A symmetry of the string background field equations

We investigate the effect of duality transformations on the geometries of a subclass of two-dimensional non-linear sigma-models. We identify the torsion which appears on dualizing such a model, initially without Wess-Zumino-Witten term, as the field strength of the gauge connection appearing in a Kałuza-Klein interpretation of the initial geometry. We show that duality preserves quantum conformal invariance at order [α′]⁰, where α′ is the string tension parameter, provided the change induced in the geometry by duality is accompanied by a shift in the dilaton field. We interpret these combined transformations as a symmetry of the order [α′]⁰ string background field equations.     

O(d,d) symmetry in quantum cosmology

We analyze the quantum cosmology of one-loop string effective models which exhibit an O(d, d) symmetry. It is shown that due to the large symmetry of these models the Wheeler-de Witt equation can completely be solved. As a result, we find a basis of solutions with well-defined transformation properties under O(d, d) and under scale factor duality in particular. The general results are explicitly applied to 2-dimensional target spaces while some aspects of higher dimensional cases are also discussed. Moreover, a semiclassical wave function for the 2-dimensional black hole is constructed as a superposition of our basis.     

Entanglement as a Semantic Resource

The characteristic holistic features of the quantum theoretic formalism and the intriguing notion of entanglement can be applied to a field that is far from microphysics: logical semantics. Quantum computational logics are new forms of quantum logic that have been suggested by the theory of quantum logical gates in quantum computation. In the standard semantics of these logics, sentences denote quantum information quantities: systems of qubits (quregisters) or, more generally, mixtures of quregisters (qumixes), while logical connectives are interpreted as special quantum logical gates (which have a characteristic reversible and dynamic behavior). In this framework, states of knowledge may be entangled, in such a way that our information about the whole determines our information about the parts; and the procedure cannot be, generally, inverted. In spite of its appealing properties, the standard version of the quantum computational semantics is strongly “Hilbert-space dependent”. This certainly represents a shortcoming for all applications, where real and complex numbers do not generally play any significant role (as happens, for instance, in the case of natural and of artistic languages). We propose an abstract version of quantum computational semantics, where abstract qumixes, quregisters and registers are identified with some special objects (not necessarily living in a Hilbert space), while gates are reversible functions that transform qumixes into qumixes. In this framework, one can give an abstract definition of the notions of superposition and of entangled pieces of information, quite independently of any numerical values. We investigate three different forms of abstract holistic quantum computational logic.     

On a2(1) reflection matrices and affine Toda theories

We construct new non-diagonal solutions to the boundary Yang-Baxter equation corresponding to a two-dimensional field theory with U_q(a₂⁽¹⁾) quantum affine symmetry on a half-line. The requirements of boundary unitarity and boundary crossing symmetry are then used to find overall scalar factors which lead to consistent reflection matrices. Using the boundary bootstrap equations we also compute the reflection factors for scalar bound states (breathers). These breathers are expected to be identified with the fundamental quantum particles in a₂⁽¹⁾ affine Toda field theory and we therefore obtain a conjecture for the affine Toda reflection factors. We compare these factors with known classical results and discuss their duality properties and their connections with particular boundary conditions.     

Quantum groups as generalized gauge symmetries in WZNW models. Part II. The quantized model

This is the second part of a paper dealing with the “internal” (gauge) symmetry of the Wess–Zumino–Novikov–Witten (WZNW) model on a compact Lie group G. It contains a systematic exposition, for G = SU(n), of the canonical quantization based on the study of the classical model (performed in the first part) following the quantum group symmetric approach first advocated by L.D. Faddeev and collaborators. The internal symmetry of the quantized model is carried by the chiral WZNW zero modes satisfying quadratic exchange relations and an n-linear determinant condition. For generic values of the deformation parameter the Fock representation of the zero modes’ algebra gives rise to a model space of U _q (sl(n)). The relevant root of unity case is studied in detail for n = 2 when a “restricted” (finite dimensional) quotient quantum group is shown to appear in a natural way. The module structure of the zero modes’ Fock space provides a specific duality with the solutions of the Knizhnik–Zamolodchikov equation for the four point functions of primary fields suggesting the existence of an extended state space of logarithmic CFT type. Combining left and right zero modes (i.e., returning to the 2D model), the rational CFT structure shows up in a setting reminiscent to covariant quantization of gauge theories in which the restricted quantum group plays the role of a generalized gauge symmetry.     

