z):=∑n∈Zy2NnqNn21−yqn,(q≡e2πiτ,y≡e2πiz),
and, for 0})$]]>,
F(N,K)(v,a;τ,z):=1N∑b∈ℤNe−2πivbNy2KaNqKa2Nf(NK)(τ,z+aτ+bN)≡∑n∈a+Nℤ(yqn)vNy2KnNqKn2N1−yqn.
We often use the abbreviation . Note that the extended discrete character of the supercoset with [13,14] is written as
χdis(N,K)(v,a;τ,z)=Θ1,(τ,z)iη(τ)3F(N,K)(v,a;τ,z).
The modular completion of is defined as [15]:
f^(N)(τ,z):=f(N)(τ,z)−12∑m∈Z2NRm,N(τ)Θm,N(τ,2z),
where we set
Rm,N(τ):=∑ν∈m+2Nℤsgn(ν+0)Erfc(πτ2N|ν|)q−ν24N≡1iπ∑ν∈m+2Nℤ∫ℝ−i0dpe−πτ2p2+ν2Np−iνq−ν24N.
Here, we denote and is the error function (A12). is a weight , index (real analytic) weak Jacobi form [15].
It is useful to rewrite (3.17) by introducing the antisymmetrization
f(N)[−](τ,z):=12[f(N)(τ,z)−f(N)(τ,−z)].
We note that
F^(N)[−](τ,z):=12[F^(N)(τ,z)−F^(N)(τ,−z)]≡F^(N)(τ,z),
since the completion gives an odd function of (see, e.g., Ref. [14]). Then, we obtain
F^(N)(τ,z)=f(N)[−](τ,z)−12∑m∈ℤ2NRm,N(τ)Θm,N[−](τ,2z)=f(N)[−](τ,z)−∑m=1N−1Rm,N(τ)Θm,N[−](τ,2z).
In the second line of (3.19), we made use of the facts that and
R−m,N(τ)=−Rm,N(τ)+2δm,0(2N),Rm+2N,N(τ)=Rm,N(τ).
We also introduce the modular completion of (3.15) [14]:
F^(N,K)(v,a;τ,z):=1N∑b∈ℤNe−2πivbNy2KaNqKa2NF^(NK)(τ,z+aτ+bN)≡F(N,K)(v,a;τ,z)−12∑j∈ℤ2KRv+Nj,NK(τ)Θv+Nj+2Ka,NK(τ,2zN).
In particular, for the cases of , the function becomes
F^(N)(v,a;τ,z)=F(N)(v,a;τ,z)−12∑j∈Z2Rv+Nj,N(τ)Θv+Nj+2a,N(τ,2zN).
Case 1 models
We propose the following elliptic genus for the case 1 models with :
Zcase 1(τ,z)=1N∑a,b∈ℤNZ[a,b](min)(τ,z)Z[a,b]SL(2)/U(1)(τ,z)=−Θ1,(τ,z)iη(τ)3∑ℓ=0N−2∑a∈ℤNchℓ,ℓ+1+2a(R˜)(τ,z)F^(N)(ℓ+1,a;τ,−z),
where we set
Z[a,b](min)(τ,z):=(−1)a+b+abqN−22Na2yN−2NaeiπN−2NabZ(min)(τ,z+aτ+b),Z(min)(τ,z):=θ1(τ,N−1Nz)θ1(τ,zN)≡∑ℓ=0N−2chℓ,ℓ+1(R~)(τ,z),
for the minimal model ( sector) [16], and
Z[a,b]SL(2)/U(1)(τ,z):=(−1)a+b+abqN+22Na2yN+2NaeiπN+2Nab×ZSL(2)/U(1)(τ,z+aτ+b),ZSL(2)/U(1)(τ,z):=Θ1,(τ,z)iη(τ)3F^(N)(τ,zN)≡Θ1,(τ,z)iη(τ)3∑v∈ℤNF^(N)(v,0;τ,z),
for the sector [14,17].
The relevant branching relation is given by (3.12), namely,
∑m∈Z2Nchℓ,m(R~)(τ,z)Θm,N(τ,−2zN)=−iΘ1,(τ,z)χℓ(k)(τ,0)=−Θ1,(τ,z)iπ∮w=0dwwΘℓ+1,N[−](τ,2w)θ1(τ,2w).
By using this identity, we can show
[nonhol. part of Zcase 1]=12Θ1,(τ,z)iη(τ)3∑ℓ=0N−2∑a∈ZN∑j∈Z2Rℓ+1+Nj,NΘℓ+1+Nj+2a,N(−2zN)chℓ,ℓ+1+2a(R~)(z)=−Θ1,(τ,z)2η(τ)31iπ∮w=0dww1iθ1(τ,2w)∑ℓ=0N−2Rℓ+1,NΘℓ+1,N[−](τ,2w).
Therefore, recalling (3.19), we find that
∂∂τ¯Zcase 1(τ)=∂∂τ¯[Θ1,(τ,z)2η(τ)31iπ∮w=0dwwF^(N)(τ,w)iθ1(τ,2w)]=∂∂τ¯[Θ1,(τ,z)2η(τ)31iπ∮w=0dwwF^(N)(τ,w)iθ1(τ,2w)e(N−2)G2(τ)w2],
where is the (unnormalized) second Eisenstein series (A7). In order to obtain the third line of (3.29), we made use of the fact that the function
∂∂τ¯F^(N)(τ,w)iθ1(τ,2w)≡∂∂τ¯F^(N),[−](τ,w)iθ1(τ,2w),
is holomorphic with respect to . It is important that the integrand
g(N)(τ,w):=F^(N)(τ,w)iθ1(τ,2w)e(N−2)G2(τ)w2
possesses the correct modular property due to the factor (see Eq. (A8)), i.e.,
g(N)(−1τ,wτ)=−−iτg(N)(τ,w).
Thus, we conclude that
Zcase 1(τ,z)=[holomorphic Jacobi form]+Θ1,(τ,z)2η(τ)31iπ∮w=0dwwF^(N)(τ,w)iθ1(τ,2w)e(N−2)G2(τ)w2.
It is easy to identify the first term because of the uniqueness of the weak Jacobi form of weight 0, index 1, i.e.,
ϕ0,1(τ,z)≡12ZK3(τ,z)≡4[(θ3(τ,z)θ3(τ,0))2+(θ4(τ,z)θ4(τ,0))2+(θ2(τ,z)θ2(τ,0))2].
We also recall that the Witten index should be
Zcase 1(τ,0)=N−1
(see, e.g., Ref. [18]), which fixes the normalization of the holomorphic term.
