I present an extended version of Riemannian geometry suitable for the description of current formulations of double field theory (DFT). This framework is based on graded manifolds and it yields extended notions of symmetries, dynamical data and constraints. In special cases, we recover general relativity with and without 1-, 2- and 3-form gauge potentials as well as DFT. We believe that our extended Riemannian geometry helps to clarify the role of various constructions in DFT. For example, it leads to a covariant form of the strong section condition. Furthermore, it should provide a useful step towards global and coordinate invariant descriptions of T- and U-duality invariant field theories.

I will give a pedagogical review of how the pure spinor formalism was used to obtain all tree-level amplitudes of the superstring, including all their alpha' corrections.

I will present a light introduction to resurgence, with applications in large N gauge theory and in string theory.

I shall review proposals that have been made to patch doubled spaces and explain the difficulties that arise in an effort to achieve consistent patching. Then I shall propose a scheme based on C-spaces that leads to a consistent patching but requires additional coordinates to those of doubled spaces. The T-dual pairs are identified as subspaces of the gerbe in C-spaces. The talk is based on work done with Paul Howe, arXiv:1612.07968 [hep-th].

Perturbing a CFT by a relevant operator on a half space and letting the perturbation flow to the far infrared we obtain an RG interface between the UV and IR CFTs. If the IR CFT is trivial we obtain an RG boundary condition. The space of massive perturbations thus breaks up into regions labelled by conformal boundary conditions of the UV fixed point. For the 2D critical Ising model perturbed by a generic relevant operator we find the assignment of RG boundary conditions to all flows. We use some analytic results but mostly rely on TCSA and TFFSA numerical techniques. We investigate real as well as imaginary values of the magnetic field and, in particular, the RG trajectory that ends at the Yang-Lee CFT. We argue that the RG interface in the latter case does not approach a single conformal interface but rather exhibits oscillatory non-convergent behaviour.

In this talk, I will report on a research programme that addresses the problem of constructing integrable relativistic quantum field theories on two-dimensional Minkowski space from their two-body S-matrix. The aims of this programme are thus identical to the form factor programme, but the tools are different: Instead of concentrating on a perturbative construction of the correlation functions of local fields, we construct a pair of "semi-local" quantum fields and use operator-algebraic tools to study local fields/observables. This leads to a construction of many models, including the Sinh-Gordon model. As another prominent example, I will also report on the status of the O(N) sigma models within this setting.

We discuss examples of relevant deformations of known CFTs placed in de Sitter spacetime, possessing AdS gravity duals with Big Crunch singularities. We point out generic aspects of such dual backgrounds, including the behaviour in the vicinity of the singularity, holographic correlators in the geodesic approximation, and retarded correlators in an analytically tractable example within the AdS_4/CFT_3 context. We briefly describe an attempt to characterise the bulk crunch and its putative resolution within the O(N) vector model in 3D.

In recent years, there has been significant process in the study of perturbative field theory scattering amplitudes using certain 2d chiral, first-order CFTs known as 'ambitwistor string theories.' After a brief review of the ambitwistor setup, I will introduce a related family of 2d CFTs, which can be viewed as constrained ambitwistor strings. These new models describe holomorphic maps from the Riemann sphere to the projective null cone in D-dimensional Minkowski space. This target space is the natural setting to describe field theories with (classical) conformal invariance in (D-2)-dimensions. Killing the anomalies associated with these models fixes critical dimensions for which three well-known space-time field theories (bi-adjoint cubic scalar, gauge theory, gravity) are conformal. Furthermore, the spectrum of each model contains all single field insertions, along with their conformal descendants, of the correct scaling dimension. Time permitting, I will also outline how the space-time 3-point functions can be obtained from the 3-point correlators on the Riemann sphere.

Berry phase is a well-known feature of quantum mechanics. In this talk I will discuss the less-explored subject of Berry phase in quantum field theory. We will see that even simple quantum field theories can exhibit non-trivial Berry phases, and will discuss an explicit example in axion electrodynamics. We will also discuss a general relation between the Berry connection in conformal field theories with non-trivial conformal manifolds and the connections previously considered by several authors in conformal perturbation theory. The implementation of this relation in 2d N=(2,2) and 4d N=2 superconformal field theories leads to a useful re-derivation of the tt* equations.

String sigma-models relevant in AdS/CFT are highly non-trivial two-dimensional field theories for which predictions at finite coupling assume integrability and/or the duality itself. In this framework, I will discuss progress on how to extract finite coupling information via the use of lattice field theory methods.

