67th North British Mathematical Physics Seminar

Abstract: The presence of integrability in planar N=4 Super Yang-Mills (SYM) was responsible for remarkable progress in our understanding of the AdS/CFT correspondence and allowed us to perform computations that would be otherwise very difficult. This topic, however, involves Hamiltonians that possess long-range interactions. In the context of integrability, such spin chains are in a sense still a young subject, and the systematic construction of the Lax operator and the R-matrix (both fundamental objects in integrability), for example, was until very recently, still unknown. In this talk, after a short introduction on spin chains and integrability, I will explain how to construct perturbative long-range spin chains while preserving integrability. In particular I will apply the method to compute the Lax operator and the R-matrix for the su(2) sector of planar N=4 SYM up to three loops and discuss other possible applications. Based on arXiv: 2206.08390, in collaboration with M. de Leeuw.

Abstract: The study of magnetic monopoles in gauge theory and the connection with algebraic geometry has been ongoing since the 1970s, and in 1983 Hitchin provided the criteria for certain algebraic curves to correspond to monopole solutions. These constraints are hard to solve, being transcendental in nature, and in the ensuing 40 years only a limited number of spectral curves have been discovered, typically requiring large symmetry groups. In this talk, I will present one approach to finding spectral curves using Nahm's equations and the standard Lax approach to integrable systems, highlighting how computer algebra software can dramatically simplify the non-linear part of the process. Finally, I will discuss how this combination of ideas has let us construct new charge-3 monopole spectral curves.

Abstract: Integrable quantum field theories in 1+1 dimensions provide beautiful examples of exactly solvable models. Relying on the existence of higher-spin conserved charges characterising these models, the axiomatic S-matrix bootstrap program was applied in the past decades to conjecture analytical expressions for the S-matrices of several integrable theories in 1+1 dimensions. Integrable theories can often also be studied perturbatively using standard Feynman diagrams, where the integrability manifests itself in a priori surprising cancellations and simplifications whose underlying mechanism is not completely understood. In this talk, I will discuss these simplifications at the tree and loop level for a famous class of integrable models, the simply-laced affine Toda field theories, and I will show how these simplifications play an important role to explain the pole structure of the S-matrices of these models.

Abstract: In this talk, we will review the recent progress in understanding accelerating black holes. In particular, we will focus the case of three-dimensional AdS gravity. Starting from a general ansatz we are able to construct three classes of geometries which can be interpreted by studying their physical parameters. From these, we construct stationary, accelerating point particles; one-parameter extensions of the BTZ family resembling an accelerating black hole; and find new solutions including a novel "accelerating BTZ geometry" not continuously connected to the BTZ black hole. We will comment on their thermodynamic description and a new way of interpreting these black holes holographically via fluid/gravity correspondence.

Abstract: Recently, a rather elegant picture has emerged for the non-perturbative topological string. In this picture the non-perturbative partition function has a piecewise-constant dependence on an additional phase, and jumps have an interpretation in terms of certain BPS states. In this meeting I will explain these ideas from the perspective of four-dimensional N=2 field theory and its lift to five dimensions, with the four-dimensional Seiberg-Witten theory and the resolved conifold as two main examples. This talk is based on 2109.14699, 2203.08249 and work in progress.