AANOMALY-FREE TQFTS FROM THE SUPER LIEALGEBRA sl (2 | NGOC PHU HA
Abstract.
It is known that the category C H of nilpotent weightmodules over the quantum group associated with the super Liealgebra sl (2 |
1) is a relative pre-modular G -category. Its modifiedtrace enables to define an invariant of 3-manifolds. In this articlewe show that the category C H is a relative modular G -categorywhich allows one to construct a family of non-semi-simple extendedtopological quantum field theories which surprisingly are anomalyfree. The quantum group associated with sl (2 |
1) is considered atodd roots of unity.
MSC: 57M27, 17B37Key words: relative (pre)modular G -category, invariant of 3-manifolds,modified trace, quantum group. Acknowledgments.
The result of the article is a part of my PhDthesis. The author would like to thank my supervisor, B. Patureau-Mirand for his useful comments and for his encouragement.1.
Introduction
By renormalization the invariant of Reshetikhin-Turaev, the authorsN. Geer and B. Patureau-Mirand presented the notion of a modifiedtrace in [7]. This notion gave them a family of link invariants (see [7, 5])and of invariant of 3-manifolds (see [3]). The invariants of 3-manifoldsin [3] have been used to define a family of non-semi-simple topologicalquantum field theories (TQFTs) in [2].A relative pre-modular G -category C is a G -graded k -linear ribboncategory having a modified trace and two abelian groups ( G, Z ) inwhich Z acts on a class of objects of C satisfying some compatibleconditions, see Definition 2.1 (see also [3]). A category of finite weightmodules over a quantization of sl (2) have been proven to be relative pre-modular in [3], and this category is used to illustrate the main results ofthe article [3]. Such categories are the ingredients to construct topolog-ical invariants N r or CGP invariants of the computable triple ( M, T, ω )where T is a C -colored ribbon graph in M and ω ∈ H ( M \ T, G ) (see[3, Theorem 4.7]). In the article [4], the author enriched the structure of a r X i v : . [ m a t h . QA ] J a n NGOC PHU HA a relative pre-modular category C by adding a relative modularity con-dition, the category C with this condition is called a relative modularcategory, see Definition 4.1. Relative modular categories are main in-gredients to construct a family of non-semisimple extended topologicalquantum field theories (ETQFTs) [4]. A model of a relative modu-lar category from the unrolled quantum group associated to the Liealgebra sl (2) was first constructed in [2].Working with the unrolled quantum group associated with the superLie algebra sl (2 | U Hξ sl (2 |
1) where ξ is a root of unity, onesee that the category C H of nilpotent weight modules over U Hξ sl (2 | G -category [8]. This fact allowed one to con-struct an invariant of CGP type. In this paper, we show for the su-per Lie algebra sl (2 |
1) that, as in the case of Lie algebras, C H is amodular G -category relative to ( G, Z ) where G = C / Z × C / Z and Z = Z × Z × Z / Z . This fact provides a new family of examples ofrelative modular categories. One specific feature with the sl (2 |
1) caseis that the anomaly of the TQFT happen to be trivial.The paper is organized in four sections. In Section 2, we recall somedefinitions about the relative pre-modular G -categories and the quan-tum group U Hξ sl (2 | C H of nilpotent finite dimensional modules over U Hξ sl (2 |
1) and its proper-ties. Finally, Section 4 proves the category C H is a relative modular G -category. 2. Preliminaries
In this section we recall some definitions and results about a relativepre-modular G -category from [3], and a quantization of the super Liealgebra sl (2 | Relative pre-modular G -categories. The categories we men-tion in the paper are ribbon. A tensor category C is a categoryequipped with a covariant bifunctor − ⊗ − : C × C → C called thetensor product, a unit object I , an associativity constraint, and left andright unit constraints such that the Triangle and Pentagon Axioms hold(see [9, XI.2]).A braiding on a tensor category C consists of a family of isomor-phisms { c V,W : V ⊗ W → W ⊗ V } , defined for each pair of objects V, W which satisfy the Hexagon Axiom [9, XIII.1 (1.3-1.4)] as wellas the naturality condition expressed in the commutative diagram [9,XIII.1.2]. We say a tensor category is braided if it has a braiding.
