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We develop a new approach to T-duality based on Courant algebroid relations which subsumes the usual T-duality as well as its various generalisations. Starting from a relational description for the reduction of exact Courant algebroids over foliated manifolds, we introduce a weakened notion of generalised isometries that captures the generalised geometry counterpart of Riemannian submersions when applied to transverse generalised metrics. This is used to construct T-dual backgrounds as generalised metrics on reduced Courant algebroids which are related by a generalised isometry. We prove an existence and uniqueness result for generalised isometric exact Courant algebroids coming from reductions. We demonstrate that our construction reproduces standard T-duality relations based on correspondence spaces. We also describe how it applies to generalised T-duality transformations of almost para-Hermitian manifolds., Comment: 68 pages; v2: clarifications and reference added; v3: definition of geometric T-duality improved, reference added

We study the geometry of foliated non-Lorentzian spacetimes in terms of the Godbillon-Vey class of the foliation. We relate the intrinsic torsion of a foliated Aristotelian manifold to its Godbillon-Vey class, and interpret it as a measure of the local spin of the spatial leaves in the time direction. With this characterisation, the Godbillon-Vey class is an obstruction to integrability of the $\mathsf{G}$-structure defining the Aristotelian spacetime. We use these notions to formulate a new geometric approach to hydrodynamics of fluid flows by endowing them with Aristotelian structures. We establish conditions under which the Godbillon-Vey class represents an obstruction to steady flow of the fluid and prove new conservation laws., Comment: 41 pages; v2: minor corrections, exposition improved, references added; Final version to be published in Journal of Physics A

We introduce T-duality invariant versions of D-branes in doubled geometry using a global covariant framework based on para-Hermitian geometry and metric algebroids. We define D-branes as conformal boundary conditions for the open string version of the Born sigma-model, where they are given by maximally isotropic vector bundles which do not generally admit the standard geometric picture in terms of submanifolds. When reduced to the conventional sigma-model description of a physical string background as the leaf space of a foliated para-Hermitian manifold, integrable branes yield D-branes as leaves of foliations which are interpreted as Dirac structures on the physical spacetime. We define a notion of generalised para-complex D-brane, which realises our D-branes as para-complex versions of topological A/B-branes. We illustrate how our formalism recovers standard D-branes in the explicit example of reductions from doubled nilmanifolds., Comment: 37 pages; v2: minor corrections, references added; v3: minor changes; Final version to appear in the Special Issue of Universe on "Dualities and Geometry"

We give a self-contained survey of some approaches aimed at a global description of the geometry underlying double field theory. After reviewing the geometry of Courant algebroids and their incarnations in the AKSZ construction, we develop the theory of metric algebroids including their graded geometry. We use metric algebroids to give a global description of doubled geometry, incorporating the section constraint, as well as an AKSZ-type construction of topological doubled sigma-models. When these notions are combined with ingredients of para-Hermitian geometry, we demonstrate how they reproduce kinematical features of double field theory from a global perspective, including solutions of the section constraint for Riemannian foliated doubled manifolds, as well as a natural notion of generalized T-duality for polarized doubled manifolds. We describe the $L_\infty$-algebras of symmetries of a doubled geometry, and briefly discuss other proposals for global doubled geometry in the literature., Comment: 56 pages; v2: minor corrections and additions; v3: minor corrections, presentation improved, reference added; Final version to appear in the Special Issue of Complex Manifolds on "Generalized Geometry"

