Representations of Motion Groups of Links via Dimension Reduction of TQFTs
aa r X i v : . [ m a t h . QA ] N ov REPRESENTATIONS OF MOTION GROUPS OF LINKSVIA DIMENSION REDUCTION OF TQFTS
YANG QIU AND ZHENGHAN WANG
Abstract.
Motion groups of links in the three sphere S are gen-eralizations of the braid groups, which are motion groups of pointsin the disk D . Representations of motion groups can be usedto model statistics of extended objects such as closed strings inphysics. Each 1-extended (3 + 1)-topological quantum field the-ory (TQFT) will provide representations of motion groups, but itis difficult to compute such representations explicitly in general.In this paper, we compute representations of the motion groupsof links in S with generalized axes from Dijkgraaf-Witten (DW)TQFTs inspired by dimension reduction. A succinct way to stateour result is as a step toward a twisted generalization (Conjecture4.1) of a conjecture for DW theories of dimension reduction from(3 + 1) to (2 + 1): DW G ∼ = ⊕ [ g ] ∈ [ G ] DW C ( g ) , where the sum runsover all conjugacy classes [ g ] ∈ [ G ] of G and C ( g ) the centralizerof any element g ∈ [ g ]. We prove a version of Conjecture 4.1 forthe mapping class groups of closed manifolds and the case of toruslinks labeled by pure fluxes. Introduction
Topological quantum field theories (TQFTs) are used as the low en-ergy effective description of topological phases of matter in physics,especially for anyon systems in two spacial dimensions. A central partof an anyon model is the description of anyon statistics by the rep-resentation of braid groups (e.g. see [12]). Braid groups are simplymotion groups of points in the disk, therefore a natural generalizationfor statistics of extended objects in higher dimensions will be motiongroups such as the motion groups of links in the three sphere S [1, 4].Representations of motion groups has been used to model statisticsof extended objects such as closed strings in physics (e.g. see [6, 10,8, 13]). Each 1-extended (3 + 1)-TQFT will provide representations ofthe motion groups, but it is difficult to compute such representations Z.W. is partially supported by NSF grants DMS 1411212 and FRG-1664351.The second author thanks C.-M. Jian, E. Samperton and K. Walker for relateddiscussions and comments. explicitly in general. In this paper, we compute representations ofthe motion groups of links in S with generalized axes from Dijkgraaf-Witten (DW) TQFTs inspired by dimension reduction. A succinct wayto state our results is a step towards a twisted generalization Conjecture4.1 of a conjecture for DW theories using dimension reduction from(3 + 1) to (2 + 1): DW G ∼ = ⊕ [ g ] ∈ [ G ] DW C ( g ) , where the sum runs overall conjugacy classes [ g ] of G and C ( g ) the centralizer of any element g ∈ [ g ] . We prove a case of the main conjecture 4.1 for mapping classgroups in Thm. 1.Dimension reduction is a simple construction in relating quantumfield theories of different dimensions. Our interest in dimension reduc-tion lies in the categorical context: if the input for an ( n + 1)-TQFT isa certain higher category, what are the resulting categories of the lowerdimensional TQFTs from dimension reduction explicitly? If known,then representations of motion groups from the ( n + 1)-TQFT mightbe reconstructed from the lower dimensional ones. In this paper, westudy representations of the motion groups from (3 + 1)-DW TQFTswith such a goal in mind. We conjecture that motion groups of linkswith generalized axes in S can be described using surface braid groups,and prove a version for the special case of torus links labeled by purefluxes in Thm. 2.Our main conjecture formulated as Conjecture 4.1 is that there is amap from labels of a 1-extended (3 + 1)-TQFTs to those of 1-extended(2 + 1)-TQFTs so that the representations of the motion groups of Our main interest is for the twisted generalization of dimension reduction. Thedirect product case of the conjecture as originally mentioned in the abstract hasbeen proved since our paper appeared on the arxiv as outlined by an anonymousreferee as below and the paper [9]. The referee pointed out: “As the authorsnote, the (n, n+1)-part of DW theory assigns to a closed n-manifold the vectorspace of functions on the finite set [Π( M ) , BG ] of isomorphism classes of functorsΠ( M ) → BG (where Π( M ) is the fundamental groupoid of M and BG is the 1-groupoid with a single object and endomorphisms G), and to an n+1 bordism W : M → M ′ the ‘linearization’ of the span of finite sets [Π( M ) , BG ] ← [Π( W ) , BG ] → [Π( M ′ ) , BG ], (or more explicitly: the linear map with coefficients at a functor F : Π( M ) → BG and a functor F ′ : Π( M ′ ) → BG given by W ) → BG such that the restrictions to Π( M ) and Π( M ′ ) agree with F and F’).Now as the authors note, there is a bijection of sets [ M × S , BG ] = [ M, [ S , BG ]]= Q conjugacy classes [ M, BC ( g )], inducing isomorphisms of vector spaces Functions([ M × S , BG ]) = P conj. classes Functions([
M, BC ( g )]. A similar computation showsthat the bordism map arising from the span ([ M × S , BG ] ← [ W × S , BG ] → [ M ′ × S , BG ])= Q conjugacy classes ([ M, BC ( g )] ← [ W, BC ( g )] → [ M ′ , BC ( g )]) isitself a direct sum of the appropriate linear maps. Hence, this gives an isomorphismof TQFTs between DW G and P conj. classes DW C ( g ) .” OTION GROUP REPRESENTATIONS 3 a link L from the (3 + 1)-TQFT decompose as direct sums of therepresentations of some surface braid groups of the fiber surface F ifthe link L has a generalized axis γ with fiber F . This conjecture isour main focus and is a twisted version of dimension reduction formotion groups. The standard dimension reduction neither touches onnon-product fibrations nor motion group representations of links in S .The link complement of a link with a generalized axis is a non-trivialfibration over the circle, hence the computation of the motion grouprepresentations does not follow from the product dimension reduction.Our conjecture is not specific to DW TQFTs and should hold for gen-eral TQFTs. The difficulty lies in the right formulation of a corre-spondence of the label sets. For (3 + 1)-DW TQFTs, the reductionof the labels for the torus boundary is related to the restriction of thehomomorphisms from Z to the meridian circle factor Z .The content of our paper is as follows. In Sec. 2, we define 1-extendedTQFTs closely following the definition of Reshetikhin-Turaev TQFTs,recall dimension reduction, and outline the theory of motion groups.In the end, we deduce presentations of the motion groups of the toruslinks from [4]. In Sec. 3, we construct the 1-extended truncation of DWTQFTs using colorings, find the label sets for the codim=2 excitations,and describe explicitly the representation spaces for motion groups oflinks labeled by pure fluxes. Sec. 3.1 is a straightforward generalizationof the results in dimension 3 from [5] to general n dimensions. Sec. 3.3mainly rephrases results from [7] in our framework. We then prove aversion of the conjecture for mapping class groups of closed manifolds.In the last Sec. 4, we focus on the motion groups of torus links andprove a special case of our conjecture for torus links.1.1. Notations.
In this subsection, we collect some notation that areused throughout the paper.1. 1-extended ( n + 1)- or ( n + 1 , − Z, V, C ) for manifolds( X n +1 , Y n , Σ n − ) of dimensions ( n + 1 , n, n − G will be denoted as ρ, α, ..., and the representa-tions that they represent as [ ρ ] , [ α ] , ..., since they are simply homomor-phims up to conjugation. Many constructions in this paper involvehomomorphisms, not just representations.3. Conjugacy class [ g ] of a group G , centralizer C G ( H ) of a subgroup H of G in G or C G ( g ) centralizer of an element g ∈ G ( G will be droppedif no confusion arises).4. The vector space of an n -manifold Y with boundary ∂Y = { Σ i } mi =1 labeled by { ([ g i , h i ] , [ α i ]) } mi =1 is denoted as V G ( Y ; ([ g i , h i ] , [ α i ])). YANG QIU AND ZHENGHAN WANG
5. The torus links
T L ( p,q ) n : for any pair of relatively prime naturalnumbers p, q ≥ T L ( p,q ) n consists of n -copies of the ( p, q )-torus knot.If the torus is regarded as a folded square, then T L ( p,q ) n is just n parallelcopies of a slope p/q curve in the square.6. The 3-sphere S is identified with the union C × S ∪ S × C oftwo open solid tori, with the identifications ( re iθ , e iζ ) ∼ ( e iθ , r e iζ ) for r >
0. Let x ′ , y ′ be the circles (0 , e iζ ) , ( e iθ , k be set of complex k -th roots of unity: Ξ k = { ξ lk } , l =0 , , ..., k − , ξ k = e πik , R t : C −→ C be the rotation by 2 πt of thecomplex plane around the origin : z → e πit z , and ι be an inclusion ofa subset of C × S into S .8. π ( Y ; B, A ) for an n -manifold Y with m boundary components { Σ i } mi =1 denotes the fundamental group of Y together with m inclu-sions of the m fundamental groups { π (Σ i , b i ) } mi =1 of the boundaries to π ( Y, b ), where B = { b, b , .., b m ) } are based points for Y and the i -thboundary component Σ i , respectively, and A = { A i } mi =1 are m arcs con-necting b to b i , respectively. We choose A i to be disjoint and embededif n ≥ π ( Y ; B, A ) to a group G is a triple( ρ, { α i } mi =1 , { g i } mi =1 ) of ρ : π ( Y, b ) → G, α i : π (Σ i , b i ) → G, g i ∈ G such that for each i , ρ = g i · α i · g − i when restricted to π (Σ i , b i ).10. The motion group of an oriented submanifold N in the interiorof an oriented ambient manifold M is denoted by M ( N ⊂ M ). Theorientation preserving diffeomorphism group of M that fixes N as anoriented submanifold is denoted as H + ( M ; N ). In the case of a surfacebraid group, we also use the notation B ( F ; P ) for a surface F and afinite collection of points in the interior of F .2. Extended TQFTs, Dimension Reduction, and MotionGroups
