Fractons with twisted boundary conditions and their symmetries

We study several exotic systems, including the X-cube model, on a flat three-torus with twist in $xy$-plane. The ground state degeneracy turns out to be sensitive function of various geometrical parameters. Starting from lattice, depending how we take continuum limit, find different values degeneracy. Yet, there is natural limit well-defined (though infinite) value that also uncover surprising global symmetry $2+1$ and $3+1$ dimensional systems. It originates underlying subsystem symmetry, but way it realized depends twist. In particular, preferred coordinate frame, modular parameter twisted two-torus $\tau = \tau_1 + i \tau_2$ has rational $\tau_1 k / m$. Then, systems based $U(1)\times U(1)$ symmetries, such as momentum winding symmetries or electric magnetic new projectively $\mathbb{Z}_m\times \mathbb{Z}_m$, which leads an $m$-fold $\mathbb{Z}_N$ like each these two $\mathbb{Z}_m$ factors replaced by $\mathbb{Z}_{\gcd(N,m)}$.