The study examines modular fusion categories with trivial Torelli group actions.
problem Characterizing modular fusion categories with trivial Torelli group actions.
method Analyzing the mapping class group representations and their kernels.
result For modular fusion categories, the Torelli group is contained in the kernel of the genus-g representation if and only if the category is pointed. A new TQFT is conjectured to extend Reshetikhin-Turaev TQFT.
problem Extending TQFT to lower dimensions.
method Defining a symmetric monoidal (4,3)-category with duals from enriched multi-fusion categories.
result A conjectured extension of 1-2-3-dimensional TQFT to dimension zero.
In this paper, we extend the notion of modular functor and fusion category to what we called G equivariant modular functor and G equivariant fusion category, where G is a finite group, and establish a correspondence between between these notions.
New 4-manifold invariant defined from trisection diagrams.
problem Defining a new 4-manifold invariant from trisection diagrams.
method Algebraic data from bimodule categories and spherical fusion categories, described diagrammatically.
result Includes Hopf algebraic invariants and modular fusion category invariants.
Generalizes string-net modular functors to non-spherical categories.
problem Extending string-net models to non-spherical categories.
method Using non-semisimple string-nets and Drinfeld centers.
result Equivalence between string-net and Lyubashenko modular functors.
Semisimplicity proven for conformal blocks representations.
problem Semisimplicity of conformal blocks representations.
method Theory of extensions in non-Abelian Hodge theory and Ocneanu rigidity.
result Semisimplicity of braid group and mapping class group representations.
The paper constructs braiding structures for a specific subfactor.
problem The challenge is to understand the braiding structures of a Jones-Wassermann subfactor.
method The approach involves constructing braiding structures on the multi-interval Jones-Wassermann subfactor planar algebra.
result The braiding structures induce a projective unitary representation of the balanced superelliptic mapping class group.
Study mapping class groups and their representations linking to algebraic structures.
problem Understanding projective representations of mapping class groups and their Morita classes.
method Defined projective representations of mapping class groups using modular fusion categories and analyzed their irreducibility.
result Irreducible representations imply unique Morita-class of simple algebras.
The center Z(C) of a spherical fusion category C (over an arbitrary commutative ring) is modular. We give an algorithm for computing the Reshetikhin-Turaev invariant defined with Z(C). It is based on Hopf diagrams and an explicit description of the structure of the coend of Z(C).
Overview of 3D TQFTs and 3-manifold invariants.
problem Quantum invariants of 3-manifolds.
method Recall and review of TQFTs, fusion categories, and recent generalizations.
result Overview of various 3D TQFTs and their invariants.
Paper connects 3D gravity averages to 2D CFT correlators.
problem Understanding 3D gravity partition functions.
method 3D topological field theories and mapping class group averages.
result Established a correspondence between 3D gravity averages and 2D CFT correlators.
Proposes a topological framework to study modular invariants and related concepts.
problem Exploring modular invariants and related concepts in topological quantum field theory.
method Topological paradigm in alterfold topological quantum field theory.
result Establishes a novel integral identity for modular invariance across multiple Morita contexts.
3D topological order linked to Seifert manifolds and gauge groups.
problem Classifying 3D topological orders using Seifert manifolds and gauge groups.
method Correspondence between topological order, Seifert manifolds, and ADE gauge groups.
result Construction of modular fusion categories from Seifert manifolds and gauge groups.
Modified invariants from quantum sl(2|1) for 3-manifolds.
problem Quantum sl(2|1) modules have vanishing quantum dimensions, complicating category construction.
method Specialize q to a root of unity, quotient by morphisms with zero modified quantum dimension, show resulting category is finite and semi-simple.
result Obtained relative G-spherical categories from modified quantum dimensions.
Innovative 2-categories create 4-manifold invariants.
problem Constructing invariants for 4-manifolds.
method Semisimple 2-categories, fusion 2-categories, and state-sum construction.
result Construct a state-sum invariant for 4-manifolds.
We present an invariant of connected and oriented closed 3-manifolds based on a coribbon Weak Hopf Algebra H with a suitable left-integral. Our invariant can be understood as the generalization to Weak Hopf Algebras of the Hennings-Kauffman-Radford evaluation of an unoriented framed link using a dual quantum-trace. Thi…
Zesting affects Reshetikhin-Turaev invariants of links and 3-manifolds.
problem Understanding how zesting affects Reshetikhin-Turaev invariants.
method Developed a local formalism to compute tangle invariants and link invariants.
result Zesting contributes to complexity-theoretic hierarchies of topological field theories.
