Duality Invariance Research Articles

Duality and the cosmological constant

A dynamical theory is studied in which a scalar field ϕ in Einstein-Minkowski space is coupled to the four-velocityN μ of a preferred inertial observer in that space. As a consistent requirement on this coupling we study a principle of duality invariance of the dynamical mass term of ϕ at some universal length in the small-distance regime. In the large-distance regime duality breaking can be introduced by giving a background value to ϕ and a background direction toN μ. It is shown that, in an appropriate approximation, duality breaking can be related to the emergence of a characteristic phase in which the condensation of the ground state allows massive excitations with a characteristic scale of squared mass which agrees with the present observational bound for the cosmological constant.

Duality invariance of black hole creation rates

Pair creation of electrically charged black holes and its dual process, pair creation of magnetically charged black holes, are considered. It is shown that the creation rates are equal provided the boundary conditions for the two processes are dual to one another. This conclusion follows from a careful analysis of boundary terms and boundary conditions for the Maxwell action.

Soft scalar-mass sum rule in gauge-Yukawa unified models and its superstring interpretation

It is shown that a certain sum rule for soft supersymmetry-breaking scalar masses, which has been recently found in a certain class of superstring models, is universal for gauge-Yukawa unified models. To explain this coincidence, we argue that the low-energy remnant of the target-space duality invariance in the effective supergravity of compactified superstrings can be identified with the (broken) scale invariance in gauge-Yukawa unified models, and that gauge-Yukawa unification which is indispensable for the sum rule to be satisfied follows from the matching of anomalies.

Wrapping the M theory five-brane on K3

Using a recently constructed M5-brane world-volume action, we demonstrate that wrapping the M5-brane on K3 gives the heterotic string in seven dimensions. To facilitate the comparison, a new version of the world-sheet action for the Narain-compactified heterotic string, with manifest T duality invariance, is formulated.

Duality, self-duality, sources and charge quantization in abelian N-form theories

We investigate duality properties of N-form fields, provide a symmetric way of coupling them to electric/magnetic sources, and check that these charges obey the appropriate quantization requirements. First, we contrast the D = 4 k case, in which duality is a well-defined SO(2) rotation generated by a Chern-Simons form leaving the action invariant, and D = 4 k + 2 where the corresponding ostensibly SO(1, 1) rotation is not only not an invariance but does not even have a generator. When charged sources are included we show explicitly in the Maxwell case how the usual Dirac quantization arises in a fully symmetric approach attaching strings to both types of changes. Finally, for D = 4 k + 2 systems, we show how charges can be introduced for self-dual (2 k)-forms, and obtain the D = 4 k models with sources by dimensional reduction, tracing their duality invariance to a partial invariance in the higher dimensions.

Electric-magnetic black hole duality

We generalize duality invariance for the free Maxwell action in an arbitrary background geometry to include the presence of electric and magnetic charges. In particular, it follows that the actions of equally charged electric and magnetic black holes are equal.

Duality constraints on supersymmetric unified models and radiative symmetry breaking

Motivated from unified models with string origin, we analyse the constraints from duality invariance on effective supergravity models with intermediate gauge symmetry. Requiring vanishing vacuum energy and invariance of the superpotential couplings, the modular weights are subject to various constraints. Further, the intermediate gauge symmetry breaking scale M U is related to the values of modular weights of the matter and Higgs fields. For certain regions of values of the latter, M U can be close to the conventional unification scale of the minimal supersymmetric standard model. We also examine particular examples where the intermediate gauge symmetry breaks down to the standard gauge group radiatively.

T-duality invariance in random lattice strings.

Preserving the $T$-duality invariance of the continuum string in its random lattice regularization uniquely determines the random matrix model potential. For $D=0$ the duality transformation can be performed explicitly on the matrix action, and replaces color with flavor; invariance, thus, requires that the color and flavor groups be the same.

Entropy of scalar fields and its duality invariance in three-dimensional spacetimes

In the brick wall model the entropy of scalar fields is calculated in the backgrounds of three-dimensional black holes, black strings, and the Einstein metric of black strings, respectively. In all these cases the entropy is proportional to the length of the horizons and is linearly divergent. In addition, we find that the entropy of scalar fields in dual solutions keeps the same form as the one in the original solutions.

Scale factor duality and hidden supersymmetry in scalar-tensor cosmology.

It is shown that spatially flat, isotropic cosmologies derived from the Brans--Dicke gravity action exhibit a scale factor duality invariance. This classical duality is then associated with a hidden $N=2$ supersymmetry at the quantum level and the supersymmetric quantum constraints are solved exactly. These symmetries also apply to a dimensionally reduced version of vacuum Einstein gravity.

Electric-magnetic duality rotations in non-linear electrodynamics

We show that there is a function of one variable's worth of Lagrangians for a single Maxwell field coupled to gravity whose equations of motion admit electric-magnetic duality. Such Lagrangians are given by solutions of the Hamilton-Jacobi equation for timelike geodesics in Witten's two-dimensional black hole. Among them are the Born-Infeld Langrangian which arises in open string theory. We investigate the effect of the axion and the dilaton in the open superstring case and we show that this theory loses its electric-magnetic duality invariance when one considers the higher order electromagnetic field terms. We discuss some implications for black holes in string theory and an extension to 2 k-forms in 4 k spacetime dimensions.

