scholarly journals Homology TQFT's and the Alexander–Reidemeister Invariant of 3-Manifolds via Hopf Algebras and Skein Theory

2003 ◽  
Vol 55 (4) ◽  
pp. 766-821 ◽  
Author(s):  
Thomas Kerler
Keyword(s):  
Moduli Spaces ◽  
Hopf Algebras ◽  
Large Family ◽  
Quantum Invariants ◽  
Topological Quantum ◽  
Irreducible Components ◽  
Skein Theory ◽  
Higher Rank ◽  
The One ◽  
Representation Varieties

AbstractWe develop an explicit skein-theoretical algorithm to compute the Alexander polynomial of a 3-manifold from a surgery presentation employing the methods used in the construction of quantum invariants of 3-manifolds. As a prerequisite we establish and prove a rather unexpected equivalence between the topological quantum field theory constructed by Frohman and Nicas using the homology ofU(1)-representation varieties on the one side and the combinatorially constructed Hennings TQFT based on the quasitriangular Hopf algebra= ℤ/2 n ⋊ Λ* ℝ2on the other side. We find that both TQFT's are SL(2; ℝ)-equivariant functors and, as such, are isomorphic. The SL(2; ℝ)-action in the Hennings construction comes from the natural action onand in the case of the Frohman–Nicas theory from the Hard–Lefschetz decomposition of theU(1)-moduli spaces given that they are naturally Kähler. The irreducible components of this TQFT, corresponding to simple representations of SL(2; ℤ) and Sp(2g; ℤ), thus yield a large family of homological TQFT's by taking sums and products. We give several examples of TQFT's and invariants that appear to fit into this family, such as Milnor and Reidemeister Torsion, Seiberg–Witten theories, Casson type theories for homology circlesà laDonaldson, higher rank gauge theories following Frohman and Nicas, and the ℤ=pℤ reductions of Reshetikhin.Turaev theories over the cyclotomic integers ℤ[ζp]. We also conjecture that the Hennings TQFT for quantum-sl2is the product of the Reshetikhin–Turaev TQFT and such a homological TQFT.

scholarly journals REPRESENTATION VARIETIES OF ALGEBRAS WITH NODES

2021 ◽  
pp. 1-31
Author(s):  
Ryan Kinser ◽  
András C. Lőrincz
Keyword(s):  
Moduli Spaces ◽  
Rational Singularities ◽  
Orbit Closures ◽  
Node Splitting ◽  
Irreducible Components ◽  
Representation Varieties

Abstract We study the behaviour of representation varieties of quivers with relations under the operation of node splitting. We show how splitting a node gives a correspondence between certain closed subvarieties of representation varieties for different algebras, which preserves properties like normality or having rational singularities. Furthermore, we describe how the defining equations of such closed subvarieties change under the correspondence. By working in the ‘relative setting’ (splitting one node at a time), we demonstrate that there are many nonhereditary algebras whose irreducible components of representation varieties are all normal with rational singularities. We also obtain explicit generators of the prime defining ideals of these irreducible components. This class contains all radical square zero algebras, but also many others, as illustrated by examples throughout the paper. We also show that this is true when irreducible components are replaced by orbit closures, for a more restrictive class of algebras. Lastly, we provide applications to decompositions of moduli spaces of semistable representations of certain algebras.

scholarly journals Pathways to relativistic curved momentum spaces: de Sitter case study

2016 ◽  
Vol 25 (02) ◽  
pp. 1650027 ◽  
Author(s):  
Giovanni Amelino-Camelia ◽  
Giulia Gubitosi ◽  
Giovanni Palmisano
Keyword(s):  
Momentum Space ◽  
Hopf Algebras ◽  
Affine Connection ◽  
Planck Scale ◽  
Test Case ◽  
De Sitter ◽  
Group Manifold ◽  
The One ◽  
Natural Test

Several arguments suggest that the Planck scale could be the characteristic scale of curvature of momentum space. As other recent studies, we assume that the metric of momentum space determines the condition of on-shellness while the momentum space affine connection governs the form of the law of composition of momenta. We show that the possible choices of laws of composition of momenta are more numerous than the possible choices of affine connection on a momentum space. This motivates us to propose a new prescription for associating an affine connection to momentum composition, which we compare to the one most used in the recent literature. We find that the two prescriptions lead to the same picture of the so-called [Formula: see text]-momentum space, with de Sitter (dS) metric and [Formula: see text]-Poincaré connection. We then show that in the case of “proper dS momentum space”, with the dS metric and its Levi–Civita connection, the two prescriptions are inequivalent. Our novel prescription leads to a picture of proper dS momentum space which is DSR-relativistic and is characterized by a commutative law of composition of momenta, a possibility for which no explicit curved momentum space picture had been previously found. This momentum space can serve as laboratory for the exploration of the properties of DSR-relativistic theories which are not connected to group-manifold momentum spaces and Hopf algebras, and is a natural test case for the study of momentum spaces with commutative, and yet deformed, laws of composition of momenta.

