Séminaire de Physique ThéoriquePré-soutenance de thèse: « Gravitation étendue aux champs de spins élevés: Aspects algébriques »
Yegor Goncharov (IDP)
Tuesday 05 September 2023 13:00 - -
In the current thesis we address the following three problems in the context of higher-spin (HS) field theories. The associated geometric and algebraic framework is sufficiently general to allow for various applications of the obtained results to other problems in mathematical physics. Firstly, we propose a generating procedure for constructing Lagrangians for symmetric HS fields on (anti-)de Sitter ((A)dS) spacetimes via radial dimensional reduction of the flat ambient space. The developed formalism promotes the known ambient-space BRST dynamics of HS fields on (A)dS to the Lagrangian level, and naturally encompasses the case of flat dimensional reduction. As a novel feature of the construction, ambient Lagrangians are n-forms in a (n+1)-dimensional ambient space. The corresponding variational principle is rigorously described in terms of jet bundles, and provides a simple explicit formula for the Euler-Lagrange derivative which leads to the correct ambient equations of motion from an ambient Lagrangian, without pulling back to the embedded (A)dS spacetime. The construction applies to a class of ambient spaces and co-dimension-one embeddings, e.g., with no requirement for an ambient space to be be flat, and for the embedding to be maximally symmetric.
Secondly, we propose an explicit and self-contained construction of the traceless projection of an arbitrary tensor. The solution of the problem relies on the representation theory of the Brauer algebra and the related Schur-Weyl-type duality, which identifies the action of the Brauer algebra on a tensor product of a vector space with the centraliser of the canonical action of the classical metric-preserving groups. As a result, the sought traceless projector is identified with a particular central element in the Brauer algebra, whose explicit expression is built from a certain operator and its spectrum. The latter is obtained by performing elementary operations with Young diagrams. Auxiliary computational techniques are developed in order to reduce the rate of growth of computational complexity related to the rank of a tensor.
Thirdly, we construct a normal form (a set of basis monomials in generators and the corresponding reduction algorithm) for walled Brauer algebras. The latter allows us to show that two equivalent monomials of minimal length are related by applying a sequence of length-preserving defining relations (an analog of Matsumoto’s lemma for Coxeter groups). We also describe the relation between the minimal length of a monomial and its diagrammatic representation.