The Yang-Mills Measure in the SU(3) Skein Module
TTHE YANG-MILLS MEASURE IN THE SU (3) SKEIN MODULE
CHARLES FROHMAN AND JIANYUAN K. ZHONG
Abstract.
Let A (cid:54) = 0 be a complex number that is not a root of unity. Let M bea compact smooth oriented 3-manifold, the SU (3)-skein space of M , S A ( M ), is thevector space over C generated by framed oriented links (including framed orientedtrivalent graphs in M ) quotient by the SU (3)-skein relations due to Kuperberg. Fora closed, orientable surface F , we construct a local diffeomorphism invariant traceon S A ( F × I ). Introduction
Throughout this paper, three manifolds and surfaces will be compact and oriented.A framed oriented trivalent graph is a space that is homeomorphic to a closedregular neighborhood of an oriented trivalent graph embedded in an orientable surface,along with an embedding of that oriented graph in the space. As these are oriented,each edge of the graph carries a direction. In diagrams, we will just draw the graphand the reader can imagine its regular neighborhood running parallel to the graph inthe plane of the paper. We always have the same “side” of the neighborhood facingup. By a framed oriented link in a three-manifold M we mean an embedding ofsuch a space in M . Two framed oriented links are equivalent if there is an isotopyof M taking one to the other that preserves the orientations of the edges. We willalso work with relative framed oriented links. These are graphs that also have somemonovalent vertices but they are exactly the points of intersection of the graph withthe boundary of the manifold. Of course, the framed graph intersects the boundaryof M in arcs so that each arc has a monovalent vertex in its interior.Let A (cid:54) = 0 be a complex number so that if A is a root of unity then A = ±
1. When A (cid:54) = ±
1, we define [ n ] = A n − A − n A − A − , and when A = ±
1, we define [ n ] = n . Let [ n ]! = [ n ][ n − · · · [1] . Finally, (cid:20) nk (cid:21) = [ n ]![ k ]![ n − k ]! . If M is a compact oriented three-manifold let L be the set of equivalence classesof framed oriented links so that all the vertices are sources or sinks . That is, at each Date : February 19, 2004.
Key words and phrases. SU (3)-skein modules, triads, fusion, Yang-Mills measure. a r X i v : . [ m a t h . G T ] J un CHARLES FROHMAN AND JIANYUAN K. ZHONG vertex either all three edges point in or they all point out. It is worth noting that theempty link is included in this collection. Let C L denote the vector space having L asa basis. Let R A be the subspace of C L spanned by the following five skein relationsfrom Kuperberg [5]: • positive crossing − A + A − • negative crossing − A − + A • square − - • bubble − [2] • trivial component D − [3] D Definition 1.
The SU (3) skein space of M at A , denoted by S A ( M ) , is the quotientspace C L /R A . There is another skein relation that can be easily derived from these, which indicateschange of framing. = A . It is convenient to note that the skein relations for the two crossings, the change offraming and the trivial component generate all the skein relations. In the case that A = ±
1, this skein module has been studied by Adam Sikora [8]. At these values of A , crossings are irrelevant as the two crossing relations reduce to show that the skeinsare equal. Consequently there is a well defined product structure. If γ and β are twoframed oriented graphs whose vertices are sources and sinks in M , we perturb them sothat they are disjoint and take their union as the product. The product structure on S ± ( M ) is the one induced by this. Sikora constructs a natural homomorphism from S ± ( M ) onto the SL ( C ) characters of π ( M ) . The kernel of this homomorphism isthe nilradical of S ± ( M ). HE YANG-MILLS MEASURE IN THE SU (3) SKEIN MODULE 3 We will also use relative skein spaces. For such, choose a collection of arcs in theboundary of M along with a sign [+] or [ − ] for each arc. The relative skein spaceis the vector space spanned by equivalence classes of relative framed oriented linkswhose vertices are sources and sinks that intersect the boundary of M in those arcsso that if the sign of the arc is [+] then edge of the graph points into M and if thesign of the is [ − ] then the edge of the graph points out of M .An especially important class of relative skein spaces are cylinders over a disk,where we have indicated a family of arcs on the boundary of the disk. You can thinkof assigning plusses and minuses to the arcs, and either by enhancing the argumentsof Sikora or imitating the work of Kuperberg. The associated relative module isisomorphic to Inv( V p ⊗ V ∗ n ), where V is the fundamental representation of SL C and V ∗ is its dual, and p is the number of positive arcs and n the number of negativearcs.If M = F × [0 , F and using the blackboard framing.In this case S A ( F × [0 , S A ( F ).In section 2, we recall some related results in SU (3) skein and study the SU (3)-skein modules of the solid torus S × D , S × S and for the connected sum of two3-manifolds. We list some results as follows. When A (cid:54) = 0 and A is not a root ofunity,(1) S A ( S × D ) has a countable basis indexed by the set of all ordered pairs ofnonnegative integers.(2) S A ( S × S ) = C ∅ , i.e., S A ( S × S ) is generated by the empty framed link.(3) S A ( M M ) ∼ = S A ( M ) ⊗ S A ( M ). This says that the SU (3)-skein moduleof the connected sum of two 3-manifolds is isomorphic to the tensor product of the SU (3)-skein modules of the manifolds.(4) From (2) and (3), we conclude that the SU (3)-skein module of the connectedsum of g copies of S × is also generated by the empty skein. S A ( g S × S ) = C ∅ . In sections 3, 4, we define and study the Yang-mills measure in a handlebody andon a closed surface. 2.
