In our article [ arxiv:1511.05226 ], we studied the commutant 𝒞'⊂Bim(R) of a unitary fusion category 𝒞 , where R is a hyperfinite factor of type II_1 , II_∞ , or III_1 , and showed that it is a bicommutant category. In other recent work [ arxiv:1607.06041 , arxiv:2301.11114 ] we introduced the notion of a (unitary) anchored planar algebra in a (unitary) braided pivotal category 𝒟 , and showed that they classify (unitary) module tensor categories for 𝒟 equipped with a distinguished object. Here, we connect these two notions and show that finite depth objects of 𝒞' are classified by connected finite depth unitary anchored planar algebras in 𝒵(𝒞) . This extends the classification of finite depth objects of Bim(R) by connected finite depth unitary planar algebras.

Zonotopal algebras of vector arrangements are combinatorially-defined algebras with connections to approximation theory, introduced by Holtz and Ron and independently by Ardila and Postnikov. We show that the internal zonotopal algebra of a cographical vector arrangement is isomorphic to the cohomology ring of a certain configuration space introduced by Moseley, Proudfoot, and Young. We also study an integral form of this algebra, which in the cographical case is isomorphic to the integral cohomology ring. Our results rely on interpreting the internal zonotopal algebra of a totally unimodular arrangement as an orbit harmonics ring, that is, as the associated graded of the ring of functions on a finite set of lattice points.

The Lie algebra of vector fields on S^1 integrates to the Lie group of diffeomorphisms of S^1. It is well known since the work of Segal and Neretin that there is no Lie group whose Lie algebra is the complexification of vector fields on S^1. A substitute for that non-existent group is provided by the complex semigroup whose elements are annuli: genus zero Riemann surfaces with two boundary circles parametrized by S^1. The group Diff(S^1) sits at the boundary of that semigroup, and can be thought of as annuli which are completely thin, i.e. with empty interior. In this paper, we consider an enlargement of the semigroup of annuli, denoted Ann, where the annuli are allowed to be partially thin: their two boundary circles are allowed to touch each other along an arbitrary closed subset. We prove that every (partially thin) annulus A∈Ann is the time-ordered exponential of a path with values in the cone of inward pointing complexified vector fields on S^1, and use that fact to construct a central extension 0→ℂ×ℤ→Ãññ→Ann→ 0 that integrates the universal (Virasoro) central extension of the Lie algebra of vector fields on S^1. In later work, we will prove that every unitary positive energy representations of the Virasoro algebra integrates to a holomorphic representation of Ãññ by bounded operators on a Hilbert space.

In our previous article [arXiv:1607.06041], we established an equivalence between pointed pivotal module tensor categories and anchored planar algebras. This article introduces the notion of unitarity for both module tensor categories and anchored planar algebras, and establishes the unitary analog of the above equivalence. Our constructions use Baez's 2-Hilbert spaces (i.e., semisimple $C^*$-categories equipped with unitary traces), the unitary Yoneda embedding, and the notion of unitary adjunction for dagger functors between 2-Hilbert spaces.

We study $\mathrm{W}^*$-categories, and explain the ways in which complete $\mathrm{W}^*$-categories behave like categorified Hilbert spaces. Every $\mathrm{W}^*$-category $C$ admits a canonical categorified inner product $\langle\,\,,\,\rangle_{\mathrm{Hilb}}\,:\,\overline C\times C\,\to\, \mathrm{Hilb}$. Moreover, if $C$ and $D$ are complete $\mathrm{W}^*$-categories there is an antilinear equivalence $$\dagger:\mathrm{Func}(C,D) \leftrightarrow \mathrm{Func}(D,C)$$ characterised by $\langle c,F^\dagger(d)\rangle_{\mathrm{Hilb}} \simeq \langle F(c),d\rangle_{\mathrm{Hilb}}$, for $c\in C$ and $d \in D$.

We study W^*-categories, and explain the ways in which complete W^*-categories behave like categorified Hilbert spaces. Every W^*-category C admits a canonical categorified inner product ⟨ , ⟩_Hilb : C× C → Hilb. Moreover, if C and D are complete W^*-categories there is an antilinear equivalence †:Func(C,D) ↔Func(D,C) characterised by ⟨ c,F^†(d)⟩_Hilb≃⟨ F(c),d⟩_Hilb, for c∈ C and d ∈ D.

