scholarly journals The homological content of the Jones representations at q = −1

2016 ◽  
Vol 25 (11) ◽  
pp. 1650062 ◽  
Author(s):  
Jens Kristian Egsgaard ◽  
Søren Fuglede Jørgensen
Keyword(s):  
Large Family ◽  
Double Cover ◽  
Braid Groups ◽  
Quantum Field ◽  
Cambridge Philos ◽  
Topological Quantum ◽  
Original Argument ◽  
Mapping Classes ◽  
Marked Points

We generalize a discovery of Kasahara and show that the Jones representations of braid groups, when evaluated at [Formula: see text], are related to the action on homology of a branched double cover of the underlying punctured disk. As an application, we prove for a large family of pseudo-Anosov mapping classes a conjecture put forward by Andersen, Masbaum, and Ueno [Topological quantum field theory and the Nielsen–Thurston classification of [Formula: see text], Math. Proc. Cambridge Philos. Soc. 141(3) (2006) 477–488] by extending their original argument for the sphere with four marked points to our more general case.

scholarly journals LIMITS OF THE QUANTUM SO(3) REPRESENTATIONS FOR THE ONE-HOLED TORUS

2012 ◽  
Vol 21 (11) ◽  
pp. 1250109 ◽  
Author(s):  
RAMANUJAN SANTHAROUBANE
Keyword(s):  
Class Group ◽  
Topological Quantum Field Theory ◽  
Mapping Class ◽  
Total Degree ◽  
Quantum Field ◽  
Homogeneous Polynomials ◽  
Cambridge Philos ◽  
Topological Quantum ◽  
The One

For N ≥ 2, we study a certain sequence [Formula: see text] of N-dimensional representations of the mapping class group of the one-holed torus arising from SO(3)-TQFT, and show that the conjecture of Andersen, Masbaum, and Ueno [Topological quantum field theory and the Nielsen–Thurston classification of M(0,4), Math. Proc. Cambridge Philos Soc.141 (2006) 477–488] holds for these representations. This is done by proving that, in a certain basis and up to a rescaling, the matrices of these representations converge as p tends to infinity. Moreover, the limits describe the action of SL2(ℤ) on the space of homogeneous polynomials of two variables of total degree N - 1.

Skeins and mapping class groups

1994 ◽  
Vol 115 (1) ◽  
pp. 53-77 ◽  
Author(s):  
Justin Roberts
Keyword(s):  
Field Theory ◽  
Mapping Class Groups ◽  
Topological Quantum Field Theory ◽  
Unitary Representations ◽  
Mapping Class ◽  
Heegaard Splittings ◽  
Quantum Field ◽  
Class Groups ◽  
Topological Quantum

AbstractThe protective unitary representations of the mapping class groups of surfaces corresponding to the Jones–Witten topological quantum field theory for SU(2) are expressed as representations in algebras of skeins in the surface. The skein-theoretic construction of the representations uses neither Kirby's surgery theorem nor a presentation of the group. Using these representations and the Reidemeister–Singer classification of Heegaard splittings gives a proof of the existence of the moduli of the Witten invariants of 3-manifolds.

scholarly journals Global anomalies on the Hilbert space

2021 ◽  
Vol 2021 (11) ◽  
Author(s):  
Diego Delmastro ◽  
Davide Gaiotto ◽  
Jaume Gomis
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Symmetry Algebra ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Strongly Coupled ◽  
Topological Quantum ◽  
Anomalous Theory

Abstract We show that certain global anomalies can be detected in an elementary fashion by analyzing the way the symmetry algebra is realized on the torus Hilbert space of the anomalous theory. Distinct anomalous behaviours imprinted in the Hilbert space are identified with the distinct cohomology “layers” that appear in the classification of anomalies in terms of cobordism groups. We illustrate the manifestation of the layers in the Hilbert for a variety of anomalous symmetries and spacetime dimensions, including time-reversal symmetry, and both in systems of fermions and in anomalous topological quantum field theories (TQFTs) in 2 + 1d. We argue that anomalies can imply an exact bose-fermi degeneracy in the Hilbert space, thus revealing a supersymmetric spectrum of states; we provide a sharp characterization of when this phenomenon occurs and give nontrivial examples in various dimensions, including in strongly coupled QFTs. Unraveling the anomalies of TQFTs leads us to develop the construction of the Hilbert spaces, the action of operators and the modular data in spin TQFTs, material that can be read on its own.

