2d CFTs have an infinite set of commuting conserved charges, known as the quantum KdV charges. There is a generalised Gibbs ensemble for these theories where we turn on chemical potentials for these charges. I will describe some partial results on calculating this partition function, both in the limit of large charges and perturbatively in the chemical potentials.

I will discuss the application of Siegel paramodular forms to constructing new examples of holography. These forms are relevant to investigate the growth of coefficients in the elliptic genus of symmetric product orbifolds at large central charge. The main finding is that the landscape of symmetric product theories decomposes into two regions. In one region, the growth of the low energy states is Hagedorn, which indicates a stringy dual. In the other, the growth is much slower, and compatible with the spectrum of a supergravity theory on $AdS_3$. I will provide a simple diagnostic which places any symmetric product orbifold in either region. The examples I will present open a path to novel realizations of $AdS_3/CFT_2$.

In the context of the AdS/CFT correspondence, charged and rotating thermal ensembles are dual to black holes with inner Cauchy horizons. We argue that an uneventful inner horizon requires certain analytic properties of correlation functions in the dual boundary ensemble which are not consistent with causality and unitarity for charged black holes and rotating black holes in $D>3$. However, they are satisfied for correlators of a holographic $2d$ CFT in a rotating thermal ensemble. This suggests that strong cosmic censorship is enforced in gravity theories with a CFT dual, with the possible exception of the rotating BTZ black hole.

This group of astronomical order is slowly yielding its secrets. It is the symmetry group of a rational conformal field theory. In this introductory talk, I will discuss the functions that constitute monstrous moonshine and explain the importance of the monster group and its connections with better established parts of mathematics.

We apply machine-learning to the study of dessins d'enfants. Specifically, we investigate a class of dessins which reside at the intersection of the investigations of modular subgroups, Seiberg-Witten curves and extremal elliptic K3 surfaces. A deep feed-forward neural network with simple structure and standard activation functions without prior knowledge of the underlying mathematics is established and imposed onto the classification of extension degree over the rationals, known to be a difficult problem. The classifications exceeded 0.93 accuracy and around 0.9 confidence relatively quickly. The Seiberg-Witten curves for those with rational coefficients are also tabulated.

We study the Fisher metrics associated with a variety of simple systems and derive some general lessons that may have important implications for the application of information geometry in holography. Some sample systems of interest are the classical 2d Ising model and the corresponding 1d free fermion theory, the instantons in 3+1d massless phi-fourth theory, and coherent states of free bosons and fermions.

We study the random geometry approach to the $T \bar T$ deformation of $2d$ conformal field theory developed by Cardy and discuss its realization in a gravity dual. In this representation, the gravity dual of the $T \bar T$ deformation becomes a straightforward translation of the field theory language. Namely, the dual geometry is an ensemble of $AdS_3$ spaces or BTZ black holes, without a finite cutoff, but instead with randomly fluctuating boundary diffeomorphisms. This reflects an increase in degrees of freedom in the renormalization group flow to the UV by the irrelevant $T \bar T$ operator.

I will first review the classical Kronecker 2^nd limit formula, viewed as a relation between partition functions and Green’s functions in orbifolds of flat space (as discussed for example in arXiv:1407.0027, appendix E). I will then discuss the generalization of this relation to orbifolds of the gravitational plane wave, a Penrose limit of AdS (dual of the BMN limit in gauge theory). This provides a natural one-parameter deformation of Kronecker-Eisenstein series, and more generally of Jacobi-Maass forms. This talk is based on arXiv:1910.02745.

I will present a formulation of gravity as a double copy of gauge theories in the context of the Becchi-Rouet-Stora-Tyutin (BRST) formalism. I will show how this gives an algorithm for consistently mapping gauge choices from Yang-Mills to gravity. Moreover, it resolves the issue of the dilaton degree of freedom arising in the double copy, thus allowing for the consistent construction of solutions in General Relativity. I will describe the perturbative construction at higher orders. I will also give a formulation of the BRST double copy in a spherical background.

We describe new boundary conditions for $AdS_2$ in Jackiw-Teitelboim gravity. The asymptotic symmetry group is enhanced to $\operatorname{Diff}(S^1) \times C^{\infty}(S^1)$, whose breaking to $\operatorname{SL}(2, \mathbb{R}) \times U(1)$ controls the near-$AdS_2$ dynamics. The action reduces to a boundary term which is a generalization of the Schwarzian theory. It can be interpreted as the coadjoint action of the warped Virasoro group. We show that this theory is holographically dual to the complex SYK model. We compute the Euclidean path integral and derive its relation to the random matrix ensemble of Saad, Shenker and Stanford. We study the flat space version of this action, and show that the corresponding path integral also gives an ensemble average, but of a much simpler nature. We explore some applications to near-extremal black holes.

