The talk reports on recent progress in amplitude computations for broad classes of $N=2$ Maxwell-Einstein and Yang-Mills-Einstein supergravities, using the framework provided by color/kinematics duality and the double-copy construction.

After a review of the main theoretical tools, we discuss the extension of the double-copy construction to a particular infinite family of $N=2$ Maxwell-Einstein theories in four and five dimensions, the so-called generic Jordan family of supergravities.

We show that the global symmetries of these theories can easily be gauged, giving the amplitudes of the corresponding Yang-Mills-Einstein supergravities. We also discuss how the construction can be modified to describe spontaneous symmetry breaking. Finally, we present an extension of the construction that captures an even larger class of $N=2$ homogeneous supergravities.

It is becoming a common wisdom that a full-fledge formulation of quantum gravity needs to prescind from at least one of the two pillars on which our current understanding of nature is based, i.e. locality and unitarity. In order to get a clue of how this particular regime is reached, it is appropriate to reformulate the physics we know, which indeed is both local and unitary, in such a way that neither locality nor unitarity are fundamental assumptions but rather they are emergent. Small steps in this direction have been done in the area of scattering amplitudes for $N=4$,  where locality and unitarity are tied to the positivity of the geometry of some polytope through which the theory can be formulated.

In this talk, I will report on some very recent progress towards a similar formulation for correlators, which holds not only in flat space but also in a cosmological set-up (dS) or in anti-de-Sitter space. Interestingly, new mathematical structures arise. I will focus on new perturbative representations for such correlators and their interpretation in terms of new polytopes.

It is known, since Witten and Kontsevich’s work regarding the description of the intersection theory of $\psi$-classes on $\overline{\mathcal{M}}_{g, n}$ in terms of the $\tau$-function of a solution to the KdV hierarchy, that the connection between Integrable Systems and 2D Topological Field Theory is a deep one.

We will begin by introducing the fundamentals of Symplectic Field Theory (SFT), a branch of symplectic topology that studies holomorphic curves with boundaries in symplectic manifolds. The potential counting these curves is interpreted as an Hamiltonian corresponding to a Quantum Integrable System. In particular, we will consider the commuting Hamiltonians obtained from the quantisation of the dispersionless KdV hierarchy, that arise naturally in the context of SFT. A complete set of common eigenvectors of these operators is found in terms of Schur polynomials and used to compute the SFT-potential of a disk.

We will finish by providing some remarks on some more recent developments in this area, namely the case of the Quantised Toda Lattice and Double Ramification Hierarchy.

We investigate measures of chaos and entanglement scrambling in rational conformal field theories in $1+1$ dimensions. First, we derive a formula for the late time value of the out-of-time-order correlators for these class of theories. Our universal result can be expressed as a particular combination of the modular $S$-matrix elements known as anyon monodromy scalar. Next, in the explicit setup of a $SU(N)_k$ WZW model, we compare the late time behaviour of the out-of-time correlators and the purity. Interestingly, in the large-$c$ limit, the purity grows logarithmically but the out-of-time-order correlators remain constant. Therefore, we find that some systems may display entanglement scrambling in the absence of chaos.

Brane tilings describe Lagrangians of 4d $\mathcal{N}=1$ supersymmetric gauge theories. These theories, written in terms of a bipartite graph on a torus, correspond to the worldvolume theories on $D3$-branes probing a toric Calabi-Yau threefold singularity. A pair of permutations compactly encapsulates the data necessary to specify a brane tiling. We show that geometric consistency for brane tilings, which ensures that the corresponding quantum field theories are well behaved, imposes constraints on the pair of permutations, restricting certain products constructed from the pair to have no one-cycles. Such permutations are known as derangements. We illustrate this formulation of consistency with known brane tilings.

I will talk about the cosmological consequences of Weyl anomalies arising from the renormalization of composite operators in a theory of gravity with a cosmological constant. Near two dimensions, the relevant anomaly can be computed explicitly using results from Liouville theory and leads to a non-local quantum effective action. The resulting quantum energy momentum tensor is also non-local and leads to a decaying vacuum energy in a homogeneous and isotropic expanding universe. I will discuss a generalization of these results to four dimensions and possible implications for inflation and de Sitter spacetime.

