# Different sectors of the Ising gauge theory

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The Hamiltonian of Quantum 2D Ising gauge theory is given by:
$$H=-\sum_p \prod_{i\in \square}\sigma^z_i -g \sum_{i\in \text{links}} \sigma^x_i$$

This $H$ is invariant under the local symmetries:
$$G_i=\prod_{l\in +}\sigma^x_l \,\,\,\,\, (i \in \text{vertices})$$
Now, we can look at $G_i$ in two different ways:

1- That $G_i$s are actual symmetries of this Hamiltonian and that star operators map different physical states to each other. Therefore we have a $2^N$ dimensional Hilbert space and the ground state (for $g\to 0$) is highly degenerate.

2- That $G_i$ are gauge transformations and therefore maps a state to a physically equivalent state and therefore we might say that our expression of $H$ has redundancy. This interpretation that I read about in Wen's book and Kogut's paper. In this case, we demand that :
$$G_i |\text{phys}\rangle = |\text{phys}\rangle$$
For every physical state. Since we build our Hilbert space out of only physical states, we can state that:
$$G_i=1$$
Which states that the electric flux is zero everywhere (The Gauss law in the absence of charge). There are two phases for $g\ll 1$ (deconfined phase) and $g \gg 1$ (confined phase). The ground state of the confined phase is non-degenerate ($\sigma^x=1$ for every link) while for the confined phase the degeneracy depends on the genus of system's manifold. Then we can couple this Ising gauge field to matter fields (of different kind), the simplest case being an Ising matter field given by $\tau^\alpha_i$ defined on the vertices of lattice via coupling:
$$H_c=-t \sum_{\langle ij \rangle}\sigma^z_{ij} \tau^z_i \tau^z_j$$
But now we have to modify the generators of gauge transformations to:
$$G_i=\tau^x_i\prod_{j}\sigma^x_{ij}$$
Now we factorize the original (large) Hilbert space according to the rule $G_i |\text{phys}\rangle = |\text{phys}\rangle$ to get:
$$\prod_{j}\sigma^x_{ij}=\tau^x_i$$
Which states that the electric flux can be non-zero in the presence of charge. So everything is well defined and elegant.

Nevertheless, I have seen that people use the Hamiltonian in conjunction with the constraint:
$$G_i|\text{phys}\rangle = \pm|\text{phys}\rangle$$

And state that the system is a $Z_2$ gauge theory for $+$ and a quantum dimer model for $-$. This, obviously, is not a gauge fixing condition for the minus sign, at least, since it alters the spectrum of the system. People say that choosing the value of $G_i$ determines the sector of the system. But to me, It does not even make sense to speak about the eigenvalue of $G_i$ beside $G_i=1$ because I understand the Hilbert space of the gauge theory as the equivalence classes defined over the original (large) Hilbert space. So every state in the original Hilbert space belongs to one of the equivalence classes which span the physical (smaller) Hilbert space. But the ground state of $H$ at $g\to \infty$ for which $\sigma^x_i=1$ on every link does not even exist if we impose $G_i=-1$ !

The same issue exists also for compact U(1) gauge theory (and probably for other gauge theories as well) where the electric flux operator $\Delta_\alpha E_\alpha(\mathbf{r})$ is the generator of gauge transformations. In this case, it is stated that  the "physical sector" is given by(in Kogut(1979) and Fradkin's book, for example):

$$\Delta_\alpha E_\alpha(\mathbf{r})|\text{phys}\rangle=0$$

which again makes sense, But like the Ising it appears that we can impose another constraint of the type:

$$\Delta_\alpha E_\alpha(\mathbf{r})=n(\mathbf{r})$$

Where $n(\mathbf{r})$ is an integer (due to compactness of U(1) field), and again we can expect the spectrum to depend on the value of $n(\mathbf{r})$.

So what is the problem here? Do people just use the Hamiltonian of Ising gauge theory and apply different constraints just to get new quantum models and this has nothing to do with the gauge theory? Should we make a distinction between the Ising gauge theory (defined on a $2^N$ dimensional Hilbert space and a Hamiltonian given above) and the $\mathbb{Z}_2$ gauge theory (with a smaller Hilbert space and the same Hamiltonian) and say that:

$$\text{Ising gauge theory} \, + \left( \, \prod_{i \in +} \sigma^x_i=1 \right) \, = \, \mathbb{Z}_2 \, \text{gauge theory}$$

$$\text{Ising gauge theory} \, + \, \left( \prod_{i \in +} \sigma^x_i=-1 \right) \, = \, \text{Quantum dimer model}$$

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