Multipole gauge theories and fractons

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I will describe an infinite set of exotic gauge theories that have recently and simultaneously emerged in several a priori unrelated areas of condensed matter physics such as self-correcting quantum memory, topological order in 3+1 dimensions, spin liquids and quantum elasticity. In these theories the gauge field is a symmetric tensor (not to be confused with higher form, which is an anti-symmetric tensor), or in more exotic situations, the gauge fields do not have a well-defined transformation properties under rotations. I will discuss a few exotic features of these theories such as (i) corresponding Gauss law constraints (ii) failure of the gauge invariance in curved space, (iii) the nature of the gauge group, etc. I will also discuss the what kind of matter such theories can couple to. It turns out that the corresponding matter must conserve electric charge and various multipole moments of the electric charge (or number) density. The conservation laws of multipole moments lead to dramatic consequences for the dynamics. I will also discuss how such theories can be obtained by gauging a global symmetry. Finally, I will discuss non-local operators in this type of theories. Remarkably, in addition to more-or-less expected Wilson line and surface operators, such theories exhibit (at least upon discretization on a lattice) non-local operators supported on a space of fractional dimension (in between line and surfaces).