Product Representation of Dyon Partition Function in CHL Models

Ashoke Sen; Dileep P. Jatkar; Justin R. David

REVIEW

Product Representation of Dyon Partition Function in CHL Models

Justin R. David, Dileep P. Jatkar, Ashoke Sen

High Energy Physics - Theory · Submitted 24 Feb 2006 · 29 references

Pith's verdict: The N=2 product formula is a genuine derivation; the N=3,5,7 cases are an honest but unproven guess, so the paper deserves a serious referee but the general-N claim is conditional.

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arXiv IDhep-th/0602254 (version 4)

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CitationAshoke Sen, Dileep P. Jatkar, Justin R. David. "Product Representation of Dyon Partition Function in CHL Models." arXiv:hep-th/0602254 (2006). https://pith.science/paper/5TVYQTK6

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Editor's note

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Short version: this paper has two very different parts. For N=2 it derives the product representation of ~Phi_6 from an orbifold CFT computation; for N=3,5,7 it constructs the product from an explicitly admitted guess for the twisted elliptic genera. The N=2 part is the real result. The rest is a plausible conditional statement.\n\nWhat is actually new: the infinite product formulas (1.4)/(7.25) for ~Phi_k and Phi_k across the CHL family with N=2,3,5,7, with exponents given by Fourier coefficients of twisted elliptic genera of K3. This generalizes the Borcherds and Gritsenko-Nikulin product for the weight-10 Igusa form. For N=2, the twisted elliptic genera in (3.13) are computed from a concrete orbifold description in Appendix A, not fitted to dyon data. The integral evaluation in Appendices B is technical but the logic is traceable, and the 31-coefficient check against the earlier construction in [7] is real evidence. The N=2 V->0 and U->\infty limits also match. Credit is earned there.\n\nSoft spots: the stress-test note is right. For N=3,5,7 the functions F^(r,s) in (7.21) are guessed. They satisfy the modular transformation law, the geometric low-order values, the elliptic genus sum rule, and the vanishing condition, but nothing proves they are the unique or correct choice. Every exponent in the product is a Fourier coefficient of these functions, so the N=3,5,7 product forms rest entirely on the unproven Ansatz. The finite-order comparisons with [7] only show that the two constructions encode the same guessed input; they cannot independently verify it. The paper also says it will not give the details of the general-N integral analysis (Section 7), so for general N the derivation is a sketch plus a guess. That is a load-bearing soft spot, but it is disclosed honestly, and it does not infect the N=2 result. I should also say that the N=2 construction is not circular: the coefficients are computed from the orbifold CFT, not fitted to the degeneracies.\n\nWho this is for: string theorists working on dyon counting, Siegel modular forms, and Borcherds products. A serious referee should look at the N=2 derivation carefully and press for a derivation or at least a uniqueness argument for the N=3,5,7 Ansatz. But the N=2 result alone justifies engaging with this paper. I would accept it for peer review and recommend the editor find a referee who knows Kawai's method and Borcherds products.

Referee report

Significance

If the proposed product formulae are correct, they give explicit, exact, S-duality-invariant degeneracies for 1/4-BPS dyons in CHL models and provide a concrete generalization of the known Borcherds/Gritsenko-Nikulin product. The N = 2 part is a genuine derivation: the input twisted elliptic genera are computed from the orbifold CFT, the integral is evaluated by an orbit decomposition, and the resulting product is compared with thirty-one coefficients of the independent construction in [7]. This is a solid, reproducible calculation and a substantial strength of the paper. The general-N claim is plausible and the guessed Ansatz satisfies the stated modular and geometric constraints, but its correctness is not established. The central identity for N = 3, 5, 7 therefore remains conditional.