In this way, we finally achieve a simple formula:
Zcase 1(τ,z)=N−112ϕ0,1(τ,z)+Θ1,(τ,z)2η(τ)31iπ∮w=0dwwF^(N)(τ,w)iθ1(τ,2w)e(N−2)G2(τ)w2.
Useful facts on Eq. (<xref ref-type="disp-formula" rid="PTW078M3_35">3.35</xref>)
We exhibit some useful computations related to the resultant formula (3.35) and make a few remarks. For convenience, we define the nonholomorphic modular form of weight 2 by the relation
Zcase 1(τ,z)=N−112ϕ0,1(τ,z)+Θ1,(τ,z)2η(τ)6H^(N)(τ).
Namely, we set
H^(N)(τ):=η(τ)3iπ∮w=0dwwF^(N)(τ,w)iθ1(τ,2w)e(N−2)G2(τ)w2.
Then, by substituting the “Poincaré series” formula [19],
F^(N)(τ,z)=i2π∑λ∈Λe−πτ2N{|λ|2+2λ¯z+z2}λ+z,(Λ≡Zτ+Z),
into (3.37), we can rewrite it as
H^(N)(τ)=14π2[NG^2(τ)+∂∂w∑λ∈Λ′e−πτ2N{|λ|2+2λ¯w+w2}λ+w|w=0]=14π2[NG^2(τ)−∑λ∈Λ′e−πτ2N|λ|2λ2{1+2πNτ2|λ|2}].
Here, the summation is taken over , and denotes the modular completion of the (unnormalized) second Eisenstein series (A9). We present an explicit derivation of (3.39) in Appendix B.
The above result (3.39) suggests that the holomorphic part would be
H(N)(τ)∼N4π2G2(τ)+14π2∂∂w∑λ=mτ+n∈Λ′qNm2e2πi(2N)mwλ+w|w=0.
However, the double series appearing in (3.40) does not converge, and thus we have to be more careful. To this end, we introduce the symbol of the “principal value”:
∑n≠0Pan:=limN→∞∑n=1N(an+a−n),∑n∈ZPan:=a0+∑n≠0Pan,
and the correct expression of should be
H(N)(τ)=N4π2G2(τ)+14π2∂∂w[∑m≠0∑n∈ZPqNm2e2πi(2N)mwλ+w+∑n≠0P1w+n]|w=0.
A rigorous derivation of (3.42) is again presented in Appendix B.
One would be interested in the -expansion of . Substituting the familiar formula (A7) as well as
∑n=1∞1n2≡ζ(2)=π26,i2π∑n∈ZP1z+n=12+y1−y=−12+11−y,
into (3.42), we readily obtain
H(N)(τ)=N−112−2N∑n=1∞nqn1−qn+2∑m=1∞qNm2[qm(1−qm)2+Nm1+qm1−qm].
Let us make the following remarks:
The constant term is anticipated. It precisely cancels the “graviton term” included in the holomorphic Jacobi form appearing in (3.35).
In fact, the term in yields the leading contribution (“graviton term”), after making the spectral flow . We thus obtain the evaluation
[graviton term]∼−N−13(θ4(τ,z)θ2(τ))2∼−N−112q−14,(τ→i∞),
by using On the other hand, the first term of (3.43) yields
N−112θ3(τ,z)2η(τ)6∼N−112q−14,(τ→i∞).
It is easily confirmed that the function has -expansion such as
12N−1H(N)(τ)=1−∑n=1∞anqn,
with integer coefficients as long as divides 24. Moreover, since the second term in (3.43) looks more dominant than the third one, we expect that all the coefficients are positive.
Amusingly, for the special case of Mathieu moonshine, we have an alternative expression for as
H^(2)(τ)=η(τ)32πi∮w=0dwwf˜^(1/2)(τ,w)iθ1(τ,w)=14π2[G2(τ)+∑λ=mτ+n∈Λ′(−1)m+n+mn2πime−π2τ2|λ|2λ].
The first line follows from the identity4F^(2)(τ,z)=θ1(τ,2z)2Θ1,(τ,z)f~^(1/2)(τ,z),
where we set
f~^(1/2)(τ,z):=F^(2)(1,0;τ,z)−F^(2)(1,1;τ,z)≡f˜(1/2)(τ,z)+[nonhol. correction],f˜(1/2)(τ,z):=∑n∈ℤ(−1)n(yqn)121−yqnynq12n2.
The second line of (3.47)5 is obtained by substituting another formula of the nonholomorphic Poincaré series:
f~^(1/2)(τ,z)=i2π∑λ=mτ+n∈Λ(−1)m+n+mne−π2τ2{|λ|2+2λ¯z+z2}λ+z.
The second line of (3.47) is again derived in Appendix B.
We can also rewrite the first line of (3.47) by replacing in the integrand with an elliptic function defined by
ξ(τ,w):=∂∂wln(θ1(τ,w)3θ2(τ,w)θ3(τ,w)θ4(τ,w)).
It is easy to see that is an elliptic function of order 4 that possesses simple poles at with the residues
Resw=0[ξ(τ,w)]=3,Resw=12[ξ(τ,w)]=Resw=τ2[ξ(τ,w)]=Resw=τ+12[ξ(τ,w)]=−1.
Then, the integrand in (3.47) becomes an elliptic function with a cubic pole and simple poles . We thus obtain by the contour deformation
H^(2)(τ)=η(τ)3Resw=0[13ξ(τ,w)f˜^(1/2)(τ,w)iθ1(τ,w)]=−13η(τ)3∑w0=12,τ2,τ+12Resw0[ξ(τ,w)f˜^(1/2)(τ,w)iθ1(τ,w)]=13η(τ)3∑w0=12,τ2,τ+12f˜^(1/2)(τ,w0)iθ1(τ,w0).
The holomorphic part of the R.H.S. of (3.53) is identical to the known expression [1] of Mathieu moonshine, which is derived using the relation among character formulas at level 1 [20–22]:
ch0(R~)(k=1,ℓ=0;τ,z)=θ1,(τ,z)iη(τ)3f~(1/2)(τ,z)=(θi(τ,z)θi(τ,0))2+θ1,(τ,z)2η(τ)3f~(1/2)(τ,wi)iθ1(τ,wi),(∀i=2,3,4),(w2≡12,w3≡τ+12,w4≡τ2).
Comments on the shadow
As a consistency check of (3.39), let us evaluate its “shadow” [15]. After a short calculation, we obtain
τ2∂∂τ¯H^(N)(τ)=iN8πτ23/2∑λ∈Λe−πτ2N|λ|2(1−2πNτ2|λ|2).
