A solution to the non-Abelian duality problem

Dualities uniquely excel at resolving non-perturbative aspects of complex phase diagrams of interacting, Landau or topologically ordered, systems. However, traditional duality transformations fail for systems like the Heisenberg model and non-Abelian gauge theories. The bond-algebraic theory of quantum and classical dualities provides a solution to this long-standing conundrum, the so-called non-Abelian duality problem, by embedding traditional dualities into a more general transformation scheme that always preserves locality in any number of dimensions. Remarkably, it turns out to be unimportant whether a model?s group of symmetries is Abelian or non-Abelian. The capability of the bond-algebraic approach to handle finite and infinite systems with arbitrary boundary conditions has recently led to the discovery of holographic symmetries  , relating topological order, edge states, and generalized order parameters. We discuss the interplay between these distinguished boundary symmetries and our solution to the non-Abelian duality problem. To illustrate our technique we present, among others, novel dualities for the SU(2)$\mathit{SU}(2)$ principal chiral field and both U(1)$U(1)$ and SU(2)$\mathit{SU}(2)$ generalizations of the planar quantum compass model of orbital ordering.     

Exact duality relations in correlated electron systems

Using gauge transformations on electron bond operators, we derive exact duality relations between various order parameters for correlated electron systems. Applying these transformations, we find two duality relations in the generalized two-leg Hubbard ladder at arbitrary filling. The relations show that unconventional density-wave orders such as staggered flux or circulating spin current are dual to conventional density-wave orders and there are direct mappings between dual phases. Several exact results on the phase diagram are also concluded.     

Quantum simulation of classical thermal states

We establish a connection between ground states of local quantum Hamiltonians and thermal states of classical spin systems. For any discrete classical statistical mechanical model in any spatial dimension, we find an associated quantum state such that the reduced density operator behaves as the thermal state of the classical system. We show that all these quantum states are unique ground states of a universal 5-body local quantum Hamiltonian acting on a (polynomially enlarged) qubit system on a 2D lattice. The only free parameters of the quantum Hamiltonian are coupling strengths of two-body interactions, which allow one to choose the type and dimension of the classical model as well as the interaction strength and temperature. This opens the possibility to study and simulate classical spin models in arbitrary dimension using a 2D quantum system.     

The spectral gap for some spin chains with discrete symmetry breaking

We prove that for any finite set of generalized valence bond solid (GVBS) states of a quantum spin chain there exists a translation invariant finite-range Hamiltonian for which this set is the set of ground states. This result implies that there are GVBS models with arbitrary broken discrete symmetries that are described as combinations of lattice translations, lattice reflections, and local unitary or anti-unitary transformations. We also show that all GVBS models that satisfy some natural conditions have a spectral gap. The existence of a spectral gap is obtained by applying a simple and quite general strategy for proving lower bounds on the spectral gap of the generator of a classical or quantum spin dynamics. This general scheme is interesting in its own right and threfore, although the basic idea is not new, we present it in a system-independent setting. The results are illustrated with a number of examples.Copyright © 1994 by the author.FFaithful reproduction of this article by any means is permitted for non-commercial purposes.     

Disorder Operators and Their Descendants

I review the concept of a disorder operator, introduced originally by Kadanoff in the context of the two-dimensional Ising model. Disorder operators acquire an expectation value in the disordered phase of the classical spin system. This concept has had applications and implications to many areas of physics ranging from quantum spin chains to gauge theories to topological phases of matter. In this paper I describe the role that disorder operators play in our understanding of ordered, disordered and topological phases of matter. The role of disorder operators, and their generalizations, and their connection with dualities in different systems, as well as with majorana fermions and parafermions, is discussed in detail. Their role in recent fermion–boson and boson–boson dualities is briefly discussed.     