Basics in SU (3) -skein theory Related results from Ohtsuki and Yamada [7] —Magic Elements.Definition 2.
A magic element of type ( n, is inductively defined by the followingformula: = CHARLES FROHMAN AND JIANYUAN K. ZHONG n = n−1 − [ n − n ] n−1 n−2 n−1 The following diagrams are called a left-Y and a right-Y: , Properties of the magic element of type ( n, : (1) When attached a left-Y to the right side or a right-Y to the left side, the magicelement of type ( n,
0) vanishes.(2) The magic element of type ( n,
0) absorbs any magic elements of type ( m, m ≤ n . n mn−m = mn−m n = n Definition 3.
A magic element of type ( n, m ) is defined by the following formula: n nm m = min( n,m ) (cid:88) k =0 ( − k (cid:20) nk (cid:21) (cid:20) mk (cid:21)(cid:20) n + m + 1 k (cid:21) k kn−km−kn nm m We illustrate the left-U and right-U as follows: , Properties of the magic element of type ( n, m ): When attached a left-Y or a left-U tothe right side, or attached a right-Y or a right-U to the left side, the magic elementof type ( n, m ) vanishes.2.2. Coloring a trivalent graph with magic elements.
The coloring of an ori-ented edge by a pair of nonnegative integers ( n, m ) is by replacing the edge inthe graph by the magic element of type ( n, m ): (n, m) = n nm m Then (n, m) = (m, n) A vertex with acceptable labels becomes a triad with three edges colored by (ac-ceptable) nonnegative integer pairs ( n, m ), ( r, s ) and (p,q). Here we illustrate a triad
HE YANG-MILLS MEASURE IN THE SU (3) SKEIN MODULE 5 with indicated choice of orientations of the edges: (n, m) (r, s)(p, q) A triad represents a skein element in the relative skein space of the disk with ( n + s + q )input points and ( m + r + p ) output points. Since there are possibly many differentways that strands can intertwine in the middle, a triad with edges colored by ( n, m ),( r, s ) and (p,q) is not uniquely defined. Therefore we introduce a label by an a ∗ insidea circle to represent a specific intertwining of strands in the middle of a triad andindicate it as (n, m) (r, s)(p, q)a * In the case that A is either ± U q ( sl )) (or U ( sl )when A = ± V be the fundamental representation of U q ( sl ) and let V ∗ be itsdual. Let V p,q be the highest weight irreducible representation in V ⊗ p ⊗ V ∗⊗ q . Thereare invariant tensors in V ⊗ V ⊗ V and V ∗ ⊗ V ∗ ⊗ V ∗ that correspond to the two triva-lent vertices. There are invariant pairings V ⊗ V ∗ → C , and V ∗ ⊗ V → C , that can beused to “stitch” two trivalent vertices together along an edge so that each “web” in adisk, that is an embedded graph with trivalent vertices in the interior of the disk andmonovalent vertices on the boundary, so that the trivalent vertices are sources andsinks, corresponds to an invariant tensor in the tensor product of copies of V and V ∗ corresponding to choosing a basepoint on the boundary of the disk and keeping trackof the arrows going in and out as you go around the disk. Modding out by the skein re-lations corresponding to removing trivial simple closed curves, bubbles and four-sidedregions yields a vector space that is isomorphic to the space of invariants. There is afurther refinement where you group bunches of edges together to form a clasped webspace. Some of these “clasped web” spaces correspond to S A ( D , ( n, m ) , ( r, s ) , ( p, q )),the relative skein space generated by the triads attached with the three magic el-ements of types ( n, m ), ( r, s ) and ( p, q ). We call ( n, m ) , ( r, s ) , ( p, q ) an admissible(acceptable) coloring of a vertex if S A ( D , ( n, m ) , ( r, s ) , ( p, q )) (cid:54) = 0. From now on, wewill only consider admissible triads. There are three conclusions that we need to drawfrom this work.