Atiyah type TQFTs do not necessarily lead to representations ofmotion groups, but fully extended ones do. In this section, we formulate1-extended ( n + 1)-TQFTs as generalizations of Reshetikhin-TuraevTQFTs, which always give rise to representations of motion groups ofcodimension=1 submanifolds of spacial n -manifolds. We are especiallyinterested in three spacial dimensions, so motion groups of links inthree manifolds.2.1. ( n + 1 , − k ) -TQFTs. A k -extended ( n + 1)- or ( n + 1 , − k )-TQFTis one that is extended from ( n + 1)-manifolds all the way to ( n − k )-manifolds. When k = 0, they are the Atiyah-type TQFTs, while OTION GROUP REPRESENTATIONS 5 k = n the fully-extended ones. Formally, they could be defined asmonoidal functors between appropriate higher categories. For explicitcalculations, a more elementary approach is preferred, and furthermore,we will allow theories with framing anomaly. Our focus is on k = 1, sowe will define 1-extended TQFTs using explicit axioms similar to theReshetikhin-Turaev (2+1)-TQFTs, which are 1-extended with framinganomaly in general. Definition 1. A -extended ( n + 1) -TQFT or ( n + 1 , − -TQFT isa triple ( Z, V, C ) , where ( Z, V ) is a projectively symmetric monoidalfunctor from the category Bord n +1 n of ( n + 1) - and n -manifolds to thecategory V ec of finitely dimensional complex vector spaces, and in ad-dition to this projective Atiyah type TQFT, an assignment of a semi-simple finite category C (Σ) to each oriented closed ( n − -manifold Σ and a finitely dimensional vector space V ( Y ; { X l } ) to each oriented n -manifold Y with parameterized and labeled boundary components by X l ∈ Π C ( ∂Y ) such that the following axioms hold: (1) Empty manifold axiom: V ( ∅ ) = 1 , C , or V ec if ∅ is regarded asa manifold of dimension= n + 1 , n, n − , respectively. (2) Disk axiom: V ( D n ; X i ) ∼ = ( C , X i = , , Otherwise , where D n is an n -diskand the tensor unit. (3) Cylinder axiom: V ( A ; X i , X j ) ≃ ( C if X i ≃ X ∗ j , otherwise , where A is the cylinder S n − × I , and X i , X j ∈ Π C ( S n − ) .Furthermore, V ( A ; X i , X j ) ∼ = C if X i ∼ = X ∗ j . The notations ≃ and ∼ = denote isomorphism as vector spaces and functorialisomorphism, respectively. The difference is necessary due toFrobenius-Schur indicators of labels . Our axioms are straightforward generalizations of the axioms in [12] for (2 + 1)dimensions. Π C is a complete set of the simple representatives of a category C . Each con-nected boundary component of Y is labeled by an object in Π C . We use ≃ forvector space isomorphism and ∼ = for functorial isomorphism. A self-dual simple object type in a spherical fusion category has a Frobenius-Schur (FS) indicator [17], which creates a subtlety that makes the original formu-lation of gluing axioms in [15] not general enough to cover theories with non-trivialFS indicators such as SU (2) k . The problem is traced back to the cylinder axiom. YANG QIU AND ZHENGHAN WANG (4)
Disjoint union axiom: V ( Y ⊔ Y ; X ℓ ⊔ X ℓ ) ∼ = V ( Y ; X ℓ ) ⊗ V ( Y ; X ℓ ) .The isomorphisms are associative, and compatible with the map-ping class group projective actions V ( f ) : V ( Y ) → V ( Y ) for f : Y → Y . (5) Duality axiom: V ( − Y ; X ℓ ) ∼ = V ( Y ; X ℓ ) ∗ , where − Y is Y with the oppositeorientation. The isomorphisms are compatible with mappingclass group projective actions, orientation reversal, and the dis-joint union axiom as follows:(i): The isomorphisms V ( Y ) → V ( − Y ) ∗ and V ( − Y ) → V ( Y ) ∗ are mutually adjoint.(ii): Given f : ( Y ; X ℓ ) → ( Y ; X ℓ ) let ¯ f : ( − Y ; X ∗ ℓ ) → ( − Y ; X ∗ ℓ ) be the induced reversed orientation map, we have h x, y i = h V ( f ) x, V ( ¯ f ) y i , where x ∈ V ( Y ; X ℓ ) , y ∈ V ( − Y ; X ∗ ℓ ) .(iii): h α ⊗ α , β ⊗ β i = h α , β ih α , β i whenever α ⊗ α ∈ V ( Y ⊔ Y ) ∼ = V ( Y ) ⊗ V ( Y ) ,β ⊗ β ∈ V ( − Y ⊔ − Y ) ∼ = V ( − Y ) ⊗ V ( − Y ) . (6) Gluing axiom: Let Y gl be the surface obtained from gluing twoboundary components Σ of a surface Y by a diffeomorphismwhich is isotopic to the orientation reversing “identity” . Then V ( Y gl ) ∼ = M X i ∈ Π C (Σ) V ( Y ; ( X i , X ∗ i )) . The isomorphism is associative and compatible with mappingclass group projective actions.
In the case with non-trivial FS indicators, the identity functor on self-dual objectsis not functorial, which leads to inconsistency.The inconsistency arose because manifolds with non-empty boundaries have thefreedom to absorb a cylindrical neighborhood of the boundary. The isomorphismof the vector spaces from a TQFT requires functoriality when tensoring the 1-dimension vector space from a cylinder. In the case with non-trivial FS indicators,the − The gluing operation should be regarded as the inverse of cutting the manifold Y gl along the glued boundary component. OTION GROUP REPRESENTATIONS 7
Moreover, the isomorphism is compatible with duality as fol-lows: Let M j ∈ Π C α j ∈ V ( Y gl ; X ℓ ) ∼ = M j ∈ Π C V ( Y ; X ℓ , ( X j , X ∗ j )) , M j ∈ Π C β j ∈ V ( − Y gl ; X ∗ ℓ ) ∼ = M j ∈ Π C V ( − Y ; X ∗ ℓ , ( X j , X ∗ j )) . Then there is a nonzero real number s j for each label j suchthat (cid:28) M j ∈ Π C α j , M j ∈ Π C β j (cid:29) = X j ∈ Π C s j h α j , β j i . In Sec. 3.3, we explicitly describe how to view (3 + 1)-DW TQFTs as1-extended ones in the sense above. All state-sum TQFTs are fully ex-tended so they should be examples of 1-extended TQFTs. More inter-esting examples are Reshetikhin-Turaev TQFTs, which are 1-extendedexamples that are not fully extended. We are not aware of any (3 + 1)-TQFTs which are 1-extended, but not fully extended. Potentiallysuch examples can be constructed analogous to the construction ofReshetikhin-Turaev TQFTs as state-sum (3 + 1)-TQFTs [16].In a 1-extended ( n + 1)-TQFT, each oriented closed ( n − C (Σ). In Crane-YetterTQFTs from ribbon fusion categories, all such categories assigned toΣ’s should come from the same input ribbon fusion category as follows:given an oriented closed 2-manifold Σ with a base point, and an inputribbon fusion category B for the Crane-Yetter TQFT, a picture cylin-drical category A (Σ × I ) can be constructed: the objects of A (Σ × I )are finitely many signed framed points colored by objects of B , andmorphisms are ribbon graphs between objects colored by morphismsof B up to framed ribbon isotopies relative to the base point in Σ × I .The representation category C (Σ) of A (Σ × I ) is a semi-simple categorythat is assigned to Σ. The label set Π B (Σ) is then the set of irreduciblerepresentations of the picture cylindrical category A (Σ × I ). For the1-truncations of fully extended (3 + 1)-TQFTs from spherical fusion2-category as in [11], all the semi-simple finite categories associated todifferent surfaces should arise as some doubles.2.2. Dimension Reduction.
Dimension reduction (DR) is a simpleconstruction that is widely used in physics that produces an (( n −
1) + 1)-TQFT from an ( n + 1)-TQFT for Atiyah type TQFTs. ButDR is not always possible in our set-up because in our definition, thevector space associated to the sphere S n is always C by the gluing YANG QIU AND ZHENGHAN WANG axiom. The condition V ( S n ) ∼ = C is a stability condition in topologicalphysics. Since the only stable (1 + 1)-TQFT in our sense is trivial,non-trivial (2 + 1)-TQFT cannot be reduced to (1 + 1) via DR. SinceDR can always be done within Atiyah type TQFTs, hence we discussDR for only TQFTs regarded as Atiyah type ones. Definition 2.
Let ( Z, V ) be an ( n + 1) -TQFT. The resulting (( n −
1) +1) -TQFT ( Z DR , V DR ) defined by V DR ( Y ) = V ( Y × S ) Z DR ( X : Y −→ Y ) = Z ( X × S : Y × S −→ Y × S ) is called the DR of ( Z, V ) . Proposition 1.
The fusion algebra V ( T ) of any -extended (2 + 1) -TQFT is a Frobenius algebra, and the reduction of a (2 + 1) -TQFT isthe (1 + 1) -TQFT given by the Frobenius algebra V ( T ) .Proof. Each 1-extended (2 + 1)-TQFT is associated with a modulartensor category in our definition and the modular S = ( s ab ) a,b ∈ L -matrixdiagonalizes the fusion rules. It follows that in the basis given by e a = P b ∈ L s ab b , the fusion algebra V ( T ) becomes the algebra A = P a ∈ L C [ e a ]. It follows that A is a Frobenius algebra of the direct sumof 1-dimensional Frobenius algebras. (cid:3) The proposition illustrates that dimension reduction in general doesnot preserve the condition that the dimension of the vector spaces froma TQFT for spheres is 1-dimensional, which is regarded as a stabilitycondition in topological phases of matter.2.3.
Motion groups.
We first recall some basic notions for motiongroups following [1, 4], then describe motion groups of links with gen-eralized axes. In the end, we derive presentations of motion groups oftorus links from [Thm 8.7 [4]].2.3.1.
Basic notions.
We work in the smooth category in this section.Let N be an oriented compact non-empty sub-manifold in the inte-rior of an oriented manifold M . A motion H ( x, t ) of N inside M is anambient isotopy H : M × I → M × I of M such that H ( x,
0) = id M and H ( x, N ) = N as an oriented sub-manifold, The orientation pre-serving diffeomorphism H ( x,
1) : M \ N → M \ N preserves the com-plement of N in M . A motion H ( x, t ) can be equally regarded as apath h t = H ( x, t ) in the orientation preserving diffeomorphism group H + ( M ) of M based at the identity. Note that isotopy classes of H + ( M )is the mapping class group of M , while motion groups concern only the OTION GROUP REPRESENTATIONS 9 transformation of non-empty sub-manifolds in M by diffeomorphismsisotopic to the identity.The composition H ◦ H of two motions H and H is defined by: H ◦ H = ( H ( x, t ) , ≤ t ≤ H ( x, t − ◦ h , ≤ t ≤ . Inversion of a motion is defined by H − ( x, t ) = H ( x, − t ) ◦ h − .A motion H ( x, t ) is stationary if H ( x, t )( N ) = N as a set for all t ∈ (0 , H ( x, N ) = N as oriented manifolds, for some t ∈ (0 , H ( x, t ) could be orientation reversing when restricted to thesub-manifold N .Two motions H , H of N ⊂ M are equivalent if H − ◦ H is homo-topic to a stationary motion. Definition 3 (Motion group) . Given oriented manifolds N ⊂ M as above, then the motion group M ( N ⊂ M ) is the group of motionsmodulo stationary ones. If N is a point b of M , then M ( N ⊂ M ) is the fundamental group π ( M, b ) of M . If N consists of n distinct points in the disk D , then M ( N ⊂ M ) is the braid group B n .When M = S , the following exact sequence holds: Z −→ M ( ⊔ ni =1 N i ⊂ M ) ∂ −→ H + ( M ; N , ..., N n ) −→ , (2.1)where ∂ is the Dahm homomorphism [Corollary 1.13 [4]].2.3.2. Links with generalized axes.