NOMALY-FREE TQFTS FROM sl (2 |
1) 3 A pivotal category is a tensor category which has duality if for eachobject V ∈ C there exits an object V ∗ and morphisms −→ ev V : V ∗ ⊗ V → I , −→ coev V : I → V ⊗ V ∗ , ←− ev V : V ⊗ V ∗ → I , ←− coev V : I → V ∗ ⊗ V satisfying the relations(Id V ⊗ −→ ev V ) ◦ ( −→ coev V ⊗ Id V ) = Id V , ( −→ ev V ⊗ Id V ∗ ) ◦ (Id V ∗ ⊗ −→ coev V ) = Id V ∗ , and( ←− ev V ⊗ Id V ) ◦ (Id V ⊗ ←− coev V ) = Id V , (Id V ∗ ⊗ ←− ev V ) ◦ ( ←− coev V ⊗ Id V ∗ ) = Id V ∗ . A twist in a braided tensor category C with duality is a family { θ V : V → V } of natural isomorphisms defined for each object V of C sat-isfying relations [9, XIV.3.1-3.2].A ribbon category is a braided tensor category with duality and atwist. We say that C is a k -linear category if for all V, W ∈ C , themorphisms Hom C ( V, W ) form a k -vector space and the compositionand the tensor product are bilinear and, End C ( I ) ∼ = k where k is afield.Let C be a k -linear ribbon category. A set of objects of C is saidto be commutative if for any pair { V, W } of these objects, we have c V,W ◦ c W,V = Id W ⊗ V and θ V = Id V . Let ( Z, +) be a commutativegroup. A realization of Z in C is a commutative set of objects { ε t } t ∈ Z such that ε = I , qdim( ε t ) = 1 and ε t ⊗ ε t (cid:48) = ε t + t (cid:48) for all t, t (cid:48) ∈ Z .A realization of Z in C induces an action of Z on isomorphismclasses of objects of C by ( t, V ) (cid:55)→ ε t ⊗ V . We say that { ε t } t ∈ Z is a freerealization of Z in C if this action is free. This means that ∀ t ∈ Z \{ } and for any simple object V ∈ C , V ⊗ ε t (cid:54)(cid:39) V . We call simple Z -orbit the reunion of isomorphism classes of an orbit for this action.We call a modified trace t on ideal Proj of projective objects of C a family of linear maps { t V : End C ( V ) → k } V ∈ Proj satisfying theconditions ∀ U, V ∈ Proj , ∀ W ∈ C , ∀ f ∈ Hom C ( U, V ) , ∀ g ∈ Hom C ( V, U ) , t V ( f ◦ g ) = t U ( g ◦ f ) ∀ f ∈ End C ( V ⊗ W ) , t V ⊗ W ( f ) = t V (ptr R ( f ))where ptr R ( f ) = (Id V ⊗ ←− ev W ) ◦ ( f ⊗ Id W ∗ ) ◦ (Id V ⊗ −→ coev W ) ∈ End C ( V ).We call d ( V ) = t V (Id V ) the modified dimension of V ∈ C . The formallinear combination Ω g = (cid:80) i d ( V i ) V i is called a Kirby color of degree g ∈ G if the isomorphism classes of the { V i } i are in one to one corre-spondence with the simple Z -orbits of C g .Recall that F is the Reshetikhin-Turaev functor from the ribboncategory R ib C of ribbon C -colored graphs to C (see [11]). The renor-malization of F is denoted by F (cid:48) (see [7]). NGOC PHU HA F Ω µ V = ∆ − Id V , F Ω µ V = ∆ + Id V Figure 1. V ∈ C g and Ω µ is a Kirby color of degree µ . Definition 2.1 ([3]) . Let ( G , × ) and ( Z, +) be two commutative groups.A k -linear ribbon category C is a pre-modular G -category relative to X with modified dimension d and periodicity group Z if (1) the category C has a G -grading { C g } g ∈ G , (2) the group Z has a free realization { ε t } t ∈ Z in C (where ∈ Gis the unit), (3) there is a Z -bilinear application G × Z → k × , ( g, t ) (cid:55)→ g • t suchthat ∀ V ∈ C g , ∀ t ∈ Z, c
