Abstract
We study the properties of a bosonization procedure based on Clifford algebra valued degrees of freedom, valid for spaces of any dimension. We present its interpretation in terms of fermions in presence of ℤ2 gauge fields satisfying a modified Gauss’ law, resembling Chern-Simons-like theories. Our bosonization prescription involves constraints, which are interpreted as a flatness condition for the gauge field. Solution of the constraints is presented for toroidal geometries of dimension two. Duality between our model and (d − 1)- form ℤ2 gauge theory is derived, which elucidates the relation between the approach taken here with another bosonization map proposed recently.
Highlights
The most well-known bosonization methods apply only to 1 + 1-dimensional systems
We provide a new interpretation of constraints present in the Γ model as the pure gauge condition for a certain Z2 gauge field
Does the Γ model with no constraints imposed provide a bosonization of a some theory of fermions coupled to a Z2 gauge field? We show that such mapping does exist
Summary
For any finite set S we let |S| be the number of elements of S. It is a classical result [37, section 4.2.1] in graph e∈E theory that Eulerian circuit exists if and only if every vertex has even degree The latter condition is equivalent to closedness of the chain ζ := e ∈ C1, i.e. to ∂ζ = 0. In some parts of this work (not essential for the main construction) we will have to assume that besides vertices and edges, the considered lattice is equipped with a set of faces F , which are polygons whose sides are identified with edges. This allows to define the space of 2-chains C2 with an obvious boundary map ∂ : C2 → C1.
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