Exact Results in Quantum Theory

World lines of particles that exchange in 2D form braids in spacetime. These braids are subject to certain universal topological relations coming from their continuous deformations. In 2D, such an approach leads to the well-known braiding relation also known as the Yang-Baxter relation. In my talk, I will show how to define counterparts of braids and derive braiding relations for particles constrained to move on planar wire networks. In particular, I will demonstrate that particles on wire networks have fundamentally different braiding properties than particles in 2D. My analysis reveals an unexpectedly wide variety of possible non-abelian braiding behaviours on networks. The character of braiding depends on the topological invariant called the connectedness of the network. As one of our most striking consequences, particles on modular networks can change their statistical properties when moving between different modules. However, sufficiently highly connected networks already reproduce braiding properties of 2D systems.In the second part of my talk, I will analyse the ways of realising the braiding of anyons on networks in a topological quantum field theory setting where anyons are allowed to braid as well as fuse. The compatibility of fusion and braiding on networks leads to new types of hexagon equations which in turn allow more general braid actions than the ones which are known from 2D physics. [1] BH An, T Maciazek, Geometric presentations of braid groups for particles on a graph, Comm. Math. Phys. 384 (2) (2021)[2] T Maciazek, BH An, Universal properties of anyon braiding on one-dimensional wire networks, Physical Review B 102 (20), 201407 (2020)[3] A Conlon, J K Slingerland, Compatibility of Braiding and Fusion on Wire Networks, arXiv:2202.08207 (2022)

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Link: https://us02web.zoom.us/j/84029651999?pwd=7tmWZrP9H5XKSwgebUoTESarH5tnjj.1 Meeting ID: 840 2965 1999 Passcode: 995804