Homotopy Classification of Bosonic String Field Theory

We prove the decomposition theorem for the loop homotopy Lie algebra of quantum closed string field theory and use it to show that closed string field theory is unique up to gauge transformations on a given string background and given S-matrix. For the theory of open and closed strings we use results in open-closed homotopy algebra to show that the space of inequivalent open string field theories is isomorphic to the space of classical closed string backgrounds. As a further application of the open-closed homotopy algebra, we show that string field theory is background independent and locally unique in a very precise sense. Finally, we discuss topological string theory in the framework of homotopy algebras and find a generalized correspondence between closed strings and open string field theories.     

Invariance operator groups for a charged particle in an external electromagnetic field

In this paper a general group theoretical approach is given for the problem of a charged particle moving in an external electromagnetic field F. From a knowledge of the symmetry transformations of the field (Galilean or Poincaré), it is possible to explicitly construct groups of operators which commute with the operators of the equations of motion (classical, quantum mechanical, Klein-Gordon or Dirac) using the concept of compensating gauge transformations together with a uniquely chosen map π: F → A fixing the gauge of the potential A. Other choices of gauges give rise to isomorphic operator groups. The general structure of the possible symmetry groups of the fields is discussed and the corresponding invariance operator groups are explicitly given for (almost) arbitrary fields. The structure of these groups is then investigated and it is shown in particular that a large class of fields give rise to non-Type I groups, i.e. to groups which have (unitary continuous) representations whose corresponding von Neumann algebras have non-discrete factors. A general criterion for these pathological cases is given. As an application, we study the problem of a Bloch electron in arbitrary constant uniform electric and magnetic fields.     

Classical spin models and the quantum-stabilizer formalism

We relate a large class of classical spin models, including the inhomogeneous Ising, Potts, and clock models of q-state spins on arbitrary graphs, to problems in quantum physics. More precisely, we show how to express partition functions as inner products between certain quantum-stabilizer states and product states. This connection allows us to use powerful techniques developed in quantum-information theory, such as the stabilizer formalism and classical simulation techniques, to gain general insights into these models in a unified way. We recover and generalize several symmetries and high-low temperature dualities, and we provide an efficient classical evaluation of partition functions for all interaction graphs with a bounded tree-width.     

Pseudo-differential equations,and the Bethe ansatz for the classical Lie algebras

The correspondence between ordinary differential equations and Bethe ansatz equations for integrable lattice models in their continuum limits is generalised to vertex models related to classical simple Lie algebras. New families of pseudo-differential equations are proposed, and a link between specific generalised eigenvalue problems for these equations and the Bethe ansatz is deduced. The pseudo-differential operators resemble in form the Miura-transformed Lax operators studied in work on generalised KdV equations, classical W-algebras and, more recently, in the context of the geometric Langlands correspondence. Negative-dimension and boundary-condition dualities are also observed.     

Duals for Non-Abelian Lattice Gauge Theories by Categorical Methods

We introduce duals for non-Abelian lattice gauge theories in dimension at least three by using a categorical approach to the notion of duality in lattice theories. We first discuss the general concepts for the case of a dual-triangular lattice (i.e., the dual lattice is triangular) and find that the commutative tetrahedron condition of category theory can directly be used to define a gauge-invariant action for the dual theory. We then consider the cubic lattice (where the dual is cubic again). The case of the gauge group SU(2) is discussed in detail. We will find that in this case gauge connections of the dual theory correspond to SU(2) spin networks, suggesting that the dual is a discrete version of a quantum field theory of quantum simplicial complexes (i.e. the dual theory lives already on a quantized level in its classical form). We conclude by showing that our notion of duality leads to a hierarchy of extended lattice gauge theories closely resembling the one of extended topological quantum field theories. The appearance of this hierarchy can be understood by the quantum von Neumann hierarchy introduced by one of the authors in previous work.     

