Study non-semisimple TQFT for Burau representation density and unitarity.
problem Density and unitarity of the Burau representation from a non-semisimple TQFT perspective.
method TQFT construction of Squier's Hermitian form on the Burau representation.
result Density of the image of braid group in unitary representations.
Proves Witten-Reshetikhin-Turaev 3-TQFT as a boundary condition of Crane-Yetter 4-TQFT.
problem Proving a boundary condition for Crane-Yetter 4-TQFT.
method Extending ideas of Crane-Yetter and Jordan, proving Crane-Yetter 4-TQFT and its non-semisimple version are once-extended TQFTs, defining a boundary condition.
result Reconstructs Witten-Reshetikhin-Turaev 3-TQFT and its non-semisimple versions using Crane-Yetter 4-TQFT.
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.
New pseudo-Hermitian models from non-semisimple TQFTs.
problem Constructing exactly solvable pseudo-Hermitian spin Hamiltonians.
method Identifying ground states on surfaces using non-semisimple TQFTs.
result Ground states depend only on spatial topology and can be assigned by non-semisimple TQFTs.
Develops Hermitian TQFTs from quantum groups, defining new topological phases.
problem Defining Hermitian non-semisimple TQFTs.
method Categorical context and representation theory of quantum groups.
result New pseudo-Hermitian topological phases from quantum group representations.
A Hermitian TQFT from non-semisimple quantum sl(2) modules.
problem Constructing a Hermitian TQFT from a non-semisimple category.
method Endowed a non-semisimple category of quantum sl(2) modules with a Hermitian structure and proved the resulting TQFT is Hermitian.
result Projective representations of the mapping class group in indefinite unitary matrices.
The paper defines new TQFTs from non-semisimple categories and proves spherical categories are chromatic.
problem Defining non-compact TQFTs from non-semisimple categories.
method Introducing admissible skein modules, chromatic categories, and using Juhász's cobordism presentation.
result Non-compact (2+1)-TQFTs can be defined from chromatic categories, extending Turaev-Viro TQFTs.
Study projective representations from non-semisimple TQFTs on surfaces.
problem Understanding projective representations of mapping class groups from non-semisimple TQFTs.
method Construct 3D TQFTs using non-semisimple modular categories and analyze projective representations of mapping class groups.
result Projective representations from non-semisimple TQFTs are equivalent to those obtained by Lyubashenko.
New skein categories for non-semisimple settings, extending existing theory.
problem Extending skein theory to non-semisimple settings.
method Introducing skein categories based on tensor ideals in linear ribbon categories.
result Skein categories coincide with factorization homology in non-semisimple settings.
Defines a new 3D TQFT from non-semisimple categories.
problem Developing a TQFT from non-semisimple categories.
method Generators and relations framework, decomposition of 3-manifolds.
result TQFT values on 3-manifolds match known invariants.
New (3+1) TQFTs created from non-semisimple categories.
problem Creating TQFTs from non-semisimple ribbon categories.
method Using skein theory and admissible skein modules, defining TQFTs with specific algebraic conditions.
result Explicit realization of a TQFT based on the cobordism hypothesis.
Constructs TQFTs for cobordisms with cohomology class decorations.
problem Creating TQFTs for cobordisms with cohomology class decorations.
method Starting from an abelian group G and a factorizable ribbon Hopf G-bialgebra H, constructs a TQFT JH for connected framed cobordisms between connected surfaces with connected boundary decorated with cohomology classes with coefficients in G. result Our functor recovers a special case of Kerler-Lyubashenko TQFTs when restricted to trivial decorations.
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.
New relation between quantum groups and BPS series.
problem Connecting quantum groups at roots of unity and generic q.
method Proposing and proving a precise relation between invariants.
result Bridging non-semisimple and semisimple TQFTs.
We show that unrolled quantum groups at odd roots of unity give rise to relative modular categories. These are the main building blocks for the construction of 1+1+1-TQFTs extending CGP invariants, which are non-semisimple quantum invariants of closed 3-manifolds decorated with ribbon graphs and cohomology classes. Whe…
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.
