From Three Dimensional Manifolds to Modular Tensor Categories
aa r X i v : . [ m a t h . QA ] J a n From Three Dimensional Manifolds to ModularTensor Categories
Shawn X. Cui , Yang Qiu , and Zhenghan Wang Department of Mathematics and Department of Physics and Astronomy, PurdueUniversity, West Lafayette, IN 47906 Department of Mathematics, University of California, Santa Barbara, CA 93106 Microsoft Station Q and Dept. of Math., University of California, Santa Barbara,CA 93106 [email protected], [email protected], [email protected]
Abstract
Using M-theory in physics, Cho, Gang, and Kim (JHEP , 115(2020) ) recently outlined a program that connects two parallel subjectsof three dimensional manifolds, namely, geometric topology and quan-tum topology. They suggest that classical topological invariants such asChern-Simons invariants of SL(2 , C )-flat connections and SL(2 , C )-adjointReidemeister torsions of a three manifold can be packaged together toproduce a (2 + 1)-topological quantum field theory, which is essentiallyequivalent to a modular tensor category. It is further conjectured thatevery modular tensor category can be obtained from a three manifold anda semi-simple Lie group. In this paper, we study this program mathemat-ically, and provide strong support for the feasibility of such a program.The program produces an algorithm to generate the potential modular T -matrix and the quantum dimensions of a candidate modular data. Themodular S -matrix follows from essentially a trial-and-error procedure. Wefind premodular tensor categories that realize candidate modular dataconstructed from Seifert fibered spaces and torus bundles over the circlethat reveal many subtleties in the program. We make a number of im-provements to the program based on our examples. Our main result is amathematical construction of the modular data of a premodular categoryfrom each Seifert fibered space with three singular fibers and a family oftorus bundles over the circle with Thurston SOL geometry. The modulardata of premodular categories from Seifert fibered spaces can be realizedusing Temperley-Lieb-Jones categories and the ones from torus bundlesover the circle are related to metaplectic categories. We conjecture thata resulting premodular category is modular if and only if the three mani-fold is a Z -homology sphere, and condensation of bosons in the resultingproperly premodular categories leads to either modular or super-modulartensor categories. ontents -manifolds to modular categories 4 k . . . . . . . . . . . . . . . . . . 303.5 Graded product of graded premodular categories . . . . . . . . . 33 There are two parallel universes in three dimensional topology for the last sev-eral decades that do not intersect much: the classical Thurston world and thequantum Jones world. One famous conjecture that hints a deep connection ofthe two worlds is the volume conjecture. Recently M-theory in physics suggestsanother surprising different connection: classical topological invariants such asChern-Simons invariants of SL(2 , C )-flat connections and SL(2 , C )-adjoint Rei-demeister torsions of a three manifold X can be packaged together to produce a(2 + 1)-topological quantum field theory (TQFT) [4], which is essentially equiv-alent to a modular tensor category [18]. It is further conjectured in [4] thatevery modular tensor category can be obtained from a three manifold and asemi-simple Lie group. In this paper, we study this program mathematically,2nd provide strong support for such a program. The program as outlined in[4] produces an algorithm to generate the potential modular T -matrix and thequantum dimensions of a candidate modular data. The modular S -matrix fol-lows from essentially a trial-and-error procedure. We find premodular tensorcategories that realize candidate modular data from Seifert fibered spaces andtorus bundles over the circle that reveal many subtleties in the program. Ourmain result is a mathematical construction of the modular data of a premodu-lar category from each Seifert fibered space with three singular fibers and sometorus bundles over the circle with Thurston SOL geometry. The modular dataof the premodular categories from Seifert fibered spaces can be realized usingTemperley-Lieb-Jones categories and the ones from torus bundles over the circleare related to metaplectic categories [19, 8]. A resulting premodular category ismodular if and only if the three manifold is a Z -homology sphere, and conden-sation of bosons in the resulting properly premodular categories leads to eithermodular or super-modular tensor categories.The program from three manifolds to modular tensor categories is a far-reaching progeny of the mysterious six-dimensional super-symmetric conformalfield theories (SCFTs) spawned by M-theory. Our strong support for the pro-gram indirectly provides evidence for these 6d SCFTs. The dimension reductionor compactification of these 6d SCFTs to 3d depends on a three manifold X , andin general the resulting theory T ( X ) is a super-conformal field theory. When X is non-hyperbolic, it is argued in [4] that T ( X ) flows to a TQFT in the infraredlimit and super-symmetry is decoupled, thus we obtain a (2+1)-TQFT labeledby X , hence a MTC B X . The program outlined in [4] centers on an algorithmto produce the quantum dimensions and topological twists of a MTC, and atrial-and-error algorithm for the modular S -matrix. The assumption on thethree manifolds X in [4] includes that X is non-hyperbolic and the SL(2 , C )representation variety of the fundamental group π ( X ) consists of finitely manyconjugacy classes that all could be conjugated into either SU (2) or SL(2 , R )subgroups of SL(2 , C ). Our examples show that all but the non-hyperbolic as-sumption can be dropped. One subtlety is that we need to use indecomposablereducible representations in our torus bundle over the circle examples. We donot know whether or not MTCs could be constructed from hyperbolic threemanifolds as the program as we formulated in this paper is more flexible. Themain difficulty for more examples lies in the explicit calculation of Chern-Simons(CS) invariant and adjoint Reidemeister torsion of flat connections.An SL(2 , C )-representation of π ( X ) is the same as a flat connection of thetrivial SL(2 , C )-bundle. There are two well-known invariants for a flat con-nection: the Chern-Simons (CS) invariant and the adjoint Reidemeister torsion.Each flat connection that satisfies certain conditions would give rise to an anyontype and the Reidemeister torsion is essentially the quantum dimension and theCS invariant is the conformal weight of the anyon.For each Seifert fibered space with three singular fibers, we define a poten-tial modular data inspired by the many examples in [4]. All those modular datacan be realized by premodular categories obtained as a Z -graded product ofTemperley-Lieb-Jones categories. We expect that our results can be easily gen-3ralized to any number of the singular fibers if the adjoint Reidemeister torsionsof the SL(2 , C ) flat connections can be calculated because the CS invariants inthis case are known. It is not clear if there are new MTCs among our examples.Going beyond Seifert fibered spaces, we analyze some torus bundles over thecircle and identify the resulting premodular categories as the integral subcat-egories of SO ( N ) for odd N . An important observation for the connectionto Temperley-Lieb-Jones categories for Seifert fibered spaces is a relation be-tween the slope of a singular fiber and the order of the Kauffman variable A inTemperley-Lieb-Jones theories [19]. Essentially the slope of a singular fiber de-termines a root of unity A , which allows us to realize all the candidate modulardata from Seifert fibered spaces with three singular fibers.The content of the paper is as follows. In Sec. 2, we outline our version ofthe program taking into account the many subtleties that we encountered in ourexamples. We also recall the definition of CS invariant and adjoint Reidemeistertorsion, and collect some known results of CS invariants of Seifert fibered spaces.In Sec. 3, we study the Seifert fibered spaces and carry out the necessarycalculations of CS and torsion invariants for our examples, and do the same fortorus bundles over the circle in Sec. 4. Finally, in Sec. 5, we discuss some futuredirections and open questions. -manifolds to modular cat-egories The proposed program in [4] from three manifolds to MTCs came from physics,and the paper provides an algorithm to produce the potential modular T -matrixand all quantum dimensions of a candidate modular data from irreducible repre-sentations of the fundamental groups of three manifolds to SL(2 , C ). Our resultsin Sec. 3 and Sec. 4 that realize candidate modular data from Seifert fiberedspaces and torus bundles over the circle reveal many subtleties in the programas outlined in [4]. In this section, we follow the overall program of [4] and makea number of improvements to reformulate mathematically the construction ofcandidate modular data from three manifolds taking into account these newsubtleties. Suppose X is an orientable connected closed 3-manifold and G is a semi-simpleLie group. The set of representations of the fundamental group π ( X ) to G con-sists of all group homomorphisms from π ( X ) to G , denoted by Hom( π ( X ) , G ),up to conjugation. The representation variety R ( X, G ) of π ( X ) to G is simplyHom( π ( X ) , G ) //G —equivalence classes of representations up to conjugation.In this paper, we will mainly consider the case G = SL(2 , C ) and its higherdimensional irreducible representations Sym j of dimension j + 1. Given such We omit the irrelevant base point. ρ : π ( X ) → SL( n, C ), its character is the function on π ( X )given by χ ρ ( x ) = Tr( ρ ( x )) for x ∈ π ( X ). The character variety χ ( X, SL( n, C ))of X consists of all such character functions. We will also denote the repre-sentation variety R ( X, SL(2 , C )) and character variety χ ( X, SL(2 , C )) simply as R ( X ) and χ ( X ). In this paper, the topology of the spaces of the representationand character varieties is not important.There are three obvious nontrivial automorhphisms of SL(2 , C ) by sendingan element g ∈ SL(2 , C ) to its complex conjuagte g ∗ , its transpose followedby inverse ( g t ) − , and the composition ( g † ) − of the previous two operations.For each representation of π ( X ) to SL(2 , C ), post-composing with one of thethree automorhphisms of SL(2 , C ) gives rise to another representation, hencerepresentations in R ( X ) come in group of four in general. Another obvious wayto change a representation ρ in R ( X, G ) is to tensor ρ with a representation of π ( X ) to the center Z ( G ) of G . Representations of π ( X ) to the center Z ( G )are in one-one correspondence with cohomology classes in the cohomology group H ( X, Z ( G )). The proposed program in [4] and in this section is to produce modular tensorcategories (MTCs) from closed three manifolds and show that each MTC canbe obtained from at least one three manifold. It is known that different threemanifolds can lead to the same MTC. As suggested in [4], we will focus on non-hyperbolic three manifolds. There are seven non-hyperbolic geometries and sixcan be realized by Seifert fibered three manifolds with the exception SOL [16].The geometry S × R is not useful for our purpose as we need representationsfrom the fundamental group to SL(2 , C ) with non-Abelian images. We willmainly consider Seifert fibered spaces in this paper, but in Sec. 4, we will alsostudy torus bundles over the circle with SOL geometry and more subtleties arise.Seifert fibered three manifolds X are those that can be foliated into disjointunion of circles and are completely enumerated [15]. In this paper, all our threemanifolds are orientable, and we will denote the Seifert fibered spaces (SFSs)by the notation X = { b ; ( o, g ); ( p , q ) , ( p , q ) , · · · , ( p n , q n ) } as explained below.The quotient space of a SFS X , called the base orbifold B , by sending each circle,called a fiber, to a point is a topological surface. The symbol ( o, g ) means thatthe base topological surface B is an orientable closed surface of genus g .Each fiber has a product neighbourhood D × S in the SFS X except n singular fibers labeled by ( p i , q i ) , i = 1 , · · · , n . The neighborhood of the i -thsingular fiber is obtained from D × [0 ,
1] by identifying the point ( x, , x ∈ D with the point ( r a i ,p i ( x ) , r a i ,p i is the rotation of the disk D by theangle 2 πa i /p i , where a i ∈ Z satisfies a i q i = 1 mod p i . The pair of coprime inte-gers ( p i , q i ) are the corresponding surgery coefficient. The fundamental group of X fits into a short exact sequence 1 → π ( F ) → π ( X ) → π orb ( B ) →
1, where π ( F ) ∼ = Z for a regular fiber F ∼ = S and π orb ( B ) is the orbifold fundamentalgroup of B (not the same as the fundamental group π ( B ) of the topologicalsurface B in general). The integer b in the notation is the obstruction class,5 q p q p n q n · · · Figure 1: Surgery link of Seifert fibered space with base S which is also the order of the generator of π ( F ) in π orb ( B ). Since we considerSFSs as three manifolds up to homeomorphism rather than as fibered spaces,we may always set b to 0.The fundamental group of X = { b ; ( o, g ); ( p , q ) , ( p , q ) , · · · , ( p n , q n ) } hasa presentation π ( X ) = h a j , b j , x i , h, j = 1 , · · · , g, i = 1 , · · · , n | [ a j , h ] = [ b j , h ] = [ x i , h ] = x p i i h q i = 1 , x · · · x n [ a , b ] · · · [ a g , b g ] = h b i . (1)In particular, the fundamental group of X = {
0; ( o, p , q ) , ( p , q ) , ( p , q ) } with base S and three singular fibers, denoted simply as { b ; ( p , q ) , ( p , q ) , ( p , q ) } sometimes, is π ( X ) = h x , x , x , h | x p i i h q i = 1 , x i h = hx i , x x x = h b i . The orientable SFS {
0; ( o, p , q ) , ( p , q ) , · · · , ( p n , q n ) } with base S and n singular fibers has a surgery diagram shown in Fig. 1. Given an orientable connected closed three manifold X , a morphism ρ of itsfundamental group π ( X ) to a semi-simple Lie group G can be identified asthe holonomy representation of a flat connection A ρ on the trivial principal G -bundle over X . Therefore, in the following we will use the terms a representation ρ and a flat connection A interchangeably via such an identification.Let X be a closed 3-manifold and ρ : π ( X ) −→ SL(2 , C ) be a holonomyrepresentation. Denote by A ρ the corresponding Lie algebra sl (2 , C )-valued 1-form on X . The Chern-Simons (CS) invariant of ρ is defined asCS( ρ ) = 18 π Z X Tr( dA ρ ∧ A ρ + 23 A ρ ∧ A ρ ∧ A ρ ) mod 1 , (2)where the integral with its coefficient in the front is well-defined up to integers.6he CS invariant CS( ρ ) depends only on the character χ ( ρ ) of ρ [11], henceit descends from the representation variety R ( X ) to the character variety χ ( X ).Auckly computed the CS invariant of SFSs for SU(2) representations in [1].The CS invariant of SFSs for SL(2 , C ) representations may be known to experts.However, to make the paper self-contained, we provide a proof to compute thatusing method from [11]. Proposition 2.1.
