The Witten-Reshetikhin-Turaev representation of the Kauffman bracket skein algebra

For A a primitive 2N–root of unity with N odd, the WittenReshetikhin-Turaev topological quantum field theory provides a representation of the Kauffman skein algebra of a closed surface. We show that this representation is irreducible, and we compute its classical shadow in the sense of [4]. The discovery of the Jones polynomial [9], and the simplification of its construction by Kauffman [10], were quickly followed by two originally unrelated developments. The first one was the introduction of the Kauffman skein module of an oriented 3–manifold by Turaev [23] and Przytycki [17]. A special case leads to the Kauffman skein algebra S(S) of an oriented surface S which, when S is connected, was later interpreted as a quantization of the character variety RSL2(C)(S), consisting of the characters of all group homomorphisms π1(S)→ SL2(C) [24, 7, 8, 18]. Another development was Witten’s interpretation [26] of the Jones polynomial within the framework of a topological quantum field theory. This topological quantum field theory point of view was formalized in mathematical terms by ReshetikhinTuraev [19, 20]. In particular, the Witten-Reshetikhin-Turaev topological quantum field theory leads, for every primitive 2N–root of unity A, to a representation ρ : S(S)→ End(VS) of the skein algebra corresponding to this parameter A. The first result of this article is the following. Theorem 1. Let S be a connected closed oriented surface. For every primitive 2N– root of unity A, the Witten-Reshetikhin-Turaev representation ρ : S(S)→ End(VS) is irreducible. This result can be compared with Roberts’s proof [22] that, when N = 2p with p prime, the action of the mapping class group of S on the Witten-ReshetikhinTuraev space VS is irreducible. In this special case, Theorem 1 can actually be deduced from some of the proofs of [22]. Our interest in the irreducibility of the Witten-Reshetikhin-Turaev representation is motivated by [3, 4, 5, 6], where we initiated the systematic study of finite-dimensional irreducible representations of the skein algebra S(S). In particular, when A is a 2N–root of unity with N odd, we associate to such an irreducible representation ρ : S(S) → End(V ) an element rρ of the character variety Date: January 13, 2015. 2010 Mathematics Subject Classification. 57M27, 57R56. This research was partially supported by grants DMS-0604866, DMS-1105402 and DMS1105692 from the National Science Foundation, and by a mentoring grant from the Association for Women in Mathematics. 1 ar X iv :1 30 9. 09 21 v2 [ m at h. G T ] 1 2 Ja n 20 15 2 FRANCIS BONAHON AND HELEN WONG RSL2(C)(S); see Theorem 13 for a precise statement. If we regard S (S) as a quantization of RSL2(C)(S) and a representation ρ as a point of this quantization, the character rρ ∈ RSL2(C)(S) is the classical shadow of ρ. With irreducibility of the Witten-Reshetikhin-Turaev representation established, we thus may inquire about the classical shadow of the best known representation of the skein algebra. The Witten-Reshetikhin-Turaev topological quantum field theory, and the associated representation of S(S), have slightly different features according to whether N is even or odd, respectively known as the SU2 and SO3 cases. The classical shadow of a representation is defined only when N is odd. Theorem 2. When A is a primitive 2N–root of unity with N odd, the classical shadow of the Witten-Reshetikhin-Turaev representation ρ : S(S) → End(VS) is the trivial character ι ∈ RSL2(C)(S), represented by the trivial homomorphism π1(S)→ SL2(C). The proof of Theorem 2 is relatively simple, although it uses deep connections in the quantum theory with both types of Chebyshev polynomials. The Chebyshev polynomial of the first type, TN (x), is used to define the classical shadow of a representation of S(S). On the other hand, the second type of Chebyshev polynomials, Sn(x), classically plays an important role in the representation theory of the quantum group Uq(sl2) and in the Witten-Reshetikhin-Turaev topological quantum field theory. We are ultimately led to Theorem 2 by exploiting these relationships and making use of some elementary relations between the two types. In [6] we construct, for every character r ∈ RSL2(C)(S), an irreducible representation ρ : S(S)→ End(VS) whose classical shadow rρ is equal to r. In particular, this associates another irreducible representation ρι to the trivial character ι. The construction of [6] is unfortunately not very explicit in the special case of the trivial character ι; however, it appears that the representation that it provides has very different features from those of the Witten-Reshetikhin-Turaev representation. 1. The Kauffman skein module The Kauffman skein module S(M) of an oriented 3–dimensional manifold M depends on a parameter A = eπi~ ∈ C − {0}, and is defined as follows: one first considers the vector space freely generated by all isotopy classes of framed links in the thickened surface S × [0, 1], and then one takes the quotient of this space by two relations: • the first relation is the skein relation that [L1] = A[L0]+A[L∞] whenever the three links L1, L0 and L∞ ⊂ S × [0, 1] differ only in a little ball where they are as represented on Figure 1; • the second relation states that [L∪O] = −(A +A−2)[L] whenever the knot O is the boundary of a disk D endowed with framing transverse to D, and the framed link L is disjoint from the disk D. In the special case where M = S × [0, 1] for an oriented surface S, we write S(S) = S ( S × [0, 1] ) and we note that this module now comes with a natural multiplication. Indeed, if [L1], [L2] ∈ S(S) are respectively represented by the framed links L1, L2, we can consider their superposition [L1] · [L2] = [L1 ∪ L2] ∈ S(S) THE WITTEN-RESHETIKHIN-TURAEV REPRESENTATION 3