Unitary anchored planar algebras

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.


Introduction
Planar algebras were introduced by Vaughan Jones in [Jon21].A planar algebra P is a collection of vector spaces P[n], called box-spaces, indexed by the nonnegative integers, along with a linear map Z(T ) : for every planar tangle T .These maps are required to be compatible with the operation of composition of tangles in the sense that Z(S • i T ) = Z(S)•(id ⊗ . ..⊗id ⊗Z(T )⊗id ⊗ . ..⊗id).See §4.1 for more details.
In [HPT23b], we internalised the notion of planar algebra to the context of a pivotal braided tensor category V, and called the resulting notion an anchored planar algebra.The box-spaces P[n] of an anchored planar algebra are now objects of our ambient category V, and we have morphisms for every anchored planar tangle.For example: The full definition is given in §4.2.
We then proceeded to establish an equivalence of categories Anchored planar algebras in V ∼ = Pointed pivotal module tensor categories over V between the category APA of anchored planar algebras in V, and the category ModTens * of pointed1 pivotal module tensor categories over V whose action functor admits a right adjoint.We refer the reader to §3.1 below for definitions of these notions.
In this paper we further specialise/generalise the notion of anchored planar algebra to the context when V is a unitary pivotal braided tensor category, and we prove an analog of the above result.This is motivated by enriched subfactor theory [JL17], bicommutant categories [Hen17b, HP17, HP23], topological orders [HBJP23], and higher unitary categories [CHPJP22,DHP22].
Let V be a braided unitary tensor category equipped with a unitary dual functor (the latter is the unitary analog of a pivotal structure [Yam04,Sel11,Pen20]; see §2.2 below for more details).A unitary module tensor category C is called pivotal if it is equipped with a unitary dual functor, compatibly with the action of V. Our main result is the unitary analog of Theorem [HPT23b, Thm.A]: Theorem A. There is an equivalence of categories2 Unitary anchored planar algebras in V ∼ = Unitary pointed pivotal module tensor3 categories over V, with chosen state .
Moreover, when V is ribbon, spherical unitary anchored planar algebras correspond under this equivalence to unitary module tensor categories whose chosen state is spherical.

Motivations for this article 1.Enriched subfactor theory
The development of planar algebras [Jon21] is intimately linked to subfactor theory.We expect a similar relation to hold between anchored planar algebras and enriched subfactor theory.In their paper [JP17], Corey Jones and the second author introduced a notion of W * -algebra object A internal to a unitary tensor category V.One could imagine formulating an analog of the notion of II 1 factor internal to V (perhaps just the condition that the neutral part of A, i.e.V(1 → A), is a II 1 factor) and, similarly, a notion of subfactor internal to V.
We conjecture that, in this context, the correct analog of the standard invariant is that of a unitary anchored planar algebra: Conjecture 1.1.The enriched standard invariant of a subfactor internal to V is a 2-shaded unitary anchored planar algebra in Z(V).Finite index, finite depth hyperfinite II 1 subfactors internal to V are classified by their enriched standard invariants4 .
As an example, in [JL17, Rem.6.1], Jaffe and Liu construct a subfactor from the inductive limit of parafermion algebras, and obtain the parafermion subfactor planar para algebra from its 'graded standard invariant.'Planar para algebras are anchored planar algebras internal to V = Vect(Z/N ) with a particular braiding (see [HPT23b, Examples 3.7 and 3.8]), and the parafermion planar para algebra corresponds to a Tambara-Yamagami module tensor category over V.
There is a classification of II 1 subfactors with index less than 4 [Jon83, GdlHJ89, Kaw95, BN91, Izu94] in terms of ADE Coxeter-Dynkin diagrams (where D odd and E 7 do not occur, and E 6 and E 8 occur twice).For subfactors enriched over super vector spaces, the non-simply laced Coxeter-Dynkin diagrams C even and F 4 also appear [ALW19, §6-7].

Internal structure of bicommutant categories
Bicommutant categories were introduced in [Hen17b], as higher categorical analogs of von Neumann algebras.The simplest example of a bicommutant category is Bim(M ), the tensor category of all bimodules over a von Neumann algebra M .Examples corresponding to unitary fusion categories were constructed in [HP17], and further studied in [HP23].Examples corresponding to conformal nets were constructed in [Hen17a].
When M = R is a hyperfinite II 1 , II ∞ , or III 1 factor, there is a well-known correspondence [Pop90, Pop95, Jon21, Tom21, BCE + 20] between conjugacy classes of finite depth R-R bimodules and isomorphism classes of unitary finite depth planar algebras5 : Finite depth R-R-bimodules conj.↔ Connected finite depth unitary planar algebras iso.
We conjecture that, under suitable assumptions, conjugacy clases of finite depth objects in a bicommutant category T are in bijective correspondence with finite depth unitary anchored planar algebras in Z(T ).
Conjecture 1.2.Let T be a bicommutant category whose Drinfeld center Z(T ) is fusion.Then, under suitable assumptions, there exists a natural bijective correspondence Finite depth objects of T conj.↔ Connected finite depth unitary anchored planar algebras in Z(T ) iso.
For bicommutant categories coming from fusion categories, Conjecture 1.2 was proven in our recent paper [HPT23a].