One of the great success of string theory is the microscopical explanation of the entropy of a class of asymptotically flat black holes. Much less is known about asymptotically AdS black holes in four dimensions or higher. In this talk I explain how to derive the entropy of a class of asymptotically AdS supersymmetric black holes in four dimensions using holography. The counting of black hole micro-states is related to a counting of states in the dual 3d gauge theory which can be explicitly performed using localization.

An interesting link between the effective string theory describing the colour confinement in Yang-Mills theories and conformal field theories perturbed by the composite operator $T \bar{T}$ was recently discovered. We will discuss various aspects of this special kind of integrable perturbation and how it affects the energy levels of a generic 2D Quantum Field Theory, through a surprising relation with the inviscid Burgers equation. As this class of models corresponds to non-Wilsonian RG trajectories, these studies may help to clarify important aspects concerning the appearance of singularities in effective QFT, and hopefully be useful in the effective string framework.

I will review the recently exposed connection between N=2 superconformal field theories in four dimensions and vertex operator algebras (VOAs). I will outline some general features of the VOAs that arise in this manner and describe the manner in which they reflect four-dimensional operations such as gauging and Higgsing. Time permitting, I will also touch on the modular properties of characters of these VOAs.

I will report on joint work with Andrea Santi outlining an algebraic reformulation of the classification problem of eleven-dimensional supergravity backgrounds. The basic object of study is the Killing superalgebra of the background, whose algebraic structure has recently been elucidated. If time permits I will also comment on work also involving Paul de Medeiros applying these techniques to the construction of rigidly supersymmetric theories in curved spaces.

Conformal symmetry relates the metric on AdS_2 x S^1 to that of S^3. This implies that under a suitable choice of boundary conditions for fields on AdS_2 the partition function of conformal field theories on these spaces must agree which makes AdS_2 \times S^1 a good testing ground to study supersymmetric localization on non-compact spaces. We evaluate the partition function of N=2 supersymmetric Chern-Simons theory on AdS_2 x S^1 using localization, where the radius of S^1 is q times that of AdS_2. With boundary conditions on AdS_2 x S^1 which ensure that all the physical fields are normalizable and lie in the space of square integrable wave functions in AdS_2, we find that the result for the partition function precisely agrees with that of the theory on the q-fold covering of S^3.

In the first part of the talk, I will revisit the question of how to decide whether two 4D N=2 superconformal field theories describe the same physics. Time permitting, I will discuss related questions regarding the classification of 2D rational conformal field theories.

In 3d gauge theories, monopole operators create and destroy vortices. I will explore this idea in the context of 3d supersymmetric gauge theories in the presence of an omega background, and explain how it leads to a finite version of the AGT correspondence.

The Riemann hypothesis asserts that the nontrivial zeros of the Riemann zeta function should be of the form 1/2 + i E_n, where the set of numbers {E_n} are real. The so-called Hilbert-Pólya conjecture assumes that {E_n} should correspond to the eigenvalues of an operator that is Hermitian. The discovery of such an operator, if it exists, thus amounts to providing a proof of the Riemann hypothesis. In 1999 Berry and Keating conjectured that such an operator should correspond to a quantisation of the classical Hamiltonian H = xp. Since then, the Berry-Keating conjecture has been investigated intensely in the literature, but its validity has remained elusive up to now. In this talk I will derive a "Hamiltonian" (a differential operator), whose classical counterpart is H = xp, having the property that with a suitable boundary condition on its eigenstates, the eigenvalues {E_n} correspond to the nontrivial zeros of the Riemann zeta function. This Hamiltonian is not Hermitian, but is symmetric under space-time reflection (PT symmetric) in a special way. A formal argument will be given for the construction of the metric operator to define an inner-product space for the eigenstates, and the formally "Hermitian" counterpart Hamiltonian. The talk is based on the work carried out in collaboration with Carl M. Bender (Washington University) and Markus P. Mueller (University of Western Ontario).

As an alternative to the Holographic Renormalization procedure in the context of AdS/CFT correspondence, we introduce a regularization scheme for AdS gravity based on the addition of boundary terms which are a given polynomial of the extrinsic and intrinsic curvatures (Kounterterms). These terms are closely related to either topological invariants or Chern-Simons densities in the corresponding dimension, they can be easily generalized to other gravity theories (Einstein-Gauss-Bonnet, Lovelock, etc.). We provide a general prescription on how to obtain standard holographic quantities at the boundary. We also briefly comment on a possible relation to supersymmetry.