Skein-Theoretic Methods for Unitary Fusion Categories
SSKEIN-THEORETIC METHODS FOR UNITARY FUSIONCATEGORIES
ANUP POUDEL § AND SACHIN J. VALERA † Abstract.
Unitary fusion categories (UFCs) have gained increased attention due toemerging connections with quantum physics. We discuss how UFCs can be understoodas fusion categories equipped with a “positive dagger structure” and apply this in agraphical context. Given a fusion rule q ⊗ q ∼ = ⊕ (cid:76) ki =1 x i in a UFC C , we extractinformation using skein-theoretic methods and a rotation operator. For instance, weclassify all associated framed link invariants when k = 1 , and C is ribbon. In particular,we also consider the instances where q is antisymmetrically self-dual. Some of this workis reformulated from the perspective of braid representations factoring through the Heckeand Temperley-Lieb algebras. Our main results follow from considering the action ofthe rotation operator on a “canonical basis”. Assuming self-duality of the summands x i ,some general observations are made e.g. the real-symmetricity of the F -matrix F qqqq .We then find explicit formulae for F qqqq when k = 2 and C is ribbon, and see that thespectrum of the rotation operator distinguishes between the Kauffman and Dubrovnikpolynomials. Finally, we apply some of our results in a physical setting (where C is aunitary modular category) and provide some worked examples: quantum entanglementis discussed using the graphical calculus, framed links are interpreted as Wilson loops,and a theorem is given on the duality of topological charges. Introduction
Fusion categories have played an important role in understanding structures arisingfrom quantum physics, and lie at the heart of quantum algebra and quantum topology.Some fusion categories can be extended to ribbon fusion categories (RFCs): these gad-gets are rich in structure, and carry a lot of information. Since ribbon categories areendowed with the topological properties of ribbon graphs, they naturally lend themselvesto investigation from a skein-theoretic perspective. For instance, it is known that one canfashion link (in fact, -manifold) invariants from RFCs: seminal work in this directionwas carried out by Reshetikhin and Turaev [1], followed by Kuperberg who used a skein-theoretic method to obtain new link invariants associated to quantum groups comingfrom Lie algebras of type A , B , C and G [2, 3]. In a similar vein, an important classof RFCs known as Temperley-Lieb-Jones categories can be understood using Kauffmanand Lins’ planar algebra of Jones-Wenzl idempotents at roots of unity [4, 5].Understanding unitary fusion categories (i.e. fusion categories with a positive daggerstructure) is crucial to developing an algebraic framework for describing topological phasesof matter (TPMs). Indeed, unitary modular tensor categories (MTCs) have proved to beuseful in the program for classifying TPMs and (2 + 1) -dimensional topological quantumfield theories (TQFTs) [6, 7, 8]. The condensed matter systems hosting these TPMs maybe repurposed as hardware for realising fault-tolerant quantum computation [9, 10, 11].The connection between link invariants and TQFTs was first observed by Witten when hegave an interpretation of the Jones polynomial in the context of Chern-Simons QFTs [12]. § Department of Mathematics, The University of Iowa, Iowa City, USA . † Selmer Center, Department of Informatics, University of Bergen, Norway . a r X i v : . [ m a t h . QA ] A ug ANUP POUDEL AND SACHIN J. VALERA
Although the classification of fusion categories is beyond our current capabilities,weaker variants of this problem can be studied by structural embellishment (e.g. impos-ing pivotality, braiding, (pre)modularity); but even with these modifications, the problemremains out of reach. It has been shown that there are finitely many braided fusion cate-gories of any given rank [13], whence there are finitely many commutative fusion algebrasthat admit categorification. The categorifications admitted by a (commutative) fusionalgebra can be explicitly calculated by solving the pentagon (and hexagon) equations:doing so recovers all of the information contained in the categories. However, solvingthese equations quickly becomes intractable as the rank grows. This motivates the ideaof determining additional general relations between unknowns, in an attempt to reducethe size of the parameter space. In this spirit, much of our exposition revolves aroundstarting with a fusion rule of the form(1.1) q ⊗ q ∼ = ⊕ k (cid:77) i =1 x i and applying skein-theoretic methods to deduce some properties of the underlying cate-gory C and the associated quantum invariants. Our work is inspired by that of Morrison,Peters and Snyder, who addressed this problem in [14, Theorems 3.1 & 3.2]: using arotation operator on End( q ⊗ ) for C ribbon and q symmetrically self-dual, they discussthe link invariants coming from q invertible and (1.1) for k = 1 , . In each case, they alsogive some relations between the eigenvalues of the R -matrix R qq .We systematically recover and extend the results of [14, Theorems 3.1 & 3.2] in Section 3.Our main contributions are contained in Section 5, where we uncover a relationshipbetween the rotation operator and the F -matrix F qqqq , thereby allowing us to deducesome general properties of said matrix, as well as some more case-specific results. Wealso hope that Sections 2 and 6 will be of use to mathematicians and physicists alike: theformer section summarises several mathematical concepts that are commonplace in thestudy of quantum symmetries, while the latter contains explicit calculations (particularlyas a means for providing a concrete introduction to some physical notions) and appliessome of the ideas presented in our main work.1.1. Outline of the paper.
In Section 2, we detail the relevant mathematical background. The concept of a“positive dagger structure” is introduced by construction of a Hermitian form on theHom-spaces of C -linear dagger fusion categories (Section 2.4): this allows us to fix acanonical orthonormal basis of “jumping jacks” (Section 2.6) that features throughoutour exposition. Proposition 2.11 describes a basic property of quantum dimensions inunitary RFCs. In Section 2.11, we define some conventions that are followed in the maindiscourse. The rotation operator (one of the tools most central to this paper) is intro-duced in Section 2.12, and a supplementary discussion is provided in Appendix A.In Section 3, we consider the action of the rotation operator on crossings in End( q ⊗ ) so as to ascertain the link invariants associated to the fusion rule (1.1) for k = 1 , . TheJones, Kauffman and Dubrovnik polynomials are recovered, and we find skein relationsfor three additional framed invariants coming from the antisymmetrically self-dual cases.In Section 3.4, we give a Lickorish-type formula [25] relating one of these invariants toKauffman’s bracket polynomial, and provide some insights into quantum invariants com-ing from antisymmetrically self-dual objects. KEIN-THEORETIC METHODS FOR UNITARY FUSION CATEGORIES 3
In Section 4, the narrative of the previous section is reframed in terms of braid grouprepresentations. In particular, observing that braid representations associated to fusionrules of the form q ⊗ ∼ = ⊕ y factor through the Iwahori-Hecke and Temperley-Lieb al-gebras, we find a skein relation for the framed HOMFLY-PT polynomial (from which werecover the quantum invariants associated to (1.1) for k = 1 ).In Section 5, we apply the rotation operator to the canonical basis of jumping jackson End( q ⊗ ) for a unitary spherical fusion category C . In doing so, we require that thesummands in (1.1) are self-dual (Remark 5.2). Theorem 5.4 determines the componentsof a “bone” morphism (a rotated jumping jack) in the canonical basis: we use this toprove some “bubble-popping” identities (Corollary 5.6) and to make some general obser-vations, a highlight of which is the real-symmetricity of F qqqq (Corollary 5.7). As a simpleapplication, we deduce the form of F qqqq when k = 1 in (1.1).We proceed to apply the results of Section 3 in order to derive explicit formulae for F qqqq when C is also ribbon and k = 2 (Theorem 5.9), and deduce that q cannot be antisymmet-rically self-dual in this instance (Corollary 5.10). It is also observed that the spectrumof the rotation operator distinguishes between the Kauffman and Dubrovnik invariants.In Section 5.3, we investigate some properties of bases for End( q ⊗ ) whose elements arepermuted (up to a sign) under the action of the rotation operator. We apply our resultsto construct such bases when k = 2 ; the diagonalisation of the rotation operator followsas an immediate consequence.In Section 6, we explore selected aspects of unitary MTCs in a physical context: here,isomorphism classes of simple objects label superselection sectors in the Hilbert space ofa (2 + 1) -dimensional condensed matter system. These sectors correspond to the topo-logical charges of quasiparticles called anyons.We mathematically clarify some physical jargon, and include a glossary of common termsin Table 1. In Section 6.1, quantum entanglement is discussed, with an emphasis on thegraphical calculus. We illustrate how “tangling” can sometimes induce entanglement, andgive some concrete examples using Fibonacci and Ising MTCs (e.g. the initialisation of aBell state via the Ising model in Example 6.4). In Section 6.2, framed, oriented links areinterpreted as Wilson loops in (2 + 1) -TQFTs: we outline how their amplitudes can beevaluated using MTCs (Procedure 6.6) and provide some examples (in which the ampli-tudes are also calculated using skein-theoretic methods). In Section 6.3, we formulate atheorem on the duality of topological charges (Theorem 6.10).In Section 7, we review the contents of our work with an eye to future extensions.1.2. Acknowledgements.
Both authors would like to thank Corey Jones and DavePenneys for organising the 2019 OSU Quantum Symmetries Summer Reseach Program(with grant support from David Penneys’ NSF grant DMS 1654159) where they met.Sachin Valera also wishes to thank Daniel R. Copeland for helpful discussions, and foracquainting him with the rotation operator and the results of [14] (at aforementionedresearch program) without which this work would not have been possible.
ANUP POUDEL AND SACHIN J. VALERA Preliminaries
We provide an overview of various concepts that are used throughout this work. Forfurther details on some of these topics, we refer the reader to [15, 16, 17, 18, 19].2.1.
Tensor categories.
Recall that a tensor category C is a k -linear, rigid monoidalcategory with End( ) ∼ = k (where denotes the unit object). We henceforth let k = C .By a simple object X ∈ C , we mean an object X such that every nonzero f ∈ End( X ) is an isomorphism. For any object X ∈ C , its left and right dual objects are respectivelydenoted by X ∗ and ∗ X . Every object X ∈ C comes with the (co) ev X and (co) ev (cid:48) X morphisms, which are the left and right (co)evaluations respectively. ev X : X ∗ ⊗ X → coev X : → X ⊗ X ∗ ev (cid:48) X : X ⊗ ∗ X → coev (cid:48) X : → ∗ X ⊗ X Dual objects are unique up to unique isomorphism [15, Proposition 2.10.5]. We henceforthidentify left and right duals, and use X ∗ to denote the dual of X .2.2. Pivotality, sphericality and quantum trace. A pivotal tensor category C is atensor category with a collection of isomorphisms (called a pivotal structure ) a X : X ∼ −→ X ∗∗ natural in X and satisfying a X ⊗ Y = a X ⊗ a Y for all objects X, Y ∈ C . For any X ∈ C and any morphism f ∈ End( X ) , the left and right quantum traces of f are defined as(2.1a) (cid:102) T r l ( f ) := ev X ◦ (id X ∗ ⊗ f ) ◦ coev (cid:48) X ∈ End( ) (2.1b) (cid:102) T r r ( f ) := ev (cid:48) X ◦ ( f ⊗ id X ∗ ) ◦ coev X ∈ End( ) If the left and right quantum traces coincide, then C is called a spherical tensor category .Hence, for a spherical tensor category, we can unambiguously define the quantum trace for any object X ∈ C and any morphism f ∈ End( X ) . That is,(2.2) (cid:102) T r ( f ) := (cid:102) T r l ( f ) = (cid:102) T r r ( f ) and the quantum dimension d X of an object X is d X := (cid:102) T r (id X ) (2.3)where we see that d X = d X ∗ .2.3. Fusion categories and trivalent vertices. A fusion category C is a semisimpletensor category with only finitely many simple objects up to isomorphism. Remark 2.1 (Skeleton of C ) . Let
Irr( C ) denote the set of representatives of isomorphismclasses of simple objects in C . Let X i ∈ Irr( C ) , where i ∈ I for some index set I ⊆ Z ≥ and X := . We also let i ∗ denote j ∈ I such that X j = X ∗ i . The cardinality of Irr( C ) iscalled the rank of C . When we restrict to working with objects in Irr( C ) , it is understoodthat we are working in the skeleton of C : this is the full subcategory of C on the subsetof objects Irr( C ) . Indeed, the skeleton is unique (up to isomorphism) and is equivalent to C . A category is called skeletal if it contains one object in each isomorphism class. Seealso Remarks 2.8 and 2.9.The so-called fusion rules for C are encoded by the fusion coefficients N ijk ∈ Z ≥ where X i ⊗ X j = (cid:77) k ∈ I N ijk X k , i, j ∈ I. (2.4)We also have (where δ ij denotes the Kronecker delta) N i j = N ij = δ ij (2.5) KEIN-THEORETIC METHODS FOR UNITARY FUSION CATEGORIES 5
There is a graphical calculus associated with morphisms for any tensor category C . Weadopt the pessimistic convention i.e. our diagrams are viewed as morphisms going from top-to-bottom . Any edge is oriented and labelled by an object X ∈ C ; and for ∈ C , theedge is either invisible or emphasised by a dotted line. Diagrams representing morphismsin the skeleton of a fusion category C have edges labelled by objects X i ∈ Irr( C ) . Trivalentvertices represent inclusion or projection morphisms e.g. we have the projections(2.6) span C (cid:40) X j X i X k µ (cid:41) N ijk µ =1 = Hom( X i ⊗ X j , X k ) where the left-hand side constitutes a basis for Hom( X i ⊗ X j , X k ) . Similarly, flippingthe trivalent vertices upside-down in (2.6), we obtain a basis of inclusion morphisms for Hom( X k , X i ⊗ X j ) . When k = 0 , the trivalent vertices are called cups and caps . It isstraightforward to check that End( X i ) ∼ = C , whence diagrammatically, we have = λf X i X i X i (2.7)where f ∈ End( X i ) and λ ∈ C .2.4. Dagger structure, inner product and unitarity.
Let C be a fusion category.Then C is called a dagger fusion category if it is equipped with an involutive, contravariantfunctor † : C → C such that it acts as the identity on objects, and for any morphisms f : X → Y , we have † ( f ) = f † : Y → X where f † is called the adjoint of f . Furthermore,for morphisms f, g ∈ C and scalars λ , λ ∈ C , the † -functor satisfies (id X ) † = id X (2.8a) ( g ◦ f ) † = f † ◦ g † (2.8b) ( f ⊗ g ) † = f † ⊗ g † (2.8c) ( λ · f + λ · g ) † = λ ∗ · f † + λ ∗ · g † (2.8d)where for (2.8b) we have f : X → Y and g : Y → Z for some objects X, Y, Z ∈ C . Notethat λ ∗ denotes the complex conjugate of λ ∈ C . Considering the skeleton of C , we have Hom( X i ⊗ X j , X k ) †∼ −→ Hom( X k , X i ⊗ X j ) (2.9)where the adjoint of a trivalent vertex is given by X j X i X k µ † (cid:55)−→ X i X j X k µ (2.10) Remark 2.2. (Multiplicity-free)
Unless stated otherwise, we henceforth assume ourfusion categories to be multiplicity-free i.e. N ijk ∈ { , } for all i, j, k . This obviates theneed to index trivalent vertices (e.g. µ can be omitted in (2.6) and (2.10) in this instance). ANUP POUDEL AND SACHIN J. VALERA
Stacking the two trivalent vertices, observe that the right-hand side of (2.11) follows bySchur’s lemma. = λ lijk · δ lk X k where λ lijk ∈ C (2.11) Proposition 2.3. λ kijk is real. Proof.