(2+1)D topological orders
Topological order is a phenomenon in (theoretical) condensed matter physics beyond Landau's symmetry breaking paradigm.In (2+1) dimensions, the low energy effective field theory of a topologically ordered phase of matter is a topological quantum field theory, and the low energy localised excitations form a unitary modular tensor category (UMTC).
Conjecture 1.3.Let X and Y be (2 + 1)D topological orders with the same anomaly, let M be a topological domain wall between them, and let m be a symmetrically self-dual point-like excitation that lives on M. Then the collection of objects

Y
(where the red dots represent m) in the unitary modular tensor category associated to X are the box-spaces of a unitary anchored planar algebra.
More generally, if U is any unitary 3-category (a notion yet to be defined) and c : v → w is a dualizable 1-morphism between dualizable objects, then V := End(1 v ) should be a unitary modular ribbon category and C := End(c) should be a unitary V-module multifusion category.Moreover, for m : c ⇒ c symmetrically self-dual, the objects Hom(1 c , m ⊗k ) ∈ V should form the box-spaces of a unitary anchored planar algebra.
Indeed, it should be the case that (2+1)D topological orders with the a given anomaly form a unitary 3-category whose 1-morphisms are topological domain walls and whose higher morphisms correspond to topological defects of higher codimension.More precisely, [HBJP23] posits that for a UMTC A whose Witt class represents an anomaly, the 3-category of A-enriched unitary fusion categories describes this putative 3-category of (2+1)D topological orders.

Unitary higher categories
Over the past decades, a theory of higher linear algebra has emerged from work of many authors, e.g., [BD95,Bae97,Lur09,HV19].The higher n-category nVect of n-vector spaces may be used as target for n-dimensional topological quantum field theories, where the fully dualizable objects give fully extended theories by the corbordism hypothesis.Until recently, definitions of nVect were bespoke, chosen to agree with well-known state-sum models.The recent breakthrough [GJF19] provides a uniform framework to construct (the fully dualizable part of) nVect via a formal inductive procedure starting from just the complex numbers.
Starting with C, a commutative algebra, we can deloop and complete to obtain the 1category Vect fd .Since Vect is a symmetric monoidal tensor category, we can deloop and complete again to obtain the 2-category Alg sep fd of finite dimensional separable algebras, bimodules, and intertwiners.Assigning to an algebra its category of modules gives an equivalence to 2Vect, the 2-category of finite semisimple categories, linear functors, and natural transformations.Since 2Vect is a symmetric monoidal 2-category, we may deloop and complete again to obtain the 3-category of (separable) multifusion categories [JF22], which is equivalent to the 3-category of semisimple 2-categories [DR18,Déc22].
For physical applications in topologically ordered phases of matter, it is important to have a version of the above construction that incorporates unitary structures at all levels.However, extending this construction to the unitary setting is quite delicate.Whereas dualizability is a property in the non-unitary setting, it is additional structure in the unitary setting [Yam04,Pen20].Thus nHilb should come with canonical involutions corresponding to duals at various levels.
Writing this article has allowed us to clarify these notions for 2Hilb.Here, a 2-Hilbert space [Bae97] is a linear category (enriched in finite dimensional vector spaces) equipped with Hilbert space structures on hom spaces satisfying As pointed out in [GMP + 23, Rem.3.6.1],2-Hilbert spaces are the natural setting for defining the unitary Yoneda embedding [JP20].In turn, we have a well-behaved notion of unitary adjunction for dagger functors, which we describe in §2.1 below.In Proposition 2.8 (see also Remark 2.9) we prove that unitary adjunction gives a canonical unitary dual functor on 2Hilb.We expect this to be a key ingredient in defining higher dagger idempotents, for the unitary analog of (1).

Outline
In §2, we provide the preliminary background for this article, together with some new contributions for unitary and involutive categories.We discuss the unitary Yoneda embedding and unitary adjunction in §2.1, and we review unitary dual functors and spherical states in §2.2.In §2.3, we use the graphical calculus for the 2-category Cat to transport involutive structures through adjunctions.
In §3, we review the notion of pivotal module tensor category over a braided pivotal category before introducing the unitary counterpart in §3.2.We briefly review the graphical calculus of strings on tubes for the categorified trace from [HPT16], and we study the adjoint of the unit map in §3.3.In the spherical setting, we prove unitarity of the traciator in §3.4.
In §4, we review the notions of planar algebra and anchored planar algebra before introducing the unitary counterparts in §4.3.We show that for a spherical unitary anchored planar algebra, the adjoint is compatible with the inside-out reflection of tangles in Proposition 4.8.Finally, in §5, we prove our main Theorem A.

Acknowledgements
The authors would like to thank Peter Huston and David Reutter for helpful conversations.David Penneys was supported by NSF DMS 1500387/1655912, 1654159, and 2154389.James Tener was supported by Australian Research Council Discovery Project DP200100067, as well as by the Max Planck Institute for Mathematics, Bonn, during the initial stages of this work.
Let Cat C be the 2-category of C-linear categories, linear and anti-linear functors, and natural transformations.We only allow natural transformations between functors if they are either both linear or both anti-linear.In this section, we shall often use the graphical calculus for Cat C , denoting linear categories by two-dimensional regions, functors by strands, and natural transformations by junctures.As in [CP22], we may identify a category A with the category of C-linear functors Vect → A. Given a functor F : A → B and objects a ∈ A and b ∈ B, this allows us to give a graphical representation e.g. for a morphism f : F (a) → b, Given an adjunction F ⊣ G between two functors F : A → B and G : B → A, recall that f : F (a) → b and g : A → G(b) are called mates if they are mapped to one another under the natural isomorphism Mates are represented in the graphical calculus as follows: where the cup and cap represent the unit and counit of the adjunction.The operations of taking the mate are natural with respect to pre-composition and postcomposition by another morphism: (M2) mate(g 2 • g 1 ) = mate(g 2 ) • F (g 1 ) for all g 1 : a 1 → a 2 and g 2 : a 2 → G(b).The above inner products satisfy

