Editing Chern–Simons theory (section)

===HOMFLY and Jones polynomials===

Consider a link ''L'' in ''M'', which is a collection of ''ℓ'' disjoint loops. A particularly interesting observable is the ''ℓ''-point correlation function formed from the product of the Wilson loops around each disjoint loop, each traced in the [[fundamental representation]] of ''G''. One may form a normalized correlation function by dividing this observable by the [[partition function (quantum field theory)|partition function]] ''Z''(''M''), which is just the 0-point correlation function.

In the special case in which M is the 3-sphere, Witten has shown that these normalized correlation functions are proportional to known [[knot polynomials]]. For example, in ''G''&nbsp;=&nbsp;''U''(''N'') Chern–Simons theory at level ''k'' the normalized correlation function is, up to a phase, equal to

:<math>\frac{\sin(\pi/(k+N))}{\sin(\pi N/(k+N))}</math>

times the [[HOMFLY polynomial]]. In particular when ''N''&nbsp;=&nbsp;2 the HOMFLY polynomial reduces to the [[Jones polynomial]]. In the SO(''N'') case, one finds a similar expression with the [[Kauffman polynomial]].

The phase ambiguity reflects the fact that, as Witten has shown, the quantum correlation functions are not fully defined by the classical data. The [[linking number]] of a loop with itself enters into the calculation of the partition function, but this number is not invariant under small deformations and in particular, is not a topological invariant. This number can be rendered well defined if one chooses a framing for each loop, which is a choice of preferred nonzero [[normal vector]] at each point along which one deforms the loop to calculate its self-linking number. This procedure is an example of the [[point-splitting]] [[regularization (physics)|regularization]] procedure introduced by [[Paul Dirac]] and [[Rudolf Peierls]] to define apparently divergent quantities in [[quantum field theory]] in 1934.

[[Sir Michael Atiyah]] has shown that there exists a canonical choice of 2-framing,<ref>{{Cite journal |last=Atiyah |first=Michael |date=1990 |title=On framings of 3-manifolds |url=https://doi.org/10.1016/0040-9383(90)90021-b |journal=Topology |volume=29 |issue=1 |pages=1–7 |doi=10.1016/0040-9383(90)90021-b |issn=0040-9383|url-access=subscription }}</ref> which is generally used in the literature today and leads to a well-defined linking number. With the canonical framing the above phase is the exponential of 2π''i''/(''k''&nbsp;+&nbsp;''N'') times the linking number of ''L'' with itself.

;Problem (Extension of Jones polynomial to general 3-manifolds)　
"The original Jones polynomial was defined for 1-links in the 3-sphere (the 3-ball, the 3-space R3). Can you define the Jones polynomial for 1-links in any 3-manifold?"

See section 1.1 of this paper<ref>
{{cite arXiv|first1=L.H |last1=Kauffman 
|first2=E |last2=Ogasa 
|first3=J |last3=Schneider 
|eprint=1808.03023|
title=A spinning construction for virtual 1-knots and 2-knots, and the fiberwise and welded equivalence of virtual 1-knots|year=2018|class=math.GT 
}} 
</ref> for the background and the history of this problem. Kauffman submitted a solution in the case of the product manifold of closed oriented surface and the closed interval, by introducing virtual 1-knots.<ref>
{{cite arXiv|first=L.E. |last=Kauffman |eprint= math/9811028
| title=Virtual Knot Theory
|year=1998 }} 
</ref> It is open in the other cases. Witten's path integral for Jones polynomial is written for links in any compact 3-manifold formally, but the calculus is not done even in physics level in any case other than the 3-sphere (the 3-ball, the 3-space '''R'''<sup>3</sup>). This problem is also open in physics level. In the case of Alexander polynomial, this problem is solved.