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

From SO/Sp instantons to W-algebra blocks

We study instanton partition functions for N=2 superconformal Sp(1) and SO(4) gauge theories. We find that they agree with the corresponding U(2) instanton partitions functions only after a non-trivial mapping of the microscopic gauge couplings, since the instanton counting involves different renormalization schemes. Geometrically, this mapping relates the Gaiotto curves of the different realizations as double coverings. We then formulate an AGT-type correspondence between Sp(1)/SO(4) instanton partition functions and chiral blocks with an underlying W(2,2)-algebra symmetry. This form of the correspondence eliminates the need to divide out extra U(1) factors. Finally, to check this correspondence for linear quivers, we compute expressions for the Sp(1)-SO(4) half-bifundamental.Comment: 83 pages, 29 figures; minor change

Construction and Deconstruction of Single Instanton Hilbert Series

Many methods exist for the construction of the Hilbert series describing the moduli spaces of instantons. We explore some of the underlying group theoretic relationships between these various constructions, including those based on the Coulomb branches and Higgs branches of SUSY quiver gauge theories, as well as those based on generating functions derivable from the Weyl Character Formula. We show how the character description of the reduced single instanton moduli space of any Classical or Exceptional group can be deconstructed faithfully in terms of characters or modified Hall-Littlewood polynomials of its regular semi-simple subgroups. We derive and utilise Highest Weight Generating functions, both for the characters of Classical or Exceptional groups and for the Hall-Littlewood polynomials of unitary groups. We illustrate how the root space data encoded in extended Dynkin diagrams corresponds to relationships between the Coulomb branches of quiver gauge theories for instanton moduli spaces and those for T(SU(N)) moduli spaces.Comment: 97 pages, 12 figure

Conjugates, Filters and Quantum Mechanics

The Jordan structure of finite-dimensional quantum theory is derived, in a conspicuously easy way, from a few simple postulates concerning abstract probabilistic models (each defined by a set of basic measurements and a convex set of states). The key assumption is that each system A can be paired with an isomorphic $\textit{conjugate}$ system, $\overline{A}$, by means of a non-signaling bipartite state $\eta_A$ perfectly and uniformly correlating each basic measurement on A with its counterpart on $\overline{A}$. In the case of a quantum-mechanical system associated with a complex Hilbert space $\mathcal H$, the conjugate system is that associated with the conjugate Hilbert space $\overline{\mathcal H}$, and $\eta_A$ corresponds to the standard maximally entangled EPR state on ${\mathcal H} \otimes \overline{\mathcal H}$. A second ingredient is the notion of a $\textit{reversible filter}$, that is, a probabilistically reversible process that independently attenuates the sensitivity of detectors associated with a measurement. In addition to offering more flexibility than most existing reconstructions of finite-dimensional quantum theory, the approach taken here has the advantage of not relying on any form of the "no restriction" hypothesis. That is, it is not assumed that arbitrary effects are physically measurable, nor that arbitrary families of physically measurable effects summing to the unit effect, represent physically accessible observables. An appendix shows how a version of Hardy's "subspace axiom" can replace several assumptions native to this paper, although at the cost of disallowing superselection rules.Comment: 33 pp. Minor corrections throughout; some revision of Appendix