V,ε t ◦ c ε t ,V = g • t Id ε t ⊗ V , (4) there exists X ⊂
G such that X − = X and G cannot be coveredby a finite number of translated copies of X , in other words ∀ g , ..., g n ∈ G , ∪ ni =1 ( g i X ) (cid:54) = G , (5) for all g ∈ G \ X , the category C g is semi-simple and its simpleobjects are in the reunion of a finite number of simple Z -orbits, (6) there exists a nonzero trace t on ideal Proj of projective objectsof C and d is the associated modified dimension, (7) there exists an element g ∈ G \ X and an object V ∈ C g suchthat the scalar ∆ + defined in Figure 1 is nonzero; similarly,there exists an element g ∈ G \ X and an object V ∈ C g suchthat the scalar ∆ − defined in Figure 1 is nonzero, (8) the morphism S ( U, V ) = F ( H ( U, V )) (cid:54) = 0 ∈ End C ( V ) , for allsimple objects U, V ∈ Proj , where H ( U, V ) = UV ∈ End C ( V ) . An example of a relative pre-modular G -category C is the categoryof finite dimensional weight modules over U Hq sl (2), see [3, Subsection6.3]. Note that in [3] the word relative modular is used instead of relative pre-modular . A relative pre-modular category is anomaly freeif ∆ + = ∆ − . In this case it is always possible to renormalize so that∆ + = ∆ − = 1. Next we consider a quantization of the super Liealgebra sl (2 |
1) from [8].2.2.
Quantum group U Hξ sl (2 | .Definition 2.2. Let (cid:96) ≥ be an odd integer and ξ = exp( πi(cid:96) ) . Thesuperalgebra U ξ sl (2 | is an associative superalgebra on C generated by NOMALY-FREE TQFTS FROM sl (2 |
1) 5 the elements k , k , k − , k − , e , e , f , f and the relations k k = k k ,k i k − i = 1 , i = 1 , ,k i e j k − i = ξ a ij e j , k i f j k − i = ξ − a ij f j i, j = 1 , ,e f − f e = k − k − ξ − ξ − , e f + f e = k − k − ξ − ξ − , [ e , f ] = 0 , [ e , f ] = 0 ,e = f = 0 ,e e − ( ξ + ξ − ) e e e + e e = 0 ,f f − ( ξ + ξ − ) f f f + f f = 0 . The last two relations are called the Serre relations. The matrix ( a ij ) is given by a = 2 , a = a = − , a = 0 . The odd generators are e , f . We define ξ x := exp( πix(cid:96) ), and denote { x } = ξ x − ξ − x . The algebra U ξ sl (2 |
1) has a structure of a Hopf algebra with thecoproduct, counit and the antipode given by (see [10])∆( e i ) = e i ⊗ k − i ⊗ e i i = 1 , , ∆( f i ) = f i ⊗ k i + 1 ⊗ f i i = 1 , , ∆( k i ) = k i ⊗ k i i = 1 , ,S ( e i ) = − k i e i , S ( f i ) = − f i k − i , S ( k i ) = k − i i = 1 , ,(cid:15) ( k i ) = 1 , (cid:15) ( e i ) = (cid:15) ( f i ) = 0 i = 1 , . We extend U ξ sl (2 |