Two dimensional lattice gauge theory based on a quantum group

In this article we analyse a two dimensional lattice gauge theory based on a quantum group. The algebra generated by gauge fields is the lattice algebra introduced recently by A.Yu. Alekseev, H. Grosse and V. Schomerus in [1]. We define and study Wilson loops. This theory is quasi-topological as in the classical case, which allows us to compute the correlation functions of this theory on an arbitrary surface.Laboratoire Propre du CNRS UPR 14     

Geometry of N = 1 dualities in four dimensions

We discuss how N = 1 dualities in four dimensions are geometrically realized by wrapping D-branes about 3-cycles of Calabi-Yau threefolds. In this setup the N = 1 dualities for SU, SO and USp gauge groups with fundamental fields get mapped to statements about the monodromy and relations among 3-cycles of Calabi-Yau threefolds. The connection between the theory and its dual requires passing through configurations which are T-dual to the well-known phenomenon of decay of BPS states in N = 2 field theories in four dimensions. We compare our approach to recent works based on configurations of D-branes in the presence of NS 5-branes and give simple classical geometric derivations of various exotic dynamics involving D-branes ending on NS branes.     

Quantum phase transitions beyond the Landau-Ginzburg paradigm and supersymmetry

We make connections between studies in the condensed matter literature on quantum phase transitions in square lattice antiferromagnets, and results in the particle theory literature on abelian supersymmetric gauge theories in 2 + 1 dimensions. In particular, we point out that supersymmetric U(1) gauge theories (with particle content similar, but not identical, to those of theories of doped antiferromagnets) provide rigorous examples of quantum phase transitions which do not obey the Landau-Ginzburg-Wilson paradigm (often referred to as transitions realizing “deconfined criticality”). We also make connections between supersymmetric mirror symmetries and condensed matter particle-vortex dualities.     

Representation Theory of Lattice Current Algebras

Lattice current algebras were introduced as a regularization of the left- and right moving degrees of freedom in the WZNW model. They provide examples of lattice theories with a local quantum symmetry . Their representation theory is studied in detail. In particular, we construct all irreducible representations along with a lattice analogue of the fusion product for representations of the lattice current algebra. It is shown that for an arbitrary number of lattice sites, the representation categories of the lattice current algebras agree with their continuum counterparts. Received: 25 April 1996 / Accepted: 14 April 1997     

Strong-weak coupling self-duality in the two-dimensional quantum phase transition of p + ip superconducting arrays

The 2D quantum phase transition that occurs in a square lattice of Josephson-coupled p +/- ip superconductors is an example of how four-body interactions in d = 2 reproduce nonperturbative effects caused by two-body interactions in d = 1. The ordered phase has an unconventional "bond order" of the local T-breaking variable. This problem can be analyzed using an exact self-duality; this duality in classical notation is the 3D generalization of the Kramers-Wannier duality of the 2D Ising model, and there are similar exact dualities in dimensions d > or = 3. We discuss the excitation spectrum and experimental signatures of the ordered and disordered phases, and the relationship between our model and previously studied behavior of 2D boson models with four-boson interactions.     

Lattice Chern-Simons gravity via the Ponzano-Regge model

We propose a lattice version of Chem-Simons gravity and show that the partition function coincides with the Ponzano-Regge model and the action leads to the Chem-Simons gravity in the continuum limit. The action is explicitly constructed by the lattice dreibein and spin connection and is shown to be invariant under lattice local Lorentz transformations and gauge diffeomorphisms. The action includes the constraint which can be interpreted as a gauge fixing condition of the lattice gauge diffeomorphism.     

Embedding Uq(sl(2)) and Sine Algebras in Generalized Clifford Algebras

We establish the connection between certain quantum algebras and generalizedClifford algebras (GCA). To be precise, we embed the quantum tori Lie algebraand U _q (sl(2)) in GCA.     

Quantum and braided group Riemannian geometry

We formulate quantum group Riemannian geometry as a gauge theory of quantum differential forms. We first develop (and slightly generalise) classical Riemannian geometry in a self-dual manner as a principal bundle frame resolution and a dual pair of canonical forms. The role of Levi-Civita connection is naturally generalised to connections with vanishing torsion and cotorsion, which we introduce. We then provide the corresponding quantum group and braided group formulations with the universal quantum differential calculus. We also give general constructions, for example, including quantum spheres and quantum planes.