Lorentzian Kac-Moody Algebras Research Articles

The [formula omitted]-tower of weak Jacobi forms and applications

We construct the tower of arithmetic generators of the bigraded polynomial ring J∗,∗w,O(Dn) of the weak Jacobi modular forms invariant with respect to the full orthogonal group O(Dn) of the root lattice Dn for 2≤n≤8. This tower corresponds to the tower of strongly reflective modular forms on the orthogonal groups of signature (2,n) which determine the Lorentzian Kac–Moody algebras related to the BCOV (Bershadsky–Cecotti–Ooguri–Vafa)-analytic torsions. We prove that the main three generators of index one of the graded ring satisfy a special system of modular differential equations. We found also a general modular differential equation of the generator of weight 0 and index 1 which generates the automorphic discriminant of the moduli space of Enriques surfaces.

Reflective modular forms and applications

The reflective modular forms of orthogonal type are fundamental automorphic objects generalizing the classical Dedekind eta-function. This article describes two methods for constructing such modular forms in terms of Jacobi forms: automorphic products and Jacobi lifting. In particular, it is proved that the first non-zero Fourier–Jacobi coefficient of the Borcherds modular form (the generating function of the so-called Fake Monster Lie Algebra) in any of the 23 one-dimensional cusps coincides with the Kac–Weyl denominator function of the affine algebra of the root system of the corresponding Niemeier lattice. This article gives a new simple construction of the automorphic discriminant of the moduli space of Enriques surfaces as a Jacobi lifting of the product of eight theta-functions and considers three towers of reflective modular forms. One of them, the tower of , gives a solution to a problem of Yoshikawa (2009) concerning the construction of Lorentzian Kac–Moody algebras from the automorphic discriminants connected with del Pezzo surfaces and analytic torsions of Calabi–Yau manifolds. The article also formulates some conditions on sublattices, making it possible to produce families of `daughter' reflective forms from a fixed modular form. As a result, nearly 100 such functions are constructed here. Bibliography: 77 titles.

Examples of lattice-polarized K3 surfaces with automorphic discriminant, and Lorentzian Kac–Moody algebras

Using our results about Lorentzian Kac--Moody algebras and arithmetic mirror symmetry, we give six series of examples of lattice-polarized K3 surfaces with automorphic discriminant.

On Brane Solutions with Intersection Rules Related to Lie Algebras

The review is devoted to exact solutions with hidden symmetries arising in a multidimensional gravitational model containing scalar fields and antisymmetric forms. These solutions are defined on a manifold of the form M = M0 x M1 x . . . x Mn , where all Mi with i >= 1 are fixed Einstein (e.g., Ricci-flat) spaces. We consider a warped product metric on M. Here, M0 is a base manifold, and all scale factors (of the warped product), scalar fields and potentials for monomial forms are functions on M0 . The monomial forms (of the electric or magnetic type) appear in the so-called composite brane ansatz for fields of forms. Under certain restrictions on branes, the sigma-model approach for the solutions to field equations was derived in earlier publications with V.N.Melnikov. The sigma model is defined on the manifold M0 of dimension d0 ≠ 2 . By using the sigma-model approach, several classes of exact solutions, e.g., solutions with harmonic functions, S-brane, black brane and fluxbrane solutions, are obtained. For d0 = 1 , the solutions are governed by moduli functions that obey Toda-like equations. For certain brane intersections related to Lie algebras of finite rank—non-singular Kac–Moody (KM) algebras—the moduli functions are governed by Toda equations corresponding to these algebras. For finite-dimensional semi-simple Lie algebras, the Toda equations are integrable, and for black brane and fluxbrane configurations, they give rise to polynomial moduli functions. Some examples of solutions, e.g., corresponding to finite dimensional semi-simple Lie algebras, hyperbolic KM algebras: H2(q, q) , AE3, HA(1)2, E10 and Lorentzian KM algebra P10 , are presented.

Uniruledness of orthogonal modular varieties

A strongly reflective modular form with respect to an orthogonal group of signature ( 2 , n ) (2,n) determines a Lorentzian Kac–Moody algebra. We find a new geometric application of such modular forms: we prove that if the weight is larger than n n , then the corresponding modular variety is uniruled. We also construct new reflective modular forms and thus provide new examples of uniruled moduli spaces of lattice polarised K 3 \mathrm {K3} surfaces. Finally, we prove that the moduli space of Kummer surfaces associated to ( 1 , 21 ) (1,21) -polarised abelian surfaces is uniruled.

