arXiv subjects

Thomas A. Wasserman

Publications and source records attributed to Thomas A. Wasserman.

10 recordsLinked to original sources

arXiv2026

A three-functor formalism for commutative von Neumann algebras

A three-functor formalism is the half of a six-functor formalism that supports the projection and base change formulas. In this paper, we provide a three-functor formalism for commutative von Neumann algebras and their modules. Using the Gelfand-Naimark theorem, this gives rise to a three-functor formalism for measure spaces and measurable bundles of Hilbert spaces. We use this to prove Fell absorption for unitary representations of measure groupoids. The three-functor formalism for commutative von Neumann algebras takes values in W*-categories, and we discuss in what sense it is a unitary three-functor formalism.

math.OA↗

arXiv2026

Solid Duality for Profinite Groups

We classify profinite groups that have a Poincaré-like duality between their homology and cohomology. Our proofs work over every profinite coefficient ring, and for profinite as well as discrete coefficients. We do not only unify and generalise existing results, but also construct two novel examples of duality groups. We prove our results via the framework of condensed mathematics. This recently developed setting is a natural home for profinite objects, and our work is among the first to leverage this for the study of the (co)homology of profinite groups. Establishing our duality results involves an investigation of homological finiteness properties in condensed mathematics.

math.GR↗

arXiv2025

Exceptional embeddings of $N=2$ minimal models

Vafa and Warner observed that the Landau-Ginzburg model associated to the potential $E_6$ (resp. $E_8$) is a product of two other models, associated to the potentials $A_2$ and $ A_3$ (resp. $A_2 $ and $ A_4$). We translate this along the Landau-Ginzburg / Conformal Field Theory correspondence to a conjecture about the unitary minimal quotients $M_d$ of the $N=2$ superconformal algebra of central charge $c_d=3-\frac{6}{d}$: there should be a conformal embedding $M_{12}\hookrightarrow M_{3} \otimes M_4$ (resp. $M_{30}\hookrightarrow M_{3} \otimes M_5$) that exhibits the product as Ostrik's $E_6$ (resp. $E_8$) algebra in the $\mathrm{Rep}(su(2)_{10})$ (resp. $\mathrm{Rep}(su(2)_{28})$) factor of the NS-sector of $\mathrm{Rep}(M_{12})$ (resp. $\mathrm{Rep}(M_{30})$). We motivate, formulate, and prove this conjecture.

math.QA↗

arXiv2025

Cohen--Macaulay Complexes, Duality Groups, and the dualizing module of ${\rm{Out}}(F_N)$

We explain how Cohen--Macaulay classifying spaces are ubiquitous among discrete groups that satisfy Bieri--Eckmann duality, and compare Bieri--Eckmann duality to duality results for Cohen--Macaulay complexes. We use this comparison to give a description of the dualizing module of ${\rm{Out}}(F_N)$ in terms of the local cohomology cosheaf of the spine of Outer space.

math.GR↗

arXiv2025

Duality for Cohen--Macaulay Complexes through Combinatorial Sheaves

We prove a duality theorem for Cohen--Macaulay simplicial complexes. This is a generalisation of Poincaré Duality, framed in the language of combinatorial sheaves. Our treatment is self-contained and accessible for readers with a working knowledge of simplicial complexes and (co)homology. The main motivation is a link with Bieri-Eckmann duality for discrete groups, which is explored in a companion paper.

math.AT↗

arXiv2024

The Reduced Tensor Product of Braided Tensor Categories Containing a Symmetric Fusion Category

We detail a construction of a symmetric monoidal structure, called the reduced tensor product on the 2-category of braided tensor categories $\mathbf{BTC}(\mathcal{A})$ containing a fixed symmetric fusion subcategory $\mathcal{A}$. The construction only depends on the braiding and monoidal structure of the categories involved. The main tool in the construction is an enriching procedure that is shown to give an equivalence between $\mathbf{BTC}(\mathcal{A})$ and a 2-category of so-called Drinfeld centre crossed braided tensor categories. As an application of the reduced tensor product, we show that it gives a pairing between minimal modular extensions of braided tensor categories containing $\mathcal{A}$ as their transparent subcategory.

math.QA↗

arXiv2022

Landau-Ginzburg/Conformal Field Theory Correspondence for $x^d$ and Module Tensor Categories

The Landau-Ginzburg/Conformal Field Theory correspondence predicts tensor equivalences between categories of matrix factorisations of certain polynomials and categories associated to the $N=2$ supersymmetric conformal field theories. We realise this correspondence for $x^d$ for any $d$, where previous results were limited to odd $d$. Our proof uses the fact that both sides of the correspondence carry the structure of module tensor categories over the category of $\mathbb{Z}_d$-graded vector spaces equipped with a non-degenerate braiding. This allows us to describe the CFT side as generated by a single object, and use this to efficiently provide a functor realising the tensor equivalence.

math.QA↗

arXiv2021

The Symmetric Tensor Product on the Drinfeld Centre of a Symmetric Fusion Category

We define a symmetric tensor product on the Drinfeld centre of a symmetric fusion category, in addition to its usual tensor product. We examine what this tensor product looks like under Tannaka duality, identifying the symmetric fusion category with the representation category of a finite (super)-group. Under this identification, the Drinfeld centre is the category of equivariant vector bundles over the finite group (underlying the super-group, in the super case). In the non-super case, we show that the symmetric tensor product corresponds to the fibrewise tensor product of these vector bundles. In the super case, we define for each super-group structure on the finite group a super-version of the fibrewise tensor product. We show that the symmetric tensor product on the Drinfeld centre of the representation category of the resulting finite super-groups corresponds to this super-version of the fibrewise tensor product on the category of equivariant vector bundles over the finite group.

math.QA↗

arXiv2021

The Drinfeld Centre of a Symmetric Fusion Category is 2-Fold Monoidal

We show that the Drinfeld centre of a symmetric fusion category is a bilax 2-fold monoidal category. That is, it carries two monoidal structures, the convolution and symmetric tensor products, that are bilax monoidal functors with respect to each other. We additionally show that the braiding and symmetry for the convolution and symmetric tensor products are compatible with this bilax structure. We establish these properties without referring to Tannaka duality for the symmetric fusion category. This has the advantage that all constructions are done purely in terms of the fusion category structure, making the result easy to use in other contexts.

math.QA↗

arXiv2019

Drinfeld Centre-Crossed Braided Tensor Categories

We introduce, for a symmetric fusion category $\mathcal{A}$ with Drinfeld centre $\mathcal{Z}(\mathcal{A})$, the notion of $\mathcal{Z}(\mathcal{A})$-crossed braided tensor category. These are categories that are enriched over $\mathcal{Z}(\mathcal{A})$ equipped with a symmetric tensor product, while being braided monoidal with respect to the usual tensor product on $\mathcal{Z}(\mathcal{A})$. In the Tannakian case where $\mathcal{A}=\mathbf{Rep}(G)$ for a finite group $G$, the 2-category of $\mathcal{Z}(\mathcal{A})$-crossed braided categories is shown to be equivalent to the 2-category of $G$-crossed braided tensor categories. A similar result is established for the super-Tannakian case where $\mathcal{A}$ is the representation category of a finite super-group.

math.QA↗