CA9 › Recherche › Cours › Advanced › Chapter 2

Chapter 2 •Advanced •~3 h •Graduate / PhD

Construction of the χ field

Note: the canonical numbers and results of this course are those of the TQNT monograph — the spacetime weave, m = √σ_χ 𝒮, G = ℏc/M_Pl², the Koide relation explained. Passages documenting older, now archived approaches are flagged as such.

The field $\chi$ is not an ordinary scalar field but a 1-form valued in the Lie algebra $\mathfrak{su}(2)$. This chapter explicitly constructs its Lagrangian, its equation of motion, and characterises the space of its classical solutions (the moduli space of flat connections modulo gauge).

1. Bundle and geometry

Let $M^3$ be an oriented 3-manifold. Consider the trivial principal bundle $P = M^3 \times \mathrm{SU}(2)$ with structure group $G = \mathrm{SU}(2)$. The field $\chi$ is a connection on $P$, i.e. locally a 1-form:

$$ \chi = \chi^a_\mu(x)\, T^a\, dx^\mu, \qquad T^a = \tfrac{i}{2}\sigma^a, \quad a = 1,2,3 $$

where the $\sigma^a$ are the Pauli matrices. The associated curvature reads:

$$ F = d\chi + \chi \wedge \chi, \qquad F^a_{\mu\nu} = \partial_\mu \chi^a_\nu - \partial_\nu \chi^a_\mu + \varepsilon^{abc}\, \chi^b_\mu\, \chi^c_\nu $$

2. Chern–Simons action

The fundamental action of the field $\chi$ is the Chern–Simons action at integer level $k$:

$$ S_{CS}[\chi] = \frac{k}{4\pi} \int_{M^3} \mathrm{Tr}\!\left(\chi \wedge d\chi + \tfrac{2}{3}\, \chi \wedge \chi \wedge \chi\right) $$

In components:

$$ S_{CS} = \frac{k}{8\pi} \int_{M^3} \varepsilon^{\mu\nu\rho}\!\left(\chi^a_\mu \partial_\nu \chi^a_\rho + \tfrac{1}{3}\, \varepsilon^{abc}\, \chi^a_\mu \chi^b_\nu \chi^c_\rho\right) d^3x $$

Why Chern–Simons?

Topological: independent of any metric on $M^3$. Gravity will emerge afterwards (P17), not upstream.
Renormalisable: quadratic plus cubic action, with no derivatives of order $\geq 2$.
Solitons = knots: the non-trivial classical configurations are indexed by the knots in $M^3$.
Integer quantisation: under large gauge transformations, $S_{CS} \to S_{CS} + 2\pi k\, n$ for $n \in \mathbb{Z}$, hence $e^{iS_{CS}}$ is well-defined iff $k \in \mathbb{Z}$.

Witten's result (1989)

The partition function of SU(2) Chern–Simons theory at level $k$ on $S^3$ with a Wilson loop along a knot $K$ equals exactly the Jones polynomial evaluated at $t = e^{2\pi i/(k+2)}$:

$$ \langle W_K \rangle_{S^3} = V_K(e^{2\pi i/(k+2)}) $$

This identity is the pivot of all of TQNT: it endows the Jones polynomial with a physical meaning as a quantum amplitude.

3. Equations of motion

Varying $S_{CS}$ with respect to $\chi$ yields:

$$ \frac{\delta S_{CS}}{\delta \chi^a_\mu} = \frac{k}{4\pi}\, \varepsilon^{\mu\nu\rho}\, F^a_{\nu\rho} = 0 \;\Longleftrightarrow\; F = 0 $$

The classical solutions are exactly the flat connections ($F = 0$). This is an algebraic constraint, not an evolution equation — the field $\chi$ is topological in the sense that its classical dynamics reduces to an integrability condition.

4. Character variety

The space of flat connections modulo gauge is in bijection with the space of representations of the fundamental group:

$$ \mathcal{M}_\flat(M^3) \;\simeq\; \mathrm{Hom}(\pi_1(M^3), \mathrm{SU}(2)) \,/\, \mathrm{SU}(2) $$

called the character variety $X(M^3)$.

The case of a knot complement

For a knot complement $M = S^3 \setminus K$, the fundamental group $\pi_1(S^3 \setminus K)$ is the knot group. Its character variety $X(S^3 \setminus K)$ is an algebraic variety of dimension $\geq 1$, and its defining polynomial is precisely the A-polynomial of $K$ (see Chapter 3).

