Modul: MAT752 Advanced Topics in Field Theory

• February 19, Giovanni Canepa
   Double BFV quantisation and application to 3d Gravity 
  
  In this talk I will extend the cohomological setting developed by Batalin, Fradkin and Vilkovisky (BFV), which produces a resolution of coisotropic reduction in terms of hamiltonian dg manifolds, to the case of nested coisotropic embeddings inside a symplectic manifold.  We will show that the data of a nested coisotropic embedding defines a natural graded coisotropic embedding, whose reduction can further be resolved using the BFV prescription. I will call this construction double BFV resolution, and I will use it to prove that "resolution commutes with reduction" for a general class of nested coisotropic embeddings. As an application, I will provide a well defined candidate space of (physical) quantum states of three-dimensional Einstein--Hilbert theory, which is thought of as a partial reduction of the Palatini--Cartan model for gravity.This talk is a based on arXiv:2410.23184,a joint work with Michele Schiavina.
  
  
  
  
• February 26, Orev Malatesta
  

  Gravity as a BV Pushforward of BF theory

  BV pushforward is a systematic way of integrating out degrees of freedom in a BV theory, yielding an effective field theory that retains a BV description. Motivated by the structural similarity between BF theory and Palatini-Cartan gravity, one can investigate whether Palatini-Cartan gravity can arise as an effective theory obtained from a suitable BF model.

  After presenting the BV pushforward construction in general terms, I will introduce a BF theory based on a Lie algebra given by a quadratic extension of so(3,1) by an orthogonal module, together with its BV extension. I will then perform the BV pushforward in this setting, which produces a (non-canonical) BV extension of Palatini-Cartan gravity, and discuss the problems that arise.
  
  
  

• March 19, Francesco Bonechi
  
  The extension to non zero codimension of Lagrangian field theories
  
  We discuss an ongoing project with G.Canepa, A.Cattaneo and M. Schiavina aiming to describe the geometry underlying the BF^kV theory in codimension k. The general philosophy is that boundary terms in codimension k determine the relevant geometry in codimension k+1. In the first part we show how the moment map arises in codimension one as the deficit of the invariance of the action on the bulk with respect to gauge transformations. In the second part we discuss how the Dirac structure in codimension two arises as the deficit of the gauge transformations to be hamiltonian. BF theory is the example that illustrates in the simplest way the main features of the construction.
  
  
  
• April 2, Marius Furter
  
  Sequential Monte Carlo in String Diagrams
  
  Sequential Monte Carlo (SMC) algorithms, also known as particle filters, iteratively infer hidden states in state-space models by maintaining a system of interacting weighted particles. In this talk, I demonstrate how abstractions from Category Theory can be used to derive, reason about, and generalize SMC. First, I introduce the category of s-finite kernels, which allows us to build and manipulate probability models using string diagrams. I then explain how this category is extended with weights via a monadic construction. Finally, I illustrate how the resulting structure facilitates intuitive reasoning about SMC algorithms.
  
  
• April 16, Leon Menger
  

  Quadratically Extended BF in Dimensions 3 and Higher

  I will present a generalisation of the quadratically extended Lie algebra structure used in [2310.01877] and show how one can use it to define a "BF+BB"-like theory in any dimension $\geq 3$. In particular I will show how this theory allows one to recover gravity in 3 dimensions by means of a BV pushforward and discuss how this relates to the existing equivalence between BF and gravity in 3 dimensions. Time permitting I will sketch how similar ideas could be used in dimensions $\geq 4$.

• April 23, Leon Menger
  

  BV Pushforward of Quadratically Extended BF in 4 Dimensions and its Relation to Gravity

  In this talk I will formulate a BV pushforward of quadratically extended BF theory (qBF) in 4 dimensions (following [2310.01877]) that yields a torsion-extension of Palatini--Cartan gravity up to quantum corrections. We will see how to massage qBF to explicitly display the diffeomorphism symmetry and then discuss the BV pushforward and the final result relating the effective action to gravity. If there is time, I will delve deeper into the very involved Feynman graph calculus behind the effective action.
  
  

• April 30, Raffaele Lomartire
  

  Logarithmic variational calculus