There is a key result in 2+1 dimensional Chern-Simons theory, which was first discussed in ref.1.: the Hilbert space of the theory, when quantised on $T^2\times\mathbb R$, is isomorphic to $$ \frac{\Lambda_w}{W\ltimes k\Lambda^\vee_r}\tag{2.14} $$ where $\Lambda_w$ is the weight lattice of the gauge group $G$ (assumed simple and simply-connected), $W$ the Weyl group of $\mathfrak g$, $k\in\mathbb Z$ the coefficient of the Chern-Simons form, and $\Lambda_r^\vee$ the co-root lattice of $G$.

Most papers nowadays simply refer to the original reference, or reproduce the argument almost verbatim. The problem is that I cannot quite follow the original reference, and the review articles follow the exact same reasoning, so I cannot follow them either. I have several questions, and I will split them into several posts.

Let $A$ be a $\mathfrak g$-valued one-form that lives in $T^2\times\mathbb R$, and decompose it into its time and spatial components, $A=A_0\mathrm dt+\tilde A$. Due to the equations of motion, $\tilde A$ is $\tilde{\mathrm d}$-flat, $\tilde{\mathrm d}\tilde A+\tilde A^2\equiv 0$, with $\mathrm d=\mathrm dt\partial_t+\tilde{\mathrm d}$. The authors claim that, being $\tilde{\mathrm d}$-flat, it can be decomposed as $$ \tilde A=-\mathrm dUU^{-1}+U\theta(t) U^{-1}\tag{2.9} $$ where $U$ is a single-valued map from $T^2\times\mathbb R$ to $G$, and where $\theta(t)$ is a $\mathfrak g$-valued one form that depends on $t$ only.

Where does this come from?

The authors offer no reference nor explanation. Is this a general result? Does it work only for the torus? What about other surfaces? What about flat connections on a higher-dimensional manifold?

I guess the problem is purely two-dimensional, and the dependence on $t$ is parametrical. In other words, we should think of this as trying to solve $\tilde D\tilde A\equiv0$ on $T^2$, where everything depends implicitly on $t$, as a parameter. Thus, the claim is that $\theta$ is constant over $T^2$.


  1. Elitzur, Moore, Schwimmer, Seiberg - Remarks on the canonical quantization of the Chern-Simons-Witten theory.
  1. Flat $G$-principal connections on $X$ modulo gauge transformations are in bijection with group homomorphisms $\pi_1(X)\to G$ modulo conjugation. The proof of this is "well-known" and can be found e.g. in Kobayashi's and Nomizu's book on differential geometry. A nice exposition is in chapter 5 of "Moduli Spaces of Flat Connections" by Daan Michiels.

    The basic idea is that a flat connection is already determined by its holonomy around non-contractible loops, since it is trivial around contractible loops due to $\oint_\gamma A = \int_S F = 0$ where $\partial S = \gamma$ for some surface $S$ and through an application of (non-Abelian) Stokes. Due to the same argument the holonomies of homotopic loops are equal, and the homotopy classes of non-contractible loops are precisely the non-trivial elements of $\pi_1(X)$. Therefore a homomorphism $\pi_1(X)\to G$ is a neat way to enumerate all non-trivial holonomies.

    Since $\pi_1(T^2) = \mathbb{Z}^2$, such a homomorphism is given by two commuting elements $\Theta_1, \Theta_2\in G$, and these correspond to the holonomies of the connection along the two basic loops in $T^2$.

  2. If $G$ is compact and simply-connected, then there are no non-trivial $G$-principal bundles over compact orientable surfaces, i.e. the connection form is globally defined, or "single valued".

Define $\theta_1 = \mathrm{e}^{\Theta_1}$ and $\theta_2 = \mathrm{e}^{\Theta_2}$ and do so for every time-slice, obtaining functions $\theta_i : \mathbb{R}\to \mathfrak{g}$. Then $$ \theta(t) = \theta_1(t) \mathrm{d}\phi_1 + \theta_2(t) \mathrm{d}\phi_2,$$ with $\mathrm{d}\phi_i$ the two basic cohomologically non-trivial 1-forms on $T^2$ is a connection on $T^2$ with holonomies $\exp(\theta_1(t)),\exp(\theta_2(t))$ at each point in time.

All that is left is to observe that the claimed general solution is then simply a general gauge transformation of this connection.

  • $\begingroup$ Nice, this actually (partially) answers what was going to be part 2. I feel the first sentence is the key, but I'm not sure where it comes from. Should it be obvious? Is it a well-known theorem? Why are we looking at homotopy instead of homology? Is it due to $\pi_1$ being abelian? $\endgroup$ – AccidentalFourierTransform Jul 5 '18 at 19:28
  • $\begingroup$ @AccidentalFourierTransform See my edit $\endgroup$ – ACuriousMind Jul 5 '18 at 19:49
  • $\begingroup$ @AccidentalFourierTransform Hurewicz' theorem states that if $X$ is path-connected then $H_1(X,\mathbb{Z})\simeq\pi_1(X)/[\pi_1(X),\pi_1(X)]$, i.e. the first holomogy group is isomorphic to the Abelianisation of the fundamental group (and since $X$ is path-connected we can omit the base point). $\endgroup$ – green.onion Jul 5 '18 at 21:00
  • $\begingroup$ @green.onion Indeed, that's why I was asking whether $\pi_1$ being abelian was the reason we were considering $\pi_1$ instead of $H$. Thanks anyway. Cheers! $\endgroup$ – AccidentalFourierTransform Jul 5 '18 at 21:14
  • 1
    $\begingroup$ @green.onion I asked why we were using $\pi_1$ instead of $H_1$ before ACM edited the answer :-) $\endgroup$ – AccidentalFourierTransform Jul 5 '18 at 23:24

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.