Major objections (3)

[Section 7, Eq. (7.21); page 3] The product formulae for N = 3, 5, 7 are conditional on the guessed twisted elliptic genera. The paper explicitly states that F^(r,s) for these cases were not evaluated directly but guessed from general considerations, and Eq. (7.21) is not derived from the orbifold conformal field theory. This is load-bearing: every exponent in the product formulae (1.4) and (7.24)-(7.25) is a Fourier coefficient of F^(r,s), so if the Ansatz is incorrect or non-unique the general-N product representation does not follow. The consistency conditions (7.9), (7.13)-(7.16) were used to select the Ansatz, and the subsequent checks in (7.27)-(7.30) and Appendix C are consequences of those same constraints, so they cannot discriminate the Ansatz from other functions satisfying the same requirements. The N=2 case is not affected because there F^(r,s) is computed in Appendix A. To make the general-N claim rigorous, the authors should either compute F^(r,s) directly for N = 3, 5, 7, ideally through an explicit orbifold realization of the Z_N action on K3, or prove uniqueness of the solution to (7.6), (7.9), (7.13)-(7.16) within the class of weak Jacobi forms used here.
[Section 7, after Eq. (7.23)] The derivation of the general-N product formulae is not supplied. After Eq. (7.23) the text states 'We shall not give the details of the analysis but write down the final expression' and then presents (7.24)-(7.25). The N=2 derivation in Sections 4-6 is a lengthy orbit-by-orbit calculation with a specific regularization and subtraction scheme, so the generalization to general N, including the new phases e^{2πir/N} and e^{2πik2r/N}, requires at least a derivation sketch. It would be important to specify the orbit decomposition, the modular-invariance argument for the integral, and the role of the subtraction terms for general N. Without this, the modular properties and product form for N=3,5,7 are asserted rather than derived, and this compounds the uncertainty coming from the guessed input.
[Appendix C and Section 1] The numerical evidence for N = 3, 5, 7 is much weaker than for N = 2. The text says the expansions were compared only up to order e^{4πiT} e^{4πiU}, and Appendix C lists only a few low-order terms for each N. For N = 2 the paper reports 31 matched coefficients. I would ask the authors either to provide many more coefficients for N = 3, 5, 7, computed independently from the two constructions, or to explain why the low-order agreement is already sensitive enough to test the Ansatz. As it stands, the finite-order comparison does not resolve the non-uniqueness of the guessed F^(r,s) raised in the first comment.

Minor objections (3)

[Eq. (3.21)] The same symbol c^{(r,s)}(4n) is used for the Fourier coefficients of both h^{(r,s)}_0 and h^{(r,s)}_1; the stated justification is that the supports differ, but the readability would improve if the two expansions were written with different dummy indices or a short remark distinguishing the two cases.
[Eqs. (7.24)-(7.25)] The ranges of the sums over r and s are not explicitly displayed in the product formulae; stating that r,s run from 0 to N-1 in both products would prevent confusion.
[Appendix C] The appendix presents the low-order expansions of Φ_k but does not state in the appendix itself the maximum order that was compared for each N; moving that information from the introduction into the appendix, or adding a small table of matched coefficients, would make the verification easier to assess.

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The paper

The paper claims

The exact dyon partition function in CHL models is the inverse of an infinite-product Siegel modular form.

The reading

This paper proposes exact product formulas for the modular forms whose inverses count quarter-BPS dyons in CHL string compactifications, a class of $\mathbb{Z}_N$ orbifold models with $N = 2, 3, 5, 7$ in which a $\mathbb{Z}_N$ shift on a torus is combined with an action on K3. For each $N$, the dyon degeneracy is a Fourier coefficient of $1/\tilde{\Phi}_k$, and the paper writes $\tilde{\Phi}_k$ as an infinite product over charges, with exponents given by Fourier coefficients of the twisted elliptic genus of K3. If correct, the formulas turn the counting of dyons in these models into one closed expression: they produce integer degeneracies, are invariant under the S-duality group, and reduce to the Bekenstein-Hawking entropy at large charges. The construction generalizes the known product representation of the weight-10 Siegel cusp form of $Sp(2,\mathbb{Z})$ to the CHL setting.

What carries the argument

The load-bearing object is the infinite product (1.4)/(7.25) for $\tilde{\Phi}_k$, with the companion product (7.24) for $\Phi_k$; these are generalized automorphic products adapted to the CHL subgroup. The exponents are the Fourier coefficients $c^{(r,s)}(4lk' - b^2)$ of the twisted elliptic genus $F^{(r,s)}(\tau,z)$, a generating function that counts BPS states of K3 in the sector twisted by the $\mathbb{Z}_N$ generator. The mechanism that produces the product is an integral $I(U,T,V)$ over the fundamental domain of a trace in the $(T^2 \times K3)/\mathbb{Z}_N$ superconformal field theory, analogous to the threshold integral used in the $N=1$ case; evaluating it by Poisson resummation and by splitting the summation into zero, degenerate, and non-degenerate orbits expresses $I$ as $-2\ln[\kappa (\det\mathrm{Im}\,\Omega)^k |\tilde{\Phi}_k|^2]$, which fixes the weight and gives the product form.

