Chern-Simons theory and the Jones Polynomial

Question

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1. What the student is really asking

The student wants to understand three rather subtle points that appear in Witten’s 1989 paper “Quantum Field Theory,  Chern‑Simons Theory, and the Jones Polynomial” :

#	Question (in plain language)
1	Why do we have to frame Wilson lines? Even though the expectation value of an unframed Wilson line in Chern‑Simons theory is finite, it depends on the background metric. Does that make the observable “invalid” or simply “unphysical”?
2	How does changing the framing give anyonic statistics? When we twist the framing of a Wilson line, the expectation value picks up a phase. Why can we interpret that phase as the statistical phase of the point‑like excitations that sit at the ends of the line?
3	Does the canonical Gauss‑law constraint already contain the framing information? In the Hamiltonian picture we impose ((\rho - \frac{k}{2\pi}\epsilon^{ij}\partial_i A_j)\,	\psi\rangle =0). Is any reference to framing hidden in this equation, or does it have to be added later?

Below we answer each question systematically, spelling out the underlying physics and the mathematics that Witten used.

2. Answer to Question 1 – Why must we frame Wilson lines?

2.1 Wilson lines in Chern‑Simons theory

In pure Chern‑Simons theory with gauge group (G) and level (k),

[ S_{\text{CS}}[A]=\frac{k}{4\pi}\int_M \operatorname{Tr}!\left(A\wedge dA+\tfrac{2}{3}A\wedge A\wedge A\right) ]

the classical equations of motion are (F=0); the gauge field is flat.
A Wilson line in representation (R) along a (oriented) curve (\gamma) is

[ W_R(\gamma)=\operatorname{Tr}R \,\mathcal{P}\exp!\left(i\int\gamma A\right) . ]

In the topological quantum field theory (TQFT) we would like the vacuum expectation value (VEV)

[ \langle W_R(\gamma)\rangle = \int \mathcal{D}A\,e^{iS_{\text{CS}}[A]}W_R(\gamma) ]

to be a topological invariant of the embedded curve (or link, when several lines are present).

2.2 The framing anomaly

When the path integral is regularised (e.g. using point‑splitting, Pauli–Villars, or lattice regularisation) a short‑distance cutoff must be introduced. Because Chern‑Simons theory is first‑order in derivatives, the regulator inevitably breaks the naïve diffeomorphism invariance: the regularised theory depends on a choice of framing – a continuous choice of a non‑vanishing normal vector field along each Wilson line.

What actually happens is that the self‑linking number (or framing number) of the curve appears in the perturbative expansion.
In perturbation theory the one‑loop diagram that contracts the Wilson line with itself yields a factor

[ \exp!\Bigl(2\pi i\,\frac{C_2(R)}{k}\, \mathrm{fr}(\gamma)\Bigr), ]

where (\mathrm{fr}(\gamma)) is the self‑linking (framing) number and (C_2(R)) is the quadratic Casimir of (R).

If we ignore the framing (i.e. set (\mathrm{fr}=0) by hand) the result is finite, but it depends on the metric that was used to define the short‑distance cutoff. Changing the metric changes the way the Wilson line is regularised, and the VEV changes by precisely the above phase.

2.3 Why metric‑dependence makes the observable “invalid” as a TQFT observable

In a topological quantum field theory the set of admissible observables must be metric‑independent after renormalisation. Otherwise the theory would not be purely topological; different choices of background geometry would give different numbers for the same knot.
The finite value you obtain for an unframed line is not a topological invariant; it is a regularisation‑scheme dependent number. In a rigorous mathematical definition of the Chern‑Simons TQFT (Reshetikhin–Turaev, Witten‑Reshetikhin–Turaev invariants) the path integral is defined only after a framing has been fixed. The resulting invariant is the framed Jones (or HOMFLY‑PT, etc.) polynomial.
One can of course choose a particular framing (e.g. the blackboard framing) and declare that to be the definition of the observable. But the choice must be recorded; otherwise the “observable” is ambiguous.

Hence the answer: Unframed Wilson lines are not part of the spectrum of the Chern‑Simons TQFT because they are not topological observables – they retain a hidden dependence on the metric introduced by the regularisation. They are perfectly fine as operators in a non‑topological regularised gauge theory, but they do not define the knot invariants that Witten was after.

3. Answer to Question 2 – Framing change ↔ anyonic statistics

3.1 Wilson lines ending on particles

In a (2+1)‑dimensional Chern‑Simons theory coupled to static point charges, the Wilson line can be thought of as the world‑line of a heavy particle that carries a representation (R) of the gauge group. The particle itself lives at the endpoints of the line (if the line is open) or, in a closed loop, there are no endpoints and the line just creates a “flux tube”.

The exchange of two such particles is represented by a braid of their world‑lines. Because the Chern‑Simons action is first order in time, the path‑integral weight of a braid is purely a phase: the theory is abelian (in the sense of “no local degrees of freedom”), but the phase can be fractional – this is the hallmark of anyons.

3.2 The framing twist

Consider a single Wilson line (\gamma) together with a chosen framing (\hat{n}(s)) (a unit normal vector along the curve). A twist of the framing by (+1) means that as we travel once around the curve the normal vector rotates by (2\pi). Topologically this is equivalent to adding a self‑linking of (+1) (the curve linked with a copy displaced along (\hat{n})).

