A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
The paper defines new TQFTs from non-semisimple categories and proves spherical categories are chromatic.
3D HQFTs constructed using graded monoidal categories.
New construction of Turaev-Viro invariants invariant under Morita equivalence.
Spider category comparison proves equivalence to Sikora's quotient category.
We introduce the notion of a relative spherical category. We prove that such a category gives rise to the generalized Kashaev and Turaev-Viro-type 3-manifold invariants defined in arXiv:1008.3103 and arXiv:0910.1624, respectively. In this case we show that these invariants are equal and extend to what we call a relativ…
Study spherical twists on K3 surfaces, compute their centers.
The category of finite dimensional module over the quantum superalgebra U_q(sl(2|1)) is not semi-simple and the quantum dimension of a generic U_q(sl(2|1))-module vanishes. This vanishing happens for any value of q (even when q is not a root of unity). These properties make it difficult to create a fusion or modular ca…
Paper connects two invariants of 3D manifolds using Hopf algebras.
Defines a new 2+1-G-HQFT using graded skein modules.
New 4-manifold invariant defined from trisection diagrams.
Studies modules over a category of Jacobi diagrams in handlebodies.
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.
Introduces admissible skein modules for non-semisimple categories.
Generalizes string-net modular functors to non-spherical categories.
It is known that every ribbon category with unimodality allows symmetrized -symbols with full tetrahedral symmetries while a spherical category does not in general. We give an explicit counterexample for this, namely the category . We define the mirror conjugate symmetry of -symbols instead and sho…
Characterizes spherical Finsler metrics satisfying a specific condition.
It is proved that the category of simplicial complete bornological spaces over carries a combinatorial monoidal model structure satisfying the monoid axiom. For any commutative monoid in this category the category of modules is also a monoidal model category with all cofibrant objects being flat. In particu…
Paper extends Schur's theorem to spherical curves via monotonicity.
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 correspondi…
We construct a state-sum type invariant of smooth closed oriented -manifolds out of a -crossed braided spherical fusion category (-BSFC) for a finite group. The construction can be extended to obtain a -dimensional topological quantum field theory (TQFT). The invariant of -manifolds generalizes s…
Defines state sum models with defects in 3-manifolds.
The Serre-Swan theorem in differential geometry establishes an equivalence between the category of smooth vector bundles over a smooth compact manifold and the category of finitely generated projective modules over the unital ring of smooth functions. This theorem is here generalized to manifolds of bounded geometry. I…
A Hermitian TQFT from non-semisimple quantum sl(2) modules.
The paper constructs semistrict monoidal 2-categories from foam evaluations.
New algebraic framework for studying surfaces in 3-manifolds.
Develops graphical calculus for monoidal categories with twisted pivotal structures.
This is the second in a series of papers intended to set up a framework to study categories of modules in the context of non-commutative geometries. In \cite{mem} we introduced the basic DG category $\Pc_{\A^\bullet}$, the perfect category of $\A^\bullet$, which corresponded to the category of coherent sheaves on a com…
Turaev-Viro invariants match for certain surface bundles.
New homological results for bordered Floer algebras derived from hypertoric categories.
Given a bundle gerbe on a compact smooth manifold or, more generally, on a compact étale Lie groupoid , we show that the corresponding category of gerbe modules, if it is non-trivial, is equivalent to the category of finitely generated projective modules over an Azumaya algebra on . This result can be seen as an …
Propose a model-independent axiomatic framework for derived skein theory.
Classifies extended Abelian Chern-Simons theories using quadratic modules.
We construct certain tensor categories that are dominated by finitely many simple objects. Objects in these categories are modules over rings of algebra integers. We show how to obtain TQFTs defined over algebra integers from these categories.
We categorify the notion of an infinitesimal braiding in a linear strict symmetric monoidal category, leading to the notion of a (strict) infinitesimal 2-braiding in a linear symmetric strict monoidal 2-category. We describe the associated categorification of the 4-term relation, leading to six categorified relations. …
Mutation graph of support τ-tilting modules over skew-gentle algebras is connected.
We show that the category of affine bundles over a smooth manifold M is equivalent to the category of affine spaces modelled on projective finitely generated C^\infty(M)-modules. Using this equivalence of categories, we are able to give an alternate proof of the main result of [13], showing that the characterization of…
A family of TQFTs parametrised by G-crossed braided spherical fusion categories has been defined recently as a state sum model and as a Hamiltonian lattice model. Concrete calculations of the resulting manifold invariants are scarce because of the combinatorial complexity of triangulations, if nothing else. Handle deco…
We give a finite presentation for the braid twist group of a decorated surface. If the decorated surface arises from a triangulated marked surface without punctures, we obtain a finite presentation for the spherical twist group of the associated 3-Calabi-Yau triangulated category. The motivation/application is that the…
We discuss nonabelian bundle gerbes and their differential geometry using simplicial methods. Associated to any crossed module there is a simplicial group NC, the nerve of the 1-category defined by the crossed module and its geometric realization |NC|. Equivalence classes of principal bundles with structure group |NC| …
Frobenius extensions play a central role in the link homology theories based upon the sl(n) link variants, and each of these Frobenius extensions may be recast geometrically via a category of marked cobordisms in the manner of Bar-Natan. Here we explore a large family of such marked cobordism categories that are releva…
We give a categorical setting in which Penrose graphical calculus naturally extends to graphs drawn on the boundary of a handlebody. We use it to introduce invariants of 3-manifolds presented by Heegaard splittings. We recover Kuperberg invariants when the category comes from an involutory Hopf algebra and Turaev-Viro …
The classical Serre-Swan's theorem defines a bijective correspondence between vector bundles and finitely generated projective modules over the algebra of continuous functions on some compact Hausdorff topological space. We extend these results to obtain a correspondence between the category of representations of an et…
New (co)homology theory for symmetric quandles developed.
In this paper we aim to understand the category of stable-Yetter-Drinfeld modules over enveloping algebra of Lie algebras. To do so, we need to define such modules over Lie algebras. These two categories are shown to be isomorphic. A mixed complex is defined for a given Lie algebra and a stable-Yetter-Drinfeld module o…
Torsion objects of von Neumann categories describe the phenomen "spectrum near zero" discovered by S. Novikov and M. Shubin. In this paper we classify Hermitian forms on torsion objects of a finite von Neumann category. We prove that any such form can be represented as a discriminant form of a degenerate Hermitian form…
In a previous work arXiv:0903.4512, we have built an homotopical Turaev-Viro invariant and an HQFT from the universal graduation of a spherical category. In the present paper, we show that every graduation of a spherical category $\C$ defines an homotopical Turaev-Viro invariant $HTV_{\C}^{(G,p)}$ and an HQFT $…
We extend the notion of an ambidextrous trace on an ideal (developed by the first two authors) to the setting of a pivotal category. We show that under some conditions, these traces lead to invariants of colored spherical graphs (and so to modified 6j-symbols).
Three definitions of graded vector bundles are shown to be equivalent.