Finite groups, spherical 2-categories, and 4-manifold invariants
Marco Mackaay
备注：
LaTeX 44 pages, 10 eps.files, accepted for publication in Adv Math. Much better introduction explaining the fundamental ideas, equivalence relation on weak mon. 2-structures defined and theorem added
In this paper we define a class of state-sum invariants of compact closed oriented piece-wise linear 4-manifolds using finite groups. The definition of these state-sums follows from the general abstract construction of 4-manifold invariants using spherical 2-categories, as we defined in a previous paper, although it requires a slight generalization of that construction. We show that the state-sum invariants of Birmingham and Rakowski, who studied Dijkgraaf-Witten type invariants in dimension 4, are special examples of the general construction that we present in this paper. They showed that their invariants are non-trivial by some explicit computations, so our construction includes interesting examples already. Finally, we indicate how our construction is related to homotopy 3-types. This connection suggests that there are many more interesting examples of our construction to be found in the work on homotopy 3-types, such as by Brown, for example.
In this paper, we propose a new approach towards the classification of spherical fusion categories by their Frobenius-Schur exponents. We classify spherical fusion categories of Frobenius-Schur exponent 2 up to monoidal equivalence. We also classify modular categories of Frobenius-Schur exponent 2 up to braided monoidal equivalence. It turns out that the Gauss sum is a complete invariant for modular categories of Frobenius-Schur exponent 2. This result can be viewed as a categorical analog of Arf's theorem on the classification of non-degenerate quadratic forms over fields of characteristic 2.
In this paper I define the notion of a non-degenerate finitely semi-simple semi-strict spherical 2-category of non-zero dimension. Given such a 2-category I define a state-sum for any triangulated compact closed oriented 4-manifold and show that this state-sum does not depend on the chosen triangulation by proving invariance under the 4D Pachner moves. This invariant of piece-wise linear closed compact oriented 4-manifolds generalizes the Crane-Yetter invariant and probably it is also a generalization of the Crane-Frenkel invariant. As an example we show how to obtain a 2-category of the right…
We show that every essentially small finitely semisimple k-linear additive spherical category in which k=End(1) is a field, is equivalent to its dual over the long canonical forgetful functor. This includes the special case of modular categories. In order to prove this result, we show that the universal coend of the spherical category with respect to the long forgetful functor is self-dual as a Weak Hopf Algebra.
A $G$-graded extension of a fusion category $\mathcal{C}$ yields a categorical action of $G$ on the center $Z(\mathcal C)$. If the extension admits a spherical structure, we provide a method for recovering its fusion rules in terms of the action. We then apply this to find closed formulas for the fusion rules of extensions of some group theoretical categories and of cyclic permutation crossed extensions of modular categories.
This paper is a study of monoidal categories with duals where the tensor product need not be commutative. The motivating examples are categories of representations of Hopf algebras and the motivating application is the definition of 6j-symbols as used in topological field theories. We introduce the new notion of a spherical category. In the first section we prove a coherence theorem for a monoidal category with duals following MacLane (1963). In the second section we give the definition of a spherical category, and construct a natural quotient which is also spherical. In the third section we d…
The first author constructed a $q$-parameterized spherical category $\sC$ over $\mathbb{C}(q)$ in [Liu15], whose simple objects are labelled by all Young diagrams. In this paper, we compute closed-form expressions for the fusion rule of $\sC$, using Littlewood-Richardson coefficients, as well as the characters (including a generating function), using symmetric functions with infinite variables.