The perturbative result quoted above tells us that adding one unit of self‑linking multiplies the expectation value by

[ \exp!\Bigl(2\pi i\,\frac{C_2(R)}{k}\Bigr) . \tag{1} ]

For the simplest case (G=SU(2)) and (R) the spin‑(j) representation, (C_2(R)=j(j+1)). Thus a single twist gives the phase

[ \theta_R = \frac{2\pi\,j(j+1)}{k}\; . ]

3.3 Interpreting the phase as a statistical angle

When two identical particles are exchanged, the braid can be decomposed (up to isotopy) into a half‑twist of each particle’s framing plus a trivial motion. More concretely, the world‑line picture of the exchange shown in the question can be continuously deformed into a picture in which each particle’s world‑line acquires half of a full framing twist.

Because the phase (1) is linear in the framing number, a half‑twist contributes half of the phase (1). Consequently the exchange of two identical particles of type (R) multiplies the total wave‑function by

[ e^{i\theta_R/2}= \exp!\Bigl(i\pi\,\frac{C_2(R)}{k}\Bigr) . \tag{2} ]

Equation (2) is precisely the anyonic statistics angle for the quasiparticles associated with representation (R). For (SU(2)_k) with (j=1/2) we get

[ \theta_{1/2}= \frac{\pi}{k}, ]

so for (k=1) the particles are fermions ((e^{i\pi}=-1)), for (k=2) they are semions ((e^{i\pi/2}=i)), etc.

3.4 Relation to the diagrammatic argument in the question

The student’s picture:

Create two particle‑antiparticle pairs.
Braid (exchange) two of the particles.
Annihilate the pairs.

When the braid is “pinched” into a small loop, the loop can be removed at the price of a framing twist. The loop removal contributes exactly the factor (\exp(2\pi i C_2(R)/k)). Since the loop is contractible it does not affect the topology of the remaining world‑lines, but the framing change remains, giving the overall phase (2).

Thus the argument is correct in principle, provided one keeps track of the fact that a full self‑linking (a closed loop) contributes the phase (1); a half linking (the exchange) contributes its square‑root, i.e. the anyonic statistical phase. The sign (-1) that the student obtained corresponds to the special case (k=1), (R) the fundamental of (SU(2)), where the anyons are ordinary fermions.

4. Answer to Question 3 – Does the canonical Gauss law know about framing?

4.1 The Gauss‑law constraint

In the Hamiltonian formulation on a spatial surface (\Sigma) (say a plane or a torus) we write the Chern‑Simons action in temporal gauge (A_0=0):

[ S_{\text{CS}}=\frac{k}{4\pi}\int dt\int_\Sigma \epsilon^{ij}\, \text{Tr}!\bigl(A_i\dot A_j\bigr)\; . ]

Varying with respect to (A_0) (which is a Lagrange multiplier) yields the Gauss law

[ \boxed{ \; \frac{k}{2\pi}\,\epsilon^{ij}F_{ij}(x) = \rho(x) \; } , \tag{3} ]

where (\rho) is the external charge density (e.g. Wilson line endpoints). In the quantum theory this becomes an operator condition on physical states:

[ \bigl(\rho(x)-\frac{k}{2\pi}\epsilon^{ij}\partial_i A_j(x)\bigr)\,|\psi\rangle =0 . ]

4.2 Where does framing appear?

Equation (3) itself is purely topological; it contains no reference to any framing or metric. The Gauss law tells us that the magnetic flux through any small region equals the charge inside, i.e. the holonomy of the gauge field is fixed by the Wilson‑line insertions.

The framing dependence enters later, when we evaluate the Wilson‑line expectation value inside the physical Hilbert space defined by (3). Concretely:

Regularisation – To define the product (W_R(\gamma)) as an operator, we must regularise the coincident point on the curve. The standard point‑splitting regularisation separates the curve from a copy displaced along a chosen normal vector field. That normal field is exactly the framing.
Operator ordering / path‑integral measure – In canonical quantisation the equal‑time commutator

[ [A_i^a(x),A_j^b(y)] = \frac{2\pi i}{k}\,\epsilon_{ij}\,\delta^{ab}\,\delta^{(2)}(x-y) ]

implies that a Wilson line operator is ill‑defined without a prescription for how to treat the self‑contraction of the line. The prescription is equivalent to a choice of framing.

Physical consequence – When we compute the matrix element (\langle\psi_f

W_R(\gamma)

\psi_i\rangle) using the Gauss‑law‑constrained Hilbert space, the result acquires the framing factor (1). Thus the Gauss law does not “know” about framing, but the representation of Wilson‑line operators on the constrained Hilbert space does, because the latter requires a regularisation that breaks diffeomorphism invariance.

4.3 Summary

The constraint (3) is metric‑free and framing‑free.
The observable (the Wilson line) is only defined after we choose a framing in order to regulate its self‑linking.
Consequently, the physical Hilbert space together with a choice of framing yields a well‑defined representation of the braid group, and the anyonic phases appear precisely from the framing choice.

5. Final Answers

#	Answer (concise

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