(1) The relative skein S A ( D , ( n, m ) , ( r, s ) , ( p, q )) is up to cyclic permutation canon-ically isomorphic to Inv( V n,m ⊗ V r,s ⊗ V p,q ). Since dim( V m,n ) ≤ mn , we see CHARLES FROHMAN AND JIANYUAN K. ZHONG that there is a polynomial p ( m, n, r, s, p, q ) so thatdim( S A ( D , ( n, m ) , ( r, s ) , ( p, q ))) ≤ p ( m, n, r, s, p, q ) . (2) The pairing S A ( D , ( n, m ) , ( r, s ) , ( p, q )) ⊗ S A ( D , ( q, p ) , ( s, r ) , ( m.n )) → C cor-responding to gluing the two disks together to form a sphere is nondegenerateas it corresponds to pairing Inv( V m,n ⊗ V r,s ⊗ V p,q ) with its dual. We can expressthis pairing in a diagram theoretic fashion as: a(cid:13) b(cid:13) * ** (n, m)(r, s)(p, q) Note that S A ( D ) = C ∅ , so it induces a pairing <, > on S A ( D , ( n, m ) , ( r, s ) , ( p, q )) ⊗ S A ( D , ( q, p ) , ( s, r ) , ( m, n )) → C by < α, β > = f ( α, β )where f ( α, β ) is the complex number such that α ⊗ β = f ( α, β ) ∅ in S A ( D ). Inthis sense, we have S A ( D , ( n, m ) , ( r, s ) , ( p, q )) ∗ = S A ( D , ( q, p ) , ( s, r ) , ( m, n )).Following [2], we choose bases { a i } for the space S A ( D , ( n, m ) , ( r, s ) , ( p, q )and dual bases { b i } for S A ( D , ( n, m ) , ( r, s ) , ( p, q )) ∗ , so that a b (n, m)(r, s)(p, q)i j = δ ji When one of the labels is (0 ,
0) then depending on the direction of the othertwo arrows, the other two labels are either the same ( n, m ) or its dual. In thiscase the skein of the disk with two clasps is one dimensional so everythingcan be written as a multiple of the skein s obtained by filling in the diskwith straight lines from one clasp to the other. Surprisingly, our choice ofnormalization leads to the peculiar realization that if our dual bases are chosenas a = αs and b = βs then α ∗ β = m,n .(3) The skein space S A ( D , ( m, n ) , ( r, s ) , ( p, q ) , ( u, v )) is isomorphic to the resultof “stitching” the sum below along the ( k, l ) and ( l, k ) factors. ⊕ S A ( D , ( m, n ) , ( r, s ) , ( k, l )) ⊗ S A ( D , ( l, k ) , ( p, q ) , ( u, v )) , HE YANG-MILLS MEASURE IN THE SU (3) SKEIN MODULE 7 where the sum is over all ( k, l ) so that Inv( V m,n ⊗ V r,s ⊗ V k,l ) is nonzero andInv( V l,k ⊗ V r,s ⊗ V u,v ) is nonzero. Using the dual bases chosen above we getthe fusion formula [2]: ( m , n ) ( m , n ) 1 2 1 2 = (cid:88) ∆ k,l aib ( m , n ) 1 1( m , n ) 1 1 ( m , n ) 2 2 ( m , n ) 2 2 ( k, l ) i . where the sum is over all admissible triples ( m , n ) , ( m , n ) , ( k, l ) and dualbases a i and b i .Let P n,m be the closure of the magic element of type ( n, m ) in the solid torus S × D : P n,m = n nmm As we show the solid torus, it is the cylinder over S × [0 , S × [0 ,
1] into the plane R induces a corresponding inclusion of cylinders. Let ∆ n,m be the complex scalar multiple of P n,m by writing P n,m = ∆ n,m ∅ in S A ( R ) = C ∅ byincluding S A ( S × [0 ,
1] into S A ( R ).The following identities hold:(1) From [7], when m, n are nonnegative integers, ∆ n,m = [ n +1][ m +1][ n + m +2] / [2].When at least one of m, n is a negative integer, we define ∆ n,m = 0. Note that if A isa real number and is not a root of unity, then ∆ n,m ≥ m, n . CHARLES FROHMAN AND JIANYUAN K. ZHONG (2) a(cid:13)i(cid:13)b(cid:13)i(cid:13) (a, b) (k, l)(m, n)(p, q) = δ p,qm,n ∆ m,n (m, n where a i and b i are dual bases for the admissible triple ( m, n ) , ( a, b ) , ( k, l ). Proof. (I) When ( m, n ) (cid:54) = ( p, q ), we can prove that the skein element on the left handside is zero. This eventually follows from the non-convexity of the basis of clasped webspace of Kuperberg [5]. To explain, we consider the clasped web space W ( C ) where C is given by the sequence [(+ · · · +) n ( − · · · − ) m (+ · · · +) p ( − · · · − ) q ] (the subscriptindicates the number of plusses and minuses in the sequence. When ( m, n ) (cid:54) = ( p, q ),the clasped web space W ( C ) is zero, as there will be a minimal cut path with lowerweight separating the clasps [(+ · · · +) n ( − · · · − ) m ] and (+ · · · +) p ( − · · · − ) q ] whichcauses convex clasps.(II) When ( m, n ) = ( p, q ), we prove the skein element on the left hand side is ascalar multiple of the magic element of type ( m, n ).According to the Lemma 3.3 of [7], if D is a diagram in the disk with 2 n + 2 m boundary points with neither biangles nor squares as shown: D =Then D satisfies at least one of the following three conditions:(i) There is a left-Y or a left-U attached to the left side,(ii) There is a right-Y or a right-U attached to the right side,(iii) The diagram D is n + m parallel lines from the left side to the right side.We can rewrite the middle part of the skein diagram of the given identity as alinear sum of diagrams fitting in the above lemma, then all three possible cases willgive a multiple of the magic element of type ( m, n ). Note that cases (i) and (ii) willcontribute 0 when attached to the magic element of type ( m, n ).