# Deformations, renormgroup, symmetries, AdS/CFT

## TL;DR

This paper explores how spacetime-dependent deformations of supersymmetric quantum field theories relate to cohomological invariants and their interpretation within the AdS/CFT correspondence, focusing on the beta-deformation in N=4 SYM.

## Contribution

It introduces a cohomological framework for understanding deformations and their renormalization in supersymmetric theories, linking these invariants to classical supergravity solutions.

## Key findings

- Cohomological invariants describe renormalization of spacetime-dependent deformations.
- A leg amputation procedure relates supergravity tree diagrams to Maurer-Cartan solutions.
- A cohomological interpretation of the beta-function's leading term is proposed.

## Abstract

We consider the deformations of a supersymmetric quantum field theory by adding spacetime-dependent terms to the action. We propose to describe the renormalization of such deformations in terms of some cohomological invariants, a class of solutions of a Maurer-Cartan equation. We consider the strongly coupled limit of $N=4$ supersymmetric Yang-Mills theory. In the context of AdS/CFT correspondence, we explain what corresponds to our invariants in classical supergravity. There is a leg amputation procedure, which constructs a solution of the Maurer-Cartan equation from tree diagramms of SUGRA. We consider a particular example of the beta-deformation. It is known that the leading term of the beta-function is cubic in the parameter of the beta-deformation. We give a cohomological interpretation of this leading term. We conjecture that it is actually encoded in some simpler cohomology class, which is quadratic in the parameter of the beta-deformation.