scholarly journals Computing the fundamental group of a higher-rank graph

2021 ◽  
pp. 1-12
Author(s):  
Sooran Kang ◽  
David Pask ◽  
Samuel B.G. Webster
Keyword(s):  
Fundamental Group ◽  
Homology Group ◽  
Klein Bottle ◽  
Fundamental Groups ◽  
The Third ◽  
Higher Rank ◽  
The One ◽  
Standard Presentation

Abstract We compute a presentation of the fundamental group of a higher-rank graph using a coloured graph description of higher-rank graphs developed by the third author. We compute the fundamental groups of several examples from the literature. Our results fit naturally into the suite of known geometrical results about higher-rank graphs when we show that the abelianization of the fundamental group is the homology group. We end with a calculation which gives a non-standard presentation of the fundamental group of the Klein bottle to the one normally found in the literature.

scholarly journals Wall-crossing in genus zero Landau–Ginzburg theory

2017 ◽  
Vol 2017 (733) ◽  
Author(s):  
Dustin Ross ◽  
Yongbin Ruan
Keyword(s):  
Moduli Spaces ◽  
Homogeneous Polynomial ◽  
Genus Zero ◽  
Quantum Invariants ◽  
Rational Parameter ◽  
The Family ◽  
Wall Crossing

AbstractWe study a family of moduli spaces and corresponding quantum invariants introduced recently by Fan–Jarvis–Ruan. The family has a wall-and-chamber structure relative to a positive rational parameter ϵ. For a Fermat quasi-homogeneous polynomial

scholarly journals Algebraic Cycles on Quadric Sections of Cubics in ℙ4 under the Action of Symplectomorphisms

2015 ◽  
Vol 59 (2) ◽  
pp. 377-392 ◽  
Author(s):  
V. Guletskiĭ ◽  
A. Tikhomirov
Keyword(s):  
Algebraic Cycles ◽  
Chow Group ◽  
Two Dimensional ◽  
Precise Description ◽  
Continuous Part ◽  
Quadric Hypersurface ◽  
Irreducible Components ◽  
Double Covers ◽  
Discriminant Curve ◽  
The One

AbstractLet τ be the involution changing the sign of two coordinates in ℙ4. We prove that τ induces the identity action on the second Chow group of the intersection of a τ-invariant cubic with a τ-invariant quadric hypersurface in ℙ4. Let lτ and Πτ be the one- and two-dimensional components of the fixed locus of the involution τ. We describe the generalized Prymian associated with the projection of a τ-invariant cubic 𝓵 ⊂ P4 from lτ onto Πτ in terms of the Prymians 𝓅2 and 𝓅3 associated with the double covers of two irreducible components, of degree 2 and 3, respectively, of the reducible discriminant curve. This gives a precise description of the induced action of the involution τ on the continuous part of the Chow group CH2 (𝓵). The action on the subgroup corresponding to 𝓅3 is the identity, and the action on the subgroup corresponding to 𝓅2 is the multiplication by —1.

scholarly journals The Algebraic de Rham Cohomology of Representation Varieties

2018 ◽  
Vol 70 (3) ◽  
pp. 702-720
Author(s):  
Eugene Z. Xia
Keyword(s):  
De Rham Cohomology ◽  
Natural Family ◽  
De Rham ◽  
The One ◽  
Representation Varieties ◽  
Algebraic De Rham Cohomology ◽  
Natural Families

AbstractThe SL(2, ℂ)-representation varieties of punctured surfaces form natural families parameterized by monodromies at the punctures. In this paper, we compute the loci where these varieties are singular for the cases of one-holed and two-holed tori and the four-holed sphere. We then compute the de Rham cohomologies of these varieties of the one-holed torus and the four-holed sphere when the varieties are smooth via the Grothendieck theorem. Furthermore, we produce the explicit Gauß-Manin connection on the natural family of the smooth SL(2, ℂ)-representation varieties of the one-holed torus.