We obtain two formulae for the higher Frobenius-Schur indicators: one for a spherical fusion category in terms of the twist of its center and the other one for a modular tensor category in terms of its twist. The first one is a categorical generalization of an analogous result by Kashina, Sommerhauser, and Zhu for Hopf algebras, and the second one extends Bantay's 2nd indicator formula for a conformal field theory to higher degree. These formulae imply the sequence of higher indicators of an object in these categories is periodic. We define the notion of Frobenius-Schur (FS-)exponent of a pivo…
Let $Λ_Q$ be the preprojective algebra of a finite acyclic quiver $Q$ of non-Dynkin type and $D^b(\mathrm{rep}^n Λ_Q)$ be the bounded derived category of finite dimensional nilpotent $Λ_Q$-modules. We define spherical twist functors over the root category $\mathcal{R}_{Λ_Q}$ of $D^b(\mathrm{rep}^n Λ_Q)$ and then realize the Weyl group associated to $Q$ as certain subquotient of the automorphism group of the Ringel-Hall Lie algebra $\mathfrak{g}(\mathcal{R}_{Λ_Q})$ of $\mathcal{R}_{Λ_Q}$ induced by spherical twist functors. We also present a conjectural relation between certain Lie subalgebras …
We prove that if two Tambara-Yamagami categories TY(A,χ,ν) and TY(A',χ',ν') give rise to the same state sum invariants of 3-manifolds and the order of one of the groups A, A' is odd, then ν=ν' and there is a group isomorphism A\approx A' carrying χto χ'. The proof is based on an explicit computation of the state sum invariants for the lens spaces of type (k,1).
We define a family of quantum invariants of closed oriented $3$-manifolds using spherical multi-fusion categories. The state sum nature of this invariant leads directly to $(2+1)$-dimensional topological quantum field theories ($\text{TQFT}$s), which generalize the Turaev-Viro-Barrett-Westbury ($\text{TVBW}$) $\text{TQFT}$s from spherical fusion categories. The invariant is given as a state sum over labeled triangulations, which is mostly parallel to, but richer than the $\text{TVBW}$ approach in that here the labels live not only on $1$-simplices but also on $0$-simplices. It is shown that a …
We introduce semisimple 2-categories, fusion 2-categories, and spherical fusion 2-categories. For each spherical fusion 2-category, we construct a state-sum invariant of oriented singular piecewise-linear 4-manifolds.
We introduce Manifold tensor categories, which make precise the notion of a tensor category with a manifold of simple objects. A basic example is the category of vector spaces graded by a Lie group. Unlike classic tensor category theory, our setup keeps track of the smooth (and topological) structure of the manifold of simple objects. We set down the necessary definitions for Manifold tensor categories and a generalisation we term Orbifold tensor categories. We also construct a number of examples, most notably two families of examples we call Interpolated Tambara-Yamagami categories and Interp…
This is a survey on spherical Hopf algebras. We give criteria to decide when a Hopf algebra is spherical and collect examples. We discuss tilting modules as a mean to obtain a fusion subcategory of the non-degenerate quotient of the category of representations of a suitable Hopf algebra.
We survey some aspects of the theory of noncommutative manifolds focusing on the noncommutative analogs of two-dimensional tori and low-dimensional spheres. We are particularly interested in those aspects of the theory that link the differential geometry and the algebraic geometry of these spaces.
We give formulae for the multiplicities of eigenvalues of generalized rotation operators in terms of generalized Frobenius-Schur indicators in a semisimple spherical tensor category $\mathcal{C}$. In particular, this implies that the entire collection of rotation eigenvalues for a fusion category can be computed from the fusion rules and the traces of rotation at finitely many tensor powers. We also establish a rigidity property for FS indicators of fusion categories with a given fusion ring via Jones's theory of planar algebras. If $\mathcal{C}$ is also braided, these formulae yield the multi…
A string-net model associates a vector space to a surface in terms of graphs decorated by objects and morphisms of a pivotal fusion category modulo local relations. String-net models are usually considered for spherical fusion categories, and in this case the vector spaces agree with the state spaces of the corresponding Turaev-Viro topological quantum field theory. In the present work some effects of dropping the sphericality condition are investigated. In one example of non-spherical pivotal fusion categories, the string-net space counts the number of r-spin structures on a surface and carri…
We classify which dual functors on a unitary multitensor category are compatible with the dagger structure in terms of groupoid homomorphisms from the universal grading groupoid to $\mathbb{R}_{>0}$ where the latter is considered as a groupoid with one object. We then prove that all unitary dual functors induce unitarily equivalent bi-involutive structures. As an application, we provide the unitary version of the folklore correspondence between shaded planar ${\rm C^*}$ algebras with finite dimensional box spaces and unitary multitensor categories with a chosen unitary dual functor and chosen …
We prove that any fusion category over $\mathbb{C}$ with exactly one non-invertible simple object is spherical. Furthermore, we classify all such categories that come equipped with a braiding.
