Over the past few years, it has been shown that, when integrating out the spacetime dependence with a certain integration measure, some four-point correlation functions in N=4 super Yang-Mills (SYM) can be computed exactly. These physical observables are often called integrated correlators, which are functions of Yang-Mills coupling \tau, and transform under S-duality of N=4 SYM. In this talk, I will review some of the recent developments regarding these integrated correlators. In particular, I will discuss the so-called Laplace-difference equations that determine the integrated correlators recursively. I will also present the generating functions of the integrated correlators that resum the ranks of the gauge group and the charges of the operators, from which we will further determine the large-N and large-charge properties of the integrated correctors.

We will study the correlation function of four superconformal primaries in N=4 super Yang-Mills (SYM) with SU(N) gauge group. Recently, very powerful methods have been developed to compute this correlator at finite coupling based on a new concept of integrated correlators, which are defined by integrating the correlator over spacetime coordinates with suitable integration measures. The integrated correlators can be computed using supersymmetric localisation. We will mostly focus on one of the integrated correlators. An exact expression was found for this integrated correlator for arbitrary values of coupling and N. The integrated correlator can be expressed as a two-dimensional lattice sum, which manifests the SL(2, Z) modular invariance of N=4 SYM. Furthermore, the result obeys an elegant Laplace-difference equation that relates the correlator of SU(N) theory with those of SU(N-1) and SU(N+1) theories. In perturbation, the formula is checked to be consistent with known results from more standard methods. Finally, one can reconstruct the unintegrated correlator with finite coupling for first few orders in large-N expansion, the results are shown to agree with known type IIB superstring amplitudes due to AdS/CFT duality. This talk will be mainly based on arxiv.org/abs/2102.09537 (a short version can be found at arxiv.org/abs/2102.08305). --- Part of the London Integrability Journal Club. Please register at integrability-london.weebly.com if you are a new participant. The link will be emailed on Tuesday.

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.

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.

Scattering amplitudes are one of most important class of physical observables in quantum field theories. Over the last decade or so, there has been a lot of activities regarding the computation of scattering amplitudes in a wide range of interesting theories, where extremely powerful new frameworks for studying scattering amplitudes have emerged, known as the modern S-matrix program. In this talk I will review some of these powerful techniques, and discuss their applications. The applications will mostly focus on effective field theories, that include twistor-like formulas for scattering amplitudes of world-volume theories of probe D-brane and M5 brane, amplitude constraints on effective field theories, such as supersymmetric non-renormalization theorems, unitarity bounds on low-energy spectra.