I will review various results related to flag manifold sigma-models, with emphasis on their integrability properties. On simpler examples, such as the $\operatorname{\mathbb{CP}}^n$-model, I will demonstrate that the trigonometrically-deformed geometries are solutions to the Ricci flow equations.

After a brief introduction to the concept of Computational Complexity, I will show how to calculate it in several theories with boundaries in two dimensions. In particular, I will consider a free boson discretized on a lattice with Dirichlet boundary conditions, and "Boundary CFTs" with a holographic dual. I will identify certain contributions in the results for the Complexity which are characteristic of the presence of boundaries. Moreover, the results in the two most popular holographic prescriptions, the so-called "CV" and "CA" prescriptions, are qualitatively different. Thus, one can obtain information on the fitness of the holographic prescriptions in describing faithfully the Complexity of the dual states.

In my talk I will discuss topologically twisted compactification of 6d $(1,0)$ theories on 4-manifolds with background flavor symmetry bundles. The effective 2d theory generically has $(0,1)$ supersymmetry and a residual flavor symmetry. Evaluation of its elliptic genus thus produces an invariant of the 4-manifold equipped with a principle bundle valued in the ring of (equivariant) modular forms. By further including torsion valued invariants of $(0,1)$ 2d theories, one obtains an invariant of 4-manifolds valued in (equivariant) topological modular forms (TMF). I will describe basic properties of this map and present a few simple examples. I will also mention some byproduct results on 't Hooft anomalies of 6d $(1,0)$ theories. The talk is based on a joint work with Gukov, Pei and Vafa.

Holographic quantum error-correcting codes have been proposed as toy models that describe key aspects of the AdS/CFT correspondence. In this work, we introduce a versatile framework of Majorana dimers capturing the intersection of stabilizer and Gaussian Majorana states. This picture allows for an efficient contraction with a simple diagrammatic interpretation and is amenable to analytical study of holographic quantum error-correcting codes. Equipped with this framework, we revisit the recently proposed hyperbolic pentagon code. Relating its logical code basis to Majorana dimers, we efficiently compute boundary state properties even for the non-Gaussian case of generic logical input. The dimers characterizing these boundary states coincide with discrete bulk geodesics, leading to a geometric picture from which properties of entanglement, quantum error correction, and bulk/boundary operator mapping immediately follow. We also elaborate upon the emergence of the Ryu-Takayanagi formula from our model, which realizes many of the properties of the recent bit thread proposal. Our work thus elucidates the connection between bulk geometry, entanglement, and quantum error correction in AdS/CFT, and lays the foundation for new models of holography.

Recent progress in understanding de Sitter spacetime in supergravity and string theory has led to the development of a four dimensional supergravity with spontaneously broken supersymmetry allowing for de Sitter vacua, also called de Sitter supergravity. One approach makes use of constrained (nilpotent) superfields, while an alternative one couples supergravity to a locally supersymmetric generalization of the Volkov-Akulov goldstino action. These two approaches have been shown to give rise to the same four dimensional action. A novel approach to de Sitter vacua in supergravity involves the generalisation of unimodular gravity to supergravity using a super-Stückelberg mechanism. We make a connection between this new approach and the previous two which are in the context of nilpotent superfields and the goldstino brane.

False theta functions have occured throughout many areas. They lack modular symmetries due to an additional sign-factor.

Jointly with Caner Nazaroglu I overcame this problem and found a modular fromework to understand these functions. This has many applications and in this talk I will discuss some of these for example asymptotics of certain partition statistics.

In this talk, we will start by a review of the salient features of the so-called BMS symmetries, which appear as asymptotic symmetries of flat spacetimes, but turn out to be also present in the near-horizon region of black holes. We will then turn to quantum aspects of BMS symmetries in conformally flat spacetimes. After presenting asymptotic and conformal Killing vectors in $d$-dimensional Minkowski and several conformally flat spacetimes, we will show that the associated quantum charges for an arbitrary CFT satisfy a closed algebra that includes the BMS as a sub-algebra (i.e. supertranslations and superrotations) plus a novel transformation we call superdilations. At the end of the talk, we will discuss possible applications of these results for holography and black hole physics.

We derive the general anomaly polynomial for a class of two-dimensional CFTs arising as twisted compactifications of a higher-dimensional theory on compact manifolds, including the contribution of its isometries. We then use the result to perform a counting of microstates for dyonic rotating supersymmetric black strings in $\operatorname{AdS}_5 \times S^5$ and $\operatorname{AdS}_7 \times S^4$. We explicitly construct these solutions by uplifting a class of four-dimensional rotating black holes. We provide a microscopic explanation of the entropy of such black holes by using a charged version of the Cardy formula.