On the other hand, due to the Poisson resummation, we find
∑r∈Z2Nθr,N(τ,zL)θr,N(τ,zR)¯=Nτ2e−πN4τ2(zL−zR¯)2∑λ∈Λe−πNτ2{|λ|2+(λ¯zL−λzR¯)},
and thus
∑r∈Z2N∂zLθr,N(τ,zL)∂zRθr,N(τ,zR)¯|zL=zR=0=πN3/22τ23/2∑λ∈Λe−πτ2N|λ|2(1−2πNτ2|λ|2).
Therefore, introducing the “unary theta function” [15]
Sr,N(τ):=12πi∂zθr,N(τ,2z)|z=0≡∑n∈r+2NZnqn24N,
we finally obtain
τ2∂∂τ¯H^(N)(τ)=i4N∑r∈Z2N|Sr,N(τ)|2.
This is the expected result. Indeed, by using the familiar property of the “R-function” of Ref. [15],
τ2∂∂τ¯Rm,N(τ)=−i2NSm,N(τ)¯.
Together with (3.19) and (3.37), we can directly evaluate the shadow of as
τ2∂∂τ¯H^(N)(τ)=−1iπ∮dwwη(τ)3iθ1(τ,2w)∑v=1N−1τ2∂∂τ¯Rv,N(τ)θv,N[−](τ,2w)=−1iπ⋅2πi⋅14πi−12πi∑v=1N−1−i2NSv,N(τ)¯∂wθv,N(τ,2w)|w=0=i4N∑r∈ℤ2N|Sr,N(τ)|2,
which coincides with (3.59).
It may also be useful to evaluate the shadow of based on the formula of the nonholomorphic Poincaré series (3.38):
F^(N)(τ,z)=i2π∑λ∈Λρ(N)(λ,z)λ+z≡i2π∑λ∈Λe−πNτ2{|λ|2+2λ¯z+z2}λ+z,
where we have introduced the notation
ρ(k)(λ,z):=e−πkτ2{|λ|2+2λ¯z+z2}≡qkm2y2kme2πikmne−πkτ2z2.
Using
∂∂τ¯ρ(N)(λ,z)=iπN2τ22(λ+z)2ρ(N)(λ,z),
we easily obtain
τ2∂∂τ¯F^(N)(τ,z)=−N4τ23/2∑λ∈Λ(λ+z)ρ(N)(λ,z).
However, by differentiating with respect to on both sides of (3.56), we find
∑r∈Z2Nθr,N(τ,2z)Sr,N(τ)¯=iN3/2τ23/2∑λ∈Λ(λ+z)ρ(N)(λ,z).
We thus obtain
τ2∂∂τ¯F^(N)(τ,z)=i4N∑r∈Z2Nθr,N(τ,2z)Sr,N(τ)¯,
which is consistent with Eqs. (3.17) and (3.60).
Elliptic genus of case 2 theory
Now, let us begin our analysis of the system of “case 2”, i.e., the theory with . According to our strategy addressed in Sect. 3.1, we propose the elliptic genus of case~2 by modifying the coupling of the variable in . The relevant branching relation is now (3.13), in other words,
2θ1,(τ,z)θ1(τ,2z)θℓ+1,N[−](τ,2z)=∑m∈Z2Nchℓ,m(R~)(τ,z)θm,N(τ,2(N−1)Nz).
This identity can be interpreted as the decomposition of the massive characters with isospin , conformal weight :
ch(R~)(N−1,h,j;τ,z)≡(−1)N−ℓqp222θ1,(τ,z)2iη(τ)3θ1(τ,2z)θℓ+1,N[−](τ,2z)=(−1)N−ℓqp22θ1,(τ,z)iη(τ)3∑m∈Z2Nchℓ,m(R~)(τ,z)θm,N(τ,2(N−1)Nz).
This formula suggests the following elliptic genus of the case 2 theory,
Zcase 2(τ,z)=θ1,(τ,z)iη(τ)3∑ℓ=0N−2∑a∈ZNchℓ,ℓ+1+2a(R~)(τ,z)F^(N)(ℓ+1,a;τ,(N−1)z),
up to an overall phase factor. This is obtained by the replacement in the function appearing in (3.22), which properly corrects the difference between the charges of cases 1 and 2, and reproduces the expected holomorphic anomaly terms expanded by the massive characters. (Recall the branching relations (3.12) and (3.13).)
Proof of an identity
We recall that the function is simply related to the massless character
ch0(R~)^(N−1,0;τ,z)≡(−1)N−12θ1,(τ,z)2iη(τ)3θ1(τ,2z)F^(N)(τ,z).
Here, denotes the modular completion of the massless character of level , isospin in the sector. This is actually the unique modular completion of massless characters since they are independent of the value of isospin , as was discussed in Ref. [19]. We shall now prove the following important identity:
2θ1,(τ,z)θ1(τ,2z)F^(N)(τ,z)=∑ℓ=0N−2∑a∈ZNchℓ,ℓ+1+2a(R~)(τ,z)F^(N)(ℓ+1,a;τ,(N−1)z).
This may be interpreted as a massless counterpart of (3.66), and it implies that (3.68) is written in terms of the modular completion .
Proof of (3.70). We set
G(R~)(τ,z):=2θ1,(τ,z)2iη(τ)3θ1(τ,2z)F^(N)(τ,z)−θ1,(τ,z)iη(τ)3∑ℓ=0N−2∑a∈ZNchℓ,ℓ+1+2a(R~)(τ,z)F^(N)(ℓ+1,a;τ,(N−1)z),
and prove .
First of all, it is obvious by definition that possesses the correct modular and spectral flow properties as a weak Jacobi form of weight 0, index .
We next discuss more nontrivial properties of the function :
(i) Holomorphicity with respect to :
By using (3.21), (3.60), and (3.66), we obtain
τ2∂∂τ¯[∑ℓ=0N−2∑a∈ZNchℓ,ℓ+1+2a(R~)(τ,z)F^(N)(ℓ+1,a;τ,(N−1)z)]=i4N∑ℓ=0N−2∑a∈ZN∑j∈Z2chℓ,ℓ+1+2a(R~)(τ,z)Sℓ+1+Nj,N(τ)¯θℓ+1+Nj+2a,N(τ,2(N−1)Nz)=i4N2θ1,(τ,z)θ1(τ,2z)∑ℓ=0N−2[Sℓ+1,N(τ)¯θℓ+1,N[−](τ,2z)+SN−ℓ+1,N(τ)¯θN−ℓ+1,N[−](τ,2z)]=i4N2θ1,(τ,z)θ1(τ,2z)∑r∈Z2NSr,N(τ)¯θr,N(τ,2z).
