We consider compactifications of heterotic string theory to four dimensions on CHL orbifolds of the type T^6 /Z_N with 16 supersymmetries. The exact partition functions of the quarter-BPS dyons in these models are given in terms of genus-two Siegel modular forms. Only the N=1,2,3 models satisfy a certain finiteness condition, and in these cases one can identify a Borcherds-Kac-Moody superalgebra underlying the symmetry structure of the dyon spectrum. We identify the real roots, and find that the corresponding Cartan matrices exhaust a known classification. We show that the Siegel modular form satisfies the Weyl denominator identity of the algebra, which enables the determination of all root multiplicities. Furthermore, the Weyl group determines the structure of wall-crossings and the attractor flows of the theory. For N> 4, no such interpretation appears to be possible.
The degeneracies of supersymmetric quarter BPS dyons in four dimensions and of spinning black holes in five dimensions in a CHL compactification are computed exactly using Borcherds lift. The Hodge anomaly in the construction has a physical interpretation as the contribution of a single M-theory Kaluza-Klein 6-brane in the 4d-5d lift. Using factorization, it is shown that the resulting formula has a natural interpretation as a two-loop partition function of left-moving heterotic string, consistent with the heuristic picture of dyons in the M-theory lift of string webs.
We study the Borcherds superalgebra obtained by adding an odd (fermionic) null root to the set of simple roots of a simple finite-dimensional Lie algebra. We compare it to the Kac-Moody algebra obtained by replacing the odd null root by an ordinary simple root, and then adding more simple roots, such that each node that we add to the Dynkin diagram is connected to the previous one with a single line. This generalizes the situation in maximal supergravity, where the E(n) symmetry algebra can be extended to either a Borcherds superalgebra or to the Kac-Moody algebra E(11), and both extensions ca…
We investigate the conjectured infinite-dimensional hidden symmetries of six-dimensional chiral supergravity coupled to two vector multiplets and two tensor multiplets, which is known to possess the $F_{4,4}$ symmetry upon dimensional reduction to three spacetime dimensions. Two things are done. (i) First, we analyze the geodesic equations on the coset space $F_{4,4}^{++}/K(F_{4,4}^{++})$ using the level decomposition associated with the subalgebra $\mathfrak{gl}(5)\oplus \mathfrak{sl}(2)$ of $F_{4,4}^{++}$ and show their equivalence with the bosonic equations of motion of six-dimensional chir…
We introduce a quantum model for the Universe at its early stages, formulating a mechanism for the expansion of space and matter from a quantum initial condition, with particle interactions and creation driven by algebraic extensions of the Kac-Moody Lie algebra $\mathbf{e_9}$. We investigate Kac-Moody and Borcherds algebras, and we propose a generalization that meets further requirements that we regard as fundamental in quantum gravity.
Borcherds algebras represent a new class of Lie algebras which have almost all the properties that ordinary Kac-Moody algebras have, and the only major difference is that these generalized Kac-Moody algebras are allowed to have imaginary simple roots. The simplest nontrivial examples one can think of are those where one adds ``by hand'' one imaginary simple root to an ordinary Kac-Moody algebra. We study the fundamental representation of this class of examples and prove that an irreducible module is given by the full tensor algebra over some integrable highest weight module of the underlying K…
We construct a Borcherds Kac-Moody (BKM) superalgebra on which the Conway group Co$_0$ acts faithfully. We show that the BKM algebra is generated by the BRST-closed states in a chiral superstring theory. We use this construction to produce denominator identities for the chiral partition functions of the Conway module $V^{s\natural}$, a supersymmetric $c=12$ chiral conformal field theory whose (twisted) partition functions enjoy moonshine properties and which has automorphism group isomorphic to Co$_0$. In particular, these functions satisfy a genus zero property analogous to that of monstrous …
We present a variant of the Theory of Lorentzian (i. e. with a hyperbolic generalized Cartan matrix) Kac-Moody algebras recently developed by V. A. Gritsenko and the author. It is closely related with and strongly uses results of R. Borcherds. This theory should generalize well-known Theories of finite Kac-Moody algebras (i. e. classical semisimple Lie algebras corresponding to positive generalized Cartan matrices) and affine Kac-Moody algebras (corresponding to semi-positive generalized Cartan matrices). Main features of the Theory of Lorentzian Kac-Moody algebras are: One should consider gen…
We study the embedding of Kac-Moody algebras into Borcherds (or generalized Kac-Moody) algebras which can be explicitly realized as Lie algebras of physical states of some completely compactified bosonic string. The extra ``missing states'' can be decomposed into irreducible highest or lowest weight ``missing modules'' w.r.t. the relevant Kac-Moody subalgebra; the corresponding lowest weights are associated with imaginary simple roots whose multiplicities can be simply understood in terms of certain polarization states of the associated string. We analyse in detail two examples where the momen…