An explicit construction of the quantum group in chiral WZW-models

It is shown how a chiral Wess-Zumino-Witten theory with globally defined vertex operators and a one-to-one correspondence between fields and states can be constructed. The Hilbert space of this theory is the direct sum of tensor products of representations of the chiral algebra and finite dimensional internal parameter spaces. On this enlarged space there exists a natural action of Drinfeld's quasi-quantum groupA _{g, t} which commutes with the action of the chiral algebra and plays the rôle of an internal symmetry algebra. TheR matrix describes the braiding of the chiral vertex operators and the coassociator gives rise to a modification of the duality property.For genericq the quasi-quantum group is isomorphic to the coassociative quantum groupU _q (g) and thus the duality property of the chiral theory can be restored. This construction has to be modified for the physically relevant case of integer level. The quantum group has to be replaced by the corresponding truncated quasiquantum group, which is not coassociative because of the truncation. This exhibits the truncated quantum group as the internal symmetry algebra of the chiral WZW model, which therefore has only a modified duality property. The case ofg=su(2) is worked out in detail.     

Duality and symmetry tests for multiparticle production models

Two-body absorptive parts generated by unitarity from multi-Regge particle production models are tested with respect to duality and symmetry structure (suppression of exotics and exchange degeneracy in output). A multi-Regge model with production of only stable particles generates exotic and non-exotic outputs of equal strength; resonances (clusters) are needed to pass these symmetry tests. Two complementary approaches are used, explicit S-matrix models and duality diagrams; the connection between dynamical assumptions and different duality diagram rules is discussed. C-parity plays a crucial role; using C-conserving duality diagrams we show that standard manipulations lead to a topological pomeron which has secondary terms; one, with C_t = ?1, cancels the topological ω₁ meson, another one, with C_t = +1, cancels the topological ?₁ meson.     

Experimental probes of emergent symmetries in the quantum Hall system

Experiments studying renormalization group flows in the quantum Hall system provide significant evidence for the existence of an emergent holomorphic modular symmetry Γ₀(2)${\Gamma }_0(2)$. We briefly review this evidence and show that, for the lowest temperatures, the experimental determination of the position of the quantum critical points agrees to the parts per mille   level with the prediction from Γ₀(2)${\Gamma }_0(2)$. We present evidence that experiments giving results that deviate substantially from the symmetry predictions are not cold enough to be in the quantum critical domain. We show how the modular symmetry extended by a non-holomorphic particle–hole duality leads to an extensive web of dualities related to those in plateau–insulator transitions, and we derive a formula relating dual pairs (B,Bd)$(B,B_d)$ of magnetic field strengths across any transition. The experimental data obtained for the transition studied so far is in excellent agreement with the duality relations following from this emergent symmetry, and rule out the duality rule derived from the “law of corresponding states”. Comparing these generalized duality predictions with future experiments on other transitions should provide stringent tests of modular duality deep in the non-linear domain far from the quantum critical points.     

Spectral duality in integrable systems from AGT conjecture

We describe relationships between integrable systems with N degrees of freedom arising from the Alday-Gaiotto-Tachikawa conjecture. Namely, we prove the equivalence (spectral duality) between the N-cite Heisenberg spin chain and a reduced gl _N Gaudin model both at classical and quantum level. The former one appears on the gauge theory side of the Alday-Gaiotto-Tachikawa relation in the Nekrasov-Shatashvili (and further the Seiberg-Witten) limit while the latter one is natural on the CFT side. At the classical level, the duality transformation relates the Seiberg-Witten differentials and spectral curves via a bispectral involution. The quantum duality extends this to the equivalence of the corresponding Baxter-Schrödinger equations (quantum spectral curves). This equivalence generalizes both the spectral self-duality between the 2 × 2 and N × N representations of the Toda chain and the famous Adams-Harnad-Hurtubise duality.     

