Towards a world‐sheet description of doubled geometry in string theory

Starting from a sigma‐model for a doubled target‐space geometry, we show that the number of target‐space dimensions can be reduced by half through a gauging procedure. We apply this formalism to a class of backgrounds relevant for double field theory, and illustrate how choosing different gaugings leads to string‐theory configurations T‐dual to each other. We furthermore discuss that given a conformal doubled theory, the reduced theories are conformal as well. As an example we consider the three‐dimensional WZW model and show that the only possible reduced backgrounds are the cigar and trumpet CFTs in two dimensions, which are indeed T‐dual to each other.     

The action of the group of bundle-automorphisms on the space of connections and the geometry of gauge theories

Some aspects of the geometry of gauge theories are sketched in this review. We deal essentially with Yang-Mills theory, discussing the structure of the space of gauge orbits and the geometrical interpretation of ghosts and anomalies. Occasionally we deal also with classical gauge theories of gravitation and in particular we study the action of the group of diffeomorphisms on the space of linear connections. Finally we comment on the mathematical interpretation of anomalies in field theories.     

Flux compactifications of string theory on twisted tori

Global aspects of Scherk‐Schwarz dimensional reduction are discussed and it is shown that it can usually be viewed as arising from a compactification on the compact space obtained by identifying a (possibly non‐compact) group manifold 𝒢 under a discrete subgroup Γ, followed by a truncation. This allows a generalisation of Scherk‐Schwarz reductions to string theory or M‐theory as compactifications on 𝒢/Γ, but only in those cases in which there is a suitable discrete subgroup of 𝒢. We analyse such compactifications with flux and investigate the gauge symmetry and its spontaneous breaking. We discuss the covariance under O(d,d), where d is the dimension of the group 𝒢, and the relation to reductions with duality twists. The compactified theories promote a subgroup of the O(d,d) that would arise from a toroidal reduction to a gauge symmetry, and we discuss the interplay between the gauge symmetry and the O(d,d,ℤ T‐duality group, suggesting the role that T‐duality should play in such compactifications.     

Gravitational and Electroweak Unification

Einstein suggested that a unified field theorybe constructed by replacing the diffeomorphisms (thecoordinate transformations of general relativity) withsome larger group. We have constructed a theory that unifies the gravitational and electroweakfields by replacing the diffeomorphisms with the largestgroup of coordinate transformations under whichconservation laws are covariant statements. Thisreplacement leads to a theory with field equations whichimply the validity of the Einstein equations of generalrelativity, with a stress-energy tensor that is justwhat one expects for the electroweak field andassociated currents. The electroweak field appears as aconsequence of the field equations (rather than as a"compensating field" introduced to secure gaugeinvariance). There is no need for symmetry breaking toaccommodate mass, because the U(1) × SU(2) gaugesymmetry is approximate from the outset. Thegravitational field is described by the space-timemetric, as in general relativity. The electroweak fieldis described by the "mixed symmetry" part of the Riccirotation coefficients. The gauge symmetry-breakingquantity is a vector formed by contracting theLevi-Civita symbol with the totally antisymmetric partof the Ricci rotation coefficients.     

Jet Extension of Finslerian Gauge Approach

The gauge field figures as a key concept in the modern theory of fundamental physical fields. Various deep and intimate relationships between the gauge field theory proper and the methods of differential geometry of fibered spaces have been recognized for a long time. Usually, such relationships are established in terms of group fibrations such that the fibres are implied to be some of Lie groups (see, e.g., [1–4]). In the present servey-article, which extends basic tools elaborated in the preceding publications [5, 6], the author makes an attempt to reach a higher step of gauge generality by getting over the proper Yang-Mills ansatz that the group character of the fibre must be a steed concept from the very beginning. Instead, it looks quite accessible to begin the gauge analysis with more primary starting point where some appropriate, and motivated in a physical sense, geometrical space is treated as a fibre. Such a generalized program proves to be quite feasible if the notion of diffeomorphisms of fibres in themselves is invoked to serve as the required generalized gauge transformations. Such way of extending the gauge transformation concept seems to be sufficiently natural and fundamental for all.     

