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 us 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 or operator representation. In particular, special dualities such as 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 ℤ2 Higgs model is dual to the extended toric code model in any number of dimensions. Non-local transformations such as dual variables and Jordan–Wigner dictionaries are algorithmically derived from the local mappings of bond algebras. This permits us to establish a precise connection between quantum dual and classical disorder variables. 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.

中文翻译：

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对偶性的键代数方法

对偶的代数理论是基于键代数​​的概念发展起来的。它以统一的方式处理经典和量子对偶，解释了量子对偶与经典统计力学（或（欧几里德）路径积分）的低温（强耦合）/高温（弱耦合）对偶之间的精确联系。其应用范围包括离散晶格、连续场和规范理论。对偶性被揭示为模型特定键代数之间的局部、结构保持映射，可以作为幺正变换实现，如果涉及规范对称性，则可以实现部分等距。这种表征允许我们在任意系统大小、维数和复杂性的量子模型中系统地搜索二元性和自二元性，以及任何允许传递矩阵或算子表示的经典模型。特别是，解决规范约束的特殊对偶性，例如精确降维、涌现和规范缩减对偶性，可以很容易地根据键代数的映射来理解。作为一个新的例子，我们证明了ℤ2 Higgs 模型在任意维数上与扩展复曲面代码模型是对偶的。非局部变换，例如对偶变量和 Jordan-Wigner 字典，是从键代数的局部映射通过算法推导出来的。这使我们能够在量子对偶变量和经典无序变量之间建立精确的联系。我们的键代数方法超越了经典对偶的标准方法，可以帮助解决获得格非阿贝尔模型对偶变换的长期问题。作为例证，我们为量子海森堡模型在任何空间维度上呈现新的对偶性。最后，我们讨论了各种应用，包括相边界的位置、光谱行为，特别是我们展示了键代数对偶性如何帮助约束和实现任意数量的空间维度中的费米化。