Exact bosonization in arbitrary dimensions

Abstract

We extend the previous results of exact bosonization, mapping from fermionic operators to Pauli matrices, in 2D and 3D to arbitrary dimensions. This bosonization map gives a duality between any fermionic system in arbitrary n spatial dimensions and a class of (n − 1)-form Z 2 gauge theories in n dimensions with a modified Gauss's law. This map preserves locality and has an explicit dependence on the second Stiefel-Whitney class and a choice of spin structure on the spatial manifold. A formula for Stiefel-Whitney homology classes on lattices is derived. In the Euclidean path integral, this exact bosonization map is equivalent to introducing a topological Steenrod square term to the space-time action.

Bosonization based on Clifford algebras and its gauge theoretic interpretation

“…Afterwards we present a certain generalization of this model, in which Gauss' operators of conventional Z 2 gauge theory are modified by including phases depending on values of the holonomies. Similar mechanism is present in the Dijkgraaf-Witten theory and has been applied in the bosonization map introduced in [13,14,34]. In contrast to Dijkgraaf-Witten models, here we are not restricting attention to topological gauge theories.…”

“…There exists a duality mapping which relates the Γ model to higher gauge theories proposed in the context of bosonization in [13,14,34]. In some aspects it resembles the classical Kramers-Wannier duality [35].…”

“…More precisely, we will dualize only operators Γ * (v) and S(e), which generate the commutant of constraint operators arising in bosonization. Composition of this map with the correspondence between the Γ model and fermions yields bosonization introduced in [13,14,34].…”

“…10 As in ordinary gauge theory, parallel transports over individual cells will turn out not to be gauge-invariant. To construct an observable one has to take the product over all Following [13,14,34], we would like to map operators Γ * (v) of the Γ model to h(v). This is possible only upon restricting to the subspace defined by the condition…”

“…We remark that in [13,14,34] a more specific construction of duality between fermions and higher gauge theory, essentially corresponding to a particular choice of the function ν, was presented. It is formulated in terms of higher cup products [44] and depends on a choice of a branching structure 11 on the dual lattice, assumed to be a triangulation.…”