The idea that gauge theory has ‘surplus’ structure poses a puzzle: in one much discussed sense, this structure is redundant; but on the other hand, it is also widely held to play an essential role in the theory. In this article, we employ category-theoretic tools to illuminate an aspect of this puzzle. We precisify what is meant by surplus structure by means of functorial comparisons with equivalence classes of gauge fields, and then show that such structure is essential for any theory that represents a rich collection of physically relevant fields that are ‘local’ in nature. 1Introduction2Theories as Categories 2.1Relations between models2.2Relations between theories3Gauge Theory as a Category 3.1Gauge theory on contractible manifolds3.2Other candidates for representing U(1) gauge theory3.3Surplus and inter-theoretical comparisons4Gauge Theory as a Functor 4.1Richness and locality4.2Richness and locality imply surplus*5ConclusionAppendix

This article investigates and resolves the question of whether gauge symmetry can display analogs of the famous Galileo's ship scenario. In doing so, it builds on and clarifies the work of Greaves and Wallace (2014) on this subject.

We provide an elegant homological construction of the extended phase space for linear Yang-Mills theory on an oriented and time-oriented Lorentzian manifold M with a time-like boundary ∂M that was proposed by Donnelly and Freidel [JHEP 1609[JHEP , 102 (2016]. This explains and formalizes many of the rather ad hoc constructions for edge modes appearing in the theoretical physics literature.

We construct a notion of teleparallelization for Newton-Cartan theory, and show that the teleparallel equivalent of this theory is Newtonian gravity; furthermore, we show that this result is consistent with teleparallelization in general relativity, and can be obtained by null-reducing the teleparallel equivalent of a five-dimensional gravitational wave solution. This work thus strengthens substantially the connections between four theories: Newton-Cartan theory, Newtonian gravitation theory, general relativity, and teleparallel gravity.Newton-Cartan theory and the teleparallel analogy-Although Newton-Cartan theory (NCT) was originally conceived as a 'geometrized' version of Newtonian gravitation [12], this theory has recently been found to have a wide and impressive range of further applications: it provides an improved model of the fractional quantum Hall effect [24,43], a geometric foundation for Horava-Lifschitz gravity [27], and a framework for non-relativistic holography [13,14], to name just a few examples. Many of these applications owe their success to two powerful techniques for analyzing NCT. First, the recentlydeveloped vielbein formalism for NCT [4,23] has played an indispensable role in understanding the hidden local Galilean invariance of NCT-akin to the hidden local Lorentz invariance of general relativity (GR)-and in analyzing the coupling of matter fields to a general Newton-Cartan background spacetime. Second, the technique of null dimensional reduction [17,30]-which allows one to formulate a D-dimensional solution of NCT as a certain (D + 1)-dimensional gravitational wave (Bargmann-Eisenhart) solution to GR-has provided an especially efficient method of using a relativistic spacetime to analyze the symmetries [18] and dynamics [19] of a large class of non-relativistic mechanical systems.Despite the obvious relevance and power of these techniques, they have not yet been applied to what is in some sense the fundamental theorem of NCT, viz., the Trautman Recovery Theorem [45], which asserts that curved Newton-Cartan gravity is 'empirically equivalent' to the standard (flat, but with forces) Newtonian theory of gravitation. Furthermore, it seems to have gone unnoticed in the literature that such a statement is highly analogous to the key idea driving teleparallel gravity (TPG), which is that generically curved models of GR are empirically equivalent (modulo subtleties-see e.g. [41,42]) to a theory with a flat but torsionful connec-

We discuss some aspects of the relation between dualities and gauge symmetries. Both of these ideas are of course multi-faceted, and we confine ourselves to making two points. Both points are about dualities in string theory, and both have the 'flavour' that two dual theories are 'closer in content' than you might think. For both points, we adopt a simple conception of a duality as an 'isomorphism' between theories: more precisely, as appropriate bijections between the two theories' sets of states and sets of quantities.The first point (Section 3) is that this conception of duality meshes with two dual theories being 'gauge related' in the general philosophical sense of being physically equivalent. For a string duality, such as T-duality and gauge/gravity duality, this means taking such features as the radius of a compact dimension, and the dimensionality of spacetime, to be 'gauge'.The second point (Sections 4, 5 and 6) is much more specific. We give a result about gauge/gravity duality that shows its relation to gauge symmetries (in the physical sense of symmetry transformations that are spacetime-dependent) to be subtler than you might expect. For gauge theories, you might expect that the duality bijections relate only gauge-invariant quantities and states, in the sense that gauge symmetries in one theory will be unrelated to any symmetries in the other theory. This may be so in general; and indeed, it is suggested by discussions of Polchinski and Horowitz. But we show that in gauge/gravity duality, each of a certain class of gauge symmetries in the gravity/bulk theory, viz. diffeomorphisms, is related by the duality to a position-dependent symmetry of the gauge/boundary theory.