Witten–Reshetikhin–Turaev Invariants of¶Seifert Manifolds
Witten–Reshetikhin–Turaev Invariants of¶Seifert Manifolds
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Seifert 流形的 Witten-Reshetikhin-Turaev 不变量
DOI：
10.1007/s002200050678
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发表时间：
1999
影响因子：
2.4
通讯作者：
L. Rozansky
中科院分区：
文献类型：
--
作者：
R. Lawrence;L. Rozansky
翻译后摘要：对于塞弗特同调球，我们推导出一个全纯函数K，其值在整数K是sl 2 Witten-Reshetikhin-Turaev不变量，ZK，在q= exp 2πi/K。该函数被表示为一个项的和，这可以自然地对应于平坦连接在Witten-Chern-Simons路径积分的定态相展开中的贡献。平凡连接贡献被发现有K−1的幂的渐近展开，对于K是奇素数幂，它K-渐近地收敛到不变量ZK在单位根处的精确总值。在合理的$K$的评价也进行了讨论。使用类似的技术，获得了环面结的有色琼斯多项式的表达式，提供了一个平凡的
连接贡献，它是颜色的解析函数。这证明了Chern-Simons-维滕理论的定相展开对于S3中的Seifert流形和环面纽结是精确的。还讨论了推广这些结果的可能性。
Abstract: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.