Imperial

In this talk, I will focus on a new construction of boundary correlators (or wavefunction coefficients in dS) that highlights simplicity at all spins and automatically imposes the conservation of boundary currents. This new construction is formulated in twistor space, a complex projective space that encodes solutions to equations of motion as holomorphic data. This is done via an isomorphism called the Penrose transform. First, I will discuss the case of AdS_3 and AdS_5, where bulk-to-boundary correlators naturally arise in minitwistor space. Then, I will show how in (A)dS₄ one can construct bulk correlation functions using only twistors, dual twistors, and the infinity twistor as building blocks. The relation to coordinate space arises now via nested Penrose transforms. The boundary limit of these correlators yields CFT correlators/wavefunction coefficients that satisfy the expected Ward identities. Finally, I will briefly discuss how this can be generalized to AdS_5 boundary correlators using ambitwistors.

Given a cyclic subgroup G of GL(2,C) acting on C^2, it was first noticed by Wunram in the 80s that there is a correspondence between certain special representations of G and the exceptional curves appearing in the minimal resolution Y of the surface singularity C^2/G. In modern terms, this was reformulated by Ishii and Ueda as the existence of a fully faithful functor from the derived category of sheaves of Y to the G-equivariant derived category of C^2. In this talk, I will describe a mirror symmetric interpretation of this which exhibits the fully faithful inclusion in algebraic geometry as a sequence of positive Lefschetz stabilizations in symplectic geometry.

Schwinger’s Closed Time Path formalism is the basis of modern treatments of cosmological field theories, hydrodynamics and open quantum systems. Its application to gauge theories at finite temperature is well studied, relying on KMS boundary conditions and complex-time contours. By contrast the discussion of gauge theories such as Yang-Mills out of equilibrium has been less well developed, in large part due to a lack of development of how to treat gauge issues and Faddeev-Popov-DeWitt ghosts on the CTP. I will show how to construct the CTP in the BRST formalism, where a single diagonal copy of BRST symmetry survives, and how to implement the boundary conditions for ghosts for arbitrary initial physical states. As an illustration I will discuss how Hard-thermal-loop EFTs can be viewed as open quantum systems, and how to construct an open EFT for a gauge theory in a Higgs phase. 

We study the clustering behavior of weakly interacting diffusions under the influence of sufficiently localized attractive interaction potentials on the one-dimensional torus. We describe how this clustering behavior is closely related to the presence of discontinuous phase transitions in the mean-field PDE. For local attractive interactions, we employ a new variant of the strict Riesz rearrangement inequality to prove that all global minimizers of the free energy are either uniform or single-cluster states, in the sense that they are symmetrically decreasing. We analyze different timescales for the particle system and the mean-field (McKean-Vlasov) PDE, arguing that while the particle system can exhibit coarsening by both coalescence and diffusive mass exchange between clusters, the clusters in the mean-field PDE are unable to move and coarsening occurs via the mass exchange of clusters. By introducing a new model for this mass exchange, we argue that the PDE exhibits dynamical metastability. We conclude by presenting careful numerical experiments that demonstrate the validity of our model.

The Friedmann--Lemaitre--Robertson--Walker family of spacetimes are the standard homogenous isotropic cosmological models in general relativity.  Each member of this family describes a torus, evolving from a big bang singularity and expanding indefinitely to the future, with expansion rate encoded by a suitable scale factor.  I will discuss a mixing effect which occurs for the Vlasov equation on these spacetimes when the expansion rate is suitably slow.

 This is joint work with Renato Velozo Ruiz (Imperial College London).

Dirac’s quantum theory of magnetic monopoles requires a Dirac string attached to each monopole, and it is important that the field equations do not depend on the positions of the Dirac strings, provided that they comply with the Dirac veto: they must not intersect the worldliness of electrically charged particles. This theory is revisited, and it is shown that it has generalised symmetries related to the freedom of moving the Dirac strings. The Dirac veto is interpreted as an anomaly and the possibility of cancelling the anomaly by embedding in a higher-dimensional theory will be discussed. 

This talk is based on arXiv:2411.18741.

We introduce a sparse and very high order hp-finite element method for the weak form of the Helmholtz equation.  The domain may be a disk, an annulus, or a cylinder. The cells of the mesh are an innermost disk (omitted if the domain is an annulus) and concentric annuli.

We demonstrate the effectiveness of this method on PDEs with radial direction discontinuities in the coefficients and data. The discretization matrix is always symmetric and positive-definite in the positive-definite Helmholtz regime. Moreover, the Fourier modes decouple, reducing a two-dimensional PDE solve to a series of one-dimensional solves that may be computed in parallel, scaling with linear complexity. In the positive-definite case, we utilize the ADI method of Fortunato and Townsend to apply the method to a 3D cylinder with a quasi-optimal complexity solve.

In this talk, I will work on a smooth projective threefold X which satisfies the Bogomolov-Gieseker conjecture of Bayer-Macrì-Toda, such as the projective space P^3 or the quintic threefold. I will show certain moduli spaces of 2-dimensional torsion sheaves on X are smooth bundles over Hilbert schemes of ideal sheaves of curves and points in X. When X is Calabi-Yau this gives a simple wall crossing formula expressing curve counts (and so ultimately Gromov-Witten invariants) in terms of counts of D4-D2-D0 branes. This is joint work with Richard Thomas.