The unitary Yoneda lemma and unitary adjunctions
and thus equip A with the structure of a 2-Hilbert space in the sense of [Bae97].Conversely, one may recover the trace from the 2-Hilbert space structure by the formula Tr a (f ) := ⟨f, id a ⟩ a→a .The notion of semisimple C * category equipped with a unitary trace is thus equivalent to the notion of 2-Hilbert space.
The first condition in (3) implies that for all c ∈ A, the functor A(− → c) : A op → Hilb is a †-functor, and the second equality implies that the unitary Yoneda embedding is a †-functor (where Fun † (A op → Hilb) denotes the †-category of †-functors A op → Hilb).The above facts were first observed in [GMP + 23, Rem.3.61 and footnote], and we shall refer to them collectively as the unitary Yoneda lemma.Note that the essential image of (4) is the same as its unitary essential image (using polar decomposition in Fun † (A op → Hilb)), so a †-functor A op → Hilb is unitarily representable if and only if the underlying functor is representable.Given a †-functor F ∈ Fun † (A op → Hilb) in the essential image of (4), a representing object is given by where d a := Tr A a (id a ), Irr(A) is a basis6 of A, and we use the notation λH to denote the Hilbert space with the same underlying vector space as H and inner product λ⟨ • , • ⟩ H .The rescaling of the inner product of F (a) ensures the unitarity of the isomorphism for b ∈ Irr(A).
Remark 2.2.If A is finitely semisimple and Hilb is taken to mean finite dimensional Hilbert spaces, then (4) is a †-equivalence.
If F and G are instead antilinear †-functors, and (7) is antiunitary, then adjunction is called an antiunitary adjunction.
Remark 2.4.In the above definition, if we had merely required F and G to be (anti)linear functors, they would nevertheless automatically be †-functors.We prove that G is a dagger functor (the argument for F is similar).For any f ∈ B(b 1 → b 2 ), the following diagram commutes: Applying † to all the arrows, by (3), the following diagram also commutes: Similarly, whenever F ⊣ G are adjoint functors between linear categories, if ( 7) is (anti)linear, then F and G are automatically (anti)linear functors.
The above facts are reminiscent of the well-known fact that if H, K are Hilbert spaces and S : H → K and T : K → H are two functions such that ⟨Sη, ξ⟩ = ⟨η, T ξ⟩ for all η ∈ H and ξ ∈ K, then S and T are automatically bounded linear maps.
Lemma 2.5.Let (A, Tr A ) and (B, Tr B ) be semisimple C * categories with unitary traces, and let F : A → B be a linear †-functor that has a right adjoint.Then F also has a unitary right adjoint.The unitary right adjoint G is unique up to unique unitary natural isomorphism, and given by The same holds true for antilinear †-functors.
Proof.The unitary right adjoint G, if it exists, sends b ∈ B to the object representing the By the unitary Yoneda Lemma, such a representing object, if it exists, is unique up to unique unitary isomorphism.We get (8) by substituting (9) into (5).If F admits a right adjoint, the functors (9) are representable, hence unitarily representable, hence F admits a unitary right adjoint.Finally, antilinear †-functors are the same thing as linear †-functors A → B, so the same results hold true for antilinear †-functors.
Lemma 2.6.If coev and ev are the unit and counit of an (anti-)unitary adjunction F ⊣ † G, then ev † and coev † are the unit and counit of an (anti-)unitary adjunction G ⊣ † F .

Proof. The adjunction is given by
The first equivalence is anti-unitary as and similarly for the third one.
If we set a = G(b), the image of id G(b) under ( 10) is ev † .So ev † is the counit of the adjunction (10).Similarly, setting b = F (a), we see that coev † is the unit of the adjunction.
Lemma 2.7.Let (A, Tr A ) be a semisimple C * category with a unitary trace.Let a ∈ A be an object, viewed as linear †-functor a : Hilb → A, and let a * : A → Hilb be its unitary adjoint.Then where ev a is the mate of id a under the unitary adjunction Hilb(a Proof. Anticipating the notion of unitary dual functor (see §2.2 below), we have the following result: Proposition 2.8.Let (A, Tr A ) be a semisimple C * category equipped with a unitary trace, and let End † d (A) be its category of dualizable linear dagger endofunctors.Then the operation which sends a dagger functor F : A → A to its unitary adjoint defines a unitary dual functor on End † d (A).The same holds true for the category of linear or antilinear †-functors from A to itself.
Proof.Given a dualizable †-functor F : A → A, we write F * for its unitary adjoint (which is unique up to unique unitary isomorphism).We must show that the canonical isomorphism F * G * ⇒ (GF ) * is unitary, and that for all θ :

and
( exhibit F * G * as a unitary adjoint of GF , as By the uniqueness statement in Lemma 2.5, the isomorphism F * G * ⇒ (GF ) * is therefore unitary.
For the second statement, we show that θ * † a = θ † * a for all a ∈ A. For all f ∈ A(G * (a) → b) and g ∈ A(F * (a) → b), we have: a , g⟩, where the second and fifth equalities hold by Lemmas 2.6 and 2.7.By the non-degeneracy of the pairing, we conclude that θ † * a = θ * † a .Remark 2.9.More generally, unitary adjoints provide a canonical unitary dual functor on the C * 2-category of semisimple C * categories, dualizable dagger functors, and bounded natural transformations.

