Gravity in three dimensions strikes a balance between models that are tractable and models that are relevant. It may exhibit black hole solutions, graviton modes and asymptotically Anti-deSitter solutions. There have been great efforts in trying to quantize 3D gravity, but the results remain unsatisfactory. In this talk we review a recent approach to 3D gravity called "chiral gravity". This model has peculiar features giving some hope that it is particularly simple to quantize. However, there has been a sometimes heated debate regarding the stability of the model. We explain why, and give an updated status report.

Chern-Simons gauge theory on a three-manifold M bears an intimate physical relationship to both the open and the closed Gromov-Witten theories. One surprising aspect of this relationship is that the latter theories are cohomological in character, whereas a corresponding cohomological description of Chern-Simons theory is not generally known (nor expected). Nevertheless, as I will explain, in the case that M admits a locally-free U(1) action, some observables in Chern-Simons theory do admit a cohomological description on the moduli space of flat connections on M.

Open-Closed String duality is one of the central ideas in String theory which is responsible for several striking conjectures such as the AdS/CFT. We will argue that Strebel differentials play a very natural role in implementing this duality bringing one to closer to a general understanding of gauge theories as string theories.

The supersymmetry constraints of minimal gauged supergravity are analysed with the purpose of finding the most general asymptotically $\operatorname{AdS}_5$ black holes with a topologically spherical horizon. We show that under general arguments, such black holes are essentially determined by a two dimensional space, whose curvature obeys a 4th order differential equation. We recast the problem of finding solutions of such equation as a variational problem, and show that an infinite class of solutions exists, with each solution being characterised by the number and strength of conical singularities as well as the mean squared curvature of the 2D space.

The OSV conjecture (2004) states that microstates of four-dimensional supersymmetric black holes are captured by the topological string, by relating the black hole partition function to the topological string partition function. Duality covariance, however, is not manifest in the OSV proposal. We show how the inclusion of a non-trivial integration measure factor restores duality covariance. Our proposal is in agreement with recent results for microstate degeneracies, and it also points to the existence of a consistent non-holomorphic deformation of special geometry.

Gauge theories with gauge potential that is an abelian 2-form or 3-form are relatively well understood and are relevant in supergravity and in string theory. We study the nonabelian version of these gauge theories, in particular their underlying geometry. We see that a relaxed version of the notion of connection one form on a principal bundle is necessary in the context of nonabelian gerbes. We then give evidence that such nonabelian gerbes are relevant for the description of a stack of M5-branes in M-theory.

The old suggestive observation that black holes often resemble lumps of fluid has recently been taken beyond the level of an analogy to a precise duality. We investigate aspects of this duality, and in particular clarify the relation between area minimization of the fluid versus area maximization of the black hole horizon, and the connection between surface tension and curvature of the fluid, and surface gravity of the black hole. We also argue that the Rayleigh-Plateau instability in a fluid tube is the holographic dual of the Gregory-Laflamme instability of a black string. Associated with this fluid instability there is a rich variety of phases of fluid solutions that we study in detail, including in particular the effects of rotation. We compare them against the known results for asymptotically flat black holes finding remarkable agreement. Furthermore, we use our fluid results to discuss the unknown features of the gravitational system. Finally, we also discuss the instability of rotating plasma balls and the dual black hole interpretation.

Supersymmetry enhancement at the horizon of BPS black holes and rings in five space-time dimensions with eight supersymmetries, imposes stringent conditions on the fields and the geometry. For example, the BPS near-horizon geometry follows irrespective of the details of the Lagrangian. From the horizon behaviour alone the entropy and the attractor equations are derived. For spinning black holes, the results only partially agree with previous work, where additional input was used beyond the near-horizon data. In particular, the results fail to agree with four-dimensional results in the presence of higher-derivative interactions. Arguments are presented for this discrepancy. For the black rings, the horizon behavior leads to results which are consistent with the four-dimensional case, although subtle issues remain.

It has been seen that semi-classical string solutions play a major role in the study of the string/gauge correspondence. Giant magons are a particular set of these solutions, which are intimately related to sine-Gordon solitons. Considered to be the basic solitons of classical strings in the $AdS_5 \times S^5$ background, giant magnons are the building blocks one can use to understand other string solutions. The theory of classical strings in both $AdS_5 \times S^5$ and $AdS_4 \times CP^3$ backgrounds is integrable, and the problem of finding the spectrum of solutions can be described by a Riemann-Hilbert problem in the so called Algebraic Curve formalism. This talk will focus on the use of the Algebraic Curve formalism to study the spectrum of these string solitons, first in the simpler case of $AdS_5 \times S^5$, and then in the much richer $AdS_4 \times CP^3$ case.

Recent progress in the theory of (super)gravity has made it possible to calculate the entropy of certain black holes beyond the area law of Bekenstein and Hawking in a large charge expansion. I will first review this in the context of string theory. I will then present my own recent and ongoing work on the following interrelated questions:

How to compute and understand exponentially suppressed contributions to the black hole entropy?

How to compute the density of states of a conformal field theory in a regime far away from the Cardy regime?

How does one understand the entropy when there is an issue of wall-crossing in the moduli space?

The answers involve an interplay of physics and mathematics - in particular, newly discovered number theoretic objects called mock modular forms.

