  The objective of the present work -- a literature review on both gapped and
gapless fractonic theories -- is to pedagogically fill in the gaps between the
research on fractons, and an undergraduate physics education (particularly
quantum and statistical mechanics). Some familiarity with classical field
theory is assumed. We will begin this work by reviewing the gauging of the
Ising model to obtain the toric code. Then, following the chronological order
of fracton research, we will establish the gauged spin model picture of gapped
fracton theories. Next (after introducing lattice gauge theory) we will cover
the developments on the tensor gauge theory front in describing gapless fracton
theories. We then explain how conservation of dipole moment is key in the
development of field-theoretic descriptions of fracton models, and conclude by
providing future plans for work on gauging such field theories. Finally, we
point out two more future plans: one being an established theoretical advance
which in fact inspired this work, and the other being exciting experimental
advances in ultracold atomic physics which could lead to the physical
realization of fracton physics. Highlights of this work include: a derivation
of the gauge field dynamics (field strength/curvature) term in the gauging of a
spin system to obtain the toric code, a reduction of the 3 terms of the
Kogut-Susskind lattice QED Hamiltonian to the 2 terms of the toric code, a
proof of the correspondence between the invariance of a gauge field and a
particular Gauss's law constraint in a tensor gauge theory context, a
demonstration that conservation of dipole moment accounts for the same
immobility/fractalization phenomenology that we also show from a gauged spin
model perspective, and proof of the necessity for polynomial shift symmetries
and higher-order spatial derivatives in a Lagrangian in order to have dipole
moment conservation.