Krąg rodziców i dzieci uprawnionych do posiadania karty dużej rodziny a konstytucyjna zasada równości wobec prawa

2015 ◽  
pp. 137-153
Author(s):  
Filip Dziedzic
Keyword(s):  
Large Family ◽  
Time Limit ◽  
The Other ◽  
Constitutional Principle ◽  
Legal Situation ◽  
The Family ◽  
The Subject ◽  
Large Families ◽  
The One ◽  
The Right

The subject of the article is the justification of the thesis that the differentiation of the legal situation of parents on the basis of the Act on the Large Family Card, who have established a family with at least three children violates the constitutional principle of equality before the law. On the one hand some parents are entitled to use the card without any time limit, and on the other hand there is a group of parents who also have large families, but are totally deprived of the right. According to the author of the article, the diversity does not represent any constitutionally protected value and the discrimination occurs due to the unlimited duration of the right to own the Card by eligible parents. The result of the above, as well as the fourth (another) child’s right to the Card depending on holding the Card by the parent, is discriminatory for the children born as the fourth (next) child in the family. The article is also an attempt to answer the question which way would be the best to remove the above-mentioned discrimination thus making it most coherent with the objective and content of the analyzed regulation.

New classes of TQC associated with self-dual, quasi self-dual and denser tessellations

2010 ◽  
Vol 10 (11&12) ◽  
pp. 956-970
Author(s):  
C. D. Albuquerque ◽  
R. Palazzo Jr. ◽  
E. B. Silva
Keyword(s):  
Minimum Distance ◽  
Bipartite Graphs ◽  
The Self ◽  
Quantum Codes ◽  
Complete Graphs ◽  
Mathematical Structures ◽  
Topological Quantum ◽  
Complete Bipartite Graphs ◽  
Compact Surfaces ◽  
The One

In this paper we present six classes of topological quantum codes (TQC) on compact surfaces with genus $g\ge 2$. These codes are derived from self-dual, quasi self-dual and denser tessellations associated with embeddings of self-dual complete graphs and complete bipartite graphs on the corresponding compact surfaces. The majority of the new classes has the self-dual tessellations as their algebraic and geometric supporting mathematical structures. Every code achieves minimum distance 3 and its encoding rate is such that $\frac{k}{n} \rightarrow 1$ as $n \rightarrow \infty$, except for the one case where $\frac{k}{n} \rightarrow \frac{1}{3}$ as $n \rightarrow \infty$.

“Victim, Perpetrator, or Just My Brother?” Sibling Sexual Abuse in Large Families: A Child Advocacy Center Study

2017 ◽  
Vol 35 (21-22) ◽  
pp. 4887-4912 ◽  
Author(s):  
Dafna Tener ◽  
Noam Tarshish ◽  
Shosh Turgeman
Keyword(s):  
Sexual Abuse ◽  
Sexual Behaviors ◽  
Family System ◽  
Large Family ◽  
Child Advocacy ◽  
Sibling Sexual Abuse ◽  
The Family ◽  
Child Advocacy Center ◽  
The One ◽  
Critical Policy

Sibling sexual abuse (SSA) is a continuum of childhood sexual behaviors that do not fit the category of age-appropriate curiosity. Although SSA may be the most prevalent and longest lasting form of intrafamilial sexual abuse—as well as the one with the worst repercussions—it is also the least reported, studied, and treated. Based on 100 mostly religious Jewish families referred to a child advocacy center (CAC) in Jerusalem from 2010 to 2015, this qualitative study examines SSA characteristics, dynamics, and perceptions of deviancy in multisibling subsystems. The findings are based on an analysis of case summaries, demographic charts, and documented conversations between social workers and siblings. Qualitative document analysis reveals two types of SSA dynamics: “identified perpetrator” and “routine relationship,” the latter being a particularly understudied dynamic that challenges common stereotypes. We also found sibling perceptions of deviancy to vary along a continuum from deviant to completely normative. These perceptions are affected by the type of dynamics as well as by factors associated with disclosure. Our findings highlight the importance of studying the lived experiences of children involved in SSA as an input with critical policy, treatment, and research implications. Interventions must be adjusted to the family system and sibling subsystem’s perceptions and needs to avoid treatment that exacerbates the crisis already experienced by the family. Common assumptions—there must be a “perpetrator”; abuse is necessarily traumatic; and treatment should focus on the trauma—are challenged by the routine type. We conclude that treatment should account for the complexity of SSA by shedding these assumptions and considering the sibling subsystem as an autonomous unit within the large family.

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