This is the live session included as part of the LonTI lecture on Witten diagrams and the Mellin transform in AdS/CFT. Please register at https://lonti.weebly.com/registration.html to receive joining instructions for this live session which will be held via Zoom.AdS/CFT duality (or more general holographic principle) represents a major advance in understanding quantum gravity, and provides powerful tools for studying strongly coupled quantum field theories. This talk will give a basic introduction to AdS/CFT duality, with the focus on the computation of Witten diagrams and their Mellin transform. Witten diagrams, which play the role of Feynman diagrams, provide the means for computing correlation functions in AdS/CFT. We will show that CFT correlation functions obtained from Witten diagrams have much simpler structures after Mellin transform. Correlators in Mellin space are very analogous to the flat-space scattering amplitudes, and they are often called Mellin amplitudes. We will demonstrate the ideas by studying a few non-trivial examples.

In perturbative amplitudes in quantum field theory and string field theory, Cutkosky rule expresses the anti-hermitian part of a Feynman diagram in terms of sum over all its cut diagrams, and this in turn is used to prove unitarity of the theory. For D-instanton contribution to a string theory amplitude, the cutting rule needed for the proof of unitarity is somewhat different; we need to sum over only those cut diagrams for which all the world-sheet boundaries ending on some particular D instanton lie on the same side of the cut. By working with the closed string effective action, obtained after integrating out the open string modes, we prove that the D-instanton amplitudes actually satisfy these cutting rules, provided the effective action is real. The violation of unitarity in the closed string sector of two dimensional string theory can be traced to the failure of this reality condition. In the critical superstring theory, multi-instanton and multi anti-instanton amplitudes satisfy the reality condition. Contribution to the amplitudes from the instanton anti-instanton sector satisfies the reality condition if we make a specific choice of integration cycle over the configuration space of string fields, whereas contribution due to the non-BPS D-instantons will need to either vanish or have an overall real normalization in order for it to give real contribution. (please email jung-wook.kim AT qmul.ac.uk for the zoom link)

(for Zoom link please email Silvia Nagy s.nagyATqmul.ac.uk There will be a pre-seminar for students at 13.30) ABSTRACT: A universal relationship between asymptotic symmetries, QFT soft theorems, and low energy observables has reinvigorated attempts at flat space holography. We will begin by introducing a map from 4d S-matrix elements to 2d correlators in a putative dual Celestial CFT, and then re-examine the IR triangle from the perspective of our Celestial CFT.

Zoom link: Please email Silvia Nagy (s.nagyATqmul.ac.uk) or Jungwook Kim (jung-wook.kimATqmul.ac.uk) for the zoom link. Abstract:In string theory SL(2,Z) invariant functions, such as modular graph functions or coefficient functions of higher derivative corrections, are ubiquitous. Using a representation in terms of Poincaré series we can combine different methods for asymptotic expansions and obtain the complete perturbative and non-perturbative expansion. In the case of the higher derivative corrections, these terms have an interpretation in terms of perturbative string loop effects and pairs of instantons/anti-instantons. There will also be a pre-seminar for the students which will begin at 2:30 pm.

Zoom link: Please email Silvia Nagy (s.nagyATqmul.ac.uk) or Jungwook Kim (jung-wook.kimATqmul.ac.uk) for the zoom link Abstract: In the first part of this talk I will briefly introduce 0-dimensional Quantum Field Theory (QFT) and Matrix Models, which are known to be a succesfull discrete geometrical, QFT approach to 2D Quantum Gravity. I will then move on to Tensor Models, seen as natural QFT generalization of the celebrated Matrix Models. In the next part of the talk I will present the Sachdev-Ye-Kitaev model, which is known to be a toy model for holography, as well as the relations of this SYK model with various Tensor Models. Finally, I will present some recent results on these topics (various diagrammatic results but also the study of the effect of non-Gaussian average over the SYK random couplings).

The Hamiltonian formulation of mechanics has many advantages, but its standard presentation destroys manifest covariance. This can be avoided by using the "covariant phase formalism" of Iyer and Wald, but until recently this formalism has suffered from several ambiguities related to boundary terms and total derivatives. In this talk I will present a new version of the formalism which incorporates boundary effects from the beginning. This eliminates all ambiguities, and leads to an algorithmic procedure for covariantly constructing the phase space and Hamiltonian of any Lagrangian field theory. It also allows us to confirm that the Poisson bracket in covariant phase space is indeed equivalent to an old proposal of Peierls for computing Poisson brackets covariantly. Along the way I'll illustrate the formalism using various examples. Based on work with Jie-qiang Wu.

There is a deep relation between classical error-correcting codes, Euclidean lattices, and chiral 2d CFTs. We show this relation extends to include quantum codes, Lorentzian lattices, and non-chiral CFTs. The relation to quantum codes provides a simple way to solve modular bootstrap constraints and identify interesting examples of conformal theories. In particular we construct many examples of physically distinct isospectral theories, examples of "would-be" CFT partition function -- non-holomorphic functions satisfying all constraints of the modular bootstrap, yet not associated with any known CFT, and find theory with the maximal spectral gap among all Narain CFTs with the central charge c=4. At the level of code theories the problem of finding maximal spectral gap reduces to the problem of finding optimal code, leading to "baby bootstrap" program. We also discuss averaging over the ensemble of all CFTs associated with quantum codes, and its possible holographic interpretation. The talk is based on arXiv:2009.01236 and arXiv:2009.01244.