The necklace link in Fig. 1 is anexample of a link with an axis: the unknot train track that the unlinkwinds around is the axis. In general, the unlink can be replaced by anybraid closure L , therefore any link L is a link with an axis. It is obviousthere is a connection between the motion group of L ∪ γ —the link L together with the axis γ —and the motion group of the intersectionpoints of the link L with the spanning disk D of the unknot γ in D .But the relation could be complicated, and also our interest is not on arelation between the motion group of L ∪ γ and the braid group, ratherthe motion group of the link L itself and some braid group. In [4],Goldsmith proved that if the unknot is generalized to a fibered knot γ as a generalized axis, then in some cases the motion group of the link L itself is the same as the motion group of L ∪ γ . One such case isthe torus link T L ( p,q ) n of n -copies of the ( p, q ) torus knot, which is ourfocus in this paper. This is not true in general as the motion group of In [4], H ( x,
1) is allowed to be orientation reversing when restricted to N , soour motion groups are the oriented ones there. the unlink—the loop braid group—is different from that of the necklacelink—the annulus braid group.A generalized axis γ for some link L is a non-trivial fibered knot γ with a fixed fibration π : S \ γ → S . Let F = π − (1) , ∈ S , withclosure ¯ F in S , then ∂ ¯ F = γ . The fibration is given by a surjectivemap f : F × I → S \ γ, I = [0 , , such that(1) f : F × (0 , → S \ γ is a diffeomorphism,(2) f extends to the closure ¯ F × I such that f =id and f is adiffeomorhpism of F with compact support, which is called themonodromy.(3) f : γ × (0 , → γ is a projection.A link L has a generalized axis γ if the link L in S is in a braidposition with respect to the fixed fibration F −→ S \ γ −→ S [4], i.e. ifa link component is parameterized, then as the parameter increases, sowill be the fibration parameter in I of F × I periodically. Set P = F ∩ L and φ : ( F, P ) −→ ( F, P ) the monodromy. There exists an exactsequence:1 −→ < [ φ ] , [ τ ] > −→ H φ ( F ; P ) eJ −→ H + ( S ; L, γ ) −→ Z , (2.2)where H φ is the centralizer of [ φ ] ∈ H + ( F ; P ) = B ( F ; P ), and [ τ ] isthe central Dehn twist on a push-off of the axis γ to a collar of F , and < [ φ ] , [ τ ] > is the subgroup generated by [ φ ] , [ τ ] [Thm 5.26 [4]].2.3.3. Presentations of the motion groups of torus links.
The motiongroups of links are interesting generalizations of the braid groups. Onlya few of them are being investigated recently, partially due to the appli-cation to statistics of loop excitations in physics. The motion groupsof the unlinks, which are loop braid groups [6, 10], and the motiongroups of the necklace links [8, 13], are the main focus. Our interestis on the motion groups of the torus link
T L ( p,q ) n — n parallel copies ofthe ( p, q )-torus knot.In [4], Goldsmith obtained a presentation of the motion groups ofthe torus link T L ( p,q ) n using the theory of motion groups of links witha generalized axis. Torus links T L ( p,q ) n are examples of links L suchthat the motion groups of L ∪ γ —the link plus the axis—is the same asthe link L itself (without the axis). We observe that the presentationin [Thm 8.7 [4]] implies that actually there are only three families ofmotions groups of the torus links T L ( p,q ) n : ( p, q ) = (1 , , p + q =odd,or p + q =even indexed by n . Hence as abstract groups, when ( p, q ) =(1 , T L ( p,q ) n depend only on the parity of p + q .Presentations of the three families of motions groups indexed by n for T L ( p,q ) n , p, q > OTION GROUP REPRESENTATIONS 11
Proposition 2. (1) If p + q with p, q ≥ is odd, then the oddmotion groups M T L n, − of the torus links has a presentation: < σ , · · · , σ n − , r , · · · , r n |{ σ j } n − j =1 satisfy the braid relations ,r · · · r n = 1 , r i r k = r k r i , r i σ j = σ j r i , ≤ i, k ≤ n, j = i − >, (2) If p + q is even but not p = q = 1 , then the even motion groups M T L n, + of the torus links has a presentation: < σ , · · · , σ n − , r , · · · , r n , r π |{ σ j } n − j =1 satisfy the braid relations ,r · · · r n = 1 , r i r k = r k r i , r i σ j = σ j r i , j = i − , r · · · r n = r π , r π = 1 >, (3) If p = q = 1 , then the motion groups M H n of the n -Hopf linkshas a presentation: < σ , · · · , σ n − , r , · · · , r n |{ σ j } n − j =1 satisfy the braid relations ,r · · · r n = 1 , r i r k = r k r i , r i σ j = σ j r i , j = i − , r = r n = 1 > . These presentations are derived from [Thm 8.7 [4]]. First our r i ’s arethe ρ i ’s there. Secondly the generator [ f ] there does not exist in ouroriented motion group as it is an orientation reversing diffeomorhpism.There is a typo in relation 7—one p is a q , and then our presentationsfollow from the existence of integers u, v such that pu − qv = 1.3. Dijkgraaf-Witten TQFTs
Given a finite group G , the untwisted DW ( n +1)-TQFTs based on G for any n ≥ DW TQFTs as Atiyah type.
Let M be an oriented compacttriangulated m -manifold with boundary ∂M . Let v be the number ofthe vertices of M and ∂v the number of the vertices on ∂M . We useboth ♯S and | S | to denote the number of elements in a set S . Definition 4.
Given a finite group G , a coloring ϕ = [ c ] of the tri-angulated manifold M is an equivalence class of assignments of anorientation ± and a group element g ∈ G to each edge of M : c : { edges of M } −→ ( ± , G ) that for any oriented triangle, the colors of the three edges satisfy: g −→ , h −→ , then the edge gh −→ . The equivalence class of a coloring isgenerated by the relation that if an oriented edge is colored by g , thenthe edge with the opposite orientation is colored by g − . For simplicity, we suppose that M is connected. A coloring ϕ es-sentially defines a flat principle G -bundle on M . After a vertex x ischosen as a base-point of M , the holonomy representation of ϕ definesa homomorphism ϕ ∗ : π ( M, x ) −→ G , which is determined in thefollowing combinatorial way. Any loop α at x can be homotopic to aloop consisting of some edges of the triangulation of M . Then ϕ ∗ ( α ) isdefined to be the product of the elements coloring the edges followingthe direction of α .Conversely, each homomorphism ρ : π ( M, x ) −→ G gives rise tocolorings as follows: pick any maximal tree T of the 1-skeleton M (1) of M and retract the tree T to the base point x , then the 1-skeleton M (1) contracts to a bouquet of circles. Each circle in the retraction receivesa group element from ρ , then coloring every edge of T by the group unitand each circle according to the image of ρ leads to a desired coloring.The set of all colorings of M will be denoted by Col( M ) and the setof all colorings of M with restriction on ∂M being a particulr color τ by Col( M, τ ). Definition 5.
For any M as above and τ a coloring of ∂M , the par-tition function Z ( M, τ ) is defined as Z ( M, τ ) = | G | ∂v − v ♯ Col ( M, τ ) . Proposition 3.
If a triangulation and a coloring τ of ∂M are fixed,then Z ( M, τ ) does not depend on the extending triangulations of M . Prop. 3 follows directly from Lemmas 1 and 2 below.
Lemma 1. ♯ Col ( M ) = | G | v − ♯ Hom ( π ( M, x ) , G ) where x is any base-point of M .Proof. The triangulation of M leads to a chart atlas for M such thateach vertex indexes an open ball and each edge means that the two openballs for the two end points intersects in an open ball. Thus for any G -principle bundle over M , a trivialization on this atlas with transitionfunctions as elements of G coloring edges defines a coloring of M . Since G is a finite group, the classifying space of G is K ( G, f ∈ Hom( π ( M, x ) , G ), there exists a G -principle bundle on M whose holonomy representation is f , which defines a coloring of M . OTION GROUP REPRESENTATIONS 13
To show that for any f ∈ Hom( π ( M, x ) , G ), there is a canonicalway to color the vertices resulting all the colorings whose holonomyrepresentation is f , we choose a vertex x as the base point and use G to color all the vertices except the base point x . Let ϕ be a coloringof M realizing f . By each coloring φ for vertices, we can change ϕ to ϕ φ as follows: a ( h ) g −→ b ( k ), where g colors the edge ab and h colors the vertex a , k colors the vertex b , is changed to be a h − gk −→ b .A straightforward check shows that ϕ φ is a still coloring of M whoseholonomy representation is f . Pairwise different φ result in pairwisedifferent ϕ φ . To show that any coloring whose holonomy representationis f can be obtained in this way, note that since the group unit 1 isused to color the base point, we can find the element coloring eachvertex from the vertices near the base point to the ones far from thebase point by the elements coloring edges. Finally, there are | G | v − colorings for vertices. (cid:3) Lemma 2.