Extended reading notes

Core claim

The central claim is that for $N = 2, 3, 5, 7$ the function $\tilde{\Phi}_k$ is a Siegel modular form of weight $k = 24/(N+1) - 2$ for a subgroup $\tilde{G}$ of $Sp(2,\mathbb{Z})$ — an automorphic form on the genus-two Siegel upper half-plane — and that it has the explicit infinite-product representation of eqs. (1.4) and (7.25). In that product each factor is $1 - e^{2\pi i(k'T + lU + bV)}$ raised to an exponent built from the coefficients $c^{(r,s)}(4lk' - b^2)$ of the twisted elliptic genus $F^{(r,s)}(\tau,z)$ of K3; a related form $\Phi_k$ is given in eq. (7.24). For $N=2$ the twisted elliptic genera are computed from the orbifold theory, while for $N = 3, 5, 7$ they are proposed from modularity and consistency requirements. The paper verifies that the product forms agree with the earlier construction of [7] in the $V \to 0$ and $U \to i\infty$ limits and in every Fourier coefficient it compares, and concludes that the two constructions define the same modular form.

Load-bearing premise

For $N = 3, 5, 7$, the twisted elliptic genera that supply every exponent in the product are guessed from consistency conditions rather than derived from the orbifold theory, so the product formulas for those models stand or fall with that guess.

Editorial extensions

If this is right

For each $N=2,3,5,7$, the degeneracies are Fourier coefficients of $1/\tilde{\Phi}_k$, so they are integers and invariant under the S-duality group of the CHL model.
At large charges, the asymptotic growth of these coefficients reproduces the Bekenstein-Hawking entropy of the corresponding dyonic black hole.
In the $V\to 0$ limit the product factorizes as $\tilde{\Phi}_k \simeq (i\sqrt{N})^{-k-2}\,4\pi^2 V^2 (\eta(T)\eta(T/N))^{k+2}(\eta(U)\eta(NU))^{k+2}$, matching the independent result of [7].
The product representation and the earlier recursive construction agree on all coefficients checked—31 coefficients for $N=2$ and coefficients up to order $e^{4\pi i T} e^{4\pi i U}$ in general—so the paper concludes they are the same modular form.
The infinite product gives a direct way to compute dyon degeneracies by expanding the product, without running the recursive algorithm of [7].

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

This suggests the same product template should extend to other $\mathbb{Z}_N$ constructions once the twisted elliptic genus is known: the weight and exponent structure are fixed by K3 data and the shift, so the formula is a template rather than an isolated identity.
This suggests the guessed twisted elliptic genera for $N=3,5,7$ could be pinned down by demanding that $\tilde{\Phi}_k$ be a Siegel modular form of the correct subgroup with the right pole structure—a finite consistency check that could turn the Ansatz into a theorem.
This suggests a representation-theoretic reading: product formulas of this type usually come from generalized Kac-Moody algebras, so the CHL dyon spectrum may admit the same algebraic interpretation as the $N=1$ case.

Original abstract

A formula for the exact partition function of 1/4 BPS dyons in a class of CHL models has been proposed earlier. The formula involves inverse of Siegel modular forms of subgroups of Sp(2,Z). In this paper we propose product formulae for these modular forms. This generalizes the result of Borcherds and Gritsenko and Nikulin for the weight 10 cusp form of the full Sp(2,Z) group.

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arXiv listing abstract v4. Results and the main claim are underlined; show every role.