We give a covariant realization of the doubled sigma-model formulation of duality-symmetric string theory within the general framework of para-Hermitian geometry. We define a notion of generalized metric on a para-Hermitian manifold and discuss its relation to Born geometry. We show that a Born geometry uniquely defines a worldsheet sigma-model with a para-Hermitian target space, and we describe its Lie algebroid gauging as a means of recovering the conventional sigma-model description of a physical string background as the leaf space of a foliated para-Hermitian manifold. Applying the Kotov-Strobl gauging leads to a generalized notion of T-duality when combined with transformations that act on Born geometries. We obtain a geometric interpretation of the self-duality constraint that halves the degrees of freedom in doubled sigma-models, and we give geometric characterizations of non-geometric string backgrounds in this setting. We illustrate our formalism with detailed worldsheet descriptions of closed string phase spaces, of doubled groups where our notion of generalized T-duality includes non-abelian T-duality, and of doubled nilmanifolds., Comment: 81 pages; v2: minor changes, references added; v3: minor corrections, references updated; Final version to be published in Reviews in Mathematical Physics

After reviewing some of the fundamental aspects of Drinfel'd doubles and Poisson-Lie T-duality, we describe the three-dimensional isotropic rigid rotator on $SL(2,\mathbb{C})$ starting from a non-Abelian deformation of the natural carrier space of its Hamiltonian description on $T^*SU(2) \simeq SU(2) \ltimes \mathbb{R}^3$. A new model is then introduced on the dual group $SB(2,\mathbb{C})$, within the Drinfel'd double description of $SL(2,\mathbb{C})=SU(2) \bowtie SB(2,\mathbb{C})$. The two models are analyzed from the Poisson-Lie duality point of view, and a doubled generalized action is built with $TSL(2,\mathbb{C})$ as carrier space. The aim is to explore within a simple case the relations between Poisson-Lie symmetry, Doubled Geometry and Generalized Geometry. In fact, all the mentioned structures are discussed, such as a Poisson realization of the $C$-brackets for the generalized bundle $T \oplus T^*$ over $SU(2)$ from the Poisson algebra of the generalized model. The two dual models exhibit many features of Poisson-Lie duals and from the generalized action both of them can be respectively recovered by gauging one of its symmetries., Comment: Contribution to the proceedings of the Workshop "Dualities and Generalized Geometries", Corfu Summer Institute on Elementary Particle Physics and Gravity, 31 August-28 September 2018, Corfu, Greece. Typos and some references corrected

The Principal Chiral Model (PCM) defined on the group manifold of SU(2) is here investigated with the aim of getting a further deepening of its relation with Generalized and Doubled Geometry. A one-parameter family of equivalent Hamiltonian descriptions is introduced, and cast into the form of Born geometries. Then O(3,3) duality transformations of the target phase space are performed and we show that the resulting dual models are defined on the group SB(2,C) which is the Poisson-Lie dual of SU(2) in the Iwasawa decomposition of the Drinfel'd double SL(2, C). Moreover, starting from the Lagrangian approach, a new kind of duality is found between the SU(2) PCM and the natural one defined on SB(2,C) which is not an isometry of the target phase space. A parent action with doubled degrees of freedom and configuration space SL(2, C) is then defined that reduces to either one of the dually related models, once suitable constraints are implemented., Comment: 41 pages, revised version published in JHEP

We survey physical models which capture the main concepts of double field theory on para-Hermitian manifolds. We show that the geometric theory of Lagrangian and Hamiltonian dynamical systems is an instance of para-Kahler geometry which extends to a natural example of a Born geometry. The corresponding phase space geometry belongs to the family of natural almost para-Kahler structures which we construct explicitly as deformations of the canonical para-Kahler structure by non-linear connections. We extend this framework to a class of non-Lagrangian dynamical systems which naturally encodes the notion of fluxes in para-Hermitian geometry. In this case we describe the emergence of fluxes in terms of weak integrability defined by the D-bracket, and we extend the construction to arbitrary cotangent bundles where we reproduce the standard generalized fluxes of double field theory. We also describe the para-Hermitian geometry of Drinfel'd doubles, which gives an explicit illustration of the interplay between fluxes, D-brackets and different polarizations. The left-invariant para-Hermitian structure on a Drinfel'd double in a Manin triple polarization descends to a doubled twisted torus, which we use to illustrate how changes of polarizations give rise to different fluxes and string backgrounds in para-Hermitian geometry., Comment: 68 pages; v2: typos corrected; Final version to be published in Fortschritte der Physik