1) to a superalgebra over C , denote by U Hξ sl (2 | h , h and the associated relations. Thismeans that U Hξ sl (2 |
1) is a C -superalgebra generated by k i , k − i , e i , f i and h i for i = 1 ,
2, and the relations in U ξ sl (2 |
1) plus the relations[ h i , e j ] = a ij e j , [ h i , f j ] = − a ij f j [ h i , h j ] = 0 , and [ h i , k j ] = 0for i, j = 1 , U Hξ sl (2 |
1) is a Hopf superalgebra where the coprod-uct ∆, the antipode S and the counit (cid:15) are determined as in U ξ sl (2 | h i ) = h i ⊗ ⊗ h i , S ( h i ) = − h i , (cid:15) ( h i ) = 0 i = 1 , . Set e = e e − ξ − e e , f = f f − ξf f . Denote B + = { e p (cid:48) e σ (cid:48) e ρ (cid:48) , p (cid:48) ∈ { , , ..., (cid:96) − } , ρ (cid:48) , σ (cid:48) ∈ { , }} , B − = { f ρ f σ f p , p ∈ { , , ..., (cid:96) − } , ρ, σ ∈ { , }} , B = { k s k s , s , s ∈ Z } and B h = { h t h t , t , t ∈ N } . NGOC PHU HA
We consider the quotients U = U ξ sl (2 | / ( e (cid:96) , f (cid:96) ), this superalgebra hasa Poincar´e-Birkhoff-Witt basis B + B B − and U H = U Hξ sl (2 | / ( e (cid:96) , f (cid:96) )which has a Poincar´e-Birkhoff-Witt basis B + B B h B − .3. Relative pre-modular G -category C H In this section we represent first the category C of nilpotent finitedimensional representations over U , then the category C H of nilpotentfinite dimensional representations over U H .3.1. Category C of weight modules over U . We consider the evencategory C of the nilpotent finite dimensional modules over U ξ sl (2 | U ξ sl (2 |
1) on which e (cid:96) = f (cid:96) = 0 and k , k are diagonalizable operators. If V, V (cid:48) ∈ C ,Hom C ( V, V (cid:48) ) is formed by the even morphisms between these two mod-ules (see [6]). Each nilpotent simple module is determined by the high-est weight µ = ( µ , µ ) ∈ C and is denoted V µ ,µ or V µ . Its highestweight vector w , , satisfies e w , , = 0 , e w , , = 0 ,k w , , = λ w , , , k w , , = λ w , , where λ i = ξ µ i for i = 1 , . For µ = ( µ , µ ) ∈ C we say that U ξ sl (2 | V µ is typical ifit is a simple module of dimension 4 (cid:96) . Other simple modules are saidto be atypical.The basis of a typical module is formed by vectors w ρ,σ,p = f ρ f σ f p w , , where ρ, σ ∈ { , } , ≤ p < (cid:96) . The odd elements are w , ,p and w , ,p ,others are even. The representation of typical U ξ sl (2 | V µ ,µ is determined by k w ρ,σ,p = λ ξ ρ − σ − p w ρ,σ,p ,k w ρ,σ,p = λ ξ σ + p w ρ,σ,p ,f w ρ,σ,p = ξ σ − p w ρ,σ,p +1 − ρ (1 − σ ) ξ − σ w ρ − ,σ +1 ,p ,f w ρ,σ,p = (1 − ρ ) w ρ +1 ,σ,p ,e w ρ,σ,p = − σ (1 − ρ ) λ ξ − p +1 w ρ +1 ,σ − ,p + [ p ][ µ − p + 1] w ρ,σ,p − ,e w ρ,σ,p = ρ [ µ + p + σ ] w ρ − ,σ,p + σ ( − ρ λ − ξ − p w ρ,σ − ,p +1 . where ρ, σ ∈ { , } and p ∈ { , , ..., (cid:96) − } .We also have V µ (cid:39) V µ + ϑ ⇔ ϑ ∈ ( (cid:96) Z ) . Remark 3.1.