In the third line, we have made use of the identities and . Combining (3.72) with (3.65), we conclude that
∂∂τ¯G(R~)(τ,z)=0.
(ii) Holomorphicity with respect to :
We next confirm the holomorphicity with respect to ; in other words, the absence of singularities in the -variable. To this aim it would be useful to rewrite the function in the form of a nonholomorphic Poincaré series (3.38):
G(R~)(τ,z)=θ1(z)22πη3θ1(2z)∑λ∈Λρ(N)(λ,z)z+λ−θ1(z)2πη3∑λ≡mτ+n∈Λ(−1)m+nyN−2NmqN−22Nm2e−2πimnNθ1(N−1N(z+λ))θ1(1N(z+λ))ρ(1N)(−λ,(N−1)z)(N−1)z−λ,
where has been defined by (3.62). The first term is obviously holomorphic.
On the other hand, the potential singularities of the second term emerge at the points
z=1N−1λ,(∀λ∈Λ).
They are, however, canceled by the simple zeros coming from the factor since
N−1N(λN−1+λ)=N−1NNλN−1=λ∈Λ.
Also, the factor gives rise to simple poles
z=−λ+Nν,(∀ν∈Λ),
which are canceled by the remaining . Therefore we have confirmed the holomorphicity of .
(iii) IR behavior of around :
We next examine the IR behavior of , which is defined by the half-spectral flow as
G(NS)(τ,z):=qN−14yN−1G(R~)(τ,z+τ+12).
The first term of (3.71) is essentially equal to , which is converted into by the half-spectral flow, yielding the IR behavior
On the other hand, the second term of (3.71) yields (up to phases)
∼θ3(z)η3∑ℓ=0N−2∑a∈ZNchℓ,ℓ+2a(NS)(z)q(N−1)24Ny(N−1)2NF^(N)(ℓ+1,a;τ,(N−1)(z+τ+12)).
The leading contribution obviously comes from the term of , , which gives the IR behavior
∼q−18−N−28N+(N−1)24N+N−12N=qN−14.
In this way, we find
G(NS)(τ,z)∼O(qN−14),(τ∼i∞).
In summary, we have shown that should be a holomorphic weak Jacobi form with weight 0, index , and that it satisfies (3.78). This is enough to conclude that because of the lemma given in Appendix C. ■
Considerations on the effective central charge
The above result for the elliptic genus of case 2 Liouville theory is summarized as
Zcase 2(τ,z)=θ1,(τ,z)iη(τ)3∑ℓ=0N−2∑a∈ZNchℓ,ℓ+1+2a(R~)(τ,z)F^(N)(ℓ+1,a;τ,(N−1)z)=(−1)N−1ch0(R~)^(N−1,0;τ,z).
Namely, we claim that the elliptic genus of case 2 should be equal to the modular completion of the massless character itself. This has been suggested from the consideration of the holomorphic anomaly (or the shadow) given in Sect. 3.1. It is similar to the case of the elliptic genus of Liouville theory ( supercoset), which is given only in terms of modular completions of massless characters [14,17].
However, one might still ask: The holomorphic anomaly remains unchanged even if we add any holomorphic Jacobi form with weight 0, index to (3.79). How can we reject this possibility? In the simplest case of , the absence of the holomorphic Jacobi form () just means the familiar fact of decoupling gravity in the noncompact models (see, e.g., Ref. [18]). However, for the cases with 2$]]>, we have a number of holomorphic Jacobi forms, and the situation becomes much more nontrivial.
To answer this question and confirm the validity of the above result (3.79), let us present a consideration about the effective central charge mentioned before. We start with refining the condition (3.1) based on the affine symmetry as the underlying structure of SCA, as we promised. To show this, we expand the elliptic genus in the NS sector in terms of the angular variable as
Z(NS)(τ,z)≡qN−14yN−1Z(τ,z+τ+12)=12∑s∈Z≥0(ys+y−s)Zs(NS)(τ).
Then, we obtain the constraint
limτ2→+∞e−2πτ2{ceff24−ℓ(ℓ+2)4N}|Zs(NS)(τ)|≡limτ2→+∞e−2πτ2{14−(ℓ+1)24N}|Zs(NS)(τ)|<∞,
where is defined by
ℓ:={|s|,(0≤|s|≤N−2)N−1,(|s|≥N−1).
In fact,
:
The leading term contributing to is composed of the highest-weight state of spin of the bosonic current , and the NS vacuum of fermions. Thus, the condition (3.81) is obviously satisfied.
:
The leading term contributing to is composed of the highest-weight state of spin of , and the level state of fermions. We thus obtain the IR evaluation as
Zs(NS)(τ)∼qα,α≥(N−2)N4N+12−ceff24≥N2−14N−ceff24.
Therefore, the condition (3.81) is still satisfied.
Now, let us expand the holomorphic part of the elliptic genus in the NS sector in terms of characters:
Zhol(NS)(τ,z)=∑ℓ=0N−1aℓch0(NS)(N−1,ℓ;τ,z)+∑j=0N−2∑n∈Z≥0bj,nch(NS)(N−1,h=n+N−24,j;τ,z).
Here, the holomorphic part has been defined so that it shows the same IR behavior as for each term of the -expansions.6
It is easy to confirm that the constraint (3.81) requires that the conformal weight of the massive representations should satisfy the inequality
h≥h(j)≡j(j+2)4N+(N−1)24N.
It is notable that the second term is equal to the mass gap due to the linear dilaton.
On the other hand, the IR behavior of the NS massless character of isospin is evaluated as :
ch0(NS)(N−1,ℓ;τ,z)∼qℓ2−N−14,(τ∼i∞).
Thus, the constraint (3.81) implies
ℓ2−N−14−(ℓ+1)24N+14≥0,⟺(ℓ+1−N)2≤0.
That is, only the maximal spin massless representation is allowed.
Consequently, (3.81) implies that
Zholcase 2(NS)(τ,z)=aN−1ch0(NS)(N−1,ℓ=N−1;τ,z)+∑j=0N−2∑n∈Z≥0,h≥h(j)bj,nch(NS)(N−1,h=n+N−24,j;τ,z),
or, equivalently,
Zholcase 2(R~)(τ,z)=a0′ch0(R~)(N−1,ℓ=0;τ,z)+∑j=1N−1∑n∈Z≥0,h≥h(j−1)+14bj,n′ch(R~)(N−1,h=n+N−14,j;τ,z).
The above result (3.79) is indeed consistent with this character expansion (3.88) (with vanishing coefficients ).