A simple mechanical system, the three-dimensional isotropic rigid rotator, is here investigated as a 0+1 field theory, aiming at further investigating the relation between Generalized/Double Geometry on the one hand and Doubled World-Sheet Formalism/Double Field Theory, on the other hand. The model is defined over the group manifold of SU(2) and a dual model is introduced having the Poisson-Lie dual of SU(2) as configuration space. A generalized action with configuration space SL(2,C), i.e. the Drinfel'd double of the group SU(2), is then defined: it reduces to the original action of the rotator or to its dual, once constraints are implemented. The new action contains twice as many variables as the original. Moreover, its geometric structures can be understood in terms of Generalized Geometry. keywords: Generalized Geometry, Double Field Theory, T-Duality, Poisson-Lie symmetry., Comment: 37 pages. Revised version to appear in JHEP

We review our recent results on the physics of quantum Hall fluids at Jain and non conventional fillings within a general field theoretic framework. We focus on a peculiar conformal field theory (CFT), the one obtained by means of the m-reduction technique, and stress its power in describing strongly correlated low dimensional condensed matter systems in the presence of localized impurities or topological defects. By exploiting the notion of Morita equivalence for field theories on noncommutative two-tori and choosing rational values of the noncommutativity parameter, we find a general one-to-one correspondence between the m-reduced conformal field theory describing the quantum Hall fluid and an Abelian noncommutative field theory. As an example of application of the formalism, we study a quantum Hall bilayer at nonconventional fillings in the presence of a localized topological defect and briefly recall its boundary state structure corresponding to two different boundary conditions, the periodic as well as the twisted boundary conditions respectively, which give rise to different topological sectors on a torus. Then we introduce generalized magnetic translation operators as tensor products, which act on the quantum Hall fluid and defect space respectively, and compute their action on the boundary partition functions: in this way their role as boundary condition changing operators is fully evidenced. From such results we infer the general structure of generalized magnetic translations in our model and clarify the deep relation between noncommutativity and non-Abelian statistics of quasi-hole excitations, which is crucial for physical implementations of topological quantum computing., Comment: 38 pages, 3 figures, review article

In a recent paper, by exploiting the notion of Morita equivalence for field theories on noncommutative tori and choosing rational values of the noncommutativity parameter $\theta $ (in appropriate units), a general one-to-one correspondence between the $m$-reduced conformal field theory (CFT) describing a quantum Hall fluid (QHF) at paired states fillings $\nu =% \frac{m}{pm+2}$ and an Abelian noncommutative field theory (NCFT) has been established . That allowed us to add new evidence to the relationship between noncommutativity and quantum Hall fluids\cite% {ncmanybody}. On the other hand, the $m$-reduced CFT is equivalent to a system of two massless scalar bosons with a magnetic boundary interaction as introduced by Callan et al., at the so called ``magic''\ points. We are then able to describe, within such a framework, the dissipative quantum mechanics of a particle confined to a plane and subject to an external magnetic field normal to it. Here we develop such a point of view by focusing on the case $m=2$ which corresponds to a quantum Hall bilayer. The key role of a localized impurity which couples the two layers is emphasized and the effect of noncommutativity in terms of generalized magnetic translations (GMT) is fully exploited. As a result, general GMT operators are introduced, in the form of a tensor product, which act on the QHF and defect space respectively, and a comprehensive study of their rich structure is performed., Comment: 18 pages, 1 figure, accepted for publication in International Journal of Modern Physics A