We generalize Jones' planar algebras by internalising the notion to a pivotal braided tensor category $\mathcal{C}$. To formulate the notion, the planar tangles are now equipped with additional `anchor lines' which connect the inner circles to the outer circle. We call the resulting notion an anchored planar algebra. If we restrict to the case when $\mathcal{C}$ is the category of vector spaces, then we recover the usual notion of a planar algebra. Building on our previous work on categorified traces, we prove that there is an equivalence of categories between anchored planar algebras in $\mathcal{C}$ and pivotal module tensor categories over $\mathcal{C}$ equipped with a chosen self-dual generator. Even in the case of usual planar algebras, the precise formulation of this theorem, as an equivalence of categories, has not appeared in the literature. Using our theorem, we describe many examples of anchored planar algebras.

A bicommutant category is a higher categorical analog of a von Neumann algebra. We study the bicommutant categories which arise as the commutant $\mathcal{C}'$ of a fully faithful representation $\mathcal{C}\to\operatorname{Bim}(R)$ of a unitary fusion category $\mathcal{C}$. Using results of Izumi, Popa, and Tomatsu about existence and uniqueness of representations of unitary (multi)fusion categories, we prove that if $\mathcal{C}$ and $\mathcal{D}$ are Morita equivalent unitary fusion categories, then their commutant categories $\mathcal{C}'$ and $\mathcal{D}'$ are equivalent as bicommutant categories. In particular, they are equivalent as tensor categories: \[ \Big(\,\,\mathcal{C}\,\,\simeq_{\text{Morita}}\,\,\mathcal{D}\,\,\Big) \qquad\Longrightarrow\qquad \Big(\,\,\mathcal{C}'\,\,\simeq_{\text{tensor}}\,\,\mathcal{D}'\,\,\Big). \] This categorifies the well-known result according to which the commutants (in some representations) of Morita equivalent finite dimensional $\rm C^*$-algebras are isomorphic von Neumann algebras, provided the representations are `big enough'. We also introduce a notion of positivity for bi-involutive tensor categories. For dagger categories, positivity is a property (the property of being a $\rm C^*$-category). But for bi-involutive tensor categories, positivity is extra structure. We show that unitary fusion categories and $\operatorname{Bim}(R)$ admit distinguished positive structures, and that fully faithful representations $\mathcal{C}\to\operatorname{Bim}(R)$ automatically respect these positive structures.

We prove that finite-index conformal nets are fully dualizable objects in the 3-category of conformal nets. Therefore, assuming the cobordism hypothesis applies, there exists a local framed topological field theory whose value on the point is any finite-index conformal net. Along the way, we prove a Peter–Weyl theorem for defects between conformal nets, namely that the annular sector of a finite defect is the sum of every sector tensored with its dual.

In this paper, we show that loop groups and the universal cover of $${{\rm Diff}_+(S^1)}$$ can be expressed as colimits of groups of loops/diffeomorphisms supported in subintervals of S1. Analogous results hold for based loop groups and for the based diffeomorphism group of S1. These results continue to hold for the corresponding centrally extended groups. We use the above results to construct a comparison functor from the representations of a loop group conformal net to the representations of the corresponding affine Lie algebra. We also establish an equivalence of categories between solitonic representations of the loop group conformal net, and locally normal representations of the based loop group.

Conformal nets provide a mathematical model for conformal field theory. We define a notion of defect between conformal nets, formalizing the idea of an interaction between two conformal field theories. We introduce an operation of fusion of defects, and prove that the fusion of two defects is again a defect, provided the fusion occurs over a conformal net of finite index. There is a notion of sector (or bimodule) between two defects, and operations of horizontal and vertical fusion of such sectors. Our most difficult technical result is that the horizontal fusion of the vacuum sectors of two defects is isomorphic to the vacuum sector of the fused defect. Equipped with this isomorphism, we construct the basic interchange isomorphism between the horizontal fusion of two vertical fusions and the vertical fusion of two horizontal fusions of sectors.

Conformal nets are a mathematical model for conformal field theory, and defects between conformal nets are a model for an interaction or phase transition between two conformal field theories. In the preceding paper of this series, we introduced a notion of composition, called fusion, between defects. We also described a notion of sectors between defects, modeling an interaction among or transformation between phase transitions, and defined fusion composition operations for sectors. In this paper we prove that altogether the collection of conformal nets, defects, sectors, and intertwiners, equipped with the fusion of defects and fusion of sectors, forms a symmetric monoidal 3-category. This 3-category encodes the algebraic structure of the possible interactions among conformal field theories.

We prove that conformal nets of finite index are an instance of the notion of a factorization algebra.This result is an ingredient in our proof that, for G = SU (n), the Drinfel'd center of the category of positive energy representations of the based loop group is equivalent to the category of positive energy representations of the free loop group.