I will present a new integrable quantum field theory describing two chiral scalars in four dimensions, in which the (generally rather obscure) origins of AdS/CFT integrability are considerably more visible. Using the integrability of this model, it is possible to compute the divergences of otherwise challenging Feynman integrals.

I will discuss the modular properties of D3-brane instantons appearing in Calabi-Yau string compactifications. I will show that the D3-instanton contribution to a certain geometric potential on the hypermultiplet moduli space can be related to the elliptic genus of $(0,4)$ SCFT. The modular properties of the potential imply that the elliptic genus associated with non-primitive divisors of Calabi-Yau is only mock modular. I will show how to construct its modular completion and prove the modular invariance of the twistorial construction of D-instanton corrected hypermultiplet moduli space.

We examine the relationship betwen classical solutions of non-Abelian gauge theory and gravity.

We discuss how the symmetries of gravitational theories may be regarded as originating from those of “Yang-Mills squared”.

Supergravity theories are naturally described in terms of certain generalised geometries, which combine diffeomorphisms with the other gauge symmetries of the model. Moreover, certain string backgrounds that defy a supergravity description can be captured by extensions of generalised geometry. I will describe the basics of these frameworks and recent results on the description of massive IIA supergravity, plus some work(s) in progress.

We present a study of $\mathcal{N}=4$ supersymmetric QED in three dimensions, on a three-sphere, with $2N$ massive hypermultiplets and a Fayet-Iliopoulos parameter. We identify the exact partition function of the theory with a conical (Mehler) function. This implies a number of analytical formulas, including a recurrence relation and a second-order differential equation. In the large $N$ limit, the theory undergoes a second-order phase transition on a critical line in the parameter space. We will discuss the critical behavior and compute the two-point correlation function of a gauge invariant mass operator.

(Joint work with Jorge G. Russo, arXiv:1610.08527)

There exists a deep correspondence between a class of physically important functions — called "on-shell functions" — and certain (cluster variety) subspaces of Grassmannian manifolds, endowed with a volume form that is left invariant under cluster coordinate transformations. These are called "on-shell varieties" (which may or may not include all cluster varieties). It is easy to prove that the number of on-shell varieties is finite, from which it follows that the same is true for on-shell functions. This is powerful and surprising, because these on-shell functions encode complete information about perturbative quantum field theory.

In this talk, I describe the details of this correspondence and how it is constructed and give the broad physics motivations for obtaining a more systematic understanding of on-shell cluster varieties. I outline a general, brute-force strategy for classifying these spaces; and describe the results found by applying this strategy to the case of $\operatorname{Gr}(3,6)$.

We will consider the $N=2$ abelian gauged supergravities in $d=4, 5$ coupled to arbitrary vector- and hypermultiplets. By choosing a static and spherical or hyperbolic ansatz for the extremal black hole in $d=4$ and for the extremal black string in $d=5$, it is possible to derive a one-dimensional effective action whose variation yields all the equations of motion. In particular, one can express the scalar potential in terms of a superpotential and write the action as a sum of squares. This procedure leads to the BPS first-order flow equations whose solution is driven by the Hamilton-Jacobi principal function of the one-dimensional dynamical system defined by the effective lagrangian. The latter squaring of the action is not unique and, by rotating the fluxes in an appropriate way, it is also possible to derive a non-BPS flow.

Furthermore, we will consider the near-horizon configurations and the so-called attractor equations, namely the near-horizon limit of the first-order flow equations. In particular, for the black string we will present the general solution for the scalars and for the radii of the horizon geometry in terms of the fluxes, the gauging parameters and the CY intersections numbers. From this result it is possible to derive the central charge of the two-dimensional CFT that describes the black string in the infrared.

Joint work with Dietmar Klemm and Marco Rabbiosi, arXiv:1602.01334, arXiv:1610.07367.