By exploiting the notion of Morita equivalence for field theories on noncommutative tori and choosing rational values of the noncommutativity parameter theta (in appropriate units), a one-to-one correspondence between an abelian noncommutative field theory (NCFT) and a non-abelian theory of twisted fields on ordinary space can be established. Starting from this general result, we focus on the conformal field theory (CFT) describing a quantum Hall fluid (QHF) at paired states fillings nu =m/pm+2, recently obtained by means of m-reduction procedure, and show that it is the Morita equivalent of a NCFT. In this way we extend the construction proposed in a previous work (V. Marotta, A. Naddeo, Nucl. Phys. B 810(2009) 575) for the Jain series nu =m/2pm+1. The case m=2 is explicitly discussed and the role of noncommutativity in the physics of quantum Hall bilayers is emphasized. Our results represent a step forward the construction of a new effective low energy description of certain condensed matter phenomena and help to clarify the relationship between noncommutativity and quantum Hall fluids., Comment: 20 pages, 1 figure; Nuclear Physics B (2010), in print

The Morita equivalence for field theories on noncommutative two-tori is analysed in detail for rational values of the noncommutativity parameter theta (in appropriate units): an isomorphism is established between an abelian noncommutative field theory (NCFT) and a non-abelian theory of twisted fields on ordinary space. We focus on a particular conformal field theory (CFT), the one obtained by means of the m-reduction procedure (V. Marotta, J. Phys. A 26 (1993) 3481; V. Marotta, Mod. Phys. Lett. A 13} (1998) 853; V. Marotta, Nucl. Phys. B 527 (1998) 717; V. Marotta, A. Sciarrino, Mod. Phys. Lett. A 13 (1998) 2863), and show that it is the Morita equivalent of a NCFT. Finally, the whole m-reduction procedure is shown to be the image in the ordinary space of the Morita duality. An application to the physics of a quantum Hall fluid at Jain fillings nu =m/2pm+1 is explicitly discussed in order to further elucidate such a correspondence and to clarify its role in the physics of strongly correlated systems. A new picture emerges, which is very different from the existing relationships between noncommutativity and many body systems (A. P. Polychronakos, arXiv:0706.1095v2)., Comment: 16 pages, 2 figures

We show how the low-energy properties of the 2-leg XXZ spin-1/2 ladders with general anisotropy parameter $\Delta $ on closed geometries can be accounted for in the framework of the m-reduction procedure developed in [1]. In the limit of quasi-decoupled chains, a conformal field theory (CFT) with central charge c=2 is derived and its ability to describe the model with different boundary conditions is shown. Special emphasis is given to the Mobius boundary conditions which generate a topological defect corresponding to non trivial single-spinon excitations. Then, in the case of the 2-leg XXX ladders we discuss in detail the role of various perturbations in determining the renormalization group flow starting from the ultraviolet (UV) critical point with c=2., Comment: 23 pages, 5 figures; J. Stat. Mech.: Theory Exp. (2008), in print

We study the Morita equivalence for field theories on noncommutative two-tori. For rational values of the noncommutativity parameter $\theta $ (in appropriate units) we show the equivalence between an abelian noncommutative field theory and a nonabelian theory of twisted fields on ordinary space. We concentrate on a particular conformal field theory (CFT), the one obtained by means of the m-reduction procedure (V. Marotta, J. Phys. A 26 (1993) 3481; Mod. Phys. Lett. A 13 (1998) 853; Nucl. Phys. B 527 (1998) 717), and show that the Morita equivalence also holds at this level. An application to the physics of a quantum Hall fluid at Jain fillings \nu =m/2pm+1 is explicitly considered in order to further elucidate such a correspondence., Comment: 12 pages, 1 figure

