Formula Quantistica

8/12/2019 Formula Quantistica

1/28

a r X i v : m a t h - p h / 0 6 1 1 0 8 4 v 2 1 5 J u n 2 0 1 2

Chern-Simons theory and the quantum Racah formula

Sebastian de Haro and Atle Hahn

June 18, 2012

Abstract

We generalize several results on Chern-Simons models on S 1 in the so-called torus gaugewhich were obtained in [ 33] (= arXiv:math-ph/0507040) to the case of general (simply-connectedsimple compact) structure groups and general link colorings. In particular, we give a non-perturbativeevaluation of the Wilson loop observables corresponding to a special class of simple but non-triviallinks and show that their values are given by Turaevs shadow invariant. As a byproduct we obtaina heuristic path integral derivation of the quantum Racah formula.

1 Introduction

In 1988 E. Witten succeeded in dening, on a physical level of rigor, a large class of new 3-manifoldand link invariants with the help of the heuristic Chern-Simons path integral, cf. [ 58]. Later a rigorousdenition of these invariants was given, cf. [ 45, 44] and part I of [52]. The approach in [ 45, 44] is basedon the representation theory of quantum groups 1 and uses surgery techniques on the base manifold. Arelated approach is the so-called shadow world approach (cf. [41, 54, 53] and part II of [52]), whichalso works with quantum groups but replaces the use of surgery operations by certain combinatorialarguments leading to nite state sums.

It is an open problem (cf., e.g., p. 2 in [ 25] and Problem (P1) in [33]) how the rigorous approachesusing quantum groups are related to Wittens path integral approach. This problem is interesting byitself. Moreover, one can expect that the solution of this problem will lead to some progress towardsthe solution of one of the central open problems in the eld, namely the question if/how one can makerigorous sense of the path integral expressions used in the heuristic treatment in [58] (cf. Sec. 7 belowfor additional comments).

The results in [33], which were obtained by extending the work in [ 12, 13, 14, 31] in a suitable way,suggest that the key for establishing a direct relationship between the CS path integral and the twoquantum group approaches mentioned above is the so-called torus gauge xing procedure, introducedin [12] for the study of CS models on base manifolds M of the form M = S 1 . Indeed, already in [ 12] itwas demonstrated that in the torus gauge setting the evaluation of the Wilson loop observables (WLOs)of special links consisting exclusively of vertical loops naturally leads to the S-matrix expressions onthe right-hand side of the so-called Verlinde formula, cf. expression ( 11) below and Remark 4 in Sec.6. In [33] it was then shown how to treat the case of general links within (a suitably modied versionof) the torus gauge setting. Moreover, it was shown that in the special case G = SU (2) the evaluation

of the Wilson loop observables of loops without double points naturally leads to the gleam factors andthe summation over (admissible) area colorings present in Turaevs formula for the shadow invariant(cf. Eq. (23) below). In the present paper we will generalize the results in [33] to general (simply-connected simple compact) groups G and to links with arbitrary colors, i.e. equipped with arbitraryrepresentations (and not only the fundamental representation as in [ 33]). As a result we will be able todemonstrate that within the torus gauge setting also the fusion coefficients (i.e. the numbers N ijl in Eq.(19)) in Turaevs formula for the shadow invariant appear naturally when links without double pointsare studied.

We mention here that Turaevs shadow invariant also appears in the evaluation of a purely two-dimensional quantum eld theory, namely q -deformed Yang-Mills theory on a Riemannian surface [20].The connection of the latter with Chern-Simons on S 1-bundles over , of which S 1 is a special case,was developed in [19, 20, 1, 21, 22, 15]. The algebraic lattice formulation of q -deformed two-dimensional

1 in fact, the approach in [ 45, 44, 52] is more general, cf. Remark 2 below

1
http://arxiv.org/abs/math-ph/0611084v2http://arxiv.org/abs/math-ph/0611084v2http://arxiv.org/abs/math-ph/0611084v2http://arxiv.org/abs/math-ph/0611084v2http://arxiv.org/abs/math-ph/0611084v2http://arxiv.org/abs/math-ph/0611084v2http://arxiv.org/abs/math-ph/0611084v2http://arxiv.org/abs/math-ph/0611084v2http://arxiv.org/abs/math-ph/0611084v2http://arxiv.org/abs/math-ph/0611084v2http://arxiv.org/abs/math-ph/0611084v2http://arxiv.org/abs/math-ph/0611084v2http://arxiv.org/abs/math-ph/0611084v2http://arxiv.org/abs/math-ph/0611084v2http://arxiv.org/abs/math-ph/0611084v2http://arxiv.org/abs/math-ph/0611084v2http://arxiv.org/abs/math-ph/0611084v2http://arxiv.org/abs/math-ph/0611084v2http://arxiv.org/abs/math-ph/0611084v2http://arxiv.org/abs/math-ph/0611084v2http://arxiv.org/abs/math-ph/0611084v2http://arxiv.org/abs/math-ph/0611084v2http://arxiv.org/abs/math-ph/0611084v2http://arxiv.org/abs/math-ph/0611084v2http://arxiv.org/abs/math-ph/0611084v2http://arxiv.org/abs/math-ph/0611084v2http://arxiv.org/abs/math-ph/0611084v2http://arxiv.org/abs/math-ph/0611084v2http://arxiv.org/abs/math-ph/0611084v2http://arxiv.org/abs/math-ph/0611084v2http://arxiv.org/abs/math-ph/0611084v2http://arxiv.org/abs/math-ph/0611084v2http://arxiv.org/abs/math-ph/0507040http://arxiv.org/abs/math-ph/0507040http://arxiv.org/abs/math-ph/0611084v2


2/28

Yang-Mills has been worked out for real q and not for q being a root of unity [18]. Although we will notfurther develop the connection to this two-dimensional theory in this paper, we note that the intermediateexpressions we obtain in our evaluation of the Chern-Simons path integral are those of q -deformed two-dimensional Yang-Mills. In turn, the path integral formulation of the simpler two-dimensional quantumeld theory may be helpful in dening the Chern-Simons path integral on non-trivial bundles over [15, 11].

The paper is organized as follows. In Subsec. 2.1 we rst recall some important concepts andconstruction from Lie theory. In Subsec. 2.2 we then introduce some concepts from Conformal FieldTheory and the theory of affine Lie algebras which played a role in [58]. In Sec. 3 we reformulateTuraevs shadow invariant for manifolds of the form S 1 using the notation from Sec. 2. In Secs.4.14.3 we recall some of the results obtained in [12, 13, 14, 31, 32, 33] on Chern-Simons models on S 1in the torus gauge and in Subsec. 4.4 we then generalize the calculations in [33] for the WLOs for linkswithout double points to the case of general (simple simply-connected compact) groups G and arbitrarylink colorings. In Sec. 5 we show that the nite state sums appearing in Sec. 4 are equivalent to the statesums in the shadow invariant. In other words, the values of the WLOs obtained in Sec. 4 agree exactlywith the values obtained by applying the shadow invariant to the corresponding links, cf. Eq. ( 82). InSec. 6 we then show that by reversing the order of arguments used in Secs. 4 and 5 one can obtain apath integral derivation of the so-called quantum Racah formula (cf. Eq. ( 15) below). We conclude the

paper with a brief outlook in Sec. 7.

2 Algebraic preliminaries

2.1 Concepts from classical Lie theory

Let G be a simply-connected and simple compact Lie group and g its Lie algebra. Moreover, let T be amaximal torus of G and t the Lie algebra of T . (We will keep G and T xed for the rest of this paper).

(, ) denotes the Killing metric on g normalized such that after making the identication t and t with the help of ( , ) we have (, ) = 2 if is a long real root.

We set r := dim( t ) = rank( g). t : g t will denote the ( , )-orthogonal projection and t the

(, )-orthogonal complement of t in g . R t will denote the set of real roots associated to ( g, t ) and R the set of real coroots, i.e. R

is given by R := { | R} t where := 2(, ) . Let t denote the real weight lattice

associated to ( g , t ), i.e. is given by

:= { t | ( ) Z for all R} (1)

R t will denote the lattice generated by the real coroots.

A Weyl chamber is a connected component of t \ R H where H := 1(0). A Weyl alcove

(or affine Weyl chamber) is a connected component of the set 2 t reg := t \ R ,kZ H ,k whereH ,k := 1 (k).

Let W denote the Weyl group (associated to g and t ), i.e. the group of isometries of t = t

generatedby the orthogonal reections on the hyperplanes H , R, dened above. W aff will denote theaffine Weyl group, i.e. the group of isometries of t = t generated by the orthogonal reections onthe hyperplanes H ,k , R, k Z , dened above 3 . For W aff we will denote the sign of bysgn( ).

In the sequel let us x a Weyl chamber C . Let P denote the unique Weyl alcove which is contained in C and has 0 t on its boundary.

Let R + denote the set of positive roots, i.e. R+ := { R | (, x ) 0 for all x C}, and let +denote the set of dominant weights, i.e. + := C .

2 note that in [31] we used the notation t reg instead of t reg .3 Equivalently, one can dene W aff as the group of isometries of t = t generated by W and the translations associated

to the coroot lattice R

2


3/28

For + let denote the (up to equivalence) unique irreducible complex representation of Gwith highest weight and the character corresponding to . The multiplicity of the globalweight associated to in will be denoted by m (), i.e. we have

(exp( b)) =

m ()e2i (,b ) for all b t (2)

will denote the half-sum of the positive roots and the unique long root in the Weyl chamber C .The dual Coxeter number cg of g is given by4

cg = 1 + ( , ) (3)

For each + we setC 2 () := ( , + 2 ) (4)

i.e., C 2 () is the second Casimir element (w.r.t. to the inner product ( , )) corresponding to theirreducible representation of g with highest weight .

For + let + denote the weight conjugated to and + the weight conjugated to

after applying a shift by . More precisely, is given by + = + .

Remark 1 Let I t denote the integral lattice, i.e. I := ker(exp | t ). From the assumption that Gis simply-connected it follows that I coincides with the lattice R generated by the real coroots so theweight lattice associated to ( g, t ) coincides with the weight lattice I of (G, T ) given by I := { t |(x) Z for all x I }.

2.2 Some concepts from CFT, the theory of affine Lie algebras, and thetheory of quantum groups

Let us x k N (the level).

We set

k+ := { + | (, ) k} (5)

Let Isom( t ) denote the group of isometries of the Euclidean vector space ( t , (, )) and let i :Isom( t ) Isom( t ) denote the automorphism of Isom( t ) given by

i( )(b) = ( k + cg ) ((b + )/ (k + cg )) (6)

for all b t and Isom( t ). We set 5

W k := i(W aff ) Isom( t ) (7)

(the ( -shifted) quantum Weyl group corresponding to the level k) and

sgn( ) := sgn( i 1( )) for W k

Let C , S , and T be the k+ k+ matrices with complex entries given by

C := , (8a)

T := eiC 2 ( )

k + c g eic12 , (8b)

S := i|R + |

(k + cg )r/ 2 |/ R |

12

wW sgn(w)e

2 ik + c g

( + ,w (+ )) (8c)

4 note that cg = 1 + ( , ) = 12 (, + 2 ) = 12 C 2 (). If we had normalized the Killing form ( , ) such that the long roots

have length 1 we would have cg = C 2 (), i.e. cg would then be the Casimir element associated to the adjoint representation.5 W k coincides with the subgroup of Isom( t ) which is generated by the orthogonal reections on the -shifted hyperplanes

H , R+ , and the hyperplane {y t | (y, ) = k + cg } = {x t | (x, ) = k + 1 }, thus W k is the same as thegroup W 0 in [47].