Spectral networks compute certain enumerative invariants associated with Hitchin systems, by focusing on the interplay of certain geometric and combinatorial data within them. In physics, spectral networks count BPS states of class S theories through 2d-4d wall crossing. I will describe a 3d-5d uplift of this based on exponential networks, that computes generalized Donaldson-Thomas invariants of toric Calabi Yau threefolds.

We propose a theoretical understanding of neural networks in terms of Wilsonian effective field theory. The correspondence relies on the fact that many asymptotic neural networks are drawn from Gaussian processes, the analog of non-interacting field theories. Moving away from the asymptotic limit yields a non-Gaussian process and corresponds to turning on particle interactions, allowing for the computation of correlation functions of neural network outputs with Feynman diagrams. Minimal non-Gaussian process likelihoods are determined by the most relevant non-Gaussian terms, according to the flow in their coefficients induced by the Wilsonian renormalization group. This yields a direct connection between overparameterization and simplicity of neural network likelihoods. Whether the coefficients are constants or functions may be understood in terms of GP limit symmetries, as expected from 't Hooft's technical naturalness. General theoretical calculations are matched to neural network experiments in the simplest class of models allowing the correspondence. Our formalism is valid for any of the many architectures that becomes a GP in an asymptotic limit, a property preserved under certain types of training.

Gravity is a fundamental theory of physics, but so weak, that we still know very little about it. A new exciting development is that the Laser Interferometer Gravitational-Wave Observatory (LIGO) can now measure the effects when massive black holes collide in the Universe. This has stimulated many new and interesting studies of gravitational interactions. I will in this talk discuss recent computational advances and discuss how to derive results for observables in general relativity from amplitudes.

Arithmetic geometry is the study *arithmetic schemes*, mathematical structures that have simultaneously an arithmetic and a geometric structure. The prototype is the so-called *spectrum of the integers* which is a geometric object on which the integers form the ring of functions. I will explain some of the difficult and classical problems that arise in their study, and how ideas of physics, especially topological quantum field theory, may be helpful.

String theory on AdS3 x S3 x T4 with one unit of NS-NS flux is argued to be exactly dual to the symmetric orbifold of T4 in the large N limit. The string theory background can be described in terms of a solvable world-sheet theory. This allows one to compute the complete single-string spacetime spectrum and thereby demonstrate that it agrees with the that of the symmetric orbifold of T4. Furthermore, the structure of the symmetric orbifold correlators can be reproduced from the world-sheet perspective.

In my talk, I will introduce concepts relevant for simulation of LHC physics. I will highlight some of the challenges when going to the level of precision necessary to match experimental accuracy.

We propose a unified description of two important phenomena: color confinement in large-N gauge theory, and Bose-Einstein condensation (BEC). We focus on the confinement/deconfinement transition characterized by the increase of the entropy from N^0 to N^2, which persists in the weak coupling region. Indistinguishability associated with the symmetry group --- SU(N) or O(N) in gauge theory, and S_N permutations in the system of identical bosons --- is crucial for the formation of the condensed (confined) phase. We relate standard criteria, based on off-diagonal long range order (ODLRO) for BEC and the Polyakov loop for gauge theory. The constant offset of the distribution of the phases of the Polyakov loop corresponds to ODLRO, and gives the order parameter for the partially-(de)confined phase at finite coupling. Furthermore we show the numerical evidence for this phenomenon at strong coupling, by using the Yang-Mills matrix model as a concrete example and solving it numerical via lattice simulation. This talk is based on a series of papers, especially "Color Confinement and Bose-Einstein Condensation" by Hanada, Shimada and Wintergerst, 2001.10459 [hep-th] and "Partial Deconfinement at Strong Coupling on a Lattice' by Bergner, Bodendorfer, Funai, Hanada, Rinaldi, Schaefer, Vranas and Watanabe to appear (should be in hep-th by the talk).

Cosmological (or de Sitter) horizons behave qualitatively different to black hole horizons and this poses a challenging problem in the context of holography. In this talk, I will discuss a novel construction to probe de Sitter horizons using the usual tools of the AdS/CFT correspondence. I will further explore ongoing efforts to reconstruct the bulk metric in two dimensions from the dual quantum mechanical correlators in the boundary.

I will explore aspects of the Coulomb-branch physics of five-dimensional superconformal field theories (SCFT). More precisely, I will consider the 5d SCFT on a circle, and describe the general structure of the Coulomb-branch BPS states as encoded in a "5d BPS quiver," which can be computed from standard string-theory geometric-engineering techniques. The interplay between 4d and 5d BPS quivers will play a central role in our story.

The information paradox can be realized in two-dimensional models of gravity. In this setting, we show that the large discrepancy between the von Neumann entropy as calculated by Hawking and the requirement of unitarity is fixed by including new saddles in the gravitational path integral. These saddles arise in the replica method as wormholes connecting different copies of the black hole.

Topological quantum field theories, which can be used to model topologically ordered phases of matter in quantum many-body systems, exhibit rich behavior in the presence of global symmetries. In this talk I will present progress over the last few years in understanding (2+1)D TQFTs in the presence of a global symmetry group G. This leads to a new mathematical structure, G-crossed braided tensor categories, that characterizes different possible patterns of symmetry fractionalization for topologically non-trivial particles and the fusion and braiding properties of symmetry defects. Two types of anomalies arise in the discussion: (1) "symmetry localization anomalies," which can also be interpreted as the theory possessing a non-trivial 2-group symmetry, and (2) symmetry fractionalization anomalies, which are equivalent to 't Hooft anomalies of the TQFT. I will describe various methods that have been developed in the past few years to compute these anomalies from the algebraic data that defines the TQFT and the symmetry action.