Let B , ..., B l be the components of ∂M and b ∈ M, b i ∈ B i be the base points of M, B i , respectively. A coloring τ of ∂M decom-poses as: τ = F li =1 τ i . Choose paths γ , ...γ l connecting b to b , ..., b l ,we obtain ♯ Col ( M, τ ) = | G | v − ∂v l Y i =1 | C G ( Im ( τ i ) ∗ ) | ♯ { f ∈ Hom ( π ( M, b ) , G ) | f ( γ i ) ∗ ( ι i ) ∗ ∼ ( τ i ) ∗ } where ι i : B i −→ ∂M is the inclusion. The induced homomorphism ( γ i ) ∗ : π ( M, b i ) −→ π ( M, b ) comes from γ i and ∼ means conjuga-tion as two homomorphisms. Here ϕ ∗ denotes the corresponding grouphomomorphism for a coloring ϕ .Proof. Any ϕ ∈ Col(
M, τ ) leads to ( ϕ ) ∗ ( γ i ) ∗ ( ι i ) ∗ = g i ( τ i ) ∗ g − , where g i colors the path γ i . Thus ϕ ∗ ( γ i ) ∗ ( ι i ) ∗ ∼ ( τ i ) ∗ .When { f ∈ Hom( π ( M, x ) , G ) | f ( γ i ) ∗ ( ι i ) ∗ ∼ ( τ i ) ∗ } is empty, there are no colorings in Col( M, τ ). When there exists f ∈{ f ∈ Hom( π ( M, x ) , G ) | f ( γ i ) ∗ ( ι i ) ∗ ∼ ( τ i ) ∗ } , there exists a coloring ϕ of M whose holonomy representations on M, B i are f, h i ( τ i ) ∗ h − i ,respectively. By coloring the base point x i of B i with h i , the method inLemma 1 can be used to modify ϕ so that its holonomy representationson M, B i are f, ( τ i ) ∗ , respectively. Then we follow the same methodby coloring vertices of ∂M to find a coloring of M whose restrictionon ∂M is τ , which is still denoted by ϕ . By the same argument inLemma 1, we see that all the colorings in Col( M, τ ) whose holonomyrepresentation is f is a modification of ϕ by coloring the vertices. Next we count the colorings in Col(
M, τ ) whose holonomy represen-tation is f . Since the restriction of colorings on B i is fixed, only theelements in C G ( Im ( τ i ) ∗ ) can be used to color the base point x i of B i and1 to color the other vertices in B i . There are | G | v − ∂ Q li =1 | C G ( Im ( τ i ) ∗ ) | colorings for vertices of M , and the proof is completed. (cid:3) To construct the ( n +1)-DW TQFT ( Z G , V G ), for any oriented closedtriangulated n -manifold Y , we first define a vector space e V ( Y ) tobe C [Col( Y )]—the vector space spanned by colorings. Given a bor-dism ζ = ( X, Y , Y , f , f ), where X is an oriented compact ( n + 1)-manifold with boundary ∂X , and Y , Y are oriented closed triangulated n -manifolds with an orientation-preserving diffeomorphism: f ⊔ f : Y ⊔ − Y −→ ∂X. Let Z ( ζ ) : e V ( Y ) −→ e V ( Y ) be Z ( ζ )( τ ) = X µ ∈ Col( Y ) Z ( X, τ ⊔ µ ) µ, where τ ∈ Col( Y ) and τ ⊔ µ is the coloring of ∂X induced by f ⊔ f . Z ( ζ ) is well-defined by Prop. 3.The proof of the following is straightforward. Lemma 3. Z ( ζ η ) = Z ( ζ ) Z ( η ) for any two bordisms ζ , η . Thus Z ( Id Y ), Id Y = ( Y × I, Y × , Y × , id, id )), is an idempotentfrom e V ( Y ) to itself. The Hilbert space V G ( Y ) is then defined to bethe image of Z ( Id Y ). The restriction of Z ( ζ ), ζ = ( X, Y , Y , f , f )) to Z ( Id Y i ) , i = 1 ,
2, defines a map from V G ( Y ) to V G ( Y ). Furthermore,equivalent bordisms define the same linear map, in other words, Z ( ζ )is defined on the equivalence class of bordisms.The DW-TQFT ( Z G , V G ) defined above leads to a natural represen-tation of the mapping class group M ( Y ) on the Hilbert space V G ( Y ) asfollows: any [ f ] ∈ M ( Y ) gives rise to a bordism ζ f = ( Y × I, Y × , Y × , f, id ). Then the linear transformation Z ( ζ f ) from V G ( Y ) to itself de-fines a representation ρ : M ( Y ) −→ GL ( V ( Y )) by ρ ([ f ]) = Z ( ζ f ). Proposition 4.
There exists a basis { e k } for V G ( Y ) in − correspon-dence with Hom ( π ( Y ) , G ) / ∼ such that the action of [ f ] ∈ M ( Y ) on V G ( Y ) is the permutation of { e k } by pre-composing f : π ( Y ) → π ( Y ) . OTION GROUP REPRESENTATIONS 15
Thus, the following diagram commutes: M ( Y ) / / ρ & & ◆◆◆◆◆◆◆◆◆◆ S N y y sssssssssss GL ( V G ( Y )) , where N = dim ( V G ( Y )) .The representation is always reducible as the conjugacy class of thetrivial homomorphism is fixed by M ( Y ) by pre-composition. Thus oneelement in { e k } is always fixed by M ( Y ) . Hence N can be reduced to N − =dim ( V G ( Y )) -1.Proof. Let v be the number of vertices of Y . By Lemma 1, dim V G ( Y ) = | G | v − ♯ Hom( π ( Y ) , G ) and a basis for V G ( Y ) is given as follows. τ i,jk ∈ Col( Y ) , where k indexes the conjugacy class of ( τ i,jk ) ∗ in Hom( π ( Y ) , G ) / ∼ , j indexes the homomorphism ( τ i,jk ) ∗ in the conjugacy class k , i indexesthe coloring τ i,jk corresponding to the homomorphism j .To compute Z ( Id Y ) : V ( Y ) −→ V ( Y ) under this basis, we triangu-late Y × I so that the restriction to boundaries Y × Y × Y . The number of vertices of the boundary of Y × I is 2 v . Let a be the number of vertices of Y × I . Lemma 2 implies Z ( Id Y )( τ i,jk )( τ i ′ ,j ′ k ′ ) = | G | v | G | a ♯ Col( Y × I, τ i,jk ⊔ τ i ′ ,j ′ k ′ )= | G | v | C G ( Im ( τ i,jk ) ∗ ) | k = k ′ k = k ′ where C G ( Im ( τ i,jk ) ∗ ) is the centralizer of Im ( τ i,jk ) ∗ in G .It follows that the transformation matrix for Z ( Id Y ) is a block-diagonalized matrix with each block labelled by the conjugation class k in Hom( π ( Y ) , G ) / ∼ , and in each block, all the entries are thesame number | G | v | C G ( Im ( τ i,jk ) ∗ ) | . Thus we can find a basis { e k } for V G ( Y ) = Im ( Z ( Id Y )) by e k = X i,j | G | v | C G ( Im ( τ i,jk ) ∗ ) | τ i,jk where k labels the conjugation class in Hom( π ( Y ) , G ) / ∼ . To compute the action of [ f ] ∈ M ( Y ) on V G ( Y ), we define an actionof [ f ] on the set of conjugacy class of Hom( π ( Y ) , G ) first. For any[ φ ] ∈ Hom( π ( Y ) , G ) / ∼ , [ f ]([ φ ]) is defined to be [ φf − ∗ ].A calculation similar to the above results in Z ( ζ [ f ] = ( Y × I, Y × , Y × , f, id ))( τ i,jk )( τ i ′ ,j ′ k ′ )= | G | v | C G ( Im ( τ i ′ ,j ′ k ′ ) ∗ ) | [ f ]( k ) = k ′ f ]( k ) = k ′ We obtain Z ( ζ [ f ] )( e k ) = Z ( ζ [ f ] )( X i,j | G | v | C G ( Im ( τ i,jk ) ∗ ) | τ i,jk )= X i,j | G | v | C G ( Im ( τ i,jk ) ∗ ) | Z ( ζ [ f ] )( τ i,jk )= X i,j | G | v | C G ( Im ( τ i,jk ) ∗ ) | X i ′ ,j ′ | G | v | C G ( Im ( τ i ′ ,j ′ [ f ]( k ) ) ∗ ) | τ i ′ ,j ′ [ f ]( k ) = X i ′ ,j ′ | G | v | C G ( Im ( τ i ′ ,j ′ [ f ]( k ) ) ∗ ) | | G | v − | G | v [ G : Z Im ( τ i,jk ) ∗ ] | C G ( Im ( τ i,jk ) ∗ ) | τ i ′ ,j ′ k ′ = X i ′ ,j ′ | G | v | C G ( Im ( τ i ′ ,j ′ [ f ]( k ) ) ∗ ) | τ i ′ ,j ′ [ f ]( k ) = e [ f ]( k ) (cid:3) Suppose ρ, ρ DR are the representations of M ( Y × S ) and M ( Y )from a TQFT ( Z, V ) and its DR ( Z DR , V DR ), respectively, then thefollowing diagram commutes: M ( Y ) i / / ρ DR ' ' ❖❖❖❖❖❖❖❖❖❖❖❖ M ( Y × S ) ρ v v ♠♠♠♠♠♠♠♠♠♠♠♠♠ GL ( V ( Y × S )) , where i is an inclusion: for any [ f ] ∈ M ( Y ), i [ f ] = [ f × id ], where f × id : Y × S −→ Y × S by f × id ( x, y ) = ( f ( x ) , y ). OTION GROUP REPRESENTATIONS 17
Apply the above diagram to DW-TQFTs, we obtain M ( Y ) i / / ρ DWDR * * ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ M ( Y × S ) ρ DW t t ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ GL ( C Hom( π ( Y × S ) , G ) / ∼ ) , where ∼ means conjugacy equivalence. ThenHom( π ( Y × S ) , G ) / ∼ = Hom( π ( Y ) × π ( S ) , G ) / ∼ = { ([ φ ] , [ g ]) | [ g ] is a conjugacy class of G, [ φ ] ∈ Hom( π ( Y ) , G ) / ∼} = [ [ g ] conjugacy class Hom( π ( Y ) , Z g ) / ∼ where g is the image of the generator of second factor Z of π ( Y ) × Z .Since the image of i preserves the second factor of π ( X ) × Z withpre-composition, we obtained the following theorem. Theorem 1. If Y is an oriented closed ( n − -manifold, then there isan intertwining map: ρ as : ⊕ [ g ] ∈ [ G ] V ((n-1)+1)-DW C G ( g ) ( Y ) −→ V (n+1)-DW G ( Y × S ) of the representations of the mapping class groups M ( Y ) and M ( Y × S ) from the DW theories associated to { C G ( g ) } and G , respectively.The assembly map ρ as is an isomorphism of the two vector spaces.More explicitly, for any f ∈ M ( Y ) and ⊕ [ g ] ∈ [ G ] v [ g ] ∈ ⊕ [ g ] ∈ [ G ] V ((n-1)+1)-DW C G ( g ) ( Y ) , we have the following identity: ρ as ( ⊕ [ g ] ∈ [ G ] f (( n − ∗ v [ g ] ) = ( f × id ) ( n +1) ∗ ρ as ( ⊕ [ g ] ∈ [ G ] v [ g ] ) . Note that this identity does not imply that the images of the two repre-sentations are necessarily the same.The subgroup C G ( g ) of G is the centralizer of g in G and the sum-mation is over all conjugacy classes [ G ] = { [ g ] } of G . The mapping class group of a closed manifold is not a motion groupas motion groups are about diffeomorphisms that are connected to theidentity, while mapping class groups are about diffeomorphisms modulothose. The theorem is an instance of our conjecture for dimension re-duction of DW theories from (3+1) to (2+1): DW G ∼ = ⊕ [ g ] ∈ [ G ] DW C ( g ) for mapping class groups. Reduction from the -torus T to T for (3 + 1) -DW. In thissubsection, G = Z n , where n is any positive integer. Those examplesserve as explicit illustrations of Thm. 1.First we characterize the images of representations for Y = T , T in(3 + 1)- and (2 + 1)-DW TQFTs as abstract groups. It is known that M ( T ) = SL (3 , Z ) and Hom( π ( T ) , Z n ) / ∼∼ = Hom( Z , Z n ) ∼ = Z n .For any ( a, b, c ) ∈ Z n and A ∈ SL (3 , Z ), A · ( a, b, c ) = ( a, b, c ) A =( a, b, c ) p ( A ), where p is the natural map from SL (3 , Z ) to SL (3 , Z n )by modulo n . Since p is surjective, it follows that Imρ (3+1) − DW Z n ,T = SL (3 , Z n ). Similarly, Imρ (2+1) − DW Z n ,T = SL (2 , Z n ).Next, as images of representations from (2 + 1) to (3 + 1) TQFTs, itis known that M ( T ) = SL (3 , Z ) is generated by S , T and M ( T ) = SL (2 , Z ) is generated by S , T , where T = , S = , T = (cid:18) (cid:19) , S = (cid:18) −
11 0 (cid:19) . The three torus T has three S factors, hence SL (2 , Z ) can be embd-eded into SL (3 , Z ) in three different ways by mapping S to S , S , S and T to T , T , T , where T = , T = , T = ,S = − , S = −
10 1 01 0 0 , S = −
10 1 0 . The relations T = T and S = S S imply that the images of the(2+1) TQFT representation by dimension reduction along the first andthird factors of T generate the image of the representation of SL (3 , Z ).The decomposition representations of SL (2 , p ) and SL (3 , p ) fromDW TQFTs into irreducibles can be found using character theory,where p is prime. The character table of SL (2 , p ) and notation canbe found from [3], and same for SL (3 , p ) from [2]. We recall the fol-lowing character tables from these two references. In TABLE 1, τ is aprimitive ( p − e = ( − p − . In TABLE 2, e p − = 1, ρ p − = 1, σ p +1 = ρ , τ p + p +1 = 1 and θ = 1. Proposition 5.