Let X = {
0; ( o, g ); ( p , q ) , ( p , q ) , · · · , ( p k , q k ) } be an SFSwith the presentation of π ( X ) given in Equation 1 with b = 0. Choose integers s j and r j such that p j s j − q j r j = 1. Suppose ρ : π ( X ) → SL(2 , C ) is non-Abelian such that Tr( ρ ( x j )) = 2 cos πn j p j , thenCS( ρ ) = k X j =1 r j n j p j mod 1 , ρ ( h ) = I k X j =1 ( r j n j p j − q j s j , ρ ( h ) = − I Remark 2.2.
The formula for the CS invariant in Proposition 2.1 differs fromthat in [1] with a negative sign. We believe this discrepancy is due to conven-tions.Before proving the proposition, we recall some facts in [11]Let T be a torus and consider χ ( T ), the character variety of T to SL(2 , C ).It is direct to see that χ ( T ) can be identified with Hom( π ( T ) , C ∗ ) / ∼ where f ∼ g if f ( · ) = g ( · ) − . We now describe a ‘coordinate-version’ of χ ( T ).Let H be a group with the presentation, H = h x, y, b | [ x, y ] = bxbx = byby = b = 1 i , and define an action of H on C by x ( α, β ) = ( α + 1 , β ) , y ( α, β ) = ( α, β + 1) , b ( α, β ) = ( − α, − β ) . Denote the image of ( α, β ) ∈ C in the quotient space C /H by [ α, β ]. Let ~v = ( v , v ) be any Z -basis of H ( T ), and define the map, f ~v : C /H → χ ( T ) , such that f ~v [ α, β ] ∈ χ ( T ) sends v e πiα , v e πiβ . It can be checked that f ~v is a homeomorphism. A representation of π ( T ) thatinduces the character f ~v [ α, β ] is given by, v (cid:18) e πiα e − πiα (cid:19) , v (cid:18) e πiβ e − πiβ (cid:19) . f ~v is natural in the following sense. Let ~w beanother basis such that ~w = ~vA for some A ∈ GL(2 , Z ) (viewing ~w and ~v as rowvectors), and define the map Φ ~v, ~w : C → C by right multiplying (row) vectorsof C by A on the right. Then Φ ~v, ~w induces a homeomorphism, still denoted byΦ ~v, ~w , from C /H to C /H , and the following diagram commutes, C /H C /Hχ ( T ) Φ ~v,~w f ~v f ~w Hence, we think of each C /H with a choice of basis ~v as a coordinate realizationof χ ( T ). In fact, χ ( T ) is isomorphic to the direct limit of { ( C /H ) ~v , Φ ~v, ~w } , χ ( T ) ≃ lim −→ ( C /H ) ~v , where ( C /H ) ~v is a copy of C /H indexed by ~v .Next, we introduce a C ∗ bundle over χ ( T ). Define an action of H on C × C ∗ lifting that on C by x ( α, β ; z ) = ( α + 1 , β ; ze πiβ ) ,y ( α, β ; z ) = ( α, β + 1; ze − πiα ) ,b ( α, β ; z ) = ( − α, − β ; z ) . The canonical projection C × C ∗ → C induces a projection p : C × C ∗ /H → C /H, which makes C × C ∗ /H a C ∗ bundle over C /H . Given two bases ~v, ~w of H ( T ) with ~w = ~vA , Φ ~v, ~w can be covered by a bundle isomorphism. Explicitly,define ˜Φ ~v, ~w : C × C ∗ /H → C × C ∗ /H which maps [ α, β ; z ] to [( α, β ) A ; z det( A ) ].Then the following diagram commutes,( C × C ∗ /H ) ~v ( C × C ∗ /H ) ~w ( C /H ) ~v ( C /H ) ~wp ˜Φ ~v,~w p Φ ~v,~w (3)Let ˜ E ( T ) be the direct limit of { ( C × C ∗ /H ) ~v , ˜Φ ~v, ~w } . Then Equation 3 inducesa map p : ˜ E ( T ) → χ ( T ) which makes ˜ E ( T ) a C ∗ bundle over χ ( T ), and thediagram below commutes, ˜ E ( T ) ( C × C ∗ /H ) ~v χ ( T ) ( C /H ) ~vp pf ~v Here all maps involved are isomorphisms, so the notion of direct limit and inverse limitdo not make a difference.
8e often represent an element of ˜ E ( T ) by a ‘coordinate’ [ α, β ; z ] ~v with respectto a basis ~v . Changing the basis to ~w = ~vA induces the equality[ α, β ; z ] ~v = [( α, β ) A ; z det( A ) ] ~w , and when the bases involved are clear from the context, we will omit them.We also need an ‘orientation-version’ of ˜ E ( T ). Now assume T is oriented ,and define E ( T ) to be the direct limit of { ( C × C ∗ /H ) ~v , ˜Φ ~v, ~w } where the limitis taken only over positive bases ~v of H ( T ), namely, those ~v such that v ∧ v matches the orientation of T . Apparently, E ( T ) and E ( − T ) are both bundlesover χ ( T ), and are both isomorphic to ˜ E ( T ). However, it will be of conceptualconvenience for latter calculations to distinguish E ( T ) from E ( − T ).There is a fiber-wise pairing h , i defined on E ( T ) × E ( − T ) as follows. Given e ∈ E ( T ) , e ′ ∈ E ( − T ) such that p ( e ) = p ( e ′ ), choose an arbitrary positive basis ~v = ( v , v ) of H ( T ) and hence ~v ′ := ( − v , v ) is a positive basis of H ( − T ), andwrite e = [ α, β ; z ] ~v , e ′ = [ − α, β ; z ′ ] ~v ′ (or e ′ = [ α, − β ; z ′ ] − ~v ′ ). Then h e, e ′ i := zz ′ .It can be checked that the pairing is well defined.Lastly, the above notions can be generalized to multiple tori in a naturalway. Let S = ⊔ ki =1 T i be a disjoint union of k oriented tori. Then χ ( S ) = χ ( T ) × · · · × χ ( T k ). The group H k acts on ( C ) k component-wise and thequotient is a ‘coordinate-version’ of χ ( S ). The action of H k can also be liftedto ( C ) k × C ∗ where the i -th component H i in H k acts on the i -th copy of C in ( C ) k times C ∗ , and E ( T ) is the quotient of ( C ) k × C ∗ by this action. For n ≤ k , similar to the pairing above, there is a generalized ‘pairing’: E ( T ⊔ · · · ⊔ T k ) × E ( − T ⊔ · · · ⊔ − T m ) → E ( T m +1 ⊔ · · · ⊔ T k ) . With the above notations, we recall several theorems in [11]. Let X be an ori-ented compact 3-manifold with toral boundaries ∂X = ⊔ ki =1 T i and ρ : π ( X ) → SL(2 , C ) be a holonomy representation. It is well-known that CS( ρ ) in Equation2 is not well defined since X has boundary. Let c X ( ρ ) = e πi CS( ρ ) . Theorem 2.3 (Theorem 3.2 of [11]) . The Chern-Simons invariant defines alifting c X : χ ( X ) −→ E ( ∂X ) of the restriction map r from the character varietyof X to the character variety of ∂X , E ( ∂X ) χ ( X ) χ ( ∂X ) pc X r Moreover, if Y = X ∪ X is a closed oriented 3-manifold such that X and X are glued along toral boundaries ∂X = − ∂X , then for χ ∈ χ ( Y ), we have e πi CS( χ ) = h c X ( χ ) , c X ( χ ) i , where χ i denotes the restriction of χ on X i .9he following theorem is also due to [11] which the authors proved for thecase of SU(2) representations (Theorem 2.7), but an almost identical proof alsoworks for SL(2 , C ) representations. Theorem 2.4.
Let X be an oriented 3-manifold with toral boundaries ∂X = ⊔ ki =1 T i and ρ ( t ) : π ( X ) → SL(2 , C ) be a path of representations. Let ( α i ( t ) , β i ( t ))be a lift of χ ◦ ρ ( t ) | T i to C with respect to some basis of H ( T i ). Suppose c X ( ρ ( t )) = [ α ( t ) , β ( t ) , · · · , α k ( t ) , β k ( t ); z ( t )]Then z (1) z (0) − = exp πi k X j =1 Z ( α j dβ j dt − β j dα j dt ) In particular, if ρ (1) is the trivial representation, then c X ( ρ (0)) = (cid:20) α (0) , β (0) , · · · , α k (0) , β k (0); exp (cid:0) − πi k X j =1 Z ( α j dβ j dt − β j dα j dt ) (cid:1)(cid:21) The following two facts are proved for SU(2) representations in [11] (Theo-rems 4.1 and 4.2, respectively). Similar methods combined with Theorems 2.3and 2.4 above show that they also hold for SL(2 , C ) representations. Fact 1
Let X be an oriented 3-manifold with toral boundaries ∂X = ⊔ ni =1 T i .Assume H ( X ) is torsion free. Choose a positive basis ( µ i , λ i ) for H ( T i ). Let { x j | j = 1 , · · · , m } be a basis of H ( X ) and µ i = P a ij x j , λ i = P b ij x j . Sup-pose that ρ : π ( X ) → SL(2 , C ) is an Abelian representation and Tr( ρ ( x j )) = e πiγ j + e − πiγ j for some γ j ∈ C . Then c X ( ρ ) = h X a j γ j , X b j γ j , · · · , X a nj γ j , X b nj γ j ; 1 i Fact 2
Let F be a genus g oriented surface with k punctures. The fundamentalgroup of F has the presentation, π ( F ) = h a , b , · · · , a g , b g , x , · · · , x k | [ a , b ] · · · [ a g , b g ] x · · · x k = 1 i , where x j corresponds to the oriented boundary (induced from F ) of the j -thpuncture. Let Y = F × S be endowed with the product orientation and let˜ h = ∗ × S be the central element of π ( Y ) corresponding to the oriented S component. Then ∂Y = ⊔ kj =1 T j with T j the torus corresponding to the j -thpuncture and ( x j , ˜ h ) is a positive basis for H ( T j ). Suppose ρ : π ( Y ) −→ SL(2 , C ) is a non-Abelian representation, which implies Tr( ρ (˜ h )) = 2 cos 2 πβ for some β ∈ { , } . Suppose Tr( ρ ( x j )) = e πiα j + e − πiα j for some α j ∈ C .Then c Y ( ρ ) = (cid:20) α , β, · · · , α n , β ; exp (cid:0) − πiβ k X j =1 α j (cid:1)(cid:21) . Note that c Y ( ρ ) does not change under the replacement of some α j by − α j .The rest of the subsection is devoted to the proof of Proposition 2.1.10 roof. Let Y = F × S be as in Fact 2 above with the chosen generators x j and ˜ h . Set h = ˜ h − . Then X is obtained from Y by gluing k solid tori wherethe j -th solid torus A j is glued along T j by sending the meridian to x p j j h q j .The generators x j and h match those as presented in Equation 1. Choose ameridian-longitude pair ( µ j , λ j ) for A j such that ( µ j , λ j ) is a positive basis of H ( ∂A j ). The gluing of A j to Y provides the transition of basis,( µ j , λ j ) = ( x j , h ) (cid:18) p j r j q j s j (cid:19) . Since ρ is non-Abelian, ρ ( h ) is ± I . By assumption,Tr( ρ ( x j )) = exp( 2 πin j p j ) + exp( − πin j p j ) , Tr( ρ ( h )) = 2 cos(2 πm ) , m = 0 , . Therefore, c Y ( ρ ) = (cid:2) n p , − m, · · · , n k p k , − m ; exp(2 πi m k X j =1 n j p j ) (cid:3) ( x , − h ; ··· ; x k , − h ) c A j ( ρ ) = [0 , r j n j p j + s j m ; 1] ( µ j ,λ j ) = [ − q j ( r j n j p j + s j m ) , r j n j + s j p j m ; 1] ( x j ,h ) = [ n j p j − s j α j , m + r j α j ; 1] , (setting α j = n j + q j m )= (cid:20) n j p j − s j α j , m ; exp (cid:0) πi ( r j α j )( n j p j − s j α j ) (cid:1)(cid:21) = (cid:20) n j p j , m ; exp (cid:0) πi ( r j α j )( n j p j − s j α j ) + 2 πi ( s j α j ) m (cid:1)(cid:21) Note that the relation x p j j h q j = 1 implies that α j must be an integer. Applyingthe pairing on c Y ( ρ ) and each c A j ( ρ ) one by one, we obtain,CS( ρ ) = k X j =1 ( r j α j n j p j + s j α j m + m n j p j )= k X j =1 (cid:0) r j n j p j + s j m ( n j + α j ) (cid:1) = k X j =1 ( r j n j p j − s j q j m ) . .4 Adjoint Reidemeister torsion The Reidemeister torsion ( R -torsion) τ ( X ) of a celluation K X of a manifold X uses the action of the fundamental group π ( X ) on the universal cover g K X tomeasure the complexity of the celluation of X . It is a topological invariant of X from determinants of matrices obtained from the incidences of the cells of g K X . The R -torsion makes essential use of the bases in the chain complex ofthe universal cover, while the homology and homotopy groups do not see thegeometric information encoded in the based chain complex. For our purpose,we need the non-Abelian generalization of R -torsion twisted by a representation ρ : π ( X ) → G for some semi-simple Lie group G , in particular the adjointReidemeister torsion for the adjoint representation of SL(2 , C ). We recall somebasics here, for more details, please refer to [13] and [17].Let C ∗ = (0 −→ C n ∂ n −→ C n − ∂ n − −→ · · · ∂ −→ C −→ C . Choosea basis c i of C i and a basis h i of the i -th homology group H i ( C ∗ ). The torsionof C ∗ with respect to these choices of bases is defined as follows. For each i , let b i be a set of vectors in C i such that ∂ i ( b i ) is a basis of Im( ∂ i ) and let ˜ h i denotea lift of h i in Ker( ∂ i ). Then the set of vectors ˜ b i := ∂ i +1 ( b i +1 ) ⊔ ˜ h i ⊔ b i is a basisof C i . Let D i be the transition matrix from c i to ˜ b i . To be specific, each columnof D i corresponds to a vector in ˜ b i being expressed as a linear combination ofvectors in c i . Define the torsion τ ( C ∗ , c ∗ , h ∗ ) := (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n Y i =0 det( D i ) ( − i +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Remark 2.5.