Unitary dual functors and spherical states
Let C be a unitary multitensor category (aka semisimple rigid tensor C* category).We recall the following definition from [Pen20]: Definition 2.10.A unitary dual functor on C is a choice of dual (c ∨ , ev c , coev c ) for each object c ∈ C (where ev c : c ∨ ⊗ c → 1, and coev c : 1 → c ⊗ c ∨ satisfy the zigzag identities), such that the corresponding functor ∨ : C → C mop is a dagger tensor functor: Remark 2.11.Unlike dual functors, unitary dual functors are not unique -see [Pen20].
A unitary dual functor ∨ on a unitary multitensor category C gives two End C (1 C )-valued traces tr L and tr R on the underlying C * category.Definition 2.12.A spherical state on a unitary multitensor category with unitary dual functor Given a spherical state ψ, we write Tr C := ψ • tr L .
(1) A unitary dual functor admits at most one spherical state.
(2) For each state ψ on End C (1 C ), there is a unique unitary dual functor with respect to which ψ is spherical.
We thus have a bijection Proof.Let U be the universal grading groupoid of C. By [Pen20, Thm.A], unitary dual functors ∨ on C are classified by groupoid homomorphisms π : for all homogeneous c ∈ 1 i ⊗ C ⊗ 1 j (homogeneous w.r.t. the U-grading) and f : c → c.Here, p 1 , . . ., p r ∈ End C (1 C ) are the projections onto the simple summands of 1 C , and gr c ∈ U is the U-grading of c.
If ∨ admits a spherical state ψ, then π factors through the 'matrix groupoid' M r with r objects and a unique isomorphism between any two objects.Indeed, applying the spherical state ψ to (12), we have λψ(p i ) = π gr(c) λψ(p j ), and thus Since i ψ(p i ) = 1, (13) completely determines ψ in terms of π.This proves part (1) of the lemma.Equation (13) also determines π in terms of ψ, so there is at most one unitary dual functor ∨ for which a given state ψ can be spherical.It remains to verify that ψ is spherical for the unitary dual functor associated to the homomrphism π defined in (13): This proves part (2) of the lemma.
Observe that (2) of Lemma 2.13 holds even when C is decomposable.Indeed, we can just restrict (and rescale) ψ to each indecomposable summand and then apply the lemma.
Definition 2.14.A unitary multitensor category with unitary dual functor (C, ∨) is called spherical if it admits a spherical state.
Warning 2.15.There exists an alternative possible definition of sphericality for a unitary multitensor category C, that we will not be using in this article: the unitary dual functor corresponding to π = 1 in (12).This is the balanced, or minimal unitary dual functor studied in [BDH14].
(where we have suppressed the tensorator of By [Pen20, Prop.3.40], when C and D are unitary multitensor categories equipped with unitary dual functors, pivotality of F is equivalent to the canonical isomorphisms χ c being unitary.
Recall that a unitary tensor category is a unitary multitensor category with simple unit.
Lemma 2.17.Let (C, ∨ C ) and (D, ∨ D ) be unitary multitensor categories equipped with unitary dual functors, with 1 C simple (i.e.C is a tensor category), and D non-zero.If D is spherical and F : C → D is a pivotal dagger tensor functor, then C is spherical.
Proof.Let c ∈ C be any object.By the pivotality of F and [Pen20, Lem.2.14], ) and tr C R (id c ) are scalars.So the two quantities in ( 14) are scalar multiples of id 1 D .Let ψ be the spherical state of D. The above two scalars are unchanged under applying ψ, so It follows that tr C L (id c ) = tr C R (id c ).We end this section by explaining how a unitary dual functor on a unitary multitensor category C induces a unitary dual functor on its unitary Drinfeld center Z † (C) (the full subcategory of the ordinary Drinfeld center Z(C) where all half-braidings are unitary): Lemma 2.18.Let ∨ = (∨, ev, coev) be a unitary dual functor on C. Then ∨ canonically induces a unitary dual functor on Z † (C).
The half-braiding σ X ∨ ,Y is unitary.The evaluation and coevaluation are morphisms in Z(C) hence in Z † (C).Finally, since the (co)evaluations for (X, σ X ) are the same as those of the underlying object X, the dual functor (X, σ X ) ∨ := (X ∨ , σ X ∨ ) is unitary.

Involutive functors
An involutive structure on a category A is an anti-linear functor • : A → A together with a coherence natural isomorphism φ : id A ⇒ • satisfying φ a = φ a for all a ∈ A. In the graphical calculus for Cat C , we denote the anti-linear functor • by a thick red strand, φ by a cap, and φ −1 by a cup.
Definition 2.19.A conjugate-linear morphism from a to b, denoted f : a ⇀ b, is a morphism a → b.
A real structure on an object a ∈ A is the data of a conjugate-linear morphism r : a ⇀ a such that r • r = φ a .
Definition 2.20.Let A and B be involutive categories.An involutive functor (F, χ) from A to B is a functor F : A → B equipped with a family of isomorphisms χ a : F (a) → F (a), for a ∈ A, satisfying In the graphical calculus for Cat C , we denote χ F by a crossing: The involutive and conjugate-natural axioms are denoted graphically by Composing the first identity with a red φ cap on the top left and a red φ −1 cup on the bottom right, we obtain the equivalent identity The involutivity of a natural transformation θ between involutive functors is represented graphically by The graphical proof of the following proposition is straightforward and left to the reader.
Proposition 2.21.Suppose (F, χ F ) : A → B is an involutive functor which admits a right adjoint G : endow G with the structure of an involutive functor.Moreover the unit and counit of the adjunction F ⊣ G satisfy Lemma 2.22.Let F : A → B be an involutive functor between involutive categories, and let G : B → A be its right adjoint (with involutive structure χ G as in Proposition 2.21).Then, for all f ∈ B(F (a) → b), we have Similarly, for all g ∈ A(a → G(b)), we have Proof.We only prove the first equality, as the other one is similar.In the graphical calculus, we have The result is an instance of the final statement in Proposition 2.21.
A bi-involutive category is an involutive category which is also a dagger category such that • is a dagger functor and φ is unitary.A bi-involutive functor F : A → B between bi-involutive categories is an involutive functor (F, χ) such that F is a dagger functor and χ is unitary.
Corollary 2.23.Let F : A → B be a dualizable functor between semisimple bi-involutive categories with unitary traces.Then its unitary adjoint G := F * is also bi-involutive, maning χ G (as defined in Proposition 2.21) is unitary.
Proof.Let C := A ⊕ B, and identify F, G with functors that equips C with the structure of a bi-involutive category.
Working in the category of dualizable linear or antiliner †-functors from C to itself (for which unitary adjoint defines a unitary dual functor by Proposition 2.8), χ G is the 2π-rotation of χ F .The result follows from the fact that χ F is unitary, and that the 2π-rotation of a unitary under a unitary dual functor is again unitary.