We study aspects of the theory of generalized Kac-Moody Lie algebras (or Borcherds algebras) and their standard modules. It is shown how such an algebra with no mutually orthogonal imaginary simple roots, including Borcherds' Monster Lie algebra $\frak m$, can be naturally constructed from a certain Kac-Moody subalgebra and a module for it. We observe that certain generalized Verma (induced) modules for generalized Kac-Moody algebras are standard modules and hence irreducible. In particular, starting from the moonshine module for the Monster group $M$, we construct a certain $\{frak gl}_2$- an…
The appearance of a generalized (or Borcherds-) Kac-Moody algebra in the spectrum of BPS dyons in N=4, d=4 string theory is elucidated. From the low-energy supergravity analysis, we identify its root lattice as the lattice of the T-duality invariants of the dyonic charges, the symmetry group of the root system as the extended S-duality group PGL(2,Z) of the theory, and the walls of Weyl chambers as the walls of marginal stability for the relevant two-centered solutions. This leads to an interpretation for the Weyl group as the group of wall-crossing, or the group of discrete attractor flows. F…
We provide evidence for the existence of a family of generalized Kac-Moody(GKM) superalgebras, G_N, whose Weyl-Kac-Borcherds denominator formula gives rise to a genus-two modular form at level N, Delta_{k/2}(Z), for (N,k)=(1,10), (2,6), (3,4), and possibly (5,2). The square of the automorphic form is the modular transform of the generating function of the degeneracy of CHL dyons in asymmetric Z_N-orbifolds of the heterotic string compactified on T^6. The new generalized Kac-Moody superalgebras all arise as different `automorphic corrections' of the same Lie algebra and are closely related to a…
We continue our study of algebraic properties of N=4 topological amplitudes in heterotic string theory compactified on T^2, initiated in arXiv:1102.1821. In this work we evaluate a particular one-loop amplitude for any enhanced gauge group h \subset e_8 + e_8, i.e. for arbitrary choice of Wilson line moduli. We show that a certain analytic part of the result has an infinite product representation, where the product is taken over the positive roots of a Lorentzian Kac-Moody algebra g^{++}. The latter is obtained through double extension of the complement g= (e_8 + e_8)/h. The infinite product i…
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.
The hyperbolic (and more generally, Lorentzian) Kac-Moody (KM) Lie algebras $\cA$ of rank $r+2 > 2$ are shown to have a rich structure of indefinite KM subalgebras which can be described by specifying a subset of positive real roots of $\cA$ such that the difference of any two is not a root of $\cA$. Taking these as the simple roots of the subalgebra gives a Cartan matrix, generators and relations for the subalgebra. Applying this to the canonical example of a rank 3 hyperbolic KM algebra, $\cF$, we find that $\cF$ contains all of the simply laced rank 2 hyperbolics, as well as an infinite ser…
Following Sen[arXiv:0911.1563], we study the counting of (`twisted') BPS states that contribute to twisted helicity trace indices in four-dimensional CHL models with N=4 supersymmetry. The generating functions of half-BPS states, twisted as well as untwisted, are given in terms of multiplicative eta products with the Mathieu group, M_{24}, playing an important role. These multiplicative eta products enable us to construct Siegel modular forms that count twisted quarter-BPS states. The square-roots of these Siegel modular forms turn out be precisely a special class of Siegel modular forms, the …
The main properties of indefinite Kac-Moody and Borcherds algebras, considered in a unified way as Lorentzian algebras, are reviewed. The connection with the conformal field theory of the vertex operator construction is discussed. By the folding procedure a class of subalgebras is obtained.
The fake monster Lie algebra is determined by the Borcherds function Phi_{12} which is the reflective modular form of the minimal possible weight with respect to O(II_{2,26}). We prove that the first non-zero Fourier-Jacobi coefficient of Phi_{12} in any of 23 Niemeier cusps is equal to the Weyl-Kac denominator function of the affine Lie algebra of the root system of the corresponding Niemeier lattice. This is an automorphic answer (in the case of the fake monster Lie algebra) on the old question of I. Frenkel and A. Feingold (1983) about possible relations between hyperbolic Kac-Moody algebra…
Tensor hierarchy algebras are infinite-dimensional generalisations of Cartan-type Lie superalgebras. They are not contragredient, exhibiting an asymmetry between positive and negative levels. These superalgebras have been a focus of attention due to the fundamental role they play for extended geometry. In the present paper, we examine tensor hierarchy algebras which are super-extensions of over-extended (often, hyperbolic) Kac--Moody algebras. They contain novel algebraic structures. Of particular interest is the extension of a over-extended algebra by its fundamental module, an extension that…
The representation theory of affine Kac-Moody Lie algebras has grown tremendously since their independent introduction by Robert V. Moody and Victor G. Kac in 1968. Inspired by mathematical structures found by theoretical physicists, and by the desire to understand the ``monstrous moonshine'' of the Monster group, the theory of vertex operator algebras (VOA's) was introduced by Borcherds, Frenkel, Lepowsky and Meurman. An important subject in this young field is the study of modules for VOA's and intertwining operators between modules. Feingold, Frenkel and Ries defined a structure, called a v…