The paper explores how the unit inclusion affects topological quantum field theories in non-semisimple categories.
problem Understanding the effects of unit inclusion in non-semisimple braided tensor categories on topological quantum field theories.
method Analyzes the dualizability of the unit inclusion morphism in Morita 4-category of braided tensor categories and applies the Cobordism Hypothesis.
result Shows that the unit inclusion in non-semisimple modular categories leads to non-compact relative 3D topological quantum field theories.
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.
We construct non-semisimple 2+1-TQFTs yielding mapping class group representations in Lyubashenko's spaces. In order to do this, we first generalize Beliakova, Blanchet and Geer's logarithmic Hennings invariants based on quantum sl2 to the setting of finite-dimensional non-degenerate unimodular ribbon H…
We develop the general theory for the construction of Extended Topological Quantum Field Theories (ETQFTs) associated with the Costantino-Geer-Patureau quantum invariants of closed 3-manifolds. In order to do so, we introduce relative modular categories, a class of ribbon categories which are modeled on representations…
New signs and gradings enable detailed comparison in Heegaard Floer theory.
problem Comparing decategorified Heegaard Floer theory with modern TQFTs.
method Added signs and gradings to interval gluing theorem over Z.
result Detailed comparison possible with modern TQFTs.
This survey covers some of the results contained in the papers by Costantino, Geer and Patureau (https://arxiv.org/abs/1202.3553) and by Blanchet, Costantino, Geer and Patureau (https://arxiv.org/abs/1404.7289). In the first one the authors construct two families of Reshetikhin-Turaev-type invariants of 3-manifolds, $\…
Innovative series invariant for knot complements, linking to existing invariants.
problem Developing a new series invariant for knot complements.
method Introducing a three-variable series FK(y,z,q) for plumbed knot complements. result Deriving a surgery formula relating FK(y,z,q) to Z^(q) invariant. Homological model for quantum representations of mapping class groups.
problem Investigate linearity of mapping class groups using quantum representations.
method Homological action on configuration space with twisted coefficients.
result Identify subrepresentation equivalent to quantum sl2 representation. Constructs maps on skein modules using non-semisimple quantum invariants.
problem Constructing maps on skein modules with specific characters.
method Uses UqHsl2 non-semisimple invariants of 3-manifolds. result Maps with any possible abelian non-central character as classical shadow.
Introduces admissible skein modules for non-semisimple categories.
problem No specific problem stated; generalization of Kauffman skein algebra.
method Introduces admissible skein modules associated to ideals in pivotal categories.
result These modules generalize Kauffman skein algebra and relate to quantum invariants.
New non-semisimple Ising anyons enable robust universal quantum computation.
problem Limitation of semisimple theories in universal topological quantum computation.
method Developed non-semisimple Ising anyon model with new anyon types indexed by α. result Robust universality of braiding persists over an open interval of α. Restricts quantum representations of mapping class groups to integral coefficients.
problem Integrality of non-semisimple quantum representations of mapping class groups.
method Exhibits explicit bases of states spaces that span Z[ζ]-lattices invariant under mapping class groups. result Restricts quantum representations to integral coefficients from Q(ζ) to Z[ζ]. New invariant calculates 4-manifolds using trisection diagrams and combings.
problem Calculating non-semisimple 4-manifold invariants.
method Using trisection diagrams and combings of the trisection surface.
result Invariant calculated for Stein nuclei, generalizing earlier semisimple version.
New proof and formula linking fusion trees to quantum knot invariants.
problem Quantum knot invariants encoding in non-semisimple TQC.
method Connection between fusion trees and Lawrence representations, using graphical calculus.
result Explicit encoding of quantum knot invariants via fusion trees.
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.
Study para-Sasakian φ-symmetric spaces using Boothby-Wang fibration.
problem Characterize para-Sasakian φ-symmetric spaces.
method Use Boothby-Wang fibration to construct and provide examples.
result Explicit construction and example of para-Sasakian φ-symmetric spaces.
Quantum invariants for fibered links determined by genus and Hopf invariant.
problem Quantum invariants of fibered links in S3. method Genus bounds and Giroux-Goodman theorem on fiber surfaces.
result Top coefficient of ADO invariant is determined by Hopf invariant.
