Edge modes without edge modes, Aldo Riello (UL Brussels)


Monday, March 1, 2021 - 21:30




This will be the first talk of a two-part mini seminar series, with the second talk the week after, March 8same time. Please find the details below.


Zoom link for the first talk is:




Meeting ID: 929 7297 9992

Passcode: 246571


Time: 9:30 pm JST/1:30pm CET/7:30am EST


Title: Edge modes without edge modes: cutting and gluing in Yang-Mills gauge theory


Abstract: In this two-part talk, I will discuss gauge theories of the Yang-Mills kind in finite regions with boundaries.

In the first talk, I will address the “cutting problem”, namely the problem of giving a definition of the gauge-invariant degrees of freedom (dof) intrinsic to a finite and bounded region. By analyzing the symplectic reduction procedure in the presence of boundaries, and how boundaries interact with the Gauss constraint, I will show that the reduced phase space is not a symplectic space, but a symplectic foliation whose leaves correspond to electric-flux superselection sectors. In each superselection sector, the relevant dof are a regional generalization of the usual radiative dof (i.e. transverse photons). Key to obtaining this result is our indiscriminate treatment of bulk and boundary gauge transformations, which means no "edge modes" are ever introduced. The absence of edge modes, however, could raise the suspicion that some crucial information is missing, e.g. in relation to one’s ability to reconstruct the global radiative dof from the regional ones. This concern will be addressed in the second talk. 

Indeed, in the second talk I will address the two-fold “gluing problem”. Whereas the first gluing problem concerns the reconstruction of the (usual) global radiative dof from the regional dof in each superselection sector, the second gluing problem is the well-known problem of explaining the non-factorizability of the gauge-invariant phase space over regions, namely the emergence, upon gluing, of new (gauge invariant) dof. Absent edge modes, only the regional radiative dof in each superselection sector are available in the discussion of the first and second gluing problems. But this raises an “edge-modes-without-edge-modes" puzzle: if all global radiative dof are reconstructible from their regional counterparts, how can new dof emerge upon gluing? My goal for this second talk will be to explain the resolution of the “edge-modes-without-edge-modes” puzzle. The resolution relies on the nonlocal nature of the regional radiative dof and on the relational nature of the new reconstructed dof. One of this new dof is the electric flux itself: understanding its reconstruction will shed further light on its superselection.


The main reference for this talk are: 2010.15894 for part 1, and section 6 of 2007.04013 (with H . Gomes) for pat 2.

Note: whereas these papers strive for generality, in the talk I will for the most part focus on Maxwell theory, which, being Abelian, allows for a much simpler treatment. Only brief references will be made to the non-Abelian theory, aimed at emphasizing its extra features.

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