We use categories of representations of finite dimensional quantum groupoids (weak Hopf algebras) to construct ribbon and modular categories that give rise to invariants of knots and 3-manifolds.
In this paper we construct invariants of 3-manifolds "à la Reshetikhin-Turaev" in the setting of non-semi-simple ribbon tensor categories. We give concrete examples of such categories which lead to a family of 3-manifold invariants indexed by the integers. We prove this family of invariants has several notable features, including: they can be computed via a set of axioms, they distinguish homotopically equivalent manifolds that the standard Reshetikhin-Turaev-Witten invariants do not, and they allow the statement of a version of the Volume Conjecture and a proof of this conjecture for an infin…
It is shown that the theory of spherical Harish-Chandra modules naturally provides the algebras of covariant, contravariant and mixed symbols on generalized flag manifolds. The general proof of the correspondence principle for all these symbol algebras is given.
We show that the renormalized quantum invariants of links and graphs in the 3-sphere, derived from tensor categories in ["Modified quantum dimensions and re-normalized link invariants", arXiv:0711.4229] lead to modified 6j-symbols and to new state sum 3-manifold invariants. We give examples of categories such that the associated standard Turaev-Viro 3-manifold invariants vanish but the secondary invariants may be non-zero. The categories in these examples are pivotal categories which are neither ribbon nor semi-simple and have an infinite number of simple objects.
We construct a pairing, which we call factorization homology, between framed manifolds and higher categories. The essential geometric notion is that of a vari-framing of a stratified manifold, which is a framing on each stratum together with a coherent system of compatibilities of framings along links between strata. Our main result constructs labeling systems on disk-stratified vari-framed $n$-manifolds from $(\infty,n)$-categories. These $(\infty,n)$-categories, in contrast with the literature to date, are not required to have adjoints. This allows the following conceptual definition: the fa…
We develop the theory of diagrammatic Temperley-Lieb categories in order to construct examples of spherical fusion categories, which we then use to construct Turaev-Viro skein modules for n-holed disks. This is a thesis submitted for the degree of Bachelor of Science with Honours at the Australian National University.
We introduce systems of objects and operators in linear monoidal categories called $\hat Ψ$-systems. A $\hat Ψ$-system satisfying several additional assumptions gives rise to a topological invariant of triples (a closed oriented 3-manifold $M$, a principal bundle over $M$, a link in $M$). This construction generalizes the quantum dilogarithmic invariant of links appearing in the original formulation of the volume conjecture. We conjecture that all quantum groups at odd roots of unity give rise to $\hat Ψ$-systems and we verify this conjecture in the case of the Borel subalgebra of quantum $sl_…
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 the choice of the framing parameter. This leads, for any rank N and level K, to a modular category \tilde H^{N,K} and a reduced invariant \tildeτ_{N,K}. If N and K are coprime, then this invariant coincides with the known PSU(N) invariant at level K. If gcd(N,K)=d>1, then we show…
Using M-theory in physics, Cho, Gang, and Kim (JHEP 2020, 115 (2020) ) recently outlined a program that connects two parallel subjects of three dimensional manifolds, namely, geometric topology and quantum topology. They suggest that classical topological invariants such as Chern-Simons invariants of $\text{SL}(2,\mathbb{C})$-flat connections and adjoint Reidemeister torsions of a three manifold can be packaged together to produce a $(2+1)$-topological quantum field theory, which is essentially equivalent to a modular tensor category. It is further conjectured that every modular tensor categor…
We show that all strongly non-degenerate trigonometric solutions of the associative Yang-Baxter equation (AYBE) can be obtained from triple Massey products in the Fukaya category of square-tiled surfaces. Along the way, we give a classification result for cyclic $A_\infty$-algebra structures on a certain Frobenius algebra associated with a pair of 1-spherical objects in terms of the equivalence classes of the corresponding solutions of the AYBE. As an application, combining our results with homological mirror symmetry for punctured tori (cf. arXiv:1601.06141), we prove that any two simple vect…