We classify all fusion categories for a given set of fusion rules with three simple object types. If a conjecture of Ostrik is true, our classification completes the classification of fusion categories with three simple object types. To facilitate the discussion we describe a convenient, concrete and useful variation o…
ETQFTs created from non-semisimple modular categories.
problem Constructing ETQFTs from non-semisimple modular categories.
method Explicitly identify linear categories and functors in the image of ETQFTs constructed from modular categories.
result The circle category of ETQFTs is equivalent to the full subcategory of projective objects of the underlying modular category, which need not be semisimple.
A modular functor is constructed from non-semisimple 3d TFTs.
problem Constructing modular functors from non-semisimple 3d topological field theories.
method Using a 3d TFT defined in [arXiv:1912.02063], a symmetric monoidal 2-functor is constructed from a 2-category of bordisms to a 2-category of finite linear categories.
result A modular functor is explicitly described as a symmetric monoidal 2-functor.
Mathematical study supports connection between 3D manifolds and modular tensor categories.
problem Connecting geometric topology and quantum topology using Chern-Simons invariants and Reidemeister torsions.
method Developed an algorithm to generate modular T-matrices and quantum dimensions from Seifert fibered spaces and torus bundles over the circle. result Mathematically constructed premodular categories from Seifert fibered spaces and torus bundles over the circle, conjecturing their modularity under specific conditions.
String-net models explore non-spherical fusion categories, revealing new spin structures and representations.
problem Investigating string-net models in non-spherical fusion categories.
method String-net models associate vector spaces to surfaces in terms of graphs decorated by objects and morphisms of a pivotal fusion category.
result String-net spaces count r-spin structures and carry representations of the mapping class group.
Recent work extends Turaev's modular categories to non-semisimple settings.
problem Building TQFTs beyond semisimplicity.
method Generalized modular categories.
result Success in extending Turaev's construction to non-semisimple settings.
Constructs a new invariant for 4-manifolds and 3+1 TQFTs.
problem Developing a new invariant for 4-manifolds and 3+1 TQFTs.
method Using a G-crossed braided spherical fusion category to construct a state-sum type invariant.
result Introduces a new invariant for 4-manifolds and 3+1 TQFTs.
3D HQFTs constructed using graded monoidal categories.
problem Constructing 3D HQFTs with specific targets.
method Using spherical χ-fusion categories and the state sum method.
result 3D HQFTs constructed with target Bχ.
We show that once-extended anomalous 3-dimensional topological quantum field theories valued in the 2-category of k-linear categories are in canonical bijection with modular tensor categories equipped with a square root of the global dimension in each factor.
We formulate the unitary rational orbifold conformal field theories in the algebraic quantum field theory framework. Under general conditions, we show that the orbifold of a given unitary rational conformal field theories generates a unitary modular category. Many new unitary modular categories are obtained. We also sh…
3D HQFT comparison proves state sum equals surgery.
problem Comparing two HQFT approaches for 3D.
method Proving isomorphism between state sum and surgery methods.
result State sum 3D HQFT isomorphic to surgery 3D HQFT.
Abstract: Mapping class groups act on cohomology of surfaces via Hochschild cohomology.
problem Understanding the action of mapping class groups on cohomology of surfaces.
method Associate cochain complexes to surfaces, with mapping class groups acting projectively on cohomology.
result Projective action of mapping class groups on Hochschild cohomology of Hopf algebras.
New invariants derived from a modular category for links and 3-manifolds.
problem Developing new invariants for links and 3-manifolds.
method Extracted invariants from a modular category with two simple objects.
result Invariants for 3-manifolds are related by a specific equation.
Defines state sum models with defects in 3-manifolds.
problem Detecting and characterizing defects in 3-manifolds.
method Turaev-Viro-Barrett-Westbury state sum models with defects labeled by bimodule categories and functors.
result State sums are triangulation-independent and can be computed using polygon diagrams.
New method calculates manifold invariants using Kirby diagrams with 3-handles.
problem Calculating manifold invariants from TQFTs is complex due to combinatorial complexity.
method Reformulated state sum model using Kirby diagrams with 3-handles and graphical calculus.
result Invariants are multiplicative under connected sum, detecting no exotic structures.
