# String Theory Seminar

## Past sessions

### Complexity in the presence of a boundary

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.

### Flag manifold sigma-models and Ricci flow

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.

### New boundary conditions for $AdS_2$

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.

### Perturbative gravity via BRST Yang-Mills2

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.

### Plane gravitational waves and Jacobi-Maass forms

I will first review the classical Kronecker 2nd 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.

### Random Boundary Geometry and Gravity Dual of $T {\bar T}$ Deformation

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.

### Information geometry in quantum field theory: lessons from simple examples

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.

### Machine-Learning Dessins d'Enfants: Explorations via Modular and Seiberg-Witten Curves

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.

### The Monster and its Moonshine Functions

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.

Video seminar @ Perimeter

### Holographic Probes of Inner Horizons

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.

### The Holographic Landscape of Symmetric Product Orbifolds

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 $AdS_3/CFT_2$.

Video seminar @ Perimeter

### Generalized Gibbs Ensemble and KdV charges in 2d CFTs

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.

### Integrability in and beyond AdS/CFT

In this talk, I am going to review some aspects of the current state of the art of Integrability in the AdS/CFT correspondence and beyond. I will first review a general nonperturbative approach to compute multipoint correlation functions of local operators in the $N=4$ SYM theory which allows us to explore the theory even beyond the planar level. In the second part, I will describe my recent work about exploring deformations of $N=4$ SYM by irrelevant operators, which revives an old attempt of generalizing the AdS/CFT correspondence. Here integrability seems to also play an important role and opens the door for its application for non-conformal field theories.

### Understanding $\operatorname{AdS}_2$: From Calogero-like models and SLE to $4d$ black hole microstate entropy

Extremal black holes show the presence of an $\operatorname{AdS}_2$ factor as an universal feature. This fact provides a strong motivation for getting a deeper understanding of $\operatorname{AdS}_2$ spacetimes. In this talk, I will argue how $\operatorname{AdS}_2$ systems have a natural description in terms of $1d$ Calogero-type models and, in turn to SLE curves, which describe the geodesic motion of particles in $\operatorname{AdS}_2$. This treatment allows to compute the dimension of the phase space of these geodesics, linking it to the leading Bekenstein-Hawking black hole entropy and the black hole degeneracy.

### A look into $3d$ modularity

Since the 1980s, the study of invariants of 3-dimensional manifolds has benefited from the connections between topology, physics and number theory. Motivated by the recent discovery of a new homological invariant (corresponding to the half-index of certain $3d$ $N=2$ theories), in this talk I describe the role of quantum modular forms, mock and false theta functions in the study of $3$-manifold invariants. The talk is based on 1809.10148 and work in progress with Cheng, Chun, Feigin, Gukov, and Harrison.

### Supersymmetric line operators and their spectral problem

I will discuss BPS invariants associated with quantum line operators in certain supersymmetric quantum field theories. Such operators can be specified via geometric engineering in the UV by assigning a path on a certain curve. In the IR they are described by representation theory data. I will discuss the associated BPS spectral problem and the relevant indices.

Note: unusual date

### Designing matrix models for zeta functions

The apparently random pattern of the non-trivial zeroes of the Riemann zeta function (all on the critical line, according to the Riemann hypothesis) has led to the suggestion that they may be related to the spectrum of an operator. It has also been known for some time that the statistical properties of the eigenvalue distribution of an ensemble of random matrices resemble those of the zeroes of the zeta function. With the objective to identify a suitable operator, we start by assuming the Riemann hypothesis and construct a unitary matrix model (UMM) for the zeta function. Our approach, however, could be termed piecemeal, in the sense that, we consider each factor (in the Euler product representation) of the zeta function to get a UMM for each prime, and then assemble these to get a matrix model for the full zeta function. This way we can write the partition function as a trace of an operator. Similar construction works for a family of related zeta functions.

Note: unusual date

### The Weak Gravity Conjecture

We discuss various versions of the weak gravity conjecture.

### Experiments with Machine Learning in Geometry & Physics

Identifying patterns in data enables us to formulate questions that can lead to exact results. Since many of the patterns are subtle, machine learning has emerged as a useful tool in discovering these relationships. We show that topological features of Calabi–Yau geometries are machine learnable. We indicate the broad applicability of our methods to existing large data sets by finding relations between knot invariants, in particular, the hyperbolic volume of the knot complement and the Jones polynomial.

### TQFTS, Orbifolds and Topological Quantum Computation

I will review basic notions and results in topological quantum field theory and discuss its orbifolds, with the aim to apply them in the context of topological quantum computation.

Unusual day and hour and room.

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Current organizers: Gabriel Lopes Cardoso, Suresh Nampuri.

Zoom access passwords are included in email announcements.