Automorphic correction of the hyperbolic Kac-Moody algebra E10

In this paper, we study automorphic correction of the hyperbolic Kac-Moody algebra E10, using the Borcherds product for O(10, 2) attached to a weakly holomorphic modular form of weight −4 for \documentclass[12pt]{minimal}\begin{document}$SL_2(\mathbb {Z})$\end{document}SL2(Z). We also clarify some aspects of automorphic correction for Lorentzian Kac-Moody algebras and give heuristic reasons for the expectation that every Lorentzian Kac-Moody algebra has an automorphic correction.

Automorphic products, generalized Kac-Moody algebras and string amplitudes

We review automorphic products and generalized Kac-Moody algebras from a physics point of view. We discuss the appearance of automorphic products in BPS-saturated one-loop quantities in heterotic string theory. At particular points in moduli space, with enhanced gauge symmetry \U0001d525, these products can be used to define a generalized Kac-Moody algebra \U0001d50a(\U0001d524++) as an automorphic correction of the Lorentzian Kac-Moody algebra \U0001d524++, which is obtained through double extension of the complement \U0001d524 = (\U0001d5228 ⊕ \U0001d5228)/\U0001d525. The root multiplicities of \U0001d50a(\U0001d524++) are then encoded in the Fourier coefficients of certain modular forms, which appear directly in the integrand of the one-loop quantities. We review particular examples of this extension for compactifications with \U0001d4a9 = 2 and \U0001d4a9 = 4 space-time supersymmetry.

Enhanced gauge groups inN=4topological amplitudes and Lorentzian Borcherds algebras

We continue our study of algebraic properties of $\mathcal{N}=4$ topological amplitudes in heterotic string theory compactified on ${\mathbb{T}}^{2}$, initiated in arXiv:1102.1821. In this work we evaluate a particular one-loop amplitude for any enhanced gauge group $\mathfrak{h}\ensuremath{\subset}{\mathfrak{e}}_{8}\ensuremath{\bigoplus}{\mathfrak{e}}_{8}$, 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 ${\mathfrak{g}}^{++}$. The latter is obtained through double extension of the complement $\mathfrak{g}=({\mathfrak{e}}_{8}\ensuremath{\bigoplus}{\mathfrak{e}}_{8})/\mathfrak{h}$. The infinite product is automorphic with respect to a finite index subgroup of the full $T$-duality group $SO(2,18;\mathbb{Z})$ and, through the philosophy of Borcherds-Gritsenko-Nikulin, this defines the denominator formula of a generalized Kac-Moody algebra $\mathcal{G}({\mathfrak{g}}^{++})$, which is an 'automorphic correction' of ${\mathfrak{g}}^{++}$. We explicitly give the root multiplicities of $\mathcal{G}({\mathfrak{g}}^{++})$ for a number of examples.

Black brane solutions related to non-singular Kac-Moody algebras

A multidimensional gravitational model containing scalar fields and antisymmetric forms is considered. The manifold is chosen in the form M = M_0 x M_1 x ... x M_n, where M_i are Einstein spaces (i > 0). The sigma-model approach and exact solutions with intersecting composite branes (e.g., solutions with harmonic functions and black brane ones) with intersection rules related to non-singular Kac-Moody (KM) algebras (e.g. hyperbolic ones) are considered. Some examples of black brane solutions are presented, e.g., those corresponding to hyperbolic KM algebras: H_2(q,q) (q > 2), HA_2^(1) = A_2^{++} and to the Lorentzian KM algebra P_{10}.

On Brane Solutions Related to Non-Singular Kac-Moody Algebras

A multidimensional gravitational model containing scalar fields and antisym- metric forms is considered. The manifold is chosen in the form M = M0 ◊ M1 ◊ ··· ◊ Mn, where Mi are Einstein spaces (i 1). The sigma-model approach and exact solutions with intersecting composite branes (e.g. solutions with harmonic functions, S-brane and black brane ones) with intersection rules related to non-singular Kac-Moody (KM) algebras (e.g. hyperbolic ones) are reviewed. Some examples of solutions, e.g. corresponding to hyperbolic KM algebras: H2(q,q), AE3, HA (1) , E10 and Lorentzian KM algebra P10 are presented.

