In the section 4.1 of Quantum Computation by Adiabatic Evolution, Farhi et al proposes a quantum adiabatic algorithm to solve the $2$-SAT problem on a ring. To compute the complexity of the algorithm the authors computed the energy gap between the ground and first excited states of the adiabatic Hamiltonian.
The adiabatic Hamiltonian is defined as $$ \tilde{H} (s) = (1-s) \sum^n_{j=1}(1-\sigma^{(j)}_x) + s \sum^n_{j=1}\frac{1}{2} (1-\sigma^{(j)}_z \sigma^{(j+1)}_z ) $$
To prove the correctness of the algorithm, the authors consider an operator which negates the value of the bits in the $z$ axis. $$ G = \prod^n_{j=1}\sigma^{(j)}_x $$
Then the authors start the steps of Jordan-Wigner transformation. The fermionic operators are defined as follows.
$$ b_j = \sigma_x^1 \sigma_x^2 \ldots \sigma_x^{j-1} \sigma_-^{j} \mathbf{ 1}^{j+1} \ldots \mathbf{ 1}^n \\ b^\dagger_j = \sigma_x^1 \sigma_x^2 \ldots \sigma_x^{j-1} \sigma_+^{j} \mathbf{ 1}^{j+1} \ldots \mathbf{ 1}^n $$ where $$ \sigma_{\pm} = \sigma_x \pm i \sigma_y $$.
Then the authors keep mentioning few more identities. Then at the beginning of page 14, they say
Since we will restrict ourselves to the $G = 1$ sector, (4.10) and (4.11) are only consistent if $b_{n+1} = −b_1$ , so we take this as the definition of $b_{n+1}$.
My questions: I understand that $G$ is an operator which commutes with the adiabatic Hamiltonian and it negates the values of the qubits in the $z$ axis. But, I don't understand what the definition of the 'sector' of $G$ is, when the sector is $G = 1$, what other sectors are possible, and finally why $b_{n+1} = b_1$ cannot be true.
My attempt: In the computational basis i.e. along $z$ axis, $\sigma^x |1\rangle = 1\cdot|0\rangle$ and $\sigma^x |0\rangle = 1\cdot|1\rangle$. So, the eigenvalue is always $1$. Is this the sector, the authors are talking about?