Gauge Dependence in the Theory of Non-Linear Spacetime Perturbations

Diffeomorphism freedom induces a gauge dependence in the theory of spacetime perturbations. We derive a compact formula for gauge transformations of perturbations of arbitrary order. To this end, we develop the theory of Taylor expansions for one-parameter families (not necessarily groups) of diffeomorphisms. First, we introduce the notion of knight diffeomorphism, that generalises the usual concept of flow, and prove a Taylor's formula for the action of a knight on a general tensor field. Then, we show that any one-parameter family of diffeomorphisms can be approximated by a family of suitable knights. Since in perturbation theory the gauge freedom is given by a one-parameter family of diffeomorphisms, the expansion of knights is used to derive our transformation formula. The problem of gauge dependence is a purely kinematical one, therefore our treatment is valid not only in general relativity, but in any spacetime theory. Received: 21 November 1996 / Accepted: 20 August 1997     

Cohomological analysis of bosonic D-strings and 2d sigma models coupled to abelian gauge fields

We analyze completely the BRST cohomology on local functionals for two-dimensional sigma models coupled to abelian world-sheet gauge fields, including effective bosonic D-string models described by Born-Infeld actions. In particular we prove that the rigid symmetries of such models are exhausted by the solutions to generalized Killing vector equations which we have presented recently, and provide all the consistent first order deformations and candidate gauge anomalies of the models under study. For appropriate target space geometries we find nontrivial deformations both of the abelian gauge transformations and of the world-sheet diffeomorphisms, and antifield-dependent candidate anomalies for both types of symmetries separately, as well as mixed ones.     

Gauging the conformal group

It is demonstrated that Kibble’s method of gauging the Poincaré group can be applied to the gauging of the conformal group. The action of the gauge transformations is the action of general spacetime diffeomorphisms (or coordinate transformations) combined with a local action of an 11-parameter subgroup of SO(4,2). Because the translational subgroup is not an invariant subgroup of the conformal group the appropriate generalisation of the derivative of a physical field is not a covariant derivative in the usual sense, but this does not lead to any inconsistencies.     

Generally covariant vs. gauge structure for conformal field theories

We introduce the natural lift of spacetime diffeomorphisms for conformal gravity and discuss the physical equivalence between the natural and gauge natural structure of the theory. Accordingly, we argue that conformal transformations must be introduced as gauge transformations (affecting fields but not spacetime point) and then discuss special structures implied by the splitting of the conformal group.     

Generalized gauge fields and applications to weak interactions

Yang-Mills' field is generalized to possess a nontrivial scalar part. The most general transformations for such a field under the 3-parameter isotopic gauge transformation is obtained. Using this generalized gauge field, a gauge invariant Lagrangian is constructed within the framework of the quark model. Interactions for spin-1 as well as for spin-0 are generated. As a further application a weak interaction theory mediated by the generalized gauge (boson) field is formulated. The entire weak interactions are generated in two halfs; the hadron-boson interaction is generated according to Yang-Mills' trick using the generalized gauge field and the other half (boson-lepton, etc.) is then generated by making use of the scalar part of the gauge fields according to the conventional pion gauge principle. The effective Lagrangian is then found to be mediated by the effective propagators which fall off as p⁻² at high momenta; the unitarity of the theory can thereby be insured. Universality in weaker sense than the usual one is applied to the intermediate bosons; our theory for β-decay then reduces to Cabibbo's at low energy.     

Noncommutative gauge fields coupled to noncommutative gravity

We present a noncommutative (NC) version of the action for vielbein gravity coupled to gauge fields. Noncommutativity is encoded in a twisted $\star $ -product between forms, with a set of commuting background vector fields defining the (abelian) twist. A first order action for the gauge fields avoids the use of the Hodge dual. The NC action is invariant under diffeomorphisms and $\star $ -gauge transformations. The Seiberg–Witten map, adapted to our geometric setting and generalized for an arbitrary abelian twist, allows to re-express the NC action in terms of classical fields: the result is a deformed action, invariant under diffeomorphisms and usual gauge transformations. This deformed action is a particular higher derivative extension of the Einstein-Hilbert action coupled to Yang-Mills fields, and to the background vector fields defining the twist. Here noncommutativity of the original NC action dictates the precise form of this extension. We explicitly compute the first order correction in the NC parameter of the deformed action, and find that it is proportional to cubic products of the gauge field strength and to the symmetric anomaly tensor $D_{IJK}$ .     