We axiomatize the defining properties of chiral WZW models. We show that such models are in almost bijective correspondence with pairs (G, k), where G is a connected Lie group and k is an element of H-+(4) (BG, Z) is a degree four cohomology class subject to a certain positivity condition. We find a couple extra models which satisfy all the defining properties of chiral WZW models, but which don't come from pairs (G, k) as above. The simplest such model is the simple current extension of the affine VOA E-8 x E-8 at level (2, 2) by the group Z(2).

Conformal nets provide a mathematical formalism for conformal field theory. Associated to a conformal net with finite index, we give a construction of the `bundle of conformal blocks', a representation of the mapping class groupoid of closed topological surfaces into the category of finite-dimensional projective Hilbert spaces. We also construct infinite-dimensional spaces of conformal blocks for topological surfaces with smooth boundary. We prove that the conformal blocks satisfy a factorization formula for gluing surfaces along circles, and an analogous formula for gluing surfaces along intervals. We use this interval factorization property to give a new proof of the modularity of the category of representations of a conformal net.

We axiomatize the defining properties of chiral WZW models. We show that such models are in almost bijective correspondence with pairs ( G , k ) (G,k) , where G G is a connected Lie group and k ∈ H + 4 ( B G , Z ) k \in H^4_+(BG,\mathbb {Z}) is a degree four cohomology class subject to a certain positivity condition. We find a couple extra models which satisfy all the defining properties of chiral WZW models, but which don’t come from pairs ( G , k ) (G,k) as above. The simplest such model is the simple current extension of the affine VOA E 8 × E 8 E_8 \times E_8 at level ( 2 , 2 ) (2,2) by the group Z 2 \mathbb {Z}_2 .

In this note, we answer the questions "What does Chern-Simons theory assign to a point?" and "What kind of mathematical object does Chern-Simons theory assign to a point?". Our answer to the first question is representations of the based loop group. More precisely, we identify a certain class of projective unitary representations of the based loop group $\Omega G$ that we locally normal representations. We define the fusion product of such representations and we prove that, modulo certain conjectures, the Drinfel'd centre of that representation category of $\Omega G$ is equivalent to the category of positive energy representations of the free loop group $LG$. The above mentioned conjectures are known to hold when the gauge group is abelian or of type $A_1$. Our answer to the second question is bicommutant categories. The latter are higher categorical analogs of von Neumann algebras: they are tensor categories that are equivalent to their bicommutant inside $\mathrm{Bim}(R)$, the category of bimodules over a hyperfinite $\mathit{III}_1$ factor. We prove that, modulo certain conjectures, the category of locally normal representations of the based loop group is a bicommutant category. The relevant conjectures are known to hold when the gauge group is abelian or of type $A_n$. Our work builds on the formalism of coordinate free conformal nets, developed jointly with A. Bartels and C. Douglas.

Bicommutant categories are higher categorical analogs of von Neumann algebras that were recently introduced by the first author. In this article, we prove that every unitary fusion category gives an example of a bicommutant category. This theorem categorifies the well known result according to which a finite dimensional *-algebra that can be faithfully represented on a Hilbert space is in fact a von Neumann algebra.

Bicommutant categories are higher categorical analogs of von Neumann algebras that were recently introduced by the first author. In this article, we prove that every unitary fusion category gives an example of a bicommutant category. This theorem categorifies the well known result according to which a finite dimensional ∗-algebra that can be faithfully represented on a Hilbert space is in fact a von Neumann algebra.

Given a braided pivotal category $\mathcal C$ and a pivotal module tensor category $\mathcal M$, we define a functor $\mathrm{Tr}_{\mathcal C}:\mathcal M \to \mathcal C$, called the associated categorified trace. By a result of Bezrukavnikov, Finkelberg and Ostrik, the functor $\mathrm{Tr}_{\mathcal C}$ comes equipped with natural isomorphisms $\tau_{x,y}:\mathrm{Tr}_{\mathcal C}(x \otimes y) \to \mathrm{Tr}_{\mathcal C}(y \otimes x)$, which we call the traciators. This situation lends itself to a diagramatic calculus of `strings on cylinders', where the traciator corresponds to wrapping a string around the back of a cylinder. We show that $\mathrm{Tr}_{\mathcal C}$ in fact has a much richer graphical calculus in which the tubes are allowed to branch and braid. Given algebra objects $A$ and $B$, we prove that $\mathrm{Tr}_{\mathcal C}(A)$ and $\mathrm{Tr}_{\mathcal C}(A \otimes B)$ are again algebra objects. Moreover, provided certain mild assumptions are satisfied, $\mathrm{Tr}_{\mathcal C}(A)$ and $\mathrm{Tr}_{\mathcal C}(A \otimes B)$ are semisimple whenever $A$ and $B$ are semisimple.