TY - JOUR

T1 - Witten-Reshetikhin-Turaev invariants of Seifert manifolds

AU - Lawrence, Ruth

AU - Rozansky, Lev

PY - 1999

Y1 - 1999

N2 - For Seifert homology spheres, we derive a holomorphic function of K whose value at integer K is the sl2 Witten-Reshetikhin-Turaev invariant, ZK, at q = exp 2πi/K. This function is expressed as a sum of terms, which can be naturally corresponded to the contributions of flat connections in the stationary phase expansion of the Witten-Chern-Simons path integral. The trivial connection contribution is found to have an asymptotic expansion in powers of K-1 which, for K an odd prime power, converges K-adically to the exact total value of the invariant ZK at that root of unity. Evaluations at rational K are also discussed. Using similar techniques, an expression for the coloured Jones polynomial of a torus knot is obtained, providing a trivial connection contribution which is an analytic function of the colour. This demonstrates that the stationary phase expansion of the Chern-Simons-Witten theory is exact for Seifert manifolds and for torus knots in S3. The possibility of generalising such results is also discussed.

AB - For Seifert homology spheres, we derive a holomorphic function of K whose value at integer K is the sl2 Witten-Reshetikhin-Turaev invariant, ZK, at q = exp 2πi/K. This function is expressed as a sum of terms, which can be naturally corresponded to the contributions of flat connections in the stationary phase expansion of the Witten-Chern-Simons path integral. The trivial connection contribution is found to have an asymptotic expansion in powers of K-1 which, for K an odd prime power, converges K-adically to the exact total value of the invariant ZK at that root of unity. Evaluations at rational K are also discussed. Using similar techniques, an expression for the coloured Jones polynomial of a torus knot is obtained, providing a trivial connection contribution which is an analytic function of the colour. This demonstrates that the stationary phase expansion of the Chern-Simons-Witten theory is exact for Seifert manifolds and for torus knots in S3. The possibility of generalising such results is also discussed.

UR - http://www.scopus.com/inward/record.url?scp=0033245033&partnerID=8YFLogxK

U2 - 10.1007/s002200050678

DO - 10.1007/s002200050678

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AN - SCOPUS:0033245033

SN - 0010-3616

VL - 205

SP - 287

EP - 314

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 2

ER -