Ground States of Quantum Antiferromagnets in Two Dimensions

We explore the ground states and quantum phase transitions of two-dimensional, spin S=1/2, antiferromagnets by generalizing lattice models and duality transforms introduced by Sachdev and Jalabert (1990, Mod. Phys. Lett. B4, 1043). The minimal model for square lattice antiferromagnets is a lattice discretization of the quantum nonlinear sigma model, along with Berry phases which impose quantization of spin. With full SU(2) spin rotation invariance, we find a magnetically ordered ground state with Néel order at weak coupling and a confining paramagnetic ground state with bond charge (e.g., spin Peierls) order at strong coupling. We study the mechanisms by which these two states are connected in intermediate coupling. We extend the minimal model to study different routes to fractionalization and deconfinement in the ground state, and also generalize it to cases with a uniaxial anisotropy (the spin symmetry groups is then U(1)). For the latter systems, fractionalization can appear by the pairing of vortices in the staggered spin order in the easy-plane; however, we argue that this route does not survive the restoration of SU(2) spin symmetry. For SU(2) invariant systems we study a separate route to fractionalization associated with the Higgs phase of a complex boson measuring noncollinear, spiral spin correlations: we present phase diagrams displaying competition between magnetic order, bond charge order, and fractionalization, and discuss the nature of the quantum transitions between the various states. A strong check on our methods is provided by their application to S=1/2 frustrated antiferromagnets in one dimension: here, our results are in complete accord with those obtained by bosonization and by the solution of integrable models.     

Duality and Hidden Symmetries in Interacting Particle Systems

In the context of Markov processes, both in discrete and continuous setting, we show a general relation between duality functions and symmetries of the generator. If the generator can be written in the form of a Hamiltonian of a quantum spin system, then the “hidden” symmetries are easily derived. We illustrate our approach in processes of symmetric exclusion type, in which the symmetry is of SU(2) type, as well as for the Kipnis-Marchioro-Presutti (KMP) model for which we unveil its SU(1,1) symmetry. The KMP model is in turn an instantaneous thermalization limit of the energy process associated to a large family of models of interacting diffusions, which we call Brownian energy process (BEP) and which all possess the SU(1,1) symmetry. We treat in details the case where the system is in contact with reservoirs and the dual process becomes absorbing.     

Are scattering amplitudes dual to super Wilson loops?

The MHV scattering amplitudes in planar N=4 SYM are dual to bosonic light-like Wilson loops. We explore various proposals for extending this duality to generic non-MHV amplitudes. The corresponding dual object should have the same symmetries as the scattering amplitudes and be invariant to all loops under the chiral half of the N=4 superconformal symmetry. We analyze the recently introduced supersymmetric extensions of the light-like Wilson loop (formulated in Minkowski space-time) and demonstrate that they have the required symmetry properties at the classical level only, up to terms proportional to field equations of motion. At the quantum level, due to the specific light-cone singularities of the Wilson loop, the equations of motion produce a nontrivial finite contribution which breaks some of the classical symmetries. As a result, the quantum corrections violate the chiral supersymmetry already at one loop, thus invalidating the conjectured duality between Wilson loops and non-MHV scattering amplitudes. We compute the corresponding anomaly to one loop and solve the supersymmetric Ward identity to find the complete expression for the rectangular Wilson loop at leading order in the coupling constant. We also demonstrate that this result is consistent with conformal Ward identities by independently evaluating corresponding one-loop conformal anomaly.     

Disorder-induced quantum-Griffiths-like features from a non-conventional renormalization group analysis near four dimensions

We investigate the role played by symmetry conserving quenched disorder on quantum criticality of a variety of d-dimensional systems with a continuous symmetry order parameter. We employ a non-standard procedure which combines a preliminary reduction to an effective classical random problem and a successive conventional renormalization group treatment. Solving the effective flow equations to first order in ε=4−d and then restoring the original coupling parameters, for d<4 we find a quantum critical point scenario exhibiting unusual features, which remind us of some predictions of the quantum Griffiths phase model.     