(III) To find the scalar multiple, we close both sides, only when a i and b i are dualbases, the closure of the left hand side is nonzero and equals 1, the closure on the HE YANG-MILLS MEASURE IN THE SU (3) SKEIN MODULE 9 right hand side of the magic element of type ( m, n ) contributes ∆ m,n . Therefore thescalar multiple is m,n , the identity holds. (cid:3) (3) Let (k,l) be a pir of nonnegative integers such that the triples ( m , n ) , ( m , n ) , ( k, l )and ( m , n ) , ( m , n ) , ( k, l ) are admissible, then the collection of elements ( m , n ) 1 1( m , n ) ( m , n )( m , n ) 2 2 ( k, l )3 3 44a bi j over all such ( k, l ) and all basis elements a i , b j of the corresponding triads, forms abasis for the skein space S A ( D , ( m , n ) , ( m , n ) , ( m , n ) , ( m , n )).On the other hand, over all pairs of nonnegative integers ( g, h ) such that the triples( m , n ) , ( m , n ) , ( g, h ) and ( m , n ) , ( m , n ) , ( g, h ) are admissible and basis ele-ments c p , d q of the corresponding triads, the collection of elements ( m , n ) 1 1( m , n ) ( m , n )( m , n ) 2 2 3 43 4cdpq ( g , h ) also forms a basis for the skein space S A ( D , ( m , n ) , ( m , n ) , ( m , n ) , ( m , n )). (4) There is a change of bases on S A ( D , ( m , n ) , ( m , n ) , ( m , n ) , ( m , n )): ( m , n ) 1 1( m , n ) ( m , n )( m , n ) 2 2 ( k, l )3 3 44a bi j = (cid:88) (cid:26) ( m , n ) , ( m , n ) ( g, h )( m , n ) ( m , n ) ( k, l ) a i , b j , c p , d q (cid:27) ( m , n ) 1 1( m , n ) ( m , n )( m , n ) 2 2 3 43 4cdpq ( g , h ) where the summation is over all bases c p , d q and admissible triples ( m , n ) , ( m , n ) , ( g, h )and ( m , n ) , ( m , n ) , ( g, h ).Similarly, we have a pairing on S A ( D , ( m , n ) , ( m , n ) , ( m , n ) , ( m , n )) ⊗ S A ( D , ( n , m ) , ( n , m ) , ( n , m ) , ( n , m )) → C induced by the bilinear form ofattaching skein elements along the boundary of D through the inclusion into S A ( D ). ( m , n ) 1 1( m , n )( m , n )( m , n ) 2 2 3 344 Definition 4.
We define
Tet (cid:26) (( m , n ) , ( m , n ) ( g, h )( m , n ) ( m , n ) ( k, l ) a i , b j , c p , d q (cid:27) HE YANG-MILLS MEASURE IN THE SU (3) SKEIN MODULE 11 to be the complex multiple of writing the following skein element as a multiple of theempty skein ∅ in S A ( D ) = C ∅ . cdpq ( g , h )( m , n ) 1 1( m , n ) ( m , n )( m , n ) 2 2 ( k, l )3 3 44a bi j Theorem 1. (cid:26) ( m , n ) , ( m , n ) ( g, h )( m , n ) ( m , n ) ( k, l ) a i , b j , c p , d q (cid:27) = Tet (cid:26) ( m , n ) , ( m , n ) ( g, h )( m , n ) ( m , n ) ( k, l ) a i , b j , c p , d q (cid:27) ∆ g,h The proof of this theorem is similar to the computation in [6].
Theorem 2. If A > and A is not a root of unity, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Tet (cid:26) ( m , n ) , ( m , n ) ( g, h )( m , n ) ( m , n ) ( k, l ) a i , b j , c p , d q (cid:27) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:112) ∆ k,l ∆ g,h Proof.
The key is using the change of bases identity twice. ( m , n ) 1 1( m , n ) ( m , n )( m , n ) 2 2 ( k, l )3 3 44a bi j = (cid:88) (cid:26) ( m , n ) , ( m , n ) ( g, h )( m , n ) ( m , n ) ( k, l ) a i , b j , c p , d q (cid:27) ( m , n ) 1 1( m , n ) ( m , n )( m , n ) 2 2 3 43 4cdpq ( g , h ) = (cid:88) α ∗ ( m , n ) 1 1( m , n ) ( m , n )( m , n ) 2 2 ( k, l )3 3 44a bi j where α ∗ = (cid:26) ( m , n ) , ( m , n ) ( g, h )( m , n ) ( m , n ) ( k, l ) a i , b j , c p , d q (cid:27) (cid:26) ( m , n ) ( m , n ) ( k, l )( m , n ) ( m , n ) ( g, h ) a i , b j , c p , d q (cid:27) Therefore, (cid:88) (cid:26) ( m , n ) , ( m , n ) ( g, h )( m , n ) ( m , n ) ( k, l ) a i , b j , c p , d q (cid:27) (cid:26) ( m , n ) , ( m , n ) ( g, h )( m , n ) ( m , n ) ( k, l ) a i , b j , c p , d q (cid:27) = 1i.e., (cid:88) Tet (cid:26) ( m , n ) , ( m , n ) ( g, h )( m , n ) ( m , n ) ( k, l ) a i b j c p d q (cid:27) ∆ k,l ∆ g,h = 1We observe that each term in the summation is positive, hence0 ≤ Tet (cid:26) ( m , n ) , ( m , n ) ( g, h )( m , n ) ( m , n ) ( k, l ) a i b j c p d q (cid:27) ∆ k,l ∆ g,h ≤ (cid:3) Corollary 1.
