We review some recent results on quaternion-Kähler geometry which were obtained in the context supergravity and string theory.

1. By proving that the supergravity \(r\)-map and \(c\)-map preserve geodesic completeness, we established a method to construct non-homogeneous geodesically complete quaternion-Kähler manifolds starting from complete special Kähler or special real manifolds.
2. Deriving the hyper-Kähler/quaternion-Kähler correspondence using conification, we have obtained a new proof that spaces in the image of the \(c\)-map are quaternion-Kähler, which extends to include the one-loop quantum deformation.
3. Time permitting we will also discuss how some of the results can be adapted to para-quaternion-Kähler manifolds.

We review some recent results on quaternion-Kähler geometry which were obtained in the context supergravity and string theory.

1. By proving that the supergravity \(r\)-map and \(c\)-map preserve geodesic completeness, we established a method to construct non-homogeneous geodesically complete quaternion-Kähler manifolds starting from complete special Kähler or special real manifolds.
2. Deriving the hyper-Kähler/quaternion-Kähler correspondence using conification, we have obtained a new proof that spaces in the image of the \(c\)-map are quaternion-Kähler, which extends to include the one-loop quantum deformation.
3. Time permitting we will also discuss how some of the results can be adapted to para-quaternion-Kähler manifolds.

High energy collisions in non-abelian gauge theories is currently a subject of great interest. In heavy ion collisions performed at RHIC and LHC the formation of a quark-gluon plasma is observed. The physics involved is presumably described by Quantum Chromodynamics (QCD), a non-conformal field theory that exhibits confinement. Nevertheless, similar collisions in conformal field theories have been investigated over recent years by exploring the AdS/CFT duality. Although the dual of QCD is not known, the analogous process in gauge theories with a gravity dual can be described via the collision of two objects that form a black hole in an asymptotically Anti-de Sitter (AdS) spacetime. This is a challenging problem that requires solving Einstein’s equations in a dynamical setting.

In this talk I will describe a first step towards extending this program to gravitational duals of confining gauge theories. We adapt the Zero Frequency Limit (ZFL) framework — a linearised approach which has been very useful in simpler contexts — to the problem under consideration, rendering it amenable to semi-analytic treatment. Our results include some universal features that are expected also for non-linear collisions.

Conformal field theories are both interesting and important. Recently there has been a renewed interest in these theories in four dimensions. After reviewing some old results about conformal field theories, I will turn to new developments such as the a-theorem and the relation between conformal and scale invariance.

We obtain a general formula for the low frequency absorption cross section of spherically symmetric black holes in \(d\) dimensions, including leading string-theoretical \(\alpha'\)-corrections. We show that for non-extremal black holes this covariant formula must be given in terms of the horizon area and temperature, while for extremal black holes it is simply given in terms of the horizon area.

Classically, this cross section equals four times the black hole entropy; by applying these formulas for the cross section to known solutions, we discuss when such relation with the entropy is preserved including the \(\alpha'\)-corrections.

In mathematics and physics the word Moonshine represents surprising and deep connections between a priori unrelated fields, such as number theory, representation theory and string theory. The most famous example is Monstrous Moonshine, which relates Fourier coefficients of modular forms with representations of the largest finite sporadic group, known as the Monster group. Recently, a new moonshine phenomenon was discovered, which connects the largest Mathieu group \(M24\) with superconformal field theories on \(K3\)-surfaces. In this talk I will describe recent progress in our understanding of this Mathieu Moonshine, and show how it is connected to the problem of counting dyonic black holes in \(N=4\) string theories.