A few remarks are in order. • The torsion, as it is denoted, does not depend on the choice of b i and thelifting of h i . • Here we define the torsion as the norm of the usual torsion, thus we donot need to deal with sign ambiguities.Let X be a finite CW-complex and ( V, ρ ) be a homomorphism ρ : π ( X ) −→ SL( V ). The vector space V turns into a left Z [ π ( X )]-module. The universalcover ˜ X has a natural CW structure from X , and its chain complex C ∗ ( ˜ X ) is afree left Z [ π ( X )]-module via the action of π ( X ) as covering transformations.View C ∗ ( ˜ X ) as a right Z [ π ( X )]-module by σ.g := g − .σ for σ ∈ C ∗ ( ˜ X ) and g ∈ π ( X ). We define the twisted chain complex C ∗ ( X ; ρ ) := C ∗ ( ˜ X ) ⊗ Z [ π ( X )] V .Let { e iα } α be the set of i -cells of X ordered in an arbitrary way. Choose a lifting˜ e iα of e iα in ˜ X . It follows that C i ( ˜ X ) is generated by { ˜ e iα } α as a free Z [ π ( X )]-module (left or right). Choose a basis of { v γ } γ of V . Then c i ( ρ ) := { ˜ e iα ⊗ v γ } is a C -basis of C i ( X ; ρ ). Definition 2.6.
Let ρ : π ( X ) −→ SL( V ) be a representation.12. We call ρ acyclic if C ∗ ( X ; ρ ) is acyclic. Assume ρ is acyclic. The torsionof X twisted by ρ is defined to be, τ ( X ; ρ ) := τ (cid:18) C ∗ ( X ; ρ ) , c ∗ ( ρ ) (cid:19) .
2. Let Adj : SL( V ) → SL( sl ( V )) be the adjoint representation of SL( V ) onits Lie algebra sl ( V ). We call ρ adjoint acyclic if Adj ◦ ρ is acyclic. Assume ρ is adjoint acyclic. Define the adjoint Reidemeister torsion of ρ to be,Tor( X ; ρ ) := τ ( X ; Adj ◦ ρ ) . Remark 2.7.
In this paper, we will only deal with the adjoint Reidemeistertorsion ρ . For that matter, we simply call it the torsion of ρ . When no confusionarises, we abbreviate Tor( X ; ρ ) as Tor( ρ ).The following tool will be useful in computing torsions. Multiplicativity Lemma
Let 0 −→ C ′∗ −→ C ∗ −→ C ′′∗ −→ C ∗ , C ′∗ , C ′′∗ are based by c ∗ , c ′∗ , c ′′∗ ,respectively, and their homology groups based by h ∗ , h ′∗ , h ′′∗ , respectively. Asso-ciated to the short exact sequence is the long exact sequence H ∗ in homology · · · −→ H j ( C ′∗ ) −→ H j ( C ∗ ) −→ H j ( C ′′∗ ) −→−→ H j − ( C ′∗ ) −→ · · · with the reference bases. For each i , identify c ′ i with its image in C i and arbi-trarily choose a preimage ˜ c ′′ i of c ′′ i in C i . If the transition matrix between thebases c i and c ′ i ⊔ ˜ c ′′ i has determinant ±
1, we call c ∗ , c ′∗ , c ′′∗ compatible. In thiscase, we have τ ( C ∗ , c ∗ , h ∗ ) = τ ( C ′∗ , c ′∗ , h ′∗ ) τ ( C ′′∗ , c ′′∗ , h ′′∗ ) τ ( H ∗ , { h ∗ ⊔ h ′∗ ⊔ h ′′∗ } ) . The modular data of an MTC or a pre-modular category consist of the modular S - and T - matrices. Given a three manifold X with certain conditions, [4]contains an algorithm for choosing the T -matrix and the first row of the S -matrix, i.e. all quantum dimensions. The next step for the full S -matrix isa trial-and-error algorithm based on finding the right loop operators for eachsimple object. When all the loop operators are given, then the modular datacan be computed. There are no general algorithms to define loop operators, butin the cases of SFSs and SOL manifolds, we find the relevant loop operatorscompletely. Each premodular category has a label set—the isomorphism classes of the simpleobjects, and a label is an isomorphism class of simple objects, so we will refer13o a label also as a simple object type. In physics, an anyon model is a unitaryMTC and a label is called an anyon type or a topological charge.A candidate label from a three manifold X and SL(2 , C ) is morally an ir-reducible representation of the fundamental group π ( X ) to SL(2 , C ). But theprecise definition is more subtle and based on our examples later, we makethe following definition. In particular, we discover that reducible but indecom-posable representations cannot be discarded and play important roles in theconstruction of premodular categories from torus bundles over the circle. Ourdefinition is specific for representations to SL(2 , C ) and we expect an appropriategeneralization is needed for other Lie groups such as SL( n, C ) , n ≥ Definition 2.8.
Let χ ∈ χ ( X ) be an SL(2 , C )-character of a three manifold X . • χ is non-Abelian if at least one representation ρ : π ( X ) → SL(2 , C ) withcharacter χ is non-Abelian, i.e. ρ has non-Abelian image in SL(2 , C ). Theset of all non-Abelian characters of X is denoted by χ nab ( X ). • A non-Abelian character χ is adjoint-acyclic if each non-Abelian represen-tation ρ : π ( X ) → SL(2 , C ) with character χ is adjoint-acyclic, namely,the chain complex associated with the universal cover ˜ X twisted by Adj ◦ ρ is acyclic (see Definition 2.6), and furthermore, the adjoint Reidemeistertorsion of all such non-Abelian representations with character χ are thesame. • A candidate label is an adjoint-acyclic non-Abelian character. • A candidate label set L ( X ) from a three manifolds X is a finite set ofadjoint-acyclic non-Abelian characters in χ ( X ) with a pre-chosen charac-ter such that the difference of the CS invariant of each character L ( X )with that of the pre-chosen character is a rational number.The pre-chosen character is the candidate tenor unit.Note that by definition, the adjoint Reidemeister torsion is well-defined foradjoint-acyclic non-Abelian characters. The CS invariant only depends on char-acters, and is hence also well-defined for such characters.In this paper, our candidate label set is in general maximal in the senseit consists of all the adjoint-acyclic non-Abelian characters of the given threemanifold. It is also true that the CS invariants of all the candidate labelsincluding the candidate tensor unit are all rational in our examples. We are notaware of any example of a candidate label set for which not all CS invariantsare rational numbers. Each simple object x of a premodular category B has a quantum dimension d x and a topological twist θ x . The set T d ( B ) := ∪ i ∈ L ( B ) { d x i , θ x i } will be calledthe twist-dimension set of B , where L ( B ) is the label set of B and { x i , i ∈ L ( B ) } form a complete representative set of simple objects of B . A candidate label14et of a three manifold X will lead to a candidate twist-dimension set in thefollowing.The choice of a tensor unit or vacuum from a collection of adjoint-acyclicnon-Abelian characters is not unique in general and it is known that differentchoices could produce different premodular categories. Once a vacuum is chosen,then the adjoint Redeimeister torsion of each character is scaled to the absolutevalue of normalized quantum dimension and the difference of the CS invariantof the character with that of the vacuum is the conformal weight of the simpleobject up to a sign .Given a three manifold X and a Lie group G , a central representation of π ( X ) is a homomorphism from π ( X ) to the center Z ( G ) of G . For G =SL(2 , C ), a central representation of π ( X ) is simply a homomorphism from π ( X ) to Z . The group of central representations can be identified with H ( X, Z ). A central representation σ ∈ H ( X, Z ) of π ( X ) naturally actson R ( X ) by tensoring ρ ∈ R ( X ), i.e. by sending ρ to ρ ⊗ σ . Moreover, this ac-tion induces an action of central representations on the character variety χ ( X ). Definition 2.9.
1. Given a candidate label set L ( X ) from a three manifold X , a central representation σ is bosonic with respect to L ( X ) if the actionof σ keeps L ( X ) invariant and preserves the CS invariant of every candi-date label. If the action of σ changes the CS invariants of all candidatelabels in L ( X ) by either 0 or , then χ is called fermionic if it is notbosonic.2. Two candidate labels are centrally related if they are in the same orbitunder the action of H ( X, Z ) and they have the same CS and torsioninvariant.Given a candidate label set L ( X ) of X that is invariant under the actionof H ( X, Z ), the candidate symmetric center s ( X ) consists of all characters in L ( X ) that are centrally related to the candidate tensor unit. Let G ( X ) be themaximal subgroup of H ( X, Z ) such that G ( X ) maps the candidate tensorunit onto s ( X ). The action of G ( X ) separates L ( X ) into orbits { O , · · · , O m } ,where each subset O i of L ( X ) consists of candidate labels that are centrallyrelated to each other, and O is the subset for the candidate vacuum.We often represent a candidate label (a character) by arbitrarily choosing arepresentative (a representation of π ( M )) for it. Definition 2.10.
A candidate label set L ( X ) = { ρ α } of a three manifold X with ρ the candidate vacuum is admissible if the following two equations holdwith the notations as above: X ρ α ∈ L ( X ) ρ α ) = 1 , (4) The sign and hence the negative sign in front of CS invariant below is not important andthe choice is made to be the same as in [4]. (cid:12)(cid:12)(cid:12)(cid:12)X α exp( − πi CS( ρ α ))2Tor( ρ α ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 1 s L p | s ( X ) | p ρ ) , (5)where s L = 1 if all central representations in G o ( X ) are bosonic and s L = √ O . In the condensed category, each subset O i will beidentified into a single composite object which has the same quantum dimen-sion as that of any simple object in O i and which splits into a number of simpleobjects of the same quantum dimension. The resulting condensed category iseither modular or super-modular depending on if there is a fermion in the can-didate Mueger center. In a particular case when X is a Z homology sphere,that is, H ( X, Z ) = 0, Equation 5 reduces to, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X α exp( − πi CS( ρ α ))2Tor( ρ α ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 1 p ρ ) . (6)Given an admissible candidate label set L ( X ) with the chosen candidatetensor unit ρ , then the candidate twist-dimension set is constructed as follows: θ α = e − πi (CS( ρ α ) − CS( ρ )) , (7) D = 2Tor( ρ ) (8) d α = D ρ α ) , (9)where D is the total dimension squared of the candidate premodular category.Next, we discuss the construction of the S -matrix. Definition 2.11.