Involutive lax monoidal functors
An involutive monoidal category [Egg11] is a monoidal category C equipped with an involutive structure ( • , φ), and coherence natural isomorphisms ν a,b : a ⊗ b → b ⊗ a and real structure r : 1 → 1 (equivalently, r : 1 ⇀ 1) which satisfy By [Pen20, §3.5], a unitary multitensor category C has a canonical involutive structure • .This involutive structure has the notable feature that for any choice of unitary dual functor ∨, the conjugate x of an object x is canonically unitarily isomorphic to its unitary dual x ∨ .Remark 2.24.Note that the tensor product of a conjugate-linear morphism with an ordinary morphism is meaningless.However, if f : a ⇀ b and g : a ′ ⇀ b ′ are two conjugate-linear morphisms, then we may define f ⊗ g : Recall that a monoidal functor F : C → D between monoidal categories involves coherences ι F : 1 D → F (1 C ), and µ and Proposition 2.26.Let (F, µ F , ι F , χ F ) : C → D be an involutive monoidal functor between involutive monoidal categories.Then its right adjoint (G, µ, ι) : D → C is involutive lax monoidal, with ι G and µ G as above, and χ G as in Proposition 2.21.
Proof.We need to check the two conditions in Definition 2.25.To prove unitality, we will argue that Taking mates under the adjunction the resulting morphisms fit in the following pasting diagram: Going right and then down is the mate of ι G • r −1 C , and going down and then right is the mate of The triangles commute by the definition of ι G .The bottom left quadrilateral commutes using the dfinition of (χ G ) −1 .And the middle pentagon is the unitality condition for F in Definition 2.25.
To prove monoidality, we argue that Taking mates under the adjunction the resulting maps fit in the following pasting diagram: Going right and then down is the mate of , while going down and then right is the mate of χ −1 . The hexagon in the top left is the monoidality condition for F in Definition 2.25.
3 Unitary module tensor categories

Module tensor categories
Let V be a braided tensor category.A module tensor category C over V is a tensor category C equipped with an "action" of V in the following sense: Definition 3.1.A module tensor category over V is a tensor category C together with a braided tensor functor Φ Z : V → Z(C) from V to the Drinfeld center of C. If V and C are pivotal and if Φ Z is a pivotal functor, then we call C a pivotal module tensor category.
Let C be a pivotal module tensor category over V, and let us write φ a : a → a ∨∨ for the pivotal structure of C. A pointing of C is a choice of object c ∈ C such that c and Φ Z (V) generate C under the operations of tensor product, direct sum, and taking direct summands.We also require that our chosen object c comes equipped with a symmetric self-duality isomorphism r Construction 3.2.Let (C, Φ Z ) be a pivotal module tensor over V. Suppose that the functor Φ := Forget Z •Φ Z : V → C admits a right adjoint, denoted Tr V (where Forget Z : Z(C) → C is the forgetful functor).In [HPT16], we showed that has a canonical structure of a categorified trace.It comes equipped with the following structure.
x 2 ) of pivotal pointed module tensor categories is a pair (G, γ) consisting of: , where r i : x i → x ∨ i is the symmetric self-duality, and χ is as in Definition 2.16.• an action coherence monoidal natural isomorphism γ : Φ 2 ⇒ G • Φ 1 satisfying the following compatibility with the half-braidings: Given two 1-morphisms (G, γ G ), (H, γ H ) between pointed module tensor categories, by [HPT23b, Lem.3.6], there is at most one monoidal natural transformation κ : (G, γ G ) ⇒ (H, γ H ) satisfying the following compatibility with the action coherence morphisms: When such a κ exists, it is necessarily invertible.Hence the 2-category of pointed pivotal module tensor categories over V is 1-truncated, i.e., equivalent to a 1-category.

Unitary module tensor categories
For the remainder of this section, we assume that (V, ∨ V ) is a braided unitary tensor category with a fixed unitary dual functor, and let tr V be the associated right pivotal trace.
Definition 3.4.Let V be as above.A unitary module (multi)tensor category over V is a unitary (multi)tensor category C with a unitary dual functor ∨ C , a faithful state ψ C on End C (1 C ), and a braided pivotal dagger tensor functor Φ Z : V → Z † (C).
We call a unitary module multitensor category spherical if ψ C is a spherical state (Definition 2.12).See Warning 2.15.Observe that, by Lemma 2.17, the existence of a non-zero spherical unitary V-module tensor category implies that V is ribbon, i.e., that the unitary dual functor on V is spherical.
The choices of ∨ C and ψ C endow C with a unitary trace Moreover, (C, Tr C ) is a pivotal module C * category (aka unitary module category) [GMP + 23, §3.6.2] for the underlying unitary tensor category of V (without the braiding).So unitary module tensor categories are, in particular, unitary module categories.We warn the reader that Tr C is a trace in the sense of Definition 2.1, whereas Tr V is a categorified trace as described in Construction 3.2.
As before, a pointing of C is a choice of object c ∈ C such that c and Φ Z (V) generate C under the operations of tensor product, orthogonal direct sum, and taking orthogonal direct summands.We also require that our symmetric self-duality r c : c → c ∨ is unitary; this is also called a real structure for c.
x 2 ) of pointed unitary module tensor categories over V is a 1-morphism of underlying pointed pivotal module tensor categories such that • G is a †-tensor functor, • the action coherence monoidal natural isomorphism γ :

3.3
The map i † and a formula for coev † In §4.3 below, we will define unitarity for an anchored planar algebra in terms of a certain pairing on P[n] from the anchored planar algebra being equal to coev † P[n] .Fix a unitary V-module multitensor category (C, ∨ C , ψ C , Φ Z ) such that Φ : V → C admits a right unitary adjoint Tr V .In this section, we will study i † : Tr V (1 C ) → 1 V , and prove the important formula (18) for coev † Tr V (c) .We represent i † diagrammatically by In Lemma 3.11, we will show that if ψ C is spherical, then all isotopies are allowed for strings on the capped tube: , hence on all branching tubes of the form , . . .