3


4/28

for all , k+ , cf. Eqs. (14.216), (14.217), and (14.229) in [23] and compare also Sec. II.3.9 in[52] where a slightly different convention is used 6 .

We remark that the factor eic12 with c := dim( g) k(k+ cg ) appearing in Eq. ( 8b) is not really

essential for the present paper. In particular, the denition of |X L | in Eq. (16) below and Theorem5.1 below (and also the computations in Sec. 6 below) are not affected 7 if we omit this factor. Theadvantage of including the factor e ic12 in Eq. (8b) is that Eq. ( 9b) below holds in the strict senseand not only in the projective sense. This point simplies the computations in our examples inSec. 3.2.One can prove (cf. Eqs. (10.206), (10.216), (14.228) and Exercise 14.14 in [ 23] and 8 Sec. II.3.9 in[52]) that

S 2 = C, (9a)(ST )3 = C (9b)

In particular, S is invertible.

For k+ we set

dim := S 0S 00

()=R +

sin ( + , )

k+ cg

sin (, )k+ cg(10)

Here () follows from S 0S 00 = A()( + )

A()( ) and the relation9 (b) = A()(b) where

A(b )(b) :=wW

sgn(w)e2i (b ,w b)

(b) :=R +

(ei (b) e i (b) ) =R +

2i sin((, b)) .

for all b, b t .

For , , k+ we dene the fusion coefficients N and N

by

N :=k+

S S S S 0

(11)

andN := N (12)

Observe that Eq. ( 9a) implies N 0 = .

Let us motivate the use of the term fusion coefficients above. Let g denote the (non-twisted) affineLie algebra corresponding to gC := g R C (cf. Eq. (14.13) in [23]) and let N be the fusion coefficientsof the modular tensor category based on the integrable representations of g at level k. Similarly, let N be the fusion coefficients in the modular tensor category constructed in [5, 6] using the representationtheory of the quantum group U q(gC ) with 10

q := e2 i

k + c g

6 the matrix C is called J in [52]. Moreover, the matrix S in [52] differs from the matrix S in Eq. (8c) by a multiplicativeconstant D, cf. the Notes at the end of Chap. II in [52]

7 cf. step ( ) in the proof of part iii) of Lemma 1 below8 in Sec. II.3.9 in [52] the expression D 1 appears. The computations in [52] imply Eq. ( 9b ) provided that D 1 =

eic

4 = ( eic12 )3 . In view of the results in [24] this is exactly what one expects

9 cf. part iii) of Theorem 1.7 in Chap. VI of [17]10 cf., e.g., [ 47, 49]. Sometimes in the literature a different convention is used for the denition of U q (gC ), which leads to

the formula q := ei

D ( k + c g ) where D is the quotient of the square lengths of the long and the short roots of g (cf., e.g., thesecond page of the introduction in [48])

4


5/28

According to the famous Verlinde(-Moore-Seiberg) formula we have

N = N (13a)

cf., e.g., [27]. Similarly, according to the quantum group analogue of the Verlinde formula (cf., e.g.,Theorem 4.5.2 in Chap. II in [52]) we have11

N = N (13b)

Moreover, in [6, 48] it was proven that

N = W k

sgn( )m ( ( )) (14)

The last formula can be considered to be a quantum analogue of the classical Racah formula. Following[48] we will call this formula the (abstract) quantum Racah formula and the formula

N = W k

sgn( )m ( ( )) , (15)

which follows from Eq. ( 14) and (13b) will be called12 the elementary quantum Racah formula.

3 The shadow invariant for links in S 1

3.1 Denition

Let be an oriented surface, let L = ( l1 , l2 , . . . , l n ), n N , be a sufficiently regular link in S 1 , andlet ljS 1 resp. l

j denote the projection of the loop lj onto the S

1-component resp. -component of theproduct S 1 . L can be turned into a framed link by picking for each loop lj the standard framingdescribed in Sec. 4 c) in [53] (this framing was called vertical framing in [ 33]). We also assume thateach loop lj is colored with an element j of k+ .

We set D(L) := ( DP (L), E (L)) where DP (L) denotes the set of double points of L, i.e. the set of points p where the loops lj , j n, cross themselves or each other, and E (L) the set of curves in into which the loops l1

, l2

, . . . , l n

are decomposed when being cut in the points of DP (L). Clearly,

D(L) can be considered to be a nite (multi-)graph. We set \ D(L) := \ ( j arc( lj )). We assume that

the set of connected components of \ D (L) has only nitely many elements Y 0 , Y 1 , Y 2 , . . . , Y , N ,which we will call the faces of \ D (L).As explained in [53] one can associate in a natural way a number gl( Y t ) Z , called gleam of Y t , toeach face Y t (for an explicit formula for the gleams in the special cases that will be relevant for us latersee Eq. (21) below). We call X L := ( D(L), (gl(Y t )t )0 t ) the shadow of L.

Let g E (L) be a xed edge of the graph D(L). Note that, as each loop lj is oriented, g is an orientedcurve in . On the other hand, as was assumed to be oriented, each face Y {Y 0 , Y 1 , Y 2 , . . . , Y } isan oriented surface and therefore also induces an orientation on its boundary Y .

There is a unique face Y , denoted by Y +g (resp. Y g ) in the sequel, such that arc( g) Y and,additionally, the orientation on arc( g) described above coincides with (resp. is opposite to) the orientationwhich is obtained by restricting the orientation on Y to g. In other words: Y +g and Y g are the two 13

faces that touch the edge g, and Y +

g (resp. Y

g ) is the face lying to the left (resp. to the right) of g, cf. Fig. 1.A mapping : {Y 0 , Y 1 , Y 2 , . . . , Y } k+ will be called an area coloring of X L (with colors in k+ )

and the set of all such area colorings will be denoted by col(X L ). We can now dene the shadow invariant| | by14

|X L | :=col( X L )

|X L |1 |X L |

2 |X L |

3 |X L |

4 (16)

11 of course Eqs. ( 13a ) and (13b) imply N = N . This is not surprising since the two modular tensor categoriesmentioned above can be shown to be equivalent, cf. [24]

12 Since for the derivation of ( 15) we used both the Verlinde formula ( 13b ) and the (abstract) quantum-Racah formulathis name might be a little bit misleading. We could equally well call ( 15) the elementary Verlinde formula

13 note that if Assumption 2 below is not fullled then possibly Y +g = Y

g , so in this case there is actually only one suchface

14 this coincides with the denition in [52] up to an overall normalization factor which will be irrelevant for our purposes

5


6/28

Y g

Y +g

g

Figure 1:

where

|X L |1 =Y

(dim((Y ))) (Y ) (17)

|X L |2 =Y

(v(Y ) )gl( Y ) (18)

|X L |3 =gE (L )

N (Y g )co(g)(Y +g )

. (19)

where N ijl and dim( ) are as in Subsec. 2.2, where co(g) denotes the color associated to the edge g (i.e.co(g) = i where i n is given by arc( li ) g) and where we have set v := T (here T is, of course,the T -matrix from Subsec. 2.2).

|X L |4 is dened in terms of quantum 6j-symbols associated to U q(gC ), cf. Chap. X, Sec. 1.2 andChap. XI, Sec. 6.3 in [52]. In view of Assumption 1 below and the consequences that this assumptionhas, cf. Eq. ( 22) below, the precise denition of |X L |4 in the general case will not be relevant in thepresent paper.

For the rest of this paper, we will restrict ourselves to the special situation where L also fullls thefollowing two assumptions.

Assumption 1 The colored link L has no double points, i.e. the projected loops l1

, l2

, . . . , l n

are non-intersecting Jordan loops in .

Assumption 2 Each lj is 0-homologous.

Assumptions 1 and 2 have the following consequences:

For each j n the set \ arc( lj ) has exactly two connected components. In the sequel R+j (resp.

Rj ) will denote the connected component to the left (resp. to the right) of lj , i.e. R

+j (resp.

Rj ) is the unique connected component containing Y +

j (resp. Y

j ) where we have set

Y j := Y

l j(20)

(i.e. Y

j = Y

g where g = lj ).

= n, i.e. \ ( j arc( lj )) has n + 1 connected components Y 0 , Y 1 , . . . , Y n

For each Y {Y 0 , Y 1 , Y 2 , . . . , Y n } we have

gl(Y ) =j with arc( l j )Y

wind( ljS 1 ) sgn(Y ; lj ) (21)

where wind( ljS 1 ) is the winding number of the loop ljS 1 and where sgn( Y ; l

j ) is given by

sgn(Y ; lj ) :=1 if Y R+j 1 if Y Rj

6


7/28

According to the general denition of the shadow invariant in Chap. X, Sec. 1.2 in [52]. Assumption1 implies |X L |4 = 1 so Eq. ( 16) reduces to

|X L | =col( X L )

|X L |1 |X L |

2 |X L |

3 (22)

Vertical framing for a loop lj in S 1 (cf. the rst paragraph of the present subsection) isequivalent to what was called horizontal framing in Subsec. 5.2 in [33]

Remark 2 1. The shadow invariant dened in [ 52] is more general than what we have dened hereabove. Our denition is the special case of Turaevs shadow invariant where the underlying modulartensor category is the one coming from the representation theory of the quantum groups U q(gC ),cf. Sec. 2.2 above.

2. In the special case G = SU (2) one has N ijk {0, 1} for all i , j, k k+ so |X L |

3 {0, 1} for each col(X L ). Let us call col(X L ) admissible iff |X L |3 = 1 and set coladm (X L ) := { col(X L ) | admissible }. Then we can rewrite Eq. ( 16) in the form

|X

L | :=

col adm (X L ) |X

L |

1 |X

L |

2 |X

L |

4 (23)

If one compares this formula with Eqs. (5.7) and (5.8) in [ 53] (and the two equations beforeTheorem 6.1 in [53]) it is easy to see that the shadow invariant that was dened in [ 53] (and usedin [33]) is the special case of the shadow invariant in the present paper which one obtains by takingG = SU (2).