Quantum gravity is a theory for the microscopic structure of spacetime. We challenge a half-century old supposition and practice that quantizing general relativity will produce such a theory. In our view general relativity is an effective theory valid only at the long wavelength, low energy limits, and the metric and connection forms are collective variables. Quantizing them will only lead to the equivalent of phonon dynamics of crystal lattices, not quantum electrodynamics of electrons and photons. The challenge of this new paradigm is to find ways to unravel the underlying microscopic structures from observed macroscopic phenomena (many to one relation), not unlike deducing the molecular constituents from hydrodynamics and kinetic theory, or universalities of microscopic theories from critical phenomena. We explore this `bottom-up’ approach, focusing on the foundational issues in quantum-classical and micro-macro interfaces, using ideas of nonequilibrium statistical mechanics and techniques from quantum field theory for strongly correlated systems, identifying the relation of gravity with matter fields.

Exploring the characteristic features of emergence and identifying how the macro phenomena could be affected by different classes of hypothetical micro-theories are the challenges we face in this new paradigm. Two central issues are nonlocality and stochasticity. Noise can seed emergent structures and determine macroscopic forms. Nonlocality appears when one tries to translate physics expressed in one set of collective variables suitable for one level of structure to another set. We begin with the notion of nonlocality commonly associated with EPR experiment. We present results in the evolution of entanglement between two oscillators analyzed in a field theoretical context. We then show how nonlocality is linked with stochasticity in nonequilibrium dynamics: Nonlocal dissipation and nonlocal fluctuations (colored noise) arise naturally in the open-system dynamics of Langevin and the effectively-open system dynamics of Boltzmann, and the dynamics of correlation (BBGKY) hierarchy. These features originate from the choice of coarse-graining measures and backreaction processes. We analyze these two key issues in the context of strongly correlated systems which we believe are generic properties in any theory for the microscopic structure of spacetime, viz, quantum gravity.

The random matrix theory can be used to solve a problem of enumerative geometry: counting maps of arbitrary genus, i.e. counting surfaces composed of polygons glued by their edges. This issue has been extensively investigated by physicists for its possible applications such as quantum gravity. Following the work of Eynard, it was possible to compute such generating functions for different matrix models using a unique recursive procedure: a topological recursion allowing to build the generating function of surfaces with fixed Euler characteristic out of generating functions of surfaces with bigger Euler characteristic. The blind application of the same procedure to more general problems of enumeration such as topological string theories, Weyl-Peterson volume of moduli space of surfaces or Kontsevich-Witten theorem, proved to be very successful: one just has to change the basis ingredient of the procedure to go from one problem to the other, a plane curve referred to as the spectral curve to reflect it originates from integrable properties of the model considered. In this non-technical talk, I will give an overview of this procedure and its applications in mathematics and physics. I will also give a foretaste of the numerous open questions arising from these works.

I will discuss a general formalism to compute non-perturbative corrections to the low energy effective action in Calabi-Yau compactifications with $N=2$ supersymmetry. I will describe how a detailed analyis of the instanton effects can support some recent conjectures on the quantum structure of the hypermultiplet moduli space.

This talk is about recent and on-going work on classical string solutions relevant for the AdS/CFT correspondence. The focus is on properties which change when translating from the old $N=4$ super-Yang-Mills case (with strings on $AdS_{5} \times S^{5}$) to the recent case involving the super-Chern-Simons-matter theory of ABJM and strings on $AdS_{4} \times CP^{3}$. The two cases have many similar features, but always with extra complications in the new one.

I describe four dimensional $N=2$ supersymmetric gauge theory in the Omega-background with the two dimensional $N=2$ super-Poincaré invariance. I explain how this gauge theory provides the quantization of the classical integrable system underlying the moduli space of vacua of the ordinary four dimensional $N=2$ theory. This four dimensional gauge theory in its low energy description has two dimensional twisted superpotential which becomes the Yang-Yang function of the integrable system. I present the thermodynamic-Bethe-ansatz like formulae for this Yang-Yang function and for the spectra of commuting Hamiltonians following the direct computation in gauge theory. Particular examples of the many-body systems include the periodic Toda chain, the elliptic Calogero-Moser system, and their relativistic versions. Gauge theory gives a complete characterization of the $L^2$-spectrum for these integrable systems.

Critical gravitational collapse has been a central subject in numerical general relativity for a long time. In this talk I will review the basic ideas of critical gravitational collapse and discuss the possible relevance of this phenomenon in the context of holography. In particular, I will discuss critical formation of trapped surfaces resulting from the collision of gravitational waves in AdS and its holographic implications.

In this work, we propose a new non-Abelian generalization of the Born-Infeld Lagrangian. It is based on a geometrical property of the Abelian Born-Infeld Lagrangian in its determinantal form. Our goal is to extend the Abelian second type Born-Infeld action to the non-Abelian form preserving this geometrical property, which permits us to compute the generalized volume element as a linear combination of the components of metric and the Yang-Mills energy-momentum tensors. Under the BPS-like condition, the action proposed reduces to that of the Yang-Mills theory, independently of the gauge group. New instanton-wormhole solution and static and spherically symmetric solution in curved spacetime for an $SU(2)$ isotopic ansatz are solved and the $N = 1$ supersymmetric extension of the model is performed. The uniqueness of the theory from the physical and geometrical point of view is discussed in the light of new results obtained.