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Résumé |
I will present a unified view of dualities and symmetries. We will start by reviewing how conventional dualities help identify topological defects in Ising, XY, and continuum elasticity theories and demonstrate how gauge invariance and conservation laws enforce kinematic glide for dislocations in elastic media. Next we will turn to general d-dimensional gauge-like symmetries which include "higher form" and "subsystem" symmetries. In quantum and classical systems, these structures can lead to dimensional reduction, constrained dynamics, and forms of fractionalization and topological order. After introducing a generalization of Elitzur's theorem, I will show how the delicate confluence of symmetries and system geometry may lead to topological degeneracies and holographic entropies (associated with exponential in boundary area multiplicities). We will introduce a general (bond-algebraic) framework for dualities which we will employ to illustrate that several models that harbor topological order as well as square lattice compass variants of the Hubbard model exhibit exact three-dimensional classical Ising transitions and further use these dualities to compute the free energy of various other systems including the fracton ``X-cube'' model. We will finally turn to to "non-invertible" symmetries and provide a generalization of Wigner theorem which will illustrate that all such symmetries may be recast as invertible ones, with the bond algebra approach providing a unifying tool. Time permitting, we will show why duality transformations are generally conformal and use this property to derive new relations in combinatorial geometry. |