Taking the adjoint of (2.11) for l = k , the result is immediate. (cid:3) We can define a sesquilinear form (cid:104) g, f (cid:105) = tr( f g † ) (2.12)where f, g ∈ Hom(
Y, X ) and f g † ∈ End( X ) for any X, Y ∈ C . Further note that (cid:104) f, g (cid:105) = (cid:104) g, f (cid:105) ∗ (2.13)whence, (2.12) actually defines a Hermitian form. Remark 2.4 ( Basis for Hom-space ) . We construct a basis for any
Hom(
X, Y ) in theskeleton of C (where at least one of X or Y is not simple). We can write Hom(
X, Y ) asa direct sum of Hom -spaces of the form
Hom( ⊗ mk =1 X i k , ⊗ nl =1 X j l ) where X i k , X j l ∈ Irr( C ) .It thus suffices to consider spaces of the form Hom( ⊗ mk =1 X i k , ⊗ nl =1 X j l ) = (cid:77) b ∈ Irr( C ) Hom( ⊗ mk =1 X i k , b ) ⊗ Hom( b, ⊗ nl =1 X j l ) (2.14)Writing V XY := Hom( X, Y ) , further note that V X i ··· X im b = (cid:77) e ,...,e m − ∈ Irr( C ) V X i X i e ⊗ V e X i e ⊗ · · · ⊗ V e m − X im − e m − ⊗ V e m − X im b (2.15a) V bX j ··· X jn = (cid:77) f ,...,f n − ∈ Irr( C ) V bf n − X jn ⊗ V f n − f n − X jn − ⊗ · · · ⊗ V f f X j ⊗ V f X j X j (2.15b)The decompositions in (2.15a) and (2.15b) correspond to a choice of fusion basis on therespective Hom-spaces. Figure 1.
By a “fusion basis”, we mean a parenthesisation of (cid:78) k X i k .Diagrammatically, a fusion basis corresponds to a full rooted binary treeon a space of the form in (2.15a) or (2.15b). The above trees illustrate thefusion bases for a space of the form V abcde . The number of distinct fusionbases for an n -fold product is given by the ( n − th Catalan number. Let η : Hom( Y, X ) ∼ → Hom(
Y X ∗ , ) . We may equivalently write (cid:104) g, f (cid:105) = η ( f )( η ( g )) † ∈ End( ) ∼ = C . KEIN-THEORETIC METHODS FOR UNITARY FUSION CATEGORIES 7
Using the basis from (2.6), and fixing fusion bases as in (2.15a) and (2.15b), we obtainthe following basis : Hom( ⊗ mk =1 X i k , ⊗ nl =1 X j l ) = span C b,e , ··· ,e m − ,f , ··· ,f n − (2.16)Let { e i } i correspond to the basis in (2.16). Then we can write g = (cid:88) i g i e i and f = (cid:88) i f i e i (2.17)where f i , g i ∈ C . Let X (cid:48) := ⊗ mi =1 X i . We write e i e † j =: K ij ˜ e ij where the value of K ij isdetermined by the following three cases:(1) e i e † j vanishes, in which case K ij := 0 .(2) e i e † j ∈ End( X (cid:48) ) and contains no loops, in which case K ij := 1 .(3) e i e † j ∈ End( X (cid:48) ) and contains loops, in which case K ij is a product of some scalars λ abca ∈ R coming from loops of the form (2.11).where in cases (2) and (3), ˜ e ij is a basis element of the form (2.16) in End( X (cid:48) ) . Then (cid:104) g, f (cid:105) = tr (cid:32)(cid:88) i,j f i g ∗ j e i e † j (cid:33) = tr (cid:32)(cid:88) i,j f i g ∗ j K ij ˜ e ij (cid:33) = (cid:88) i f i g ∗ i K ii Remark 2.5 ( Positive dagger structure ) . Note that (cid:104) f, f (cid:105) = (cid:88) i | f i | K ii (2.18)Hence, given K ii > , our Hermitian form defines a Hermitian inner product. This isensured by setting λ kijk > in (2.11). Under this constraint, our category is said to havea positive dagger structure . Furthermore, this means that C is a unitary fusion category(see also Remarks 2.8 and 2.9). Also note that basis in (2.16) is orthogonal with respectto this inner product. If C is also spherical, viewing the quantum dimension of an objectas an inner product immediately shows that it must be positive. Throughout this paper,we assume that any category we work with possesses a positive dagger structure. Note that inpermissible values of the indices do not contribute to the basis.
ANUP POUDEL AND SACHIN J. VALERA
Frobenius-Schur indicator.Remark 2.6.
A unitary fusion category admits a unique pivotal structure [20, Prop. 8.23].Let C be a unitary pivotal fusion category, and relax the multiplicity-free assumption.Following [21, Proposition 3.9], we identify zig-zag morphisms with the pivotal structure: X X ∗ X ∗∗ = a X XX ∗∗ µ µ (cid:48) (2.19)Thus, passing to the skeleton yields X i X i = X i t i X ∗ i µ µ (cid:48) (2.20)where t i ∈ C × is called a pivotal coefficient . It can be shown [18, Lemma E.3] that(2.20) implies (2.21), whence the indices on the trivalent vertices in (2.19) and (2.20)can be dropped. N ij = N ji = δ ij ∗ (2.21)It can also be shown [18, 19] that(2.22) | t i | = 1 , t i ∗ = t ∗ i , t = 1 • If X i is non self-dual, we will assume that t i = 1 . This choice is always possiblethrough a unitary transformation of the trivalent vertices in (2.20). • If X i is self-dual, then t i is called the Frobenius-Schur indicator and is written κ i ;this quantity is invariant under any unitary transformations of trivalent vertices,and is therefore a fixed property of X i . Furthermore, (2.22) tells us that κ i = ± .The object X i is said to be (anti)symmetrically self-dual when κ i is ( − or) +1 .Following (2.20), we can make the identification(2.23) X ∗ k = X k which allows us to slide arrows around cups and caps.2.6. Normalisation and partial trace.
Let C be a unitary fusion category. Remark 2.7.
We henceforth use labels i ∈ I to denote objects X i ∈ Irr( C ) .We adopt a normalisation convention where trivalent vertices as in (2.6) are normalisedthrough a scaling of factor (cid:113) d k d i d j . KEIN-THEORETIC METHODS FOR UNITARY FUSION CATEGORIES 9
Under this normalisation, observe that(2.24) λ kijk = (cid:114) d i d j d k in (2.11). Following Remark 2.4, we have a canonical (orthonormal) basis (2.25) Hom( i ⊗ j, l ⊗ m ) = span C (cid:40) (cid:32) (cid:115) d k d i d j d l d m (cid:33) l mi jk (cid:41) k ∈ I : N ijk N lmk (cid:54) =0 where we call the graphical components of the basis diagrams jumping jacks or jackmorphisms . Using the canonical basis for End( i ⊗ j ) , we have the decomposition id i ⊗ j = i j = (cid:88) k ∈ I : N ijk (cid:54) =0 (cid:115) d k d i d j i ji jk (2.26)For any morphism f ∈ Hom( i ⊗ i · · · ⊗ i n , j ⊗ j · · · ⊗ j n ) , one can define a right partialtrace if i n = j n , and a left partial trace if i = j . f f... ...... ...i i n − i n j j n − i i n j j n i Figure 2.
The right and left partial traces of f .Now suppose that C is also spherical. We define the phi-net (2.27) Φ( i, j, k ) := (cid:102) T r = = where the final diagram corresponds to the left and right partial trace of a basis jack in
End( i ∗ ⊗ k ) . Following (2.24), we know that(2.28) Φ( i, j, k ) = (cid:114) d i d j d k · (cid:102) T r (id k ) = (cid:112) d i d j d k Given a, b, c self-dual, we define the theta-net (2.29) Θ( a, b, c ) := where we have left the edges unoriented (since the labels are self-dual). Note that Θ( a, b, c ) = Φ( a, b, c ) . Applying the left and right partial traces to (2.26), we get(2.30) d i d j = (cid:88) k ∈ I : N ijk (cid:54) =0 (cid:115) d k d i d j Φ( i ∗ , k, j ) = (cid:88) k N ijk d k F -matrices. Recall that a monoidal category C has associativity isomorphisms α X,Y,Z : ( X ⊗ Y ) ⊗ Z ∼ −→ X ⊗ ( Y ⊗ Z ) for any objects X, Y, Z ∈ C . These isomorphismssatisfy compatibility conditions given by the pentagon and triangle axioms.For a skeletal fusion category, after making a choice of basis for each
Hom( i ⊗ j, k ) where i, j, k ∈ I , we obtain a block-diagonal matrix A abc corresponding to each associativityisomorphism α a,b,c where a, b, c ∈ I . Each block in A abc is called an F - matrix , and iswritten F abcd (where d indexes each block). As a map, F abcd represents the isomorphism(2.31) and can be interpreted as a change of (fusion) basis on Hom( a ⊗ b ⊗ c, d ) . F abcd : (cid:77) e Hom( a ⊗ b, e ) ⊗ Hom( e ⊗ c, d ) ∼ −→ (cid:77) f Hom( a ⊗ f, d ) ⊗ Hom( b ⊗ c, f ) (2.31)where A abc = (cid:76) d F abcd . In the graphical calculus,(2.32) = (cid:80) f [ F a bc d ] f e a b c d f a b c d e The entries of an F -matrix are called F - symbols (or j -symbols). In terms of the fusioncoefficients, associativity is expressed as (cid:88) e N abe N ecd = (cid:88) f N afd N bcf (2.33) Remark 2.8 ( Skeletal data I ) . Given a fusion category C , its skeletal data is given bythe set of all fusion coefficients and F -symbols; this data completely characterises C . The F -symbols satisfy the pentagon equation coming from the pentagon axiom. If C has apositive dagger structure, it is easy to check that all associated F -matrices will be unitary(and so C is called unitary).2.8. Braided tensor categories.
Recall that for any two objects X and Y in a braidedtensor category C , a braiding is a natural isomorphism c X,Y : X ⊗ Y ∼ −→ Y ⊗ X which iscompatible with the associativity isomorphisms: this is ensured by the hexagon axioms,and the braidings consequently satisfy the Yang-Baxter equation (2.34) ( c Y,Z ⊗ id X ) ◦ (id Y ⊗ c X,Z ) ◦ ( c X,Y ⊗ id Z ) = (id Z ⊗ c X,Y ) ◦ ( c X,Z ⊗ id Y ) ◦ (id X ⊗ c Y,Z ) for any X, Y, Z ∈ C . This affords us braid isotopy in the graphical calculus.(a)
X Y X Y = (b) X Y Z X Y Z = Figure 3. (a) ( c X,Y ) − ◦ c X,Y = id X ⊗ Y , (b) Yang-Baxter equation. KEIN-THEORETIC METHODS FOR UNITARY FUSION CATEGORIES 11
For a skeletal braided fusion category, after making a choice of basis for each
Hom( i ⊗ j, k ) where i, j, k ∈ I , we obtain a block-diagonal matrix R ij corresponding to each braidingisomorphism c i,j . In the multiplicity-free case, each block is a × matrix denoted by R ijk (where k indexes each block) whose entry is called an R - symbol . By abuse of notation,we will use R ijk to denote the × matrix and the R -symbol interchangeably. In thegraphical calculus, the R -symbols are given by(2.35) ki j := R ijk ki j whence in the graphical calculus, the R - matrix is given by R ij := i j (2 . = (cid:88) k ∈ I : N ijk (cid:54) =0 R ijk (cid:115) d k d i d j j ii jk (2.36)Thus, the R -matrix is diagonal; specifically, we have R ij = (cid:77) k ∈ I : N ijk (cid:54) =0 R ijk (2.37)In the presence of a braiding, all fusion coefficients clearly satisfy N ijk = N jik (2.38) Remark 2.9 ( Skeletal data II ) . Given a braided fusion category C , its skeletal data isgiven by the set of all fusion coefficients, F -symbols and R -symbols; this data completelycharacterises C . The F -symbols and R -symbols satisfy the hexagon equations comingfrom the hexagon axioms. If C has a positive dagger structure, then the category iscalled unitary: we know that all associated F -matrices will be unitary; furthermore, allassociated R -matrices must also be unitary, since every admissible braiding on a unitaryfusion category must also be unitary [22, Theorem 3.2].2.9. Ribbon structure.
A spherical braided fusion category C is called a ribbon fusion (or premodular ) category . This is a braided fusion category with a ribbon structure , whichis given by a natural isomorphism θ X : X ∼ → X called the twist that satisfies θ X ⊗ Y = c Y,X ◦ c X,Y ◦ ( θ X ⊗ θ Y ) (2.39a) ( θ X ) ∗ = θ X ∗ (2.39b)for all X, Y ∈ C , and where ∗ denotes the dual functor on the left-hand side of (2.39b).Graphically, the twist is defined as follows for a skeletal ribbon category:(2.40) (a) i θ i (cid:55)−→ i = ϑ i i , (b) i θ − i (cid:55)−→ i = ϑ − i i where ϑ i ∈ C × . Note that (2.40b) follows from (2.40a), since by planar isotopy(2.41) = i i The skeletal data of a ribbon fusion category or modular tensor category is also given by this set.
Further note that(2.42) i ∗ iθ i i ∗ i ( θ i ) ∗ i i ∗ i ∗ i θ i ∗ i i ∗ i ∗ i ∗ i ∗ (2 . a ) = = = (2 . b ) = whence we obtain (2.43a). Equation (2.43b) follows similarly.(2.43) (a) i = i , (b) i = i From (2.42), we also have that(2.44) ϑ i = ϑ i ∗ Taking the left and right partial traces for the crossing i i , note that(2.45) = = = ϑ i d i Resolving the crossing in the first diagram of (2.45) using (2.36), it easy to check that(2.46) ϑ i = 1 d i (cid:88) k R iik d k It can also be shown [18, 19] that for i self-dual,(2.47) ϑ i = κ i (cid:0) R ii (cid:1) − In the graphical calculus for ribbon categories, edges may be promoted from lines toribbons, and twists are π clockwise self-rotations of a ribbon. A labelled edge is assumedto be oriented from top-to-bottom. For instance, (2.44) can be observed from(2.48) = where the twist is pushed around the closed ribbon. For any x, y, z ∈ Irr( C ) , (2.39a) maybe illustrated via the action of the monodromy on a basis element of Hom( x ⊗ y, z ) :(2.49) ( R yx ◦ R xy ) = = = ϑ z ϑ x ϑ y where we have relaxed the multiplicity-free assumption. Thus,(2.50) (cid:88) λ [ R yxz ] µλ [ R xyz ] λν = ϑ z ϑ x ϑ y δ µν Graphically, ribbon structure affords our diagrams equivalence under regular isotopy onthe -sphere. We henceforth refer to regular isotopy on the -sphere as framed isotopy . KEIN-THEORETIC METHODS FOR UNITARY FUSION CATEGORIES 13
Remark 2.10. (i) The Anderson-Moore-Vafa theorem [23, 24] tells us that the twist factor ϑ i is aroot of unity for all i ∈ I . For a proof, we refer the reader to [18, Theorem E.10].(ii) A unitary braided fusion category admits a unique ribbon structure [22]. Proposition 2.11.
