77th North British Mathematical Physics Seminar

Abstract: In this talk, I will present recent results on the definition and properties of partner systems in free quantum field theories in Gaussian states, both pure and mixed. I will then show how this framework can be used to characterize the distribution of quantum correlations in Minkowski spacetime and in the cosmological patch of de Sitter spacetime. These results provide new insights into how entanglement is distributed across spacetime and into the extent to which such correlations may be operationally accessible.

Abstract: The Temperley-Lieb (TL) relations first appeared as a set of physical relations between local operators used to construct transfer matrices for lattice models in statistical mechanics, thus reducing a key physical computation to the study of “transfer matrix algebras”, i.e. the abstract algebras generated by said operators, in particular, their representation theory. A key assumption underpinning such models is that of homogeneity, that is, all sites of the lattice are subject to the same physics.
In 2019, Kadar, Martin, and Yu, introduced a discrete family of diagram categories/algebras with the aim of capturing inhomogeneity in transfer matrix algebras. Their construction introduces the notion of “height” which is a topological condition on the crossings of a Brauer diagram. The result is a family of nested algebras interpolating between the Temperley-Lieb (TL) and Brauer algebras, setting up a fascinating program in representation theory. In this talk, I will review the construction of these “Kadar-Martin-Yu” (KMY) algebras and present the firstfruits of the latter program. A key result on semi simplicity establishes an analogy between the TL and KMY algebras. Namely, for each integer partition, there exists a Chebyshev series of polynomials which controls (some of) the non-semisimple cases of the KMY algebras; this can be seen as a direct generalisation of the classical “root of unity” criteria for TL algebras.

Abstract: I will introduce/review Lagrangian multiform theory as a variational framework to describe integrable systems, more precisely their hierarchies. Then, I will focus on the question of how to construct them, specifically in the case of Lagrangian one-forms which deals with finite dimensional integrable systems, and show how this new criterion for integrability is intimately related to older, well-established criteria such as classical r-matrix integrability and Lax pair integrability. Via a short detour on the variational formulation of Hamiltonian Lie group actions that multiforms afford, I will finally explain how to gauge Noether symmetries of a Lagrangian one-form and show how this can be applied to obtain a variational description of Hitchin's integrable system in the form of a mixed topological-holomorphic gauge theory called mixed 3d BF theory. Special famous specialisations include (elliptic) Gaudin and spin Calogero models.

Abstract: Boundary energy-momentum tensors (EMTs) are a useful tool for encoding symmetries and associated charges in gravitational theories. In this talk, I will consider spacetimes in 3+1D that are asymptotically flat near future null infinity (I⁺). The asymptotic data specifying such spacetimes consists of a boundary conformal Carrollian structure and the Bondi shear. I will present a generalisation of the Bondi-Sachs gauge that respects boundary Carroll covariance and allows us to construct the asymptotic solution space in terms of a boundary covariant radial expansion. By subsequently varying a suitably renormalised on-shell action with respect to the boundary Carrollian data and shear, I will obtain a relevant generalisation of a boundary EMT called a boundary EMT-News complex. The invariance of the on-shell action under boundary diffeomorphisms results in Ward identities for the EMT-News complex which generalise the standard Bondi loss equations. The remaining residual bulk gauge transformations that preserve asymptotic flatness and respect the gauge conditions reduce to Weyl transformations and Carroll boosts at I⁺. I will demonstrate that the latter are anomalous and evaluate the associated anomaly.

Abstract: Due to the presence of extended observables in a U(1) gauge theory, reconstructing the algebra of observables from those localised in small regions is a challenging task. This is known as the cutting and gluing problem.
Using a relative differential cohomology approach, I will present a construction of such an algebra satisfying a gluing axiom. I will discuss its physical motivation and the link with k-form electromagnetism, the quantisation of such a model and its associated superselection sectors. The talk is based on an ongoing work with D. Janssen and K. Rejzner.