A single scalar with higher-derivative kinetic term and derivative quartic interactions is a toy model for higher derivative gravity. The Functional Renormalization Group shows that this theory is asymptotically free both in the UV and in the IR. The physical implications of this result are clarified in part by by computing the scattering amplitudes.
I will introduce quartic melonic tensor field theories, a class of field theories built using a non-local quartic interaction term. These resemble the more well-known φd4 models but behave differently with regards to power-counting and the structure of their divergences. In particular, these models are conjectured to be non-trivial in their critical dimension, in contrast with φ44. I will then report on recent joint work with Ajay Chandra where we use stochastic analysis methods to construct the φ24 and φ34 analogues of these models.
We consider the scalar φ4 model on the 4-dimensional noncommutative Moyal space. This is the critical dimension where the model becomes just-renormalisable. At the self-dual point, this model breaks down to a matrix model, where the noncommutativity of the underlying space is related to the size N of the matrix. Assuming a formal expansion in 1/N, the Dyson-Schwinger equations (after applying Ward identities) decouple which leads to (non-)linear integral equations at each order in 1/N. We will present and discuss from different perspectives the leading order (genus g = 0) result of the 2-point function, which is a resummation of infinitely many Feynman diagrams. We will also discussion the Hopf-algebraic renormalision of this model in the sense of Connes-Kreimer, which has the same complexitiy as an ordinary just-renormalisable QFT.
I will describe a joint work with Bailleul, Ferdinand and To where we construct the φ34 quantum field theory measure on compact Riemannian 3-manifolds, as invariant Gibbs measure of some stochastic partial differential equation.
I will discuss how to use tools from Gaussian analysis and operator semigroups together with some commutator estimates to construct Markov semigroups for some singular SPDEs. This yields in particular uniqueness for Goncalves-Jara-Gubinelli type energy solutions. The method applies to some critical equations and, in finite dimensions, even for some supercritical equations. In infinite dimensions we get Markov semigroups for supercritical equations but we lack a uniqueness result for supercritical energy solutions in infinite dimensions. The main SPDE examples where this works are of Burgers type: quadratic, divergence-free nonlinearity and Gaussian quasi-invariant measure.
In this talk, I will describe a synergy between the renormalization group (RG) in the form of Polchinski's equation and the stochastic quantisation in the form of a forward-backward stochastic differential equation (FBSDE). This approach can be used for constructing subcritical Grassmann Gibbsian measures and is based on controlling the solution of the FBSDE by means of a flow equation with respect to a scale parameter. However, unlike the standard RG approach, we only need to solve Polchinski’s equation in an approximate way, resulting in a great simplification of the analysis.
In this talk I will talk about our recent work on a class of singular SPDEs via convex integration method. In particular, we establish global-in-time existence and non-uniqueness of probabilistically strong solutions to the three dimensional Navier–Stokes system driven by space-time white noise. In this setting, solutions are expected to have space regularity at most −1/2 − κ for any κ > 0. Consequently, the convective term is ill-defined analytically and probabilistic renormalization is required. Up to now, only local well-posedness has been known. With the help of paracontrolled calculus we decompose the system in a way which makes it amenable to convex integration. By a careful analysis of the regularity of each term, we develop an iterative procedure which yields global non-unique probabilistically strong paracontrolled solutions. Our result applies to any divergence free initial condition in L2 ∪ B−1+κ∞,∞, κ > 0, and implies also non-uniqueness in law. Finally I will show the existence, non-uniqueness, non-Gaussianity and non-unique ergodicity for singular quasi geostrophic equation in the critical and supercritical regime.
We introduce a theory of non-commutative Lp spaces suitable for non-commutative probability in a non-tracial setting and use it to develop stochastic analysis of Grassmann-valued processes, including martingale inequalities, stochastic integrals with respect to Grassmann Itô processes, Girsanov’s formula and a weak formulation of Grassmann SDEs. We apply this new setting to the construction of several unbounded random variables, including a Grassmann analog of the φ24 Euclidean QFT in a bounded region.
I will discuss the application of some constructive field theory inspired techniques to the study of resurgence in the 0-dimensional O(N) model and its small N limit. This is the first step in the program of applying such techniques to fully fledged higher-dimensional quantum field theory.
The Markov property is an important property of Random Fields that allows to use them to construct a Quantum Field Theory. It is closely connected to Segal's Axioms, which describe how to assemble Random Fields on a bigger manifold from Fields on smaller pieces. In this talk I will describe how to establish these properties for the φ34 model on cylinders.
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