String organization of field theories: Duality and gauge invariance.

String theories should reduce to ordinary four-dimensional field theories at low energies. Yet the formulation of the two are so different that such a connection, if it exists, is not immediately obvious. With the Schwinger proper-time representation, and the spinor helicity technique, it has been shown that field theories can indeed be written in a string-like manner, thus resulting in simplifications in practical calculations, and providing novel insights into gauge and gravitational theories. This paper continues the study of string organization of field theories by focusing on the question of local duality. It is shown that a single expression for the sum of many diagrams can indeed be written for QED, thereby simulating the duality property in strings. The relation between a single diagram and the dual sum is somewhat analogous to the relation between a old- fashioned perturbation diagram and a Feynman diagram. Dual expressions are particularly significant for gauge theories because they are gauge invariant while expressions for single diagrams are not.

Cosmological and wormhole solutions in low-energy effective string theory.

We derive and study a class of cosmological and wormhole solutions of low-energy effective string field theory. We consider a general four-dimensional string effective action where moduli of the compactified manifold and the electromagnetic field are present. The cosmological solutions of the two-dimensional effective theory obtained by dimensional reduction of the former are discussed. In particular we demonstrate that the two-dimensional theory possesses a scale-factor duality invariance. Euclidean four-dimensional instantons describing the nucleation of the baby universes are found and the probability amplitude for the nucleation process is given.

Duality invariance of the Hawking temperature and entropy.

We consider solutions to low energy string theory which have a horizon and a spacelike symmetry. Each of these solutions has a geometrically different dual description. We show that the dual solution has a horizon with exactly the same Hawking temperature (surface gravity) and entropy (area) as the original solution.

On elliptic rotations and invariants of the electromagnetic field

Duality transformations and duality invariants of the electromagnetic field are considered in the context of the Gibbs concept of elliptic rotations and a simple Riemann space for the plane of rotation. The results demonstrate the unifying role played by the metric of this space with its associated matrix and eigenvalues. The duality invariants obtained here emerge as properties of the matrix.

Dependence of Yukawa couplings on the axionic background moduli of ZN orbifolds.

The metrical and axionic background moduli which determine a general symmetric ${\mathit{Z}}_{\mathit{N}}$ orbifold model are chosen in such a way that the rotational twist leaves the underlying \ensuremath{\sigma} model action invariant. A thorough analysis of this condition is given. We notice that it plays a key role in the evaluation of the four-point correlation functions of ground states which belong to the lowest twisted sectors. Having fixed the normalization of these functions we factorize them with respect to the twisted intermediate channel. This method yields the moduli-dependent part of the twisted sector Yukawa couplings of an orbifoldized heterotic string model. We then perform various discrete mappings (axionic shifts, duality) on the space of background moduli and recognize that the induced linear transformations of the Yukawa couplings are essentially independent of the choice of a specific background. If compensating unitary redefinitions of the twist fields are applied then orbifold models whose backgrounds are related by one of the above mappings cannot be distinguished. For many twist orders we arrive at an explicit form of the phase factors needed to redefine twist fields in order that a general discrete axionic shift can be undone. The requirement of duality invariance is sufficient to determine the moduli dependence of the Yukawa couplings. Hence one may even bypass the evaluation of instanton actions.

Duality invariance in four-dimensional N=2 supergravity.

The consequences of duality invariance for {ital N}=2 supergravity coupled to vector multiplets are explored. A nonlinear equation for the prepotential is derived.

A Green-Schwarz mechanism for D = 4, N = 1 supergravity anomalies

A method is presented for cancelling the superconformal-Weyl-U(1) anomaly in four-dimensional N = 1 supergravity theories coupled to matter. This method is explicitly demonstrated for both the new minimal and old minimal formulations of supergravity. An anomaly in Kähler transformations is shown to exist in such theories and we present a mechanism for rendering these theories quantum Kähler invariant. Finally, the relationship to duality invariance in low-energy effective superstring theory is discussed.

Monopoles without strings: a conflict between the one-photon condition and duality invariance

We prove a new “no-go” theorem in the Dirac-algebra formulation of generalized electromagnetic theory, which includes magnetic monopoles and uses two potentialsA andM : It is impossible to construct a Lagrangian which is duality invariant and satisfies the one-photon assumption, from which Maxwell's equations and the equations of motion can be derived. Such a Lagrangian can be found only if either duality invariance or the one-photon assumption is sacrificed. These constraints as well as others discussed here are based on recently published results on monopoles without strings in the Dirac algebra, but they do not arise from any artificial restrictions in the Dirac-algebra formulation.

All-loop gauge couplings from anomaly cancellation in string effective theories

We derive, to all orders in perturbation theory, the E 8 gauge coupling and the modified dilaton-axion Kähler potential for the effective theories of a class of d = 4, N = 1 heterotic string models. The derivation relies on an extended version of the Green-Schwarz anomaly cancellation mechanism, and exploits target-space duality invariance. Although we deal with field-dependent effective gauge couplings and scales in a non-renormalizable supergravity theory, we derive for them a renormalization group equation as a relation among dynamical fields. When expectation values of these fields are considered, our results agree with those previously obtained in renormalizable theories with N = 1 global supersymmetry. We finally comment on possible generalizations of the present results.