Given a three manifold X , a primitive loop operator of X isa pair ( a, R ), where a is a conjugacy class of the fundamental group π ( X ) of X and R a finite dimensional irreducible representation of SL(2 , C ).Given an SL(2 , C )-representation ρ of π ( X ) and a primitive loop opera-tor ( a, R ), then the weight of the loop operator ( a, R ) with respect to ρ is W ρ ( a, R ) := Tr R ( ρ ( a )). Denote by Sym j the unique ( j + 1)-dimensional irre-ducible representation of SL(2 , C ). Then W ρ ( a, Sym j ) can be computed fromthe Chebyshev polynomial ∆ j ( t ) defined recursively by,∆ j +2 ( t ) = t ∆ j +1 ( t ) − ∆ j ( t ) , ∆ ( t ) = 1 , ∆ ( t ) = t. (10)Explicitly, W ρ ( a, Sym j ) = ∆ j ( t ) , t = W ρ ( a, Sym ) = Tr( ρ ( a )) . (11)From the above two equations, it follows that W ρ ( a, Sym j ) only depends on thecharacter χ induced by ρ . It is direct to check that,∆ j (2 cos θ ) = sin(( j + 1) θ ) / sin θ, ∆ j ( − t ) = ( − j ∆ n ( t ) . (12)16 fundamental assumption in constructing the S -matrix is that each candi-date label ρ α should correspond to a finite collection of primitive loop operators: ρ α
7→ { ( a κα , R κα ) } κ . (13)Obtaining the above correspondence involves a guess-and-trial process as fol-lows. With a guess in hand and a choice ǫ = ±
1, we define the W -symbols W β ( α ) := Y κ W ǫ ρ β ( a κα , R κα ) = Y κ Tr R κα ( ǫ ρ β ( a κα )) , ρ α , ρ β ∈ L ( X ) . (14)The W -symbols and the un-normalized S -matrix ˜ S = D S are related by, W β ( α ) = ˜ S αβ ˜ S β or ˜ S αβ = W β ( α ) W ( β ) , (15)where 0 denotes the tensor unit ρ . In particular, the quantum dimension d α = W ( α ) (16)Hence, we can try to guess a correspondence between candidate labels andloop operators so that the quantum dimension computed by Equation 16 matches(in absolute value) with that computed by Equation 9.We expect that the resulting modular data corresponds to a MTC if and onlyif H ( X, Z ) = 0. Note that, this is purely a topological condition, independentof the choice of loop operators. Hence, if H ( X, Z ) = 0, we can also validate achoice of the loop operators by checking whether the resulting S and T matricesdefine a representation to SL(2 , Z ). In this section, we consider SFSs with three singular fibers and construct mod-ular data associated with premodular categories. Throughout the section, set M = {
0; ( o, p , q ) , ( p , q ) , ( p , q ) } , where each pair ( p k , q k ) are co-prime.So the underlying 2-manifold of the orbit surface Σ has genus 0 and both M and Σ are orientable. For M = {
0; ( o, p , q ) , ( p , q ) , ( p , q ) } , its fundamental group has the fol-lowing presentation, π ( M ) = h x , x , x , h | x p k k h q k = 1 , x k h = hx k , x x x = 1 , k = 1 , , i We look for all non-Abelian characters of π ( M ) to G = SL(2 , C ).17et ρ : π ( M ) → G be a non-Abelian representation. Since h is in the centerof π ( M ) and ρ is non-Abelian, ρ ( h ) must be ± I . It follows that each ρ ( x k )has finite order, and is diagonalizable in particular. Moreover, any ρ ( x k ) doesnot commute with another ρ ( x j ). This implies neither ρ ( x k ) can be ± I . Up toconjugation, we assume ρ ( x k ) take the following form (writing ρ ( x k ) simply as x k ), x = (cid:18) e iα e − iα (cid:19) , x = (cid:18) a bc d (cid:19) ∼ (cid:18) e iα e − iα (cid:19) , x ∼ (cid:18) e iα e − iα (cid:19) (17)where 0 < α k < π , ad − bc = 1, and b and c are not simultaneously zero. Wehave the following linear equations for a and d .Tr( x ) = e iα + e − iα = a + d (18)Tr( x ) = e iα + e − iα = ae iα + de − iα (19)Hence, given the α ′ k s, or equivalently Tr( x k ), a and d are uniquely determined,and a = ¯ d . Moreover, when | a | 6 = 1 implying bc = 0, this also determines ρ upto conjugacy. When | a | = 1 implying bc = 0, there are precisely two conjugacyclasses with x = (cid:18) a
10 ¯ a (cid:19) or x = (cid:18) a
01 ¯ a (cid:19) (20)It can be checked that these two representations are complex conjugate to eachother up to conjugacy, and that their characters take real values. They giverise to the same character. There are two types of non-Abelian representations.One type is irreducible satisfying b, c = 0. Characters of representations ofthis type one-to-one correspond to conjugacy classes of representations [6]. Theother type is reducible with exactly one of b, c zero. Each character of this typecorresponds to two conjugacy classes.To summarize, the triple ( α , α , α ) and Tr( h ) uniquely determine the char-acter. Next, we find all possible such triples.If h = I , each e iα k is a p k -th root of 1. If h = − I , then e iα k is a p k -th root of 1 if q k is even, and a p k -th root of − q k is odd. We claim alltriples satisfying the above conditions can be realized by some representations.Indeed, given such a triple ( α , α , α ), we define ρ ( x ) and ρ ( x ) as in Equation17 and let ρ ( x ) := ( ρ ( x ) ρ ( x )) − . Equations 18, 19 determine a and d , andwe arbitrarily choose b and c such that ad − bc = 1. Again, Equations 18, 19guarantee that ρ ( x k ) so defined has eigenvalues e ± iα k , and therefore they satisfyall the relations in the presentation of π ( M ).Set α k = πn k p k and ρ ( h ) = e πiλ I , λ = 0 , . If λ = 0 or if λ = and q k is even, then n k is an integer strictly between 0 and p k . If λ = and q k isodd, then n k is a proper half integer strictly between 0 and p k . The quadruple( n , n , n , λ ) completely characterizes a character.For an integer p >
0, denote by [0 · · · p ] the set of integers { , , · · · , p } , andby [0 · · · p ] e (resp. [0 · · · p ] o ) the subset of even (resp. odd) integers in [0 · · · p ].18he non-Abelian character variety of M is given as follows, χ nab ( M ) = (cid:26)(cid:18) j + 12 , j + 12 , j + 12 , (cid:19) | j k ∈ [0 · · · p k − ǫ k (cid:27) ⊔ (cid:26)(cid:18) j + 12 , j + 12 , j + 12 , (cid:19) | j k ∈ [0 · · · p k − o (cid:27) , (21)where ǫ k = ‘ e ’ if q k is odd, and ǫ k = ‘ o ’ otherwise. For ( n , n , n , λ ) ∈ χ nab ( M ),a corresponding representation ρ has e ± πinkpk as the eigenvalue of ρ ( x k ) and ρ ( h ) = e πiλ I .The size of χ nab ( M ) is | χ nab ( M ) | = ⌊ p ⌋⌊ p ⌋⌊ p ⌋ + ⌊ p − ⌋⌊ p − ⌋⌊ p − ⌋ , where ⌊ x ⌋ is the greatest integer less than or equal to x .For instance, if all the q ′ k s are odd, then χ nab ( M ) can also be written as, χ nab = (cid:26)(cid:0) j + 12 , j + 12 , j + 12 , ( j + 1) mod 22 (cid:1) | j k ∈ [0 · · · p k − , j = j = j mod 2 (cid:27) Freed computed torsions of Brieskorn homology spheres for the adjoint represen-tations of irreducible SU(2) representations in [7]. Kitano computed torsions ofSFSs for irreducible SL(2 , C ) representations in [12]. However, we need to com-pute torsions of SFSs for the adjoint representations of nonAbelian SL (2 , C )representations containing both irreducible and reducible ones. This may beknown to experts, but we did not find a reference for explicitly doing so. Tomake the paper self-contained, we provide a detailed derivation of these torsions,generalizing the work of [7] and [12].Let X be the SFS {
0; ( o, p , q ) , ( p , q ) , ( p , q ) } . Decompose X as ∪ i =0 A i ∪ B along ∪ i =0 T i where B = ( S − pts ) × S , and B , B i ( i = 1 , , B by index 1 , p i q i along T , T i , respectively. Let ρ : π ( X ) −→ SL(2 , C ) be a non-Abelian representation, V = sl (2 , C ) be theadjoint representation of ρ with the basis e = (cid:18) (cid:19) , e = (cid:18) − (cid:19) , e = (cid:18) (cid:19) From Section 3.1, ρ is parametrized by ( n , n , n , h ) where 0 < n i < p i , n i ∈ Z , h = 0 , . Assume that r i , s i ∈ Z , such that p i s i − r i q i = 1. Proposition 3.1.
When ρ is nonAbelian, C ∗ ( X ) ⊗ Z [ π ( X )] V is acyclic andTor( X ; ρ ) = p p p Q i =1 πr i n i p i roof. Denote C ∗ ⊗ Z [ π ] V by C ∗ ,ρ , twisted homology by H ∗ , and the matrix ofelement in π under ρ by the same letter.Given CW structure on X , we have the following exact chain sequence0 −→ M i =0 C ∗ ,ρ ( T i ) −→ M i =0 C ∗ ,ρ ( A i ) ⊕ C ∗ ,ρ ( B ) −→ C ∗ ,ρ ( X ) −→ −→ H ( T i ) −→ H ( A i , B ) −→ H ( X ) −→ · · ·−→ H ( T i ) −→ H ( A i , B ) −→ H ( X ) −→ C ( B ) = < v B >, C ( T i ) = < v T i >, C ( A i ) = < v A i >C ( B ) = < x , x , x , h >, C ( T i ) = < m i , l i >, C ( A i ) = < b i >C ( B ) = < u ,B , u ,B , u ,B >, C ( T i ) = < u T i > where v ∗ are base points of connected spaces, x i generate π ( S − pts ), h = ∗× S ∈ π ( S − pts × S ), m i , l i are meridians and longitudes of T i respectively, b i are longitudes of boundary of A i , u i,B are squares with boundary x i hx − i h − , u T i are squares with boundary m i l i m − i l − i . T i ( i = 1 , ,
3) are attached to x i × h by identity map and boundary of A i by (cid:18) s i − q i − r i p i (cid:19) . T is attached to x x x × h and boundary of A by identity map. x , x , x , h generate π ( X )as follows. π ( X ) = < x , x , x , h | x p i h q i = 1 , x i h = hx i , x x x = 1 > For matrix under ρ , we have x i ∼ ζ i ζ − i , h = I where ζ i is a p i -th root of unity. m i = x i , b i = x r i i , l i = h . Here we use 1-cellwith ends points attached as element in π .The work of [7] can be generalized to irreducible representations of SL (2 , C ).Thus we focus on reducible and nonAbelian representations. According to 20,taking upper triangular ones for example, they have the following form. x = (cid:18) a a − (cid:19) , x = (cid:18) a a − (cid:19) , x = (cid:18) a − a − − a a a (cid:19) a , a , a = a − a − are roots of 1 or − x = a − a , x = a − a − −
10 1 − a a x = a a − a − a − a − a − a − a − (22)Let w ± i be the eigenvectors of x i for eigenvalue ζ i = a − i = e πinipi , ζ − i = e − πinipi respectively and w i be the eigenvector of x i for eigenvalue 1. Then w ± i are the eigenvectors of x ri for ζ r i i and w i be the eigenvector of x r i i for 1. Byscaling, assume that | [ w ± i w i ] | = 1 in V . According to 22 , w ± , w − is a basis of V . Similarly, for lower triangular ones in 20, w ± , w +2 is a basis of V .For T i ( i = 1 , , −→ C ,ρ ( T i ) ∂ −→ C ,ρ ( T i ) ∂ −→ C ,ρ ( T i ) −→ ∂ = (cid:18) Ox i − I (cid:19) , ∂ = (cid:0) x i − I O (cid:1)