The module V µ is typical if [ µ − p +1] (cid:54) = 0 ∀ p ∈ { , ..., (cid:96) − } ( µ (cid:54) = p − (cid:96) Z ∀ p ∈ { , ..., (cid:96) − } ) and [ µ ][ µ + µ + 1] (cid:54) = 0 ( µ (cid:54) = (cid:96) Z , µ + µ (cid:54) = − (cid:96) Z ) (see [1]) . NOMALY-FREE TQFTS FROM sl (2 |
1) 7
Relative pre-modular G -category C H . Let V µ ,µ be an objectof C . We define the actions of h i , i = 1 , V µ ,µ by h w ρ,σ,p = ( µ + ρ − σ − p ) w ρ,σ,p , h w ρ,σ,p = ( µ + σ + p ) w ρ,σ,p . With these actions V µ ,µ is a module over U H .We consider the even category C H of nilpotent finite dimensional U H -modules, that means e (cid:96) = f (cid:96) = 0, and for which ξ h i = k i asdiagonalizable operators with i = 1 ,
2. The categories C and C H arepivotal in which the pivotal structure is given by g = k − (cid:96) k − (see [8]).Thus V µ ,µ is a weight module of C H . A module in C H is saidto be typical if, seen as a U ξ sl (2 | V we denote V the same module with the opposite parity.We set G = C / Z × C / Z and for each µ ∈ G we define C Hµ as thesubcategory of weight modules which have their weights in the coset µ (modulo Z × Z ). So { C Hµ } µ ∈ G is a G -graduation (where G is anadditive group): let V ∈ C Hµ , V (cid:48) ∈ C Hµ (cid:48) , then the weights of V ⊗ V (cid:48) are congruent to µ + µ (cid:48) (modulo Z × Z ). Furthermore, if µ (cid:54) = µ (cid:48) thenHom C H ( V, V (cid:48) ) = 0 because a morphism preserves weights.It is shown that the category C H is braided, pivotal and have a twist(see [8]). Thus we have the proposition. Proposition 3.2. C H is a ribbon category. The pivotal structure is given by g = k − (cid:96) k − and the braiding isdetermined by R = ˇ RK where(1) ˇ R = (cid:96) − (cid:88) i =0 { } i e i ⊗ f i ( i ) ξ ! (1 − e ⊗ f )(1 − e ⊗ f ) , in which (0) ξ ! = 1 , ( i ) ξ ! = (1) ξ (2) ξ · · · ( i ) ξ , ( k ) ξ = − ξ k − ξ and(2) K = ξ − h ⊗ h − h ⊗ h − h ⊗ h . In addition, by [8, Theorem 3.17] the subcategory C Hα is semi-simplefor α ∈ G \ X where X = (cid:26) , (cid:27) × C / Z ∪ C / Z × (cid:26) , (cid:27) ∪ (cid:26) ( µ , µ ) : µ + µ ∈ (cid:26) , (cid:27)(cid:27) . By [8, Theorem 4.4], there exists a modified trace t on the ideal ofprojective modules of C H . It is checked that these datas satisfy theconditions of Definition 2.1 and we have the statement. Proposition 3.3. C H is pre-modular G -category relative to ( G , Z ) where G = C / Z × C / Z and Z = Z × Z × Z / Z . Furthermore, it isanomaly free, i.e., ∆ + = ∆ − = 1 . NGOC PHU HA Ω µ V i V j Figure 2.
Representation of the morphism f µij Note that the third component of the group Z sets up the parity ofmodules ε n , n ∈ Z in the realization of Z .4. Relative modular G -category C H In this section we show C H is relative modular G -category. Thiscategory allows one to construct an invariant of 3-manifolds which is amain ingredient of the constructions of ETQFTs (see [4]). Definition 4.1 ([4]) . A pre-modular G -category C relative to X withmodified dimension d and periodicity group Z is said a modular G -category relative to ( G , Z ) if it satisfies the modular condition, i.e., itexists a relative modularity parameter ζ ∈ C ∗ such that d ( V i ) f µij = (cid:40) ζ ( −−→ coev V i ◦ ←− ev V i ) if i = j if i (cid:54) = j for all µ, ν ∈ G \ X and for all i, j ∈ ν which V i , V j are not in the same Z -orbit, where f µij is the morphism determined by the C -colored ribbontangle depicted in Figure 2 under Reshetikhin-Turaev functor F . Theorem 4.2.