When
A > and A is not a root of unity, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Tet (cid:26) ( m , n ) , ( m , n ) ( g, h )( m , n ) ( m , n ) ( k, l ) a i , b j , c p , d q (cid:27) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:112) ∆ k,l HE YANG-MILLS MEASURE IN THE SU (3) SKEIN MODULE 13 and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Tet (cid:26) ( m , n ) , ( m , n ) ( g, h )( m , n ) ( m , n ) ( k, l ) a i , b j , c p , d q (cid:27) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:112) ∆ g,h Proof.
This follows from the fact that ∆ g,h ≥ k,l ≥ g, h ) and ( k, l ). (cid:3) Some fundamental examples.
Assume A is not a root of unity. Proposition 1. [7] S A ( S × D ) has a basis given by the collection { P n,m | n, m are nonnegative integers } . Theorem 3. S A ( S × S ) = C ∅ . Proof. As S × S can be obtained from the solid torus S × D by adding a 2-handle.There is an epimorphism S A ( S × D ) → S A ( S × S ) induced by embedding S × D into S × S . Adding a 2 handle results in adding relations to the generators. Herewe prove it suffices to consider only the following sliding relation: L = L the equality holds in S A ( S × S ), where L is any skein element in S A ( S × D ). Weonly need to consider the sliding relation on the generators P n,m of S A ( S × D ). From[7], we have the following skein relation: n(cid:13)m(cid:13) m(cid:13)n(cid:13) = ( A n +2 m +6 + A − n +2 m + A − n − m − ) n nm m While the product of P n,m with a trivial component is [3] P n,m = ( A + A − + 1) P n,m ,we conclude that ( A n +2 m +6 + A − n +2 m + A − n − m − − A − A − − P n,m = 0 in S A ( S × S ). On the other hand, it’s easy to obtain the following skein relationsimilar to the above: nm mn = ( A m +2 n +6 + A − m +2 n + A − m − n − ) n nm m Then ( A m +2 n +6 + A − m +2 n + A − m − n − − A − A − − P n,m = 0 in S A ( S × S ). When A is not a root of unity and m, n are not both zero, one can prove that ( A n +2 m +6 + A − n +2 m + A − n − m − − A − A − −
1) and ( A m +2 n +6 + A − m +2 n + A − m − n − − A − A − −
1) are not both zero. Therefore P n,m = 0 in S A ( S × S ) when n, m are notboth zero. While P , = ∅ doesn’t involve the sliding relation, it survives. Hence S A ( S × S ) = < ∅ > . (cid:3) Theorem 4. S A ( M M ) = S A ( M ) ⊗ S A ( M )The proof follows the same outline as [3]. Corollary 2. S A ( g S × S ) = C ∅ . The Yang-Mills measure in S A ( F × [0 , with ∂F (cid:54) = ∅ Let F be a compact oriented surface and I = [0 , A is not a root of unity.We denote the skein algebra of S A ( F × I ) by S A ( F ) to emphasize that the algebrastructure depends on F . Notice that F × I is a handlebody. If you choose a family K of proper arcs on F that cut it down to a disk, then K × I is a family of disks that cut F × I into a ball. The double of F × I , denoted by D ( F × I ), is the result of gluing twocopies of F × I together using the identity map on their boundary. The disks K × I in each copy are glued together to form a system of spheres in D ( F × I ) that cut itdown to a punctured ball. Therefore D ( F × I ) is homeomorphic to a connected sumof copies of S × S . From the Preliminaries in 2.4, S A ( D ( F × I )) = C ∅ . This inducesa linear functional YM : S A ( F × I ) → C by the inclusion of F × I into D ( F × I ),i.e., if α ∈ S A ( F × I ) ⊂ S A ( D ( F × I )), we can write α = f ( α ) ∅ for some complexnumber f ( α ) in S A ( D ( F × I )), then YM ( α ) = f ( α ) . Proposition 2. YM ( α ∗ β ) = YM ( β ∗ α ) Proof.
If you remove ∂F × I from F × I , then the double of the resulting objectis homeomorphic to ˚ F × S where the product structure coincides with the product˚ F × I on each half. Also F × I ⊂ ˚ F × S ⊂ D ( F × I ) , we can perturb representations of α and β so that they miss ∂F . Now it is clear that α ∗ β and β ∗ α are isotopic in ˚ F × S . (cid:3) Let Γ be an oriented trivalent spine of F . An admissible coloring of Γ is by attachingthe magic elements of types ( m i , n i ) along the edges and acceptable labels at thevertices, such a coloring of Γ is an element of S A ( F ). Theorem 5.