On the other hand, as shown in Ref. [2], any holomorphic Jacobi form can never be written in the form (3.88); i.e., we need additional contributions from the massless characters with , or the massive characters with conformal weight “below the mass gap” in order to construct a holomorphic Jacobi form. In this way, we conclude that there is no room for adding extra holomorphic Jacobi forms to (3.79).
Some remarks
We add a few remarks on the analyses in this subsection:
(i) It should be emphasized that case 2 is not equivalent to
SU(2)N/U(1)⊗[N=2 Liouville]Q=−(N−1)2N|ZN−orbifold,
which is a superconformal system of the type studied in Ref. [23]. The same value of linear dilaton is expected for our case 2, but the Liouville potential does not preserve the superconformal symmetry except for the special case (). In fact, we have
c^hid(N,(N−1)2)(v,a;τ,z)≡Θ1,(τ,z)iη(τ)3F^(N,(N−1)2)(v,a;τ,z)
as suitable building blocks for the Liouville theory with , which is equivalent to the supercoset. The elliptic genus of (3.89) is obtained as
Z(τ,z)=−Θ1,(τ,z)iη(τ)3∑ℓ=0N−2∑a∈ZNchℓ,ℓ+1+2a(R~)(τ,z)F^(N,(N−1)2)(ℓ+1,a;τ,−z),
which is similar to (3.22). On the other hand, (3.79) can be rewritten as
Zcase 2(τ,z)=−Θ1,(τ,z)iη(τ)3∑ℓ=0N−2∑a∈ZNchℓ,ℓ+1+2a(R~)(τ,z)×1N−1∑λ∈Λ/(N−1)Λsλ((N−1)2N)⋅F^(N,(N−1)2)(ℓ+1,a;τ,−z),
where denotes the spectral flow operator defined by (A10). Stated physically, the orbifolding (or the Eichler--Zagier operator [24])
W(N−1)≡1N−1∑λ∈Λ/(N−1)Λsλ(∗)
reduces the radius of the asymptotic cylindrical region of the Liouville theory so as to be compatible with the superconformal symmetry.
(ii) An important consequence of the “ condition” (3.81), which physically means the normalizability of a spectrum, is the inevitable emergence of the holomorphic anomaly in as long as we require its good modular property.
It is also worthwhile to note the fact that the formulas of modular -transformations of any superconformal characters in the sector are schematically written as
[N=4 character]|S=Ach0(R~)(ℓ=0)+∑j∫h≥h(j)[N=4 massive character],
where the coefficient on the R.H.S. does not vanish when the character of the L.H.S. is any massless character. Namely, we find the following facts for the representations on the R.H.S. of (3.93):
Only the massless character can appear.
Only massive characters with conformal weight satisfying (“above the mass gap”) appear.
These representations are precisely identical to those satisfying the condition (3.81). This feature is indeed anticipated, since the modular invariant partition function (and thus the elliptic genus) should only include the normalizable spectrum that contributes to the net degrees of freedom.
“Duality” in <inline-formula><tex-math notation="LaTeX" id="ImEquation324"><![CDATA[${\cal N}=4$]]></tex-math></inline-formula> Liouville theory and umbral moonshine
In the previous sections we studied two systems possessing superconformal symmetry:
case 1: ()] The elliptic genus is given as (3.22):
Zcase 1(τ,z)=−Θ1,(τ,z)iη(τ)3∑r=1N−1∑a∈ZNchr−1,r+2a(R~)(τ,z)F^(N)(r,a;τ,−z),
case 2: ()]
The elliptic genus is given as (3.79):
Zcase 2(τ,z)=Θ1,(τ,z)iη(τ)3∑r=1N−1∑a∈ZNchr−1,r+2a(R~)(τ,z)F^(N)(r,a;τ,(N−1)z)=(−1)N−1ch0(R~)^(N−1,0;τ,z).
As we have emphasized several times, these two systems have the same degrees of freedom even though the central charges differ from each other, and are expected to be dual in the sense of the correspondence. We can explicitly observe a very simple correspondence:
F^(N)(v,a;τ,−z)\bf for case 1⟷F^(N)(v,a;τ,(N−1)z)\bf for case 2.
Now, we present some comments on the relation with the analyses on “umbral moonshine” [2–4,25]. In Ref. [4], the authors studied (the holomorphic part of) the extension of (3.22) with general modular coefficients determined by the simply laced root system corresponding to each Niemeier lattice. We have by definition; let be the Coxeter number of . A Niemeier lattice is explicitly expressed as
X=∐iXi,∑irankXi=24,
where each is the irreducible component of the root system possessing the common Coxeter number .
We can schematically write
ZX[c^=2](τ,z)≡∑iZcase 1(Xi)(τ,z):=−Θ1,(τ,z)iη(τ)3∑i∑ri,si=1N−1∑ai∈ZNNri,siXichri−1,si+2ai(R~)(τ,z)F^(N)(si,ai;τ,−z)=−Θ1,(τ,z)iη(τ)3∑r,s=1N−1∑a∈ZNNr,sXchr−1,s+2a(R~)(τ,z)F^(N)(s,a;τ,−z)
where denotes the modular invariant coefficients of associated with the simply laced root system , and we set . One may identify as the elliptic genus of the ALE space associated with the simple singularity of type . In Ref. [4] it was suggested that the root system should be identified as the geometrical data of various K3 singularities.
Since we assume , we can rewrite (4.3) as
ZX[c^=2](τ,z)=2ϕ0,1(τ,z)−Θ1,(τ,z)2η(τ)3h^X(τ),
where is the completion of the mock modular form of weight characterized by the shadow
τ2∂∂τ¯h^X(τ)=−i2N∑r,s=1N−1Nr,sXχr−1(N−2)(τ,0)Ss,N(τ)¯≡−i2N1η(τ)3∑r,s=1N−1Nr,sXSr,N(τ)Ss,N(τ)¯.
This is derived from (4.3) with the help of the branching relation (3.27). (We recall that the space of the weight 0, index 1 holomorphic Jacobi form is 1D and is spanned by ).
Let us next consider the type- generalization of (3.79), which is related to (4.3) via the “duality correspondence” (4.1):
ZX[c^=2(N−1)](τ,z):=Θ1,(τ,z)iη(τ)3∑r,s=1N−1∑a∈ZNNr,sXchr−1,s+2a(R~)(τ,z)F^(N)(s,a;τ,(N−1)z).
Another useful realization of the duality correspondence is given as the natural extension of the identity (3.35):
ZX[c^=2](τ,z)=2ϕ0,1(τ,z)+Θ1,(τ,z)212πi∮w=0dwwe(N−2)G2(τ)w2θ1(τ,w)2ZX[c^=2(N−1)](τ,w)≡2ϕ0,1(τ,z)+Θ1,(τ,z)2η(τ)618π3i∮w=0dwwe(N−1)G2(τ)w2σ(τ,w)2ZX[c^=2(N−1)](τ,w),
where we have introduced the Weierstrass -function (A6) in the second line. In fact, one can straightforwardly confirm that the second term possesses the correct modular property and reproduces the expected shadow (4.5) in a manner similar to the derivation of (3.35).