Category C H of nilpotent weight modules over U H ismodular G -category relative to ( G , Z ) . To prove the theorem we need the lemmas below.
Lemma 4.3.
For ν ∈ G \ X , let V i ∈ C Hν . Then any vector y in theimage of f µii is U H -invariant.Proof. Let V k ∈ C Hν , by [3, Lemma 4.9] we can do a handle-slide moveon the circle component of the graph representing f µii ⊗ Id V k to obtainthe equalities c W,V k ◦ ( f µii ⊗ Id V k ) = c − V k ,W ◦ ( f µ + νii ⊗ Id V k ) = c − V k ,W ◦ ( f µii ⊗ Id V k )where W = V i ⊗ V ∗ i . The braidings c W,V k , c − V k ,W : W ⊗ V k → V k ⊗ W are given by c W,V k = τ s ◦ R and c − V k ,W = R − ◦ τ s where R = ˇ RK , seeEquations (1) and (2).Let x (cid:54) = 0 be a weight vector of weight 0 of W and v ∈ V k be an evenweight vector of weight ν = ( ν , ν ), set y = f µii ( x ) ∈ W .Let W (cid:48) + be the vector space generated by { e i e i e i y | i + i + i > ≤ i ≤ (cid:96) − , ≤ i , i ≤ } , W (cid:48) − be the vector space generated NOMALY-FREE TQFTS FROM sl (2 |
1) 9 by { f i f i f i v | i + i + i > ≤ i ≤ (cid:96) − , ≤ i , i ≤ } , V (cid:48) + be the vector space generated by { e i e i e i v | i + i + i > ≤ i ≤ (cid:96) − , ≤ i , i ≤ } and V (cid:48) − be the vector space generated by { f i f i f i y | i + i + i > ≤ i ≤ (cid:96) − , ≤ i , i ≤ } . Becausethe weight of x is 0 then K ( y ⊗ v ) = y ⊗ v . Hence c W,V k ( y ⊗ v ) = τ s ◦ ˇ RK ( y ⊗ v )= v ⊗ y + ( ξ − ξ − ) f v ⊗ e y + f v ⊗ e y + f v ⊗ e y + W (cid:48) − ⊗ W (cid:48) + and, c − V k ,W ( y ⊗ v ) = R − ◦ τ s ( y ⊗ v ) = ( S ⊗ Id U H )( R )( v ⊗ y )= ( S ⊗ Id U H ) (cid:16) v ⊗ y + ( ξ − ξ − ) e v ⊗ f y − e v ⊗ f y − e v ⊗ f y + V (cid:48) + ⊗ V (cid:48) − (cid:17) = v ⊗ y − ( ξ − ξ − ) k e v ⊗ f y + k k e v ⊗ f y + k e v ⊗ f y + S (cid:16) V (cid:48) + (cid:17) ⊗ V (cid:48) − . Identify the two right hands of the equations above, on gets e y = f y =0 and e y = f y = 0. By the relations e f − f e = k − k − ξ − ξ − , e f + f e = k − k − ξ − ξ − , it implies that k i y = y for i = 1 , k i act as ξ h i and the weights of W are in Z × Z × Z / Z , we have that the eigenvaluesof k i are in ξ Z which does not contain − (cid:96) is odd). Thus k i y = y for i = 1 , y is an invariant vector of W . (cid:3) Recall that S (cid:48) ( V, W ) is the number complex determined by S (cid:48) ( V, W ) = (cid:42) VW (cid:43) where the bracket of a diagram T is defined by F ( T ) = < T > Id W with the simple module W . Following are some properties of S (cid:48) ( V, W )and the modified dimension d where d ( µ ) := d ( V µ ) (for details, see [8]). Lemma 4.4.
