Meeting 5th of September 2022, Nottingham
11:30 - 12:30 Severin Bunk (Oxford): Differential cohomology, gerbes, and functorial field theories
12:30 - 14:00 Lunch
14:00 - 14:30 Diego Vidal Cruz Prieto (York): Hadamard State Extension in Conformally Ultrastatic Spacetimes
14:30 - 15:00 Giorgio Musante (Genova): Green hyperbolic complexes on Lorentzian manifolds
15:00 - 16:00 Ana Ros Camacho (Cardiff): Module tensor categories and the Landau-Ginzburg/conformal field theory correspondence
16:00 - 17:00 Informal discussions
Severin Bunk: Differential cohomology, gerbes, and functorial field theories
In this talk I will argue that a differential cocycle on a space of fields gives rise to a (bordism-type) functorial field theory (FFT). I will discuss some background on smooth FFTs and then focus on the two-dimensional case. Here, gerbes with connection (categorified line bundles) provide a geometric model for differential cocycles, and I will present a construction of two-dimensional smooth FFTs on background manifolds from gerbes with connection, which are related to WZW theories. If time permits, I will comment on an extension of this construction which produces open-closed field theories.
Diego Vidal Cruz Prieto: Hadamard State Extension in Conformally Ultrastatic Spacetimes
It is known that Hadamard states defined on a region of spacetime might not have extensions (as Hadamard states) to larger regions. In this talk, I will present a method to extend said states to a larger region by sacrificing a "safety margin", while preserving them as Hadamard states. Then, I will show this is possible to do this in bounded regions of conformally ultrastatic spacetimes. Work in progress with Chris Fewster.
Giorgio Musante: Green hyperbolic complexes on Lorentzian manifolds
Green hyperbolic operators play a distinguished role in field theory since they enjoy desirable properties and allow for relevant constructions, such as Poisson structures on the space of linear observables of the theory. Unfortunately, they cannot be compatible with gauge symmetry.
In this talk I will present a homological generalization of Green hyperbolic operators, called Green hyperbolic complexes, which is meant to be applied to linear gauge field theories in Lorentzian signature. Green hyperbolic complexes are defined through a generalization of retarded and advanced Green’s operators, called retarded and advanced Green’s homotopies, and are such that homological generalizations of the most relevant features of Green hyperbolic operators hold, namely the retarded-minus-advanced cochain map, generalizing the retarded-minus-advanced propagator, is a quasi-isomorphism and a differential pairing, generalizing the usual fiber-wise metric, on a Green hyperbolic complex allows for covariant and fixed-time Poisson structures with which the retarded-minus-advanced cochain map is suitably compatible.
Ana Ros Camacho: Module tensor categories and the Landau-Ginzburg/conformal field theory correspondence
The Landau-Ginzburg/conformal field theory correspondence is a physics result from the late 80s and early 90s predicting some relation between categories of representations of vertex operator algebras and categories of matrix factorizations. At present we lack an explicit mathematical statement for this result, yet we have examples available. The only example of a tensor equivalence in this context was proven back in 2014 by Davydov-Runkel-RC, for representations of the N=2 unitary minimal model with central charge 3(1-2/d) (where d integer bigger than 2) and matrix factorizations of the potential x^d-y^d. This equivalence was proven back in the day only for d odd, and in this talk we explain how to generalize this result for any d, realising these categories as module tensor categories enriched over Z_d-graded vector spaces. Joint work with T. Wasserman (University of Oxford).
How to reach the venue
Some general info is available here: https://www.nottingham.ac.uk/about/visitorinformation/mapsanddirections/universityparkcampus.aspx
The talks will take place in the lecture theatre UP-PHYS-B21+ in the School of Physics. The latter is located directly next to the School of Mathematical Sciences. See also the map below: