This is the live session included as part of the LonTI lecture on "From spin chain to AdS/CFT with Mathematica". Please register at https://lonti.weebly.com/registration.html to receive joining instructions for this live session which will be held via Zoom. ​In this introductory lecture we describe the XXX Heisenberg spin chain, study its spectrum, wavefunctions and discuss integrability of the system. Some examples are given with simple Mathematica code. We also discuss applications to AdS/CFT correspondence.

The release of the youtube video for the London Theory institute seminar series https://youtu.be/jTRQTwBX9ek. In this introductory lecture we describe the XXX Heisenberg spin chain, study its spectrum, wavefunctions and discuss integrability of the system. Some examples are given with simple Mathematica code. We also discuss applications to AdS/CFT correspondence. Some derivations are formulated in the form of step-by-step exercises. They can be solved either with Mathematica or by hand. A quick introduction to Mathematica is provided in a separate video.

We discuss the applicability of the Quantum Spectral Curve approach (the most advanced and precise method initially developed for the spectrum of anomalous dimensions of planar N=4 SYM) to the problem of computing structure constants. We give a pedagogical introduction to the QSC formalism for the anomalous dimensions and then present our new results about a more general class observables.

TBA

We compute three-point functions of single trace operators in planar N = 4 SYM. We consider the limit where one of the operators is much smaller than the other two. We find a precise match between weak and strong coupling in the Frolov-Tseytlin classical limit for a very general class of classical solutions. To achieve this match we clarify the issue of back-reaction and identify precisely which three-point functions are captured by a classical computation.

The Y-system, originating from integrability, is a tool describing the exact spectrum in 4D N=4 Super-Yang-Mills theory in a planar limit. We describe the construction and discuss predictions and tests for weak, strong and intermediate couplings.

Recently Kazakov, Vieira and the author conjectured the Y system set of equations describing the planar spectrum of AdS/CFT. In this paper we solve the Y system equations in the strong coupling scaling limit. We show that the quasiclassical spectrum of string moving inside AdS3 x S1 matches precisely with the prediction of the Y system. Thus the Y system, unlike the asymptotic Bethe ansatz, describes correctly the spectrum of one-loop string energies including all exponential finite size corrections. This gives a very non-trivial further support in favor of the conjecture. We also discuss how the generalization to the full AdS5 x S5 can be easily constructed using the PSU(2,2 4) symmetry of the problem.