We have H ( T i ) = < ˜ u T i ⊗ w i >H ( T i ) = < ˜ m i ⊗ w i , ˜ l i ⊗ w i >H ( T i ) = < ˜ v T i ⊗ w i > Choose preference basis h ∗ for H ∗ ( T i ) as above and similarly with others.Without confusion, we omit h ∗ in the expression as c ∗ . τ ( C ∗ ,ρ ( T i )) = | [˜ l i ⊗ ( x i − I ) w ± i , ˜ m i ⊗ w i , ˜ l i ⊗ w i , ˜ m i ⊗ w ± i ][˜ u T i ⊗ w i , ˜ u T i ⊗ w ± i ][˜ v T i ⊗ w i , ˜ v T i ⊗ ( x i − I ) w ± i ] | = | [˜ l i ⊗ ( ζ ± i − w ± i , ˜ m i ⊗ w i , ˜ l i ⊗ w i , ˜ m i ⊗ w ± i ][˜ u T i ⊗ w i , ˜ u T i ⊗ w ± i ][˜ v T i ⊗ w i , ˜ v T i ⊗ ( ζ ± i − w ± i ] | = 1 (23)For T , we have ∂ = 0 , ∂ = 0. H ( T ) = < ˜ u T ⊗ e i > ( i = 1 , , H ( T ) = < ˜ m ⊗ e i , ˜ l ⊗ e i >H ( T ) = < ˜ v T ⊗ e i >τ ( C ∗ ρ ( T )) = 1 (24)21or A i ( i = 1 , , −→ C ,ρ ( A i ) −→ C ,ρ ( A i ) −→ ∂ = x r i i − I .We have H ( A i ) = < ˜ b i ⊗ w i >H ( A i ) = < ˜ v A i ⊗ w i >τ ( C ∗ ,ρ ( A i )) = | [˜ b i ⊗ w i , ˜ b i ⊗ w ± i ][˜ v A i ⊗ ( x r i i − I ) w ± i , ˜ v A i ⊗ w i ] | = | [˜ b i ⊗ w i , ˜ b i ⊗ w ± i ][˜ v A i ⊗ ( ζ ± r i i − w ± i , ˜ v A i ⊗ w i ] | = 1 | ζ r i i − || ζ − r i i − | (25)For A , we have ∂ = 0. H ( A ) = < ˜ b ⊗ e i > ( i = 1 , , H ( A ) = < ˜ v A ⊗ e i >τ ( C ∗ ρ ( A )) = 1 (26)For B , we have 0 −→ C ,ρ ( B ) ∂ −→ C ,ρ ( B ) ∂ −→ C ,ρ ( B ) −→ ∂ = O O OO O OO O Ox − I x − I x − I , ∂ = (cid:0) x − I x − I x − I O (cid:1)
We have H ( B ) = < ˜ u i,B ⊗ w i , (˜ u ,B + ˜ u ,B x + ˜ u ,B x x ) ⊗ e i > ( i = 1 , , H ( B ) = < ˜ x i ⊗ w i , (˜ x + ˜ x x + ˜ x x x ) ⊗ e i > ( C ∗ ,ρ ( B ))= | [˜ u i,B ⊗ w i , ˜ u ⊗ e i , ˜ u ,B ⊗ w ± , ˜ u ,B ⊗ w − ] − [˜ v B ⊗ ( x − I ) w ± , , ˜ v B ⊗ ( x − I ) w − ] − [˜ x i ⊗ w i , ˜ x ⊗ e i , ˜ h ⊗ ( x − I ) w ± , ˜ h ⊗ ( x − I ) w − , ˜ x ⊗ w ± , ˜ x ⊗ w − ] | = | [˜ u i,B ⊗ w i , ˜ u ⊗ e i , ˜ u ,B ⊗ w ± , , ˜ u ,B ⊗ w − ] − [˜ v B ⊗ ( ζ ± − w ± , ˜ v B ⊗ ( ζ − − w − ] − [˜ x i ⊗ w i , ˜ x ⊗ e i , ˜ h ⊗ ( ζ ± − w ± , ˜ h ⊗ ( ζ − − w − , ˜ x ⊗ w ± , ˜ x ⊗ w − ] | = | [˜ u i,B ⊗ w i , ˜ u ⊗ e i , ˜ u ,B ⊗ w ± , , ˜ u ,B ⊗ w − ] − [˜ v B ⊗ w ± , ˜ v B ⊗ w − ] − [˜ x i ⊗ w i , ˜ x ⊗ e i , ˜ h ⊗ w ± , ˜ h ⊗ w − , ˜ x ⊗ w ± , ˜ x ⊗ w − ] | = 1 (27)where ˜ x = ˜ x + ˜ x x + ˜ x x x , ˜ u = ˜ u ,B + ˜ u ,B x + ˜ u ,B x x .In the long exact sequence for twisted homology group, we have isomor-phisms 0 −→ H ∗ ( T i ) −→ H ∗ ( A i , B ) −→ C ∗ ,ρ ( X ) is acyclic as follows.We have 0 −→ M i =0 H ( T i ) −→ M i =0 H ( A i ) −→ ∂ (˜ v T i ⊗ w i ) = ˜ v A i ⊗ w i , ∂ (˜ v T ⊗ e i ) = ˜ v A ⊗ e i , det ( ∂ ) = 1.0 −→ M i =0 H ( T i ) −→ M i =0 H ( A i ) ⊕ H ( B ) −→ ∂ ( ˜ m i ⊗ w i ) = (˜ x i − ˜ b i Q i ) ⊗ w i , ∂ (˜ l i ⊗ w i ) = ˜ b i P i ⊗ w i , ∂ ( ˜ m ⊗ e i ) =(˜ x + ˜ x x + ˜ x x x ) ⊗ e i , ∂ (˜ l ⊗ e i ) = ˜ b ⊗ e i , Q i = P q i j =1 x − jr i , P j = P p i − j =0 x jr i , det ( ∂ ) = p p p . 0 −→ M i =0 H ( T i ) −→ H ( B ) −→ ∂ (˜ u T i ⊗ w i ) = ˜ u i,B ⊗ w i , ∂ (˜ u ⊗ e i ) = (˜ u ,B + ˜ u ,B x + ˜ u ,B x x ) ⊗ e i , det ( ∂ ) = 1.According to Multiplicativity lemma, Equations 23, 24, 25, 26, 27 and thecalculations about homology above, we haveTor( C ∗ ,ρ ( X )) = p p p Q i =1 πr i n i p i .3 Modular data from Seifert fibered spaces We will show that the modular data constructed from 3-component SFSs arerelated to the Temperley-Lieb-Jones categories at root of unit. So let us collectsome basic facts about those. For references, see for instance [19].Let A be a complex number such that A = 1. For an integer n , define thequantum integer [ n ] A = A n − A − n A − A − . So [0] A = 0 , [1] A = 1 , [2] A = A + A − .For each A , usually called the Kauffman variable, such that A is a primitive r -th root of unity for some integer r ≥
2, there is an associated premodularcategory, called the Temperley-Lieb-Jones category and denoted by TLJ( A ).The category has the label set (simple objects) [0 · · · p −
2] where the label 0 isthe unit object. For i, j ∈ [0 · · · p − d j ( A ) = ( − j [ j + 1] A = ( − j A j +2 − A − j − A − A − , the twist is θ j ( A ) = ( − A ) j ( j +2) , and the (un-normalized) S -matrix is˜ S ij ( A ) = ( − i + j [( i + 1)( j + 1)] A . The total dimension can be computed directly, D ( A ) = √ r | A − A − | . Denote by TLJ( A ) (resp. TLJ( A ) ) the subcategory linearly spanned by even(resp. odd) labels. We call TLJ( A ) and TLJ( A ) the even and odd subcategoryof TLJ( A ), respectively.The even and odd subcategory has the same dimension,both equal to D ( A ) √ .It is well known that if A is a primitive 4 r -th root of unity, then TLJ( A )is non-degenerate. If r is odd and A is a primitive 2 r -th root of unity, thenTLJ( A ) is degenerate, but the even subcategory TLJ( A ) is non-degenerate.Now we consider the construction of modular data. As before, set M = {
0; ( o, p , q ) , ( p , q ) , ( p , q ) } . Here each pair ( p k , q k ) are co-prime. Chooseintegers s k and r k such that p k s k − q k r k = 1. If q k is odd, set c k = p k q k s k − r k .Otherwise, set c k = p k q k s k − r k ( p k − . Let A k = − exp( πi p k c k ). Note thatwhile c k depends on the choice of s k and r k , A k does not. Moreover, A k is aprimitive 4 p k -th root of unity if q k is odd, a primitive 2 p k -th root of unity if q k = 0 mod 4, and a primitive p k -th root of unity if q k = 2 mod 4. In thelatter two cases, p k clearly must be odd. Hence, in all cases, A k is a primitive p k -th root of unity.If some q ′ k s are even, we re-arrange the elements of χ nab ( M ) as follows. For( p, q ) co-prime, j ∈ [0 · · · p − n p,q ( j ) = ( p − − j , q even and j even j +12 , otherwise24hen from Equation 21, χ nab ( M ) can also be written as (cid:26) ( n p ,q ( j ) , n p ,q ( j ) , n p ,q ( j ) ,
12 ) | j k ∈ [0 · · · p k − e , k = 1 , , (cid:27) ⊔ (cid:26) ( n p ,q ( j ) , n p ,q ( j ) , n p ,q ( j ) , | j k ∈ [0 · · · p k − o , k = 1 , , (cid:27) (28)Thus, the elements of χ nab ( M ) are indexed by ~j ∈ Q k =1 [0 · · · p k − e ⊔ Q k =1 [0 · · · p k − o . Given such a ~j = ( j , j , j ), denote a corresponding repre-sentation by ρ ~j . (The choice of a representative is irrelevant.)Proposition 3.1 shows that all non-Abelian characters of M are adjointacyclic and Proposition 2.1 shows that the CS invariants of non-Abelian char-acters are all rational. We choose the candidate label set L ( M ) to be χ nab ( M ).We propose the correspondence between L ( M ) and loop operators by thefollowing map, ρ ~j (cid:8) ( x c k k , Sym j k ) | k = 1 , , (cid:9) . (29)Moreover, we designate ρ ~ = ρ (0 , , as the unit object, which of course corre-sponds to the loop operator = ρ ~ (cid:8) ( x c k k , Sym ) | k = 1 , , (cid:9) . (30)The following two lemmas are direct consequences of Proposition 2.1 andProposition 3.1, respectively. Lemma 3.2.
Let
M, c k , A k be given as above. For each ~j = ( j , j , j ) ∈ Q k =1 [0 · · · p k − e ⊔ Q k =1 [0 · · · p k − o with ρ ~j a corresponding representation,then CS( ρ ~j ) = X k =1 − c k p k ( j k + 1) . (31)As a consequence, e − πi CS( ρ ~j ) = Y k =1 ( − A k ) ( j k +1) = ( − A A A ) Y k =1 θ j k ( A k ) . (32) Lemma 3.3.
Let
M, c k , A k be given as above and let D = D ( A ) D ( A ) D ( A ) / ~j = ( j , j , j ) ∈ Q k =1 [0 · · · p k − e ⊔ Q k =1 [0 · · · p k − o with ρ ~j acorresponding representation, thenTor( ρ ~j ) = Y k =1 p k ( πr k ( j k +1) p k ) , (33)and hence, (cid:0) ρ ~j ) (cid:1) − = 2 Y k =1 (cid:12)(cid:12)(cid:12)(cid:12) d j k ( A k ) D ( A k ) (cid:12)(cid:12)(cid:12)(cid:12) = | Q k =1 d j k ( A k ) | D . (34)25he main result of the section is the following theorem.
Theorem 3.4.
Let M = {
0; ( p , q ) , ( p , q ) , ( p , q ) } and { A k } k =1 , , be givenas above. With the operators and tensor unit given in Equations 29 and 30,respectively, the modular data constructed from M matches that of the followingpre-modular category, B := (cid:0) ⊠ k =1 TLJ( A k ) (cid:1) M (cid:0) ⊠ k =1 TLJ( A k ) (cid:1) Proof.
Since A k is a primitive p k -th root of unity, the label set for B is clearly L := Q k =1 [0 · · · p k − e ⊔ Q k =1 [0 · · · p k − o , the same index set for L ( M ).The modular data of B can be easily expressed in terms of that of the individualTLJ( A k ). For ~i,~j ∈ L , d ~j = Y k =1 d j k ( A k ) , θ ~j = Y k =1 θ j k ( A k ) , ˜ S ~i~j = Y k =1 ˜ S i k j k ( A k ) . Also, the total dimension of B is D = D ( A ) D ( A ) D ( A ) / e − πi CS( ρ ~j ) = θ ~j , and Lemma 3.3 shows that the torsion matches the absolute value of the nor-malized quantum dimension, (cid:0) ρ ~j ) (cid:1) − = d ~j D .
Lastly, We check the S -matrix computed from local operators. Given ~i =( i , i , i ) , ~j = ( j , j , j ) ∈ L , we have (choosing ǫ = − W ~i ( ~j ) = Y k =1 Tr Sym jk ( − ρ ~i ( x c k k )) . Note that, Tr (cid:0) ρ ~i ( x c k k ) (cid:1) = 2 cos 2 n p k ,q k ( i k ) πc k p k = 2 cos ( i k + 1) πc k p k , where the second equality holds irrelevant of the parity of q k . Combining theprevious two equations, we get W ~i ( ~j ) = Y k =1 ∆ j k ( − i k + 1) πc k p k ) = Y k =1 ( − j k sin ( i k +1)( j k +1) πc k p k sin ( i k +1) πc k p k , j k ( · ) is the Chebyshev polynomial (see Equation 12). Therefore, the( ~j,~i )-entry of the potential un-normalized S matrix is, W ~i ( ~j ) W ~ ( ~i ) = Y k =1 ( − i k + j k sin ( i k +1)( j k +1) πc k p k sin πc k p k = Y k =1 ˜ S ( A k ) j k i k , which is precisely ˜ S ~j~i of B .The premodular category produced in the previous theorem may not be mod-ular in general, and it depends crucially on the topology of the three manifold.For a three-component SFS M , it is a Z homology sphere, i.e., H ( M, Z ) = 0,if and only if p p p ( q p + q p + q p ) ∈ Z + 1 Lemma 3.5.
Assume that r is odd. Suppose that T ( p, j, l, ∗ ) = X m ∈ [ p ] ∗ (cid:16) e ( j + l ) mr πp i − e ( j − l ) mr πp i − e ( − j + l ) mr πp i + e ( − j − l ) mr πp i (cid:17) where ∗ = 1 ,
0, and [ p ] ∗ denotes the set of odd integers from 1 to p − ∗ is 1and the set of even integers in the same range otherwise.When p is odd, j = l , j + l is odd, T ( p, j, l, ∗ ) = ( j + l = p ( − ∗ p j + l = p When p is odd, j = l , j + l is even, T ( p, j, l, ∗ ) = 0When p is odd, j = l , T ( p, j, l, ∗ ) = − p When p is even, j = l , j + l is odd, T ( p, j, l, ∗ ) = 0When p is even, j = l , j + l is even, T ( p, j, l, ∗ ) = ( j + l = p ( − ∗ p j + l = p p is even, j = l , T ( p, j, l,
0) = ( − p j + l = p j + l = pT ( p, j, l,
1) = ( − p j + l = p − p j + l = p Proof.
We prove the lemma by direct computation.When p is odd, j = l , j + l is odd, T ( p, j, l,
1) = p − X m =1 ,m odd ( e ( j + l ) mr πp i − e ( j − l ) mr πp i + e ( j − l )( p − m ) r πp i − e ( j + l )( p − m ) r πp i )= p − X m =1 ,m odd ( e ( j + l ) mr πp i − e ( j − l ) mr πp i ) + p − X m =2 , even ( e ( j − l ) mr πp i − e ( j + l ) mr πp i )= − p − X m =1 ( − e ( j + l ) r πp i ) m + p − X m =1 ( − e ( j − l ) r πp i ) m = ( j + l = p − p j + l = p = − T ( p, j, l, Proposition 3.6.
Given a three-component SFS M , the premodular category B M produced in Theorem 3.4 is modular if and only if M is a Z homologysphere. Proof.