Let χ be the character of the representation of SL (2 , p ) , and χ be the character of the representation of SL (3 , p ) coming from DW theories, respectively. OTION GROUP REPRESENTATIONS 19 ψ ζ i ξ ξ p p + 1 ( p + 1) ( p + 1) z p ( − i ( p + 1) e ( p + 1) e ( p + 1) a l τ il + τ − il ( − l ( − l b m -1 0 0 0 c (1 + √ ep ) (1 − √ ep ) d (1 − √ ep ) (1 + √ ep ) Table 1. SL (2 , p ) χ p ( p +1) χ ( i ) p + p +1 C k p ( p + 1) ( p + p + 1) ω ik C k p ( p + 1) ω ik C k ,l ω ik C k p + 1 ( p + 1)( e ik + e − ik ) C k e ik + e − ik C k ,l ,m e ik + e il + e im C k e ik C k -1 0 Table 2. SL (3 , p )(1) The representation image of M ( T ) from (3+1) -DW-TQFTs isgenerated by the representation images from (2 + 1) -DW TQFTrepresentations by dimensional reduction along the first andthird circles of T . (2) χ = 2 SL (2 ,p ) + ψ + 2 p − X i =1 ζ i + ξ + ξ . When p = 2 , the last three summations on the right side vanish.And in this case, SL (2 ,p ) + ψ is the natural representation of S on the 3-dimensional space by permutation. (3) χ = 2 χ + χ p ( p +1) + p − X i =1 χ ( i ) p + p +1 . When p = 2 , the last summation vanishes. Here Proof.
When p > SL (2 , p ) has p +4 conjugation classes: , z , a l , b m , c , d , zc , zd ,where = (cid:18) (cid:19) , z = (cid:18) − − (cid:19) , a = (cid:18) v v − (cid:19) , c = (cid:18) (cid:19) , d = (cid:18) v (cid:19) , and b is the element of order ( p + 1), which is not diagonalizable over Z p and v generates the multiplicative group of Z p and 1 ≤ l ≤ p − and1 ≤ m ≤ p − .For any conjugation class [ g ] in SL (2 , p ), χ ([ g ]) = ♯ { elements in Z p fixed by g } = p D , where D = dimension of eigenspace of g for eigenvalue 1. Thus χ ( ) = p χ ( c, d ) = pχ ( z, a l , b m , zc, zd ) = 1If χ ([ g ]) = RHS([ g ]), then the proof completes, which follows fromstraightforward check using the character table of SL (2 , p ). SL (3 , p ) has p + p conjugation classes: C ( k )1 = ω k ω k
00 0 ω k , C ( k )2 = ω k ω k
00 0 ω k ,C ( k ,l )3 = ω k θ l ω k θ l ω k , C ( k )4 = ρ k ρ k
00 0 ρ − k ,C ( k )5 = ρ k ρ k
00 0 ρ − k , C ( k ,l ,m )6 = ρ k ρ l
00 0 ρ m ,C ( k )7 = ρ k σ − k
00 0 σ − pk , C ( k )8 = τ k τ pk
00 0 τ p k . For any conjugation class [ g ] in SL (3 , p ), χ ([ g ]) = ♯ { elements in Z p fixed by g } = p D OTION GROUP REPRESENTATIONS 21 where D = dimension of eigenspace of g for eigenvalue 1 . Thus χ ( C (0)1 ) = p χ ( C (0)2 ) = p χ ( C (0 ,l )3 , C ( p − )4 , , C ( k ,l ,p − , C ( n ( p − ) = pχ (otherwise) = 1It is sufficient to prove χ ([ g ]) = RHS([ g ]), which follows from adirect comparison using the character tables. (cid:3) DW TQFTs as -extended. In a 1-extended ( n + 1)-TQFT,vector spaces are assigned to n -manifolds Y when the connected com-ponents of their boundaries ∂Y are labeled. Physically, the labelsare types of topological charges for some extended excitations. When n = 2, they are the so-called anyon types. We are interested in n = 3,then besides pointed excitations in the three manifold Y , there are 1-dimensional excitations supported around any embeded graph Γ in Y .The shape of such an extended excitation is the boundary surface Σof a closed neighborhood N (Γ) of Γ. Therefore, we need to find thelabel set for each of the genus= g surface Σ g , where g = 0 correspondsto pointed excitations and g = 1 loop excitations.Our description of the label set, vector spaces, and gluing formulasfor (3 + 1)-DW TQFTs is essentially a reformulation of results in [7].Afterwards, we describe representations of the motion groups of linksin the three sphere.3.3.1. Label sets for DW TQFTs.
For (2 + 1)-DW theories ( Z G , V G ),the shape of a pointed excitation is a circle—the boundary of the smalldisk neighborhood of a point. It is known that the labels are pairs([ g ] , α ), where [ g ] is a conjugacy class, and α an irreducible represen-tation (irrep) of the centalizer C G ( g ) of g ∈ G . This is a special caseof the general description of labels for DW TQFTs: for a shape M excitation of a manifold Y , the label set is a pair ( ρ, α ), where ρ is arepresentation of π ( M ) to G , and α an irrep of the centralizer of theimage of ρ ( π ( M )) in G . Proposition 6.
The label set for a boundary manifold Σ consists ofpairs { ([ ρ ] , α ) } , where [ ρ ] is a conjugacy class of maps ρ : π (Σ) −→ G ,and α is an irreducible representation of the centralizer C G ( Im ( ρ )) ofthe image of ρ in G (homomorphisms up to conjugation ). There is an involution ∗ on { ([ ρ ] , α ) } by ∗ ([ ρ ] , α ) = ([ ρ ] , α ∗ ) where α ∗ is the dual representation of α . [ ρ ] C G ( Im ( ρ )) α C G ( Im ( ρ )) [1] S α S , α S , α S , [(12)] Z α Z , α Z [(123)] Z α Z , α Z , α Z Table 3. ((2 + 1) , G = S )[ ρ ] C G ( Im ( ρ )) α C G ( Im ( ρ )) [1] S α S , α S , α S , Table 4. ((3 + 1) , two sphere , G = S )As an analogue to (2 + 1)-discrete gauge theory, we will refer toa label ([ ρ ] , [ α ]) for a boundary in DW theories as a pure flux if therepresentation α is trivial.As an example, for G = S , there are 8 types of different labels for(2 + 1)-TQFTs as shown in TABLE 3. where α iS are all the irreduciblerepresentations of S up to conjugation and similarly for α i Z , α i Z .For (3 + 1)-TQFTs with G = S , there are 3 types for the two sphereas shown in TABLE 4. And there are 21 types for the torus.We follow the notation of [7] in this sub-section. For any two cat-egories C , D , [ C , D ] denotes the category of functors from C to D asobjects and natural transformations as morphisms. For any compact m -manifold M ( m ≥
0) with base points b i on the connected compo-nents M i ( i = 1 , ..., k ) of M , let Π ( M ) be the category with { b i } as theset of objects, the fundamental group π ( M i , b i ) as the morphism setof b i , and the empty set as morphism space between any two differentbase points.Given any finite group G , we have [Π ( M ) , G ], where G is the cate-gory with only one object whose morphism set is G . An explicit descrip-tion of [Π ( M ) , G ] is as follows. Each object is a k -tuples ( ρ , ..., ρ k )where ρ i is a homomorphsim from π ( M i , b i ) to G . The morphism setbetween ( ρ , ..., ρ k ) , ( ρ , ..., ρ k ) is Q i G i , where G i = { g ∈ G | gh = h g, h j ∈ Im ρ ji , j = 1 , } . A presentation for the skeleton of [Π ( M ) , G ]follows from this description. Each object of the skeleton is a k -tuples([ ρ ] , ..., [ ρ k ]) , where [ ρ i ] is conjugacy class and the morphism set is Q i C G (Im ρ i ). From now on, [Π ( M ) , G ] denotes the skeleton.The representation category of [Π ( M ) , G ] is defined as the functorcategory [[Π ( M ) , G ] , V ec ] , where V ec is the category of finite dimen-sional vector spaces and linear transformations. Next we describe theirreducible representations, i.e. irreducible objects of [[Π ( M ) , G ] , V ec ]. OTION GROUP REPRESENTATIONS 23
For any ([ ρ ] , ..., [ ρ k ]) and a functor object α ∈ Obj([[Π ( M ) , G ] , V ec ]),the image α ([ ρ ] , ..., [ ρ k ]) of α is a finite dimensional vector space and α (Aut(([ ρ ] , ..., [ ρ k ]))) = α ( Q i C G (Im ρ i )) defines a Q i C G (Im ρ i )-actionon the vector space α ([ ρ ] , ..., [ ρ k ]). By the additive structure inducedfrom V ec , the irreducible representations of [Π ( M ) , G ], i.e. the irre-ducible objects of [[Π ( M ) , G ] , V ec ], are irreducible representations α of the group Q i C G (Im ρ i ) for some ([ ρ ] , ..., [ ρ k ]) and representatives ρ i .Let L M be the set of irreducible representations of [Π ( M ) , G ]. Spe-cializing to the boundary closed ( n − L Σ for Σ as in Prop. 6.3.3.2. Vector spaces.
