Abstract:
We survey recent results on quantum corrections to the hypermultiplet moduli space M in type IIA/B string theory on a compact Calabi-Yau threefold X, or, equivalently, the vector multiplet moduli space in type IIB/A on X x S^1. Our main focus lies on the problem of resumming the infinite series of D-brane and NS5-brane instantons, using the mathematical machinery of automorphic forms. We review the proposal that whenever the low-energy theory in D=3 exhibits an arithmetic "U-duality" symmetry G(Z) the total instanton partition function arises from a certain unitary automorphic representation of G, whose Fourier coefficients reproduce the BPS-degeneracies. For D=4, N=2 theories on R^3 x S^1 we argue that the relevant automorphic representation falls in the quaternionic discrete series of G, and that the partition function can be realized as a holomorphic section on the twistor space Z over M. We also offer some comments on the close relation with N=2 wall crossing formulae.

Abstract:
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 is automorphic with respect to a finite index subgroup of the full T-duality group SO(2,18;Z) and, through the philosophy of Borcherds-Gritsenko-Nikulin, this defines the denominator formula of a generalized Kac-Moody algebra G(g^{++}), which is an 'automorphic correction' of g^{++}. We explicitly give the root multiplicities of G(g^{++}) for a number of examples.

Abstract:
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 an older version of generalized Mathieu moonshine proposed by Mason in the '90s. Surprisingly, some of the Siegel modular forms we find coincide with the multiplicative (Borcherds) lifts of Jacobi forms in umbral moonshine.

Abstract:
Understanding the implications of SL(2,Z) S-duality for the hypermultiplet moduli space of type II string theories has led to much progress recently in uncovering D-instanton contributions. In this work, we suggest that the extended duality group SL(3,Z), which includes both S-duality and Ehlers symmetry, may determine the contributions of D5 and NS5-branes. We support this claim by automorphizing the perturbative corrections to the "extended universal hypermultiplet", a five-dimensional universal SL(3,R)/SO(3) subspace which includes the string coupling, overall volume, Ramond zero-form and six-form and NS axion. Using the non-Abelian Fourier expansion of the Eisenstein series attached to the principal series of SL(3,R), first worked out by Vinogradov and Takhtajan 30 years ago, we extract the contributions of D(-1)-D5 and NS5-brane instantons, corresponding to the Abelian and non-Abelian coefficients, respectively. In particular, the contributions of k NS5-branes can be summarized into a vector of wave functions \Psi_{k,l}, l=0... k-1, as expected on general grounds. We also point out that for more general models with a symmetric moduli space G/K, the minimal theta series of G generates an infinite series of exponential corrections of the form required for "small" D(-1)-D1-D3-D5-NS5 instanton bound states. As a mathematical spin-off, we make contact with earlier results in the literature about the spherical vectors for the principal series of SL(3,R) and for minimal representations.

Abstract:
We consider four dimensional CHL models with sixteen spacetime supersymmetries obtained from orbifolds of type IIA superstring on K3 x T^2 by a Z_N symmetry acting (possibly) non-geometrically on K3. We show that most of these models (in particular, for geometric symmetries) are self-dual under a weak-strong duality acting on the heterotic axio-dilaton modulus S by a "Fricke involution" S --> -1/NS. This is a novel symmetry of CHL models that lies outside of the standard SL(2,Z)-symmetry of the parent theory, heterotic strings on T^6. For self-dual models this implies that the lattice of purely electric charges is N-modular, i.e. isometric to its dual up to a rescaling of its quadratic form by N. We verify this prediction by determining the lattices of electric and magnetic charges in all relevant examples. We also calculate certain BPS-saturated couplings and verify that they are invariant under the Fricke S-duality. For CHL models that are not self-dual, the strong coupling limit is dual to type IIA compactified on T^6/Z_N, for some Z_N-symmetry preserving half of the spacetime supersymmetries.

Abstract:
We present a systematic study of the cosmological billiard structures of Einstein-p-form systems in which all spatial directions are compactified on a manifold of nontrivial topology. This is achieved for all maximally oxidised theories associated with split real forms, for all possible compactifications as defined by the de Rham cohomology of the internal manifold. In each case, we study the Coxeter group that controls the dynamics for energy scales below the Planck scale as well as the relevant billiard region. We compare and contrast them with the Weyl group and fundamental domain that emerge from the general BKL analysis. For generic topologies we find a variety of possibilities: (i) The group may or may not be a simplex Coxeter group; (ii) The billiard region may or may not be a fundamental domain. When it is not a fundamental domain, it can be described as a sequence of pairwise adjacent chambers, known as a gallery, and the reflections in the billiard walls provide a non-standard presentation of the Coxeter group. We find that it is only when the Coxeter group is a simplex Coxeter group, and the billiard region is a fundamental domain, that there is a correspondence between billiard walls and simple roots of a Kac-Moody algebra, as in the general BKL analysis. For each compactification we also determine whether or not the resulting theory exhibits chaotic dynamics.

Abstract:
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 E_10, 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.

Abstract:
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.

Abstract:
When formulated in twistor space, the D-instanton corrected hypermultiplet moduli space in N=2 string vacua and the Coulomb branch of rigid N=2 gauge theories on $R^3 \times S^1$ are strikingly similar and, to a large extent, dictated by consistency with wall-crossing. We elucidate this similarity by showing that these two spaces are related under a general duality between, on one hand, quaternion-Kahler manifolds with a quaternionic isometry and, on the other hand, hyperkahler manifolds with a rotational isometry, further equipped with a hyperholomorphic circle bundle with a connection. We show that the transition functions of the hyperholomorphic circle bundle relevant for the hypermultiplet moduli space are given by the Rogers dilogarithm function, and that consistency across walls of marginal stability is ensured by the motivic wall-crossing formula of Kontsevich and Soibelman. We illustrate the construction on some simple examples of wall-crossing related to cluster algebras for rank 2 Dynkin quivers. In an appendix we also provide a detailed discussion on the general relation between wall-crossing and the theory of cluster algebras.

Abstract:
Motivated by string theory scattering amplitudes that are invariant under a discrete U-duality, we study Fourier coefficients of Eisenstein series on Kac-Moody groups. In particular, we analyse the Eisenstein series on $E_9(R)$, $E_{10}(R)$ and $E_{11}(R)$ corresponding to certain degenerate principal series at the values s=3/2 and s=5/2 that were studied in 1204.3043. We show that these Eisenstein series have very simple Fourier coefficients as expected for their role as supersymmetric contributions to the higher derivative couplings $R^4$ and $\partial^{4} R^4$ coming from 1/2-BPS and 1/4-BPS instantons, respectively. This suggests that there exist minimal and next-to-minimal unipotent automorphic representations of the associated Kac-Moody groups to which these special Eisenstein series are attached. We also provide complete explicit expressions for degenerate Whittaker vectors of minimal Eisenstein series on $E_6(R)$, $E_7(R)$ and $E_8(R)$ that have not appeared in the literature before.