Let C be a unitary ribbon fusion category. Then for any x ∈ Irr( C ) ,(2.51) d x ∈ { } ∪ [ √ , ∞ ) Proof.
We know that d x > by unitarity. Using (2.30), we have d x d = d x , whence d = 1 . It will be useful to classify x according to whether it satisfies the property (P0) (cid:80) z N xyz = 1 for all y ∈ Irr( C ) Claim: x satisfies (P0) if and only if x is invertible (i.e. x ⊗ x ∗ = ).If x satisfies (P0), then x is clearly invertible. If x is invertible, then x ∗ ⊗ x ⊗ y = y forall y ∈ Irr( C ) . Thus, (cid:80) z N x ∗ zy N xyz = 1 , whence (cid:80) z N xyz = 1 for all y . This shows theclaim. It immediately follows that if x satisfies (P0), then so does x ∗ . Now,(i) If x satisfies (P0), then d x d x ∗ = d x = d = 1 , whence d x = 1 .(ii) If x does not satisfy (P0), then d x d x ∗ = d x = d + (cid:80) y (cid:54) = N xx ∗ y d y > , whence d x > . The lower bound is attained when x ⊗ x ∗ = ⊕ y for some y satisfying(P0). Thus, d x ≥ √ . (cid:3) Modularity.
Let C be a braided fusion category. An object X in C such that(2.52) c Y,X ◦ c X,Y = id X ⊗ Y for all objects Y in C is called transparent . If all transparent objects in C are isomorphicto , then the braiding is called non-degenerate .Further assume that C is ribbon. We define the matrix ˜ S where(2.53) [ ˜ S ] xy := x y , x, y ∈ Irr( C ) i.e. the left and right partial trace of R y ∗ x ◦ R xy ∗ . The S-matrix is S := 1 D ˜ S where(2.54) D := (cid:88) x ∈ Irr( C ) (cid:112) d x is called the total quantum dimension of C . A ribbon fusion category C is called a modular tensor category (MTC) if it has a non-degenerate braiding (or equivalently, ifthe associated S -matrix is invertible).2.11. Additional conventions.
Throughout much of this paper, we consider a fusioncategory C containing a fusion rule of the form q ⊗ q = ⊕ k (cid:77) i =1 x i (2.55)where q, x i ∈ Irr( C ) and objects x i are distinct. In this context, we fix some conventions: • Unlabelled, unoriented edges are understood to represent edges labelled by theself-dual object q . In a field-theoretic context, (P0) characterises the abelianity of a quasiparticle (see Section 6). We caution the reader that conventions for the orientation of (2.53) vary in the literature. • Greek indices (e.g. λ ) will be used to denote elements in I for which N qqλ (cid:54) = 0 .Latin indices (e.g. i ) will be used to denote elements in I \ { } for which N qqi (cid:54) = 0 .For instance, (2.36) may be written as follows for i = j = q : = (cid:88) λ R qqλ √ d λ d q λ = R qq d q + (cid:88) i R qqi √ d i d q i (2.56)where we call the jumping jacks on the right-hand side of (2.56) i -jacks .2.12. Rotation operator.
We follow the conventions of Section 2.11, and further assumethat C is pivotal . Let f ∈ End( q ⊗ ) and f (cid:48) := id q ⊗ f ⊗ id q ∈ End( q ⊗ ) . We define the rotation operator ϕ : f (cid:55)→ f (cid:48) (cid:55)→ (id q ⊗ id q ⊗ ev q ) (cid:124) (cid:123)(cid:122) (cid:125) ∈ Hom( q ⊗ ,q ⊗ ⊗ ) ◦ f (cid:48) ◦ (coev q ⊗ id q ⊗ id q ) (cid:124) (cid:123)(cid:122) (cid:125) ∈ Hom( ⊗ q ⊗ ,q ⊗ ) (2.57)Hence, ϕ ( f ) ∈ Hom( ⊗ q ⊗ , q ⊗ ⊗ ) = End( q ⊗ ) . Graphically, ϕ acts as an anticlockwise π -rotation on a morphism in End( q ⊗ ) :(2.58) f ϕ (cid:55)−→ f = f By C -linearity of C and bilinearity of the bifunctor “ ⊗ ” on morphisms, note that ϕ isa C -linear operator. Further note that ϕ ( f ) = f (as demonstrated in (2.59), where thefinal diagram can be straightened to the first diagram).(2.59) ϕ (cid:55)−→ ϕ (cid:55)−→ ϕ (cid:55)−→ ϕ (cid:55)−→ f f f f f To the knowledge of the authors, the first instance where the rotation operator was usedin a categorical context was in [14]. See Appendix A for a supplementary excursion onthe rotation of morphisms.3.
Some Framed Invariants from Ribbon Categories
Let C be a unitary ribbon fusion category containing a fusion rule of the form(3.1) q ⊗ q = ⊕ k (cid:77) i =1 x i where q, x i ∈ Irr( C ) and objects x i are distinct. Framed, oriented links whose componentsare labelled by elements of Irr( C ) can be thought of as morphisms in End( ) ; the valuein C to which any such link evaluates is invariant under framed isotopy. Restated, givenan oriented link diagram D whose components are labelled as such, there is a complex-valued function whose value is constant on the framed isotopy class of D . Such a function In the instance where all labels are self-dual, D is an unoriented diagram. KEIN-THEORETIC METHODS FOR UNITARY FUSION CATEGORIES 15 is called a framed link invariant .Let Λ C ,q denote the framed link invariant for oriented links with all components labelledby q ∈ Irr( C ) . Our goal is to extract information pertaining to Λ C ,q when q satisfies(3.1); in particular, we use the rotation operator ϕ to find relations amongst d q and theeigenvalues of R qq . We do this for the trivial case (i.e. q ⊗ = ) and then for cases k = 1 , . When q is symmetrically self-dual (i.e. κ q = 1 ), it is easy to see that(3.2) ˜Λ C ,q ( L ) = ϑ − w ( D ) q Λ C ,q ( D ) is a link invariant (where L is the link for which D is a diagram, and w is the writhe).Much of the exposition in this section is already well-established and has been presentedin [14] (see Theorems 3.1 & 3.2) where a broader discussion may be found. However, wechoose to include this material for its relevance to Section 5. Furthermore, our approachdiffers slightly to that taken in [14] and we also treat the instances where q is antisym-metrically self-dual (i.e. κ q = − ), which leads to an extended discussion in Section 3.4.We follow the conventions fixed in Section 2.11. R qq is diagonalisable in the canonical basis, so we may resolve crossings as follows:(3.3) = (cid:88) λ R qqλ √ d λ d q λ and = (cid:88) λ ( R qqλ ) − √ d λ d q λ Also,(3.4) ϕ (cid:16) (cid:17) = = κ q Trivial case.
Here, the fusion rule is given by(3.5) q ⊗ q = Let α := R qq . Then from (3.3), = αd q and = α − d q (3.6)whence ϕ (cid:16) (cid:17) = αd q . Using (3.4) and comparing coefficients, we get α = κ q α − = ⇒ α = κ q . and so we have the following two possibilities:(1) For κ q = 1 , R qq = ± with skein relation(3.7) = and = 1 (2) For κ q = − , R qq = ± i with skein relation(3.8) = − and = 1 k=1. Now our fusion rule is of the form(3.9) q ⊗ q = ⊕ x Let α := R qq and β := R qqx . Using (3.3) and (2.26), we resolve the crossings in the basis (cid:110) , (cid:111) to get = 1 d q ( α − β ) + β (3.10a) = 1 d q ( α − − β − ) + β − (3.10b)whence(3.11) ϕ (cid:16) (cid:17) = β + 1 d q ( α − β ) Then using (3.4) and comparing coefficients with (3.10b), we have : κ q β − = 1 d q ( α − β ) = ⇒ α = κ q d q β − + β (3.12a) : κ q β = 1 d q ( α − − β − ) = ⇒ α − = κ q d q β + β − (3.12b)which can be solved to get(3.13) d q = − κ q ( β + β − ) , α = − β − We thus have the following two possibilities:(1) For κ q = 1 , R qq = diag( − β − , β ) with skein relation(3.14) = β + β − and = − ( β + β − ) i.e. the Kauffman bracket .(2) For κ q = − , R qq = diag( − β − , β ) with skein relation(3.15) = β − β − and = β + β − k=2. Our fusion rule is of the form(3.16) q ⊗ q = ⊕ x ⊕ y Let α := R qq , β := R qqx and γ := R qqy . Using (3.3) and (2.26), we resolve the crossings inthe basis (cid:110) , , x (cid:111) to get = A + B x + C (3.17a) = A (cid:48) + B (cid:48) x + C − (3.17b)where A := 1 d q ( α − γ ) , B := √ d x d q ( β − γ ) , C := γA (cid:48) := 1 d q ( α − − γ − ) , B (cid:48) := √ d x d q ( β − − γ − ) KEIN-THEORETIC METHODS FOR UNITARY FUSION CATEGORIES 17
Remark 3.1.
We henceforth set β (cid:54) = γ . The case B, B (cid:48) = 0 is treated in Section 3.3.1.Eliminating the x -jacks and rearranging yields = (cid:18) A − A (cid:48) BB (cid:48) (cid:19) + (cid:18) C − C − BB (cid:48) (cid:19) + BB (cid:48) (3.18a) = ⇒ ϕ (cid:16) (cid:17) = (cid:18) A − A (cid:48) BB (cid:48) (cid:19) + (cid:18) C − C − BB (cid:48) (cid:19) + κ q BB (cid:48) (3.18b)Using (3.17a) to express ϕ (cid:16) (cid:17) in our chosen basis, ϕ (cid:16) (cid:17) = (cid:18) A − A (cid:48) BB (cid:48) + κ q CBB (cid:48) (cid:19) + (cid:18) C − C − BB (cid:48) + κ q ABB (cid:48) (cid:19) + κ q B B (cid:48) x Applying (3.4) and comparing coefficients with (3.17b), we have x : B (cid:48) = ± B (3.19a) : A (cid:48) = κ q C + BB (cid:48) ( A − κ q C − ) (3.19b) : A (cid:48) = κ q C + B (cid:48) B ( A − κ q C − ) (3.19c)Thus,(3.20) B (cid:48) = ± B , A (cid:48) = κ q ( C ∓ C − ) ± A Remark 3.2 ( Caveat ) . When κ q = − , there is a difference in sign betweeen (vertical)twists and their “horizontal” counterparts (i.e. a π -rotated version). This is taken intoaccount when solving for Cases and below. For instance, = u = ⇒ = u = ⇒ u = κ q ϑ q There are now four cases to examine (for B (cid:48) = ± B and κ q = ± ).Case 1: Let B (cid:48) = B and κ q = 1 . Then (3.18a) becomes(3.21) − = D (cid:16) − (cid:17) where D := C − C − . Stacking on top of and using (3.21) , = D (cid:16) − (cid:17) (cid:16) − (cid:17) + D (cid:16) − + − (cid:17) += D (cid:16) − − d q + (cid:17) + D ( ϑ q − ϑ − q ) + D (cid:16) − (cid:17) + (3 . = D (cid:2) (2 − d q ) D + ϑ q − ϑ − q (cid:3) − D + D (cid:16) − (cid:17) += D (cid:2) (1 − d q ) D + ϑ q − ϑ − q (cid:3) + whence by Reidemeister-II,(i) D = ϑ q − ϑ − q d q − or (ii) D = 0 For (i), note that D is well-defined since d q > . Then(3.22) d q = α − − αγ − γ − + 1 = α − − αβ − β − + 1 where the second equality follows from B (cid:48) = B . Since D (cid:54) = 0 , we have β, γ (cid:54) = ± and so(3.22) is also well-defined.Case 2: Let B (cid:48) = − B and κ q = 1 . Then (3.18a) becomes + = K (cid:16) + (cid:17) (3.23)where K := C + C − . Similarly, we get(i) K = ϑ q + ϑ − q d q + 1 or (ii) K = 0 where for (i),(3.24) d q = α − + αγ + γ − − α − + αβ + β − − Case 3: Let B (cid:48) = B and κ q = − . Then (3.18a) becomes − = D (cid:16) + (cid:17) (3.25)Similarly, we get (i) D = ϑ q − ϑ − q d q + 1 or (ii) D = 0 where for (i),(3.26) d q = α − α − γ − γ − − α − α − β − β − − Case 4: Let B (cid:48) = − B and κ q = − . Then (3.18a) becomes + = K (cid:16) − (cid:17) (3.27)Similarly, we get (i) K = ϑ q + ϑ − q d q − or (ii) K = 0 where for (i),(3.28) d q = − α − + αγ + γ − + 1 = − α − + αβ + β − + 1 Heuristically, this equality also follows by the symmetry of the construction in β and γ . That is, wecould have alternatively used the y -jack in our chosen basis. KEIN-THEORETIC METHODS FOR UNITARY FUSION CATEGORIES 19
Remark 3.3.