It has been conjectured that every (2+1)-TQFT is a Chern-Simons-Witten (CSW) theory labelled by a pair (G,λ), where G is a compact Lie group, and λ∈H4(BG;Z) a cohomology class. We study two TQFTs constructed from Jones' subfactor theory which are believed to be counterexamples to this conjecture: one is the…
Classifies 2D TQFTs for orientable cobordisms using additional data.
problem Classifying 2D TQFTs for orientable cobordisms.
method Describes an intermediate framework using an involution and Möbius strip value.
result Intermediate classification of 2D TQFTs for orientable cobordisms.
This paper categorifies Quinn's TQFTs and computes them for specific omega-groupoids.
problem Constructing and computing finite total homotopy TQFTs.
method Direct homotopy theoretical construction, categorification of Quinn's TQFTs, explicit computation for omega-groupoids.
result Categorification and explicit computation of Quinn's TQFTs for omega-groupoids.
Defines extended TQFTs using handle attachments.
problem Constructing extended topological quantum field theories (TQFTs).
method Finite presentation of cobordism symmetric monoidal bicategory using handle attachments and relations.
result Constructs a once extended TQFT from categorified TQFT and handle 2-morphisms.
Paper uses Turaev-Viro TQFT to estimate 3-manifold genus.
problem Estimating the Heegaard genus of 3-manifolds.
method Turaev-Viro state sum TQFT and unitary modular category.
result Provides a lower bound for Heegaard genus using TQFT.
Almost integral TQFTs were introduced by Gilmer [Duke Math. J. 125 (2004) 389--413]. The aim of this paper is to modify the TQFT of the category of extended 3-cobordisms given by Turaev (in his book: Quantum invariants of knots and 3-manifolds) to obtain an almost integral TQFT.
Chern-Simons and Reshetikhin-Turaev theories are shown equivalent for U(1) gauge group.
problem Equivalence between U(1) Chern-Simons and Reshetikhin-Turaev TQFTs. method Proof of natural isomorphism between theories for finite quadratic modules.
result Extended (2+1)-dimensional TQFTs are naturally isomorphic. New TQFTs distinguish torus bundles and lens spaces.
problem Existence of faithful (2+1)-TQFTs.
method Subcategory of 3-cobordisms containing disjoint unions of tori and simplest cobordisms between them, defined TQFTs.
result TQFTs can distinguish torus bundles and lens spaces.
New invariants derived from Kauffman bracket for 3-manifolds.
problem Quantum invariants of 3-manifolds, especially non-semisimple ones.
method Combinatorial methods using Temperley-Lieb algebras and Kauffman bracket polynomials.
result Recovery of invariants from small quantum group of sl2. Abstract TQFT for sutured manifolds using Floer homology.
problem Classical Frohman-Nicas TQFT for Alexander polynomial.
method Decategorification of bordered sutured Heegaard Floer homology.
result Generalization to arbitrary cobordisms between surfaces.
Quantum modularity proved for SU(2) TQFT signature on genus 2 surfaces.
problem Proving quantum modularity of SU(2) TQFT signature for genus 2 surfaces.
method Using quantum modularity of generalized Dedekind sums associated with modular forms and trigonometric sum expressions.
result Quantum modularity of SU(2) TQFT signature on genus 2 surfaces proved.
We find bases for naturally defined lattices over certain rings of integers in the SU(2)-TQFT-theory modules of surfaces. We consider the TQFT where the Kauffman's A variable is a root of unity of order four times an odd prime. As an application, we show that the Frohman Kania-Bartoszynska ideal invariant for 3-manifol…
TQFT invariants are either easy or hard to compute, depending on the TQFT type.
problem Computing TQFT invariants on closed 3-manifolds.
method Application of a dichotomy result for weighted constraint satisfaction problems over C.
result TQFT invariants are either solvable in polynomial time or #P-hard. Develops a TQFT framework to compute Z^ invariants of three-manifolds.
problem Understanding the TQFT structure of Z^ invariants of three-manifolds. method Decorated Spin-TQFTs, novel quantization of SL(2,C) Chern-Simons theory, and algebra of observables. result Explicit closed-form expressions for Z^ invariants of various three-manifolds. We construct families of TQFT's over the finite field Z/pZ starting from an integral TQFT obtained by Frohman and Nicas. These TQFT's are likely to describe the constant order contributions of the cyclotomic integer expansions of the Reshetikhin Turaev Ohtsuki theories. Their modular structure is intimately related to …