Let Y be a connected n -manifold with boundarycomponents (Σ i , b i ) , i = 1 , .., k labeled by { ([ ρ i ] , α i ) } and with basedpoints b i . We also need to choose a base point b of Y and a fixed col-lection of arcs A i in Y such that A i connecting b to b i . Then for any ρ : π ( Y, b ) −→ G , A i induces a homomorphism ρ | Σ i ,b i : π (Σ i , b i ) −→ G byfirst mapping into π ( Y, b i ) then using A i to send the image to π ( Y, b ),and finally composing with ρ . For any ρ such that [ ρ | Σ i ,b i ] = [ ρ i ], thecentralizer C G (Im ρ ) is conjugacy to one subgroup of C G (Im ρ i ) by τ i .Then an representation of C G (Im ρ ) can be defined as follows. Givenlabels ([ ρ i ] , α i ), we have a representation ⊗ i α i of Q i C G (Im ρ i ). Thehomomorphism Q i τ i : C G (Im ρ ) −→ Q i C G (Im ρ i ) induces a represen-tation ⊗ α i of C G (Im ρ ). Definition 6.
Define the vector space as V ( Y ; ([ ρ i ] , α i )) = M [ ρ ] T ρ , where ρ : π ( Y, b ) −→ G , and the sum is over all the conjugacy classes [ ρ ] such that [ ρ | Σ i ,b i ] = [ ρ i ] , and T ρ is the trivial component of therepresentation ⊗ i α i of C G ( Im ρ ) for some representative ρ . Note that first the space T ρ depends on the choice of representativesof the conjugacy class and the paths A i . Moreover, when α i are allpure fluxes, the vector space is isomorphic to C { [ ρ ] | [ ρ | Σ i ,b i ] = [ ρ i ] } .A label ([ ρ ] , [ α ]) with α the trivial representation is referred to as apure flux. For pure fluxes, V ( Y ; [ ρ i ] , ) = V ( Y ; [ ρ i ]) = C { [ ρ ] | [ ρ | S i ] =[ ρ i ] } . For future computation, V ( Y ; [ ρ i ]) can be reformulated as follows.Given labels [ ρ i ], choose a representative ρ i for each conjugacy class [ ρ i ],then we define the vector space V ( Y ; ρ i ) to be spanned by { ( ρ, a i ) | ρ : π ( Y, b ) −→ G, a i ( ρ | Σ i ,A i ) a − i = ρ i } / ∼ , where ( ρ, a i ) ∼ ( ρ ′ , a ′ i ) iff [ ρ ] =[ ρ ′ ]. Then V ( Y ; [ ρ i ]) is isomorphic to V ( Y ; ρ i ). Disk, annulus, and gluing axioms.
Suppose that Y is an orientedcompact n -manifold with boundary ∂Y = Σ = ⊔ i Σ i . The manifold Y can be regarded as a bordism between Σ and the empty manifold.By [7], Y defines a functor [[Π ( Y ) , G ] , V ec ] from [[Π (Σ) , G ] , V ec ] to[[Π ( ∅ ) , G ] , V ec ] = V ec . The functor [[Π ( Y ) , G ] , V ec ] corresponds toa (1 × l )-matrix with vector spaces as entries and the columns in-dexed by L Σ , where l = | L Σ | = | Q i L Σ i | . Then we assign to anyoriented compact n -manifold Y with boundary components Σ i labelledby irreducible representations in L Σ i the vector space in the matrix[[Π ( Y ) , G ] , V ec ] indexed by the same irreducible representation in L Σ .A direct computation proves that the vector space is the same as the V ( Y ; ([ ρ i ] , α i )) in Def. 6.Let D be an n -ball with boundary labelled by ([ ρ ] , α ) where ρ : π ( ∂D ) −→ G . Since π ( D ) is trivial, to make V ( D ; [ ρ ] , α ) nonzero, [ ρ ]has to be the trivial map. Thus T ρ is the irreducible representation of G . Since T ρ is the trivial component, V ( D ; [ ρ ] , α ) = C when ([ ρ ] , α ) =( , ) and 0 otherwise.Let Σ be any closed n − × I by([ ρ i ] , α i )( i = 1 , π (Σ × I ) = π (Σ), to make V (Σ × I ; ([ ρ i ]) , α i )nonzero, [ ρ ] = [ ρ ] and thus we suppose that ρ = ρ and α i are theirreducible representations of C G ( Im ( ρ )). Then T ρ = Hom G (1 , α ⊗ α ) = Hom G ( α ∗ , α )Since α i are irreducible, T ρ = C when α ∗ = α and 0 otherwise.To verify the gluing axiom, let Y gl be the n -manifold obtained fromgluing two boundary components { Σ , − Σ } of an n -manifold Y . Sup-pose the other boundary components Σ i of Y are labelled by ([ ρ i ] , α i ).Let Y be the bordism from ⊔ i Σ i to Σ ⊔ − Σ, Σ × I be the bordismfrom Σ ⊔ − Σ to the empty manifold. Then Y gl is the composition ofthe two bordisms Y, Σ × I , which is from ⊔ i Σ i to the empty manifold,and Thm. 10 in [7] implies the gluing axiom.3.3.4. Linear isomorphisms for extended bordisms.
Extended bordismsare bordisms between bordisms with corners. Suppose the followingdiagram commutes as in [7]: Σ τ (cid:15) (cid:15) τ / / Y θ (cid:15) (cid:15) Y θ / / X OTION GROUP REPRESENTATIONS 25 such that τ i : Σ −→ ∂Y i , θ ⊔ θ : Y ⊔ τ (Σ)= τ (Σ) Y −→ ∂X . Then[[ X, G ] , V ec ] defines a linear isomorphism between the vector spaces inthe matrices [[ Y i , G ] , V ec ] with the corresponding labels.3.4. Pure flux representations of the motion groups of linksfrom DW TQFTs.
From now on, we just consider pure flux labelsand labels can be regarded as conjugacy classes of homomorphisms.Suppose that L = ⊔ i L i is an oriented link in S , where L i the connectedcomponents and N ( L i ) a tubular neighbourhood. Any f ∈ H + ( S ; L )induces an isotopy class in H + ( S \ ∪ i N ( L i ) , ∪ i ∂N ( L i )). Then thefollowing diagram commutes as in Sec. 3.3.4: ∂N ( L ) × { , } id ⊔ id / / id ⊔ f | ∂N ( L ) (cid:15) (cid:15) ( S \ N ( L )) × { , } id ⊔ f (cid:15) (cid:15) ∂N ( L ) × I id / / ( S \ N ( L )) × I .
Choose base points b, b i for S \ N ( L ) , ∂N ( L i ), respectively. Each L i is labeled by a representation ρ i , where ρ i : π ( ∂N ( L i ) , b i ) −→ G .More concretely, the representation ρ i can be parameterized by a pairof commuting group elements, so each L i is labeled by ( g i , h i ) ∈ G ,where g i h i = h i g i and g i , h i correspond to the meridian and longitude of ∂N ( L i ) at b i , respectively. Any motion h t in M ( L ⊂ S ) defines a linearisomorphism between V G ( S \ N ( L ); ρ i ) and V G ( S \ N ( L ); f | ∂N ( L ) ( ρ i )),where f ( ρ i ) comes from pre-composition and f = h . Then the actionof h t ∈ M ( L ⊂ S ) is defined by ∂ ( h t ) = h ∈ H + ( S ; L ) as above.When [ ρ i ] = [ h ( ρ i )] for any h t , we obtain a representation of M ( L ⊂ S ) on the vector space V G ( S \ N ( L ); ρ i ).4. Dimension reduction conjecture for representationsof motion groups of links with generalized axes
In this section, we focus on representations of the motion groups oflinks with generalized axes from (3 + 1)-DW TQFTs labeled by purefluxes. The main technical theorems are Goldsmith’s presentation ofsuch motion groups and dimension reduction. We conjecture that therepresentations of such motion groups in general can be reduced torepresentations of certain surface braid groups.4.1.
Main conjecture for representations of motion groups oflinks with generalized axes.
Given a link L with a generalized axis γ and fiber F . Since F −→ S \ γ −→ S is the fibration and L inter-sects with each fiber transversely in P , we have an induced fibration F \ P −→ S \ ( γ ∪ L ) −→ S , and the following exact sequence1 −→ π ( F \ P ) i −→ π ( S \ ( γ ∪ L )) π −→ π ( S ) −→ . Choose base points p, p i , p γ on F \ P ⊂ S \ ( γ ∪ L ) for S \ γ ∪ L, L i , γ ,and arcs A i , A γ on F \ P ⊂ S \ ( γ ∪ L ) connecting p to p i , p γ . Label L i , γ by [ g i , h i ] , [ g, h ] such that [ g i , h i ] are consistent with the action of H + ( S , γ ∪ L ) on L i . Next the corresponding labels for F \ P are givenas follows. The intersection points between L i and F are labelled by[ g i ] and ∂F by [ h ]. Then for any [ ρ ] ∈ V G ( S \ ( γ ∪ L ); [ g, h ] , [ g i , h i ]), i ∗ ([ ρ ]) ∈ V G ( F \ P ; [ g i ] , [ h ]), where i ∗ induced by i ∗ : π ( F \ P, p ) −→ π ( S \ ( γ ∪ L ) , p ). And for any h ∈ H φ ( F, P ), we have i ∗ eJ ( h )([ ρ ]) = h ( i ∗ ([ ρ ])).Our main conjecture is as follows. Conjecture . There is a map from labels of a 1-extended (3 + 1)-TQFTs to those of 1-extended (2 + 1)-TQFTs so that the representa-tions of the motion groups of a link L from the (3+1)-TQFT decomposeas direct sums of the representations of some surface braid groups ofthe fiber surface F if the link L has a generalized axis γ with fiber F .Equations (2.1) (2.2) suggest a form of the decomposition of the mo-tion group representations or rather an organization of some surfacebraid group representations. The non-trivial monodromy map of thegeneralized axis knot complement fibration induces a map of surfacebraid group representations. The non-trivial fibfation should manifestitself in that the motion group representation should be essentially thesub-representation of a direct sum of surface braid group representa-tions consisting of commutants of this monodromy map.Thm. 1 is the analogous statement of this conjecture for the mappingclass groups. In the following, we will provide further evidence bystudying the motion groups of the necklace links and the torus links.Our main Thm. 2 is formulated using the fundamental group of a2-complex, which is not a fiber surface. But our theorem should beequivalent to one using a braid group of some fiber surface.4.2. Representations of the motion groups of the necklace links,and cablings of the Hopf link.