3.2 Some examples

Example 1 Let = S 2 and let L = (( l1 , l2 , l3), (,, )) be a colored link in S 1 such that wind( liS 1 ) =1 for all i = 1 , 2, 3 and such that the projection of L onto the surface looks like in the following gure.Let, for i {1, 2, 3}, Y i denote the face enclosed by l i and let Y 0 denote the remaining face. Clearly,we have (Y i ) = 1 for i {1, 2, 3} and (Y 0 ) = 2 2g 3 = 1 and gl(Y i ) = 1 for i {1, 2, 3} andgl(Y 0) = 3. So we obtain

Y Y Y1 32

Y0

Figure 2:

|X L | =1 2 3 0k+

dim( 1 )dim( 2 )dim( 3)(dim( 0)) 1 N 0 1 N 0 2 N

0 3 T 1 1 T 2 2 T 3 3 T

3 0 0

= T T T

T 300 S 200N . (24)

In deriving the last line, we used the following equation three times

k+

dim( ) T N = 1T 00 S 00

(T ST ) (25)

(Eq. ( 25) follows from (9) and ( 11)).

7


8/28

Y

Y

Y

Y

0

1

2

3

Figure 3:

Example 2 Let again = S 2 and let L = (( l1 , l2 , l3), (,, )) be a colored link in S 1 such thatwind( liS 1 ) = 1 for all i = 1 , 2, 3 and such that the projection of L onto the surface looks like in Fig. 3.Then we have (Y 1) = (Y 3 ) = 1, (Y 0 ) = (Y 2) = 0 and gl( Y 1 ) = gl( Y 3) = 1, gl( Y 0) = 2, gl(Y 2) = 0where the faces Y 0 , Y 1 , Y 2 , Y 3 are given as in Fig. 3. One obtains

|X L | = 1 2 3 0k+

dim( 1) dim( 3 ) N 2 1 N 0 2 N

0 3 T 1 1 T 3 3 T

2 0 0 . (26)

The sums over 1 and 3 can be performed right away using twice Eq. ( 25). We get

|X L | = T T T 200 S 200 2 0

T 2 2 T 10 0 S 2 S 0 N 0 2 . (27)

Now observe that

2 0

T 2 2 T 1 0 0 S 2 S 0 N

0 2

()=

1T 0

T 1 0 0 T 1 S 0 S

10 S S

1S 0

()=

T T

N (28)

Here step () follows from Eq. (11) and ST S = T 1ST 1 (which in turn follows from Eq. ( 9)) and step() follows from Eq. ( 11) and ST 1S 1 = T ST . From Eqs. ( 27) and ( 28) we nally get

|X L | = T 2

T 200 S 200N . (29)

The next example generalizes the rst two examples above.

Example 3 Let = S 2 and let L = ( l1 , l2, . . . , l m + n +1 ) be a colored link in S 1 consisting of m + n +1loops with colors 1 , 2 , , m , , and 1 , 2 , , n such that wind( liS 1 ) = 1 for all i = 1 , 2, . . . , m + n +1and such that the projection of L onto the surface looks like in the following gure.

Figure 4:

Let X 1 , X 2 , , X m be the faces encircled by the rst m loops with colors 1 , 2 , , m . Loop lm +1has color and encircles the rst m loops. The face inside this loop (subtracting the (closure of the)faces of the m loops from it) is X 0 . Outside this group of loops are n more loops with colorings1 , 2 , , n encircling the faces Y 1 , Y 2 , , Y n , respectively.

8


9/28

The Euler characters are as follows:

(Y 1 ) = (Y 2) = = (Y n ) = (X 1) = (X 2) = (X m ) = 1 (30a) (Y 0 ) = 1 n (30b) (X 0 ) = 1 m (30c)

and the gleams:

gl(X 1) = gl( X 2) = = gl( X m ) = gl( Y 1 ) = gl( Y 2) = = gl( Y n ) = 1 , (31a)gl(Y 0) = n 1 (31b)gl(X 0) = m + 1 . (31c)

The value of |X L | is obtained by summing over all possible colorings of the faces. Using the summationvariables 0 , 1 , , n , 0 , 1 , . . . , m k+ we obtain

|X L | = 0 n , 0 m

dim1dim2 n dim 1 dim m (dim 0 )1 n (dim )1 m

N 0 1 1 N 0 2 2 N

0 n n N

01 1 N

0 1 1 N

0 m m N

0 0

T 1 1 T 2 2 T n n T n 10

0

T 1 1 T 2 2 T m m T 1 m 0

0

(32)

We use (25) to remove all of the fusion coefficients except one:

k+

dim T N = 1

T 00 S 00(T ST ) (33)

therefore:

|X L | =0 0

(dim 0)1 n (dim) 1 m N 0 T n 1 0 0 T

1 m 0 0

1(T 00 S 00 )n + m

(T ST )1 0 (T ST )2 0 (T ST )n 0 (T ST ) 1 0 (T ST ) 2 0 (T ST ) m 0 (34)

Collecting the common factors of T this equals:

|X L | = T 1 1 T n n T 1 1 T m m(T 00 S 00 )n + m 0 0(dim 0 )1 n (dim 0 )1 m

N 0 0 T 10 0 T 0 0 S 1 0 S n 0 S 1 0 S m 0 (35)

By lling in the denition of the fusion coefficients and the quantum dimensions dim( ) we can rewritethis as

|X L | = T 1 1 T n n T 1 1 T m m

T n + m00 S 200 0 0 (S 0 0 )

1 n (S 0 0 )1 m S

10 S 0 S

S 0

T 10 0 T 0 0 S 1 0 S n 0 S 1 0 S m 0 (36)

For example, in the special case m = 2 , n = 1 we have

|X L | = T 1 1 T 1 1 T 2 2

T 300 S 200 0 0 S 10 S 0 S T 10 0 T 0 0 S 1 0 S 1 0 S 2 0

S 0 0S 0. (37)

which using ST 1S 1 = T ST can be reduced to:

|X L | =T 2 1 1 T 1 1 T 2 2

T 300 S 200 0

1S 0 0S 0

S 0 S S 1 S 1 0 S 2 0 T T 0 0 . (38)

Example 4 Note that X L is also dened if L is the empty link . In this case one has

|X | =k+

(dim )2 2g . (39)

where g is the genus of the surface .

9


10/28

4 State sums from the Chern-Simons path integral in the torusgauge

4.1 Chern-Simons models

Let M be an oriented compact 3-manifold and A the space of smooth g

-valued 1-forms on M . Withoutloss of generality we can assume that the group G xed in Subsec. 2.1 above is a Lie subgroup of U (N ),N N . The Lie algebra g of G can then be identied with the obvious Lie subalgebra of the Lie algebrau(N ) of U (N ).

The Chern-Simons action function S CS associated to M , G, k (with k as in Subsec. 2.2) is given by

S CS (A) = k4 M Tr( A dA + 23 A A A), A Awith Tr := c Tr Mat( N, C ) where the normalization constant c is chosen15 such that 16

(A, B ) = 14 2 Tr( A B ) A, B g (40)

holds, cf. e.g. [46, 50] where the same normalization is used.

Example 5 If G = SU (N ) then c = 1 so in this case Tr coincides with Tr Mat( N, C ) .

From the denition of S CS it is obvious that S CS is invariant under (orientation-preserving) diffeo-morphisms. Thus, at a heuristic level, we can expect that the heuristic integral (the partition function)Z (M ) := exp(iS CS (A))DA is a topological invariant of the 3-manifold M . Here DA denotes theinformal Lebesgue measure on the space A .

Similarly, we can expect that the mapping which maps every sufficiently regular colored link L =((l1 , l2 , . . . , l n ), ( 1 , 2 , . . . , n )) in M to the heuristic integral (the Wilson loop observable associatedto L)

WLO( L) := 1Z (M ) i Tr i P exp l i A exp( iS CS (A))DA (41)

is a link invariant (or, rather, an invariant of colored links). Here we have set i := i i n, (cf. Subsec.2.1), Tr i is the trace in the representation i , and P exp l i A denotes the holonomy of A around theloop li .

Let us now consider the special case M = S 1 where is a closed oriented surface. Due to thewell-known equivalence of Wittens invariants and the Reshetikhin/Tureav invariants (cf., e.g., [57])and the equivalence of the Reshetikhin/Tureav invariants with the shadow invariant (cf. Theorem 3.3in Chap. X in [52]) one can conclude that in this situation WLO( L) should coincide with |X L | up toa multiplicative constant (independent of the link). The value of this constant can be determined bylooking at the special case L = , i.e. where L is the empty link. As WLO( ) = 1 one can concludethat WLO( L) = 1|X | |X L | should hold. One of the goals of this paper is to show this formula directly(for the special situation where the link L fullls Assumptions 1 and 2 above) by applying a suitablegauge xing procedure to the Chern-Simons path integral. This generalizes 17 the treatment in [33].

4.2 Torus gauge xing applied to Chern-Simons modelsIn the present section and in Sec. 4.3 below we will give a short summary of those results from [33] whichwill be relevant later. Our presentation will not be totally self-contained, so the reader will probably ndit helpful to have a look at [33] for more details.

During the rest of this paper we will set M := S 1 where is a closed oriented surface. Moreover,we will x an arbitrary point 0 and an arbitrary 18 point t0 S 1 .

15 such a normalization is always possible because by assumption g is simple so all Ad-invariant scalar products on g areproportional to the Killing metric

16 Here is, of course, the standard multiplication in Mat( N, C ) and the wedge product appearing in Eq. ( 40) is theone for Mat( N, C )-valued forms.

17 in [33] only for the case where G = SU (2) and where each j was the highest weight of the fundamental representationthe full path integral was evaluated explicitly

18 in order to simplify the notation somewhat we will later restrict ourselves to the special case where t 0 = iS

1 (0) = 1

10


11/28

By A (resp. A , t ) we will denote the space of smooth g-valued (resp. t -valued) 1-forms on . twill denote the vector eld on S 1 which is induced by the curve iS 1 : [0, 1] t e2it S 1 C and dtthe 1-form on S 1 which is dual to t . We can lift

t and dt in the obvious way to a vector eld resp. a

1-form on M , which will also be denoted by t resp. dt. Every A A can be written uniquely in theform A = A + A0 dt with A A and A0 C (M, g) where A is dened by

A := {A A | A( t ) = 0 } (42)

We say that A A is in the T -torus gauge if A A {Bdt | B C ( , t )}.By computing the relevant Faddeev-Popov determinant 19 one obtains 20 for every gauge-invariant

function : A C

A (A)DA = const. C ( , t ) A (A + Bdt )DA det 1t exp(ad( B )) | t DB (43)where DA denotes the (informal) Lebesgue measure on A and DB the (informal) Lebesgue mea-sure on C ( , t ).

In the special case where (A) =

i Tr i P exp

l i A exp( iS CS (A)) we then get

WLO( L) C ( , t ) A i Tr i P exp l i A + Bdt exp( iS CS (A + Bdt ))DA det 1t exp(ad( B )) | t DB (44)

Here and in the sequel denotes equality up to a multiplicative constant independent of L. Now

S CS (A + Bdt ) = k4 M Tr( A dA) + 2Tr( A Bdt A) + 2Tr( A dB dt)so S CS (A + Bdt ) is quadratic in A for xed B , which means that the informal (complex) measure

exp( iS CS (A + Bdt ))DA appearing above is of Gaussian type. This increases the chances of makingrigorous sense of the right-hand side of Eq. ( 44) considerably.