The admissible colorings of Γ form a spanning set for S A ( F ) .Proof. Let H g be the handlebody F × I , let D be a separating meridian disk of H g : .... D HE YANG-MILLS MEASURE IN THE SU (3) SKEIN MODULE 15 Let V D = [ − , × D be a regular neighborhood of D in H g , V D can be projectedinto a disk D p = [ − , × [0 , α be a framed link in H g in general position to V D , let α (cid:48) = α ∩ V D , α (cid:48) = kl In the next lemma, we show that we can write α (cid:48) as a linear sum of skein elements in V D which have the magic elements in the middle such as the following: diagram diagram kk ll where k, l are some nonnegative integers.By induction on the number of separating meridian discs of H g , we can write α as alinear sum of skein elements in S A ( H g ) which have the magic elements at the regularneighborhood of each separating disk. Note that such an element corresponds to anadmissible coloring of the trivalent spine of H g . Therefore, the admissible coloringsof Γ spans S A ( H g ). (cid:3) Lemma 1.
An element in the relative skein module of the cylinder with n parallelstrands going to the right and m parallel strands going to the left can be written as alinear sum of elements in the form diagram diagram kk ll with ( k + l ) ≤ ( m + n ) .Proof. We proceed by induction on n + m .(1) When n + m = 1, i.e, n = 1 , m = 0 or m = 1 , n = 0, it is trivial.When n + m = 2, there are three cases: (i) n = m = 1; (ii) n = 2 , l = 0; (iii) m = 2 , n = 0. We illustrate cases (i) and (ii) by the following. Case (iii) is similar tocase (ii). = + [1][2] = + [1][2] (2) Assume the result is true for ( n + m ) ≤ k for some natural number k , we provethe result is true for the case ( n + m ) = ( k + 1). By induction, the result is true forall nonnegative integers n , m .When n + m = k + 1, without loss of generality, we can assume that m ≥
1. Bythe induction assumption, the result is true for n + ( m −
1) = k , i.e., we can considerthe part with n parallel strands going to the right and ( m −
1) parallel strand goingto the left and write it as a linear sum of elements of the assumed form with the sizeof the magic elements in the middle of ≤ k ; now it suffices to prove that the followingelement is a linear sum of the assumed form, ab where ( a + b ) = k .By the definition of the magic element of type ( a, b ), ab = min( a,b ) (cid:88) i =0 ( − i (cid:20) ai (cid:21) (cid:20) bi (cid:21)(cid:20) a + b + 1 i (cid:21) a-i aa b-ib bi i1 Note that all terms in the summation corresponding to i ≥ k times, so by the induction assumption, these can be writtenas the sum of the assumed forms. Therefore we only need to worry about the firstterm which corresponds to i = 0, i.e., ab1 Now use the definition of the magic element of type (0 , b ), ab1 = a(b+1) + [ b ][ b + 1] b b-1 ba Similarly, the second term intersects the separating disk at ( a + b ) = k times, so weonly need to consider the first term on the right hand side, again we use the definition HE YANG-MILLS MEASURE IN THE SU (3) SKEIN MODULE 17 of the magic element of type ( a, b + 1): a(b+1) = a(b+1) − min( a,b ) (cid:88) i =1 ( − i (cid:20) ai (cid:21) (cid:20) b + 1 i (cid:21)(cid:20) a + ( b + 1) + 1 i (cid:21) a-i aa b+1-i(b+1) (b+1)i i We observe again that each term in the summation has i ≥
1, so the skein elementsintersect the separating disk at ≤ k times; by the induction assumption, these can bewritten as a linear sum of elements which have the magic elements at the middle ofsize ≤ k . Therefore we conclude that the result is true for n + m = k + 1. (cid:3) Lemma 2. If k is a properly embedded arc in F , then S A ( F ) is spanned by trivalentcolored graphs so that each graph intersects k in one transverse point of intersectionin the interior of one of its edges. The preceeding lemmas give a method of computing the Yang-Mills measure. Let k be a proper arc in F . By Lemma 1 any skein can be written as a colored trivalentgraph intersecting k in a single point of transverse intersection in the interior of anedge. By the argument in Theorem 3 the Yang-Mills measure of such a graph is zerounless that edge carries the label (0 , F into a family of disks. Using Lemma 2 repeatedly write the skein as a colored trivalentgraph that intersects each of the arcs at most once in a point of transverse intersectionin the interior of an edge. Throw out all terms where a graph carries a nonzero labelon an edge intersecting one of the arcs. Next erase the edges and renormalize to takeinto account the peculiarity of the normalization. Finally, evaluate the invariant ofthe remaining skeins in the disks.We formalize this: Proposition 3. Locality
Let F be a compact oriented surface and let k be a properarc. Let F (cid:48) be the result of cutting F along k . If s is a skein that is represented bya sum of colored trivalent graphs that each intersects k in at most a single point oftransverse intersection in the interior of an edge. Let s be sum of all terms wherethe graph is either disjoint from k or intersects in an edge labeled (0 , . The skein s corresponds to a skein s (cid:48) in the image of the inclusion of S A ( F (cid:48) ) → S A ( F ) . Then, YM ( s ) = YM ( s (cid:48) ) . (cid:3) The Yang-Mills measure on a closed surface
