740 research outputs found
Enriched Stone-type dualities
A common feature of many duality results is that the involved equivalence
functors are liftings of hom-functors into the two-element space resp. lattice.
Due to this fact, we can only expect dualities for categories cogenerated by
the two-element set with an appropriate structure. A prime example of such a
situation is Stone's duality theorem for Boolean algebras and Boolean
spaces,the latter being precisely those compact Hausdorff spaces which are
cogenerated by the two-element discrete space. In this paper we aim for a
systematic way of extending this duality theorem to categories including all
compact Hausdorff spaces. To achieve this goal, we combine duality theory and
quantale-enriched category theory. Our main idea is that, when passing from the
two-element discrete space to a cogenerator of the category of compact
Hausdorff spaces, all other involved structures should be substituted by
corresponding enriched versions. Accordingly, we work with the unit interval
and present duality theory for ordered and metric compact Hausdorff
spaces and (suitably defined) finitely cocomplete categories enriched in
The Bond-Algebraic Approach to Dualities
An algebraic theory of dualities is developed based on the notion of bond
algebras. It deals with classical and quantum dualities in a unified fashion
explaining the precise connection between quantum dualities and the low
temperature (strong-coupling)/high temperature (weak-coupling) dualities of
classical statistical mechanics (or (Euclidean) path integrals). Its range of
applications includes discrete lattice, continuum field, and gauge theories.
Dualities are revealed to be local, structure-preserving mappings between
model-specific bond algebras that can be implemented as unitary
transformations, or partial isometries if gauge symmetries are involved. This
characterization permits to search systematically for dualities and
self-dualities in quantum models of arbitrary system size, dimensionality and
complexity, and any classical model admitting a transfer matrix representation.
Dualities like exact dimensional reduction, emergent, and gauge-reducing
dualities that solve gauge constraints can be easily understood in terms of
mappings of bond algebras. As a new example, we show that the (\mathbb{Z}_2)
Higgs model is dual to the extended toric code model {\it in any number of
dimensions}. Non-local dual variables and Jordan-Wigner dictionaries are
derived from the local mappings of bond algebras. Our bond-algebraic approach
goes beyond the standard approach to classical dualities, and could help
resolve the long standing problem of obtaining duality transformations for
lattice non-Abelian models. As an illustration, we present new dualities in any
spatial dimension for the quantum Heisenberg model. Finally, we discuss various
applications including location of phase boundaries, spectral behavior and,
notably, we show how bond-algebraic dualities help constrain and realize
fermionization in an arbitrary number of spatial dimensions.Comment: 131 pages, 22 figures. Submitted to Advances in Physics. Second
version including a new section on the eight-vertex model and the correction
of several typo
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