Recently a one-dimensional closed ladder of Josephson junctions has been studied (G. Cristofano et al., Phys. Lett. A 372 (2008) 2464) within a twisted conformal field theory (CFT) approach (G. Cristofano et al., Mod. Phys. Lett. A 15 (2000) 1679; Nucl. Phys. B 641 (2002) 547) and shown to develop the phenomenon of flux fractionalization (G. Cristofano et al., Eur. Phys. J. B 49 (2006) 83). That led us to predict the emergence of a topological order in such a system (G. Cristofano et al., JSTAT (2005) P03006). In this letter we analyze the ground states and the topological properties of fully frustrated Josephson junction arrays (JJA) arranged in a Corbino disk geometry for a variety of boundary conditions. In particular minimal configurations of fully frustrated JJA are considered and shown to exhibit the properties needed in order to build up a solid state qubit, protected from decoherence. The stability and transformation properties of the ground states of the JJA under adiabatic magnetic flux changes are analyzed in detail in order to provide a tool for the manipulation of the proposed qubit., Comment: 24 pages, 4 figures

We show how to realize a ``protected'' qubit by using a fully frustrated Josephson Junction ladder (JJL) with Mobius boundary conditions. Such a system has been recently studied within a twisted conformal field theory (CFT) approach (Mod. Phys. Lett. A 15 (2000) 1679; Nucl. Phys. B 641 (2002) 547) and shown to develop the phenomenon of flux fractionalization (Eur. Phys. J. B 49 (2006) 83). The relevance of a ``closed'' geometry has been fully exploited in relating the topological properties of the ground state of the system to the presence of half flux quanta and the emergence of a topological order has been predicted (JSTAT (2005) P03006). In this letter the stability and transformation properties of the ground states under adiabatic magnetic flux change are analyzed and the deep consequences on the realization of a solid state qubit, protected from decoherence, are presented., Comment: 21 pages, 2 figures, in print in Phys. Lett. A

Following a suggestion given in Phys. Lett. B 571(2003) 621, we show how a bilayer Quantum Hall system at fillings nu =1/p+1 can exhibit a point-like topological defect in its edge state structure. Indeed our CFT theory for such a system, the Twisted Model (TM), gives rise in a natural way to such a feature in the twisted sector. Our results are in agreement with recent experimental findings (Phys. Rev. B 72 (2005) 041305) which evidence the presence of a topological defect in the transport properties of the bilayer system., Comment: 10 pages, 4 figures; talk given by A. Naddeo at "X Training Course in the Physics of Correlated Electron Systems and High-Tc Superconductors, Vietri sul Mare (SA),Italy, 3-14 October 2005

Following a suggestion given in Nucl. Phys. B 300 (1988)611,we show how the U(1)*Z_{2} symmetry of the fully frustrated XY (FFXY) model on a square lattice can be accounted for in the framework of the m-reduction procedure developed for a Quantum Hall system at "paired states" fillings nu =1 (cfr. Cristofano et al.,Mod. Phys. Lett. A 15 (2000)1679;Nucl. Phys. B 641 (2002)547). The resulting twisted conformal field theory (CFT) with central charge c=2 is shown to well describe the physical properties of the FFXY model. In particular the whole phase diagram is recovered by analyzing the flow from the Z_{2} degenerate vacuum of the c=2 CFT to the infrared fixed point unique vacuum of the c=3/2 CFT. The last theory is known to successfully describe the critical behavior of the system at the overlap temperature for the Ising and vortex-unbinding transitions., Comment: 18 pages, 1 figure, to appear in JSTAT

We propose a CFT description for a closed one-dimensional fully frustrated ladder of quantum Josephson junctions with Mobius boundary conditions (see cond-mat/0503555; we show how such a system can develop topological order thanks to flux fractionalization. Such a property is crucial for its implementation as a "protected" solid state qubit., Comment: 19 pages, 3 figures, to appear in Proceedings of PI-2005, 8-th International Conference on Path Integrals. From Quantum Information to Cosmology