So far we have ignored the following two subtleties

1. When one tries to nd a rigorous meaning for the informal measure resp. the corresponding integralfunctional in Eq. ( 44) above one encounters certain problems which can be solved by introducinga suitable decomposition 21 A = A Ac , which we will describe now :Let us make the identication A = C (S 1 , A ) where C (S 1 , A ) denotes the space of allsmooth functions : S 1 A , i.e. all functions : S 1 A with the property that everysmooth vector eld X on the function S 1 (, t ) (t)(X ) is smooth.The decomposition A = A Ac is dened by22

A := {A C (S 1 , A ) | A , t (A(t0 )) = 0 }, (45)

Ac := {A C (S 1 , A ) | A is constant and A , t -valued } (46)

where A , t : A A , t is the projection onto the rst component in the decomposition A =A , t A , t . It turns out that S CS behaves nicely under this decomposition. More precisely, wehave

S CS ( A + Ac + Bdt ) = S CS ( A + Bdt ) +

k2 Tr( dAc B ) (47)

19 cf. Sec. 2.3, Sec. 2.4, and Appendix C in the latest version of [ 33], i.e. [arXiv:math-ph/0507040 v7]. We remark thatthe print version of [33] contains an error in Sec. 2.3 (cf. footnote 10 in the latest version of [ 33]). Moreover, Appendix Cis missing in the print version of [33]

20 cf. Eq. (2.23) in Sec. 2.4 of [33] for a variant of this equation where C ( , P ) instead of C ( , t ) appears. Observethat Eq. (2.23) in [ 33] assumes to be non-compact. The compact case is covered by Eq. (2.28) in [33]

21 for a detailed motivation of this decomposition, see Sec. 8 in [ 31]22 note that the space A depends on the choice of the point t0 , and so will some expressions appearing later, see e.g.

Eq. ( 54) below

11
http://arxiv.org/abs/math-ph/0507040http://arxiv.org/abs/math-ph/0507040


12/28

Using this and setting dB ( A) := 1Z (B ) exp(iS CS (

A+ Bdt ))D A where Z (B ) := exp(iS CS ( A+

Bdt ))D A we obtain

WLO( L) C ( , t ) A c A i Tr i P exp l i ( A + Ac + Bdt ) dB ( A) exp( i k2 Tr( dAc B ))DAc det 1t exp(ad( B ) | t ) Z (B ) DB (48)

A more careful analysis shows that in the formula above one can replace t by t reg or, alternatively,by the Weyl alcove P . This amounts to including the extra factor 1 C ( , t reg ) or 1C ( ,P ) in theintegral expression above. In the sequel we will use the factor 1 C ( ,P ) .

2. If one studies the torus gauge xing procedure more closely for compact one nds that due tocertain topological obstructions (cf. [14, 32, 33]) in general a 1-form A can be gauge-transformedinto a 1-form of the type A + Bdt only if one uses a gauge transformation which has a certain(mild) singularity and if one allows A to have a similar singularity. Concretely, in [ 32, 33] weworked with gauge transformations of the type = smooth sing (h) C (( \{ 0}) S 1 , G)with smooth C ( S 1 , G) and sing (h) C ( \{ 0}, G) C (( \{ 0}) S 1 , G) where0 is the point xed above and where the parameter h is an element of [ ,G/T ], i.e. ahomotopy class of mappings from to G/T . sing (h) is obtained from h by xing a representativeg(h) C ( ,G/T ) of h and then lifting the restriction g(h) | \{ 0 } : \{ 0} G/T to a mapping \{ 0} G. In other words: sing (h) C ( \{ 0}, G) is a xed mapping with the propertythat G/T sing (h) = g(h) | \{ 0 } where G/T : G G/T is the canonical projection.The use of the singular gauge transformations sing (h) gives rise to an extra summation h[ ,G/T ]and to extra terms Asing (h) := t (sing (h) 1 dsing (h)), i.e. in Eq. ( 48) above we have to includea summation h[ ,G/T ] and we have to replace A

c by Ac + Asing (h) (for a detailed description

and justication of all this, see [32, 33]).

Taking into account these two subtleties we obtain

WLO( L)

h[ ,G/T ] A c C ( , t ) 1C ( ,P ) (B ) A i Tr i P exp l i ( A + Ac + Asing (h) + Bdt ) dB ( A) exp( i k2 Tr( dAsing (h) B ))det 1t exp(ad( B ) | t ) Z (B )

exp( i k2 Tr( dAc B ))( DAc DB ) (49)where

Tr( dAsing (h) B ) := lim0 \ B (0 ) Tr( dAsing (h) B )Here B(0) is the closed -ball around 0 with respect to an arbitrary but xed Riemannian metric on

.Remark 3 The mapping n : [,G/T ] t given by n (h) = lim 0 \ B ( 0 ) dA

sing (h) is independent of

the special choice of g(h) and sing (h), cf. [32, 33]. Moreover, and this will be important in Subsec. 4.4below one can show that n is a bijection from [ ,G/T ] onto I = ker(exp | t ) (cf. also [14] for a similarresult).

4.3 Some comments regarding a rigorous realization of the r.h.s. of Eq. (49)In [33, 34] it is explained how one can make rigorous sense of the path integral expression appearing onthe right-hand side of Eq. ( 49) using results/constructions from White Noise Analysis (cf., e.g., [38]),certain regularization techniques like loop smearing and framing, and a suitable regularization of theexpression det 1t exp(ad( B ) | t ) Z (B ) appearing above. We do not want to repeat the discussion in[33, 34] in the present paper. Let us just remark the following:

12


13/28

1. In view of the results [12] it is clear how to make sense of the factor det 1t exp(ad( B ) | t ) Z (B )appearing Eq. ( 49) in the special case where B is a constant function (this was the only casewhich was relevant in [12]). More precisely, using the same -function regularization as the onedescribed in Sec. 6 in [12] one comes to the conclusion that in this special case of constant B bthe expression det 1t exp(ad( B ) | t ) Z (B ) should be replaced by 23

exp( i cg2 Tr( dAc B )) det reg 1t exp(ad( B ) | t ) (50)where

det reg 1t exp(ad( B ) | t ) := det 1t exp(ad( b)) | t () / 2

In [33] it was suggested that in the more general situation where B is a step function of the formB = t =0 bt 1Y t one should again replace det 1t exp(ad( B ) | t ) Z (B ) by expression ( 50) wherenow

det reg 1t exp(ad( B ) | t ) :=

t =0det 1t exp(ad( bt )) | t

(Y t ) / 2 (51)

Moreover, it was suggested that one should include a exp( i cg2

Tr( dA

sing (h) B ))-factor in the

expression ( 50) above. Incorporating these changes into ( 49) one obtainsWLO( L)

h[ ,G/T ] A c C ( , t ) 1C ( ,P ) (B ) A i Tr i P exp l i ( A+ Ac + Asing (h)+ Bdt ) dB ( A) exp( i k+ cg2 Tr( dAsing (h) B ))det reg 1t exp(ad( B ) | t ) d ((Ac , B)) (52)

where we have introduced the heuristic complex measure d given by

d ((Ac , B)) := exp( ik+ cg

2 Tr( dAc B ))( DAc DB )2. The Gauss type integral functionals A d

B ( A) resp. A c C ( , t ) d ((A

c , B)) appear-

ing above can be realized rigorously as Hida distributions B resp. on suitable extensions of the spaces A and Ac C ( , t ). Moreover, also the space C ( , P ) appearing in the indicatorfunction 1 C ( ,P ) must be replaced by a larger space. (The fact that one has to extend the originalspaces of smooth functions by larger spaces consisting of less regular functions is a usual phenomenonin Constructive Quantum Field Theory.) The details regarding the extensions of the spaces A andAc C ( , t ) have been or will be discussed elsewhere 24 and they are not relevant if one is onlyinterested in a heuristic evaluation of the r.h.s of Eq. ( 49) resp. ( 52). By contrast, the question of how to extend the space C ( , P ) appearing in the indicator function 1 C ( ,P ) is more subtle evenif one is only interested in a heuristic treatment. One might think that if one replaces C ( , P )by the space P of all P -valued functions on this should be enough. In fact that was the ansatzused in [33] and in the special case where all the link colors i are (minimal) fundamental weightsthis ansatz works. However, it turns out that in the case of general link colors i the space P istoo small. In order to nd the correct space note that 1 C ( ,P ) (B ) = 1 C ( , t reg ) (B )1P (B (0 )).This suggests that one might try to replace C ( , t reg ) by ( t reg ) . As the computations in thenext subsections show the second ansatz is the correct one. Of course, it would be desirable tond a thorough justication for using the second ansatz which is independent of the results in therest of this paper.

23 the rst factor in Eq. ( 50) gives rise to the so-called charge shift k k + cG . Let us mention that the claim thatsuch a charge shift appears is contested by some authors, cf. Remark B.2 in [36]. If one does not believe that such a chargeshift will appear one will have to omit the rst factor in Eq. (50) and the analogous terms in the equations below

24 the extension of A is described in Sec. 8 in [31], see also Sec. 4 in [ 33]; the extension of Ac C ( , t ) was described

in [34]

13


14/28

3. For the implementation of the framing procedure in [ 33, 34] a suitable family ( s )s> 0 of diffeo-morphisms of S 1 fullling certain condition (see list below) was xed. For each diffeomorphisms a deformation B, s resp. s of

B resp. was then introduced and used to replace B and

in the original formula. Later the free parameter s > 0 in the resulting formulas was eliminatedby taking the limit s 0.

Among others ( s )s> 0 was assumed to fulll the following conditions s idM as s 0 uniformly w.r.t. to an arbitrary Riemannian metric on M . (s )A = A for all s > 0. This condition implies that each s , s > 0, is of the form

s (, t ) = ( s (), vs (, t )) (, t ) S 1

for a uniquely determined diffeomorphism s : and vs C ( S 1 , S 1). (s )s> 0 is horizontal in the sense that 25 it can be obtained by integrating a smooth vector

eld X on S 1 , which for all i n, u [0, 1] is orthogonal to the tangent vector l i (u) (i.e.X (li (u)) li (u)) and, at the same time, horizontal in li (u) (i.e. dt(X (li (u))) = 0).

4.4 Explicit heuristic evaluation of the WLOs

As mentioned above we will not go into details concerning a rigorous realization of the r.h.s. of ( 52) butgive a short heuristic treatment instead. As the starting point for this heuristic treatment we use thefollowing modication 26 of Eq. (52) above.

WLO( L)

h[ ,G/T ] 1( t reg ) (B )1P (B (0)) Ac A i Tr i P exp l i ( A + Ac + Asing (h) + Bdt ) dB ( A) exp( i k+ cg2 Tr( dAsing (h) B ))det reg 1t exp(ad( B ) | t )

exp( i k+ cg2 Tr( dAc B ))DAc DB (53)Let, for xed j n, u1 , u2 , . . . u n j be the solutions of the equation ljS 1 (u) = t0 , i.e. those curveparameters in which ljS 1 hits t 0 . For m n

j we set jm := lj (um ), and

jm := 1 resp. jm := 1 resp.

jm := 0 if l

jS 1 crosses t0 in um from below resp. from above resp. only touches t0 in um .

