We establish a precise correspondence between the ABC Conjecture and N=4 super-Yang-Mills theory. This is achieved by combining three ingredients: (i) Elkies' method of mapping ABC-triples to elliptic curves in his demonstration that ABC implies Mordell/Faltings; (ii) an explicit pair of elliptic curve and associated Belyi map given by Khadjavi-Scharaschkin; and (iii) the fact that the bipartite brane-tiling/dimer model for a gauge theory with toric moduli space is a particular dessin d'enfant in the sense of Grothendieck. We explore this correspondence for the highest quality ABC-triples as well as large samples of random triples. The Conjecture itself is mapped to a statement about the fundamental domain of the toroidal compactification of the string realization of N=4 SYM.

We study the web of correspondences linking the exceptional Lie algebras E8,7,6 and the sporadic simple groups Monster, Baby and the largest Fischer group. We will survey some old observations from the perspective of Moonshine and representation theory and present some new ones from that of congruence groups and enumerative geometry. Based on joint work with John McKay.

5d theories can be at fixed points, in many cases exhibiting rather amusing properties. In this talk we will study several properties of these fixed point theories. In particular, we will describe how to put them supersymmetrically in curved spaces. As a consequence, we will discuss when a deformation to a Yang-Mills theory is possible. The latter typically exhibit a smaller global symmetry that that of the fixed point theory, which, from the YM point of view, appears as an enhanced symmetry. In this enhancement instantons play a pivotal role, and we will discuss several aspects of these.

Counting operators in supersymmetric field theories is very interesting, as the corresponding generating functions encode properties of the field theory. In this lecture we will mostly concentrate on 4d SUSY field theories. After reviewing some basic notions about SUSY field theories leading to the notion of chiral ring, we will introduce a generating function enumerating operators in the ring. This function is the so-called Hilbert series. In the case of SCFTs with a gravity dual, we will show how this function recovers geometric properties of the dual gravity background. The uses of the Hilbert series go beyond this, since, as we will see, being a counting of operators, it is of relevance in the computation of instanton partition functions for pure gauge theories. In the second part of the lecture we will briefly touch upon these issues.

We study the general formulation of gauged supergravity in seven dimensions with sixteen supercharges keeping duality covariance by means of the embedding tensor formalism. We first classify all inequivalent duality orbits of consistent deformations. Secondly, we analyse the complete set of critical points in a systematic way. Interestingly, we find the first examples of stable de Sitter solutions within a theory with such a large amount of supersymmetry.

Physical systems with unexpected, or `hidden,’ symmetries have often played an important role in physics, beginning with the classical Kepler problem whose Laplace-Runge-Lenz vector ensures the closure of planetary orbits, and degeneracies of the Hydrogen spectrum. I will describe how precisely the same symmetry governs a unique four-dimensional quantum field theory, a maximally supersymmetric (`N=4') cousin of the strong-interaction Yang-Mills theory. After reviewing progress in recent years in using these symmetries to solve this model, I will describe novel applications involving massive particles. Combining the Laplace-Runge-Lenz vector with relativity then yields a novel way to calculate the spectrum of its Hydrogen-like bound states, including relativistic corrections. Based on 1408.0296.

Inspired by the multiplicative nature of the Ramanujan modular discriminant, Delta, we consider physical realizations of certain multiplicative products over the Dedekind eta-function in two parallel directions: the generating function of BPS states in certain heterotic orbifolds and elliptic K3 surfaces associated to congruence subgroups of the modular group. We show that they are, after string duality to type II, the same K3 surfaces admitting Nikulin automorphisms. In due course, we will present some identities arising from q-expansions as well as relations to the sporadic Mathieu group M24.

Abelian gerbes play a ubiquitous role in string and supergravity theories. It is conjectured that non-abelian gerbe theories exist, but a satisfactory explicit construction of such a theory has not yet been achieved. Inspired by the possibility that a non-abelian theory of gerbes may be possible on non-conventional spacetimes, I will consider a toy model on a lattice that allows one to make some progress in this problem. In this talk, I will show how abelian gerbe theories may be described naturally on the lattice. I will outline a non-abelian generalisation and show how, under dimensional reduction, such theories give rise to lattice Yang-Mills.