Similarly to (4.4), the R.H.S. of (4.6) can be decomposed as
ZX[c^=2(N−1)](τ,z)=Φ0,N−1X(τ,z)−Θ1,(τ,z)2η(τ)3∑r=1N−1h^rX(τ)χr−1(N−2)(τ,2z)≡Φ0,N−1X(τ,z)−2Θ1,(τ,z)2iη(τ)3θ1(τ,2z)∑r=1N−1h^rX(τ)Θr,N[−](τ,2z).
In the above expression, is a weak Jacobi form of weight 0, index , which is holomorphic with respect to , but generically meromorphic with respect to . is the affine character of level , isospin , and are the completions of vector-valued mock modular forms of weight 1/2, whose shadow is given as
τ2∂∂τ¯h^rX(τ)=−i2N∑s=1N−1Nr,sXSs,N(τ)¯.
Equation (4.9) again follows from the definition (4.8) and the identity (3.66).
We note a subtlety in the decomposition (4.8). is not necessarily determined only from the shadow (4.9). This is in contrast to in (4.4), which is indeed determined uniquely. For a sufficiently large , there exist nontrivial holomorphic weak Jacobi forms of weight 1, index , which we denote here by, say, . Then, we may add to the first term of (4.8), while subtracting the same function from the second term. We can thus modify in (4.8) while keeping the shadow (4.9) unchanged. To avoid this ambiguity, we should impose the optimal growth condition:
limτ→i∞q14N|h^rX(τ)|<∞,(∀r=1,…,N−1),
according to Ref. [3]. In fact, the above can be expanded by theta functions as
ψ1,N[−](τ,z)=∑r=1N−1αr(τ)Θr,N[−](τ,2z),
and the holomorphic coefficients cannot satisfy the condition (4.10) due to Theorem {\bf 9.7} of Ref. [26]. Consequently, we can remove the ambiguity in the decomposition (4.8), and can determine as well as uniquely.
Now, substituting the decompositions (4.4), (4.8) into Eq. (4.7), we find
h^X(τ)=∑r=1N−1h^rX(τ)χr−1(N−2)(τ,0)≡1η(τ)3∑r=1N−1h^rX(τ)Sr,N(τ),
since the contour integral should be a holomorphic modular form of weight 2, and thus vanishes.7 This is the duality relation between the expansion coefficients of massive representations of and . In the case of Mathieu moonshine (), one has the self-dual situation . In general, holomorphic parts of should reproduce the mock modular form of umbral moonshine on which the umbral group should act [2].
Let us discuss the relation of the present analysis to a closely related consideration given in Ref. [4]. For this purpose it is convenient to introduce the meromorphic Jacobi form with weight 1, index defined by
Ψ1,NX(τ,z):=iη(τ)3θ1(τ,2z)2Θ1,(τ,z)2Φ0,N−1X(τ,z)≡F^(1)(τ,z)Φ0,N−1X(τ,z),
following Refs. [2,3]. Then, (4.8) can be rewritten as
Ψ1,NX(τ,z)=F^(1)(τ,z)ZX[c^=2(N−1)](τ,z)+∑r=1N−1h^rX(τ)Θr,N[−](τ,2z),
where the first and second terms on the R.H.S. correspond to the polar and finite parts in the terminology of Refs. [2,3].
If includes only the -type components, our results are manifestly consistent with those given in Ref. [4]. Namely, both the polar and finite parts in (4.14) (the “case 2 theory” with ) separately correspond to the first and second terms in (4.4) (the “case 1 theory” with ) under the transformation given in Ref. [4]:
φ1,N(τ,z)↦φ′0,N2(τ,z):=θ1(τ,(N−1)z)θ1(τ,Nz)iη(τ)3θ1,(τ,z)φ1,N(τ,z)↦φ″0,1(τ,z):=1N∑λ∈Λ/NΛsλ(N2)⋅φ0,N2′(τ,z/N).
Here, we have denoted a (holomorphic or nonholomorphic) weak Jacobi form of weight and index by the symbol “”, and the second line is identified as the Eichler--Zagier operator ( orbifolding). In fact, the first term on the R.H.S. of (4.14) just becomes
F^(1)(τ,z)ZX[c^=2(N−1)](τ,z)=24N−1F^(N)(τ,z),
by substituting the identity (3.70) into (4.6), and the map (4.15) converts (4.14) into the definition of itself. (Recall the formula for the elliptic genus of the minimal model (3.24).) Furthermore, the correspondence of finite parts by (4.15) is found to be equivalent to the relation (4.12) due to the branching relation (3.27). The fact (4.16) also implies that here coincides with the “umbral Jacobi form” constructed in Ref. [3]. It is also obvious that (4.16) is consistent with our duality relation (4.7) because of Eq. (3.35).
In the cases when includes - or -type components, however, generally differs from the umbral Jacobi form. Indeed, according to the explicit construction of the umbral Jacobi form given in Ref. [3], it would contain -torsion points with created by the Eichler--Zagier operator written schematically as
WX≡∑ni|NciW(ni).
On the other hand, , given above, possesses -torsion points with , which is inherited from the functions in the expression (4.6).8
Nevertheless, if we replace with the umbral Jacobi form of Ref. [3] in the “elliptic genus” , the duality relation (4.7) is still satisfied, because the difference between these meromorphic Jacobi forms just yields a vanishing contribution to the contour integral.
How about the correspondence (4.15)? Indeed, one can show that the claim of Ref. [4] still holds even when we replace the umbral Jacobi form with . Namely, and still correspond to and by the map (4.15), as shown by the following arguments:
(i) It turns out that the finite part of (4.14) is mapped to that of , and the relation (4.12) still holds.
(ii) When considering the map (4.15), all the -torsion points included in are canceled out with the zeros of the factor .9 We thus find that is always mapped to the unique holomorphic Jacobi form with some constant coefficient .
(iii) This coefficient is, however, just determined from the finite part so that the contribution of “graviton representation” should be canceled out. In fact, it is straightforward to confirm that (4.6) cannot include such a term after converting it into the system by the transformation (4.15). This is quite similar to the argument given in Sect. 3.3.1 (see the discussions around (3.44), (3.45)). On the other hand, the finite part of (4.6) is mapped to as noted above, yielding the behavior in the NS sector as in (3.45). Therefore, the cancellation of the graviton term implies that
α=24N−1⋅N−112=2,
which completes the proof of our statement.
