Week 15.04.2024 – 21.04.2024

3d mirror symmetry for theories with eight supercharges is understood in terms of Hanany-Witten brane setups and plays an important role in many areas of supersymmetric qft’s. The generalization to theories with four supercharges, in the non-Abelian case, has been a long standing open problem. In this talk, based on work with Riccardo Comi and Sara Pasquetti, we focus on brane setups with NS and D5’ branes, proposing that the related quiver gauge theories involve ‘improved bifundamentals’, that is strongly coupled SCFT's which are ancestors of the well known T[SU(N)] theories. Our proposal leads to 3d mirror dualities that can be exactly proven, reducing them to known Seiberg-like dualities. This gives strong support to the proposal. The simplest example is the duality between adjoint SQCD with F flavors, and a quiver with F-1 nodes and F-2 improved bifundamentals.

These lectures will summarise mathematical aspects of classical General Relativity that are helpful in understanding current developments in the field. Lecture I will focus on Lorentzian-signature geometry, with an emphasis on causal structure. Some topological notions will also be introduced. In Lecture II we will go on to study the behaviour of geodesics in General Relativity and derive the famous Raychaudhuri equation. The null version of this equation, due to Sachs, will also be derived. Lecture III will focus on the "Hawking singularity theorem", namely that cosmological spacetimes with positive local Hubble constant are geodesically incomplete in the past under suitable conditions. In Lecture IV we will discuss the "Penrose singularity theorem" for black holes.

These lectures will summarise mathematical aspects of classical General Relativity that are helpful in understanding current developments in the field. Lecture I will focus on Lorentzian-signature geometry, with an emphasis on causal structure. Some topological notions will also be introduced. In Lecture II we will go on to study the behaviour of geodesics in General Relativity and derive the famous Raychaudhuri equation. The null version of this equation, due to Sachs, will also be derived. Lecture III will focus on the "Hawking singularity theorem", namely that cosmological spacetimes with positive local Hubble constant are geodesically incomplete in the past under suitable conditions. In Lecture IV we will discuss the "Penrose singularity theorem" for black holes.

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In this talk we use integrability data to bootstrap correlation functions of planar maximally supersymmetric Yang- Mills theory. Focusing on four-point correlation function of stress-tensor, we first introduce a set of sum rules that are only sensitive to single-traces in the OPE expansion (this is advantageous because this data is available from integrability). We then discuss how these sum rules can be employed in numerical bootstrap to nonperturbatively bound planar OPE coefficients. We show rigorous bounds for the OPE coefficient of the Konishi operator at various t’Hooft couplings outside the perturbative regime. The talk is based on an ongoing work and 2207.01615.