Let V µ be a typical module and V (cid:48) µ (cid:48) be a simple modulewith respective highest weight µ and µ (cid:48) . Then d ( µ ) = { µ + 1 } (cid:96) { (cid:96)µ }{ µ }{ µ + µ + 1 } = { α } (cid:96) { (cid:96)α }{ α }{ α + α } where ( α , α ) = ( µ − (cid:96) + 1 , µ + (cid:96) ) . S (cid:48) ( V µ , V (cid:48) µ (cid:48) ) := S (cid:48) ( µ, µ (cid:48) ) = ξ − α α (cid:48) − α α (cid:48) + α α (cid:48) ) { (cid:96)α (cid:48) }{ α (cid:48) }{ α (cid:48) + α (cid:48) }{ α (cid:48) } where ( α (cid:48) , α (cid:48) ) = ( µ (cid:48) − (cid:96) + 1 , µ (cid:48) + (cid:96) ) . d ( µ (cid:48) ) S (cid:48) ( µ, µ (cid:48) ) = d ( µ ) S (cid:48) ( µ (cid:48) , µ ) . Proof of Theorem 4.2.
By Proposition 3.3, the category C H is pre-modular G -category relative to ( G , Z ). Now we show that this categoryis a relative modular G -category. It is necessary to verify the relativemodularity condition. We consider the morphism f which representsby the diagram as in Figure 3. By the handle-slide the circle colored Ω µ V k V i V j Figure 3.
Representation of the morphism f Ω µ V k V i V j . = Ω µ + ν V k V i V j . = Ω µ + ν V k V i V j Figure 4.
Sliding of the circle colored by V k along theclosed component of f µij by V k along the circle of f µij and an isotopy we have two equalities givenby the diagrams as in Figure 4. It follows that S (cid:48) ( V k , V i ) f µij = S (cid:48) ( V k , V j ) f µ + νij for all V k ∈ C Hν . It implies f µ + νij = S (cid:48) ( V k , V i ) S (cid:48) ( V k , V j ) f µij = S (cid:48) ( V k , V i ) S (cid:48) ( V k , V j ) f µij for V k , V k ∈ C Hν . We denote the highest weights of V i , V j , V k and V k by ( ν + i , ν + i ) , ( ν + j , ν + j ) , ( ν + s , ν + s ) and ( ν + t , ν + t ) for 0 ≤ i , i , j , j , s , s , t , t ≤ (cid:96) −
1. By Lemma 4.4 we have S (cid:48) ( V k , V i ) = ξ − ν + s )( ν + i ) − ν + s )( ν + i )+( ν + s )( ν + i )) (cid:96)d ( V i ) ,S (cid:48) ( V k , V j ) = ξ − ν + s )( ν + j ) − ν + s )( ν + j )+( ν + s )( ν + j )) (cid:96)d ( V j ) ,S (cid:48) ( V k , V i ) = ξ − ν + t )( ν + i ) − ν + t )( ν + i )+( ν + t )( ν + i )) (cid:96)d ( V i ) ,S (cid:48) ( V k , V j ) = ξ − ν + t )( ν + j ) − ν + t )( ν + j )+( ν + t )( ν + j )) (cid:96)d ( V j ) . Hence S (cid:48) ( V k , V i ) S (cid:48) ( V k , V j ) = ξ − ν + s )( i − j ) − ν + s )( i − j )+( ν + s )( i − j )) d ( V j ) d ( V i ) ,S (cid:48) ( V k , V i ) S (cid:48) ( V k , V j ) = ξ − ν + t )( i − j ) − ν + t )( i − j )+( ν + t )( i − j )) d ( V j ) d ( V i ) . NOMALY-FREE TQFTS FROM sl (2 |
1) 11
We see that S (cid:48) ( V k , V i ) S (cid:48) ( V k , V j ) . S (cid:48) ( V k , V j ) S (cid:48) ( V k , V i ) = ξ − s − t )( i − j ) − s − t )( i − j )+( s − t )( i − j )) and the term − s − t )( i − j ) − s − t )( i − j ) + ( s − t )( i − j ))is determined by a symmetric bilinear non-degenerate B from ( Z /(cid:96) Z ) × ( Z /(cid:96) Z ) to Z /(cid:96) Z which has the matrix B = ( b ij ) × where b =0 , b = b = − b = −