Consider the finite dimensional abelian C
By unitary adjunction, we have an isomorphism hence the right hand side is also equipped with the structure of an abelian C * -algebra.The multiplication and * -structure on the right hand side of (16) are given by To see that the isomorphism (16) intertwines the two * -structures, i.e., The third equality follows from involutivity of Tr V (Definition 2.20), and the final equality uses the unitality axiom χ 1 C • Tr(r C ) • i = ī • r V from Definition 2.25, which holds by Proposition 2.26.
where the first equality holds by combining (15) with the definition of ⟨• , •⟩ (Definition 2.1), and the second equality is the unitarity of the adjunction (16), using that mate(i) = id 1 C .We will use the following lemma to get our equality of pairings in Proposition 3.9 below.Lemma 3.8.Two maps p, q : v ⊗ v → 1 V are equal if and only if for all u ∈ V and f, g Proof.The forward direction is trivial.Considering only simple u ∈ Irr(V), the equality (17) holds iff This is true for all u iff p = q.
where we've suppressed the inverse of χ Tr V c from the diagram.
Proof.Fix v ∈ V, f, g : v → Tr V (c), and consider the morphism Its mate under ( 16) is We claim that the above morphism is equal to Indeed, in the diagrammatic notation of [HPT16], we check: . Now, by Lemma 3.7, i † • (19) = ψ C ((20)).By (15) and the definition of ⟨• , •⟩, this is equal to The result follows by Lemma 3.8.Note that the first and third tubes above represent the same morphism regardless of whether ψ C is spherical or not, because the following two pictures are isotopic: = .
Proof.By the above discussion, it is enough to prove the first equality.Two maps f, g ) and wish to show that By Lemma 3.7, this is the same as showing that

Unitarity of the traciator
Suppose (C, ∨ C , ψ C , Φ Z ) is a unitary module multitensor category over V and Tr V : C → V is the right unitary adjoint of Φ : V → C. Recall that each object Φ(v) comes with a unitary half-braiding e Φ(v),x : Φ(v) ⊗ x → x ⊗ Φ(v).In this section, we explore an extra coherence between these unitary half-braidings and the involutive structure of the categorified trace.
Lemma 3.12.We have the following coherence between the half-braiding e Φ(v),• and χ Φ v : , so the right hand side in the statement of the lemma simplifies to The following lemma expresses the coherence from Lemma 3.12 in terms of traciators.Lemma 3.13.We have the following compatibility of the tracitator with the involutive structure of Tr V : Proof.We prove this equality after talking inverses on both sides, and after taking mates under the adjunction The mate of the inverse of the left hand side of (22 where the third equality holds by the last equation in Proposition 2.21.The mate of the inverse of the right hand side of ( 22) is mate(τ x,y )•Φ(Tr V (ν −1 x,y ))•Φ(χ −1 y⊗x ).Expanding, we get the map going right and then down in the pasting diagram below.To save space, we omit tensor symbols and subscripts.

Φ(Tr
It remains to prove that the map going down and then right in the above pasting diagram is equal to y⊗x) .Equivalently, we must show that By the coherences for ( • , ν), the above equality is equivalent to which is exactly [HPT16, Lem.4.14].
In Jones' planar algebras [Jon21], unitarity of the rotation plays a crucial role.Under the equivalence between anchored planar algebras and pivotal module tensor categories, the analog of rotation is given by the traciator.The next result says that traciators, and thus rotations in an anchored planar algebra, are unitary assuming sphericality.Proposition 3.14.If the state ψ C is spherical, then the traciator τ a,b is unitary.

Planar algebras
We present here an abridged introduction to planar tangles, the planar operad, and planar algebras, and refer the reader to [HPT23b, §2.1] for an extended treatment.
An planar tangle consists of a collection of (round) parametrized disc D 1 , . . ., D n inside some biger parametrised disc D 0 , along with a collection of non-intersecting paths in D 0 \ (D 1 ∪ . . .∪ D k ) called strings that start and end at the boundary of one of the circles, or are themselves closed loops.Here, a disc D i is called parametrised if D i = φ i (D) for some chosen affine linear maps φ i : D → D i from the standard unit disc.The points φ i (1) ∈ ∂D i are called anchor points, and the strands are required to not start or end on the anchor points.
Here is an example of a planar tangle, where we have marked the anchor points in red: There is a composition operation for planar tangles when the number of external string boundary points of T agrees with the number of internal string boundary points in the i-th disk of S. We shrink the tangle T , insert it into the i-th input disk of S using the map ϕ i , and match up the boundary points of the strings.For an explicit example, see [HPT23b, Ex. 2.2].
The collection of isotopy classes of planar tangles with the operation of tangle composition is called the planar operad.A planar algebra is an algebra for this operad.Unpacking, we have a vector space P[n] for each n ∈ N ≥0 , and each (isotopy class of) planar tangle T gives a linear map Z(T ) : Here, n i for 1 ≤ i ≤ k is the number of boundary points of strings on the i-th input disk of T , and n 0 is the number of boundary points of strings on the output disk of T .For future convenience, we abbreviate these properties by saying T has type (n 1 , . . ., n k ; n 0 ).Given planar tangles S, T of types (m 1 , . . ., m j ; m 0 ), (n 1 , . . ., n k ; n 0 ) respectively with n 0 = m i , composition of the linear maps Z(S), Z(T ) must be compatible with composition of planar tangles: We also require that the identity tangle must act as the identity linear map.Finally, letting T σ be the tangle obtained by renumbering the input disks of T by some permutation σ, we should have Z(T σ ) = Z(T ) • σ (where we also use σ to denote the symmetric braiding of the vector spaces P[n i ] corresponding to the permutation σ).
Remark 4.1.The notion of a planar algebra makes sense in any symmetric tensor category.