Cases D = 0 and K = 0 for κ q = ± are covered in Section 3.3.1.For B (cid:48) = B we have β − β − = γ − γ − = D whence sin(arg β ) = sin(arg γ ) . Since β (cid:54) = γ ,we have arg β + arg γ = π . Thus,(3.29) βγ = − and D = β + γ , B (cid:48) = B For B (cid:48) = − B we have β + β − = γ + γ − = K whence cos(arg β ) = cos(arg γ ) . Since β (cid:54) = γ , we have arg β + arg γ = 2 π . Thus,(3.30) βγ = +1 and K = β + γ , B (cid:48) = − B Following the notation in [14], we let z := β + γ and a := ϑ q . Summarising these fourcases (where β (cid:54) = γ and z (cid:54) = 0 ),(1) For B (cid:48) = B and κ q = 1 , R qq = diag( α, β, − β − ) with skein relation(3.31) − = z (cid:16) − (cid:17) and = a − a − z + 1 i.e. the framed Dubrovnik polynomial .(2) For B (cid:48) = − B and κ q = 1 , R qq = diag( α, β, β − ) with skein relation(3.32) + = z (cid:16) + (cid:17) and = a + a − z − i.e. the framed Kauffman polynomial .(3) For B (cid:48) = B and κ q = − , R qq = diag( α, β, − β − ) with skein relation(3.33) − = z (cid:16) + (cid:17) and = a − a − z − (4) For B (cid:48) = − B and κ q = − , R qq = diag( α, β, β − ) with skein relation(3.34) + = z (cid:16) − (cid:17) and = a + a − z + 1 Special cases. (1) Suppose
B, B (cid:48) = 0 (i.e. β = γ ). Then (3.17a) and (3.17b) become (3.10a) and(3.10b). That is, the crossings lie in the subspace span (cid:110) , (cid:111) . Thus, κ q = 1 κ q = − Skein relation (3.14) (3.15) R qq diag( − β − , β, β ) diag( − β − , β, β ) (2) Suppose D = 0 with κ q = ± . Then ( A (cid:48) , B (cid:48) , C − ) = ( A, B, C ) = ⇒ α, β, γ ∈ {± } and using (3.17a),(3.17b) and β (cid:54) = γ we get(3.35) R qq = ( α, ± , ∓ and = , α ∈ {± } (3) Suppose K = 0 with κ q = ± . Then ( A (cid:48) , B (cid:48) , C − ) = ( − A, − B, − C ) = ⇒ α, β, γ ∈ {± i } and using (3.17a),(3.17b) and β (cid:54) = γ we get(3.36) R qq = ( α, ± i, ∓ i ) and = − , α ∈ {± i } Quantum invariants coming from κ q = − .Theorem 3.4. Let L denote a link, D a corresponding diagram, and w the writhe.(i) Let Λ (1)( a,z ) and Λ (2)( a,z ) respectively denote the framed Dubrovnik (3.31) and framedKauffman (3.32) polynomial. Then(3.37) Λ (1)( a,z ) ( D ) = i − w ( D ) ( − c ( L )+1 Λ (2)( ia, − iz ) ( D ) where L is possibly oriented, and w denotes the writhe.(ii) Let (cid:104)·(cid:105) β and [ · ] β respectively denote the Kauffman bracket (3.14) and framedinvariant (3.15). Then(3.38) (cid:104) D (cid:105) β = i −W ( D ) ( − c ( L )+1 [ D ] iβ where W denotes the local writhe , and D is oriented such that it is equivalent(under framed isotopy) to a diagram ˜ D for which w ( ˜ D ) = W ( ˜ D ) . Part (i) was proved by Lickorish [25], while (ii) is a new result whose proof follows in asimilar fashion (see also Remark 3.8).
Corollary 3.5.
Given any link diagram D as defined in Theorem 3.4(ii),(3.39) (cid:104) D (cid:105) β = ω [ D ] β , ω = 1 Proof.
From (3.38), we have(3.40) [ D ] β = i W ( D ) ( − c ( L )+1 (cid:104) D (cid:105) − iβ Recalling that the difference of any two exponents in (cid:104) D (cid:105) β is a multiple of , the resultimmediately follows upon inspection of (3.40). (cid:3) Since invariants Λ C ,q coming from κ q = − are derived from a setting where an isotopyof the form ϕ −→ = − introduces a difference in sign, it is natural to askwhether the invariant Λ C ,q carries such a sensitivity. Problem 3.6.
Let D and D (cid:48) be link diagrams that are equivalent under framed isotopy,and let k := |W ( D ) − W ( D (cid:48) ) | . Is it always true that(3.41) Λ C ,q ( D (cid:48) ) = ( − k Λ C ,q ( D ) given Λ C ,q for some q with κ q = − ? The local writhe W of a link diagram D is defined as the sum over the signs of all crossings in D ,where and respectively contribute +1 and − . Note that such a choice of orientation is always possible: by Alexander’s theorem, we can alwayschoose ˜ D to be the diagram given by the Markov trace of some braid. Then we endow all componentswith an identical orientation. KEIN-THEORETIC METHODS FOR UNITARY FUSION CATEGORIES 21
Understanding the invariant (3.15).
The skein relation (3.15) must be applied lo-cally i.e. the link must be without orientation, and the relation should be applied tocrossings precisely as they appear.For instance, while a twist of the form would usually be resolved as a positive crossing,we only take into account the local form of the crossing (which is negative here). Indeed,with the notation from (3.38),(3.42) β = β , (cid:34) (cid:35) β = − β whence we see that the categorical distinction between horizontal and vertical twists(Remark 3.2) carries over to the invariant (3.15). Of course, this observation is subsumedby the following: Proposition 3.7.
The answer to Problem 3.6 is positive when q ⊗ = ⊕ x . Proof.
In this case, we know that
Λ := Λ C ,q is given by (3.15). In the following, we assumethat all diagrams are projected onto the plane. It suffices to consider an isotopy D → ˇ D which is one of: (i) a Reidemeister- move (i.e. an ambient isotopy) such that only onecrossing has its sign flipped under the deformation (so k = 1 ); (ii) a Reidemeister-IImove; (iii) a Reidemeister-III move.(i) Given a link diagram D containing some crossing , we let D and D ∞ re-spectively denote the same diagram but with the crossing smoothed to and. We consider Λ( D ) and Λ( ˇ D ) , first applying the skein relation to the crossingwhose sign was flipped: suppose (without loss of generality) that the crossing in D was . Then Λ( D ) = β · Λ( D ) − β − · Λ( D ∞ )Λ( ˇ D ) = β − · Λ( ˇ D ) − β · Λ( ˇ D ∞ ) Label the boundary points of the crossing in D as , and so in ˇ D saidcrossing is either or . Then it is easy to see that the smoothings D → D and ˇ D → ˇ D ∞ are locally identical, whence the diagrams D and ˇ D ∞ arealso equivalent under ambient isotopy. It follows that Λ( D ) = Λ( ˇ D ∞ ) . Similarly, Λ( ˇ D ) = Λ( D ∞ ) . Thus, Λ( ˇ D ) = − Λ( D ) .(ii) Two crossings have their signs flipped under a Reidemeister-II move (so k = 2 ).Thus, (7.1) respects the invariance of Λ under Reidemeister-II i.e. Λ( ˇ D ) = Λ( D ) .(iii) Either zero or two crossings have their signs flipped under a Reidemeister-IIImove (so k = 0 or ). It follows that (7.1) respects the invariance of Λ underReidemeister-III i.e. Λ( ˇ D ) = Λ( D ) .Since D → D (cid:48) in Problem 3.6 is a composition of moves (i)-(iii), the result follows. (cid:3) Thus, [ D ] β is invariant under framed isotopy ( up to a sign , which depends on W ( D ) ) e.g.(3.43) (cid:34) (cid:35) β = − β = β From Figure 4, we see that (3.43) is equal to β d − d − d + β − d (where d := β + β − is the value of the loop). Figure 4.
When we apply the skein relation (3.15), we must take care toaccount for any minus signs accumulated if we choose to rotate crossingsmid-evaluation e.g. if the parenthesised route is taken above (for which theresulting minus signs are highlighted in red).
Remark 3.8.
Given ˜ D as in Theorem 3.4(ii), we may write(3.44) [ ˜ D ] β = i w ( ˜ D ) ( − c ( L )+1 (cid:104) ˜ D (cid:105) − iβ where W ( ˜ D ) = w ( ˜ D ) . Given any diagram D that is equivalent to ˜ D under framedisotopy, let k := (cid:104) W ( D ) − W ( ˜ D ) (cid:105) . Then W ( D ) = w ( ˜ D ) + 2 k , whence (3.38) implies [ D ] β = i w ( ˜ D )+2 k ( − c ( L )+1 (cid:104) D (cid:105) − iβ = ( − k [ ˜ D ] β and so (3.38) is consistent with (7.1).In summary, we have seen that while (3.15) is invariant under Reidemeister-II and IIImoves, it retains sensitivity to pivotality at some level: when a link diagram is isotoped ina way that locally rotates crossings, this introduces “internal zig-zags” that are detectedby the quantum invariant. Remark 3.9.
Throughout this work, we refer to invariants associated to q antisymmet-rically self-dual as “framed link invariants”. However, in light of the above, this could beconsidered an abuse of terminology. We therefore propose the term pivotal framed linkinvariant for (3.15). Indeed, if the answer to Problem 3.6 is positive, then we propose thatall invariants associated to κ q = − should be termed as such. When a pivotal framedlink invariant is normalised as in (3.2), the sensitivity to framing is removed but thepivotal discrimination remains: we therefore propose that the resulting invariant shouldsimilarly be called a pivotal link invariant . KEIN-THEORETIC METHODS FOR UNITARY FUSION CATEGORIES 23 Unitary Representations of the Braid Group
We now review part of the exposition in Section 3 from a slightly different perspective.While much of the discourse here is well-known, we feel that it would be amiss to excludethis material from our presentation. The n -strand braid group is given by(4.1) B n = (cid:28) σ , . . . , σ n − σ i σ i +1 σ i = σ i +1 σ i σ i +1 σ i σ j = σ j σ i , | i − j | ≥ (cid:29) whose graphical interpretation is given in Figure 5. We also have the symmetric group(4.2) S n = (cid:42) s , . . . , s n − s i = es i s i +1 s i = s i +1 s i s i +1 s i s j = s j s i , | i − j | ≥ (cid:43) σ i = ... ... i − i i + 1 i + 2 n...... σ − i = ... ... i − i i + 1 i + 2 n...... Figure 5.
Braid words are read from right-to-left. Braids are drawn andcomposed from top-to-bottom in accord with our pessimistic convention.There is an epimorphism ψ : B n → S n where ψ ( σ ± i ) = s i . The pure braid group P B n is a normal subgroup of B n given by ker ψ . That is, B n (cid:30) P B n ∼ = S n . There is a closelyrelated quotient for the algebra C [ B n ] ; namely, we take the ideal Q ( σ i ) generated by ( σ i − r )( σ i − r ) where r , r ∈ C × . Then(4.3) C [ B n ] (cid:30) Q ( σ i ) ∼ = H n ( r , r ) where H n ( r , r ) is called the Iwahori-Hecke algebra . Indeed, H n ( ± , ∓ ∼ = C [ S n ] (and sothe Iwahori-Hecke algebra can be thought of as a deformation of C [ S n ] ). Let T , . . . , T n − be the generators of the H n ( r , r ) . The generators satisfy relations ( T i − r )( T i − r ) = 0 (4.4a) T i T i +1 T i = T i +1 T i T i +1 (4.4b) T i T j = T j T i , | i − j | ≥ (4.4c)where (4.4a) is called the Hecke relation . Viewing the Iwahori-Hecke algebra as a vec-tor space, we have dim C ( H n ) = n ! . The generalised Hecke algebra H n ( Q, k ) is given bythe quotient of C [ B n ] by the ideal Q ( σ i ) which is now generated by Π kj =1 ( σ i − r j ) where r j ∈ C × and k ≥ . H n ( Q, k ) has the same presentation as the Iwahori-Hecke algebraexcept that (4.4a) is now replaced with the generalised Hecke relation Π kj =1 ( T i − r j ) = 0 .Let C be a ribbon fusion category and take some q ∈ Irr( C ) . Then(4.5) End (cid:0) q ⊗ n (cid:1) = (cid:77) X Hom (cid:0) q ⊗ n , X (cid:1) ⊗ Hom (cid:0)
X, q ⊗ n (cid:1) where X indexes all the simple objects appearing in the decomposition of q ⊗ n . Fixing afusion basis on Hom ( q ⊗ n , x ) for some X = x defines a linear representation(4.6) ρ : B n → U ( V q n x ) , V q n x := Hom (cid:0) q ⊗ n , x (cid:1) where U ( V q n x ) denotes the group of unitary matrices on V q n x . Let n ≥ . There existsat least one i such that ρ ( σ i ) = R , where R is a diagonal matrix whose eigenvalues aresome subset of the eigenvalues of R qq (eigenvalues are counted without multiplicity here).Let { r , . . . , r k } denote the eigenvalues of R where the r i ∈ U (1) are distinct and mayappear in R with arbitrary nonzero multiplicity. We define(4.7) p ( Z ) = ( Z − r I s ) · . . . · ( Z − r k I s ) where k ≤ s := dim( V q n x ) , I s is the s × s identity matrix and Z is an s × s matrix withentries in C . It is clear that p ( R (cid:48) ) = 0 (where R (cid:48) is any matrix similar to R ); this is aninstance of the Cayley-Hamilton theorem. It follows that p ( ρ ( σ i )) = 0 for all i , whence (4.8) ρ : C [ B n ] → H n ( Q, k ) → U ( V q n x ) i.e. ρ factors through the generalised Hecke algebra H n ( Q, k ) .In Section 3.1, we considered a fusion rule q ⊗ q = . Clearly, = z for some z ∈ C × . Applying (3.4), we see that z = ± for κ q = ± . For κ q = +1 , (4.8) becomes ρ ( u ) : C [ B n ] → C [ S n ] → U (1) σ j (cid:55)−→ s j (cid:55)−→ u , u = ± (4.9)For κ q = − , (4.8) becomes ρ ( u ) : C [ B n ] → C [ S n ] → U (1) σ j (cid:55)−→ ± is j (cid:55)−→ u , u = ± i (4.10)In Section 3.2, we considered a fusion rule q ⊗ q = ⊕ y . This means that our crossingscan be written = a + b (4.11a) = c + d (4.11b)where a, b, c, d ∈ C × . This motivates the idea that the homomorphism (4.8) should alsofactor through some algebra of cup-cap diagrams and non-intersecting strands (for n ≥ );this is precisely the Temperley-Lieb algebra
T L n ( δ ) : an associative A -algebra (where A is a commutative ring) with generators U , . . . , U n − satisfying relations U i = δU i , δ ∈ A (4.12a) U i U j U i = U i , | i − j | = 1 (4.12b) U i U j = U j U i , | i − j | ≥ (4.12c) U i = δ = ... ... i − i i + 1 i + 2 n...... Figure 6.
Diagrams run from top-to-bottom. The identity element isgiven by n vertical strands. In (4.8), ρ is a C -linear extension of ρ in (4.6). Through an abuse of notation, we implicitly assumethat the representation in (4.8) is restricted to B n so as to coincide with (4.6). KEIN-THEORETIC METHODS FOR UNITARY FUSION CATEGORIES 25
T L n ( δ ) is a free A -module of rank C n where C n denotes the ( n − th Catalan number.Following (4.11a)-(4.11b), we construct a C -linear map ζ : C [ B n ] → T L n ( δ ) σ i (cid:55)→ a + bU i σ − i (cid:55)→ c + dU i , A = C [ a ± , b ± , c ± , d ± ] (4.13) Proposition 4.1. ζ defines an algebra homomorphism when c = a − , d = b − and δ = − ( ab − + a − b ) .We henceforth assume that c, d and δ are as in Proposition 4.1. Since our representation(4.14) ρ : C [ B n ] ζ → T L n ( δ ) → U ( V q n x ) is unitary, the conditions in Proposition 4.2 (adapted from [28, p.237]) must hold. Proposition 4.2.