A Topos for Algebraic Quantum Theory

The aim of this paper is to relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces. Motivated by Bohr’s idea that the empirical content of quantum physics is accessible only through classical physics, we show how a noncommutative C*-algebra of observables A induces a topos \({\mathcal{T}(A)}\) in which the amalgamation of all of its commutative subalgebras comprises a single commutative C*-algebra \({\underline{A}}\) . According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum \({\underline{\Sigma}(\underline{A})}\) in \({\mathcal{T}(A)}\) , which in our approach plays the role of the quantum phase space of the system. Thus we associate a locale (which is the topos-theoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a Heyting algebra) to a C*-algebra (which is the noncommutative notion of a space). In this setting, states on A become probability measures (more precisely, valuations) on \({\underline{\Sigma}}\) , and self-adjoint elements of A define continuous functions (more precisely, locale maps) from \({\underline{\Sigma}}\) to Scott’s interval domain. Noting that open subsets of \({\underline{\Sigma}(\underline{A})}\) correspond to propositions about the system, the pairing map that assigns a (generalized) truth value to a state and a proposition assumes an extremely simple categorical form. Formulated in this way, the quantum theory defined by A is essentially turned into a classical theory, internal to the topos \({\mathcal{T}(A)}\).These results were inspired by the topos-theoretic approach to quantum physics proposed by Butterfield and Isham, as recently generalized by Döring and Isham.     

Renormalization of Lorentz non-invariant actions and manifest T-duality

We study general two-dimensional σ-models which do not possess manifest Lorentz invariance. We show how demanding that Lorentz invariance is recovered as an emergent on-shell symmetry constrains these σ-models. The resulting actions have an underlying group-theoretic structure and resemble Poisson–Lie T-duality invariant actions. We consider the one-loop renormalization of these models and show that the quantum Lorentz anomaly is absent. We calculate the running of the couplings in general and show, with certain non-trivial examples, that this agrees with that of the T-dual models obtained classically from the duality invariant action. Hence, in these cases solving constraints before and after quantization are commuting operations.     

Structure of reversible computation determines the self-duality of quantum theory

Predictions for measurement outcomes in physical theories are usually computed by combining two distinct notions: a state, describing the physical system, and an observable, describing the measurement which is performed. In quantum theory, however, both notions are in some sense identical: outcome probabilities are given by the overlap between two state vectors--quantum theory is self-dual. In this Letter, we show that this notion of self-duality can be understood from a dynamical point of view. We prove that self-duality follows from a computational primitive called bit symmetry: every logical bit can be mapped to any other logical bit by a reversible transformation. Specifically, we consider probabilistic theories more general than quantum theory, and prove that every bit-symmetric theory must necessarily be self-dual. We also show that bit symmetry yields stronger restrictions on the set of allowed bipartite states than the no-signalling principle alone, suggesting reversible time evolution as a possible reason for limitations of nonlocality.     

Langlands Duality for Finite-Dimensional Representations of Quantum Affine Algebras

We describe a correspondence (or duality) between the q-characters of finite-dimensional representations of a quantum affine algebra and its Langlands dual in the spirit of Frenkel and Hernandez (Math Ann, to appear) and Frenkel and Reshetikhin (Commun Math Phys 197(1):1?C32, 1998). We prove this duality for the Kirillov?CReshetikhin modules and their irreducible tensor products. In the course of the proof we introduce and construct ??interpolating (q, t)-characters?? depending on two parameters which interpolate between the q-characters of a quantum affine algebra and its Langlands dual.     

Quantum Noether method

We present a general method for constructing perturbative quantum field theories with global symmetries. We start from a free non-interacting quantum field theory with given global symmetries and we determine all perturbative quantum deformations assuming the construction is not obstructed by anomalies. The method is established within the causal Bogoliubov-Shirkov-Epstein-Glaser approach to perturbative quantum field theory (which leads directly to a finite perturbative series and does not rely on an intermediate regularization). Our construction can be regarded as a direct implementation of Noether's method at the quantum level. We illustrate the method by constructing the pure Yang-Mills theory (where the relevant global symmetry is BRST symmetry), and the N = 1 supersymmetric model of Wess and Zumino. The whole construction is done before the so-called adiabatic limit is taken. Thus, all considerations regarding symmetry, unitarity and anomalies are well defined even for massless theories.