In this talk I will review recent progress in understanding aspects of 5d gauge theories. Under certain circumstances these can be at fixed points admitting a large N limit, whose gravity dual we will discuss. Such gravity dual suggests a yet more general class of quiver-type fixed point theories which can sometimes be understood in terms of S-duality. To this matter the braneweb construction of the theories will prove very helpful, yielding new insights which we will test using the 5d superconformal index.

In this talk I will review recent progress in understanding aspects of 5d gauge theories. Under certain circumstances these can be at fixed points admitting a large N limit, whose gravity dual we will discuss. Such gravity dual suggests a yet more general class of quiver-type fixed point theories which can sometimes be understood in terms of S-duality. To this matter the braneweb construction of the theories will prove very helpful, yielding new insights which we will test using the 5d superconformal index.

How many H2O molecules are needed to form water? While the precise answer is not known, it is clear that the answer should be a finite number rather than infinity. We revisit with care the ideal Bose gas confined in a cubic box which is discussed in most statistical physics textbooks. We show that the isobar of the ideal gas zigzags on the temperature-volume plane featuring a `boiling-like' discrete phase transition, provided the number of particles is equal to or greater than a particular value: 7616. This demonstrates for the first time how a finite system can feature a mathematical singularity and realize the notion of `Emergence', without resorting to the thermodynamic limit. ref: arXiv:1310.5580

Chern-Simons theories coupled to vector matter exhibit interesting phenomena. In the planar limit, these theories are conjectured to be holographically dual to generalized theories of gravity, involving high-spin fields. This is a weak-weak holographic duality that is in some aspects very simple, and may serve as a toy model for deepening our understanding of both holography and string theory. On the CFT side, exact calculations performed in the planar limit, along with constraints imposed by a ‘slightly-broken’ high-spin symmetry, have led to many exact results. These have uncovered the details of a 3D bosonization duality, relating theories with bosonic matter to theories with fermionic matter. I will present dynamical evidence for this duality.

To the full order in fermions, we construct D = 10 type II supersymmetric double field theory. We spell the precise N = 2 supersymmetry transformation rules as for 32 supercharges. In terms of a stringy differential geometry beyond Riemann, the constructed action unifies type IIA and IIB supergravities in a manifestly covariant manner with respect to O(10, 10) T-duality and a ‘pair’ of local Lorentz groups, or Spin(1, 9) × Spin(9, 1), besides the usual general covariance of supergravities or the generalized diffeomorphism. The distinction of IIA and IIB may arise after a diagonal gauge fixing of the Lorentz groups. They are identified as two different types of ‘solutions’ rather than two different theories. References: arXiv:1210.5078 (N=2) arXiv:1206.3478 (bosonic N=2) arXiv:1112.0069 (N=1)

An exact solution representing black holes in an expanding universe is found. The black holes are maximally charged and the universe is expanding with arbitrary equation of state. It is an exact solution of the Einstein-scalar-Maxwell system, in which we have two Maxwell-type U(1) fields coupled to the scalar field. The potential of the scalar field is an exponential. The solution depends on two parameters, the charge Q and one parameter (the ratio of the energy density of U(1) fields to that of the scalar field). We find a regular horizon, which is static because of the balance on the horizon between gravitational attractive force and U(1) repulsive force acting on the scalar field. We also calculate the black hole temperature. For the case without a potential, we can derive such a solution from a time-dependent intersecting M-brane solution in eleven dimensions by the dimensional reduction.

We establish a correspondence between generalized quiver gauge theories in four-dimensions and congruence subgroups of the modular group, hinging upon the trivalent graphs which arise in both. The gauge theories and the graphs are enumerated and their numbers are compared. The correspondence is particularly striking for genus zero torsion-free congruence subgroups which are crucial to Moonshine. We analyze in detail the case of index 24, where modular elliptic K3 surfaces emerge: here, the elliptic j-invariants can be recast as dessins d'enfant which dictate the Seiberg-Witten curves.