Anchored planar algebras
We present here an abridged introduction to the anchored planar operad and anchored planar algebras, and we refer the reader to [HPT23b, §2.2] for an extended treatment.An anchored planar tangle is a thing like this: It differs from a planar tangle in that each internal anchor point is connected to the external anchor point by a red anchor line which stays in D 0 \ {D 1 , . . ., D k }.These anchor lines are transparent to the strings, but they may not intersect each other.
Like before, we say an anchored planar tangle T has type (n 1 , . . ., n k ; n 0 ) if T has k input disks, the i-th input disk of T meets n i string boundary points for i ≤ 1 ≤ k, and the output disk of T meets n 0 string boundary points.
The anchored planar operad is the collection of isotopy classes of anchored planar tangles, with the operation of tangle composition.Here, given anchored planar tangles S, T of type (m 1 , . . ., m j ; m 0 ), (n 1 , . . ., n k ; n 0 ) respectively with n 0 = m i , the composition operation S • i T differs from the previous operation in that we must also give new anchor lines.We do so by replacing the i-th anchor line of T by k parallel lines, and composing each one of them with the i-th anchor line of S, as illustrated in the following picture: Recall from [HPT16, Appendix A.2] that a braided pivotal tensor category is also rigid balanced.8Hence our pivotal structure φ V induces a twist by (By [HPT16, Prop.A.4], a braided unitary tensor category (V, β, ∨) is ribbon if and only if the φ induced by ∨ is a spherical structure.)Definition 4.2.Let (V, β, φ V ) be a braided pivotal tensor category.An anchored planar algebra over V is an algebra in V over the anchored planar operad.Unpacking, we have a sequence P = (P[n]) n≥0 of objects of V, along with operations for every isotopy class of anchored planar tangle T of type (n 1 , . . ., n k ; n 0 ), subject to the following axioms: • (identity) the identity anchored tangle acts as the identity morphism • (composition) if S and T are anchored planar tangles of type (m 1 , . . ., m j ; m 0 ) and (n 1 , . . ., n k ; n 0 ), and if n 0 = m i , then • (anchor dependence) the following relations hold: (Here, a little number n next to a string to indicates n parallel strings.)We call P[n] the n th box object of the anchored planar algebra P.
Notation 4.3.In the sections below, we use the notation of an anchored planar tangle T inserted into a coupon to denote the map Z(T ) in V afforded by an anchored planar algebra.The strings in these diagrams are usually drawn horizontally for convenience, and we read them left to right.For example: where T is the tangle in the coupon.

Unitary anchored planar algebras
Let (V, ∨, β) be a unitary tensor category equipped with a chosen unitary dual functor ∨ and a unitary braiding β = {β u,v : u ⊗ v → v ⊗ u} u,v∈V satisfying the usual axioms.Using the canonical pivotal structure (11) induced by ∨, the formula (24) for the twist simplifies to Definition 4.5.A unitary anchored planar algebra in V is a triple (P, r, ψ P ) consisting of an anchored planar algebra P in V, a †-structure r, which is a family of real structures r n : P[n] ⇀ P[n] for each n ≥ 0, and a morphism ψ P : P[0] → 1 V called a faithful state satisfying (P1) For every anchored planar tangle T of type (n 1 , . . ., n k ; n 0 ), where T is the reflection of T , and (Suppressing the †-structure r, this axiom reads Z(T ) = Z(T ).) (P2) For every n ≥ 0, we have an equality of pairings If (V, ∨, β) is moreover ribbon (so that ∨ is spherical), we say that a unitary anchored planar algebra (P, r, ψ P ) is spherical if .
In this paper, we do not always assume our anchored planar algebras to be spherical.Definition 4.6.A morphism H : (P 1 , r 1 , ψ 1 ) → (P 2 , r 2 , ψ 2 ) of unitary anchored planar algebras is a morphism H of anchored planar algebras satisfying for all n ≥ 0, and When V = Hilb fd , the definition of unitary anchored planar algebra reduces to the definition of a unitary planar algebra (also known as a semisimple C * planar algebra) from [GMP + 23], but equipped with a faithful state.(The planar algebra of a bipartite graph [Jon00] is an example of a C * planar algebra that is naturally equipped wiht a faithful state.) The following lemma is straightfoward and left to the reader.
Lemma 4.7.If a unitary anchored planar algebra is spherical, then for every n ≥ 0 we have Proposition 4.8.Let (P, r, ψ P ) be a spherical unitary anchored planar algebra.Let A be an annular anchored planar tangle, and denote the inside-out reflection by A † .Then Z(A) † = Z(A † ).
Proof of Proposition 4.8.Suppose A has m input strings and n output strings.Since ψ P is spherical, the morphism Indeed, this can be checked directly using sphericality of ψ P by writing A as a composite of generating annular anchored planar tangles (see §5.1 below), and 'pulling them over' one at a time.By (26), the above equality simplifies (after a 90 • rotation of the string diagrams) to .
Precomposing with id ⊗ ev † P[m] , we get the desired equality

Extending the equivalence
We now prove Theorem A, extending the equivalence from [HPT23b, Thm.A] to an equivalence Unitary anchored planar algebras in V ∼ = Pointed unitary module multitensor categories over V .
Recall that, by definition (see §3.2), a unitary module multitensor category comes with a state ψ C and requires that the V-action Φ Z : V → Z † (C) is pivotal.