In [33] it is shown how one can evaluate (the rigorous realization of) the heuristic expression

A i Tr i P exp l i ( A + Ac + Asing (h) + Bdt ) dB ( A)explicitly (for links fullling Assumption 1 and 2) and that by doing so one obtains the expression

n

j =1

Tr j exp( l j Ac )exp( l j Asing (h))exp(n j

m =1

jm B (

jm ))

By plugging the last expression into Eq. ( 53) we obtainWLO( L)

h 1( t reg ) (B )1P (B (0 )) A cn

j =1

Tr j exp( l j Ac ) exp( l j Asing (h)) exp( nj

m =1

jm B (

jm ))

exp( i k+ cg2 Tr( dAsing (h) B ))det reg 1t exp(ad( B ) | t ) exp( i k+ cg2 Tr( dAc B ))DAc DB (54)

25 in fact the denition of the term horizontal in [33] was somewhat broader but also more complicated26 clearly, the modication consists in replacing the integration d by the two separate integrations DB and

A

c DA

c and in the use of 1

( t reg ) (B ) 1P (B (0 )) instead of 1 C ( ,P ) (B ) = 1 C ( , t

reg) (B )1P (B (0 ))

14


15/28

Let us now x an auxiliary Riemannian metric g on for the rest of this paper. Let g denotethe corresponding volume measure on and the Hodge star operator induced by g . Moreover, letL2t ( , dg ) denote obvious 27 L2-space. Then we have (cf. Eq. ( 40))

Tr( dAc B ) =

Tr(dAc B )dg = 4

2 dAc , B L 2t ( ,d g )

From Stokes Theorem we obtain

l j Ac = R +j dAc = R +j dAc dg = dAc 1R +j dgwhich implies

(, l j Ac ) = dAc , 1R +j L 2t ( ,d g ) (55)for every t . Here 1R +j denotes the indicator function of the region R

+j dened in Subsec. 3.1 above.

Note that Eq. ( 55) also holds if we replace 1R +j by

1shiftR

+j

:= 1R

+j

1R

+j

(0) (56)

We will use this modied version of Eq. ( 55) in the sequel. Finally, we take into account that

Tr j (exp( b)) = j (exp( b)) =

m j ()e2i (,b ) b t (57)

for suitable 28 m j () N 0 , cf. Eq. (2) above. Setting

:= 2, (58)

for each we obtain from Eq. ( 57) and the modied version of Eq. ( 55)

Tr j exp(

l j

Ac ) exp(

l j

Asing (h))exp(m

jm B (jm ))

=

m j ()exp( i(, m jm B (

jm )))) exp( i l j (, Asing (h))) exp( i dAc , 1shiftR +j L 2t ( ,d g ) )

(Here (, Asing (h)) denotes the obvious real-valued 1-form). Plugging this into Eq. ( 54) above we get

WLO( L)

h 1( t reg ) (B )1P (B (0 )) A cn

j =1 j m j ( j )exp( i l j ( j , Asing (h))) exp( idAc , j 1shiftR +j L 2t ( ,d g ) ) exp( i( j ,

m

jm B (

jm ))) det reg 1t exp(ad( B ) | t ) exp( i

k+ cg2

Tr( dAsing (h) B ))

exp( 2i (k + cg ) dAc , B L 2t ( ,d g ) )DAc DB

=h 1 ,..., n

n

j =1

m j ( j ) 1( t reg ) (B )1P (B (0 ))n

j =1

exp( i l j ( j , Asing (h))) exp( i( j , m jm B ( jm ))) det reg 1t exp(ad( B ) | t ) exp( i

k+ cg2 Tr( dAsing (h) B ))

A c

exp( i dAc ,n

j =1

j 1shiftR +j 2(k + cg )B L 2t ( ,d g ) )DA

c DB

27 the inner product , L 2t ( ,d g ) is given by B 1 , B 2 L 2t ( ,d g ) = (B 1 ( ) , B 2 ( )) d g ( ) where ( , ) is theinner product on g t xed above.

28 m j

( ) is just the multiplicity of the weight in the character j

15


16/28

()=

h 1 , 2 ,..., n

n

j =1

m j ( j ) 1( t reg ) (B )1P (B (0 ))exp( i k+ cg2 Tr( dAsing (h) B )) det reg 1t exp(ad( B ) | t )

n

j =1

exp( i l j ( j , Asing (h))) exp( i( j , m jm B (jm )))) d

n

j =1

j 1shiftR +j 2(k + cg )B DB (59)

Here, in step () we have used the informal equation

Ac

exp( i dAc ,n

j =1

j 1shiftR +j 2(k + cg )B L 2t ( ,d g ) )DA

c

dn

j =1 j 1shiftR +j 2(k + cg )B (60)

which is a kind of innite dimensional analogue of the well-known informal equation

R exp(i x, y )dx

(y). In fact, as nj =1 j 1shiftR +j 2(k + cg )B is in general not smooth (not even continuous) we shouldbe a little bit more careful. Instead of using the delta-function d nj =1 j 1

shiftR +j

2(k + cg )B in

Eqs. ( 59) and ( 60) above we should rather use the superposition 29

t B b + 12 (k+ cg )n

j =1 j 1shiftR +j db

of delta-functions. Then we obtain

WLO( L)

h[ ,G/T ] 1 , 2 ,..., n

n

j =1

m j ( j )

t

db exp( i k+ cg2

Tr( dAsing (h) b))

1( t reg ) (B )1P (B (0 ))det reg 1t exp(ad( B ) | t )n

j =1

exp( i( j ,m

jm B (

jm )))

| B = b+ 12 (k+ cg )n

j =1 j 1shift

R +j

exp( ij l j ( j , Asing (h))) exp( i k+ cg2 Tr dAsing (h) 12 (k+ cg )

n

j =1

j 1shiftR +j

()=

1 ,..., n

n

j =1

m j ( j )h[ ,G/T ] t db exp( i2(k + cg )(n(h) , b))

1( t reg ) (B )1P (B (0 ))det reg 1t exp(ad( B ) | t )n

j =1

exp( i( j ,m

jm B (

jm )))

| B = b+ 12 (k+ cg )n

j =1 j 1shift

R +j

1

()=

1 , 2 ,..., n

n

j =1

m j ( j ) b 1k + c g 1( t reg ) (B )1P (B (0 ))

det reg 1t exp(ad( B ) | t )n

j =1

exp(2 i ( j , m jm B (

jm )))

| B = b+ 1k+ cgnj =1 j 1shiftR +j

29 one can equally well use the superposition t B 12 ( k + c g ) b +nj =1 j 1

shiftR +j

) db or t B 12 ( k + c g ) b +nj =1 j 1shiftR +j

) db the nal result will be the same, which is not surprising since, heuristically, d nj =1 j 1shiftR +j

2 (k + cg )B d 2 (k + cg )B nj =1 j 1shiftR +

j

d B 12 ( k + c g )nj =1 j 1

shiftR +

j

16


17/28

In step () we have used the denition of n(h) and Eq. ( 40). Moreover, we have used

exp i k+ cg2 Tr dAsing (h) 12 (k+ cg )n

j =1

j 1shiftR +j = exp ij l j ( j , Asing (h))

Step () follows, informally by interchanging h and

t db and then using

xI

exp(2i (k + cg )(x, b)) =b 1k + c g I

(b b )

which is an informal version of the Poisson summation formula (moreover, one has to take into accountRemark 3 and the relation I = , cf. Remark 1).

The framing procedure mentioned above which has to be used for a rigorous treatment can also beimplemented in the heuristic setting we work with in the present paper. This amounts to replacing (byhand) the expressions B ( jm ) appearing above by 30

12 B (s (

jm )) + B( 1s ( jm )) . Accordingly, one can

expect that in the rigorous treatment where WLO( L) is dened and computed rigorously one has

WLO( L) = C 1 S tCS (L) (61)

where C 1 is a suitable constant independent of L (see Eq. (63) below) and where S tCS (L) is the rigorousnite state sum (called the Chern-Simons state sum of L in horizontal framing in the sequel) given by

S tCS (L) := 1 , 2 ,..., n

b 1k + c g

n

j =1

m j ( j ) 1( t reg ) (B )1P (B (0 ))det reg 1t exp(ad( B ) | t )

n

j =1

exp(2i ( j ,m

jm

12 B (s (

jm )) + B (

1s (

jm )) ))


(62)

where s > 0 is chosen small enough 31 .

In the special case n = 0, i.e. the case where the link L is empty, it follows from the heuristicdenition of WLO( L) that we must have WLO( L) = 1. From this and Eqs. ( 61), (62), and ( 51) we cantherefore conclude

C 1 = bP 1k + c g det 1t exp(ad( b)) | t

1 g 1

(63)

where g is the genus of . In Sec. 5 below we will give a somewhat more explicit expression for C 1 .

5 Equivalence of the Chern-Simons state sums and those in theshadow invariant

Theorem 5.1 Let L be the colored link in S 1 which we have xed above. Then

S tCS (L) = K 2 2g |X L | (64)

where g is the genus of and K :=

R +

2sin (, )k+ cg (65)

30 in a rigorous treatment where the Hida distributions s , s > 0, are used instead of the heuristic integral functional

exp( i k + c g2 Tr( dA

c B ))( DA

c DB ) a suitably regularized version of this linear combination 12 B (s (jm )) +

B ( 1s (jm )) appears naturally as a result of the application of the polarization identity for quadratic forms.

31 From Lemma 1 iii) below it follows that the right-hand side of Eq. (62) as a function of s is stationary as s 0, soit is clear what small enough means. Moreover, Lemma 1 iii) below shows that S t CS (L ) does not depend on the specialchoice of the horizontal framing ( s )s> 0 .

17


18/28

Before we prove Theorem 5.1 we will rst introduce some notation and then state and prove twolemmas. The proof of Theorem 5.1 will be given after the proof of Lemma 2 below.

For each sequence ( i )0 i n of elements of we set

B ( i ) i := 1k+ cg 0 +

n

j =1

j 1shiftR +j .

Then we can rewrite Eq. ( 62) as

S tCS (L) =( i ) i n +1

n

j =1

m j ( j ) 1P (B ( i ) i (0))1 ( t reg ) (B ( i ) i )det reg 1t exp(ad( B ( i ) i ) | t )

n

j =1

exp(2 i ( j ,m

jm

12 B ( i ) i (s (

jm )) + B ( i ) i (

1s (

jm )) ) (66)

Each B ( i ) i gives rise to an area coloring ( i ) i : {Y 0 , Y 1 , . . . , Y n } given by

( i ) i (Y t ) := ( k + cg )B ( i ) i (Y t ) = 0 +n

j =1

j 1shiftR +j (Y t ) (67)

where Y t is an arbitrary point of Y t . Note that so ( i ) i is well-dened.