In this section F will be a closed oriented surface of genus greater than 1. Furtherwe suppose that A is a positive real number not equal to 1. We prove that there is a linear functional YM : S A ( F ) → C . that is a trace in the sense that YM ( α ∗ β ) = YM ( β ∗ α ) for all α, β ∈ S A ( F ).If we remove an open disk with nice boundary from F we get a compact surfacewith one boundary component F (cid:48) which is a subsurface of F . The inclusion map i : F (cid:48) → F induces a surjective map, S A ( F (cid:48) ) → S A ( F ) . Let ∂ ( m,n ) be the skein that is the result of coloring a framed knot that is parallel to ∂F with the ( m, n ) type magic element. We define YM : S A ( F ) → C . If α ∈ S A ( F )then choose α (cid:48) to be a skein in S A ( F (cid:48) ) that gets mapped onto α by the inclusion. Wedenote the Yang-Mills measure on F (cid:48) by YM (cid:48) F and define,(*) YM ( α ) = lim N →∞ m + n ≤ N (cid:88) ( m,n ) ∆ m,n YM F (cid:48) ( ∂ ( m,n ) ∗ α (cid:48) ) . We need to prove two things. The first is that the series we gave above is convergentand the second is to prove that is independent of the choice of α (cid:48) . To prove convergencewe just need to prove it converges on a collection of skeins that spans S A ( F ). Luckilywe have such a family, admissibly colored spines.Let s c denote a trivalent spine of a compact oriented surface F with one boundarycomponent that has been colored admissibly. If the spine has v i vertices and e j edges,then v i = − χ ( F ); e j = − χ ( F ) . where χ ( F ) is the Euler characteristic of F .We need a global estimate on YM ( ∂ m,n ∗ s c ). Let ( p j , q j ) be the label on the j thedge of s c and let a i be the skein in the i th vertex. To compute this we fuse along thehandles, that look like this ( m, n )( p j , q j )( m, n )We compute by fusing one ( m, n ) strand with the ( p i , q i ) strand, then fusing with theother ( m, n ) strand, and throwing out everything that the central edge is not labeled(0 , V ( n,m ) in V ( m,n ) ⊗ V ( p i ,q i ) .We call the set of pairs of skeins that appear in the two vertices along this edge, A j .We then need to erase the (0 ,
0) edges, which entails dividing by ∆ m,n for each edge.
HE YANG-MILLS MEASURE IN THE SU (3) SKEIN MODULE 19 We are then just computing the value of sum of the products of some tetrahedralcoefficients. The result is (cid:88) A j (cid:89) v i Tet (cid:26) ( m, n ) ( m, n ) ( m, n )( p i , q i ) ( p i , q i ) ( p i , q i ) a j , b j , c j , d j (cid:27) The a j , b j , c j , d j are skeins in vertices coming from the fusions along the edges. Thenumber of terms is less than or equal to a polynomial evaluated on the labels, and thesize of each term in the sum is less than (cid:81) i √ ∆ m,n . Therefore we have the followingProposition. Proposition 4.
There is a polynomial in variables ( m, n ) that only depends on thecolors assigned to the edges, p ( m, n ) so that YM ( ∂ ( m,n ) ∗ s c ) ≤ p ( m, n )∆ − χ ( F ) m,n , where χ ( F ) is the Euler characteristic of F . Proposition 5.
The formula given by the equation (*) for the Yang-Mills measureconverges.Proof.
The proof is by comparison with the series (cid:80) m,n p ( m,n )∆ / m,n . We know from itsformula that ∆ m,n grows exponentially in m and n . Hence the series we just mentionedconverges. By the estimate given by the previous proposition and since the eulercharacteristic of F (cid:48) is ≤ − YM ( s c ) arebounded in absolute value by the series we just gave. (cid:3) The final step of the argument is to show that YM ( α ) is independent of the choiceof α (cid:48) . Since we can pass from any skein α (cid:48) to any other skein α (cid:48)(cid:48) that is sent to α under S A ( F (cid:48) ) → S A ( F )by handleslides. By fusing, we can reduce this to check this is true for the result ofsliding one string of a trivalent colored spine of F (cid:48) across the boundary disk. Withoutloss of generality, let α (cid:48) = s c be the trivalent spine of a compact oriented surface F with one boundary component that has been colored admissibly, let ∂ m,n be theframed knot corresponding to ∂F oriented with the boundary orientation from F colored with the ( m, n ) magic element. Locally ∂ ( m,n ) ∗ α (cid:48) looks like ( p , q) ( p, q − 1) ( p, q )( m, n )( m, n )1 Let α (cid:48)(cid:48) be the skein obtained from α (cid:48) = s c by sliding one strand over the added disk,locally the diagram ∂ ( m,n ) ∗ α (cid:48)(cid:48) looks like ( p , q) ( p, q − 1) ( p, q )( m, n )1 1( m, n ) In the following, we will show that the Yang-Mills measure defined by the equation(*) is well-defined by provinglim N →∞ m + n ≤ N (cid:88) ( m,n ) ∆ m,n ( YM F (cid:48) ( ∂ ( m,n ) ∗ α (cid:48) ) − YM F (cid:48) ( ∂ ( m,n ) ∗ α (cid:48)(cid:48) )) = 0 . Lemma 3. m + n ≤ N (cid:88) ( m,n ) ∆ m,n (( ∂ ( m,n ) ∗ α (cid:48) ) − ( ∂ ( m,n ) ∗ α (cid:48)(cid:48) )) = m + n = N (cid:88) ( m,n ) ∆ m,n (∆ m +1 ,n s − ∆ n,m +1 s ) HE YANG-MILLS MEASURE IN THE SU (3) SKEIN MODULE 21 where s illustrated below is the skein element which is almost the same as ∂ ( m,n ) ∗ α (cid:48) except where it’s shown, s = ( p , q) ( p, q − 1) ( p, q )( m, n )1 1( m , n ) ( m , n) ( m + 1 , n ) s is the skein element which is almost the same as ∂ ( m,n +1) ∗ α (cid:48)(cid:48) except where it’sshown, s = ( p , q) ( p, q − 1) ( p, q )1 1( m , n )( m , n + 1 ) ( m , n + 1 ) ( m , n + 1 ) . The proof follows from the following lemma.