5. Quantisation

One quantises the theory on $\Sigma \times \mathbb{R}$ with $\Sigma$ a Riemann surface. The physical Hilbert space $\mathcal{H}_\Sigma$ is finite-dimensional:

$$ \dim \mathcal{H}_\Sigma = N_{g,k} = \left(\frac{k+2}{2}\right)^{g-1} \sum_{j=0}^{k/2} \left(\sin\frac{(2j+1)\pi}{k+2}\right)^{2-2g} $$

where $g$ is the genus of $\Sigma$. For $\Sigma = S^2$ (genus 0), this is the space of conformal blocks of the affine algebra $\widehat{\mathfrak{su}}(2)_k$.

For an oriented knot $K \subset S^3$: Wilson loops

$$ W_R[K, \chi] = \mathrm{Tr}_R\, \mathcal{P} \exp\!\oint_K \chi $$

where $R$ is an irreducible representation of SU(2) (spin $j$, dimension $2j+1$). For $j = 1/2$, the expectation value yields the Jones polynomial; for generic $j$, one obtains the coloured Jones polynomial $J_n(K, q)$ with $n = 2j$.

6. Level $k = 135$ and the fine-structure constant

The key identification of TQNT is:

$$ \alpha_{EM} = \frac{1}{k+2} $$

With $\alpha_{EM} = 1/137.035999$, one obtains $k+2 = 137.036$. Since $k$ must be an integer, $k = 135$ is the unique compatible value — the fractional part $0.036$ is interpreted as a loop correction (see Ch. 4):

$$ k_{\mathrm{eff}}(\mu) = k + \Delta k_{\mathrm{1-loop}}(\mu), \quad \Delta k \sim O(\alpha_{EM}) $$

7. From Chern–Simons to quantum mechanics

The construction proceeds in five steps (justified in Ch. 3):

Pick $K \subset S^3$ — identify it with a particle through P4.
Compute the coloured Jones polynomial $J_n(K, q)$ with $q = e^{2\pi i/(k+2)}$.
Extract the A-polynomial $A_K(\ell, m)$ as the quantisation class of $J_n$ in the limit $n \to \infty$.
Promote $(\ell, m)$ to operators $\hat{L}, \hat{M}$ satisfying $\hat{M}\hat{L} = q\, \hat{L}\hat{M}$.
Define the AJ operator $\hat{A}_K(\hat{L}, \hat{M}; q)$ and write $\hat{A}_K\, \Psi = 0$ — the emergent Schrödinger equation.

Core insight

The quantisation of the A-polynomial produces a differential operator on the representation space of $\pi_1(S^3 \setminus K)$ — which acts as a Schrödinger equation on the wave function of the associated particle. All of the quantum mechanics of TQNT arises from this procedure.

8. Matter coupling (charge, current)

To introduce a charge $Q$, one adds a minimal-coupling term to the action:

$$ S_{\mathrm{matière}} = \int_{M^3} J^\mu_K\, A^{EM}_\mu\, d^3x $$

with $J^\mu_K$ the topological current density (P5) of the knot $K$ and $A^{EM}$ the usual U(1) electromagnetic connection. The coupling is topological in the sense that $J^\mu_K$ is automatically conserved by knot closure.

9. Continuum limit and emergence of the classical fields

In the hydrodynamic limit (lengths large compared to $\lambda_C$), the amplitude density $|\Psi|^2$ of a collection of Wilson loops satisfies a continuity equation:

$$ \partial_t |\Psi|^2 + \nabla \cdot \vec{j} = 0, \quad \vec{j} = \frac{\hbar}{2mi}(\Psi^*\nabla\Psi - \Psi\nabla\Psi^*) $$

— this is the standard quantum continuity equation, recovered from a purely topological theory. The mass $m$ appears as a dynamical scale tied to the mean circulation around $K$.

10. Regularisation and discretisation

For numerical computations the code theory/chern_simons.py uses:

q-deformation: $q = e^{2\pi i/(k+2)}$, q-integers $[n]_q = (q^{n/2}-q^{-n/2})/(q^{1/2}-q^{-1/2})$
Complexified representations: SL(2,$\mathbb{C}$) instead of SU(2) to obtain an algebraic variety
Adaptive mesh: local refinement around the tubular neighbourhood $N_\epsilon(K)$

End-of-chapter quiz

Four questions on the construction of the χ field.

1. What are the classical solutions of the Chern–Simons action?

Exactly. The equation of motion $F=0$ is an algebraic condition, not an evolution equation.

2. Why must the Chern–Simons level $k$ be an integer?

Right. This is the topological quantisation in the Wess–Zumino–Witten sense.

3. What is the partition function of SU(2) Chern–Simons at level $k$ on $S^3$ with a Wilson loop along $K$?

Correct — this is the founding result of Witten 1989.

4. Which property makes the field $\chi$ a natural candidate for quantum mechanics without introducing a fundamental spacetime?

Exactly. The topological invariance of Chern–Simons is what allows TQNT to invert the hierarchy: metric = by-product, not datum.