Lemma 1 For each ( i )0 i n n +1 we have

i) j = ( i ) i (Y +

j ) ( i ) i (Y

j ) for 1 j n

ii) det reg 1t exp(ad( B ( i ) i ) | t ) = K 2 2g Y (dim(( i ) i (Y ))) (Y )

iii) nj =1 exp(2i ( j , m

jm

12 B ( i ) i (s (

jm )) + B ( i ) i ( 1s (jm )) )) = Y v ( i ) i (Y )

gl( Y )

Proof of i): By Assumption 1 the loops l1 , l2 , . . . , l n do not intersect. This means that for each j nwith j = j the two faces Y +j and Y

j are either both inside l

j

or both outside lj

. More precisely, we

have either Y +

j , Y

j R+j

or Y +

j , Y

j Rj

. From Eq. ( 67) we therefore obtain ( i ) i (Y +

j ) ( i ) i (Y

j ) = j 1shiftR +j (Y +

j) j 1shiftR +j (Y

j

) = j 1R +j (Y +j ) j 1R +j (Y j ) = j (1 0) = j .

Proof of ii): For every b t we have

det(1 t exp(ad( b)) | t ) =R +

(1 e2i (b) )(1 e 2i (b) ) =R +

(e2i (b) / 2 e 2i (b) / 2)2

=R +

(ei (,b ) e i (,b ) )2 =R +

(2i sin((, b))) 2 =R +

4sin((, b))2 (68)

Taking into account Eqs. ( 51), (67), and the relation Y (Y ) = () = 2 2g we obtain

det reg 1t exp(ad( B ( i ) i ) | t ) =n

t =0 R +

2 sin((, B ( i ) i (Y t )))2 (Y t ) / 2

=n

t =0 R +

2sin((, 1k+ cg ( i ) i (Y t ) + )) (Y t ) =

Y R +

2 sin( k+ cg (, ( i ) i (Y ) + ) (Y )

= K ()

Y R +

sin( (, ( i ) i (Y )+ )k+ cg )

sin( (, )k+ cg )

(Y )

= K 2 2g

Y

(dim(( i ) i (Y ))) (Y ) (69)

Proof of iii): Recall that the framing ( s )s> 0 was assumed to be horizontal. Thus, for xed j and m,exactly one of the two points s (jm ) and 1s (jm ) will lie in Y

+j and the other one in Y

j and we have

for sufficiently small s > 0

B (i)

i(s (j

m)) + B (

i)

i( 1

s (j

m)) = 1

k+ cg((

i)

i(Y +

j ) + (

i)

i(Y

j ) + 2 ) (70)

18


19/28

Let us setj :=

m n j

jm = wind( l

jS 1 ) (71)

Then, taking into account part i) of the Lemma we get (for small s > 0)

jexp(2i ( j , m jm 12 B ( i ) i (s (jm )) + B ( i ) i ( 1s ( jm )) ))

=j

exp(2 im

jm

12

1k+ cg ( i ) i (Y

+j ) ( i ) i (Y

j ),( i ) i (Y

+j ) + ( i ) i (Y

j ) + 2

=j

exp ik+ cg j (( i ) i (Y +

j ),( i ) i (Y +

j ) + 2 ) (( i ) i (Y

j ),( i ) i (Y

j ) + 2 )

=j

exp ik+ cg j sgn(Y +

j ; lj )C 2(( i ) i (Y

+j )) + sgn( Y

j ; l

j )C 2 (( i ) i (Y

j ))

=Y

exp ik+ cg j with arc( l j )Y j sgn(Y ; lj ) C 2 (( i ) i (Y )

()=

Y exp ik+ cg gl(Y ) C 2(( i ) i (Y ) =

Y v ( i ) i (Y )

gl( Y )

Y e

ic12 gl( Y )

()=

Y

v ( i ) i (Y )gl( Y )

Here step () follows from Eq. (21) and Eq. ( 71). Moreover, also step ( ) follows from Eq. (21) whichclearly implies Y gl(Y ) = 0.Recall that c ol(X L ) denotes the set of mappings {Y 0 , Y 1 , . . . , Y n } k+ . In the sequel let c ol (X L )denote the set of mappings {Y 0 , Y 1 , . . . , Y n } ((k + cg )t reg ) and let ( W k ){Y 0 ,Y 2 ,...,Y n } , or simply,(W k )n +1 denote the space of functions from {Y 0 , Y 1 , . . . , Y n } with values in W k .

Lemma 2 The mappings

: {( i )0 i n n +1 | 1P (B ( i ) i ) = 0 } ( i )0 i n ( i ) i col(X L )

: {( i )0 i n n +1 | 1( t reg ) (B ( i ) i ) = 0 } ( i )0 i n ( i ) i col (X L )

are well-dened bijections and we have

col (X L ) = { | col(X L ), (W k )n +1 } (72)

where (W k )n +1 is given by ( )(Y ) = (Y ) (Y ) for all Y {Y 0 , Y 1 , . . . , Y n }.

Proof.

1. is well-dened and surjective: Clearly, we have {( i ) i | ( i )0 i n n +1 } = {Y 0 ,Y 1 ,...,Y n } .

On the other hand, for xed ( i )0 i n n +1

the relation 1 ( t reg )

(B ( i ) i ) = 0 is equivalentto Image( B ( i ) i ) = Image( 1k+ cg (( i ) i + )) t reg which is equivalent to Image( ( i ) i ) (k +cg )t reg . The assertion now follows.

2. is well-dened and surjective: For xed ( i )0 i n n +1 the relation 1 P (B ( i ) i ) = 0 is equiv-alent to Image( B ( i ) i ) = Image(

1k+ cg (( i ) i + )) P which is equivalent to Image( ( i ) i )

(k + cg )P . Thus the assertion follows if we can show that

((k + cg )P ) = k+ (73)

19


20/28

In order to prove this equation note that P = C { t | (, ) < 1} so we have

((k + cg )P )= C { t | (, ) < k + cg } = (C ) { t | ( + , ) < k + cg }

()= C { t | ( + , ) < k + cg }= + { t | (, ) < k + cg (, )}()= { + | (, ) < k + 1 }

()= { + | (, ) k} = k+

Here step () follows because for each , + is in the open Weyl chamber C iff is in theclosure C , i.e. we have (C ) = C = + (cf. the last remark in Sec. V.4 in [17]). Step ()follows from cg = 1 + ( , ) and step ( ) from (, ) Z for each .

3. Formula (72) holds: This follows from the fact that both the mapping W aff P (, b) b t regand the mapping i : W aff W k in Subsec. 2.2 are bijections.

4. and

are injective: Let (i )i , (

i )i {( i )0 i n

n +1

| 1( t reg ) (B ( i ) i ) = 0 } such that( i ) i = ( i ) i . From Lemma 1 i) it then follows immediately that

i = i for i {1, 2, . . . , n }.

Moreover, from Eq. ( 67) and Eq. ( 56) we get 0 = ( i ) i (Y 0 ) + = ( i ) i (Y 0 ) + = 0 where

Y 0 denotes the face which contains the point 0 .

Proof of Theorem 5.1 : Applying Lemma 1 to Eq. ( 66) we obtain

S tCS (L) = K 2 2g

( i ) i n +11P (B ( i ) i (0 ))1 ( t reg ) (B ( i ) i )

n

j =1

m j (( i ) i (Y +

j ) ( i ) i (Y

j ))

Y

(dim(( i ) i (Y ))) (Y )

Y

(v ( i ) i (Y ))gl( Y ) (74)

Without loss of generality we can assume that 0 Y 0 . Then we obtain from Lemma 2

S tCS (L) = K 2 2g

col (X L )

1P ((k + cg ) ((Y 0) + ))n

j =1

m j ((Y +j ) (Y

j ))

Y

(dim((Y ))) (Y )Y

(v(Y ) )gl( Y ) (75)

Now observe that for all W k , b ((k + cg )t reg ) we have

v b = vb (76)dim( b) = sgn( )dim( b) (77)

Moreover, 1 P ((k + cg )( (Y 0) + )) = 1 =1 for col(X L ), W k . Thus we obtain from Eq. ( 75)and Eq. ( 72) in Lemma 2

S tCS (L) = K 2 2g

col (X L ) (W k ) n +11 0 =1

n

j =1

m j ( (Y +j ) (Y +

j ) (Y

j ) (Y

j )

Y

sgn( (Y )) (Y )

Y

(dim((Y ))) (Y )

Y

(v(Y ) )gl( Y )

= K 2 2g

col (X L ) 0 , 1 ,... n W k

1 0 =1n

t =0sgn( t )

(Y t )n

j =1

m j ( t ( j, +) (Y +j ) t ( j, ) (Y

j )

Y

(dim((Y ))) (Y )Y

(v(Y ) )gl( Y ) (78)

20


21/28

where t( j, +) resp. t( j, ) is the unique index t {0, 1, 2, . . . , n } such that Y t = Y +j resp. Y t = Y

jholds. Each m j is invariant under the (classical) Weyl group W . From (7) and ( 6) and the fact thateach W aff can be written as the product of a translation and an element of W it easily follows that

m j ( t ( j, +) (Y +j ) t ( j, ) (Y

j ) = m j ((Y +j )

1t ( j, +) t ( j, ) (Y

j ) (79)

Accordingly, let us set j := 1t ( j, +) t ( j, ) . Clearly, we have

n

t =0sgn( t )

(Y t ) ()=n

t =0sgn( t )

# { j n | arc( l j )Y t } =n

j =1

sgn( t ( j, +) ) sgn( t ( j, ) ) =n

j =1

sgn( j ) (80)

(here step ( ) follows from (Y t ) = 2 # { j n | arc( lj ) Y t }). On the other hand the expressions j sgn( j )

nj =1 m j ((Y

+j ) j (Y

j )) , j n, are exactly of the form of the expression on the

right-hand side of formula ( 15) so we have

1 , 2 ,... n W k

n

j =1

sgn( j )n

j =1

m j ((Y +j ) j (Y

j ) =n

j =1

N (Y j )

j (Y +j )

Combining this with Eqs. ( 78)(80) we obtain

S tCS (L) = K 2 2g

col (X L )

n

j =1

N (Y j )

j (Y +j )Y

(dim((Y ))) (Y )

Y

(v(Y ) )gl( Y )

= K 2 2g

col( X L )

|X L |1 |X L |

2 |X L |

3

= K 2 2g |X L | q.e.d.

From Theorem 5.1 and Eq. ( 61) above we can conclude that WLO( L) coincides with |X L | up to a mul-tiplicative constant (independent of L). We can easily determine this multiplicative constant explicitly.According to Eqs. ( 63), (68), and Eq. ( 10) we have (cf. Eq. ( 73) and Example 4 above)

C 1 = 1

k+(K dim())2 2g

= 1

K 2 2g1

|X | (81)

so from Eq. ( 61) and Theorem 5.1 we nally obtain

WLO( L) = |X L ||X |

(82)

This agrees exactly with the formula appearing at the end of Subsec. 4.1 above.