On Gauge Invariant Theories of Gravitation

Theories of gravitation are called gauge invariant if the invariance of the gravitational field lagrangian with respect to gauge transformations of the gravitational field variables is independend of the invariance of this lagrangian with respect to the Einstein group of general coordinate transformations. They are bimetric theories because the coordinate covariance is ensured by constructing scalar densities relative to a globally flat background metric. Such a theory is represented by the PAUL-FIERZ equations for massless spin 2 particles. But this theory is inconsistent if nongravitational matter is enclosed as a source. All attempts to overcome this inconsistancy preserving gauge invariance lead to Einstein's GRT. We review this problem and compare the situation with a theory proposed by LOGUNOV showing that he overcomes the inconsistency of linear Einstein's equations by replacing the field variables by a gauge invariant combination of new ones, which turns out to be the first order form of v. FREUD'S superpotential.     

Electroweak Symmetry on the Tangent Bundle

Various symmetries of elementary particles can be represented by gauge transformations acting on a fiber of the tangent bundle. These are diffeomorphisms of linear groups which act on vertical vector fields. It is shown how the electroweak vector boson potentials and a corresponding Kaluza-Klein-like metric can be obtained by application of SU(2) × U(1) to a tangent fiber. This geometry gives a more unified approach to gravitation and gauge symmetries.     

Lattice spinor gravity

We propose a regularized lattice model for quantum gravity purely formulated in terms of fermions. The lattice action exhibits local Lorentz symmetry, and the continuum limit is invariant under general coordinate transformations. The metric arises as a composite field. Our lattice model involves no signature for space and time, describing simultaneously a Minkowski or euclidean theory. It is invariant both under Lorentz transformations and euclidean rotations. The difference between space and time arises from expectation values of composite fields. Our formulation includes local gauge symmetries beyond the generalized Lorentz symmetry. The lattice construction can be employed for formulating models with local gauge symmetries purely in terms of fermions.     

Quantum geometry from coordinate transformations relating quantum observers

The relativity principle that the law of propagation for light has the same form for all macroscopic observers is extended to include quantum observers; i.e., observers who may be large, but not infinitely large, by comparison with quantum mechanical systems. This leads to the extension of the covariance group from the diffeomorphisms to the conservation group (which is the largest group of coordinate transformations under which conservation laws are covariant statements) and, thus, to the quantum geometry and quantum unified field theory considered in a previous paper.     

Braid group, gauge invariance, and topological order

Topological order in two-dimensional systems is studied by combining the braid group formalism with a gauge invariance analysis. We show that flux insertions (or large gauge transformations) pertinent to the toroidal topology induce automorphisms of the braid group, giving rise to a unified algebraic structure that characterizes the ground-state subspace and fractionally charged, anyonic quasiparticles. Minimal ground-state degeneracy is derived without assuming any relation between quasiparticle charge and statistics. We also point out that noncommutativity between large gauge transformations is essential for the topological order in the fractional quantum Hall effect.     

Necessary and sufficient conditions for the existence of a Lagrangian in field theory. I. Variational approach to self-adjointness for tensorial field equations

In this paper we identify some of the most significant references on the inverse problem of the calculus of variations for single integrals and initiate the study of the generalization of the underlying methodology to classical field theories. We first classify Lorentz-covariant tensorial field equations into nonlinear, quasi-linear, and semilinear forms, and then introduce their systems of equations of variation and adjoint systems. The necessary and sufficient conditions for the self-adjointness of class $\text{C}$², regular, tensorial, nonlinear, quasi-linear and semilinear forms are worked out. We study the Lagrange equations, their system of equations of variations (Jacobi equations) and their adjoint system by proving that, for class $\text{C}$⁴ and regular Lagrangian densities, they are always self-adjoint. We then introduce a concept of analytic representation which occurs when the Lagrange equations coincide with the field equations up to equivalence transformations and refine the definition by particularizing it as direct or indirect and ordered or nonordered. Some of the conventional cases of tensorial fields are considered and we prove, in particular, that the conventional representation of the complex scalar field in interaction with the electromagnetic field is of the ordered indirect type. For the objective of identifying our program we recall the two classes of equivalence transformations of the Lagrangian densities which are primarily used nowadays, namely, the Lorentz (coordinate) transformations and the gauge transformations (transformations of fields within a fixed coordinate system), and postulate the existence of a third class, which we term isotopic transformations of the Lagrangian density and which consist of equivalence transformations within a fixed coordinate system and gauge. We finally outline the objectives of our program, which essentially consist of the identification of the necessary and sufficient conditions for the existence of a Lagrangian in field theories and their first application to the transformation theory within the framework of our variational approach to self-adjointness.