Homotopy Quantum Field Theories (HQFTs) generalize more familiar Topological Quantum Field Theories (TQFTs). In generalization of the surgery construction of 3-dimensional TQFTs from modular categories, we use surgery to derive 3-dimensional HQFTs from G-modular categories.
Skein theory classifies UFCs with specific fusion rules.
problem Classifying unitary fusion categories with specific fusion rules.
method Graphical calculus and rotation operator action on a canonical basis.
result Explicit formulae for Fqqqq when k=2 and C is ribbon. Internalizes Turaev's construction for TQFTs using ribbon categories.
problem Constructing TQFTs from ribbon categories.
method Using a special kind of tangle and morphisms defined between tensorial products of the coend, an internal TQFT is constructed.
result An internal TQFT is a lift of Turaev's construction, equivalent to vector spaces when modular.
Turaev-Viro invariants match for certain surface bundles.
problem Matching Turaev-Viro invariants for surface bundles.
method Profinite completions and spherical fusion categories.
result Invariants coincide for specific surface groups.
The paper extends asymptotic faithfulness to skein quantum representations of mapping class groups.
problem Lack of explicit fusion spaces with multiplicities for skein quantum representations.
method Generalizes asymptotic faithfulness to skein quantum representations of mapping class groups.
result Conjectures asymptotic faithfulness for skein quantum G representations when G is a simply-connected simple Lie group. The article is devoted to the qR-conformal modular functors, which being ``deformations'' of the conformal modular functor (the projective representation of the category Train(Diff+(S1)), the train of the group Diff+(S1) of all orientation preserving diffeomorphisms of a circle) in the class of all projectiv…
Classical and quantum Chern-Simons with gauge group U(1)N were classified by Belov and Moore in \cite{belov_moore}. They studied both ordinary topological quantum field theories as well as spin theories. On the other hand a correspondence is well known between ordinary (2+1)-dimensional TQFTs and modular te…
We show that there exist infinitely many pairs of non-homeomorphic closed oriented SOL torus bundles with the same quantum (TQFT) invariants. This follows from the arithmetic behind the conjugacy problem in SL(2,Z) and its congruence quotients, the classification of SOL (polycyclic) 3-manifold groups and an elementa…
New mapping class group actions on Hochschild complexes for modular categories.
problem Understanding actions of mapping class groups on Hochschild complexes of modular categories.
method Construction of a symmetric monoidal functor with excision property.
result Homotopy coherent projective action of mapping class groups on Hochschild complexes.
Modular categories are a well-known source of quantum 3-manifold invariants. In this paper we study structures on modular categories which allow to define refinements of quantum 3-manifold invariants involving cohomology classes or generalized spin and complex spin structures. A crucial role in our construction is play…
New modular data from torus bundles via particle-hole equivariantization.
problem Constructing modular tensor categories from 3-manifolds.
method Using Chern-Simons invariants and adjoint Reidemeister torsions, and performing Z2-equivariantization. result Modular data from torus bundles realized by Z2-equivariantization of premodular categories. We prove that the category of abelian gerbes with connection over a smooth manifold is equivalent to a certain category of principal bundles over the free loop space. These bundles are equipped with a connection and with a "fusion" product with respect to triples of paths. The equivalence is established by explicit fun…
A p-periodic 3-manifold is a 3-manifold that admits a Z_{p}-action whose fixed point set is a circle. We give a congruence relates the quantum invariant of a p-periodic 3-manifold associated to any modular category over an integrally closed ground ring and the corresponding quantum invariant of its orbit space.
New 3D TQFTs derived from non-semisimple categories.
problem Constructing topological invariants from non-semisimple categories.
method Using modified traces and Lyubashenko's invariants, with additional assumptions for factorizability.
result Produces new 2+1-TQFTs and monoidal extensions of representations.
Discoveries new symmetries in 3d topological and physical systems.
problem Identifying hidden symmetries in 3d topological and physical systems.
method Analysis of modular forms, Weil representations, and chiral algebras.
result Identification of new modular structures in 3d theories.
We construct modular categories from Hecke algebras at roots of unity. For a special choice of the framing parameter, we recover the Reshetikhin-Turaev invariants of closed 3-manifolds constructed from the quantum groups U_q sl(N) by Reshetikhin-Turaev and Turaev-Wenzl, and from skein theory by Yokota. We then discuss …