In M-theory, the only \( AdS_7 \) supersymmetric solutions are \( AdS_7 \times S^4 \) and its orbifolds. In this talk, I will first describe a classification of \( AdS_7 \) supersymmetric solutions in type II supergravity. While in IIB none exist, in IIA with Romans mass (which does not lift to M-theory) there are many new ones. Pure spinor methods determine the metric and fluxes without the need for any Ansatz, up to solving a system of ODEs. The internal space \( M_3 \) is an \( S^2 \) fibration over an interval. I will then propose the holographically dual \( (1,0) \) CFTs in six dimensions. It is conjectured that the \( AdS_7 \) duals represent near-horizon limits of NS5-D6-D8 systems in flat space. The new field theories arise as endpoints of RG flows triggered by expectation values on the Higgs branch of the \( (1,0) \) theory describing multiple M5 branes transverse to an orbifold singularity.

Non-geometric string backgrounds were proposed to be related to a non-associative deformation of the space-time geometry. In this talk, using the flux formulation of double field theory (DFT), the structure of non-associative deformations is analyzed in detail. It is argued that on-shell there should not be any violation of associativity in the effective DFT action. I discuss two possible non-associative deformations of DFT featuring two different ways how on-shell associativity can still be kept.

The pioneering work of Bekenstein and Hawking in the 70's produced a universal area law for black hole entropy valid in the infinite size limit.  Quantum corrections to the gravitational action induce finite size corrections to the black hole entropy.  I shall report on progress in the computation of the exact quantum entropy of supersymmetric black holes in supergravity, using localization techniques.  In simple examples in string theory, one has a solvable dual microscopic description as an ensemble of microscopic excitations. I shall describe how the gravity functional integral leads to the microscopic integer degeneracies of this black hole, and its associated number theoretic properties.

We will introduce and discuss special Kähler manifolds, which appear as target spaces of $4D, N = 2$ supergravity coupled to vector multiplets. Through dimensional reduction over a spacelike or timelike circle, a special Kähler manifold can be mapped to a quaternion Kähler or para-quaternion Kähler manifold, respectively. This construction will be reviewed, and we shall see how stationary supergravity solutions appear as interesting submanifolds of certain para-quaternion Kähler manifolds.

I will present recent results on the computation of finite N effects in supergravity in the context of \(AdS_2/CFT_1\) and \(AdS_4/\)ABJM holography. I will show how to use localisation to compute all perturbative and nonperturbative charge corrections to the entropy of supersymmetric black holes including complicated number theoretic objects called Kloosterman sums. These are essential to recover an integer which can be identified as the number of black hole ground states. I will then explain how these techniques can be used on M-theory on \(AdS_4 \times S^7\) to compute the exact perturbative \(AdS_4\) partition function, the Airy function, as predicted from ABJM theory on a three sphere.

Using the attractor mechanism for extremal solutions in \(\mathcal{N} = 2\) gauged supergravity, we construct a \(c\)-function that interpolates between the central charges of theories at ultraviolet and infrared conformal fixed points corresponding to anti-de Sitter geometries. The \(c\)-function we obtain is couched purely in terms of bulk quantities and connects two different dimensional CFTs at the stable conformal fixed points under the RG flow.

In this talk I will propose a general correspondence which associates a non-perturbative quantum mechanical operator to a toric Calabi-Yau manifold, and I will propose a conjectural expression for its spectral determinant. As a consequence of these results, I will derive an exact quantization condition for the operator spectrum. I will give a concrete illustration of this conjecture by focusing on the example of local $P^2$ and local $P^1 \times P^1$. This approach also provides a non-perturbative Fermi gas picture of topological strings on toric Calabi-Yau manifolds and suggests the existence of an underlying theory of M2 branes behind this formulation.

The theory of resurgence is the unified framework where to understand why many results in perturbation theory are asymptotic and divergent, how to relate this to nonperturbative effects, and how to go from a formal (trans)series to a proper function. Topological string theory is at the center of a net of dualities, equivalences, and connections to string and gauge theories, and to complex and enumerative geometry. The perturbative free energy is an asymptotic series in the string coupling constant that can be computed to high order. The nonperturbative completion is an open problem that has seen great advance this year from connections to the refined topological string and from the perspective of resurgence. In this talk I present the basic aspects of resurgence and asymptotics, and I explain how to apply them to reconstruct the nonperturbative (trans)series that represents the closed topological string free energy.