Spacelike Singularities and Hidden Symmetries of Gravity.

We review the intimate connection between (super-)gravity close to a spacelike singularity (the “BKL-limit”) and the theory of Lorentzian Kac-Moody algebras. We show that in this limit the gravitational theory can be reformulated in terms of billiard motion in a region of hyperbolic space, revealing that the dynamics is completely determined by a (possibly infinite) sequence of reflections, which are elements of a Lorentzian Coxeter group. Such Coxeter groups are the Weyl groups of infinite-dimensional Kac-Moody algebras, suggesting that these algebras yield symmetries of gravitational theories. Our presentation is aimed to be a self-contained and comprehensive treatment of the subject, with all the relevant mathematical background material introduced and explained in detail. We also review attempts at making the infinite-dimensional symmetries manifest, through the construction of a geodesic sigma model based on a Lorentzian Kac-Moody algebra. An explicit example is provided for the case of the hyperbolic algebra E10, which is conjectured to be an underlying symmetry of M-theory. Illustrations of this conjecture are also discussed in the context of cosmological solutions to eleven-dimensional supergravity.

COSMOLOGICAL SINGULARITIES, EINSTEIN BILLIARDS AND LORENTZIAN KAC–MOODY ALGEBRAS

The structure of the general, inhomogeneous solution of (bosonic) Einstein-matter systems in the vicinity of a cosmological singularity is considered. We review the proof (based on ideas of Belinskii–Khalatnikov–Lifshitz and technically simplified by the use of the Arnowitt–Deser–Misner Hamiltonian formalism) that the asymptotic behavior, as one approaches the singularity, of the general solution is describable, at each (generic) spatial point, as a billiard motion in an auxiliary Lorentzian space. For certain Einstein-matter systems, notably for pure Einstein gravity in any spacetime dimension D and for the particular Einstein-matter systems arising in String Theory, the billiard tables describing asymptotic cosmological behavior are found to be identical to the Weyl chambers of some Lorentzian Kac–Moody algebras. In the case of the bosonic sector of supergravity in 11 dimensional spacetime the underlying Lorentzian algebra is that of the hyperbolic Kac–Moody group E10, and there exists some evidence of a correspondence between the general solution of the Einstein-three-form system and a null geodesic in the infinite dimensional coset space E10/K(E10), where K(E10) is the maximal compact subgroup of E10.

E11 as E10 representation at low levels

The Lorentzian Kac–Moody algebra E 11, obtained by doubly overextending the compact E 8, is decomposed into representations of its canonical hyperbolic E 10 subalgebra. Whereas the appearing representations at levels 0 and 1 are known on general grounds, higher level representations can currently only be obtained by recursive methods. We present the results of such an analysis up to height 120 in E 11 which comprises representations on the first five levels. The algorithms used are a combination of Weyl orbit methods and standard methods based on the Peterson and Freudenthal formulae. In the appendices we give all multiplicities of E 10 occurring up to height 340 and for E 11 up to height 240.

Very extended E8 and A8 at low levels, gravity and supergravity

We define a level for a large class of Lorentzian Kac–Moody algebras. Using this we find the representation content of very extended AD−3 and E8 (i.e., E11) at low levels in terms of AD−1 and A10 representations, respectively. The results are consistent with the conjectured very extended A8 and E11 symmetries of gravity and maximal supergravity theories given respectively in preprints hep-th/0104081 and hep-th/0107209. We explain how these results provided further evidence for these conjectures.

A class of Lorentzian Kac–Moody algebras

We consider a natural generalisation of the class of hyperbolic Kac–Moody algebras. We describe in detail the conditions under which these algebras are Lorentzian. We also construct their fundamental weights, and analyse whether they possess a real principal so(1,2) subalgebra. Our class of algebras include the Lorentzian Kac–Moody algebras that have recently been proposed as symmetries of M-theory and the closed bosonic string.

