Sundance Bilson-Thompson, Jonathan Hackett, Lou Kauffman, Lee Smolin
We develop the idea that the particles of the standard model may arise from excitations of quantum geometry. A previously proposed topological model of preons is developed so that it incorporates an unbounded number of generations. A condition is also found on quantum gravity dynamics necessary for the interactions of the standard model to emerge.
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An overview of microlocality in braided ribbon networks is presented. Following this, a series of definitions are presented to explore the concept of microlocality and the topology of ribbon networks. Isolated substructure of ribbon networks are introduced, and a theorem is proven that allows them to be relocated. This is followed by a demonstration of microlocal translations. Additionally, an investigation into macrolocality and the implications of invariants in braided ribbon networks are presented.
We present a consistent definition for braided ribbon networks in 3-dimensional manifolds, unifying both three and four valent networks in a single framework. We present evolution moves for these networks which are dual to the Pachner moves on simplices and present an invariant of this evolution. Finally we relate these results back to previous work in the subject.
Braided quantum field theories proposed by Oeckl can provide a framework for defining quantum field theories having Hopf algebra symmetries. In quantum field theories, symmetries lead to non-perturbative relations among correlation functions. We discuss Hopf algebra symmetries and such relations in braided quantum field theories. We give the four algebraic conditions between Hopf algebra symmetries and braided quantum field theories, which are required for the relations to hold. As concrete examples, we apply our discussions to the Poincare symmetries of two examples of noncommutative field th…
We review and present a few new results of the program of emergent matter as braid excitations of quantum geometry that is represented by braided ribbon networks. These networks are a generalisation of the spin networks proposed by Penrose and those in models of background independent quantum gravity theories, such as Loop Quantum Gravity and Spin Foam models. This program has been developed in two parallel but complimentary schemes, namely the trivalent and tetravalent schemes. The former studies the braids on trivalent braided ribbon networks, while the latter investigates the braids on tetr…
We demonstrate that the covariance of the algebra of quantum NC fields under quantum-deformed Poincare symmetries implies the appearence of braided algebra of fields and the notion of braided locality in NC QFT. We briefly recall the historical development of NC QFT which was firstly formulated in the framework using classical relativistic symmetries but further it was described as generated by the quantum-deformed symmetries. We argue that consistent covariant quantum-deformed formalism requires "braiding all the way", in particular braided commutator of deformed field oscillators as well as …
We give a pedagogical introduction to $L_\infty$-algebras and their uses in organising the symmetries and dynamics of classical field theories, as well as of the conventional noncommutative gauge theories that arise as low-energy effective field theories in string theory. We review recent developments which formulate field theories with braided gauge symmetries as a new means of overcoming several obstacles in the standard noncommutative theories, such as the restrictions on gauge algebras and matter fields. These theories can be constructed by using techniques from Drinfel'd twist deformation…
Ribbon categories of weight modules for affine $\mathfrak{sl}_2$ at admissible levels
该研究证明容许水平下仿射$\frak{sl}_2$的单顶点算子代数有限生成权模的辫子张量范畴是刚性ribbon范畴，借助嵌入定理与Adamović逆量子哈密顿约化代数推导，并得到N=2超Virasoro顶点算子超代数相关范畴刚性的推论。
We define a new homotopy algebraic structure, that we call a braided $L_\infty$-algebra, and use it to systematically construct a new class of noncommutative field theories, that we call braided field theories. Braided field theories have gauge symmetries which realize a braided Lie algebra, whose Noether identities are inhomogeneous extensions of the classical identities, and which do not act on the solutions of the field equations. We use Drinfel'd twist deformation quantization techniques to generate new noncommutative deformations of classical field theories with braided gauge symmetries, …
We describe the cyclic $L_{\infty}$-algebra formulation of classical general relativity without matter fields in the Einstein-Cartan-Palatini formalism. Using Drinfel'd twist deformation techniques, we define a noncommutative version of the theory with braided gauge symmetries. We introduce a notion of braided $L_{\infty}$-algebra, and use it to encode the symmetries, field content, field equations and Noether identities of noncommutative gravity.