6 A path integral derivation of the quantum Racah formula

In [12] WLO(L) was evaluated in the torus gauge approach in the special case where the link L consistsexclusively of 3 vertical loops with colors , , k+ (cf. Remark 4 below). The result of this evaluationis the expression on the right-hand side of Eq. ( 11) above. As we showed in Secs. 45 when evaluatingthe WLOs of loops without double points in the torus gauge approach the expressions on the right-hand side of Eq. (15) arise naturally. In other words: both the left-hand side and the right-handside of Eq. (15) appear naturally in the torus gauge approach when computing the WLOs of suitablelinks. One can therefore hope to obtain a path integral derivation of Eq. ( 15) by considering linksthat contain both vertical loops and loops without double points. Accordingly, let us now generalizesome of the results obtained in Secs. 4 and 5 to this more general situation where the (colored) linkL = (( l1 , l2 , . . . , l N ), ( 1 , 2 , . . . , N )) is allowed to contain vertical loops. More precisely, we assume thatthe sub link ( l1 , l2, . . . , l n ), n N , is admissible and each loop lk for k {n + 1 , . . . , N } is a verticalloop above the point k , i.e. lk is a constant mapping taking only the value k . For each of the vertical loops lk , k {n + 1 , . . . , N }, we will use a canonical framing, i.e. a framing which fullls

21


22/28

the following condition: if the framing is represented 32 in terms of a vector eld X on arc (lk ), then theprojection of each vector X lk ( s ) , s [0, 1], onto T k coincides with some xed vector v T k .

Finally, we will assume for simplicity that wind( lkS 1 ) = 1 for each k {n + 1 , . . . , N }.Then, using similar arguments as in Subsec. 4.4 we can again derive Eq. ( 61) where S tCS (L) is now

given by

S tCS (L) := 1 , 2 ,..., n

b 1k + c g

n

j =1

m j ( j ) 1( t reg ) (B )1P (B (0 ))det reg 1t exp(ad( B ) | t )

N

k= n +1

k (exp( B (k )))

n

j =1

exp(2 i ( j , k jk

12 B (s (

jk )) + B (

1s (

jk )) ))


(83)

for sufficiently small s > 0. Recall that k is the character associated to the dominant weight k .Also the computations in the proof of Theorem 5.1 can be generalized in a straightforward way. One

obtains

S tCS (L) = K 2 2g

col (X L )

n

j =1

M (Y j )

j (Y +j )

N

k= n +1

k (exp( 1k+ cg ((Y k ) + )))

Y

(dim((Y ))) (Y )Y

(v(Y ) )gl( Y ) (84)

where Y k , k {n + 1 , . . . , N }, denotes the face in which k lies and where we have set

M := W k

sgn( )m ( ( )) (85)

(According to Eq. ( 15) we have M = N

so we could replace M

by N

above. But as we want togive a path integral derivation of Eq. ( 15) which is based on Eq. ( 84) we avoid this replacement here.)

Now observe that (exp( 1k+ cg ( + ))) =

S S 0

(86)

Eq. (86) follows from the denition of the S-matrix in Subsec. 2.2 if one takes into account Weylscharacter formula. Combining Eqs. ( 61), (81) (84), and ( 86) we nally obtain

WLO( L)

= 1|X |

col (X L )

n

j =1M (Y j )

j (Y +j )

N

k= n +1

S k (Y k )S 0(Y k ) Y

(dim((Y ))) (Y )

Y

(v(Y ) )gl( Y ) (87)

In the special case n = 0, i.e. in the case where there are only vertical loops, there is only one faceY 0 = and we have gl( Y 0) = 0, (Y 0) = () = 2 2g so Eq. (87) then reduces to

WLO( L) = 1|X | k+

N

k=1

S k S 0

dim( )2 2g (88)

Remark 4 In the special case G = SU (2) the last equation is equivalent to formula (7.27) in [12]. Weremark that for vertical loops the inner integral in Eq. ( 49) is trivial, so for the derivation of Eq. ( 88)one does not need the general formula ( 49) but can work with the simpler formulas appearing in [ 12], cf.

32 alternatively, if we represent the framing in terms of another loop l k which is sufficiently close to lk (cf. e.g. Sec. 2.1and Fig. 3b and Fig. 3c in [58]) then a canonical framing is one where (not only lk but also) l k is vertical loop

22


23/28


24/28

Figure 6:

Let us now deform L using an orientation preserving diffeomorphism : S 2

S 1

S 2

S 1

of theform (, t ) = t (), t for S 2 , t S 1 where : S 1 = SO(2) Diff(S 2) is the group homomorphismcorresponding to the rotation of S 2 in R 3 around a suitably chosen rotation axis a R 3 (and where wehave set t := (t) for t S 1). Let L be the link obtained from L by the deformation . For a suitable

Figure 7:

choice of the rotation axis a the projection of L onto = S 2 will look like in Fig. 7. From the invarianceproperties of the Chern-Simons path integral we conclude at a heuristic level that

WLO( L) = WLO( L) = 1|X |

1S 200

N (92)

On the other hand, after performing a suitable change of framing 34 for each of the two loops l2 and l3 (i.e.the two vertical loops with colors and in Fig. 5) the link L becomes isotopic to L. According to

Eq. (2.33) in [58] (and the paragraph after Eq. (4.40) in Sec. 4.5 in [ 58] where the conformal weight happearing in (2.33) in [58] is related to the matrix T ) each of these two changes of framing 35 alters thevalue of the WLO by a factor T T 00 and

T T 00

1, i.e. we have

WLO( L) = T T

WLO( L) = 1|X |

1S 200

T T

N (93)

Conclusion: By combining the two equations ( 91) and ( 93) we obtain

T T

N = 0 , 1

M 1 0 S 0 S 1 T 0 0 T 1 1 1 (94)

On the other hand according to Eq. ( 28) in Example 2 above we have

T T

N = 0 , 1

N 1 0 S 0 S 1 T 0 0 T 1 1 1 (95)

Eq. ( 94) and Eq. ( 95) imply 0 , 1 M 1 0 S 0 S 1 T 0 0 T

1 1 1 = 0 , 1 N

1 0 S 0 S 1 T 0 0 T

1 1 1 . This

holds for arbitrary , , k+ so using the fact that S-matrix and the T-matrix are invertible (cf. Eqs.(9) above) we indeed obtain M = N (for all , , k+ ).

34 alternatively, we can replace the two changes of framing by two simple surgery operations. The rst surgery operationinvolves a suitable tubular neighborhood of the vertical loop l2 in Fig. 5 (cf. Sec. 4.2 in [ 58]); the second surgery operationsinvolves a similar tubular neighborhood of the vertical loop l3

35 similarly, according to Sec. 4.5 in [58] the two surgery operations mentioned in the previous footnote will alter the valueof the WLO by the factor T T

1 , so also by using the surgery argument we arrive at Eq. ( 93)

24


25/28

Remark 5 Wittens argument from Sec 4.5 in [58] which we used in the paragraph preceding Eq. ( 93)is based on ideas from conformal eld theory. So if we want to give a (complete) path integral derivationof Eq. (15) we will have to derive the rst equality in Eq. ( 93) using only path integral methods, cf.[37] for partial results in this direction. On the other hand, if one is happy with mixing argumentsfrom conformal eld theory and arguments based on the CS path integral then the derivation of the(elementary) quantum Racah formula Eq. ( 15) which we have just given is ne and by combiningEq. ( 15) with the Verlinde formula derived in Remark 4 (using Eq. (4.36) in [ 58]) one nally obtainsN = W k sgn( )m ( ( )), which is the affine Lie algebra version of the abstract quantumRacah formula appearing at the end of Subsec. 2.2.7 Outlook

In the introduction we mentioned one of the most important open questions in the theory of 3-manifoldquantum invariants, the question whether and how one can make rigorous sense of Wittens heuristicpath integral expressions for the Wilson loop observables of Chern-Simons theory, cf. the r.h.s. of Eq.(41). A related and probably less difficult question is whether and how one can make rigorous sense of those path integral expressions that arise from the r.h.s. of Eq. ( 41) after choosing a suitable gaugexing. Until recently Lorentz gauge xing was the only 36 gauge xing procedure for which the relevantpath integral expressions have been evaluated completely for general groups, links and manifolds, cf.[29, 9, 10, 7, 16, 8, 4]. The nal result of this evaluation is a complicated innite series whose termsinvolve integrals over (high-dimensional) conguration spaces, cf. [ 16, 4]. The heuristic path integralexpressions which appear during the intermediate computations are even more complicated and it shouldbe hard to nd a rigorous realization of these path integrals 37 .

It is therefore desirable to nd other gauges for which the WLOs can also be evaluated explicitly. Agauge which leads to the expressions appearing in Turaevs shadow world approach to the 3-manifoldquantum invariants would be particularly desirable. This is because the expressions appearing in theshadow world approach involve only nite sums, which are dened in a purely combinatorial way. Thesenite combinatorial sums are considerably less complicated than the innite series of conguration spaceintegrals mentioned above. Accordingly, it is reasonable to believe that for such a gauge xing also thecorresponding path integral expressions and the heuristic arguments used for their evaluation will be less

complicated than those for Lorentz gauge xing.The results in [ 33] and the present paper suggest that for manifolds M of the form M = S 1 torusgauge xing is a gauge xing with the desired properties. Moreover, it is reasonable to expect that thepath integral expressions for the WLOs in the torus gauge, i.e. the r.h.s. of Eq. ( 49) above, admit arigorous treatment, either in a continuum setting (cf. Secs. 89 in [31], Sec. 46 in [33], and [34] forpartial results) or in a suitable simplicial setting (cf. [35, 36] for ongoing work in this direction).

Acknowledgements: A. H. would like to thank Ambar N. Sengupta and Stephen Sawin for usefuldiscussions and the Alexander von Humboldt foundation for the generous nancial support during theperiod Oct. 2005 Sept. 2006.

References

[1] M. Aganagic, H. Ooguri, N. Saulina and C. Vafa, Black holes, q-deformed 2d Yang-Mills, andnon-perturbative topological strings, Nucl. Phys. B 715 (2005) 304 [ arXiv:hep-th/0411280] .

[2] S. Albeverio and A.N. Sengupta. A Mathematical Construction of the Non-Abelian Chern-SimonsFunctional Integral. Commun. Math. Phys. , 186:563579, 1997.

[3] S. Albeverio and I. Mitoma. Asymptotic expansion of perturbative Chern-Simons theory via Wienerspace. Bull. Sci. Math. , 133(3): 272-314, 2009.

36 for CS models on the special manifold M = R 3 there is an alternative approach based on light-cone gauge xing anda suitable complexication of the manifold, cf. [26], [40]. However, in this approach certain correction factors have to beinserted by hand in the course of the computations. At present the origin of these correction factors is not clear

37 There are, however, some very interesting partial results in this direction, cf. [3]

25
http://arxiv.org/abs/hep-th/0411280http://arxiv.org/abs/hep-th/0411280


26/28

[4] D. Altschuler and L. Freidel. Vassiliev knot invariants and Chern-Simons perturbation theory to allorders. Comm. Math. Phys. , 187:261287, 1997.

[5] H. H. Andersen. Tensor products of quantized tilting modules. Commun. Math. Phys. , 149:149159,1992.

