Triangle Seminars

Abstract:
Thermal correlators admit two complementary descriptions. At short times they are controlled by local operator data through the thermal OPE, while at late times they are governed by collective excitations, or quasinormal modes. Underlying these two descriptions is a richer analytic structure: the continuation of thermal correlators to complex time can encode information about bulk causal structure and the black-hole interior through complex-time singularities. We show that the thermal OPE and the quasinormal-mode expansion overlap in a common domain of complex time and therefore constrain one another through analyticity. This relation can be used constructively in both directions. Starting from OPE data, we recover low-lying quasinormal modes by analytic continuation and determine large-overtone frequencies and residues from the singularity structure of the correlator. In Mellin space, the relation becomes a set of spectral sum rules connecting the OPE spectrum to QNM data and constraining the allowed spectral structure. I will illustrate these ideas for holographic thermal correlators, in particular scalar perturbations of the Schwarzschild-AdS_5 black brane. More broadly, this suggests a bootstrap perspective on real-time thermal response in which microscopic operator data, causality, collective spectra, and the analytic structure associated with black-hole geometry are treated as mutually constrained.

Abstract:
I'll discuss a connection between topological field theories and topological phases of lattice models, which shows how partition functions of field theories can inspire order parameters for lattice models. After some generalities, I'll use this connection to develop, out of lens space partition functions, a novel order parameter for distinguishing two-dimensional symmetry protected topological (SPT) phases. The resulting procedure takes a wavefunction, such as a ground state of a lattice model, and detects its SPT invariants as expectation values of finitely supported operators, similar to how string order parameters work for 1D SPT phases.

Abstract:
Could the apparently continuous world of mathematics and physics be an approximation to a fundamentally finite universe? Boris Zilber explores this question through model theory, a branch of mathematics that studies theories and their models.

The lectures introduce the notion of global approximation: rather than viewing a finite model simply as a local discretisation of a continuous theory, one asks whether a single large finite structure can approximate an entire continuous structure globally. Zilber will describe how this perspective leads to compact complex structures and discuss its possible implications for spacetime and fundamental physics. Examples will include finite cyclic models of spacetime, Lorentz symmetry, quantum mechanics and string-theoretic structures. The lectures will also consider the distinction between the global mathematical structure and the local coordinates and measurements reconstructed by an observer.

Boris Zilber is Professor Emeritus of Mathematical Logic at the University of Oxford. His research is in model theory, geometry and their connections with other areas of mathematics and physics. He is one of the founders of geometric stability theory and Zariski geometries, and of the study of transcendental functions by model-theoretic methods.