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It is well known that topological $\theta$-terms in gauge theories are total derivatives and vanish after integration over the Lagrangian (or Hamiltonian) density, unless there are nontrivial boundary terms corresponding to a nontrivial topology of the theory's vacuum. For example, the $\theta$-term of quantum electrodynamics (QED) in 3+1D dimensions vanishes after integration, because the vacuum of the $U(1)$ gauge theory is trivial, i.e., there are no instantons.

To find out whether a vacuum is nontrivial, we usually compute the $j$th homotopy group $\pi_j$ of the gauge group $G$, where $j$ is the spatial dimension of the theory. If $\pi_j(G)=0$, then the vacuum is trivial and the $\theta$-term vanishes. For example, $\pi_3(U(1))=0$ and $\pi_3(SU(3))=\pi_3(SU(2))=\mathbb{Z}$, so we have no $\theta$-term in 3+1D QED but $\theta$-terms in 3+1D quantum chromodynamics (QCD) and electroweak theory (however, the electroweak $\theta$-term can be eliminated by a chiral rotation).

Now I've stumbled across some seemingly inconsistent arguments in various different books (including "Advanced Topics in Quantum Field Theory" by Shifman) about which homotopy groups we need to consider in different dimensions. (All these books are not available online, but Tong's lecture notes on Gauge Theory briefly discusses these issues in section 7.)

The arguments go as follows:

(i) 3+1D QCD contains instantons because of $\pi_3(SU(3))\neq 0$ (see Eq. (20.6) in Shifman's book),

(ii) 1+1D QED contains instantons because of $\pi_1(U(1))\neq 0$ (see Eq. (33.38)),

(iii) the 1+1D $CP^1$ model contains instantons because of $\pi_2(CP^1)\neq 0$ (see Eq. (29.6) in Shifman's book or Tong's arguments below Eq. (7.28)).

However, when applying the arguments (i) and (ii) to (iii), I would expect that there are no instantons and therefore no $\theta$-term in the 1+1D $CP^1$ model because of $\pi_1(CP^1)=0$. Shifman and others argue that we can treat the time direction as a compactified spatial direction in the 1+1D $CP^1$ model, but they only use this argument for the $CP^1$ case! Thus, they consider Minkowski space in the cases (i) and (ii) but Euclidean space in case (iii).

Aren't these arguments inconsistent? Apart from that, even if they somehow hold true in the Lagrangian formalism, shouldn't they break down in the Hamiltonian formalism, where we cannot compactify the time dimension (see, e.g., above Eq. (3.5) in arXiv:1301.4158)?

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These arguments are consistent with each other. The difference is the kind of field they handle.

Since the arguments are about instantons, let's forget about $\theta$-terms, which are just a distraction. An instanton is a field configuration which is a local minimum of the action. The reason topology comes up when thinking about instantons is that is that topological quantities are invariant under continuous deformations, and thus under any infinitesimal deformations. So a field configuration that minimizes the action for a given "topological charge" is automatically an instanton.

The application (3) is just the usual and simplest way of classifying topologically nontrivial field configurations. You just take some topological property of the whole field configuration, after imposing appropriate boundary conditions.

The applications (1) and (2) look different because the relevant field is a gauge field. In these cases, looking at $\pi^4(SU(3))$ and $\pi^2(U(1))$ wouldn't make any sense, because a gauge field is not a map from spacetime to the gauge group $G$. Instead, it is a connection on a $G$-bundle over spacetime. These bundles are characterized by the transition functions between different patches over spacetime, so the standard thing to do is to compactify spacetime to $S^d$, cover the two hemispheres with patches, and consider the transition function between them defined on the overlap $S^{d-1}$. This transition function is a map from $S^{d-1}$ to $G$, so instantons associated with gauge fields are classified by $\pi_{d-1}(G)$.

In other words, we're treating spacetime the exact same way in all three arguments you listed. The only difference is that we need an extra step for gauge fields, which is ultimately due to gauge redundancy.

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  • $\begingroup$ Thanks for this great answer! Do these arguments also hold true for the Hamiltonian formalism, even though we cannot compactify the time dimension there? $\endgroup$
    – anon
    Jan 29 '20 at 1:01
  • $\begingroup$ @Thomas Hmm, I'm not sure what that would mean. An instanton is a tool in the Lagrangian formalism, used to compute tunneling rates. In the Hamiltonian formalism, you just wouldn't use them to begin with. $\endgroup$
    – knzhou
    Jan 29 '20 at 2:04
  • $\begingroup$ Yes, that's why I was wondering whether there is a theta-term in the Hamiltonian formalism. The term is known to exist for the Hamiltonian of e.g. 1+1D QED (where the term is simply a background electric field) but I haven't found a similar statement for the 1+1D $CP^1$ model. $\endgroup$
    – anon
    Jan 30 '20 at 1:16
  • $\begingroup$ Do you have any insights on that? So whether there's a theta-term in the Hamiltonian formulation of the 1+1D $CP^1$ model? The standard argument of having a theta-term in that model is that you can treat the time direction as a compactified spatial direction, but I've only seen this argument for the Lagrangian formulation and I'm wondering whether it stops holding true for the Hamiltonian formulation. $\endgroup$
    – anon
    Jan 31 '20 at 21:44

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