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We exposit the construction of Rademacher sums in arbitrary weights and describe their relationship to mock modular forms. We introduce the notion of Rademacher series and describe several applications, including the determination of coefficients of Rademacher sums and a very general form of Zagier duality. We then review the application of Rademacher sums and series to moonshine both monstrous and umbral and highlight several open problems. We conclude with a discussion of the interpretation of Rademacher sums in physics., Comment: 46 pages

We describe surprising relationships between automorphic forms of various kinds, imaginary quadratic number fields and a certain system of six finite groups that are parameterised naturally by the divisors of twelve. The Mathieu group correspondence recently discovered by Eguchi-Ooguri-Tachikawa is recovered as a special case. We introduce a notion of extremal Jacobi form and prove that it characterises the Jacobi forms arising by establishing a connection to critical values of Dirichlet series attached to modular forms of weight two. These extremal Jacobi forms are closely related to certain vector-valued mock modular forms studied recently by Dabholkar-Murthy-Zagier in connection with the physics of quantum black holes in string theory. In a manner similar to monstrous moonshine the automorphic forms we identify constitute evidence for the existence of infinite-dimensional graded modules for the six groups in our system. We formulate an umbral moonshine conjecture that is in direct analogy with the monstrous moonshine conjecture of Conway-Norton. Curiously, we find a number of Ramanujan's mock theta functions appearing as McKay-Thompson series. A new feature not apparent in the monstrous case is a property which allows us to predict the fields of definition of certain homogeneous submodules for the groups involved. For four of the groups in our system we find analogues of both the classical McKay correspondence and McKay's monstrous Dynkin diagram observation manifesting simultaneously and compatibly., Comment: 124 pages, 75 tables; updated references and minor editing in v.2; minor editing and two new theorems in v.3

We review the relationship between the largest Mathieu group and various modular objects, including recent progress on the relation to mock modular forms. We also review the connections between these mathematical structures and string theory on K3 surfaces., Comment: 32 pages, 7 tables, 1 figure, submitted to the AMS Proceedings of Symposia in Pure Mathematics (String-Math 2011)

Using experiments and simulations, we investigate the clusters that form when colloidal spheres stick irreversibly to -- or "park" on -- smaller spheres. We use either oppositely charged particles or particles labeled with complementary DNA sequences, and we vary the ratio $\alpha$ of large to small sphere radii. Once bound, the large spheres cannot rearrange, and thus the clusters do not form dense or symmetric packings. Nevertheless, this stochastic aggregation process yields a remarkably narrow distribution of clusters with nearly 90% tetrahedra at $\alpha=2.45$. The high yield of tetrahedra, which reaches 100% in simulations at $\alpha=2.41$, arises not simply because of packing constraints, but also because of the existence of a long-time lower bound that we call the "minimum parking" number. We derive this lower bound from solutions to the classic mathematical problem of spherical covering, and we show that there is a critical size ratio $\alpha_c=(1+\sqrt{2})\approx 2.41$, close to the observed point of maximum yield, where the lower bound equals the upper bound set by packing constraints. The emergence of a critical value in a random aggregation process offers a robust method to assemble uniform clusters for a variety of applications, including metamaterials., Comment: 24 pages, 6 figures

Recently a conjecture has been proposed which attaches (mock) modular forms to the largest Mathieu group. This may be compared to monstrous moonshine, in which modular functions are attached to elements of the Monster group. One of the most remarkable aspects of monstrous moonshine is the following genus zero property: the modular functions turn out to be the generators for the function fields of their invariance groups. In particular, these invariance groups define genus zero quotients of the upper half plane. It is therefore natural to ask if there is an analogue of this property in the Mathieu case, and at first glance the answer appears to be negative since not all the discrete groups arising there have genus zero. On the other hand, in this article we prove that each (mock) modular form appearing in the Mathieu correspondence coincides with the Rademacher sum constructed from its polar part. This property, inspired by the AdS/CFT correspondence in physics, was shown previously to be equivalent to the genus zero property of monstrous moonshine. Hence we conclude that this "Rademacher summability" property serves as the natural analogue of the genus zero property in the Mathieu case. Our result constitutes further evidence that the Rademacher method provides a powerful framework for understanding the modularity of moonshine, and leads to interesting physical questions regarding the gravitational duals of the relevant conformal field theories., Comment: 34 pages with 6 appendices

We consider branes in refined topological strings. We argue that their wave-functions satisfy a Schr\"odinger equation depending on multiple times and prove this in the case where the topological string has a dual matrix model description. Furthermore, in the limit where one of the equivariant rotations approaches zero, the brane partition function satisfies a time-independent Schroedinger equation. We use this observation, as well as the back reaction of the brane on the closed string geometry, to offer an explanation of the connection between integrable systems and N=2 gauge systems in four dimensions observed by Nekrasov and Shatashvili., Comment: 64 pages, 2 figures

We give a non-perturbative completion of a class of closed topological string theories in terms of building blocks of dual open strings. In the specific case where the open string is given by a matrix model these blocks correspond to a choice of integration contour. We then apply this definition to the AGT setup where the dual matrix model has logarithmic potential and is conjecturally equivalent to Liouville conformal field theory. By studying the natural contours of these matrix integrals and their monodromy properties, we propose a precise map between topological string blocks and Liouville conformal blocks. Remarkably, this description makes use of the light-cone diagrams of closed string field theory, where the critical points of the matrix potential correspond to string interaction points., Comment: 36 pages

