BKM Lie superalgebras from dyon spectra in Z_N-CHL orbifolds for composite N
TL;DR: In this article, Cheng and Dabholkar showed that the genus-two Siegel modular forms of the Z_N-CHL orbifolds of the heterotic string on T^6 are given by multiplicative eta-products.
Abstract: We show that the generating function of electrically charged 1/2-BPS states in N=4 supersymmetric Z_N-CHL orbifolds of the heterotic string on T^6 are given by multiplicative eta-products. The eta-products are determined by the cycle shape of the corresponding symplectic involution in the dual type II picture. This enables us to complete the construction of the genus-two Siegel modular forms due to David, Jatkar and Sen [arXiv:hep-th/0609109] for Z_N orbifolds when N is non-prime. We study the Z_4 CHL orbifold in detail and show that the associated Siegel modular forms, \Phi_3(Z) and \widetilde{\Phi}_3(Z), are given by the square of the product of three even genus-two theta constants. Extending work by us[arXiv:0807.4451] as well as Cheng and Dabholkar[arXiv:0809.4258], we show that their `square roots' appear as the denominator formulae of two distinct Borcherds-Kac-Moody (BKM) Lie superalgebras. The BKM Lie superalgebra associated with the generating function of 1/4-BPS states, i.e., \widetilde{\Phi}_3(Z) has a parabolic root system with a light-like Weyl vector and the walls of its fundamental Weyl chamber are mapped to the walls of marginal stability of the 1/4-BPS states.
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TL;DR: In this article, the authors discussed the possibility of Mathieu group M 24 acting as symmetry group on the K3 elliptic genus as proposed recently by Ooguri, Tachikawa and one of the present authors.
173 citations
Cites background from "BKM Lie superalgebras from dyon spe..."
...Sen and his collaborators [5, 16] (also [15]) in connection with the counting problem of 1 4 BPS monopoles and dyons....
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TL;DR: In this paper, the quantum entropy function formalism was used to calculate logarithmic corrections to the entropy of black holes in the near horizon geometry of N = 4 supergravity.
Abstract: We evaluate the one loop determinant of matter multiplet fields of N=4 supergravity in the near horizon geometry of quarter BPS black holes, and use it to calculate logarithmic corrections to the entropy of these black holes using the quantum entropy function formalism. We show that even though individual fields give non-vanishing logarithmic contribution to the entropy, the net contribution from all the fields in the matter multiplet vanishes. Thus logarithmic corrections to the entropy of quarter BPS black holes, if present, must be independent of the number of matter multiplet fields in the theory. This is consistent with the microscopic results. During our analysis we also determine the complete spectrum of small fluctuations of matter multiplet fields in the near horizon geometry.
170 citations
TL;DR: In this paper, it has been observed that the elliptic genus of K3 has a natural interpretation in terms of the dimensions of representations of the largest Mathieu group M24, and a new "moonshine" for M24 is connected to an earlier observation on a moonshine of M24 through the 1/4-BPS spectrum of k3xT^2-compactified type II string theory.
Abstract: A close relationship between K3 surfaces and the Mathieu groups has been established in the last century. Furthermore, it has been observed recently that the elliptic genus of K3 has a natural interpretation in terms of the dimensions of representations of the largest Mathieu group M24. In this paper we first give further evidence for this possibility by studying the elliptic genus of K3 surfaces twisted by some simple symplectic automorphisms. These partition functions with insertions of elements of M24 (the McKay-Thompson series) give further information about the relevant representation. We then point out that this new "moonshine" for the largest Mathieu group is connected to an earlier observation on a moonshine of M24 through the 1/4-BPS spectrum of K3xT^2-compactified type II string theory. This insight on the symmetry of the theory sheds new light on the generalised Kac-Moody algebra structure appearing in the spectrum, and leads to predictions for new elliptic genera of K3, perturbative spectrum of the toroidally compactified heterotic string, and the index for the 1/4-BPS dyons in the d=4, N=4 string theory, twisted by elements of the group of stringy K3 isometries.
147 citations
TL;DR: In this paper, it was shown that the supersymmetry-preserving automorphisms of any non-linear σ-model on K3 generate a subgroup of the Conway group Co1.
Abstract: It is shown that the supersymmetry-preserving automorphisms of any non-linear σ-model on K3 generate a subgroup of the Conway group Co1. This is the stringy generalisation of the classical theorem, due to Mukai and Kondo, showing that the symplectic automorphisms of any K3 manifold form a subgroup of the Mathieu group M23. The Conway group Co1 contains the Mathieu group M24 (and therefore in particular M23) as a subgroup. We confirm the predictions of the Theorem with three explicit CFT realisations of K3: the T 4 /Z2 orbifold at the self-dual point, and the two Gepner models (2) 4 and (1) 6 . In each case we demonstrate that their symmetries do not form a subgroup of M24, but lie inside Co1 as predicted by our
146 citations
TL;DR: For BPS black holes with at least four unbroken supercharges, the macroscopic entropy can be used to compute an appropriate index, which can be then compared with the same index computed in the microscopic description.
Abstract: For BPS black holes with at least four unbroken supercharges, we describe how the macroscopic entropy can be used to compute an appropriate index, which can be then compared with the same index computed in the microscopic description. We obtain exact results incorporating all higher order quantum corrections in the limit when only one of the charges, representing momentum along an internal direction, approaches infinity keeping all other charges fixed at arbitrary finite values. In this limit, we find that the microscopic index is controlled by certain anomaly coefficients whereas the macroscopic index is controlled by the coefficients of certain Chern-Simons terms in the effective action. The equality between the macroscopic and the microscopic index then follows as a consequence of anomaly inflow. In contrast, the absolute degeneracy does not have any such simple expression in terms of the anomaly coefficients or coefficients of Chern-Simons terms. We apply our analysis to several examples of spinning black holes in five dimensions and non-spinning black holes in four dimensions to compute the index exactly in the limit when only one of the charges becomes large, and find perfect agreement with the result of exact microscopic counting. Our analysis resolves a puzzle involving M5-branes wrapped on a 5-cycle in K3 × T
3.
