The paper generalizes TQFTs to fermionic systems and classifies SPTs and SETs.
problem Classifying fermionic SPTs and SETs with finite group symmetries.
method Formulating fermionic TQFTs, gauging SPTs, using bordism groups, and constructing anomalous boundary states.
result Explicit classification of fermionic SPTs and SETs, including new anomalous boundary states.
Extends RT TQFT to include surface defects and line defects.
problem Distinguishing non-isotopic embeddings of surfaces in 3D bordisms.
method Uses modular tensor categories, symmetric Frobenius algebras, and equivariant multi-modules.
result Invariant distinguishes non-isotopic embeddings of 2-tori but not 2-spheres.
The goal of the present paper is the calculation of the equivariant twisted K-theory of a compact Lie group which acts on itself by conjugations, and elements of a TQFT-structure on the twisted K-groups. These results are originally due to D.S.Freed, M.J.Hopkins and C.Teleman. In this paper we redo their calculations i…
Define web algebras for annular SL(2) and SL(3) using foam TQFTs.
problem Define invariant algebras for tangles in the annulus.
method Use annular foam TQFTs to assign complex of bimodules to tangles.
result Establish properties of algebras and bimodules, and provide a bijection between webs and paths.
We 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…
Constructs a TQFT for 3-manifolds and extends it to 4-dimensional 2-handlebodies.
problem Building a TQFT for 3-manifolds and extending it to 4D.
method Using a Frobenius algebra and skein relations, constructing a TQFT for 3-manifolds and extending it to 4D using an inductive state-sum construction.
result Extends a TQFT to 4-dimensional 2-handlebodies.
Study invariants for branched G-covers of surfaces, focusing on stable cases.
problem Classifying equivalence of homomorphisms on surfaces with punctures.
method Classifying space for (framed) C-branched G-covers and branched Schur invariants.
result Provide a stable answer to equivalence of homomorphisms for surfaces with enough genus and punctures.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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. 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.
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 …
Goosen extends TQFTs using generators and relations for a simple example.
problem Developing a formalism for 3D TQFTs with generators and relations.
method Combining string-net and generators-and-relations formalisms for a specific example.
result Provides an elementary string-net description of linear maps and mapping class group action.
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.
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.
Researchers compute TQFT representation for sphere with 4 punctures.
problem Computing the representation of mapping class group for a sphere with 4 punctures.
method Non semi-simple TQFT approach, focusing on sphere with 4 punctures.
result The representation is faithful and compared with braid groups.
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.
Teichmüller TQFT upgraded to level N, linked to quantum Chern-Simons Theory.
problem Constructing a Teichmüller TQFT at level N.
method Using results from [AK1] and [AK3], relating to quantum Chern-Simons Theory.
result Teichmüller TQFT at level N is connected to quantum Chern-Simons Theory.
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.
New non-semisimple TQFTs derived from quantum Hennings invariants.
problem Constructing non-semisimple 2+1-TQFTs with mapping class group representations.
method Generalized Hennings invariants, modified traces, and modified TQFT construction.
result Construction of non-semisimple TQFTs yielding mapping class group representations.
TQFT signatures linked to trace fields of knots.
problem Relationship between TQFT signatures and knot trace fields.
method Analysis of Frobenius algebras and TQFTs at specific roots.
result TQFT signatures equal to trace fields of two-bridge knots.
The paper constructs TQFTs and Schrödinger representations for Heisenberg group.
problem Constructing TQFTs and Schrödinger representations for Heisenberg group.
method Using Lagrangian correspondences and q-deformation of U(1).
result Normalization of Schrödinger bimodule action reproduces abelian TQFT.
Paper connects TQFTs and SKK invariants through a group homomorphism.
problem Relating TQFTs and SKK invariants for smooth manifolds.
method Investigates a group homomorphism between invertible TQFTs and SKK invariants.
result All positive real-valued SKK invariants can be realized as TQFT restrictions.
Decorated TQFTs compute invariants with additional structures.
problem Computing topological invariants with additional structures.
method Cutting and gluing to obtain decorated invariants, proposing Hilbert spaces for two-dimensional surfaces.
result Proposal for Hilbert spaces assigned to surfaces in decorated TQFTs.
We construct a cohomology theory for oriented links using singular cobordisms and a special type of 2-dimensional Topological Quantum Field Theory (TQFT), categorifying the quantum sl(2) invariant. In particular, we give a description of the universal dot-free sl(2) foam cohomology for links via a TQFT.
We find two bases for the lattices of the SU(2)-TQFT-theory modules of the torus over given rings of integers. We use variant of the bases defined in [GMW]for the lattices of the SO(3)-TQFT-theory modules of the torus. Moreover, we discuss the quantization functors (V_{p},Z_{p}) for p=1, and p=2. Then we give concrete …
We examine the anomalies arising in instanton calculus as detailed by Damiano Anslemi in 1994. Whereas Anselmi uses BRST theory, we use the ADHM construction to arrive at the same conclusions from a differential-geometric way. We observe that Anselmi's TQFT is similar to Donaldson Theory applied to charge 1 instantons …
We construct maps on hat Heegaard Floer homology for cobordisms decorated with graphs. The graph TQFT allows for cobordisms with disconnected ends. Our construction uses Juhász's sutured Floer TQFT. We compute the maps for several elementary graph cobordisms. As an application, we compute the action of the fundamental …
Defines a universal state sum construction for various TQFTs.
problem No specific problem stated; universal construction for TQFTs.
method Defines a universal state sum construction using n-categories with specific conditions.
result Produces state sums from n-categories and handle decompositions of n+1-manifolds.
Study on knot invariants from Teichmüller TQFT, proving equivalence and new calculations.
problem Calculating knot invariants from Teichmüller TQFT.
method Formal stationary phase analysis, explicit isomorphism, and equivalence proofs.
result Two formulations of Teichmüller TQFT give equivalent knot invariants.
New categories from TQFTs interpret skein relations.
problem Interpreting skein relations in TQFTs.
method Constructing half-braided algebras and their bimodules.
result Stated skein relations correspond to a TQFT.
In this paper we show how one can extend Turaev-Viro invariants, defined for an arbitrary spherical fusion category C, to 3-manifolds with corners. We demonstrate that this gives an extended TQFT which conjecturally coincides with the Reshetikhin-Turaev TQFT corresponding to the Drinfeld center Z(C). In the present…