Given ρ as in (4.14), we have U † i = U i and | a | = | b | = 1 .From (4.8) we know that ρ must also factor through H n ( Q, k ) . Since we are consideringa fusion rule of the form q ⊗ = y ⊕ z , we have that(4.15) ρ : C [ B n ] → H n ( r , r ) → U ( V q n x ) It is easy to check that (4.14) is compatible with (4.15). Following Proposition 4.2, U † i = U i ⇐⇒ (cid:2) b − ( ρ ( σ i ) − a ) (cid:3) † = b − ( ρ ( σ i ) − a ) ⇐⇒ b ( ρ ( σ i )) † − a ∗ b = b ∗ ρ ( σ i ) − b ∗ a ⇐⇒ (cid:0) ρ ( σ i ) + a ∗ b (cid:1) ( ρ ( σ i ) − a ) = 0 whence(4.16) ρ : C [ B n ] → H n ( − a ∗ b , a ) → T L n ( δ ) → U ( V q n x ) Specifically, we have the following commutative diagram of linear homomorphisms: C [ B n ] H n ( − a − b , a ) T L n ( δ ) U ( V q n x ) φζ ηρ (cid:48) a, b ∈ U (1) with δ = − ( ab − + a − b ) ρ = ρ (cid:48) ◦ ζ where φ ( σ i ) = T i such that ker φ is generated by ( σ i + a − b )( σ i − a ) , and η ( T i ) = a + bU i .We may thus resolve crossings using skein relations = a + b = a − + b − , = − ( ab − + a − b ) (4.17)Note that the resolution of the crossings in (4.17) implies(4.18) − b = ( a − a − b ) which is simply the Hecke (skein) relation for H n ( − a − b , a ) . To recover the boxed resultin Section 3.2, we consider relations (4.17) in the setting of a ribbon category. Note that(4.19) ϑ q d q = = a + b = − a b − whence ϑ q = − a b − . But we also have that R qq = a − b whence (2.47) tells us that b = ± a − for κ q = ± . Another way of seeing this is by applying (3.4) to (4.17). Remark 4.3.
The skein relations (4.17) correspond to the framed
HOMFLY-PT poly-nomial. In order to see this, consider the well-known Lickorish-Millet presentation [29]of the HOMFLY-PT skein relations,(4.20) l + l − + m = 0 , = 1 Then setting(4.21) l = ± ib − , m = ∓ i ( ab − − a − b ) recovers (4.18). Finally, with l and m as in (4.21), l + l − + m = 0 (4 . = ⇒ − a b − l − a − bl − + m = 0= ⇒ − ( ab − + a − b ) = whence we have the rescaled loop value = − ( ab − + a − b ) . Let ˜ H denote the framedHOMFLY-PT polynomial, L be a link and D a corresponding link diagram. Then the(unframed) HOMFLY-PT polynomial H is simply(4.22) H ( L ) = ( − a − b ) w ( D ) ˜ H ( D ) where w denotes the writhe. The HOMFLY-PT invariant can be derived by applying anormalised Markov trace to the Iwahori-Hecke algebra; this trace is characterised by itsaction on the basis elements of the HOMFLY-PT skein algebra of the annulus [26, 27].We omit an analogous discussion for the fusion rule q ⊗ q = ⊕ x ⊕ y , and solely remarkthat since there must exist p , p , p , p nonzero such that(4.23) p + p + p + p = 0 this indicates that the representation should factor through the Temperley-Lieb algebra,which in turn motivates the construction of a linear map ζ (cid:48) : C [ B n ] → T L n ( δ ) σ i + aσ − i (cid:55)→ b + cU i (4.24)where a, b, c ∈ C × and δ ∈ C [ a ± , b ± , c ± ] are such that ζ (cid:48) defines a homomorphism. Itis also clear that the representation should factor through H n ( Q, . Given H n ( r , r ) , the HOMFLY-PT skein algebra H n ( r , r ) is obtained by joining the ends of thestrands (where the i th top is respectively connected to the i th bottom). We have H n ∼ = H n (cid:30) [ · , · ] (i.e. thequotient by the ideal generated by the commutator) and dim C ( H n ) is given by the n th partition number. KEIN-THEORETIC METHODS FOR UNITARY FUSION CATEGORIES 27 Main Results
Let C be a unitary spherical fusion category containing a fusion rule of the form(5.1) q ⊗ q = ⊕ (cid:77) i x i where q, x i ∈ Irr( C ) , objects x i are distinct and where N := dim (End( q ⊗ )) ≥ .We write f µν := (cid:2) F qqqq (cid:3) µν . For any simple object x in the decomposition of q ⊗ , thesymmetries of the fusion coefficients give N qqx = N xqq = N qxq = N qqx ∗ . Firstly, this tellsus that the indices of f µν run over and { x i } i . Secondly, this tells us that the set { x i } i is closed under taking duals: this allows us to define a (charge) conjugation matrix C := δ µµ ∗ where µ indexes and { x i } i . We follow the conventions from Section 2.11 andlet σ ( A ) denote the spectrum of a linear operator A .5.1. Rotation operator in the canonical basis.Lemma 5.1. (i) f λ = κ q √ d λ d q (ii) δ λ = κ q (cid:80) ρ √ d ρ d q f ρλ (iii) q µλ qq = κ q d q f µλ (iv) µ qq ρq = κ q d q δ µλ (cid:80) ρ f ρλ Proof. (5.2) λ = (cid:80) ρ f ρλ ρ (i) Capping off the rightmost pair of leaves in (5.2) gives λ = (cid:80) ρ f ρλ ρ = ⇒ κ q λ = (cid:80) ρ f ρλ δ ρ d q = ⇒ κ q √ d λ d q = (cid:88) ρ f ρλ δ ρ . (ii) Capping off the leftmost pair of leaves in (5.2) gives = (cid:80) ρ f ρλ = ⇒ δ λ d q ρ = κ q (cid:80) ρ f ρλ λ ρ q = ⇒ δ λ = κ q (cid:80) ρ (cid:112) d ρ d q f ρλ (iii) Stacking the adjoint tree of the right-hand side on (5.2) gives λ µ ρµ = (cid:80) ρ f ρλ = ⇒ κ q µλ = f µλ µµ = d q (cid:112) d µ f µλ µ = d q f µλ (iv) Stacking the adjoint tree of the left-hand side on (5.2) gives λµ ρµ = (cid:80) ρ f ρλ = ⇒ δ µλ d q √ d λ = κ q (cid:80) ρ f ρλ µλ ρ = ⇒ κ q d q δ µλ µ ρ = (cid:80) ρ f ρλ (cid:3) Note that plugging the adjoint of (iii) into (iv) yields (cid:80) ρ f ρλ f ∗ ρµ = δ λµ , which agrees withthe unitarity of F qqqq . KEIN-THEORETIC METHODS FOR UNITARY FUSION CATEGORIES 29
Remark 5.2. (Duality)
In the following, we wish to consider the action of rotation operator ϕ on our canonicalbasis. Expanding some arbitrary h ∈ End( q ⊗ ) in this basis, it is clear that ϕ = C . Whenconsidering the image of an x -jack under ϕ , the directed edge calls for extra caution. For x (cid:54) = we have(5.3) xx = ϕ where we call the right-hand side a bone morphism. Observing that the jack morphismmay equivalently be represented with a slant, x xϕ =(5.4a) x x ∗ ϕ =(5.4b)For the x -bone to be well-defined, we must be able to identify (5.4a), (5.4b) and (5.3).The adjunction of (5.4a) yields † (cid:103) −→ x x and since the bone is taken to be self-adjoint, we require that x = x ∗ . When considering ϕ in the canonical basis we must therefore assume that C = id i.e. { x i } i are self-dual in(5.1). This obviates the need to direct any edges in our diagrams. Lemma 5.3. (5.5) R qqµ = R qqµ ∗ Proof. = (cid:88) µ R qqµ µ ϕ = ⇒ = (cid:88) µ R qqµ µ ∗ (cid:3) In particular, this tells us that if R qq has no repeated eigenvalues then C = id . Similarly,one can show that R abc = R a ∗ b ∗ c ∗ . Theorem 5.4. (Bones via jacks)
Given fusion rule (5.1) with x i self-dual, we have(5.6) λ = √ d λ d q + κ q (cid:88) i f iλ i Proof.
Expanding the bone in the canonical basis,(5.7) λ = a λ + (cid:88) i b λi i Given a morphism h ∈ End( q ⊗ ) , let h (cid:48) := id q ⊗ h ∈ End( q ⊗ ) . Then we define the linearmap Ω : h (cid:55)→ h (cid:48) (cid:55)→ (ev q ⊗ id q ) ◦ h (cid:48) . Applying Ω to (5.7), we get + κ q (cid:80) i b λi = κ q a λ λ i = ⇒ = a λ i + (cid:80) i b λi λ From (2.32), we see that ( a λ , b λi ) = ( κ q f λ , κ q f iλ ) . The result follows from Lemma 5.1(i). (cid:3) Corollary 5.5.
Let D denote the matrix representation of a rotation operator ϕ in thecanonical basis. Then(i) D = κ q F qqqq (ii) F qqqq is self-inverse(iii) f λ = f λ (iv) The parity of all entries in σ ( ϕ ) cannot be the same Proof. (i) Follows directly from Theorem 5.4.(ii) D = C where C = id (since { x i } i are self-dual), whence the result follows by (i).(iii) For λ = 0 , note that (5.6) is (2.26) and so f i = κ q √ d i d q . The result follows fromLemma 5.1(i).(iv) Since ϕ is an involution for C = id , its spectrum can only consist of ± s. Observethat | tr( F qqqq ) | < N since | f ii | ≤ and | f | = d q < . By (i), tr( ϕ ) = κ q tr( F qqqq ) whence (iv) follows. (cid:3) Corollary 5.5(ii) can also be shown by applying linear map Ω (cid:48) : h (cid:55)→ h (cid:48)(cid:48) (cid:55)→ (id q ⊗ ev q ) ◦ h (cid:48)(cid:48) to (5.7), where h ∈ End( q ⊗ ) and h (cid:48)(cid:48) := h ⊗ id q .Stated differently, Corollary 5.5(iv) says that there are strictly less than N linearly inde-pendent formal diagrams in End( q ⊗ ) that are (anti)symmetric under rotation. Corollary 5.6. (Bubble-popping) (i) = √ d i i (ii) ji = κ q d q √ d i f ij i (iii) ji = (cid:112) d i d j d q + (cid:80) k d q √ d k f ki f kj k (iv) ij = κ q d q f ij KEIN-THEORETIC METHODS FOR UNITARY FUSION CATEGORIES 31 (v) ij = (cid:112) d i d j + d q (cid:80) k f ki f kj See also (5.10).
Proof. (i) Cup off the bottom of an i -bone and use (5.6) to get = √ d i i = √ d i d q Indeed, capping off both sides agrees with Θ( q, i, q ) = Φ( q, i, q ) = d q √ d i .(ii) Stacking a j -bone on top of an i -jack and using (5.6), we get(5.8) j = κ q f ij ii = κ q d q √ d i f ij ii (iii) Stack an i -bone on top of a j -bone and use (5.6).(iv) ijj i = κ q d q √ d i f ij = ⇒ = κ q d q √ d i f ij ii whence the result follows from Φ( q, i, q ) = d q √ d i . Alternatively, this identity co-incides with taking the quantum trace of Lemma 5.1(iii) for ( µ, λ ) = ( i, j ) .(v) Take the left and right partial traces of (iii) and plug in Φ( q, k, q ) . (cid:3) Corollary 5.7. F qqqq is real-symmetric. Proof.
Corollary 5.5(ii) tells us that F qqqq is Hermitian. It thus suffices to show that F qqqq is one of (a) real or (b) symmetric ; nonetheless, we will show both explicitly. Applyingthe left and right partial traces to (5.8), we obtain(5.9) ij = κ q d q f ij (a) Inverting the pretzel in Corollary 5.6(iv) via adjunction and comparing the resultto (5.9), we see that f ij = f ∗ ij . We know that entries f λ and f λ are also real fromLemma 5.1(i) and Corollary 5.5(iii).(b) Note that (5.9) can be deformed to the quantum trace of Lemma 5.1(iii) for ( µ, λ ) = ( j, i ) . Comparing scalars, we see that f ij = f ji . We also know that f λ = f λ from Corollary 5.5(iii). (cid:3) In light of Corollary 5.7, we may further simplify Corollary 5.6(v) to(5.10) ij = (cid:112) d i d j + δ ij d q Computing some F -symbols. We now turn our attention to calculating F qqqq for q self-dual using the rotation operator. If q ⊗ q = then F qqqq = [ f ] = (cid:104) κ q d q (cid:105) . In the case q ⊗ q = ⊕ x , we have(5.11) F qqqq = κ q d q (cid:112) d q − d q (cid:112) d q − d q − d q Since x is necessarily self-dual, we may apply the corollaries of Theorem 5.4. Indeed,(5.11) follows almost immediately from Lemma 5.1(i) and Corollary 5.5(iii); all that re-mains is to find f xx . Applying Corollaries 5.5(i) and (iv), we have tr (cid:0) F qqqq (cid:1) = κ q tr( ϕ ) = 0 whence f xx = − κ q d q .If we promote C to be ribbon, we may also determine f xx by combining the skein theoryfrom Section 3.2 with Theorem 5.4. Resolving as in (3.10a) and rotating, ϕ (cid:16) (cid:17) = 1 d q α + √ d x d q β x = 1 d q α (cid:18) d q + √ d x d q x (cid:19) + √ d x d q β (cid:18) √ d x d q + κ q f xx x (cid:19) = α + βd x d q + √ d x d q (cid:18) αd q + κ q βf xx (cid:19) x Comparing coefficients with the -crossing, the cup-cap component corresponds to(2.46), while the x -jack component yields f xx = β − − κ q d q αβ − . Plugging in the valuesfrom (3.13), we get f xx = − κ q d q . Remark 5.8.
Another approach to extracting information via the rotation operatoris to stack a crossing on its image under ϕ and then solve for the equation levied byReidemeister-II. This approach is equivalent to the one taken above i.e. solving(5.12) ϕ (cid:16) (cid:17) = κ q is clearly equivalent to solving ϕ (cid:16) (cid:17) = κ q . Of course, this is solved with respectto some choice of basis B . For the case q ⊗ q = ⊕ x , it is interesting to observe thatfixing B = (cid:110) , (cid:111) gave us information pertaining to R qq , while fixing B canonicalgave us information about F qqqq . Moreover, the information extracted via the latter basisrelied on that found using the former basis. KEIN-THEORETIC METHODS FOR UNITARY FUSION CATEGORIES 33
Now suppose that q ⊗ q = ⊕ x ⊕ y where C = id and C is ribbon. Following (3.17a), ϕ (cid:16) (cid:17) = A + B x + C Thm. . = A + (cid:18) √ d x d q B + C (cid:19) + κ q f xx B x + κ q f yx B y (2 . = (cid:32) A + κ q d q (cid:112) d y f yx B (cid:33) + (cid:32) √ d x d q B + C − κ q (cid:112) d y f yx B (cid:33) + κ q (cid:32) f xx B − (cid:115) d x d y f yx B (cid:33) x Solving ϕ (cid:16) (cid:17) = κ q with respect to basis (cid:110) , , x (cid:111) , we match coefficientswith (3.17b) to obtain : A = κ q (cid:32) C − − d q (cid:112) d y f yx B (cid:33) (5.13a) : A (cid:48) = κ q (cid:18) C + √ d x d q B (cid:19) − (cid:112) d y f yx B (5.13b) x : B (cid:48) = B (cid:32) f xx − (cid:115) d x d y f yx (cid:33) (5.13c)Suppose B, B (cid:48) (cid:54) = 0 . Recall from (3.20) that B (cid:48) = ± B . Also by Lemma 5.1(i) & (ii),(5.14) − κ q √ d i d q = (cid:88) j (cid:112) d j f ji whence for i = x ,(5.15) f xx = − (cid:32) κ q d q + (cid:114) d y d x f yx (cid:33) Combining (5.13c) and (5.15) eventually yields(5.16) ( f xx , f yx ) = (cid:32) ∓ d x ( d q ∓ κ q ) d q ± , ∓ (cid:112) d x d y ( d q ∓ κ q ) d q (cid:33) , B (cid:48) = ± B Setting i = y in (5.14) gives f yy = − ( κ q d q + (cid:113) d x d y f xy ) . By Corollary 5.7, we have f xy = f yx whence(5.17) f yy = ∓ d y ( d q ∓ κ q ) d q ± , B (cid:48) = ± B Theorem 5.9.
