Some Chern-Simons theory

In this article (on the way to describing anyons as quantum knots) we explain a few things related to quantum field theory and cohomology. Specifically, we need the fields equations of Chern-Simons theory and indicate why the Chern form is telling us something about topology. Conceptually, if there are holes in spacetime this quantum field can notice it. It generalizes, in a way, the Aharonov-Bohm effect we have described in a previous article.

The maths below is at the same time extremely standard (any textbook on QFT contains this) and extremely magic if you are only used to classic computer science. If you need help, have a look at recent books like Quantum field theory for the gifted amateur or Quantum field theory in a nutshell.

Let $\mathcal{M}$ be a manifold, $G$ a gauge group and $A$ a Lie algebra valued field on the manifold:

$$A = A^\mu dx^\mu = A^a_\mu T_a dx^\mu$$

with $T_a$ the generators of the Lie algebra satisfying the usual

$$[T_a,T_b] = f_{ab}^cT_c.$$

The gauge group acts on the connection $A$ as

$$A\mapsto A^U = U^{-1}AU + U^{-1}dU$$
and the curvature (or field strength)

$$F = dA + A^2$$

changes accordingly as

$$F \mapsto F^U = dA^U + (A^U)^2 = U^{-1}FU.$$

The trace

$$P_{2n}(F):= \text{Tr}(F^n)$$

is invariant under the gauge transformation due to the cyclic property of the trace. It’s a $2n$-form and is called the Chern form on the manifold. Besides being invariant the form is also closed (aka the Chern-Weil theorem). To prove this one notes that the covariant derivative changes to the normal one when moving outside the trace:

\begin{align}\text{Tr}(DF) &= \text{Tr}(dF + [A,F])\ &=d\text{Tr}(F) + \text{Tr}(AF – (-)^{1×2}FA)\ &= d\text{Tr}(F)\end{align},

as well as the well-known Bianchi identity:

\begin{align} DF &= dF+[A,F] \\ &= d(dA +A^2) + [A, dA + A^2] \\ &= 0. \end{align}

Hence, the Chern-Weil statement becomes always trivial

$$dP_{2n}(F) = \text{Tr}(DF\wedge F\ldots+ F\wedge DF\wedge\ldots + \ldots +F\wedge\ldots DF) = 0$$

and it shows that it sits in the cohomology class $H^{2n}(\mathcal{M})$. It’s well-defined in the sense that when one shifts the connection the result is an exact form and thus remains in the same cohomology class:

\begin{align}\delta P_{2n}(F) & =\text{Tr}(\delta F\wedge F\ldots+ F\wedge \delta F\wedge\ldots + \ldots +F\wedge\ldots \delta F) \\ &= n \text{Tr}(\delta F\wedge\ldots\wedge F) \\ &= n \text{Tr}(D\delta A\wedge\ldots\wedge F) \\ &= n \text{Tr}(D(\delta A\wedge\ldots\wedge F)) \\ &= n d\,\text{Tr}(\delta A\wedge\ldots\wedge F). \end{align}

If we now focus on a linear offset defined as

$$\delta A = A’ – A,\;A’ = A+t\delta A$$
the difference is exact as well
\begin{align}P_{2n}(F’)- P_{2n}(F) &= \text{Tr}(F’^n) – \text{Tr}(F^n) \\ &=\int_0^1\frac{d}{dt}(\text{Tr}(F_t^n))\,dt \\ &=n\, \int_0^1 d\text{Tr}(\delta A\wedge F^{n-1}_t)\,dt \\ &= d\left(n\, \int_0^1 \text{Tr}(\delta A\wedge F^{n-1}_t)\,dt\right) \end{align}

One can thus define the so-called Chern class $c_n(E)$ of the vector bundle over the manifold. The computation above works locally but not globally and it’s the topological deficit which shows up in the character. In fact, because of the Poincare lemma one can locally find a form such that

$$P_{2n}(F) = dQ_{2n-1}(F).$$

If this would be globally true the Chern character would automatically be zero obviously since in that case

$$\int_\mathcal{M} P_{2n}(F) = \int_\mathcal{M} dQ_{2n-1}(F) = \int_{\partial\mathcal{M}} Q_{2n-1}(F)=0.$$

The $(2n-1)$-form $Q_{2n-1}(F)$ is called the Chern-Simons form. For $n=1$ one has

$$P_2(F) = \text{Tr}(F) = \text{Tr}(dA+A^2) = d\text{Tr}(A)$$

and the Chern-Simons form is hence $Q_1 = \text{Tr}(A)$. For $Q_2$ one starts with $A_t = t\,A$ and

$$F_t = dt\wedge A + t\,dA + t^2\,A^2.$$

The Chern-Simons form becomes

\begin{align}\text{Tr}(F^2) &= \int_0^1 \frac{d}{dt} \text{Tr}(F^2_t) \,dt \\ &= d\text{Tr}(A\,dA + \frac{2}{3} A^3) \end{align}

from which you can deduce that the Chern-Simons 3-form is

$$\text{Tr}(A\,dA + \frac{2}{3} A^3).$$
An explicit calculation shows that the form, when differentiated, indeed does give the Chern character.

In the context of topological quantum computing one usually looks at the 3D Chern-Simons form with $U(N)$ as a gauge group and a field action defined as

$$\mathcal{S}_{CS} := \kappa\int \text{Tr}(\,A\,dA + \frac{2}{3}A\wedge A \wedge A).$$
If you prefer, the non-Abelian Chern-Simons Lagrangian in components form is

$$\mathcal{L}_{CS} = \kappa\epsilon^{\mu\nu\sigma}\text{Tr} (A_{\mu} \partial_{\nu} A_{\sigma} + \frac{2}{3} A_{\mu} A_{\nu} A_{\sigma}).$$

Through this the field equations can be found

$$\delta\mathcal{S}_{CS} = \kappa \int \epsilon^{\mu\nu\sigma}\text{Tr} (\delta A_{\mu} \,F_{\nu\sigma} )$$

which means that the curvature vanishes

$$F = 0.$$

Classically the dynamics of the Chern-Simons field is trivial. The theory is called a topological theory not just because of the link to topological invariants but also because the Lagrangian does not depend on the metric

$$\frac{\delta \mathcal{L}_{CS}}{\delta g^{\mu\nu}} = 0$$

the field does not ‘feel’ any gravitational forces. It only feels topological deformations and singularities.

Much can be said at this point about how Chern-Simons articulates loop quantum gravity, topological string theories, holographic duals and whatnot. While a lot of fun it would also take me too far away from quantum computing. So, let’s just say that the overlap of quantum computing and field theory is a rich domain where lots of fertile ideas come together. Including consciousness and Penrose-like theories of awareness.