4. It follows that for all i (cid:54) = j ∈ ( Z /(cid:96) Z ) it exists k (cid:54) = k ∈ ( Z /(cid:96) Z ) such that B ( i − j, k − k ) (cid:54) = 0. Thus forall i (cid:54) = j ∈ ν it exists k (cid:54) = k ∈ ν such that S (cid:48) ( V k , V i ) S (cid:48) ( V k , V j ) (cid:54) = S (cid:48) ( V k , V i ) S (cid:48) ( V k , V j ) ,it implies that f µij = 0 if i (cid:54) = j .If i = j we have f µii = f µ + νii for µ, ν ∈ G \ X . We see by Lemma 4.3that f µii ∈ End U H ( V i ⊗ V ∗ i ) factors through invariant vectors of W = V i ⊗ V ∗ i . As Hom U H ( V i ⊗ V ∗ i , C ) (cid:39) Hom U H ( C , V i ⊗ V ∗ i ) (cid:39) End U H ( V i ) (cid:39) C Id V i then these imply that two morphisms f µii and −−→ coev V i ◦ ←− ev V i areproportional, i.e., there is a λ ∈ C ∗ such that f µii = λ (cid:16) −−→ coev V i ◦ ←− ev V i (cid:17) . Next we compute λ from this equality. We consider the value F (cid:48) ofthe braid closure of the graphs associated with this equality. The valueassociated with f µii is F (cid:48) Ω µ V i V i = (cid:88) k F (cid:48) d ( V k ) V k V i V i = (cid:88) k F (cid:48) d ( V k ) V k V i V k V i = (cid:88) k F (cid:48) V k V i F (cid:48) V k V i = (cid:88) k d ( V i ) S (cid:48) ( V k , V i ) d ( V i ) S (cid:48) ( V ∗ k , V i )= (cid:88) k d ( V i ) S (cid:48) ( V k , V i ) S (cid:48) ( V ∗ k , V i ) where Ω µ = (cid:80) k ∈ µ d ( V k ) V k and the second equality by F (cid:48) ( L V L ) = d − ( V ) F (cid:48) ( L ) F (cid:48) ( L ). Furthermore, S (cid:48) ( V ∗ k , V i ) = ξ ν + s )( ν + i )+2(( ν + s )( ν + i )+( ν + s )( ν + i )) (cid:96)d ( V i ) , it implies that F (cid:48) Ω µ V i V i = (cid:96) − (cid:88) s ,s =0 (cid:96) = 1 . For the graph of −−→ coev V i ◦ ←− ev V i , the value F (cid:48) of its closure is F (cid:48) V i = F (cid:48) V i = d ( V i ) . Hence λ = d − ( V i ) and it follows that d ( V i ) f µii = −−→ coev V i ◦ ←− ev V i . (cid:3) We see that the relative modularity parameter ζ = ∆ − ∆ + = 1. By[4, Theorem 1.1], one gets the corollary. Corollary 4.5.
There exists a family of ETQFTs from category C H . Remark 4.6.
The anomaly free specificity of this theory implies thatthe TQFT does not depend on the framing of cobordisms nor on theLagrangian on surfaces (see [4] ). It produces linear (and not only pro-jective as in usual TQFT from quantum groups) representations of themapping class group.
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Hung Vuong University, Faculty of Natural Science, Nong Trang,Viet Tri, Phu Tho, Viet Nam
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