Since the structure from Section 3 respects the change of parametrizationof Seifert fiber space, it suffices to verify the following 5 cases for ( p q , p q , p q ).( oddodd , oddodd , oddodd ) , ( oddodd , oddodd , evenodd ) , ( oddodd , evenodd , evenodd ) , ( evenodd , evenodd , evenodd ) , ( oddodd , oddodd , oddeven )The first two cases correspond to Z -homology sphere. In the following, we willexplicitly calculate S , which directly implies the proposition.When q , q , q are odd, j = j = j mod 2, l = l = l mod 2.Up to a scalar, S ( j ,j ,j ) , ( l ,l ,l ) = ( − j + l Y k =1 sin j k l k r k πp k S ) ( j ,j ,j ) , ( l ,l ,l ) = X ( m ,m ,m ) ( − j + m + m + l Y k =1 sin j k m k r k πp k sin m k l k r k πp k = ( − j + l X ( m ,m ,m ) 3 Y k =1 −
14 ( e ( j k + l k ) m k r k πpk i − e ( j k − l k ) m k r k πpk i − e ( − j k + l k ) m k r k πpk i + e ( − j k − l k ) m k r k πpk i )= ( − j + l ( X ( m ,m ,m ) ,m i odd + X ( m ,m ,m ) ,m i even ) ... = ( − j + l ( Y k =1 T ( p k , j k , l k ,
1) + Y k =1 T ( p k , j k , l k , p , p , p are odd,( S ) ( j ,j ,j ) , ( l ,l ,l ) = j , j , j ) = ( l , l , l ) p p p
32 ( j , j , j ) = ( l , l , l )When p , p are odd, p is even,( S ) ( j ,j ,j ) , ( l ,l ,l ) = j , j , j ) = ( l , l , l ) p p p
32 ( j , j , j ) = ( l , l , l )Thus S = cI for the above two cases.When p is odd, p , p are even,( S ) (1 , , , ( l ,l ,l ) = p p p
32 ( l , l , l ) = (1 , , , (1 , p − , p − S ) (1 ,p − ,p − , ( l ,l ,l ) = p p p
32 ( l , l , l ) = (1 , , , (1 , p − , p − S ) (1 , , = ( S ) (1 ,p − ,p − When p , p , p are even,( S ) (1 , , , ( l ,l ,l ) = p p p
32 ( l , l , l ) = (1 , , , (1 , p − , p − S ) (1 ,p − ,p − , ( l ,l ,l ) = p p p
32 ( l , l , l ) = (1 , , , (1 , p − , p − S ) (1 , , = ( S ) (1 ,p − ,p − S is degenerate for above two cases.When q , q are odd, q is even, j = j mod 2, l = l mod 2, j = 0 mod 2, l = 0 mod 2.( S ) ( j ,j ,j ) , ( l ,l ,l ) = Y k =1 T ( p k , j k , l k , T ( p , j , l ,
0) + Y k =1 T ( p k , j k , l k , p , p , p are odd,( S ) (1 , , , ( l ,l ,l ) = − p p p
32 ( l , l , l ) = (1 , , , ( p − , p − , S ) ( p − ,p − , , ( l ,l ,l ) = − p p p
32 ( l , l , l ) = (1 , , , ( p − , p − , S is degenerate.It is worth noting even if every TLJ( A k ) appearing in the construction of B M in Theorem 3.4 is not modular, B M could still be modular. For instance, for theSFS M = (0; ( o, , , (3 , , (5 , A = − e iπ , A = − e iπ , A = − e iπ . It is direct to see that TLJ( A ) ismodular, but TLJ( A ) and TLJ( A ) are not. However, M is a Z homologysphere, by Proposition 3.6, B M is modular, a rank-8 MTC. (2) k Here we study a special class of SFSs with three components, namely, M ( r ) := {
0; ( o, , , (3 , , ( r, } . We show explicitly that different choice of charac-ters as the unit object may lead to different theories. In fact, it will be provedthat from M ( r ) we can construct either the MTC SU(2) r − or TLJ( e πi r ).For each integer r ≥
2, there is a unitary MTC, usually denoted by SU(2) r − [3], which is closely related to the Temperley-Lieb-Jones categories. Here r − e πi r ), butdiffers from it in modular data by some signs. Explicitly, setting A = e πi r , themodular data for SU(2) r − is given as follows, θ j = A j ( j +2) = e πi j ( j +2)4 r , ˜ S ij = [( i + 1)( j + 1)] A = sin ( i +1)( j +1) πr sin πr . In particular, its quantum dimensions are all positive (since it is unitary), d j = [ j + 1] A = sin ( j +1) πr sin πr , D = r r πr . Note that d j = | d j ( A ) | and D = D ( A ), where d j ( A ) and D ( A ) are the quantumdimension of j and total dimension of TLJ( A ), respectively.We will use notations from Section 3.1 and 3.3. The non-Abelian charactersof M ( r ) is given by χ nab ( M ( r )) = (cid:26)(cid:18) , , j + 12 , (cid:19) | (0 , , j ) ∈ { } × { } × [0 · · · r − e (cid:27) ⊔ (cid:26)(cid:18) , , j + 12 , (cid:19) | (1 , , j ) ∈ { } × { } × [0 · · · r − o (cid:27) . (35)Thus, each j ∈ [0 · · · r −
2] corresponds to a non-Abelian character indexedby ( j mod 2 , j mod 2 , j ). We denote the corresponding representation by ρ j (instead of using the triple as the subscript). The eigenvalues of ρ j ( x ) are e ± ( j +1) πir . The eigenvalues of ρ j ( x ) and those of ρ j ( x ) are both e ± ajπi , where a j = 1 if j even and a j = 2 otherwise.Also, it is direct to see that c = c = c = 1, and A = A = − e πi , A = − e πi r .In Section 3.3, we chose the candidate label set L ( M ( r )) to be χ nab ( M ( r )),and defined the following map from χ nab ( M ( r )) to local operators, ρ j = (cid:8) ( x , Sym j mod 2 ) , ( x , Sym j mod 2 ) , ( x , Sym j ) (cid:9) . (36)It can be checked directly that for i, j ∈ [0 · · · r − ρ i ( x )) = Tr( ρ i ( x )) = ±
1, and it follows that, W i ( j ) = Tr Sym j mod 2 ( − ρ i ( x )) Tr Sym j mod 2 ( − ρ i ( x )) Tr Sym j ( − ρ i ( x ))= Tr Sym j ( − ρ i ( x )) . Hence, we may as well choose a simplified map to local operators, ρ j
7→ { ( x , Sym j ) } . (37)The unit object was chosen to be ρ which corresponds to the local operator( x , Sym ). By Theorem 3.4, the modular data match that of the premodularcategory, B M ( r ) = (cid:0) ⊠ k =1 TLJ( A k ) (cid:1) M (cid:0) ⊠ k =1 TLJ( A k ) (cid:1) . (38)Note that TLJ( A ) = TLJ( − e πi ) has label set { , } , the twists θ = 1, θ = i , and un-normalized S -matrix,˜ S = (cid:18) − − − (cid:19) . B M ( r ) has the same twists for even labels and S -matrix asTLJ( A ). The twists for odd labels differ by a minus sign between the twotheories. Let A ( r ) = − A = e πi r . Note that a change of the Kauffman vari-able from A to − A does not change the S -matrix. It follows that B M ( r ) andTLJ( A ( r )) has the same modular data. In fact, they are isomorphic.Therefore, by using the local operator correspondence in Equation 37 andletting ρ be the unit object, we recover the MTC TLJ( A ( r )).Now we examine an alternative choice of the unit object. Since M ( r ) isa Z homology sphere, a potential unit object ρ α can be determined by theequation, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ρ ∈ χ nab ( M ( r )) exp( − πi CS( ρ ))2Tor( ρ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (2Tor( ρ α )) − . (39)Such a ρ α would have quantum dimension in absolute value equal to 1 in anyMTC produced by M ( r ). Since we already know that we can produce TLJ( A ( r ))from M ( r ) and the only non-unit object in TLJ( A ( r )) whose quantum dimensionis 1 in absolute value is ρ r − , we can choose ρ r − as the unit object in a newtheory.In this case, we reverse the previous order of the simple objects. Denoteby ˜ ρ j := ρ r − − j , j ∈ [0 · · · r − ρ = ρ r − as the unit object. Thecorrespondence between characters and local operators is now defined as,˜ ρ j ( x , Sym j ) . (40)We claim that with above choice of unit object and local operators, the modulardata produced from M ( r ) matches that of SU(2) r − where ˜ ρ j corresponds to j in the label set of SU(2) r − . See Section 3.3 for a collection of facts aboutSU(2) r − .Firstly, by Lemma 3.2, up to an irrelevant phase factor,CS( ρ j ) = − j ( j + 2)4 r + 1 − ( − j . (41)Then rewriting above equation in terms of ˜ ρ j , we get, again up to an irrelevantfactor, CS(˜ ρ j ) = − j ( j + 2)4 r mod 1 . (42)Thus, e − πi CS(˜ ρ j ) = e πi j ( j +2)4 r (43)is the twist θ j of SU(2) r − .Next, we check the S -matrix. W ( j ) = Tr Sym j ( − ˜ ρ ( x )) = ∆ j (2 cos πr ) = sin ( j +1) πr sin πr , (44)32nd the ( j, i )-entry of the potential S -matrix is, W i ( j ) W ( i ) = Tr Sym j ( − ˜ ρ i ( x )) W ( i ) (45)= ∆ j (2 cos ( i + 1) πr )∆ i (2 cos πr ) (46)= sin ( i +1)( j +1) πr sin πr , (47)which is ˜ S ji of SU(2) r − .Lastly, by Lemma 3.3, (cid:0) ρ j ) (cid:1) − = (cid:0) ρ r − − j ) (cid:1) − = | d r − − j ( A ) | D ( A ) , (48)where we used the fact that in TLJ( A ) = TLJ( A ), the two simple objects havequantum dimensions ± D ( A ) = √
2. Also note that A = − e πi r , then | d r − − j ( A ) | = | d j ( A ) | and D ( A ) areequal to the quantum dimension d j and the total dimension D , respectively,in SU(2) r − . Hence, the torsion invariant computes the normalized quantumdimension, (cid:0) ρ j ) (cid:1) − = d j D . (49)To summarize, for the SFS M ( r ), two choices of the unit object togetherwith appropriate definition of local operators produce the MTCs TLJ( e πi r ) andSU(2) r − , with the former non-unitary and the latter unitary. In Section 3.3, we have seen that the premoduar category resulting from three-component SFSs is formed from three Temperley-Lieb-Jones categories, by tak-ing the Deligne product of the even sectors, that of the odd sectors, and sumingthem up. Here we generalize the operation.
Definition 3.7.
Let C = ⊕ g ∈ G C g and D = ⊕ g ∈ G D g be two G -graded pre-modular tensor categories for some finite group G (which must be Abelian).The graded product of C and D is again a G -graded premodular category C ⊠ gr D = ⊕ g ∈ G ( C ⊠ gr D ) g such that ( C ⊠ gr D ) g := C g ⊠ D g .The monoidal and braiding structure on C ⊠ gr D is defined in the obviousway which make it into a premodular category. Another way to see this isthat C ⊠ gr D is a full subcategory of the premodular category C ⊠ D and isclosed under tensor product and braiding. The graded product operation ⊠ gr is associative up to canonical equivalence.For a Kauffman variable A , TLJ( A ) is a Z -graded premodular category withTLJ( A ) spanned by even labels and TLJ( A ) odd labels. Hence, Theorem 3.433tates that, for a three-component SFS M = {
0; ( o, p , q ) , ( p , q ) , ( p , q ) } with A k , k = 1 , , M is B M = TLJ( A ) ⊠ gr TLJ( A ) ⊠ gr TLJ( A ).The graded product operation provides method to construct new premodularcategories from old ones. A very interesting question is when the graded productof two pre-modular categories is modular. For instance, take A = − e iπ , A = − e − iπ . Here A is a primitive 12-th root of unity and A a primitive 5-th rootof unity. Hence TLJ( A ) is modular of rank 2 and TLJ( A ) is none modular ofrank 4. Their S -matrices are given by,˜ S ( A ) = (cid:18) − − − (cid:19) , ˜ S ( A ) = ϕ ϕ ϕ − − ϕϕ − − ϕ ϕ ϕ , (50)where ϕ = (1 − √ S -matrix of TLJ( A ) ⊠ gr TLJ( A ) with itssimple objects ordered as { ⊠ , ⊠ , ⊠ , ⊠ } is,˜ S = ϕ − ϕ − ϕ − − ϕ − ϕ − ϕ − − ϕ − ϕ − , (51)which can be checked straightforwardly to be non-degenerate. Thus TLJ( A ) ⊠ gr TLJ( A ) is modular.We leave the question of when the graded product of two arbitrary graded(and more generally multiple) premodular categories is modular as a futuredirection. In the rest of this section, we focus on the case where the group is Z and study a special class of Z -graded modular categories, namely SU(2) k . Forbasic facts, see Section 3.4.Let C = C ⊕ C be a Z -graded MTC. Denote by I the label set of C and partition I = I ⊔ I where I α consists of objects of I that are in the C α sector. To avoid confusion, when there is more than one MTC present, we write I ( C ) , ˜ S ( C ), etc. Proposition 3.8.
Let C and D be two Z -graded MTCs. Then C ⊠ gr D is aproper (i.e., degenerate) premodular category if and only if there exist i ∈ I ( C ), j ∈ I ( D ), scalars c ( C ) , c ( C ) , c ( D ) , and c ( D ), such that,1. i and j belong to sectors of the same parity;2. the following equations concerning S -entries hold:˜ S ( C ) ik = ( c ( C ) d k ( C ) k ∈ I ( C ) c ( C ) d k ( C ) k ∈ I ( C ) ˜ S ( D ) jk = ( c ( D ) d k ( D ) k ∈ I ( D ) c ( D ) d k ( D ) k ∈ I ( D )3. c ( C ) /c ( C ) = c ( D ) /c ( D ) = 1. 34 roof. The main idea is to show that the conditions presented in the statementof the proposition are equivalent to the property that in the S -matrix of C ⊠ gr D ,the row corresponding to the object i ⊠ j is proportional to the first row (i.e.,the row corresponding to the unit object). Remark 3.9.
In the above proposition, the conditions c ( C ) /c ( C ) = 1 and c ( D ) /c ( D ) = 1 are used to eliminate the trivial case where i and j are boththe unit object. When neither of i nor j is the unit object, those conditionsautomatically hold since otherwise the S -matrix of C or D would be degenerate.Also, note that if either C or D is non-degenerate, then i and j must be in thesector of odd parity.For m ≥
0, SU(2) m is a Z -graded MTC with (SU(2) m ) spanned by evenlabels and (SU(2) m ) by odd labels. Theorem 3.10.