We review the main results of the effective description of the Quantum Hall fluid for the Jain fillings, nu=m/2pm+1, and the non-standard ones nu=m/pm+2 by a conformal field theory (CFT) in two dimensions. It is stressed the unifying character of the m-reduction procedure to construct appropriate twisted CFT models, called Twisted Models (TM), which by construction reproduce the Quantum Hall topological properties at those fillings. Indeed for the Jain plateaux we find that the different descriptions given in the literature fall into different sectors of the TM for the torus topology. Other interesting aspects are explicitly seen for the m=2 non standard filling nu=1/p+1 (the pairing case) as the merging of non-Abelian statistics or the instability of the TM model (c=2) versus the Moore-Read one (c=3/2). Furthermore by using Boundary CFT techniques the presence of localized impurities and/or dissipation is shown to be closely connected with the twisted sector of the TM, whose presence assures the consistency of the construction and whose role in describing non trivial global properties of 2D quantum condensed matter systems is still under study., Comment: Review article to appear on Recent Research Developments in Physics-Transworld Research Network, Kerala, India (2005)

Following a suggestion given in Phys. Lett. B 571 (2003) 250, we show how a bilayer Quantum Hall system at fillings nu =m/pm+2 can exhibit a point-like topological defect in its edge state structure. Indeed our CFT theory for such a system, the Twisted Model (TM), gives rise in a natural way to such a feature in the twisted sector. Our results are in agreement with recent experimental findings (cond-mat/0503478) which evidence the presence of a topological defect in the bilayer system., Comment: 9 pages, 3 figures

We found new identities among the Dedekind eta-function, the characters of the W_{m} algebra and those of the level 1 affine Lie algebra su(m)_{1}. They allow to characterize the Z_{m}-orbifold of the m-component free bosons u(1)_{K_{m,p}} (our theory TM) as an extension of the fully degenerate representations of W_{1+infty}^{(m)}. In particular, TM is proven to be a Gamma _{theta}-RCFT extension of the chiral fully degenerate W_{1+infty}^{(m)}., Comment: 37 pages, minor revisions, one definition added, no formula or result has been modified

We analyze the modular properties of the effective CFT description for Jain plateaux corresponding to the fillings nu=m/(2pm+1). We construct its characters for the twisted and the untwisted sector and the diagonal partition function. We show that the degrees of freedom entering the partition function go to complete a Z_{m}-orbifold construction of the RCFT U(1)xSU(m)$ proposed for the Jain states. The resulting extended algebra of the chiral primary fields can be also viewed as a RCFT extension of the U(1)xW(m) minimal models. For m=2 we prove that our model, the TM, gives the RCFT closure of the extended minimal models U(1)xW(2)., Comment: 27 pages, Latex, JHEP style, no figures

Crystal tensor operators, which tranform under U_q->0(sl(2)), in analogous way as the vectors of the crystal basis, are introduced. The Wigner-Eckart theorem for the crystal tensor is defined: the selection rules depend on the initial state and on the component of the tensor operator; the transition amplitudes to the states of the same final irreducible representation are all equal., Comment: 11 pages, Latex file, no figures

I discuss a realization of stress-tensor for parafermion theories following the generalized Frenkel-Kac construction for higher level Kac-Moody algebras. All the fields are obtained from $d$=rank free bosons compactified on torus. This gives an alternative realization of Virasoro algebra in terms of a non-local correction of a free field construction which does not fit the usual background charge of Feigin-Fuchs approach., Comment: Comments and references added, 8 pages, Revtex

W_k structure underlying the tensverse realization of SU(2) at level k is analyzed. Extension of the equivalence existing between covariant and light-cone gauge realization of affine Kac-Moody algebra to W_k algebras is given. Higher spin generators related to parafermions are extracted from the operator product algebra of the generators and are showed to be written in terms of only one free boson compactified on a circle., Comment: 20 pages, Latex

The correlation function of the product of N generalized vertex operators satisfies an infinite set of Ward identities, related to a W_{\infty} algebra, whose extention out of the mass shell gives rise to equations which can be considered as a generalization of the compactified Calogero-Sutherland (CS) hamiltonians. In particular the wave function of the ground state of the compactified CS model is shown to be given by the value of the product of N vertex operators between the vacuum and exitated state. The role of vertex algebra as underlying unifying structure is pointed out., Comment: 28 pages, Latex