Lemma 4. m + n ≤ N (cid:88) ( m,n ) ∆ m,n ( − ) = m + n = N (cid:88) ( m,n ) ∆ m,n (∆ m +1 ,n − ∆ m,n +1 ) Proof.
We consider the tensor product of the magic element of type ( m, n ) and themagic element of type (1 , V m,n ⊗ V , = V m,n − ⊕ V m − ,n +1 ⊕ V m +1 ,n . Similarly, V m,n ⊗ V , = V m − ,n ⊕ V m +1 ,n − ⊕ V m,n +1 . These give the corresponding fusion identities in the SU (3)-skein. When we applythese fusion identities to the left hand side of the identity in the lemma, almost allterms are canceled except the terms left on the right hand side. (cid:3) Theorem 6. lim N →∞ m + n = N (cid:88) ( m,n ) ∆ m,n ∆ m +1 ,n YM F (cid:48) ( s ) = 0 Proof.
To compute the Yang-Mills measure of the skein s , we fuse to isolate thevertices. Notice that fusing s will require two more cross cuts than that of ∂ ( m,n ) ∗ α (cid:48) .After throwing out everything that the central edge is not labeled (0 ,
0) and erasingthe (0 ,
0) edges, the Yang-Mills measure of s is the product of(I) (cid:88) A (cid:48) j m,n ∆ m +1 ,n Tet (cid:26) ( m, n ) ( m, n ) ( m + 1 , n )(1 ,
0) ( p, q −
1) ( p, q ) a j , b j , c j , d j (cid:27) × T et (cid:26) ( m, n ) ( m + 1 , n ) ( m, n )(1 ,
0) ( p, q ) ( p, q − a (cid:48) j , b (cid:48) j , c (cid:48) j , d (cid:48) j (cid:27) (the a i , b j , c j , d j and a (cid:48) i , b (cid:48) j , c (cid:48) j , d (cid:48) j are skeins in vertices coming from the fusions alongthe edges.)with the standard product of fusion on ∂ ( m,n ) ∗ α (cid:48) ,(II) (cid:88) A j (cid:89) v i of α (cid:48) Tet (cid:26) ( m, n ) ( m, n ) ( m, n )( p i , q i ) ( p i , q i ) ( p i , q i ) a i , b j , c j , d j (cid:27) First the product in (I) is less than or equal to (cid:88) A j m,n ∆ m +1 ,n (cid:112) ∆ m,n (cid:112) ∆ m +1 ,n ≤ p (cid:48) ( m, n )∆ − m,n ∆ − m +1 ,n HE YANG-MILLS MEASURE IN THE SU (3) SKEIN MODULE 23 where p (cid:48) ( m, n ) is another polynomial depending on the colors ( m, n ).Secondly, by a previous proposition, there exist polynomials p ( m, n ) in variables( m, n ) that depend only on the colors assigned to the edges so that the above standardproduct is less than or equal to p ( m, n )∆ − χ ( F ) m,n . Therefore YM F (cid:48) ( s ) ≤ p (cid:48) ( m, n ) p ( m, n )∆ − m,n ∆ − m +1 ,n ∆ χ ( F ) m,n , andlim N →∞ m + n = N (cid:88) ( m,n ) ∆ m,n ∆ m +1 ,n YM F (cid:48) ( s ) = lim N →∞ m + n = N (cid:88) ( m,n ) p (cid:48) ( m, n ) p ( m, n )∆ − + χ ( F ) m,n ∆ − m +1 ,n = 0 (cid:3) Corollary 3. lim N →∞ m + n = N (cid:88) ( m,n ) ∆ m,n ∆ n,m +1 YM F (cid:48) ( s ) = 0 Corollary 4. lim N →∞ m + n ≤ N (cid:88) ( m,n ) ∆ m,n ( YM F (cid:48) ( ∂ ( m,n ) ∗ α (cid:48) ) − YM F (cid:48) ( ∂ ( m,n ) ∗ α (cid:48)(cid:48) )) =lim N →∞ m + n = N (cid:88) ( m,n ) ∆ m,n (∆ m +1 ,n YM F (cid:48) ( s ) − ∆ n,m +1 YM F (cid:48) ( s ) = 0 References [1] A. K. Aiston and H. R. Morton,
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Quantum invariants of -manifolds associated with classical simpleLie algebras , Internat. J. Math., 4 (1993) 323-358. Department of Mathematics, The University of Iowa, Iowa-City, Iowa 52242
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