For m, n ≥
0, SU(2) m ⊠ gr SU(2) n is an MTC if and only if thepair ( m, n ) have different parity. In particular, SU(2) m ⊠ gr SU(2) m is alwaysdegenerate. Proof.
In SU(2) m , the un-normalized S -matrix is given by,˜ S ab = sin ( a +1)( b +1) πm +2 sin πm +2 . Hence, ˜ S mb = ( − b ˜ S b = ( − b d b . For ( m, n ) with the same parity, withthe notation from the statement of Proposition 3.8, we choose i = m, j = n .Then the relevant constants are c (SU(2) m ) = c (SU(2) n ) = 1, c (SU(2) m ) = c (SU(2) n ) = − m ⊠ gr SU(2) n is degenerate. For the converse direction, it can beseen that the only non-unit simple object in SU(2) m for which c (SU(2) m ) and c (SU(2) m ) exist is the object m . Therefore, if ( m, n ) have different parity, theonly pair of indexes for ( i, j ) is ( m, n ) which contradicts the first condition ofProposition 3.8. This implies that SU(2) m ⊠ gr SU(2) n is non-degenerate. Example 3.11.
By Theorem 3.10, SU(2) ⊠ gr SU(2) is an MTC of rank 6. Itsun-normalized S -matrix and T -matrix are given by,˜ S = (cid:0) √ (cid:1) (cid:0) √ (cid:1) √ √ √ (cid:0) √ (cid:1) − (cid:0) √ (cid:1) − −√ √ √ (cid:0) √ (cid:1) (cid:0) √ (cid:1) − √ √ −√ (cid:0) √ (cid:1) − (cid:0) √ (cid:1) − √ − √ √ √ √ −√ − √ √ √ √ √ √ −√ − √ √ = e iπ − − e iπ e iπ
00 0 0 0 0 − ie iπ Since SU(2) ⊠ gr SU(2) contains the even part of SU(2) as a subcategorywhich is itself an MTC (Fibonacci), SU(2) ⊠ gr SU(2) must split. In fact,SU(2) ⊠ gr SU(2) ≃ Fib ⊠ TLJ( − ie πi ). One of the non-hyperbolic geometries is SOL and some examples of closed man-ifolds are torus bundles over the circle with Anosov monodromy maps.Let M be a torus bundle over S with the monodromy map (cid:18) a bc d (cid:19) ∈ SL(2 , Z ) where | a + d | >
2. Its fundamental group has the presentation, π ( M ) = h x, y, h | x a y c = h − xh, x b y d = h − yh, xyx − y − = 1 i , (52)where x and y are the meridian and longitude, respectively, on the torus, and h corresponds to a loop around the S component. We consider non-Abeliancharacters of M to SL(2 , C ). Let ρ : π ( M ) → SL(2 , C ) be a non-Abelianrepresentation.First, we consider the case where ρ ( x ) is diagonalizable. Up to conjugation,assume ρ ( x ) is diagonal. Since y commutes with x , ρ ( y ) is also diagonal, andmoreover, ρ ( x ) and ρ ( y ) cannot be both contained in the center {± I } . (Oth-erwise, the image of ρ would be Abelian.) If ρ ( x ) = ± I , it follows from therelation x a y c = h − xh that ρ ( h ), up to conjugation, simply permutes the twoeigenvectors of ρ ( x ). The same conclusion is obtained if ρ ( y ) = ± I . Hence, wemay assume ρ takes the following form (abbreviating ρ ( x ) simply as x ), x = (cid:18) α α − (cid:19) , y = (cid:18) β β − (cid:19) , h = (cid:18) − (cid:19) , (53)where Im( α ) ≥ α = ± β = ±
1. The presentation of π ( M )yields the following equations for ρ , α a +1 β c = α b β d +1 = 1 , (54)from which we deduce the relations, α a + d +2 = β a + d +2 = 1 . (55)36et N = | a + d + 2 | . Hence α and β are both N -th root of unity. Set α = e πi kN , β = e πi lN such that 0 ≤ k ≤ N , ≤ l < N , and either k = 0 , N or l = 0 , N . Then, Equation 54 can be equivalently written as,( a + 1) k + c l = 0 mod Nb k + ( d + 1) l = 0 mod N (56)The solutions to Equation 56 depend on a number of conditions involving a, b, c, and d . When at least one of a +1 , c, b, d +1 is co-prime to N , there is a compactform to organize all the solutions. For instance, when ( c, N ) are co-prime, thesolutions are simply given by, l = − ˜ c ( a + 1) k mod N, k = 1 , · · · , ⌊ N − ⌋ , (57)where ˜ c is the multiplicative inverse of c in Z N . The representations thus ob-tained are all irreducible.Now we consider the case where ρ ( x ) is not diagonalizable. Then neitheris ρ ( y ) diagonalizable. Up to conjugation, we may assume that ρ ( x ) and ρ ( y )are both upper triangular, each have a single eigenvalue +1 or − ρ ( h ) is diagonal. Thus, ρ takes the form, x = ( − ǫ x (cid:18) (cid:19) , y = ( − ǫ y (cid:18) u (cid:19) , h = (cid:18) v v − (cid:19) , (58)where ǫ x , ǫ y ∈ { , } and u = 0. From the presentation of π ( M ), we deducethe equations to be satisfied,( a + 1) ǫ x + c ǫ y = 0 mod 2 b ǫ x + ( d + 1) ǫ y = 0 mod 2 (59) c u + ( a − d ) u − b = 0 , v = 1 cu + a . (60)Equation 60 is equivalent to,( v + v − ) = a + d + 2 , u = v − − ac . (61)From Equation 61, we see that for each fixed ǫ x and ǫ y , there are four inequiv-alent representations, but only two characters. We choose a representative foreach character by setting, u = d − a + p ( a + d ) − c , v = 1 cu + a = a + d − p ( a + d ) − . (62)The solution set to Equation 59 depends on the parity of the entries ofthe monodromy matrix. Let P be the quadruple that records the parity ofthe entries ( a, d ; b, c ) and we use ‘ e ’ to denote for ‘even’ and ‘ o ’ for ‘odd’. Forinstance, P = ( e, e ; o, e ) means b is odd and the rest are even. The solutionscontain the following possible values for ǫ x and ǫ y ,37 ǫ x = 0 , ǫ y = 0; • ǫ x = 1 , ǫ y = 1, only if P = ( e, e ; o, o ) or P = ( o, o ; e, e ); • ǫ x = 0 , ǫ y = 1, only if P = ( o, o ; o, e ) or P = ( o, o ; e, e ); • ǫ x = 1 , ǫ y = 0, only if P = ( o, o ; e, o ) or P = ( o, o ; e, e ).Note that the last three cases above all imply that N = | a + d + 2 | is even andall possible configurations of P that have N even are contained in one (or more)of the last three cases.To summarize, the non-Abelian characters of M contain two types, the ir-reducible and the reducible ones. The irreducible characters take the form ofEquation 53 and are determined by Equation 56. The reducible characters takethe form of Equation 58 and are determined by Equation 62 and the possiblevalues of ǫ x and ǫ y discussed above. In this subsection, we compute the torsion and Chern-Simons invariant for thetorus bundle over the circle M with the monodromy map (cid:18) a bc d (cid:19) ∈ SL(2 , Z )where | a + d | >
2. Its fundamental group has a presentation given in Equation52. Construct a cell structure for M as follows. See Figure 2. The cell structurecontains, • a single 0-cell v ; • three 1-cells corresponding to the generators x, y , and h in the presenta-tion of π ( M ); • three 2-cells corresponding to the three relations in the presentation of π ( M ). Explicitly, denote them by s , s and s such that ∂s = yxy − x − , ∂s = h − xh ( x a y c ) − , and ∂s = h ( x b y d ) h − y − . Graphically, s , s and s correspond to the top face, the back face, and the left face, respectively,in Figure 2 with the induced orientation of the cube. • a single 3-cell t . Think of a 3-cell as a cube. Then the attaching map isdetermined by the identification of faces described in Figure 2.Let V be a representation ρ : π ( M ) → GL ( V ), and let { v j | j = 1 , , · · · } be an arbitrary basis of V . We now construct the chain complex. For simplicity,assume that a, b, c, d ≥ a ≥ c , b ≥ d . Other cases can be dealt similarly. Fixan arbitrary preimage ˜ v of v . For each other cell σ , fix a lifting ˜ σ starting atthe base point ˜ v . We have the following chain complex,0 −→ C ∂ −→ C ∂ −→ C ∂ −→ C −→ a y c x b y d Φ = (cid:18) a bc d (cid:19) v xy h
Figure 2: A cell structure for the torus bundle with monodromy matrix Φ Forconvenience but no other purposes, mark the vertical edges green, the horizontalon the top face red, and the 45 o -slope edges on the top face blue. Edges of thesame color and the same arrow are identified. The front and back faces areidentified by the obvious map, and so are the left and right side faces. Thebottom face is identified to the top via the monodromy map Φ. Hence, thesingle-arrow edge and the double-arrow edge at the bottom face are homotopicto x a y c and x b y d , respectively.where C i = C i ( f M ) ⊗ Z [ π ( M )] V . As a vector space, C i has the following basis, C = span { ˜ t ⊗ v j | j = 1 , , · · · } , C = span { ˜ s i ⊗ v j | i = 1 , , , j = 1 , , · · · } , C = span { ˜ σ ⊗ v j | σ = x, y, h, j = 1 , , · · · } , C = span { ˜ v ⊗ v j | j = 1 , , · · · } .We present the boundary map ∂ i as a block matrix with each entry a dim( V ) × dim( V ) block. Also, denote S : Z [ π ( M )] → Z [ π ( M )] the antipode map thatsends a group element g ∈ π ( M ) to its inverse g − and linearly extends to thewhole ring. Lastly, for a matrix A with entries in Z [ π ( M )], ρ ◦ S ( A ) is meantapplying ρ ◦ S to every entry of A . With the above conventions, the boundarymap is given by, ∂ = ρ ◦ S − hw ( x, y )1 − y − x ∂ = ρ ◦ S y − − h P a − i =1 x i h P b − i =1 x i − x − hx a P c − i =1 y i hx b P d − i =1 y i − x − − y ∂ = ρ ◦ S (cid:0) x − y − h − (cid:1) where w is a polynomial of x, y with the sum of its coefficients equal to 1.For each of the non-Abelian characters of π ( M ) to SL(2 , C ), we will computeits torsion below and show (implicitly) that the associated chain complex isalways acyclic and the torsion does not depend on the representation chosen inthe equivalence class of a character. 39or an irreducible representation ρ given in Equation 53 that satisfies Equa-tion 56, its adjoint representation has the form, x = α α − , y = β β − , h = − − − Denote by I and O and 3 × A = , B = . Define the block matrices, K = AOB , K = O AI OO B , K = (cid:0) I (cid:1) . It can be checked directly that the columns (as vectors in C i − ) of ∂ i K i is abasis of Im( ∂ i ). Set K = K to be the empty matrix. Now for i = 0 , , ,
3, let A i = (cid:0) ∂ i +1 K i +1 K i (cid:1) , then the columns of A i give a basis for C i . By direct calculations, we obtainthe torsion, Tor( ρ ) = (cid:12)(cid:12)(cid:12)(cid:12) det( A ) det( A )det( A ) det( A ) (cid:12)(cid:12)(cid:12)(cid:12) = | a + d + 2 | . Now we compute the torsion of the reducible representations ρ given inEquation 58. The associated adjoint representation takes the form, x = − −
10 1 10 0 1 , y = − u − u u , h = v v , which are clearly independent on the sign terms ǫ x and ǫ y . Let, A = , B = , C = ,D = , E = , F = . Define the block matrices, K = EOF , K = A OB CO D , K = (cid:0) I (cid:1) . K i have the same properties as outlined in the case of irreduciblerepresentations above, and in the same way define the matrices A i . It can becomputed that, Tor( ρ ) = (cid:12)(cid:12)(cid:12)(cid:12) det( A ) det( A )det( A ) det( A ) (cid:12)(cid:12)(cid:12)(cid:12) = | a + d + 2 | . Some details for the derivation are as follows, where the condition cu + ( a − d ) u − b = 0 is used to simplify expressions,Tor( ρ ) = | (2 cu + a − d )( b − u + du )( a − b + 1 + ( c − d − u ) u (1 − cu − a ) ( u − | = | (2 cu + a − d )( b − u + du )( a − b + 1 + ( c − d − u )( cu + ( a − u )( cu + ( a − − c ) u − a + 1) | = | (2 cu + a − d )( b − u + du )( a − b + 1 + ( c − d − u )(( d − u + b )(( d − c − u + b − a + 1) | = | ( d − c − u + b − a + 12( c − d − cu + (2 c ( a − b + 1) + ( a − d )( c − d − u + ( a − d )( a − b + 1) | = | (2 c ( a − b + 1) − ( a − d )( c − d − u + ( a − d )( a − b + 1) + 2 b ( c − d − d − c − u + b − a + 1 | = | ( a + d + 2)(( d − c − u + b − a + 1)( d − c − u + b − a + 1 | = | a + d + 2 | . Now, we compute the CS invariant of M . Any irreducible representation of π ( M ) to SL(2 , C ) can be conjugated to one into SU(2) (see Equation 53), andKirk and Klassen computed its CS invariant in [10]. Here we use methods inSection 2.3 to compute the CS invariant of both irreducible and reducible butindecomposable ones, the latter of which can not be conjugated to SU(2).Let T i ( i = A, B ) be two copies of the torus, and I be the interval [0 , M is obtained by gluing the two T i × I such that T B × { } is glued to T A × { } via the identity map and T B × { } is glued to T A × { } via the map (cid:18) a bc d (cid:19) . Let ( µ i , λ i ) be a positive basis of H ( T i ) so that, under the embedding T i × I ֒ → M , µ i and λ i are sent to x and y , respectively. For κ = 0 ,