[6] H. H. Andersen. and J. Paradowski. Fusion categories arising from semisimple Lie algebras. Comm.Math. Phys. , 169(3):563588, 1995.

[7] S. Axelrod and I.M. Singer. Chern-Simons perturbation theory. In Catto, Sultan et al., editor, Dif- ferential geometric methods in theoretical physics. Proceedings of the 20th international conference,June 3-7, 1991, New York City, NY, USA , volume 1-2, pages 345. World Scientic, Singapore,1992.

[8] S. Axelrod and I.M. Singer. Chern-Simons perturbation theory. II. J. Differ. Geom. , 39(1):173213,1994.

[9] D. Bar-Natan. Perturbative Chern-Simons theory. J. Knot Th. Ram. , 4:503547, 1995.

[10] D. Bar-Natan. On the Vassiliev knot invariants. Topology , 34:423472, 1995.

[11] C. Beasley and E. Witten, Non-abelian localization for Chern-Simons theory. J. Differential Geom. ,70(2): 183-323, 2005. [arXiv:hep-th/0503126]

[12] M. Blau and G. Thompson. Derivation of the Verlinde Formula from Chern-Simons Theory and theG/G model. Nucl. Phys. , B408(1):345390, 1993.

[13] M. Blau and G. Thompson. Lectures on 2d Gauge Theories: Topological Aspects and Path IntegralTechniques. In E. Gava et al., editor, Proceedings of the 1993 Trieste Summer School on High Energy Physics and Cosmology , pages 175244. World Scientic, Singapore, 1994.

[14] M. Blau and G. Thompson. On Diagonalization in Map (M, G ). Commun. Math. Phys. , 171:639660,1995.

[15] M. Blau and G. Thompson. Chern-Simons Theory on S 1-Bundles: Abelianisation and q-deformed

Yang-Mills Theory, J. High Energy Phys. , no. 5, 003, 35 pp, 2006. [arXiv:math-ph/0601068 ][16] R. Bott and C. Taubes. On the self-linking of knots. J. Math. Phys. , 35(10):52475287, 1994.

[17] Th. Brocker and T. tom Dieck. Representations of compact Lie groups , volume 98 of Graduate Texts in Mathematics . Springer-Verlag, New York, 1985.

[18] E. Buffenoir and P. Roche, Two-dimensional lattice gauge theory based on a quantum group,Commun. Math. Phys. 170 (1995) 669 [ arXiv:hep-th/9405126] .

[19] S. de Haro. Chern-Simons theory on lens spaces from 2d Yang-Mills on the cylinder. J. High. Energy Phys. , 41(8), 2004.

[20] S. de Haro. A Note on Knot Invariants and q -Deformed 2 d Yang-Mills. Phys. Lett. B , 634 (2006)78-83 [arXiv:hep-th/0509167 ]

[21] S. de Haro, Chern-Simons theory, 2d Yang-Mills, and Lie algebra wanderers, Nucl. Phys. B 730(2005) 312 [arXiv:hep-th/0412110] .

[22] S. de Haro and M. Tierz, Discrete and oscillatory matrix models in Chern-Simons theory, Nucl.Phys. B 731 (2005) 225 [arXiv:hep-th/0501123 ].

[23] F. Di Francesco, P. Mathieu and D. Senecal, Conformal Field Theory, Springer-Verlag, 1997

[24] M. Finkelberg. An equivalence of fusion categories. Geom. Funct. Anal. , 6:249267, 1996.

[25] D. S. Freed. Quantum groups from path integrals. Preprint, 1995 [arXiv:q-alg/9501025]

[26] J. Frohlich and C. King. The Chern-Simons Theory and Knot Polynomials. Commun. Math. Phys. ,126:167199, 1989.

26
http://arxiv.org/abs/hep-th/0503126http://arxiv.org/abs/math-ph/0601068http://arxiv.org/abs/hep-th/9405126http://arxiv.org/abs/hep-th/0509167http://arxiv.org/abs/hep-th/0412110http://arxiv.org/abs/hep-th/0501123http://arxiv.org/abs/q-alg/9501025http://arxiv.org/abs/q-alg/9501025http://arxiv.org/abs/hep-th/0501123http://arxiv.org/abs/hep-th/0412110http://arxiv.org/abs/hep-th/0509167http://arxiv.org/abs/hep-th/9405126http://arxiv.org/abs/math-ph/0601068http://arxiv.org/abs/hep-th/0503126


27/28

[27] J. Fuchs and C. Schweigert. A representation theoretic approach to the WZW Verlinde formula.Preprint, 1997 [arXiv:hep-th/9707069 ]

[28] J. Fuchs, and P. van Driel. WZW fusion rules, quantum groups, and the modular matrix S. Nucl.Phys. B, 346:632648, 1990.

[29] E. Guadagnini, M. Martellini, and M. Mintchev. Wilson Lines in Chern-Simons theory and Linkinvariants. Nucl. Phys. B , 330:575607, 1990.

[30] A. Hahn. The Wilson loop observables of Chern-Simons theory on R 3 in axial gauge. Commun.Math. Phys. , 248(3):467499, 2004.

[31] A. Hahn. Chern-Simons models on S 2 S 1 , torus gauge xing, and link invariants I. J. Geom.Phys. , 53(3):275314, 2005.

[32] A. Hahn. Chern-Simons models on S 2 S 1 , torus gauge xing, and link invariants II. J. Geom.Phys. , 58:11241136, 2008.

[33] A. Hahn. An analytic Approach to Turaevs Shadow Invariant. J. Knot Th. Ram. , 17(11): 13271385,2008 [see arXiv:math-ph/0507040 v7 for the most recent version]

[34] A. Hahn. White noise analysis in the theory of three-manifold quantum invariants. In A.N. Sen-gupta and P. Sundar, editors, Innite Dimensional Stochastic Analysis , volume XXII of Quantum Probability and White Noise Analysis , pages 201225. World Scientic, 2008.

[35] A. Hahn. From simplicial Chern-Simons theory to the shadow invariant I, Preprint, 2012[arXiv:1206.0439]

[36] A. Hahn. From simplicial Chern-Simons theory to the shadow invariant II, Preprint, 2012[arXiv:1206.0441]

[37] A. Hahn Surgery operations on the Chern-Simons path integral via conditional expectations, inpreparation.

[38] T. Hida, H.-H. Kuo, J. Potthoff, and L. Streit. White Noise. An innite dimensional Calculus .Dordrecht: Kluwer, 1993.

[39] L. Kauffman. Knots . Singapore: World Scientic, 1993.

[40] L. Kauffman. Functional integration, Kontsevich integral and formal integration. J. Korean Math.Soc. , 38(2):437468, 2001.

[41] A. N. Kirillov and N.Y. Reshetikhin. Representations of the algebra U q(sl2), q -orthogonal polynomi-als and invariants of links. In V.G. Kac et al., editor, Innite Dimensional Lie Algebras and Groups ,Vol. 7 of Advanced Ser. in Math. Phys. , pages 285339, 1988.

[42] G. Kuperberg. Quantum invariants of knots and 3-manifolds (book review). Bull. Amer. Math. Soc. ,33(1):107110, 1996.

[43] M. Polyak and N. Reshetikhin. On 2D Yang-Mills Theory and Invariants of Links. In Sternheimer D.et al, editor, Deformation Theory and Symplectic Geometry , pages 223246. Kluwer Academic Pub-lishers, 1997.

[44] N.Y. Reshetikhin and V.G. Turaev. Ribbon graphs and their invariants derived from quantumgroups. Commun. Math. Phys. , 127:126, 1990.

[45] N.Y. Reshetikhin and V.G. Turaev. Invariants of three manifolds via link polynomials and quantumgroups. Invent. Math. , 103:547597, 1991.

[46] L. Rozansky. A contribution of the trivial connection to Jones polynomial and Wittens invariant of 3d manifolds. I, Commun.Math.Phys. 175:275296, 1996. [arXiv:hep-th/9401061]

[47] S. Sawin. Jones-Witten invariants for non-simply connected Lie groups and the geometry of theWeyl alcove. Adv. Math. , 165(1):134, 2002. [arXiv: math.QA/9905010 ]

27
http://arxiv.org/abs/hep-th/9707069http://arxiv.org/abs/math-ph/0507040http://arxiv.org/abs/1206.0439http://arxiv.org/abs/1206.0441http://arxiv.org/abs/hep-th/9401061http://arxiv.org/abs/math/9905010http://arxiv.org/abs/math/9905010http://arxiv.org/abs/hep-th/9401061http://arxiv.org/abs/1206.0441http://arxiv.org/abs/1206.0439http://arxiv.org/abs/math-ph/0507040http://arxiv.org/abs/hep-th/9707069


28/28

[48] S. Sawin. Quantum groups at roots of unity and modularity. J. Knot Th. Ram., 15(10):1245-1277,2006. [arXiv:math.QA/0308281]

[49] S. Sawin. Closed subsets of the Weyl alcove and TQFTs. Pacic Journal of Mathematics , 238305-324, 2006. [arXiv:math.QA/0503692 ]

[50] J. Sawon. Perturbative expansion of Chern-Simons theory. Geom. Topol. Monogr. 8:145166, 2006[arXiv:math/0504495]

[51] M. Spiegelglas. Spin sums, fusion rules and correlators by lling. Phys. Lett. B, 247:3640, 1990.

[52] V. G. Turaev, Quantum Invariants of Knots and 3-Manifolds, 1994, De Gruyter

[53] V. G. Turaev, Shadow Links and Face Models of Statistical Mechanics, J. Differential Geometry36 (1992) 35-74

[54] V.G. Turaev and O. G. Viro. State sum invariants of 3-manifolds and quantum 6j-symbols. Topology ,31(4):865902, 1992.

[55] V.G. Turaev and H. Wenzel. Quantum invariants of 3-manifolds associated with classical simple Liealgebras. Internat. J. Math , 4:323358, 1993.

[56] M. Walton. Fusion rules in Wess-Zumino-Witten models Nucl. Phys. B, 340:777790, 1990.

[57] K. Walker On Wittens 3-manifold invariants, preprint, 1991

[58] E. Witten. Quantum Field Theory and the Jones Polynomial. Commun. Math. Phys. , 121:351399,1989.

Amsterdam University College, University of AmsterdamPlantage Muidergracht 14, 1018 TV Amsterdamand

Institute for Theoretical Physics, University of AmsterdamScience Park 904, Postbus 94485, 1090 GL Amsterdam, The NetherlandsE-mail address: [email protected]

Grupo de Fsica Matem atica da Universidade de LisboaAv. Prof. Gama Pinto, 2PT-1649-003 Lisbon, PortugalE-mail address: [email protected]
http://arxiv.org/abs/math/0308281http://arxiv.org/abs/math/0503692http://arxiv.org/abs/math/0504495http://arxiv.org/abs/math/0504495http://arxiv.org/abs/math/0503692http://arxiv.org/abs/math/0308281

Formula Quantistica

Documents

Transcript of Formula Quantistica