Let C be a unitary ribbon fusion category containing a fusion rule(5.18) q ⊗ q = ⊕ x ⊕ y where x, y, q ∈ Irr( C ) . If R qqx (cid:54) = R qqy then R qqx R qqy = ± where • If R qqx R qqy = − then d q = κ q (cid:32) ϑ q − ϑ − q R qqx + R qqy + 1 (cid:33) and Λ C ,q is given by (a) the framedDubrovnik polynomial (3.31) for κ q = 1 and (b) the framed link polynomial (3.33)for κ q = − . • If R qqx R qqy = 1 then d q = κ q (cid:32) ϑ q + ϑ − q R qqx + R qqy − (cid:33) and Λ C ,q is given by (c) the framedKauffman polynomial (3.32) for κ q = 1 and (d) the framed link polynomial (3.34)for κ q = − .If x and y are self-dual then(i) For R qqx R qqy = − , we have(5.19) F qqqq = κ q d q √ d x d q (cid:112) d y d q √ d x d q − d x ( κ q d q − d q + 1 − (cid:112) d x d y ( κ q d q − d q (cid:112) d y d q − (cid:112) d x d y ( κ q d q − d q − d y ( κ q d q − d q + 1 (ii) For R qqx R qqy = 1 , we have(5.20) F qqqq = κ q d q √ d x d q (cid:112) d y d q √ d x d q d x ( κ q d q + 1) d q − (cid:112) d x d y ( κ q d q + 1) d q (cid:112) d y d q (cid:112) d x d y ( κ q d q + 1) d q d y ( κ q d q + 1) d q − Corollary 5.10.
For a unitary ribbon fusion category C containing a fusion rule of theform (5.18) with x and y self-dual, we have(i) q is symmetrically self-dual(ii) σ ( ϕ ) = (cid:40) { +1 , +1 , − } , Λ C ,q is the framed Dubrovnik polynomial { +1 , − , − } , Λ C ,q is the framed Kauffman polynomial Proof. (i) tr (cid:0) F qqqq (cid:1) = ± for R qqx R qqy = ∓ when κ q = − . The result follows by Corollaries5.5 (i) and (iv).(ii) By (i), Λ C ,q is either the framed Dubrovnik or Kauffman polynomial. For theformer, tr( F qqqq ) = 1 and for the latter tr( F qqqq ) = − . The result follows byCorollary 5.5 (i). (cid:3) KEIN-THEORETIC METHODS FOR UNITARY FUSION CATEGORIES 35
Some new bases.
As an application of the results thus far, we establish some newbases for
End( q ⊗ ) where(5.21) q ⊗ q = ⊕ x ⊕ y in a unitary ribbon fusion category C with x and y self-dual. First, let us make someobservations for q such that(5.22) q ⊗ q = ⊕ k (cid:77) i =1 x i with q, x i ∈ Irr( C ) and distinct self-dual objects x i . We restrict our search to bases B satisfying the following property : (P1) The elements of B are permuted under the action of ϕ (up to a sign for -cycles). We will see that bases satisfying this property are closely related to the eigenbasis of ϕ .Clearly, the matrix representation of ϕ in such a basis is a symmetric permutation ma-trix with − ’s permitted along the diagonal. The permutation consists of -cycles andsigned -cycles. Then +1 ’s and − ’s along the matrix diagonal respectively correspondto ‘positive’ and ‘negative’ -cycles. Let N := dim (cid:0) End( q ⊗ ) (cid:1) , n := { cycles } b := { positive -cycles } , f := { negative -cycles } Assume B satisfying (P1) exists. Then write B = { D ij } ( n,l ( i ))( i,j ) where i indexes the n cycles and l ( i ) is the length of the i th cycle. We have ϕ ( D ij ) = D i,j +1 , l ( i ) = 2 D ij , i indexes a positive -cycle − D ij , i indexes a negative -cycle(5.23)where j denotes j modulo . Recall that σ ( ϕ ) must consist of a mixture of ± ’s. Let V and V − respectively denote the +1 and − eigenspaces of ϕ . Then(5.24) dim( V ) = n − f , dim( V − ) = n − b whence(5.25) N = 2 n − b − f and (cid:24) N (cid:25) ≤ n ≤ N where the upper bound is realised when B is an eigenbasis for ϕ . Example 5.11.
We can use the above to determine the possible actions (as a signedpermutation) of ϕ on B given σ ( ϕ ) . We will denote such an action by the signed cycletype ( a , . . . , a n ) where | a i | = l ( i ) and a i = ± encodes the sign of a -cycle. In thefollowing, we exclude the instances where n = N (i.e. eigenbases).(i) N = 2 : ( n, b, f ) (1 , , σ ( ϕ ) { +1 , − } Cycle type (2) Recall that the method employed for determining Λ C ,q in Section 3 relied on expressing a crossingas a linear combination of morphisms that were invariant under the action of the rotation operator (upto permutation). This motivates the study of bases satisfying (P1) . (ii) N = 3 : ( n, b, f ) (2 , ,
0) (2 , , σ ( ϕ ) { +1 , +1 , − } { +1 , − , − } Cycle type (2 ,
1) (2 , − (iii) N = 4 : ( st instance where there are two distinct cycle types for the same σ ( ϕ ) ). ( n, b, f ) (3 , ,
1) (3 , , (3,0,2) (2,0,0) σ ( ϕ ) { +1 + 1 , − , − } { +1 , +1 , +1 , − } { +1 , − , − , − } { +1 , +1 , − , − } Cycle type (2 , , −
1) (2 , ,
1) (2 , − , −
1) (2 , etc. • We already encountered basis (cid:110) , (cid:111) corresponding to Example 5.11(i). • We show by construction that there exist bases corresponding to Example 5.11(ii).We define(5.26) J X := X + X and J (cid:48) X := X − X Observe that for (5.21), + = √ d x d q − J x + (cid:112) d y d q − J y (5.27a) − = − √ d x d q + 1 J (cid:48) x − (cid:112) d y d q + 1 J (cid:48) y (5.27b)whence (cid:112) d x J x + (cid:112) d y J (cid:48) y = ( d q − (cid:16) + (cid:17) − (cid:112) d y y (5.28a) (cid:112) d x J x − (cid:112) d y J (cid:48) y = ( d q + 1) (cid:16) − (cid:17) + 2 (cid:112) d x x (5.28b)Recall from Corollary 5.10(i) that κ q = 1 . Expanding in the canonical basis, (cid:112) d x J x + (cid:112) d y J (cid:48) y = (cid:18) d q − d q (cid:19) + (cid:112) d x (cid:18) − d q (cid:19) x − (cid:112) d y (cid:18) d q (cid:19) y (5.29a) (cid:112) d x J x − (cid:112) d y J (cid:48) y = (cid:18) d q − d q + 2 d x d q (cid:19) + (cid:112) d x (cid:18) d q + 2 f xx (cid:19) x (5.29b) + (cid:20)(cid:112) d y (cid:18) d q (cid:19) + 2 (cid:112) d x f yx (cid:21) y where in (5.29b) we used (5.6). Let(5.30) J + xy := (cid:112) d x J x + (cid:112) d y J (cid:48) y , J − xy := (cid:112) d x J x − (cid:112) d y J (cid:48) y Lemma 5.12. J + xy and J − xy are linearly independent. Proof. ϕ ( J + xy ) = J − xy . Suppose J − xy = z J + xy for some z ∈ C . Then ϕ ( J + xy ) = z J + xy whence J + xy = ±J − xy . For z = +1 and z = − we respectively get J (cid:48) y = 0 and J x = 0 , both ofwhich yield a contradiction. (cid:69) KEIN-THEORETIC METHODS FOR UNITARY FUSION CATEGORIES 37
Theorem 5.13.
Let q be defined as in (5.21). Then(i) (cid:110) J + xy , J − xy , + (cid:111) defines a basis for End( q ⊗ ) when Λ C ,q is the framedDubrovnik polynomial.(ii) (cid:110) J + xy , J − xy , − (cid:111) defines a basis for End( q ⊗ ) when Λ C ,q is the framedKauffman polynomial.Note that the bases in the above theorem satisfy (P1) (see Example 5.11(ii)), and thatwe can permute labels x and y by the symmetry of our construction. Proof.
By Lemma 5.12, it suffices in each case to show that the final basis element is nota linear combination of the first two. For c , c ∈ C , c J + xy + c J − xy = a + a x + a y where a := 1 d q (cid:2) (1 − d q )( c − c ) + 2 c d x (cid:3) , a := (cid:112) d x (cid:20)(cid:18) d q (cid:19) c + (cid:18) − d q (cid:19) c + 2 c f xx (cid:21) a := (cid:112) d y (cid:18) d q (cid:19) ( c − c ) + 2 c (cid:115) d x d y f yx Recall from (5.13c) that(5.31) (cid:115) d x d y f yx = f xx − r where r = 1 , − when Λ C ,q is the framed Dubrovnik and framed Kauffman polynomialrespectively. Thus,(5.32) a = (cid:115) d y d x a − (cid:112) d y ( c + rc ) (i) Suppose there exist c and c such that c J + xy + c J − xy = + . Then ( a , a , a ) = (cid:32) d q , √ d x d q , (cid:112) d y d q (cid:33) . Setting r = 1 , (5.32) gives c = − c . Nowcomparing values for a yields c = 12 (cid:18) d q d y (cid:19) , c = − (cid:18) d q d y (cid:19) whence comparing values for a yields f xx = − d y d q (1 + d q ) − d q where we can manipulate the right-hand side to get − d y d q (1 + d q ) − d q = − d x d q ( d q − − d x d q ( d q − (5 . = − f xx − d x d q ( d q − implying that f xx = − d x d q ( d q − . Rearranging the expression for f xx from (5.19), f xx = 1 − d x ( d q − d q = d q ( d q − − d x ( d q + 1) d q ( d q −
1) = d q d y − d x d q ( d q − whence we arrive at a contradiction since d q , d y (cid:54) = 0 . (cid:69) (ii) Suppose there exist c and c such that c J + xy + c J − xy = − . Then ( a , a , a ) = (cid:32) d q − , √ d x d q , (cid:112) d y d q (cid:33) . Setting r = − , (5.32) gives c = c . Nowcomparing values for a yields c = c = 12 (cid:18) − d q d x (cid:19) whence comparing values for a yields f yx = (cid:112) d x d y ( d q − d q From (5.20), f yx = (cid:112) d x d y ( d q + 1) d q whence we arrive at a contradiction since d q (cid:54) = 0 . (cid:69) Corollary 5.14. (Diagonalising ϕ ) Following the notation from (5.24), we have
End( q ⊗ ) = V ⊕ V − .(i) If Λ C ,q is the framed Dubrovnik polynomial then(5.33) V = span (cid:110) J x , + (cid:111) , V − = span (cid:8) J (cid:48) y (cid:9) (ii) If Λ C ,q is the framed Kauffman polynomial then(5.34) V = span {J x } , V − = span (cid:110) J (cid:48) y , − (cid:111) Proof.
Given b , b , b ∈ C , we have ϕ (cid:16) b J + xy + b J − xy + b (cid:16) ± (cid:17)(cid:17) = b J − xy + b J + xy + b (cid:16) ± (cid:17) The result then easily follows from solving b J − xy + b J + xy + b (cid:16) ± (cid:17) = ± (cid:16) b J + xy + b J − xy + b (cid:16) ± (cid:17)(cid:17) (cid:3) Remark 5.15. (i) For the N = 2 case, (cid:110) + , − (cid:111) trivially defines an eigenbasis for ϕ .(ii) By symmetry of our construction, we may permute labels x and y in (5.33) and(5.34). For e.g. (5.33), this tells us that J y ∈ V and J (cid:48) x ∈ V − . Recovering theprecise linear relations is a straightforward task. KEIN-THEORETIC METHODS FOR UNITARY FUSION CATEGORIES 39 Some Physical Remarks
It is conjectured that every unitary MTC corresponds to a Chern-Simons-Wittentopological quantum field theory (TQFT); these QFTs describe the low-energy be-haviour of condensed matter systems confined to two spatial dimensions. Such systemsare expected to host point-like quasiparticle excitations (called anyons ) which have exoticexchange statistics. Throughout this section, we will assume that C is a unitary MTC,and that any trivalent vertices implicitly carry the normalisation described in Section 2.6. Unitary MTC C Anyonic system
Simple objects Anyons/quasiparticlesDual object AntiparticleTrivial object/ VacuumLabel x ∈ Irr( C ) Topological charge/anyon ‘type’Triangular space Fusion/splitting Hilbert spaceUnitary transformations on triangular spaces Gauge transformationsMorphisms Physical processes/operators/worldlinesNonzero elements of
Hom( (cid:78) ni =1 x i , x ) Fusion statesParenthesisation of x ⊗ · · · ⊗ x n Choice of fusion basis/orderAssociator/ F -matrix Change of fusion basisBraiding/ R -matrix Particle exchangeEvaluation/coevaluation Annihilation/creationTwist factor ϑ x Topological spin of x Coloured braid groupoid representation Exchange statistics
Table 1.
Dictionary of terms adapted from [19, Table 6.1]A charge x ∈ Irr( C ) is called abelian (or an abelion ) if (cid:80) z N xyz = 1 for all y ∈ Irr( C ) ;else, it is called non-abelian (or a non-abelion ). If q is a self-dual abelion, then we knowfrom Sections 3.1 and 4 that q is always one of the following:(i) q is called a boson if ϑ q = +1 (ii) q is called a fermion if ϑ q = − (iii) q is called a semion if ϑ q = ± i where in the above, ϑ q = R qq = [ R qq ] . In particular, (i) and (ii) correspond to (3.7) and(4.9), while (iii) corresponds to (3.8) and (4.10). Semions q always have κ q = − . They are ‘topological’ in the sense that propagators do not depend on the spacetime metric. I.e. a Hom-space spanned by trivalent vertices. A trivalent vertex corresponds to the ‘fusion’ or‘splitting’ of anyons. This terminology is rooted in the associated exchange statistics; namely, all R -matrices involving x will be scalar (and will thus commute). Caveat:
This is not a true fermion since it is not transparent; true fermions are identified with ina MTC. 2d electron liquids constitute the most important class of topological phases of matter: in orderto fully describe such systems, a refinement of MTCs called spin MTCs are required [30].
Remark 6.1.