A formula for the exact partition function of 1/4 BPS dyons in a class of CHL models has been proposed earlier. The formula involves inverse of Siegel modular forms of subgroups of Sp(2,Z). In this paper we propose product formulae for these modular forms. This generalizes the result of Borcherds and Gritsenko and Nikulin for the weight 10 cusp form of the full Sp(2,Z) group.

background contribution method result interpretation main claim

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	A formula for the exact partition function of 1/4 BPS dyons in a class of CHL models has been proposed earlier.	background 1.00	0.04	0.04	asserted 1.00
s1	The formula involves inverse of Siegel modular forms of subgroups of Sp(2,Z).	background 1.00	0.08	0.04	asserted 1.00
s2	In this paper we propose product formulae for these modular forms.	contribution 0.96	0.93	0.74	asserted 0.98
s3	This generalizes the result of Borcherds and Gritsenko and Nikulin for the weight 10 cusp form of the full Sp(2,Z) group.	interpretation 0.82	0.55	0.20	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.97.

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References

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Forward citations

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74 later papers cite it

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1 later paper by other authors is listed below: 1 used its method or tool and 1 compared its own results with it. It appears under more than one label. 2 more papers by the same authors are listed: 2 used its method or tool.

Uses its method or tool 3 of 10 papers with this label · 2 by the same authors

Used the relations given in this paper to calculate the sums over j and p.

Borcherds Algebras and N=4 Topological Amplitudes (2011) · 0.91 · paraphrased: the licence does not allow quoting

By the same authors

Followed this paper in analyzing the contribution from the other orbits.

Discrete Information from CHL Black Holes (2010) (same authors) · 0.91 · paraphrased: the licence does not allow quoting
Followed the methods of this paper for the details of the evaluation of the threshold integral.

Properties of dyons in ${\cal N}=4$ theories at small charges (2018) (same authors) · 1.00 · paraphrased: the licence does not allow quoting

Compares with it 1 of 8 papers with this label

Found its former contribution identical to the one found in this paper, using the same regularisation.

Borcherds Algebras and N=4 Topological Amplitudes (2011) · 0.87 · paraphrased: the licence does not allow quoting

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72 arXiv papers cite this in 158 sentences

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Papers citing it · 94 sentences without a confident label

Builds on it 3 papers

We extend this correspondence to argue for the existence of new GKM superalgebras, $𝒢_N$, whose denominator formulae give rise to modular forms with character, $Δ_k/2(𝐙)$, of suitable groups, $G_0(N)⊂Sp(2,)$, with the property [equation] (Note that the $Φ_k(𝐙)$ used here differs from the one defined by David-Jatkar-Sen [ref] by an overall sign.) This is the main assumption of this paper.

Generalized Kac-Moody Algebras from CHL dyons · Denominator formulae for $Δ_k/2(𝐙)$ · 0.94
Explicit formulae for these are available for $N∈{1,…,5}$ in the literature [ref] and we give the formulae for $N=6$ in section [ref].

BKM Lie superalgebras from counting twisted CHL dyons -- II · Product Formulae · 0.89
This would generalise the “threshold integrals” previously calculated by Sen and collaborators for a class of CHL models [ref] .

Fricke S-duality in CHL models · Conclusions · 0.94

Uses its method or tool 10 papers

The original formula for the degeneracy was first proposed in [ref] for the special case of heterotic string theory compactified on $T^6$, and extended to more general models in [ref] .

Black Hole Entropy Function, Attractors and Precision Counting of Microstates · Non-supersymmetric extremal black holes · 0.81
Then in section [ref] we compute the Gopakumar Vafa invariants from the new supersymmetric index following the previous works of [ref] .

Gravitational couplings in ${\cal N}=2$ heterotic compactifications with Wilson lines · Introduction · 0.93
Following the manipulations outlined in [ref] following earlier work of [ref] one can show that (ρ,σ, v) = -2ln[ (Im Ω)^k ] - 2lnΦ(ρ,σ, v) - 2lnΦ̌(ρ̅,σ̅,v̅) +constant where $Φ$ has been defined in ek7 and[Since the exponent $a$ in ek7, ek7check is a real number (in fact an integer) we have $Φ̌(ρ̅,σ̅, v̅) = Φ(ρ, σ,v)$.] Φ̌(ρ̅,σ̅,v̅) = e^-2πi (αρ̅+γσ̅+ βv̅) ∏_b=0^1 ∏_r=0^N-1∏_r'=0^M-1 ∏_k∈+r'M, l∈, j∈2+bk,l≥0, j<0 for k=l=0 [ 1 - e^-2πi r/N e^-2πi (kσ̅+lρ̅+ j v̅) ]^a a ≡ ∑_s=0^N-1 ∑_s'=0^M-1 e^-2 πi (s'l / M + rs/N) c_b^(0,s;r',s')(4kl - j^2) , k = 12 ∑_s=0^N-1∑_s'=0^M-1 c_0^(0,s;0,s')(0) .