1, denote by µ κi the element of H ( T i × { κ } ) that corresponds to µ i in H ( T i × I ), and by λ κi in a similar way. Then ( µ i , λ i ) is a positive basis for H ( T i × { } ) and ( − µ i , λ i )is a positive basis for H ( T i × { } ). These basis are identified as follows,( µ B , λ B ) = ( µ A , λ A ) , ( µ B , λ B ) = ( µ A , λ A ) (cid:18) a bc d (cid:19) . Set N = | a + d + 2 | . For an irreducible representation ρ in Equation 53 where41 = e πi kN and β = e πi lN , we have c T i × I ( ρ ) = [ kN , lN , kN , lN ; 1] ( µ i ,λ i ) , ( µ i ,λ i ) = [ kN , lN , − kN , lN ; 1] ( µ i ,λ i ) , ( − µ i ,λ i ) Hence, c T A × I ( ρ )= [ kN , lN , kN , lN ; 1] ( µ A ,λ A ) , ( µ A ,λ A ) = [ kN , lN , ak + clN , bk + dlN ; 1] ( µ A ,λ A ) , ( µ B ,λ B ) = [ kN , lN , − kN , bk + dlN ; exp(2 πi ( − ν ) bk + dlN )] , ( ν := ( a + 1) k + clN )= [ kN , lN , − kN , − lN ; exp(2 πi ( − ν ) bk + dlN + 2 πi ( − µ ) kN )] , ( µ := bk + ( d + 1) lN )= [ kN , lN , − kN , lN ; exp(2 πif )] ( µ A ,λ A ) , ( − µ B ,λ B ) where, f = ν bk + dlN + µ kN = kµ − lνN + µν. Note that, by Equation 56, µ and ν are both integers. Also, c T B × I ( ρ ) = [ kN , lN , − kN , lN ; 1] ( µ B ,λ B ) , ( − µ B ,λ B ) = [ kN , lN , − kN , lN ; 1] ( µ B ,λ B ) , ( − µ A ,λ A ) By taking the pairing on c T A × I ( ρ ) and c T B × I ( ρ ), we obtain that,CS( ρ ) = f = kµ − lνN . (63)For reducible representations ρ ǫ x ,ǫ y in Equation 58 depending on the valuesof ǫ x and ǫ y (see Section 4.1), the computation of the CS invariant proceedsin the exactly the same way as for irreducible representations by making thesubstitution, kN → ǫ x , lN → ǫ y . Consequently, by setting ν = ( a + 1) ǫ x + cǫ y , µ = bǫ x + ( d + 1) ǫ y ,
42e obtain that, CS( ρ ǫ x ,ǫ y ) = ǫ x µ − ǫ y ν ǫ x µ + ǫ y ν
2= ( a + d + 2) ǫ x ǫ y + bǫ x + cǫ y ρ ǫ x ,ǫ y ) ∈ Z . In this subsection, let M be a torus bundle over the circle with the monodromymap given by a matrix, (cid:18) a bc d (cid:19) ∈ SL(2 , Z ) . We assume that N := a + d + 2 > c, N ) are co-prime. It isdirect to see that b and c are both odd, while a and d have different parity. Set N = 2 r + 1. Denote by ˜ c ∈ Z N the multiplicative inverse of c in Z N .The non-Abelian character variety of M to SL(2 , C ) consists of the repre-sentations χ nab ( M ) = { ρ + , ρ − , ρ k , k = 1 , · · · , r } which are defined as follows.For ρ ± , x (cid:18) (cid:19) , y (cid:18) u (cid:19) , h (cid:18) v ± v − ± (cid:19) (65)where u = d − a + p ( a + d ) − c , v ± = ± √ cu + a . (66)For ρ k , k = 1 , · · · , r , x e πikN e − πikN ! , y e − πi ˜ c ( a +1) kN e πi ˜ c ( a +1) kN ! , h (cid:18) − (cid:19) (67)In Section 4.1, we computed the adjoint torsion and CS of representationsof π ( M ). In particular, it implies that all non-Abelian characters are adjoint-acyclic and their CS invariants are all rational numbers. As with the exampleof SFSs, we choose the candidate label set L ( M ) = χ nab ( M ). According toSection 4.1, the torsion of these representations are given byTor( ρ ± ) = N, Tor( ρ k ) = N . (68)The Chern-Simons invariant of ρ ± is 0 by Equation 64. Lemma 4.1.
For k = 1 , · · · , r , the Chern-Simons invariant of ρ k is given by,CS( ρ k ) = − ˜ ck N . (69)
Proof.
This can be derived from Equation 63.43e will show below that the premodular categories obtained from the torusbundles are related to quantum group categories associated with so r +1 .For an odd integer N = 2 r + 1 >
0, let so N (Type B ) be the Lie algebra ofSO( N ). Given q = e mπi N such that q is a primitive 2 N -th root of unity (thus m is odd and ( m, N ) are co-prime), there is an associated premodular category C ( so N , q, N ) of rank r + 4. See [14] and references therein. When m = 1, thecorresponding category is always an MTC, and is denoted by SO( N ) in physicsliterature. The MTC has the label set, { , Z } ⊔ { Y , · · · , Y r } ⊔ { X , X } . (70)We will mainly be interested in the (adjoint) monoidal subcategory C ( so N , q, N ) ad linearly spanned by the objects , Z, Y , · · · , Y r . So only modular data on thissubcategory is given below.The twists are, θ = θ Z = 1 , θ Y k = q Nk − k ) , k = 1 , · · · , r. (71)The un-normalized S -matrix is,˜ S αβ = ( α ∈ { , Z } , β ∈ { , Z } α ∈ { , Z } , β ∈ { Y , · · · , Y r } (72)˜ S kj := ˜ S Y k Y j = 2( q kj + q − kj ) = 4 cos 2 πm kjN . (73)In particular, there are only two values for quantum dimensions, d = d Z = 1and d k := d Y k = 2. The total dimension is D = √ N . Note that C ( so N , q, N ) ad is a proper premodular category of rank r + 2. Remark 4.2.
The label set as ordered in Equation 70 correspond to the labels { , λ , λ , · · · , λ r − , λ r , λ r , λ r + λ } in [14]. Although the S -matrix in [14] isonly given for the root q = e πi N , the case for other roots can be easily deducedby either applying a Galois action to the original S -matrix or using the formula˜ S λµ = θ − λ θ − µ X ν N νλ ∗ µ θ ν d ν . Now, for the torus bundle defined at the beginning of the subsection, recallthat N = a + d + 2 is odd, and ˜ cc = 1 ∈ Z N . Let m = − c − N ∈ Z which iswell defined up to multiples of 2 N . For clarity, fix an arbitrary representativefor m , and let q = e mπi N . Note that m is odd and co-prime to 2 N . Hence q isa primitive 2 N -th root of unity.We propose the following correspondence between χ nab ( M ) and local oper-ators, ρ ± ( x, Sym ) ,ρ k ( x mk , Sym ) . (74)and designate ρ + as the unit object, ρ + = ( x, Sym ) . (75)44 heorem 4.3. Let M be the torus bundle over the circle with the monodromymatrix (cid:18) a bc d (cid:19) such that N = a + d + 2 > c, N ) are co-prime.With the choice of local operators and unit object in Equations 74 and 75,respectively, and q as above, the modular data constructed from M matchesthat of C ( so N , q, N ) ad , the adjoint subcategory of C ( so N , q, N ). Proof.
For convenience, we also write ρ ± and ρ k simply as ± and k , respectively.The correspondence between χ nab ( M ) and label set of C ( so N , q, N ) ad is, ρ + ↔ , ρ − ↔ Z, ρ k ↔ Y k , k = 1 , · · · , r. We first check the twists. By Equation 71, θ Y k = q Nk − k ) = e − πiN Nk − k (2˜ c + N ) = e πi ˜ ck N . Note that in the last equality, we used the fact that (
N k − k ) / θ Y k = e − πi CS( ρ k ) . Of course, for ρ ± , a similar relation to the above holds trivially.Next, we verify quantum dimension. W + ( ± ) = 1 , W + ( k ) = Tr Sym ( ρ + ( x mk )) = 2 . (76)This means that the total dimension is D = √ N (equal to the dimension of C ( so N , q, N ) ad ), and by Equation 68, for each ρ ∈ χ nab ( M ), the normalizedquantum dimension matches the torsion, W + ( ρ ) D = (2Tor( ρ )) − . Lastly, for the S -matrix computed from the W matrix,˜ S αβ = 1 , α, β ∈ { + , −} . ˜ S αk = W k ( α ) W + ( k ) = 2 , ˜ S kα = W α ( k ) W + ( α ) = 2 , α ∈ { + , −} . ˜ S kj = W j ( k ) W + ( j ) = 2 Tr Sym ( ρ j ( x mk )) = 4 cos 2 πm kjN , k, j = 1 , · · · , r. This matches the S -matrix of C ( so N , q, N ) ad in Equations 72 and 73. Remark 4.4.
In this subsection, we restricted ourselves to the case where N = a + d + 2 > c, N ) are co-prime. In other cases, it seemsless straightforward to derive the character variety and the structure of thecharacter variety depends on the parity of N (among other factors). This isexpected, since we conjecture in the general case the corresponding premodularcategory is also related to the adjoint subcategory of some C ( so N , q, l ) whosestructure varies dramatically depending on the parity of N and the value of N modulo 4 in the case of even N . We leave this as a future direction.45 Full data of modular categories and beyond
The structure theory of MTCs is naturally divided into two parts: one is theclassification of modular data (MD), and the other is for a fixed modular data,the classification of modular isotopes (MIs) . The missing steps in the programfrom three manifolds to MTCs are then an algorithm to define loop operatorsfor an admissible candidate label set, hence a candidate MD, and the F -matricesfor the fusion structures beyond MD.Physics point to a framework that is a generalization of gauging finite groupsymmetries [2, 5] to continuous non-Abelian Lie group symmetries such asSU(2). One hint from physics is the form of the primitive loop operators inthis paper: a pair ( a, R ), where a is a conjugacy class of the fundamental group,some kind of flux, and R is an irreducible representation of SU(2), some chargeof the SU(2) symmetry. The F -matrices are difficult to find, so we wonder ifthey depend on more than topology: some geometric information of the giventhree manifolds. The identification of a simple object type with a non-Abelian character is basedon the relation between a simple object type and a loop operator in the solidtorus. In a (2 + 1)-TQFT, the rank of an MTC is the same as the dimension ofthe vector space V ( T ) associated to the torus T from the TQFT. One basis { e a } of the vector space V ( T ) consists of labeled core curves of a solid torusby a complete representative set of simple objects { a } . Then each basis element e a can be obtained as the image of a loop operator O a on e —the basis elementassociated to the vacuum, i.e. | e a > = O a | e > .Suppose a non-Abelian character corresponds to a primitive loop operator( a, R ) of the three manifold X . Then a can be represented by a knot K a in X . The knot complement of K a in X determines a vector in V ( T ) from thereduction of 6d SCFT onto X , which should be related to e a , hence a simpleobject type eventually. One possible relation between pentagon equations and flatness is that the flat-ness of SL(2 , C )-connections corresponding to the fundamental group represen-tations can be translated into pentagon equations for the F -matrices. It isknown that pentagon equations can be interpreted as flatness equations for bi-unitary connections on finite graphs (see e.g. [9]). A terminology due to C. Delaney: distinct MTCs with the same MD are called modularisotopes of each other. .2 Towards gauging SU (2) R-symmetry
An R-symmetry of a super-symmetric theory is an outer automorphism of thesuper-Poincare group that fixes the Poincare group. It is pointed out in [4] thatthe R-symmetries in infrared could be different from those in ultra-violet. Hencewe could have an SU(2) R-symmetry for the residual topological theory in in-frared, which is probably often trivial. We believe that the MTCs obtained fromthree manifolds in this program are actually the results of gauging such SU(2)R-symmetries of the residual topological theory in infrared, which generalizesgauging of finite group symmetries [2, 5].
There are many other interesting open questions in this program. One obviousone is to extend our results to more examples such as Seifert fibered spaces withmore than three fibers and the remaining cases of our torus bundles over thecircle examples. It is also not clear how to obtain MTCs which are not self-dual.As mentioned in Sec. 2, representations of SL(2 , C ) come in group of four anda natural guess is that one of the four is the dual anyon type. If so, then whichone? The dual representation is a candidate. Another general direction is whatoperations on MTCs that standard topological constructions of three manifoldssuch as connected sum and torus decomposition correspond to. Connect sumshould correspond to Deligne product.Our adjoint-acyclic condition for a representation ρ is closely related to H ( π, Adj ◦ ρ ) = 0. Are they equivalent? It should be equivalent for irre-ducible representations, but for the indecomposable reducible ones, it is notclear. Acknowledgments.
Y. Q. and Z.W. are partially supported by NSF grantFRG-1664351 and CCF 2006463. S.-X. C. is partially supported by NSF CCF2006667. The research is also partially supported by ARO MURI contractW911NF-20-1-0082. The third author thanks Dongmin Gang for helpful com-munications, who pointed out that the Seifert fibered spaces (3 , , r ) would giverise to modular tensor categories related to SU (2) k . The first author thanksEric Rowell for clarifying some facts about C ( so , q, l ). References [1] David R. Auckly. Topological methods to compute Chern-Simons invari-ants.
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