Discussions
We conclude that our ansatz (4.6) for should reproduce the expected expansion coefficients whose holomorphic parts give the coefficients of massive representations of Mathieu and umbral moonshine. Therefore, we naturally consider that there are two world-sheet descriptions of the moonshine phenomena based on the theories and , depending on the choice of symmetry. Mathieu moonshine sits at the self-dual point.
Our observation is related to Ref. [4] (and also to Ref. [25]). Without invoking the idea of duality, the authors of Ref. [4] introduced a transformation rule between the corresponding Jacobi forms of the two theories that seem to fit our results very well. A possible geometrical interpretation of the umbral moonshine based on particular types of singular K3 is also discussed. As we briefly discussed in Sect. 4, the correspondence (4.15) proposed in Ref. [4] is likely to be consistent with our “duality relation” (4.7).
Some of the results in this paper appear correct by symmetry arguments (modularity, etc.) but are not verified explicitly: We have not computed the holomorphic parts of functions except for in the case of Mathieu moonshine. In subsequent work we want to fill these gaps. We also mainly discussed the -type modular invariant and not the and types.
One should keep in mind that the Jacobi form given here is at most meromorphic except for the purely -type models, since generally includes torsion points, as we have mentioned. Consequently, it would be subtle whether one can strictly interpret as the elliptic genus of a well-defined superconformal system. Of course, we have no such subtlety for the “dual” -realization (4.3) for an arbitrary . We would like to further discuss this point in future work.
Acknowledgements
We thank the organizers of the LMS--EPSRC Durham Symposium on “New Moonshines, Mock Modular Forms and String Theory” at Durham University, 3--12 August 2015, where part of this work was done, for their kind hospitality. The research of T.E. is supported in part by JSPS KAKENHI grant nos. 25400273, 22224001 and 23340115. The research of Y.S. is supported in part by JSPS KAKENHI grant no. 23540322.
Funding
Open Access funding: SCOAP3.
Notations and useful formulas
In this appendix we summarize the notations adopted in this paper and useful formulas related to them. We assume that , 0$]]> and set , :
We use the abbreviations (), . We also set
η(τ)=q1/24∏n=1∞(1−qn).
The spectral flow properties of the theta functions are summarized as follows:
θ1(τ,z+mτ+n)=(−1)m+nq−m22y−mθ1,(τ,z),θ2(τ,z+mτ+n)=(−1)nq−m22y−mθ2(τ,z),θ3(τ,z+mτ+n)=q−m22y−mθ3(τ,z),θ4(τ,z+mτ+n)=(−1)mq−m22y−mθ4(τ,z),θa,k(τ,2(z+mτ+n))=e2πinaq−km2y−2kmθa+2km,k(τ,2z).
We introduce the Weierstrass -function:
σ(τ,z):=e12G2(τ)z2θ1,(τ,z)2πη(τ)3≡z∏ω∈Λ′(1−zω)ezω+12(zω)2.(Λ′≡Λ−{0}),
where is the (unnormalized) second Eisenstein series:
G2(τ):=∑n∈ℤ−{0}1n2+∑m∈ℤ−{0}∑n∈ℤ1(mτ+n)2≡π23[1−24∑n=1∞nqn1−qn].
It is useful to note the anomalous -transformation formula of :
G2(−1τ)=τ2G2(τ)−2πiτ.
We also set
G^2(τ):=G2(τ)−πτ2,
which is a nonholomorphic modular form of weight 2.
Spectral flow operator (see also Ref. [24]):
sλ(κ)⋅f(τ,z):=e2πiκτ2λ2(λ+2z)f(τ,z+λ)≡qκα2y2καe2πiκαβf(τ,z+ατ+β),(λ≡ατ+β,∀α,β∈R).
An important property of the spectral flow operator is the modular covariance, which precisely means the following:
Assume that is an arbitrary function with the modular property
f(τ+1,z)=f(τ,z),f(−1τ,zτ)=e2πiκτz2ταf(τ,z);
then, we obtain for sλ(κ)⋅f(τ+1,z)=sλ(κ)⋅f(τ,z),sλτ(κ)⋅f(−1τ,zτ)=e2πiκτz2ταsλ(κ)⋅f(τ,z).
(ii) double quasiperiodicity:
Φ(τ,z+mτ+n)=(−1)2r(m+n)q−rm2y−2rmΦ(τ,z).
In this paper, we shall use this terminology in a broader sense. We allow a half-integral index , and, more crucially, allow nonholomorphic dependence on , while we keep the holomorphicity with respect to .10
Derivation of Eqs. (<xref ref-type="disp-formula" rid="PTW078M3_39">3.39</xref>), (<xref ref-type="disp-formula" rid="PTW078M3_42">3.42</xref>), and (<xref ref-type="disp-formula" rid="PTW078M3_47">3.47</xref>)
In this appendix, we derive Eqs. (3.39), (3.42), and (3.47).
Derivation of (3.39):
We start with the definition of (3.37). Substituting (3.38) into (3.37), we obtain ()
H^(N)(τ)=12π2∮w=0dww2[wη(τ)3iθ1(τ,2w)e(N−2)G2(τ)w2∑λ∈Λe−πτ2N{|λ|2+2λ¯w+w2}λ+w].
It would be easiest to carry out this residue integral by introducing the Weierstrass -function defined in (A6). Namely,
H^(N)(τ)=18π3i∮w=0dww2[2wσ(τ,2w)eNG2(τ)w2∑λ∈Λe−πτ2N{|λ|2+2λ¯w+w2}λ+w]=18π3i∮w=0dww2[2wσ(τ,2w)eNG2(τ)w2∑λ∈Λ′e−πτ2N{|λ|2+2λ¯w+w2}λ+w]+18π3i∮w=0dww32wσ(τ,2w)eNG^2(τ)w2.(Λ′≡Λ−{0}).
Moreover, since we have
2wσ(τ,2w)=1+O(w4),
due to (A6), (B2) just becomes
H^(N)(τ)=18π3i∮w=0dww2∑λ∈Λ′e−πτ2N{|λ|2+2λ¯w+w2}λ+w+18π3i∮w=0dww3eNG^2(τ)w2=14π2[∂∂w∑λ∈Λ′e−πτ2N{|λ|2+2λ¯w+w2}λ+w|w=0+NG^2(τ)]−=14π2[−∑λ∈Λ′e−πτ2N|λ|2λ2{1+2πNτ2|λ|2}+NG^2(τ)].
We have thus obtained (3.39).