From unitary module tensor categories to unitary anchored planar algebras
Let (C, Φ Z , ∨ C , ψ C ) be a pointed unitary module multitensor category over V, with real generator (x, r x ) ∈ C, and Tr V : C → V the unitary right adjoint of Φ = Forget Z •Φ Z : V → C. We may then construct an ordinary anchored planar algebra, as in [HPT23b, Thm.5.1].This anchored planar algebra is completely determined by the following information: • x ⊗i x ⊗j

Adding the dagger structure
It remains to construct the †-structure r and the state ψ P on P, satisfying the conditions listed in Definition 4.5.
Definition 5.1.For n ≥ 0, we define r n : P[n] → P[n] to be the composite Proof.Going right and then down in the following commutative diagram is r n This is equal to the curved arrow φ Tr V (x ⊗n ) = φ P[n] .
Proposition 5.3.The pair (P, r) satisfies (P1): Proof.This essentially follows from the fact that Tr V is involutive lax monoidal by Proposition 2.26.It is enough to check (P1) on each of the following generating tangles: First, since u = u, by unitality, Since m i,j = m j,i , by monoidality, The third equality above uses the lax involutive property of Tr V , and the fourth equality uses naturality of µ.Since a i = a n−i (when Z(a i ) : Tr V (x ⊗n+2 ) → Tr V (x ⊗n )), we see The proof for the a † i is similar and omitted.(It also follows formally from the proof for a i as Tr V is a dagger functor and χ Tr V , r, ν, and r x are all unitary.)Finally, for the tangles t i,j , we have t i,j = t −1 i,j , and by Lemma 3.13, Definition 5.4.We define a state ψ P := i † : P[0] → 1 V as in §3.3.
Proof.If ψ C is spherical, then ) be the unitary anchored algebra constructed in the previous subsection for i = 1, 2. In [HPT23b, §5.3], we obtained a map of ordinary anchored planar algebras Λ(G) : P 1 → P 2 as follows.

From unitary anchored planar algebras to unitary module tensor categories
Given a unitary anchored planar algebra (P, r, ψ P ) in V, we begin by constructing an ordinary pivotal module tensor category (C, Φ Z ) as in [HPT23b,§6].First, we construct a full subcategory C 0 , and we obtain C by taking the Cauchy completion.
• Objects in C 0 are formal symbols "Φ(v) ⊗ x ⊗n " for v ∈ V and n ≥ 0.
Morphisms are represented graphically by f .
• Composition is given by • f .
Moreover, the identity map witnesses the adjunction Φ ⊣ Tr V .
• Tensor product is given by ⊗ g where we use the braiding axiom of an anchored planar algebra in the final equality.Finally, the associators and unitors are visibly unitary.
It remains to show that Ψ lifts to a unitary natural isomorphism upstairs.To do so, it is enough (in view of the fact that Ψ is an isomorphism) to show that f → Ψ(f ) is norm-preserving, i.e., f V(u→v⊗P[i+j]) = Ψ(f ) C(Φ(u)⊗x ⊗i →Φ(v)⊗x ⊗j ) .
By replacing f by ev v •(1 v ⊗ f ) (i.e.folding the v strand to the left), we may assume without loss of generality that v = 1 V .Similarly, by replacing f ∈ C ′ "Φ(u) ⊗ x ⊗i " → "Φ(v) ⊗ x ⊗j " by the corresponding morphism in C ′ "Φ(u) ⊗ x ⊗0 " → "Φ(v) ⊗ x ⊗i+j " , we may assume without loss of generality that i = 0. We may now proceed as in the proof of Proposition 3.9 to show: as desired.
In this section, we work with semisimple C * categories.(Note that every Cauchy complete C * category with finite dimensional hom spaces is semisimple [GMP + 23, §3.1.1].)Definition 2.1.A unitary trace on a semisimple C * category A is a collection of linear maps Tr a : A(a → a) → C, for all a ∈ A, satisfying • Tr a (f • g) = Tr b (g • f ) for all f : a → b and g : b → a, and • The sesquilinear forms ⟨f, g⟩ a→b := Tr a (g † • f ) are positive definite.

Definition 2. 3 .
Let (A, Tr A ) and (B, Tr B ) be semisimple C * categories equipped with unitary traces.A unitary adjunction, denoted F ⊣ † G, consists of linear †-functors F : A → B and G : B → A and a family of unitary and natural isomorphisms ψ a,b : B(F (a) → b) Definition 2.16.Suppose (C, φ C ) and (D, φ D ) are pivotal categories.A monoidal functor F : C → D is called pivotal if the canonical isomorphism F a,b : F (a) ⊗ F (b) → F (a ⊗ b).Definition 2.25.Let C and D be involutive monoidal categories.An involutive (lax) monoidal functor (F, µ, ι, χ) : C → D is an involutive functor (F, χ) : C → D such that (F, µ, ι) : C → D is (lax) monoidal, and such that the following additional conditions hold:

Remark 3. 5 .:
Suppose (C, ∨ C , ψ C , Φ Z ) is a unitary module multitensor category over V.If Φ admits a right adjoint, then by Lemma 2.5, Φ admits a right unitary adjoint Tr V which is automatically bi-involutive lax monoidal by Proposition 2.26 and Corollary 2.23.Thus Tr V comes equipped with canonical unitary isomorphisms {χ Tr V c Tr V (c) → Tr V (c)} c∈C satisfying the axioms in Definitions 2.20 and 2.25.

Remark 3. 10 .
The problem of showing the pairing on the right hand side in the statement of Proposition 3.9 is non-degenerate was left open in [HPT16, Rem.5.4].As it is equal to coev † Tr V (c) (up to suppressing χ), this open question is now resolved in the unitary setting.Lemma 3.11.If ψ C is spherical, then the following maps Tr V (c ⊗ c) → 1 V are equal: = = .

,
Definition 5.8.We define a dagger structure on C 0 by to verify that f * * = f for all morphisms f .To check the remainder of involutivity, we see that * on the composite"Φ(u) ⊗ x ⊗k " f − → "Φ(v) ⊗ x ⊗n "which is exactly the composite of f * and g * .We now observe that and only if f = 0. Proposition 5.10 ((C1)).The dagger structure (29) on C 0 is C * .
positive linear functional, i.e., ϕ(M * M ) ≥ 0 with equality if and only if M = 0. Indeed, ϕ(M * M ) is a sum of four terms, each being positive by Lemma 5.9, and zero iff each term is individually zero.Remark 5.11.Observe that the unitary Cauchy completion of C 0 is the same as the ordinary Cauchy completion of C 0 [GMP + 23, Rem.3.6].In light of Proposition 5.10, C is a unitary multitensor category.