Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2101.04154

Microscopic Models for Fusion Categories

Ramona Wolf
View original source ↗Read source document ↗Export metadata ↓

Abstract

This is a PhD Thesis on the connection between subfactors (more precisely, their corresponding fusion categories) and Conformal Field Theory (CFT). Besides being a mathematically interesting topic on its own, subfactors have also attracted the attention of physicists, since there is a conjectured correspondence between these and CFTs. Although there is quite a persuasive body of evidence for this conjecture, there are some gaps: there exists a set of exceptional subfactors with no known counterpart CFT. Hence, it is necessary to develop new techniques for building a CFT from a subfactor. Here, it is useful to study the underlying mathematical structure in more detail: The even parts of every subfactor give rise to two Unitary Fusion Categories (UFCs), and it is a promising direction to study quantum spin systems constructed from these categories to find a connection to CFTs. The simplest example that requires new techniques for building a CFT is the Haagerup subfactor, since it is the smallest subfactor with index larger than 4. In this thesis, we investigate the question whether there is a CFT corresponding to the Haagerup subfactor via lattice models in one and two dimensions. The first task here is to find the F-symbols of the fusion category since these are crucial ingredients for the construction of a physical model in all of the models we consider in this thesis. We then investigate microscopic models such as the golden chain model and the Levin-Wen model in order to find evidence for a corresponding CFT. We find that there is no evidence for a corresponding CFT from the investigation of the UFCs directly and it is necessary to expand these studies to the corresponding unitary modular tensor category, which can, for instance, be obtained via the excitations of the Levin-Wen model.

Explore related subjects

math-phcond-mat.str-elmath.CTmath.MPquant-ph

Keep this discovery

BibTeXRIS

Ramona Wolf. 2021-01-11. Microscopic Models for Fusion Categories. https://doi.org/10.15488/10324

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Why we should condition denoising diffusion generative models on windows of past observations

Data assimilation (DA) is, traditionally, a cycling process that relies on time-dependent priors to propagate information from past observations to future cycles. Using denoising diffusion generative modeling for DA is challenging because standard approaches use a fixed training data set, which in turn leads to a static prior that ignores information from past observations. Because past observations are ignored, DA systems with static priors lead to larger posterior errors than cycling DA systems. Incorporating time-dependent priors into generative models, however, requires expensive and frequent retraining. Motivated by linear systems theory - where the dependence of a prediction of a Kalman filter on past observations decays exponentially - we condition diffusion models on short windows of past observations. Specifically, we describe training procedures for two frameworks: a diffusion DA system predicting the current state given a set of past observations, and a diffusion ``direct observation prediction'' (DOP) system, predicting future observations given a set of past observations. Using a canonical linear system, we show that both systems can achieve the minimal posterior error characteristic of a fully-cycled DA/DOP system, without re-training, provided the time windows are long enough. The linear setup ensures analytical tractability, avoids confounding neural network training errors, and confirms that conditioning on windows of past observations is required for efficient and accurate diffusion-based DA or DOP.

math-ph↗
arXiv2026

The kinematic structures and the inertial geometry of a moving charge

We ask how much of the geometry a charged particle moves in is fixed by its motion, and how much a particle must bring. We ask of a symplectic structure only that it relate velocity to momentum as Hamilton's equations do, and we ask it of every energy at once. In particular, we show that the structures meeting that demand are the canonical one and its twists by a closed two-form of the base. A field provides the two-form, a particle the multiplier before it, which we identify constitutively with its charge. Thus, a single energy governs a family of structures, and each particle takes the one its charge fixes. We then ask what a particle must bring to be given a momentum, and we show that the degree of that map settles the degree at which a field enters Newton's Second Law. An antisymmetric bilinear form returns no Lorentz force, whilst a Randers metric returns one --- a length whose difference from a Riemannian one is linear in the velocity. Moreover, we find that metric already within the twisted structure, as its primitive over a level of the free energy, and its law of transport to be nonlinear, no affine connection being known to serve. Under an indefinite signature the length parts from the dynamics, and the extremals turn from shortest to longest. On the round sphere a monopole flux leaves no such metric, whilst the transport remains and prequantisation, given a unit of action, restricts the charge to a lattice. In this manner, we conclude that each charge-to-mass ratio receives a geometry of its own, so that by a functionalist criterion none of them is the spacetime of a charged particle.

math-ph↗