CONSTRUCTING TQFTS FROM MODULAR FUNCTORS

2001 ◽  
Vol 10 (08) ◽  
pp. 1085-1131 ◽  
Author(s):  
JAKOB GROVE
Keyword(s):  
The Other ◽  
Field Theories ◽  
Careful Study ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Other Hand ◽  
Topological Quantum ◽  
Extended Surfaces ◽  
Modular Functor ◽  
Marked Points

We prove in this paper that any 2 dimensional modular functor satisfying that S1,1≠0 induces a family of 2+1 dimensionally topological quantum field theories. We do this for two kinds of modular functors namely a modular functor on the category of extended surfaces and a modular functor on the category of extended surfaces with marked points and directions. We follow the ideas of M. Kontsevich, [21], and K. Walker, [32] but we give proofs and provide details left out in [21] and [32]. Careful study also shows that more choices are needed to define the TQFT than it is revealed in [21] and [32]. On the other hand, relations found in [32] turns out not to be needed here.

Pseudo-Anosov Theory

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Braid Groups ◽  
Intersection Numbers ◽  
Branched Covers ◽  
Algebraic Integers ◽  
Dehn Twists ◽  
Mapping Classes ◽  
Dynamical Properties

This chapter focuses on the construction as well as the algebraic and dynamical properties of pseudo-Anosov homeomorphisms. It first presents five different constructions of pseudo-Anosov mapping classes: branched covers, constructions via Dehn twists, homological criterion, Kra's construction, and a construction for braid groups. It then proves a few fundamental facts concerning stretch factors of pseudo-Anosov homeomorphisms, focusing on the theorem that pseudo-Anosov stretch factors are algebraic integers. It also considers the spectrum of pseudo-Anosov stretch factors, along with the special properties of those measured foliations that are the stable (or unstable) foliations of some pseudo-Anosov homeomorphism. Finally, it describes the orbits of a pseudo-Anosov homeomorphism as well as lengths of curves and intersection numbers under iteration.

QFT and unification of knot theories

2014 ◽  
Vol 29 (24) ◽  
pp. 1430025
Author(s):  
Alexey Sleptsov
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Knot Theory ◽  
Topological Quantum Field Theory ◽  
Quantum Field ◽  
Knot Invariants ◽  
Topological Quantum

We discuss relation between knot theory and topological quantum field theory. Also it is considered a theory of superpolynomial invariants of knots which generalizes all other known theories of knot invariants. We discuss a possible generalization of topological quantum field theory with the help of superpolynomial invariants.

GENERAL COVARIANCE, TOPOLOGICAL QUANTUM FIELD THEORIES AND FRACTIONAL STATISTICS

1992 ◽  
Vol 07 (02) ◽  
pp. 209-234 ◽  
Author(s):  
J. GAMBOA
Keyword(s):  
Hamiltonian Formalism ◽  
Field Theories ◽  
General Covariance ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Fractional Statistics ◽  
Multiply Connected ◽  
Topological Quantum ◽  
Gauge Fixing ◽  
Topological Quantum Field Theories

Topological quantum field theories and fractional statistics are both defined in multiply connected manifolds. We study the relationship between both theories in 2 + 1 dimensions and we show that, due to the multiply-connected character of the manifold, the propagator for any quantum (field) theory always contains a first order pole that can be identified with a physical excitation with fractional spin. The article starts by reviewing the definition of general covariance in the Hamiltonian formalism, the gauge-fixing problem and the quantization following the lines of Batalin, Fradkin and Vilkovisky. The BRST–BFV quantization is reviewed in order to understand the topological approach proposed here.

TOWARDS A QUANTUM ALGORITHM FOR THE PERMANENT

2007 ◽  
Vol 05 (01n02) ◽  
pp. 223-228 ◽  
Author(s):  
ANNALISA MARZUOLI ◽  
MARIO RASETTI
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Quantum Computation ◽  
Quantum Computer ◽  
Quantum Algorithm ◽  
Topological Quantum Field Theory ◽  
Quantum Field ◽  
Topological Quantum

We resort to considerations based on topological quantum field theory to outline the development of a possible quantum algorithm for the evaluation of the permanent of a 0 - 1 matrix. Such an algorithm might represent a breakthrough for quantum computation, since computing the permanent is considered a "universal problem", namely, one among the hardest problems that a quantum computer can efficiently handle.

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