In this subsection, we obtain resultsfor the representations of the motion groups of several families of linksfrom DW TQFTs labeled by pure fluxes: the motion groups of thenecklace links, which are used to study statistics of loops in physics [8],and the motion groups of the n -Hopf links. OTION GROUP REPRESENTATIONS 27
The necklace links.
Let L be necklace links as in Fig.1 where L i ( i = 1 , ..., n ) are the components of the unlink and L c is the trackcircle that links them. Its fundamental group is π ( S \ L, P ) = F ( x ) × F ( x , ..., x n ) , where F ( x ) is the infinite cyclic group generated by the loop for themeridian of L c and F ( x , ..., x n ) is the free group generated by theloops for the meridians of L i . The motion groups M ( L ⊂ S ) aregenerated by σ i ( i = 1 , ..., n −
1) and p , where σ i interchanges L i and L i +1 , and p permutes L i along the counterclockwise. Since the motiongroup permutes all L i , we require the labels on L i to be the same. Thus( g, h ) , ( g c , h c ) are used to label L i , L c such that gh = hg , g c h c = h c g c .The images of x, x i in G are also denoted by x, x i . Then V G ( S \ L ; ( g, h ) , ( g c , h c )) = C { [( x, x i , a i , a c )] } satisfying compatible conditions x = a c g c a − c , x i = a i ga − i , x = a i ha − i , x · · · x n = a c h c a − c , xx i = x i x. To have the above vector space nonzero, h has to be a conjugacy of g c . Then choosing the labels ( g c , h c ) , ( g, g c ), we obtain the action of themotion group as follows. σ i ( x ) = xσ i ( x i ) = x i x i +1 x − i , σ i ( x i +1 ) = x i σ i ( x j ) = x j , j = i, i + 1 p ( x ) = xp ( x i ) = x i − n Let D be the disk bounded by L c in S . Then L i intersects with D in P i transversely. We define an bijection T from V C G ( g c ) ( D \{ P i } ; a − c, a i, ga − i, a c, , h c ) to V G ( S \ L ; ( g, g c ) , ( g c , h c )) by T ([ x i , a i , a c ]) = [ g c , x i , a i , a c ] , where [ x , x i, , a i, , a c, ] ∈ { [ x, x i , a i , a c ] } ( g,g c ) , ( g c ,h c ) . It induces a bi-jection ˜ T on their automorphism spaces. Let i be the inclusion of B ( D, n pts) into M ( L ⊂ S ). The above computation results in Proposition 7.
The following diagram commutes: M ( L ⊂ S ) ρ (3+1) − DW (cid:15) (cid:15) B ( D, n pts ) i o o ρ (2+1) − DW (cid:15) (cid:15) Aut ( V G ( S \ L ; ( g, g c ) , ( g c , h c ))) Aut ( V C G ( g c ) ( D \{ P i } ; a − c, a i, ga − i, a c, , h c )) ˜ T o o Moreover Im ( ρ (3+1) − DWG ) = ˜ T ( Im ( ρ (2+1) − DWC G ( g ) )) . L L L n P P P n L c · · · P Figure 1.
The necklace link4.2.2.
The n -Hopf links. Let L be the link with n components consist-ing of n fibers of the Hopf fibration π : S → S . It is also the toruslink H n = T L (1 , n . The north and south hemispheres of the base S are the two meridian disks of the two solid tori of S . Taking one of thecore circle of one solid torus as an axis, the median disk of the othersolid torus is the fiber surface F .The fundamental group of the n -Holf link similar to Lemma 4 is: π ( S \ L ) = F ( y ) × F ( x , ..., x n )and by choosing K δ to be the disk formed by the meridian disk and O as in Fig. 2, labeling the components of H n by ( g, h ) satisfying gh = hg , we have the following commuting diagram: Proposition 8. M ( H n ⊂ S ) ρ DW (cid:15) (cid:15) B ( disk , n pts ) i o o ρ DW (cid:15) (cid:15) Aut ( V G ( S \ H n ; ( g, h ))) Aut ( V C G ( h ) ( K δ \ n pts ; labeling as above )) T [ y ]=[ h ] − o o Representations of the motion groups of the torus links.
Let
T L ( p,q ) n be the n -component torus link of type ( p, q ) in S for anycoprimes ( p, q ) , p, q ≥
2. Using the parameterization of S as in Sec.1.1 or [4], we have T L ( p,q ) n = n [ j =1 ι ( { ( 1 j R qtp Ξ p , e πit ) | ≤ t ≤ } ) . Fig. 2 illustrates the meridian disk D m of C × S for case n = 4 andFig. 3 illustrates the meridian disk D m ′ of S × C for case n = 4. The OTION GROUP REPRESENTATIONS 29 S S S P O ′ B B B B B B B B B A A A A A A A A A A A A O Figure 2. D m ˜ S ˜ S ˜ S OB B B bC C C ˜ A ˜ A ˜ A ˜ A ˜ A ˜ A ˜ A ˜ A O ′ Figure 3. D m ′ points { A ij } , { ˜ A kj } are the intersections between the j -th component ofthe torus link and the meridian disks of the first factor of C × S andsecond factor of S × C , respectively, where i = 1 , , k = 1 , Lemma 4.
Let b be a base point of S \ T L (3 , n , then π ( S \ T L (3 , n , b ) has the following presentation: π ( S \ T L (3 , n , b ) = < x, y, u i , v i | x = y = u i v i = v i u i , i = 1 , ..., n − > = < u i , x, y | x = y , y u i = u i y > Proof.
First we show that the link complement E L = S \ T L (3 , n re-tracts to a 2-complex E ′ L .For each j = 1 , ..., n −
1, a torus T j in E L is constructed as follows. T j = ι ( { ( 1 j + e iθ , e iζ ) | ≤ θ, ζ ≤ π } ) B i +1 x ′ y ′ B i B i B i C i OO ′ b Figure 4. K T Then any meridian disk m of C × S intersects T j at the circle S j asshown in Fig. 2.Next a 2-complex K T is constructed: K T = [ j> ι ( { ( 1 j R t e − πi Ξ , e πit ) | ≤ t ≤ } ) ∪ x ′ ∪ y ′ Note that K T is a 2-complex consisting of a continuous family of (3 , T L (3 , n and form an cylinder, and thetwo core circles x ′ , y ′ as shown in Fig. 4. For any meridian disk m of C × S , the intersection between K T and m is just the union of the threesegments pointing to O as shown in Fig. 2. And Fig. 3 illustrates theintersection between K T and the meridian disk of S × C . Therefore, K T is the 2-complex obtained by attaching the two boundaries of thecylinder to the two core circles using z z and z z .Set E ′ L = S n − i =1 T i ∪ K T . Retracting in each meridian disk of C × S as shown in Fig. 2 gives rise to a retraction of E L to E ′ L .To compute the fundamental group of E ′ L , we choose b to be the basepoint as shown in Fig. 4, loops a i = bB i S i bB i and b i = bB i ˜ S i B i b tobe generators for the fundamental group of T i , and loops x = bOx ′ OP and y = bO ′ y ′ O ′ b to be the generators of the fundamental group of K T .The direction of each S i here is chosen to be counterclockwise.Since K T ∩ T i is a (3 , T i , by Van-Kampen’s theorem, wearrive at π ( S \ T (2 , n , b ) = < x, y, a i , b i | x = y = a i b i , a i b i = b i a i , i = 1 , ..., n − > . Setting u i = a i b i , v i = a i b i leads to the desired presentation. (cid:3) OTION GROUP REPRESENTATIONS 31
O OO OD m D m K δ K δ σ i ˜ σ i ρ i ˜ ρ i Figure 5.
Motion of the (3,2)-torus linkTo describe the action of M ( T L (3 , n ⊂ S ) on π ( S \ T L (3 , n , b ),we use the presentation of M ( T L (3 , n ⊂ S ) in Thm. 8.7 [4]: thegenerating motions for M ( T L (3 , n ⊂ S ) are { σ i } n − i =1 , and { ρ i } ni =17 ,where σ i interchanges the i -th and ( i + 1)-th components of the toruslink, while ρ i rotates the i -th component about the x ′ -axis by e iqπp asin Fig. 3. Proposition 9.
The action of M ( T L (3 , n ⊂ S ) on π ( S \ T L (3 , n , b ) by pre-composition is as follows. Let u = y, u n = x formally, then σ i ( u j ) = ( u i − u − i u i +1 j = iu j j = iσ i ( x ) = xσ i ( y ) = yρ i ( u j ) = ( ( u i − u − i ) u j ( u i − u − i ) − j ≥ iu j j < i ; ρ i ( x ) = ( u i − u − i ) x ( u i − u − i ) − ρ i ( y ) = y Our r i in Prop. 2 x ′ O B B B b O ′ y ′ P P P P + − K K K Figure 6. K δ Proof.