134 citations
Cites background from "BKM Lie superalgebras from dyon spe..."
...In a class of supersymmetric string theories with sixteen or more unbroken supercharges we now have a near complete understanding of the spectrum of BPS states [1–35]....
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References
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TL;DR: In this paper, the authors proved an identity that equates the elliptic genus partition function of a supersymmetric sigma model on the N-fold symmetric product M N /SN of a manifold M to the partition function for a second quantized string theory on the space M × S 1, where the generating function of these elliptic genera is shown to be an automorphic form for O(3,2, Z).
Abstract: In this note we prove an identity that equates the elliptic genus partition function of a supersymmetric sigma model on the N-fold symmetric product M N /SN of a manifold M to the partition function of a second quantized string theory on the space M × S 1 . The generating function of these elliptic genera is shown to be (almost) an automorphic form for O(3,2, Z). In the context of D-brane dynamics, this result gives a precise computation of the free energy of a gas of D-strings inside a higher-dimensional brane.
594 citations
"BKM Lie superalgebras from dyon spe..." refers background in this paper
...Let us write f(ρ, σ, v) as f(ρ, σ, v) = [ ÊS∗(K3/ZN )(ρ, σ, v)× ETN(ρ, v)× g(ρ) ]−1 , (2.11) where [ ES∗(K3/ZN )(ρ, σ, v) ]−1 ≡ ∑ Q1,L,J ′ (−1)J ′dD1(Q1, L, J ′ ) e2πi(σQ1/N+ρL+vJ ′) , [ ETN(ρ, v) ]−1 ≡ 1 4 ∑ l0,j0 (−1)j0dCM(l0, j0) e2πil0ρ+2πij0v , [ g(ρ) ]−1 ≡ 1 16 ∑ l′0 dKK(l ′ 0) e 2πil′0ρ ....
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...ES∗(K3/ZN )(ρ, σ, v) is the second-quantized elliptic genus of K3/ZN [19]....
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TL;DR: In this paper, the authors describe recent progress in understanding the attractor mechanism and entropy of extremal black holes based on the entropy function formalism, and compare the statistical entropy of these dyons, expanded in inverse powers of electric and magnetic charges, with a similar expansion of the corresponding black hole entropy.
Abstract: In these lecture notes we describe recent progress in our understanding of attractor mechanism and entropy of extremal black holes based on the entropy function formalism. We also describe precise computation of the microscopic degeneracy of a class of quarter BPS dyons in \({\mathcal{N}=4}\) supersymmetric string theories, and compare the statistical entropy of these dyons, expanded in inverse powers of electric and magnetic charges, with a similar expansion of the corresponding black hole entropy. This comparison is extended to include the contribution to the entropy from multi-centred black holes as well.
545 citations
"BKM Lie superalgebras from dyon spe..." refers background or methods in this paper
...The computations in the appendices of David and Sen in [11] provide a microscopic understanding of the three different sources....
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...David and Sen further show that f(ρ, σ, v) = 1 Φ̃k(ρ, σ, v) = 1 Φ̃k(σ/N,Nρ, v) , (2.13) in the process obtaining a product representation for the generating function of 1 4 -BPS states, Φ̃k(Z)....
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...In this subsection, we will discuss the microscopic counting of dyon degeneracies carried out by David and Sen [11, 13]....
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...(See the review by Sen [13] and references therein for details....
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TL;DR: In this article, the degeneracy of dyons in four-dimensional N = 4 string theory has been studied in terms of a generalized super Kac-Moody algebra.
471 citations
"BKM Lie superalgebras from dyon spe..." refers background in this paper
...More than a decade ago, Dijkgraaf, Verlinde and Verlinde (DVV) proposed a microscopic index formula for the degeneracy of 1 4 -BPS dyons in heterotic string theory compactified on a six-torus [1]....
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360 citations
"BKM Lie superalgebras from dyon spe..." refers background in this paper
...Mukai then showed that if G is a finite group of symplectic automorphisms of K3, then (i) G acts as a permutation on the generators of H∗(K3,Z) and can be embedded as a subgroup of M23....
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...Mukai showed that any finite group of symplectic automorphisms of a K3 surface is a subgroup of the Mathieu group, M23 [24]....
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...Then, M23 is the subgroup of M24 that preserves one element of the set....
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...He observed that the number of fixed points, ε(n) (which depends only on the order of σ) is given by ε(n) = 24 n ∏ p|n(1 + 1 p ) , and happens to match the number of fixed points for a similar element of the Mathieu group, M23....
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...The embedding of G into M23 ⊂ M24 enables one to use known properties of M24....
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Book•
01 Jan 2007TL;DR: Modular forms Modular forms of level $1$ Modular form of weight $2$ Dirichlet characters Eisenstein series and Bernoulli numbers Dimension formulas Linear algebra General modular symbols Computing with new forms Computing periods Solutions to selected exercises Appendix A: Computing in higher rank Bibliography Index.
Abstract: Modular forms Modular forms of level $1$ Modular forms of weight $2$ Dirichlet characters Eisenstein series and Bernoulli numbers Dimension formulas Linear algebra General modular symbols Computing with newforms Computing periods Solutions to selected exercises Appendix A: Computing in higher rank Bibliography Index.
357 citations
"BKM Lie superalgebras from dyon spe..." refers background in this paper
...It is known that all Eisenstein series in this normalization have integral coefficients except for the constant term [52]....
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