Throughout the remainder of this section, we will reference the Fibonacciand Ising models in some examples.(i) The
Fibonacci model is given by a set of charges { , τ } with nontrivial fusion rule τ ⊗ τ = ⊕ τ where τ is called the Fibonacci anyon. This model is realised by two distinctunitary MTCs , both of which have κ τ = 1 . We have d τ = φ where φ := 1 + √ is the golden ratio.(ii) The Ising model is given by a set of charges { , ψ, σ } with nontrivial fusion rules σ ⊗ σ = ⊕ ψ , σ ⊗ ψ = ψ ⊗ σ = σ , ψ ⊗ ψ = where σ is called the Ising anyon and ψ is a fermion. This model is realised by distinct unitary MTCs (of which half have κ σ = 1 and the other half κ σ = − ).We have d ψ = 1 and d σ = √ .6.1. Quantum entanglement.
If the total (topological) charge of a collection of adjacentquasiparticles is fixed (Figure 7), this charge is called the superselection sector of thecharges; else, the quasiparticles exist in a superposition of total charges (i.e. a super-position of fusion states).
Remark 6.2.
We write V a ...a m b ...b n := Hom( (cid:78) mi =1 a i , (cid:78) nj =1 b j ) . Note that ‘triangular spaces’are of the form V abc and V cab . Figure 7. f ∈ V xx ...x n and f ∈ V x ...x n x (cid:48) . The system of particles x , . . . , x n lies in superselection sector x (cid:48) = x by Schur’s lemma. This is interpretedas the conservation of (topological) charge. Figure 8. f ∈ V xx ...x m , g ∈ V yy ...y n and p ∈ V x ...x m y ...y n x (cid:48) y (cid:48) . (cid:45) k denotes k lines carrying some permissible labels and orientations. ν j is a (linearcombination of) permissible diagram(s) of oriented braided strands andpossibly trivalent vertices. The skeletal data of one MTC can be obtained from the other via Hermitian conjugation.
KEIN-THEORETIC METHODS FOR UNITARY FUSION CATEGORIES 41
Now take two adjacent systems of quasiparticles x , . . . , x m and y , . . . , y n that are ini-tialised in superselection sectors x and y respectively (Figure 8). We consider somedifferent cases:(1) k = l = 0 . Then ν ∈ V x ...x m x , ν ∈ V y ...y n y and ν ∈ C .(2) Only one of k and l is nonzero. Then if k is zero, we have ν ∈ V x ...x m x ; and if l is zero, we have ν ∈ V y ...y n y .(3) where k and l are nonzero.In the trivial cases above, the systems lie in superselection sectors x and y respectively.(4) where k and l (cid:48) are nonzero, ν is a ( k (cid:48) + l (cid:48) ) -braid suchthat it cannot be partitioned as in (3); and the incoming and outgoing strands ona given side are connected. There are two possibilities:(a) There is only one permissible label for each of x (cid:48) and y (cid:48) . Then the systemswill lie in superselection sectors x (cid:48) and y (cid:48) post-braiding.(b) Else, ν is an example of an entangling operator in the sense that we can nolonger write the vector p ∈ V x ...x m y ...y n x (cid:48) y (cid:48) as p = f (cid:48) ⊗ g (cid:48) for some f (cid:48) ∈ V x ...x m x (cid:48) and g (cid:48) ∈ V y ...y n y (cid:48) . Here, ‘tangling’ between systems results in the entanglementof their fusion states.For instance,(6.1) = (cid:88) c N byc ϑ c ϑ b ϑ y This shows that the left-hand side is not an inner product for y nontrivial, and so x (cid:48) isnot necessarily x (i.e. the total charge of a and b may no longer be constrained to a singlepossibility, in which case a and b are entangled with y ). Figure 9.
Extension of Figure 8. h ∈ V zxy , h (cid:48) ∈ V x (cid:48) y (cid:48) z (cid:48) and p (cid:48) ∈ V x ...x m y ...y n z . E.g. this could happen if m = n = 1 ; or if each system is composed of abelian anyons; or if V x ...x m y ...y n z is -dimensional in Figure 9. In all these examples, ( x (cid:48) , y (cid:48) ) = ( x, y ) . If ν is as in cases (1)-(4a) for Figure 9, then we may write p (cid:48) = f (cid:48) ⊗ g (cid:48) ⊗ h (cid:48) for some f (cid:48) ∈ V x ...x m x (cid:48) and g (cid:48) ∈ V y ...y n y (cid:48) . That is, p (cid:48) has a nonzero component in precisely onesummand of (6.2). Note that we are guaranteed ( x (cid:48) , y (cid:48) ) = ( x, y ) in cases (1)-(3).(6.2) V x ...x m y ...y n z = (cid:77) x (cid:48) ,y (cid:48) V x ...x m x (cid:48) ⊗ V y ...y n y (cid:48) ⊗ V x (cid:48) y (cid:48) z If ν is as in case (4b), then the two systems no longer belong to distinct superselectionsectors post-braiding: they lie in the joint superselection sector z . In particular, p (cid:48) hasnonzero components in more than one summand of (6.2), and is thus an entangled state.Let us consider some concrete examples. Example 6.3.
We choose the Ising MTC for which(6.3) F := F σσσσ = 1 √ (cid:18) − (cid:19) , R := R σσ = e − i π (cid:18) i (cid:19) where κ σ = 1 and the matrices act on a basis in the order , ψ . Take the process(6.4)and consider the space V σσσσψ with the fusion basis fixed as follows:(6.5) V σσσσψ = (cid:77) x (cid:48) ,y (cid:48) V σσx (cid:48) ⊗ V σσy (cid:48) ⊗ V x (cid:48) y (cid:48) ψ = span C {| x (cid:48) y (cid:48) (cid:105)} x (cid:48) ,y (cid:48) : N σσx (cid:48) N σσy (cid:48) N x (cid:48) y (cid:48) ψ (cid:54) =0 Note that ( x (cid:48) , y (cid:48) ) takes values ( , ψ ) or ( ψ, ) . From (6.4), the system is initialised instate | ψ (cid:105) . Performing a trivial change of basis, (cid:0) F σσ ψ ⊗ id V σσ (cid:1) | ψ (cid:105) = ∈ V σσψ ⊗ V σσσσ It suffices to consider V σσσσ to compute the action of the braiding. We have V σσσσ = span C , =: span C {| (cid:105) , | ψ (cid:105)} Write (cid:18) ab (cid:19) = a | (cid:105) + b | ψ (cid:105) . The action of the braiding is given by F R F − (cid:18) (cid:19) = e − iπ (cid:18) (cid:19) (cid:18) (cid:19) = e − iπ | ψ (cid:105) Post-braiding, we thus have ∈ V σσψ ⊗ V σσσσ KEIN-THEORETIC METHODS FOR UNITARY FUSION CATEGORIES 43
Switching back to the original basis in (6.4), (cid:16) F σσψψ (cid:17) − ⊗ id V σσψ (cid:104) F σσψψ (cid:105) − = (cid:104) F σσψψ (cid:105) − | ψ (cid:105) whence (6.4) is(6.6) =Thus, the monodromy of the middle pair of Ising anyons results in the teleportation ofthe fermion from the left pair to the right pair. This is an example of case (4a). Example 6.4. (Bell state via Ising model)
We fix the same Ising MTC with F and R as in Example 6.3. Take the process(6.7)and consider the space V σσσσ with the fusion basis fixed as follows:(6.8) V σσσσ = (cid:77) x (cid:48) ,y (cid:48) V σσx (cid:48) ⊗ V σσy (cid:48) ⊗ V x (cid:48) y (cid:48) = span C {| x (cid:48) y (cid:48) (cid:105)} x (cid:48) ,y (cid:48) : N σσx (cid:48) N σσy (cid:48) N x (cid:48) y (cid:48) (cid:54) =0 Note that ( x (cid:48) , y (cid:48) ) takes values ( , ) or ( ψ, ψ ) . From (6.7), the system is initialised instate | (cid:105) . Performing a trivial change of basis, (cid:16) id V σσ ⊗ (cid:2) F σσ (cid:3) − (cid:17) | (cid:105) = ∈ V σσσσ ⊗ V σσ It suffices to consider V σσσσ to compute the action of the braiding. We have V σσσσ = span C , =: span C {| (cid:105) , | ψ (cid:105)} Write (cid:18) ab (cid:19) = a | (cid:105) + b | ψ (cid:105) . The action of the braiding is given by F − R − F (cid:18) (cid:19) = e iπ (cid:18) − i i i − i (cid:19) (cid:18) (cid:19) = e iπ √ (cid:18) e − iπ e iπ (cid:19) Post-braiding, we thus have √ + i ∈ V σσσσ ⊗ V σσ Switching back to the original basis in (6.7), id V σσ ⊗ F σσ , id V σσψ ⊗ F ψσσ where F σσ = F ψσσ = [1] . Note that the change of basis on the left is trivial. It followsthat (6.7) is(6.9) = √ + i · where the fusion state is(6.10) √ | (cid:105) + i | ψψ (cid:105) ) ∈ V σσσσ This is a
Bell state : the fusion states of the left and right pairs of Ising anyons are maximally entangled . Thus, the anticlockwise exchange of the middle pair of anyonsrealises an example of an entangling operator. We can concretely interpret this exchangeas an entangling quantum gate:
Figure 10. (i) b is a -braid whose strands are labelled by Ising anyons.Letting | (cid:105) and | ψ (cid:105) correspond to logical basis states | (cid:105) and | (cid:105) respectively,(i) can be interpreted as a quantum circuit as in (ii). (ii) is a -qubitquantum circuit initialised in state | (cid:105) ⊗ | (cid:105) , and with C representing someconfiguration of quantum gates corresponding to the action of b .(iii) b = σ − . (iv) The quantum circuit realised by b = σ − .If the orientation of the exchange is reversed in (iii), we simply append a Pauli- Z gate tothe bottom wire in (iv). Example 6.5.
We choose the Fibonacci MTC for which(6.11) F := F ττττ = (cid:18) φ − φ − / φ − / − φ − (cid:19) , R := R ττ = (cid:18) e − i π e i π (cid:19) where κ σ = 1 and the matrices act on a basis in the order , τ . Take the process(6.12) KEIN-THEORETIC METHODS FOR UNITARY FUSION CATEGORIES 45 and consider the space V ττττ with the fusion basis fixed as follows:(6.13) V ττττ = (cid:77) x (cid:48) ,y (cid:48) V ττx (cid:48) ⊗ V ττy (cid:48) ⊗ V x (cid:48) y (cid:48) = span C {| x (cid:48) y (cid:48) (cid:105)} x (cid:48) ,y (cid:48) : N ττx (cid:48) N ττy (cid:48) N x (cid:48) y (cid:48) (cid:54) =0 Note that ( x (cid:48) , y (cid:48) ) takes values ( , ) or ( τ, τ ) . From (6.12), the system is initialised instate | (cid:105) . Performing a trivial change of basis, (cid:16) id V ττ ⊗ (cid:2) F τττ (cid:3) − (cid:17) | (cid:105) = ∈ V ττττ ⊗ V ττ It suffices to consider V ττττ to compute the action of the braiding. We have V ττττ = span C , =: span C {| (cid:105) , | τ (cid:105)} Write (cid:18) ab (cid:19) = a | (cid:105) + b | ψ (cid:105) . The action of the braiding is given by F − R F (cid:18) (cid:19) = e − i π (cid:18) φ − + φ − e i π φ − / (1 − e i π ) φ − / (1 − e i π ) φ − + φ − e i π (cid:19) (cid:18) (cid:19) = (cid:32) − φ − φ − / (cid:16) e − i π − e − i π (cid:17)(cid:33) Post-braiding, we thus have φ − + φ − / (cid:16) e − i π − e i π (cid:17) ∈ V ττττ ⊗ V ττ Switching back to the original basis in (6.12), id V ττ ⊗ F ττ , id V τττ ⊗ F τττ where F ττ = F τττ = [1] . Note that the change of basis on the left is trivial. It followsthat (6.12) is(6.14) = φ − · + φ − / (cid:16) e − i π − e i π (cid:17) · where the fusion state is(6.15) φ − | (cid:105) + φ − / (cid:16) e − i π − e i π (cid:17) | τ τ (cid:105) ∈ V ττττ Thus, the monodromy of the middle pair of Fibonacci anyons entangles the fusion statesof the left and right pair. This is an example of case (4b), where ‘tangling’ two systemsresults in their entanglement.
Wilson loops.
In the field-theoretic context, link diagrams in
End( ) are called Wilson loops : physically, they correspond to a process where anyons are pair-createdfrom the vacuum, braided, and then fused straight back to the vacuum. Given sucha diagram D , the expectation value (cid:104) W ( D ) (cid:105) of the Wilson loop can be interpreted asthe amplitude of said process, and may be evaluated via a functional integral using theChern-Simons action (see e.g. [31, Section 3]). Alternatively, we can evaluate (cid:104) W ( D ) (cid:105) via the skeleton of the corresponding MTC C , or by using a skein-theoretic method. Ourdiscussion will focus on the latter two approaches.Note that (cid:104) W ( D ) (cid:105) is a framed, oriented link invariant at a root of unity. In orderfor (cid:104) W ( D ) (cid:105) to be a physically meaningful quantity, it must be gauge-invariant: in thecategorical setting, this means that (cid:104) W ( D ) (cid:105) should be invariant under any unitary trans-formations on triangular spaces. Procedure 6.6 ( (cid:104) W ( D ) (cid:105) via skeleton of C ) . (1) Isotope D (using only Reidemeister-II and III moves) so that we obtain somebridge representation D (cid:48) . That is,(6.16) D planar isotopy D (cid:48) = where b is a n -braid, and κ & κ are some configurations of n non-intersectingcaps and cups respectively.(2) κ corresponds to a normalised element of Hom( , (cid:78) nk =1 X i k ) . Take the adjointmorphism | ψ (cid:105) ∈ Hom( (cid:78) nk =1 X i k , ) .(3) Using some operator F , transform | ψ (cid:105) into a fusion basis compatible with theconfiguration κ (labels do not matter).(4) Transform F | ψ (cid:105) using the operator ρ ( b ) where ρ is a unitary linear action of thebraid groupoid on n strands.(5) κ corresponds to a normalised element | ψ (cid:48) (cid:105) ∈ Hom( (cid:78) nk =1 X j k , ) . (cid:104) W ( D ) (cid:105) isfound by applying the linear form (cid:104) ψ (cid:48) | to the fusion state obtained in (4).More succinctly, Procedure 6.6 tells us that(6.17) (cid:104) W ( D ) (cid:105) = ( − z (cid:104) ψ (cid:48) | ρ ( b ) F | ψ (cid:105) where | ψ (cid:105) and | ψ (cid:48) (cid:105) are called vacuum states, and z is the total number of zig-zagsstraightened along components spanned by antisymmetrically self-dual anyons in (1);note that there may be more than one fusion basis compatible with each of κ and κ . The categorical notion of ‘gauge-invariance’ does not coincide with that used by physicists; however,categorical gauge-invariance is necessary for physical gauge-invariance. For instance, (cid:104) W ( D ) (cid:105) can bewritten in the form re iθ where r ∈ [0 , : the phase e iθ should not be regarded as physically significant(since it is subject to global gauge transformations), but r may be interpreted as the probability ofobserving the Wilson loop. In cases where there are components of D spanned by antisymmetrically self-dual anyons, we alsoneed to account for any signs accumulated from straightening zig-zags along these components. KEIN-THEORETIC METHODS FOR UNITARY FUSION CATEGORIES 47
Figure 11.