We study a family of Siegel modular forms that are constructed using Jacobi forms that arise in Umbral moonshine. All but one of them arise as the Weyl-Kac-Borcherds denominator formula of some Borcherds-Kac-Moody (BKM) Lie superalgebras. These Lie superalgebras have a $\widehat{sl(2)}$ subalgebra which we use to study the Siegel modular forms. We show that the expansion of the Umbral Jacobi forms in terms of $\widehat{sl(2)}$ characters leads to vector-valued modular forms. We obtain closed formulae for these vector-valued modular forms. In the Lie algebraic context, the Fourier coefficients …
We study T^{11-D-q}xT^q/Z_n orbifold compactifications of 11D supergravity and M-theory by a purely algebraic method. Using the mapping between scalar fields of toroidally compactified maximal supergravity and generators of the U-duality symmetry, we express the orbifold action as a finite order inner automorphism and compute the residual real U-duality algebra surviving the orbifold projection for all dimensions D=1,...,10-q. In D=1, these invariant subalgebras are shown to be described by Borcherds and Kac-Moody algebras with a degenerate Cartan matrix, modded out by their centres and deriva…
We discuss a set of heterotic and type II string theory compactifications to 1+1 dimensions that are characterized by factorized internal worldsheet CFTs of the form $V_1\otimes \bar V_2$, where $V_1, V_2$ are self-dual (super) vertex operator algebras. In the cases with spacetime supersymmetry, we show that the BPS states form a module for a Borcherds-Kac-Moody (BKM) (super)algebra, and we prove that for each model the BKM (super)algebra is a symmetry of genus zero BPS string amplitudes. We compute the supersymmetric indices of these models using both Hamiltonian and path integral formalisms.…
The D1-D5-KK-p system naturally provides an infinite dimensional module graded by the dyonic charges whose dimensions are counted by the Igusa cusp form, Phi_{10}(Z)$. We show that the Mathieu group, M_{24}, acts on this module by recovering the Siegel modular forms that count twisted dyons as a trace over this module. This is done by recovering Borcherds product formulae for these modular forms using the M_{24} action. This establishes the correspondence (`moonshine') proposed in arXiv:0907.1410 that relates conjugacy classes of M_{24} to Siegel modular forms. This also, in a sense that we ma…
The square-root of Siegel modular forms of CHL Z_N orbifolds of type II compactifications are denominator formulae for some Borcherds-Kac-Moody Lie superalgebras for N=1,2,3,4. We study the decomposition of these Siegel modular forms in terms of characters of two sub-algebras: one is a $\widehat{sl(2)}$ and the second is a Borcherds extension of the $\widehat{sl(2)}$. This is a continuation of our previous work where we studied the case of Siegel modular forms appearing in the context of Umbral moonshine. This situation is more intricate and provides us with a new example (for N=5) that did no…
We study the problem of quadruple extensions of simple Lie algebras. We find that, adding a new simple root $α_{+4}$, it is not possible to have an extended Kac-Moody algebra described by a Dynkin-Kac diagram with simple links and no loops between the dots, while it is possible if $α_{+4}$ is a Borcherds imaginary simple root. We also comment on the root lattices of these new algebras. The folding procedure is applied to the simply-laced triple extended Lie algebras, obtaining all the non-simply laced ones. Non- standard extension procedures for a class of Lie algebras are proposed. It is show…
We propose that Borcherds' Fake Monster Lie algebra is a broken symmetry of heterotic string theory compactified on $T^7 \times T^2$. As evidence, we study the fully flavored counting function for BPS instantons contributing to a certain loop amplitude. The result is controlled by $Φ_{12}$, an automorphic form for $O(2, 26, \mathbb{Z})$. The degeneracies it encodes in its Fourier coefficients are graded dimensions of a second-quantized Fock space for this large symmetry algebra. This construction provides a concrete realization of Harvey and Moore's proposed relationship between Generalized Ka…
We give an analog of a Chevalley-Serre presentation for the Lie superalgebras W(n) and S(n) of Cartan type. These are part of a wider class of Lie superalgebras, the so-called tensor hierarchy algebras, denoted W(g) and S(g), where g denotes the Kac-Moody algebra A_r, D_r or E_r. Then W(A_{n-1}) and S(A_{n-1}) are the Lie superalgebras W(n) and S(n). The algebras W(g) and S(g) are constructed from the Dynkin diagram of the Borcherds-Kac-Moody superalgebras B(g) obtained by adding a single grey node (representing an odd null root) to the Dynkin diagram of g. We redefine the algebras W(A_r) and …
We study the second quantized version of the twisted twining genera of generalized Mathieu moonshine, and prove that they give rise to Siegel modular forms with infinite product representations. Most of these forms are expected to have an interpretation as twisted partition functions counting 1/4 BPS dyons in type II superstring theory on K3\times T^2 or in heterotic CHL-models. We show that all these Siegel modular forms, independently of their possible physical interpretation, satisfy an "S-duality" transformation and a "wall-crossing formula". The latter reproduces all the eta-products of a…