A Theory of Lorentzian Kac--Moody Algebras

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 generalized Kac-Moody algebras introduced by Borcherds. Denominator function should be an automorphic form on IV type Hermitian symmetric domain (first example of this type related with Leech lattice was found by Borcherds). The Kac-Moody algebra is graded by an integral hyperbolic lattice $S$. Weyl group acts in the hyperbolic space related with $S$ and has a fundamental polyhedron $\Cal M$ of finite (or almost finite) volume and a lattice Weyl vector. There are results and conjectures which permit (in principle) to get a ``finite'' list of all possible Lorentzian Kac-Moody algebras. Thus, this theory looks very similar to Theories of finite and affine Kac-Moody algebras but is much more complicated. There were obtained some classification results on Lorentzian Kac-Moody algebras and many of them were constructed.

K3 surfaces with interesting groups of automorphisms

By the fundamental result of I. I. Piatetsky-Shapiro and I. R. Shafarevich (1971), the automorphism group Aut(X) of aK3 surfaceX over ℂ and its action on the Picard latticeS X are prescribed by the Picard latticeS X . We use this result and our method (1980) to show the finiteness of the set of Picard latticesS X of rank ≥ 3 such that the automorphism group Aut(X) of theK3 surfaceX has a nontrivial invariant sublatticeS 0 inS X where the group Aut(X) acts as a finite group. For hyperbolic and parabolic latticesS 0, this has been proved by the author before (1980, 1995). Thus we extend these results to negative sublatticesS 0. We give several examples of Picard latticesS X with parabolic and negativeS 0. We also formulate the corresponding finiteness result for reflective hyperbolic lattices of hyperbolic type over purely real algebraic number fields. We give many examples of reflective hyperbolic lattices of the hyperbolic type. These results are important for the theory of Lorentzian Kac-Moody algebras and mirror symmetry.

AUTOMORPHIC FORMS AND LORENTZIAN KAC–MOODY ALGEBRAS PART I

Using the general method which was applied to prove finiteness of the set of hyperbolic generalized Cartan matrices of elliptic and parabolic type, we classify all symmetric (and twisted to symmetric) hyperbolic generalized Cartan matrices of elliptic type and of rank 3 with a lattice Weyl vector. We develop the general theory of reflective lattices T with 2 negative squares and reflective automorphic forms on homogeneous domains of type IV defined by T. We consider this theory as mirror symmetric to the theory of elliptic and parabolic hyperbolic reflection groups and corresponding hyperbolic root systems. We formulate Arithmetic Mirror Symmetry Conjecture relating both these theories and prove some statements to support this Conjecture. This subject is connected with automorphic correction of Lorentzian Kac–Moody algebras. We define Lie reflective automorphic forms which are the most beautiful automorphic forms defining automorphic Lorentzian Kac–Moody algebras and formulate finiteness Conjecture for these forms. Detailed study of automorphic correction and Lie reflective automorphic forms for generalized Cartan matrices mentioned above will be given in Part II.

AUTOMORPHIC FORMS AND LORENTZIAN KAC–MOODY ALGEBRAS PART II

We give variants of lifting construction, which define new classes of modular forms on the Siegel upper half-space of complex dimension 3 with respect to the full paramodular groups (defining moduli of Abelian surfaces with arbitrary polarization). The data for these liftings are Jacobi forms of integral and half-integral indices. In particular, we get modular forms which are generalizations of the Dedekind eta-function. Some of these forms define automorphic corrections of Lorentzian Kac–Moody algebras with hyperbolic generalized Cartan matrices of rank three classified in Part I of this paper. We also construct many automorphic forms which give discriminants of moduli of K3 surfaces with conditions on Picard lattice.

Siegel automorphic form corrections of some Lorentzian Kac-Moody Lie algebras

We find automorphic form corrections which are generalized Lorentzian Kac-Moody superalgebras without odd real simple roots for two elliptic Lorentzian Kac-Moody algebras of rank 3 with a lattice Weyl vector, and calculate multiplicities of their simple and arbitrary roots. These Kac-Moody algebras are defined by hyperbolic symmetrized generalized Cartan matrices [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] of rank 3. Both these algebras have elliptic type (i.e., their Weyl groups have fundamental polyhedra of finite volume in corresponding hyperbolic spaces) and have a lattice Weyl vector. The correcting automorphic forms are Siegel modular forms. The form corresponding to G 1 is the classical Siegel cusp form of weight 5 which is the product of ten even theta-constants. In particular we find an infinite product formula for this modular form.