Discrete Information from CHL Black Holes · Introduction · 0.84

Uses its data or values 2 papers

For $=K3$ and prime values of $N$ ($N=$ 1, 2, 3, 5, 7) the functions $F^(r,s)(τ)$ are known explicitly and are given by [ref] fifthn F^(0,0)(τ, z) = 8N A(τ, z) , F^(0,s)(τ, z) = 8N(N+1) A(τ, z) -2N+1 B(τ, z) E_N(τ) for $1≤s≤(N-1)$ , F^(r,rk)(τ, z) = 8N(N+1) A(τ, z) + 2N(N+1) E_N(τ+kN) B(τ, z) , for $1≤r ≤(N-1)$, $0≤k≤(N-1)$ , where A(τ, z) = [ ϑ_2(τ,z)^2 ϑ_2(τ,0)^2 + ϑ_3(τ,z)^2ϑ_3(τ,0)^2 + ϑ_4(τ,z)^2ϑ_4(τ,0)^2] , B(τ, z) = η(τ)^-6 ϑ_1(τ, z)^2 , and E_N(τ) = 12 iπ(N-1) _τ[ lnη(τ) -lnη(Nτ)]= 1 + 24N-1 ∑_n_1,n_2≥1n_1 0 mod N n_1 e^2πi n_1 n_2 τ .

Black Hole Entropy Function, Attractors and Precision Counting of Microstates · The full partition function · 0.91
For example, in the case of the $pA$ orbifolds with $p =1,2, 3, 5, 7$, they are given by [ref] .

Horizon states and the sign of their index in ${\cal N}=4$ dyons · Horizon states for the $1/4$ BPS dyon · 0.93

Compares with it 8 papers

In two previous papers [ref] we used the quantum entropy function to compute logarithmic corrections to the entropy of 1/8 BPS and 1/4 BPS black holes in $=8$ and $=4$ supersymmetric string theories respectively and found results in perfect agreement with the microscopic results of [ref] .

Logarithmic Corrections to N=2 Black Hole Entropy: An Infrared Window into the Microstates · Introduction · 0.82
The partition function for the twisted primitive DT invariants on $X$ is conjecturally given by the negative reciprocal of the Borcherds lift of the corresponding twisted-twined elliptic genera, [equation] and thus agrees with the quarter-BPS counting function obtained in [ref] , which is (possibly up to a multiplicative constant) the function $2^-4Φ_6,3^-1$.

Lost Chapters in CHL Black Holes: Untwisted Quarter-BPS Dyons in the $\mathbb{Z}_2$ Model · Comparison to results from Donaldson-Thomas theory · 0.92
Finally due to the same reasons we see that the only contributions to $F^(1, 1)$ arise from [equation] Again summing up the contributions leads to [equation] To conclude, from ([ref]), ([ref]), ([ref]) and ([ref]) we see that the twisted elliptic genus is identical to the class $2A$ first evaluated in [ref] using $K3$ in the $T^4/ℤ_2$ orbifold limit.

${\cal N}=2$ heterotic string compactifications on orbifolds of $K3\times T^2$ · Twisted elliptic genus · 0.92

Cites it as background 34 papers

Introduction: Since the original proposal of Dijkgraaf, Verlinde and Verlinde [ref] for quarter BPS dyon spectrum in heterotic string theory compactified on $T^6$, there has been extensive study of dyon spectrum in a variety of $=4$ supersymmetric string theories [ref] and also in $=8$ and $=2$ supersymmetric string theories [ref] .

Partition Functions of Torsion >1 Dyons in Heterotic String Theory on T^6 · Introduction · 0.99
The partition function of quarter BPS dyons is now known in a variety of $=4$ supersymmetric string theories [ref] .

Generalities of Quarter BPS Dyon Partition Function and Dyons of Torsion Two · Introduction · 0.98
We now have a good understanding of the exact spectrum of a class of quarter BPS dyons in a variety of $=4$ supersymmetric string theories [ref] .

Three String Junction and N=4 Dyon Spectrum · Introduction · 0.96

Discussion

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