Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userLecture notes on 2-dimensional defect TQFT
70
0
0.0
(
0
)
Authors
Nils Carqueville
Ask ChatGPT about the research
No Arabic abstract
These notes offer an introduction to the functorial and algebraic description of 2-dimensional topological quantum field theories `with defects, assuming only superficial familiarity with closed TQFTs in terms of commutative Frobenius algebras. The generalisation of this relation is a construction of pivotal 2-categories from defect TQFTs. We review this construction in detail, flanked by a range of examples. Furthermore we explain how open/closed TQFTs are equivalent to Calabi-Yau categories and the Cardy condition, and how to extract such data from pivotal 2-categories.
rate research
Read More
We introduce the notion of $n$-dimensional topological quantum field theory (TQFT) with defects as a symmetric monoidal functor on decorated stratified bordisms of dimension $n$. The familiar closed or open-closed TQFTs are special cases of defect TQFTs, and for $n=2$ and $n=3$ our general definition recovers what had previously been studied in the literature. Our main construction is that of generalised orbifolds for any $n$-dimensional defect TQFT: Given a defect TQFT $mathcal{Z}$, one obtains a new TQFT $mathcal{Z}_{mathcal{A}}$ by decorating the Poincare duals of triangulated bordisms with certain algebraic data $mathcal{A}$ and then evaluating with $mathcal{Z}$. The orbifold datum $mathcal{A}$ is constrained by demanding invariance under $n$-dimensional Pachner moves. This procedure generalises both state sum models and gauging of finite symmetry groups, for any $n$. After developing the general theory, we focus on the case $n=3$.
We initiate a systematic study of 3-dimensional `defect topological quantum field theories, that we introduce as symmetric monoidal functors on stratified and decorated bordisms. For every such functor we construct a tricategory with duals, which is the natural categorification of a pivotal bicategory. This captures the algebraic essence of defect TQFTs, and it gives precise meaning to the fusion of line and surface defects as well as their duality operations. As examples, we discuss how Reshetikhin-Turaev and Turaev-Viro theories embed into our framework, and how they can be extended to defect TQFTs.
These informal lecture notes describe the progress in semiconductor spintronics in a historic perspective as well as in a comparison to achievements of spintronics of ferromagnetic metals. After outlining motivations behind spintronic research, selected results of investigations on three groups of materials are presented. These include non-magnetic semiconductors, hybrid structures involving semiconductors and ferromagnetic metals, and diluted magnetic semiconductors either in paramagnetic or ferromagnetic phase. Particular attention is paid to the hole-controlled ferromagnetic systems whose thermodynamic, micromagnetic, transport, and optical properties are described in detail together with relevant theoretical models.
These lecture notes have been developed for the course Computational Social Choice of the Artificial Intelligence MSc programme at the University of Groningen. They cover mathematical and algorithmic aspects of voting theory.
Using the fact that the algebra M(3,C) of 3 x 3 complex matrices can be taken as a reduced quantum plane, we build a differential calculus Omega(S) on the quantum space S defined by the algebra C^infty(M) otimes M(3,C), where M is a space-time manifold. This calculus is covariant under the action and coaction of finite dimensional dual quantum groups. We study the star structures on these quantum groups and the compatible one in M(3,C). This leads to an invariant scalar product on the later space. We analyse the differential algebra Omega(M(3,C)) in terms of quantum group representations, and consider in particular the space of one-forms on S since its elements can be considered as generalized gauge fields.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