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., Comment: 34 pages, references added, minor update on the multiplier system of two weak jacobi forms

The simplest gravity duals for quantum critical theories with z=2 `Lifshitz' scale invariance admit a marginally relevant deformation. Generic black holes in the bulk describe the field theory with a dynamically generated momentum scale Lambda as well as finite temperature T. We describe the thermodynamics of these black holes in the quantum critical regime where T >> Lambda^2. The deformation changes the asymptotics of the spacetime mildly and leads to intricate UV sensitivities of the theory which we control perturbatively in Lambda^2/T., Comment: 1+27 pages, 12 figures

We consider Yukawa couplings generated by a configuration of intersecting seven-branes in F-theory. In configurations with a single interaction point and no fluxes turned on, the Yukawa matrices have rank one. This is no longer true when the three-form H-flux is turned on, which is generically the case for F-theory compactifications on Calabi-Yau fourfolds. In the presence of H-fluxes, the Yukawa coupling is computed using a non-commutative deformation of holomorphic Chern-Simons theory (and its reduction to seven-branes) and subsequently the rank of the Yukawa matrix changes. Such fluxes give rise to a hierarchical structure in the Yukawa matrix in F-theory GUTs of the type which has recently been proposed as a resolution of the flavor hierarchy problem., Comment: v2: 78 pages, reference added

Recently, using supergravity analysis, a hyperbolic reflection group was found to underlie the structure of wall-crossing, or the discontinuous moduli dependence of the supersymmetric index due to the presence of walls of marginal stability, of the BPS dyons in the N=4, d=4 compactification. In this paper we work in the regime where four-dimensional gravity decouples and we show how the presence of such a group structure can be easily understood as a consequence of the supersymmetry of a system of (p,q) five-brane network, or equivalently the holomorphicity of the Riemann surface wrapped by the appropriate M5 branes in the Euclidean M-theory frame., Comment: 37 pages, 5 figures

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., Comment: 44 pages, 1 figure

In this thesis we study the spectra of supersymmetric states in string theory compactifications with eight and sixteen supercharges, with special focus placed on the quantum states of black holes and the phenomenon of wall-crossing in these theories. A self-contained introduction to the relevant background material is included., Comment: PhD Thesis, 220 pages, 17 figures

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. Furthermore we propose an equivalence between a "second-quantized multiplicity" of a charge- and moduli-dependent highest weight vector and the dyon degeneracy, and show that the wall-crossing formula following from our proposal agrees with the wall-crossing formula obtained from the supergravity analysis. This can be thought of as providing a microscopic derivation of the wall-crossing formula of this theory., Comment: This is a contribution to the Special Issue on Kac-Moody Algebras and Applications, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

The dyonic 1/4-BPS states in 4D string theory with N=4 spacetime supersymmetry are counted by a Siegel modular form. The pole structure of the modular form leads to a contour dependence in the counting formula obscuring its duality invariance. We exhibit the relation between this ambiguity and the (dis-)appearance of bound states of 1/2-BPS configurations. Using this insight we propose a precise moduli-dependent contour prescription for the counting formula. We then show that the degeneracies are duality-invariant and are correctly adjusted at the walls of marginal stability to account for the (dis-)appearance of the two-centered bound states. Especially, for large black holes none of these bound states exists at the attractor point and none of these ambiguous poles contributes to the counting formula. Using this fact we also propose a second, moduli-independent contour which counts the "immortal dyons" that are stable everywhere., Comment: 27 pages, 2 figures; one minus sign corrected

In this note we construct families of asymptotically flat, smooth, horizonless solutions with a large number of non-trivial two-cycles (bubbles) of N=1 five-dimensional supergravity with an arbitrary number of vector multiplets, which may or may not have the charges of a macroscopic black hole and which contain the known bubbling solutions as a sub-family. We do this by lifting various multi-center BPS states of type IIA compactified on Calabi-Yau three-folds and taking the decompactification (M-theory) limit. We also analyse various properties of these solutions, including the conserved charges, the shape, especially the (absence of) throat and closed timelike curves, and relate them to the various properties of the four-dimensional BPS states. We finish by briefly commenting on their degeneracies and their possible relations to the fuzzball proposal of Mathur et al., Comment: 36 pages, Latex; JHEP version, one appendix added, references added

The microstates of 4d BPS black holes in IIA string theory compactified on a Calabi-Yau manifold are counted by a (generalized) elliptic genus of a (0,4) conformal field theory. By exploiting a spectral flow that relates states with different charges, and using the Rademacher formula, we find that the elliptic genus has an exact asymptotic expansion in terms of semi-classical saddle-points of the dual supergravity theory. This generalizes the known "Black Hole Farey Tail" of [1] to the case of attractor black holes., Comment: 36 pages, 3 figures, note added

We derive necessary conditions for the spinorial Witten-Nester energy to be well-defined for asymptotically locally AdS spacetimes. We find that the conformal boundary should admit a spinor satisfying certain differential conditions and in odd dimensions the boundary metric should be conformally Einstein. We show that these conditions are satisfied by asymptotically AdS spacetimes. The gravitational energy (obtained using the holographic stress energy tensor) and the spinorial energy are equal in even dimensions and differ by a bounded quantity related to the conformal anomaly in odd dimensions., Comment: 36 pages, 1 figure; minor corrections, JHEP version