For a process given by a link diagram D as in (i), we isotopethis into a diagram D (cid:48) (as in step (1) of Procedure 6.6) illustrated in (ii),where we have outlined the areas of D (cid:48) corresponding to κ , κ and b .We now consider some concrete examples. For each of these, we assume that eachcomponent of the Wilson loop is spanned by a self-dual anyon, and that the topologicalcharge of anyons spanning distinct components is the same. We will also have that κ isa reflection (in the horizontal) of κ , so F = id and | ψ (cid:48) (cid:105) = | ψ (cid:105) in Procedure 6.6. Hence,(6.18) (cid:104) W ( D ) (cid:105) = ( − z (cid:104) ψ | ρ ( b ) | ψ (cid:105) where ρ : B n → U ( V q n ) is a unitary linear representation of the braid group as in (4.6),and q is the topological charge of each component. In each example, we compute (cid:104) W ( D ) (cid:105) using Procedure 6.6, and also skein-theoretically. For the latter approach, we have(6.19) (cid:104) W ( D ) (cid:105) = ( − z d −B ( L ) q Λ C ,q ( D ) where L is the link for which D is a diagram, B ( L ) is the bridge number of L , and Λ C ,q isthe framed link invariant associated to the fusion rule for q ⊗ . The factor d −B ( L ) q accountsfor the normalisation of d − / q assigned to caps and cups, of which only B ( L ) need re-scaling (owing to our choice of normalisation). In the examples that follow, it is easy to see that (cid:104) W ( D ) (cid:105) is a gauge-invariant quantity:when Λ C ,q is given by one of the framed link invariants (3.14)-(3.15) or (3.31)-(3.34), thecoefficients in the corresponding skein relation are gauge-invariant. Example 6.7.
Let C be the Fibonacci MTC for which F and R are as in Example 6.5,and let D be given by the right-handed trefoilThen, | ψ (cid:105) = ∈ V ττττ ⊗ V ττ In the context of Procedure 6.6, note that B ( L ) = n . This normalisation ensures that the probability of measuring fusion outcomes is isotopy-invariant.
It suffices to consider V ττττ to compute the action of the braiding, which is F − R F = e − i π F (cid:18) e i π (cid:19) F = e − i π (cid:18) φ − + e i π φ − φ − / (1 + e − i π ) φ − / (1 − e i π ) φ − − e − i π φ − (cid:19) whence (cid:104) W ( D ) (cid:105) = e − i π φ − + e − i π φ − . Alternatively, we know that Λ := Λ C ,τ is theKauffman bracket (3.14). Then, Λ( D ) = β (cid:42) (cid:43) + β − (cid:42) (cid:43) (= β ˜ S ττ + β − ϑ − τ d τ ) where the Kauffman bracket of the Hopf link is β + β + β − + β − , whence(6.20) (cid:104) D (cid:105) = β + β + β − − β − Recalling that β = R τττ here, (cid:104) D (cid:105) = e − i π + e − i π φ . Then using (6.19), (cid:104) W ( D ) (cid:105) = d − τ (cid:104) D (cid:105) (which agrees with our skeletal calculation). Example 6.8.
Let C be the Ising MTC for which(6.21) F := F σσσσ = − √ (cid:18) − (cid:19) , R := R σσ = e i π (cid:18) − i (cid:19) where κ σ = − and the matrices act on a basis in the order , ψ . Let D be given by D = planar isotopy = D (cid:48) (i.e. the left-handed trefoil). Then, | ψ (cid:105) = ∈ V σσ ⊗ V σσσσ It suffices to consider V σσσσ to compute the action of the braiding, which is F RF − R − F RF − = − √ e − i π (cid:18) − (cid:19) whence (cid:104) W ( D ) (cid:105) = √ e − i π . Alternatively, Λ := Λ C ,σ is the framed invariant (3.15). Then, Λ( D ) = β − · Λ − β · Λ (cid:32) (cid:33) (= β − ˜ S σσ − βϑ σ d σ ) where Λ = β · Λ (cid:18) (cid:19) − β − · Λ (cid:32) (cid:33) (= βϑ q d q − β − κ q ϑ − q d q )= β (cid:34) β · Λ (cid:16) (cid:17) − β − · Λ (cid:32) (cid:33)(cid:35) − β − (cid:34) β · Λ (cid:32) (cid:33) − β − · Λ (cid:32) (cid:33)(cid:35) = β + β + β − + β − KEIN-THEORETIC METHODS FOR UNITARY FUSION CATEGORIES 49 whence Λ( D ) = − β + β + β − + β − . Recalling that β = R σσψ here, Λ( D ) = √ e − i π .Then using (6.19), (cid:104) W ( D ) (cid:105) = d − σ Λ( D ) (which agrees with our skeletal calculation). Example 6.9.
Let C be the ( A , MTC, for which partial skeletal data can be foundin [6]. We have
Irr( C ) = { , α, β } and nontrivial fusion rules α ⊗ α = ⊕ β , α ⊗ β = α ⊕ β , β = ⊕ α ⊕ β where κ α = κ β = 1 , d α = 2 cos( π ) , d β = 2 cos( π )+1 , and R ββ = diag( e − i π , e − i π , e − i π ) acts on a basis in the order , α, β . Let R := R ββ , F := F ββββ , and let D be given byThen, | ψ (cid:105) = ∈ V ββββ ⊗ V ββ It suffices to consider V ββββ to compute the action of the braiding, which is given by F − R F . Using Lemma 5.1 (i) and the symmetry of F (Corollary 5.7), we see that (cid:104) W ( D ) (cid:105) = (cid:104) ¯ ψ | F − R F | ¯ ψ (cid:105) = (cid:16) d β √ d α d β √ d β (cid:17) (cid:16) e − i π d β e − i π √ d α d β e i π √ d β (cid:17) T = e − i π d β + e − i π d α d β + e i π d β where | ¯ ψ (cid:105) denotes the component of | ψ (cid:105) in V ββββ . Alternatively, we know that Λ := Λ C ,β is the framed Dubrovnik polynomial (3.31). Then, Λ − Λ (cid:18) (cid:19) = z (cid:34) Λ (cid:18) (cid:19) − Λ (cid:32) (cid:33)(cid:35) whence Λ( D ) = d β + d β z ( a − a − ) . Recalling that a = ϑ β and z = R ββα + R βββ here, weget Λ( D ) = − d α (= ˜ S ββ ) . Then using (6.19), we get (cid:104) W ( D ) (cid:105) = d − β Λ( D ) , which agreeswith our skeletal calculation.6.3. Duality.
Let q be some self-dual topological charge. Then for a model of anyonswhose superselection sectors are determined by the (particle) exchange operator,(6.22) V qq = (cid:77) X : N qqX (cid:54) =0 V qqX is the eigenspace decomposition into superselection sectors under R qq . We have(6.23) R qq = (cid:77) X N qqX (cid:77) i =0 [ R qqX ] Another way to calculate this is using Theorem 3.4 (ii), which tells us that Λ( D ) = ( i ) − (cid:104) D (cid:105) − iβ .Since D is the mirror knot of the trefoil in Example 6.7, (cid:104) D (cid:105) β is obtained by setting β → β − in (6.20).Then one can check that Λ( D ) = i (cid:104) D (cid:105) − iβ = (cid:104) D (cid:105) β . where scalars R qqX ∈ U (1) are the eigenvalues of R qq . It follows that there are no repeatedvalues in the set { R qqX } X , whence Lemma 5.3 gives us Theorem 6.10. Theorem 6.10 ( Self-duality of topological charges ) . Given a self-dual topologicalcharge q in a model whose superselection sectors are determined by the exchange operator,we have that all charges X for which N qqX (cid:54) = 0 are also self-dual.6.4. Applications to quantum computing.
The fusion space of non-abelian anyonscould be used for encoding and processing quantum information in a way that is innatelyprotected from the usual sources of noise: this scheme is known as topological quantumcomputation [9, 10, 11]. The braiding of anyons realises the role of quantum gates, whilethe measurement of fusion outcomes corresponds to the readout. It can be shown thatthe Fibonacci model is universal for quantum computation , i.e. ρ : B → U ( V ττττ ) isdense in SU(2) ; whereas the Ising model is not, i.e the image of ρ : B → U ( V σσσσ ) isa finite subgroup of SU(2) . Abelian anyons can also be used for implementing robustquantum memories (e.g. toric code). The detection and control of anyons is underway.For instance, Ising and Fibonacci anyons are respectively expected to manifest in the ν = and ν = fractional quantum Hall states. Concluding Remarks and Outlook
Using the rotation operator, we exploited the graphical calculus as a tool for exploringunitary spherical fusion categories (and their braided counterparts). We also used thisapproach to learn more about the link invariants associated to fusion rules of a particularform. Below, we summarise some of the highlights of the paper and discuss some possibledirections for future work.7.1.
Quantum invariants.
Using the rotation operator, we extended [14, Theorems 3.1 & 3.2] to cover the anti-symmetrically self-dual cases. This produced skein relations (3.15), (3.33) and (3.34): tothe knowledge of the authors, these have not previously appeared in the literature. InSection 3.4, we briefly investigated some properties of quantum invariants associated toantisymmetrically self-dual objects. Lickorish found a formula [25] relating the Dubrovnikand Kauffman polynomials: in Theorem 3.4, we presented a similar formula relating theKauffman bracket (3.14) to the invariant (3.15).
Problem 1.
Let D and D (cid:48) be link diagrams that are equivalent under framed isotopy, andlet k := |W ( D ) − W ( D (cid:48) ) | . Is it always true that (7.1) Λ C ,q ( D (cid:48) ) = ( − k Λ C ,q ( D ) given Λ C ,q for some q with κ q = − ? This is Problem 3.6. Recall that W denotes the “local writhe” (which was defined beneathTheorem 3.4). In Proposition 3.7, we showed that the answer to this question is positivewhen q ⊗ = ⊕ x . We believe that more work is required to understand quantum invariantsassociated to antisymmetrically self-dual objects. Here, this means that any unitary can be realised with accuracy (cid:15) via braiding; see e.g. [32]. See “Property F conjecture” : a charge q supports universal computation if and only if d q (cid:54)∈ Z [33]. Ising and Fibonacci theories are also respectively referred to as
SU(2) and SU(2) theories in theliterature (where SU(2) k denotes a “level k ” Chern-Simons theory). KEIN-THEORETIC METHODS FOR UNITARY FUSION CATEGORIES 51
We determined the framed link invariants Λ C ,q associated to (3.1) for k = 1 , . Naturally: Problem 2.
What is Λ C ,q when k ≥ ? Partial results are known when k = 3 . If q ⊗ = ⊕ x ⊕ y ⊕ q , then Λ C ,q is said to beKuperberg’s G invariant in most “nontrivial” cases [14]. See also [34].We narrowed our focus to discussing skein-theoretic methods for evaluating link diagramsin End( ) when all components are labelled by the same self-dual element q ∈ Irr( C ) . Moregenerally, one could ask the same question but for(i) “Polychromatic” link diagrams (i.e. components may have distinct labels), or(ii) When the labels are not necessarily self-dual (so that orientation matters).For instance, if ∈ C when C is a Temperley-Lieb-Jones (TLJ) category, then any poly-chromatic link diagram can be evaluated as an element of the Kauffman bracket skein al-gebra: each component is replaced by the corresponding closed Jones-Wenzl idempotent,and the diagram is evaluated via skein relations (3.14). In TLJ categories, all objects aresymmetrically self-dual. An important class of TQFTs known as Jones-Kauffman theo-ries are described by TLJ categories (e.g. Ising and Fibonacci MTCs): here, Jones-Wenzlidempotents may be reinterpreted as anyons [28].By studying unitary representations of C [ B n ] that factor through the Iwahori-Hecke andTemperley-Lieb algebras, we found a skein relation (4.17) for the framed HOMFLY-PTpolynomial. This specialised to (3.14)-(3.15) in the context of a RFC C , since b = ± a − when κ q = ± . Similarly: Problem 3.
Is there some -variable link polynomial that specialises to (3.31)-(3.34)? At the end of Section 4, we see that the representation of C [ B n ] associated to Λ C ,q for q ⊗ = ⊕ x ⊕ y should factor through the cubic Hecke algebra H n ( Q, and the Temperley-Lieb algebra. This motivates Problem 3 by analogy with the q ⊗ = ⊕ x exposition.7.2. F-Symbols.
In Section 5, we considered the action of the rotation operator ϕ on a basis of jumpingjacks for End( q ⊗ ) . This was for a unitary spherical fusion category C containing a fusionrule of the form q ⊗ = ⊕ (cid:76) i x i with all the x i self-dual. We deduced that ϕ = κ q F qqqq (Theorem 5.4) and that F qqqq is real-symmetric (Corollary 5.7). For instances where C admits a braiding and q ⊗ = ⊕ x ⊕ y (with x and y self-dual), we found formulae for F qqqq in terms of the quantum dimensions (Theorem 5.9), and concluded that κ q (cid:54) = − (Corollary 5.10). We also saw that the spectrum of the rotation operator distinguishesbetween the Dubrovnik and Kauffman invariants (Corollary 5.10). Obvious extensions ofthis work would entail relaxing various assumptions: Problem 4.
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Appendix A. Rotated Morphisms
The rotation operator ϕ as defined in Section 2 has a simple generalisation to Hom( q ⊗ m , q ⊗ n ) (where q is still assumed to be self-dual, and m, n ∈ Z > ). Let at least one of m or n begreater than one, and f ∈ Hom( q ⊗ m , q ⊗ n ) . We have(A.1)where the left and right diagrams respectively illustrate ϕ ( f ) for an (anti)clockwise rota-tion and where ≤ l ≤ min { m, n } . A further variant is studied in [35]. Example A.1.
Let m = n = 3 and l = 1 with ϕ anticlockwise. Then ϕ = id . Let, f = , f = , f = f = , f = , f = Observe that ϕ ( f k ) = κ q f k +1 where k denotes a residue modulo , and that f k +3 = f k .Suppose there exist α, β ∈ C such that f = αf + βf . Applying ϕ , we get f = αf + βf = αf + β ( αf + βf )= ⇒ f = 1 − αβα + β f where in the final line we assume that β (cid:54) = − and α (cid:54) = β − . Thus, f , f and f areeither (a) linearly independent, (b) linearly dependent with f = αf + βf such that β = − and α = β − or (c) collinear. In the collinear case, note that f k is an eigenvectorof ϕ ; coupling this with the fact that ϕ ( f k ) = κ q f k , we have that ϕ ( f k ) = ωf k where ω is a rd root of κ q . It follows that f k +1 = κ q ωf k (and so all of the morphisms are relatedto one another by a scaling of some thth