Representations of a group $G$ in vector spaces over a field $K$ form a category. One can reconstruct the given group $G$ from its representations to vector spaces as the full group of monoidal automorphisms of the underlying functor. This is a special example of Tannaka-Krein theory. This theory was used in recent years to reconstruct quantum groups (quasitriangular Hopf algebras) in the study of algebraic quantum field theory and other applications. We show that a similar study of representations in spaces with additional structure (super vector spaces, graded vector spaces, comodules, braid…
We describe the role of Rational Hopf Algebras as the symmetries of rational field theories and discuss their relation with algebraic field theory, braided monoidal categories and modular fusion rule algebras.
Bias in Local Spin Measurements from Deformed Symmetries
研究发现，在量子群对称性下，传统张量因子局域性无法保持，需用编织局域性来实现一致的局部测量，从而消除测量偏差。
The role of quantum groups and braid groups in the description of Standard Model particles is discussed. Some recent results on the use of the quantum group $SU_q(3)$ as a flavour symmetry are reviewed and a connection between two descriptions of Standard Model symmetries, one based on the normed division algebras and the other describing elementary matter as braided objects, is presented.
We introduce the free quantum noncommutative fields as described by braided tensor products. The multiplication of such fields is decomposed into three operations, describing the multiplication in the algebra M of functions on noncommutative space-time, the product in the algebra H of deformed field oscillators, and the braiding by factor Psi_{M,H} between algebras M and H. For noncommutativity generated by the twist factor we shall employ the star-product realizations of the algebra M in terms of functions on standard Minkowski space. The covariance of single noncommutative quantum fields und…
We discuss the obstruction to the construction of a multiparticle field theory on a $κ$-Minkowski noncommutative spacetime: the existence of multilocal functions which respect the deformed symmetries of the problem. This construction is only possible for a light-like version of the commutation relations, if one requires invariance of the tensor product algebra under the coaction of the $κ$-Poincaré group. This necessitates a braided tensor product. We study the representations of this product, and prove that $κ$-Poincaré-invariant N-point functions belong to an Abelian subalgebra, and are ther…
The setting of Braided Ribbon Networks is used to present a general result in spin-networks embedded in manifolds: the existence of an infinite number of species of conserved quantities. Restricted to three-valent networks the number of such conserved quantities in a given network is shown to be invariant barring a single case. The implication of these conserved quantities is discussed in the context of Loop Quantum Gravity.
Multiparticle states in braided lightlike $κ$-Minkowski noncommutative QFT
本研究在类光κ-闵可夫斯基非交换时空构建1+1维自由复标量量子场论，通过量子R矩阵定义保持κ-庞加莱不变性的辫代数，解决多粒子态定义难题，发现P、T破缺而PT对称，全同粒子可区分并讨论泡利原理命运。
We examine the notion of symmetry in quantum field theory from a fundamental representation theoretic point of view. This leads us to a generalization expressed in terms of quantum groups and braided categories. It also unifies the conventional concept of symmetry with that of exchange statistics and the spin-statistics relation. We show how this quantum group symmetry is reconstructed from the traditional (super) group symmetry, statistics and spin-statistics relation. The old question of extending the Poincare group to unify external and internal symmetries (solved by supersymmetry) is reexa…
Fractional quantum Hall (FQH) states are examples of symmetry-enriched topological states (SETs): in addition to the intrinsic topological order, which is robust to symmetry breaking, they possess symmetry-protected topological invariants, such as fractional charge of anyons and fractional Hall conductivity. In this paper we develop a comprehensive theory of symmetry-protected topological invariants for FQH states with spatial symmetries, which applies to Abelian and non-Abelian topological states, by using a recently developed framework of $G$-crossed braided tensor categories ($G\times$BTCs)…
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