Derivation of (3.42):
We next evaluate the holomorphic part of (3.39) or (B3). Let us derive the expression of the holomorphic function given in (3.42).
Recalling (3.28), the nonholomorphic part of can be rewritten as
ΔZcase 1(τ,z)≡θ1,(τ,z)2η(τ)6ΔH^(N)(τ),ΔH^(N)(τ)=2η(τ)3limw→0[F^(N)(w)−f(N)[−]iθ1(2w)]=12πi∂∂w[F^(N)(w)−f(N)[−]]|w=0.
Moreover, we note the formula
F^(N)(w)−f(N)[−](w)=i2π∑λ∈Λsλ(N)⋅[e−πτ2Nw2−1w],
where denotes the spectral flow operator (A10).
Substituting (B5) into (B4), we obtain
Here, we have included the convergence factor to make the second line well-defined, and the symbol “” denotes the principal value defined in (3.41):
∑n≠0Pan:=limN→∞∑n=1N(an+a−n),∑nPan:=a0+∑n≠0Pan.
Then, by comparing (B6) with (B3), and recalling we find that the holomorphic part is indeed given by Eq. (3.42).
Derivation of (3.47):
Finally, let us derive the formula given in the second line of (3.47) for the case. To this end, it is again convenient to make use of the -function (A6). Substituting the identity (3.51) into the first line of (3.47), we obtain
In the last line, we made use of the identity
∑λ=mτ+n∈Λ(−1)m+n+mne−π2τ2|λ|2=0.
In fact,
∑λ∈Λ(−1)m+n+mne−π2τ2|λ|2(∑m,n∈2ℤ−∑m∈2ℤ+1orn∈2ℤ+1)e−π2τ2|λ|2∑λ∈Λe−2πτ2|λ|2−{∑λ∈Λe−π2τ2|λ|2−∑λ∈Λe−2πτ2|λ|2}2∑λ∈Λe−2πτ2|λ|2−∑λ∈Λe−π2τ2|λ|2;
the last line vanishes due to the Poisson resummation with respect to both and .
To proceed further, we set
g(τ):=−∑λ∈Λ′(−1)m+n+mne−π2τ2|λ|2λ2≡−∑λ∈Λ′(−1)m+nq12m2e−π2τ2λ2λ2.
Then, we obtain
∂∂τ¯g(τ)=−∑λ∈Λ′(−1)m+n+mne−π2τ2|λ|2iπ4τ22=iπ4τ22=−12∂∂τ¯(πτ2).
Here we have again used (B8). By the definition (B9), should be a nonholomorphic modular form of weight 2 for . Thus is uniquely determined by (B10) as
g(τ)=12G^2(τ).
Combining (B9), (B11) with (B7), we finally obtain
H^(2)(τ)=14π2[∑λ∈Λ′(−1)m+n+mn2πime−π2τ2|λ|2λ+G^2(τ)+πτ2]=14π2[∑λ∈Λ′(−1)m+n+mn2πime−π2τ2|λ|2λ+G2(τ)],
which is the second line of (3.47).
A lemma for the weak Jacobi form
In this appendix, we present a simple lemma that is necessary for the proof of (3.70).
Let be any holomorphic weak Jacobi form of weight 0 and index 0})$]]>. Then, the “NS counterpart” has the following behavior around (for a generic value of ):
Φ(NS)(τ,z)∼qα,−d4≤α≤d12,(τ∼i∞).
The proof is straightforward. Let us recall that bases of holomorphic weak Jacobi forms of weight 0 and index 0})$]]> consist of functions defined by
ϕ(n1,n2,n3)(τ,z)∝∑σ∈S3(θ2(τ,z)θ2(τ))2nσ(1)(θ3(τ,z)θ3(τ))2nσ(2)(θ4(τ,z)θ4(τ))2nσ(3),ϕ(n1,n2,n3)(τ,0)=1,∀(n1,n2,n3)∈S(d),
with
S(d):={(n1,n2,n3)∈Z≥03;n1≥n2≥n3≥0,n1+n2+n3=d}.
Now, we define the NS counterpart of the function by the spectral flow :
ϕ(n1,n2,n3)(NS)(τ,z):=qd4ydϕ(n1,n2,n3)(τ,z+τ+12).
Then, by using the evaluations around ,
(θ4(τ,z)θ2(τ))2∼q−14,(θ1,(τ,z)θ3(τ))2∼q14,(θ2(τ,z)θ4(τ))2∼q14,
we can easily find that the behavior of becomes
ϕ(n1,n2,n3)(NS)(τ,z)∼q−14n1+14(n2+n3)=q−12(n1−d2),(τ∼i∞).
Since we are assuming that and , we find
d3≤n1≤d.
Therefore, we obtain
ϕ(n1,n2,n3)(NS)(τ,z)∼qα,−d4≤α≤d12,(τ∼i∞),
for . ■
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1We are here using a slightly different convention from that of Refs. [5,6]. Our , , correspond to , , and in Refs. [5,6].
2Here, we may have a subtlety, since could be nonholomorphic due to the existence of modular completions, the correction terms of which show a continuous spectrum. Thus, the Cardy-type argument in this context truly means that the asymptotic growth of the coefficients of -expansion of the holomorphic part, which only includes a discrete spectrum, is governed by the IR behavior of the total.
3 describe free fermions in the NS (Ramond (R)) sector.
4The holomorphic part of the identity (3.48) essentially means the familiar equivalence between the massless character of level 1 and the spectral flow sum of the massless matter characters with [20,21], which is also presented in Ref. [4]. The identity (3.48) claims that this equivalence still holds after taking the modular completions, and it is surely a nontrivial identity. One of its proofs is obtained by setting in the identity (3.70), which we will prove. Note that the minimal character just reduces to the mod 2 Kronecker delta: in the case of ().
5This is equivalent to the identity that Zagier gave in his lecture at the Durham workshop, August 2015.
6In fact, the coefficients of massless characters can be uniquely determined from by this assumption, while the massive coefficients would not be necessarily unique. However, this ambiguity does not affect the following discussions.
7We remark that the optimal growth condition is not necessary for the purpose of proving the identity (4.12). In other words, the ambiguity of mentioned above does not spoil this relation; always holds for the coefficients appearing in (4.11).
8Such torsion points are canceled out in the cases when is made up only of -type components, as we illustrated in Sect. 3.4.1.
9On the other hand, the -torsion points appearing in the umbral Jacobi form are canceled out by the factor .
10According to the original terminology of Ref. [24], the “weak Jacobi form” of weight and index () means that should be Fourier expanded as in addition to the conditions (A14) and (A15). It is called the “Jacobi form” if it further satisfies the condition for s.t. .