First we construct another 2-complex K δ in S as follows. K δ = [ j> ι ( { ( 1 j R t e − πi Ξ , e πit ) | ≤ t ≤ } ) ∪ x ′ ∪ y ′ The 2-complex K δ consists of a continuous family of (1 , x ′ , y ′ . The intersection between K δ and the meridian disk D m in Fig. 2 is just the segment bO . Thus K δ is a cylinder with x ′ , y ′ as two boundaries.Note that the i -th component of T L (3 , n is on the torus T ′ i , where T ′ i = ι ( { i e iθ , e iζ ) | ≤ θ, ζ ≤ π } ) . Since the intersection number between the (1 ,
1) and (3 ,
2) knotson the same torus is 1, each of the i -th component of T L (3 , n inter-sects with K δ exactly at 1 point P i transversely on the torus T ′ i , thus K δ \ T L (3 , n can be represented by Fig. 6, where the circles K i arethe (1 ,
1) knots on the torus T i and P i are the intersections between T L (3 , n and K δ .Choose b to be the base point of S \ T L (3 , n . Set ˜ u i to be the loops bB i K i B i b . Since K i is the (1 ,
1) knot on T i , it follows that ˜ u i = a i b i = u i . Thus it suffices to describe the action of M + ( T L (3 , n ⊂ S ) on˜ u i , x, y .Consider the restriction of the action of σ i , ρ j as shown in Fig. 5 on K δ . For the interchange σ i , we choose an isotopy H t such that eachcomponent is always on the same torus during the isotopy. Since theintersection number between (1 ,
1) and (3 ,
2) on the same torus is 1, σ i induces ˜ σ i on K δ which interchanges the i -th and ( i + 1)-th points as OTION GROUP REPRESENTATIONS 33 shown in Fig. 5. For ρ i , according to Fig. 5, it induces ˜ ρ i on K δ whichrotates the i -th point by 2 π along the direction of K i .Next we compute the action of ˜ ρ i on ˜ u i , x, y . For simplicity, thedirection of arcs is shown by + and − on Fig. 6. The path formed bythe composition of arcs along points X , ..., X k is denoted by ( X ...X k ).For uniformity of notation, we set ˜ u to be y , and ˜ u n to be x .For any i = 1 , ..., n −
1, if j = i , then the motion ˜ σ i does not touch˜ u j . Thus ˜ σ i (˜ u j ) = ˜ u j . If j = i , then˜ σ i (˜ u i ) = ( bB i − + − B i − B i − + B i B i +1 + − B i +1 b )= ( bB i − + − B i − B i − + B i b ) · ( bB i +1 + − B i +1 b )= ( bB i − + − B i − b ) · ( bB i − + B i b ) · ( bB i +1 + − B i +1 b )= ˜ u i − ˜ u − i ˜ u i +1 Since ˜ σ i does not touch x, y for any i = 1 , ..., n −
1, hence˜ σ i ( x ) = x, ˜ σ i ( y ) = y For any i = 1 , ..., n , when j < i , ˜ ρ i does not touch ˜ u j . Thus ˜ ρ i (˜ u j ) = ˜ u j .If j ≥ i , then˜ ρ i (˜ u j ) = ( bB i − + − B i − B i − + B i B j + − B j b )= ( bB i − + − B i − B i − + B i P ) · ( P B j + − B j b ) · ( bB i + − B i B i − − + B i − b )= ˜ u i − ˜ u − i u j (˜ u i − ˜ u − i ) − For any i = 1 , ..., n , since ρ i does not touch y , so ˜ ρ i ( y ) = y .For x , similarly ˜ ρ i ( x ) = ˜ u i − ˜ u − i x (˜ u i − ˜ u − i ) − . (cid:3) By Sec. 3.3, to describe explicitly the representations of the mo-tion groups of the torus links, it suffices to compute the action of therepresentatives of the generating motions { σ i , ρ i } in [4].In the following, for easiness of notation, u i is identified with ˜ u i .Given pure flux labels ( g, h ) on T L (3 , n such that g , h correspond tothe meridian m i and longitude l i , respectively, on the boundary, andthe paths A i connecting b to P i on the boundary of S \ T L (3 , n , thenthe following holds: A i m i A − i = u i − u − i , A i l i A − i = x . Thus for any [( ρ, a , ..., a n )] ∈ V G ( S \ T L (3 , n ) ; ( g, h )), we can find arepresentative ( x, y, u i , a i ) satisfying the following conditions. For sim-plicity, elements are identified with their images of ϕ . x = y , yu − = a ga − ,u u − = a ga − , · · · , u n − x − = a n ga − n , y = a i ha − i . Consider the n -punctured cylinder M = K δ \{ P i } as constructed above.For any finite group H , labeling the boundary of K δ \{ P i } by ( g i , g x , g y ),we obtain V H ( K δ \{ P i } ; ( g i , g x , g y )) = C { [(˜ x, ˜ y, ˜ u i , ˜ a i , ˜ a x , ˜ a y )] } , where (˜ x, ˜ y, ˜ u i , ˜ a i , ˜ a x , ˜ a y ) satisfy the following conditions.˜ x = ˜ a x ˜ g x ˜ a − x , ˜ y = ˜ a y ˜ g y ˜ a − y , ˜ y ˜ u − = ˜ a ˜ g ˜ a − , ˜ u ˜ u − = ˜ a ˜ g ˜ a − , · · · , ˜ u n − ˜ x − = ˜ a n ˜ g n ˜ a − n and [ − ] denotes the conjugation class of homomorphisms.Next we construct a bijection Ψ [ x ] , [ y ] for [ x ] , [ y ] ∈ [ G ], where [ G ] isset of the conjugation class of G , from S [ x ] , [ y ] = { [( x, y, u i , a i )] | x ∈ [ x ] , y ∈ [ y ] } S \ T L (2 , n ; G ;( g,h ) to F [ x ] , [ y ] = { [(˜ x, ˜ y, ˜ u i , ˜ a i , ˜ a x , ˜ a y )] } K δ \{ P i } ; Z ( y );( a i, ga − i, ,x ,y ) , where [( x , y , u i, , a i, )] ∈ S [ x ] , [ y ] and Z ( y ) is the centralizer C G ( y ).Since y, y ∈ [ y ], any element in S [ x ] , [ y ] can be represented by [( x, y , u i , a i )].Then we defineΨ [ x ] , [ y ] ([( x, y , u i , a i )]) = ([( x, y , u i , a i a − i, , g x , , where x = g x x g − x as x, x ∈ [ x ]. Lemma 5. Ψ [ x ] , [ y ] is well-defined and bijective.Proof. First we show T [ x ] , [ y ] is well-defined.Since x = g x x g − x , so y = x = g x x g − x = g x y g − x . Thus g x ∈ Z ( y ). Since a − i y a i = a − i, y a i, , hence a i, a − i y a i a − i, = y . Thus a i a − i, ∈ Z ( y ). For any [( x ′ , y , u ′ i , a ′ i ] = [( x, y , u i , a i )], ( x ′ , y , u ′ i ) = g ( x, y , u i ) g − . Thus g ∈ Z ( y ). It follows that their images are thesame.Now we show that T [ x ] , [ y ] is injective. For any [( x, y , u i , a i )] , [( x ′ , y , u ′ i , a ′ i )]such that [( x, y , u i , a i a − i, , g x , x ′ , y , u ′ i , a ′ i a − i, , g x ′ , g ∈ Z ( y ) such that ( x, y , u i ) = g ( x ′ , y , u ′ i ) g − . Thus[( x, y , u i , a i )] = [( x ′ , y , u ′ i , a ′ i )]. OTION GROUP REPRESENTATIONS 35
Finally we show T [ x ] , [ y ] is surjective. For any [( x, y, u i , a i , a x , a y )] ∈ A [ x ] , [ y ] , we have y = a y y a − y . Choose a representative ( x, y , u i , a i , a x , x, y , u i , a i a i, ), we directly check that ( x, y , u i , a i a i, )satisfies the compatible conditions for S [ x ] , [ y ] . It follows that T ([( x, y , u i , a i a i, )]) = [( x, y , u i , a i , a x , , which completes the proof. (cid:3) By Prop. 9, S [ x ] , [ y ] is preserved by the motion group of T L (3 , n in S .Thus C S [ x ] , [ y ] is invariant under the action of the motion group fromDW theory. Furthermore, the following diagram commutes. Proposition 10. (1) M ( T L (3 , n ⊂ S ) ρ (3+1) − DW (cid:15) (cid:15) B ( cylinder ; n pts ) i o o ρ (2+1) − DW (cid:15) (cid:15) GL ( C S [ x ] , [ y ] ) GL ( V Z ( y ) ( K δ \{ P i } ; labeling as above )) Ψ − x ] , [ y ] o o , where i is the inclusion and ρ DW is the representation definedas above. Moreover, the images of the representations are thesame: Im ( ρ (3+1) − DWG ) = Ψ − x ] , [ y ] ( Im ( ρ (2+1) − DWZ ( y ) ))(2) As representations of the motion groups from the DW TQFTsassociated to groups G and C G ( y ) , respectively, V G ( S \ T L (3 , n ; ( g, h )) = M [ x ] , [ y ] C S [ x ] , [ y ] = M [ x ] , [ y ] V C G ( y ) (( cylinder \ n pts ); ( a i, ga − i, , x , y ))These results can be generalized to general coprimes ( p, q ) , p, q ≥ Theorem 2.
Let
T L ( p,q ) n be the torus link of n copies of the ( p, q ) -torusknot in S with a presentation for π ( S \ T L ( p,q ) n , b ) : π ( S \ T L ( p,q ) n , b ) = < x, y, u i | x p = y q , y q u i = u i y q , i = 1 , , ..., n > . Suppose the n components of T L ( p,q ) n are labeled by { (( g, h ) , ) } ni =1 suchthat gh = hg . Then the representation of motion group of T L ( p,q ) n in S from (3 + 1) -DW TQFT decomposes as: V G ( S \ T L ( p,q ) n ; { (( g, h ) , ) } ni =1 ) = M [ x ] , [ y ] V C G ( y q ) (( K u,v \{ b i } ni =1 ); ( a i, ga − i, , x , y )) , where u, v are positive integers such that pv − qu = 1 , and K u,v = [ j> ι ( { ( 1 j R vtu e − iπp Ξ u , e πit ) | ≤ t ≤ } ) ∪ x ′ ∪ y ′ is the -complex obtained by attaching the two boundaries of the cylin-der to two circles by z z u , z z v . References [1] David Dahm. “A generalization of braid theory”. In:
Princeton Ph. D. thesis (1962).[2] William A Simpson and J Sutherland Frame. “The character tables for SL (3,q), SU (3, q 2), PSL (3, q), PSU (3, q 2)”. In:
Canadian Journal of Mathematics25.3 (1973), pp. 486–494.[3] James E Humphreys. “Representations of SL (2, p)”. In:
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Mathematica Scan-dinavica (1982), pp. 167–205.[5] Michihisa Wakui. “On Dijkgraaf-Witten invariant for 3-manifolds”. In:
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Dimensional Reduction, Extended Topological FieldTheories and Orbifoldization. arXiv preprint arXiv:2004.04689 (2020).[10] Zolt´an K´ad´ar et al. “Local representations of the loop braid group”. In:
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Communications in Mathematical Physics (2019), pp. 1–25[14] V. G. Turaev, ”Quantum invariants of knots and 3-manifolds”, Vol. 18. Walterde Gruyter Co., 2020.[15] K. Walker. ”On Witten’s 3-manifold Invariants”, 1991 TQFT notes. ( K u,v \{ b i } ni =1 ) is not a manifold if u, v = 1, but we can define the vector spaceas the span of the representations of the fundamental group as in the DW TQFTs. OTION GROUP REPRESENTATIONS 37 [16] Kevin Walker and Zhenghan Wang. “(3+ 1)-TQFTs and topological insula-tors”. In:
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Department of Mathematics, University of California, Santa Bar-bara, CA 93106, USA
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