Finite Group Discretization of Yang-Mills and Einstein Actions

Discrete versions of the Yang-Mills and Einstein actions are proposed for any finite group. These actions are invariant respectively under local gauge transformations and under the analogues of Lorentz and general coordinate transformations. The case Z_n×Z_n×···×Z_n is treated in some detail, recovering the Wilson action for Yang-Mills theories and a new discretized action for gravity.     

Superpotentials for Gauge and Conformal Supergravity Backgrounds

Effective superpotentials obtained by integrating out matter in super Yang-Mills and conformal supergravity backgrounds in SUSY theories are considered. The pure gauge and supergravity contributions (generalizing Veneziano-Yankielowicz terms) are derived by considering the case with matter fields in the fundamental representation of the gauge group. These contributions represent quantum corrections to the tree-level Yang-Mills and conformal supergravity actions. The classical equations of motion following from the conformal supergravity action require the background to be (super)conformally flat. This condition is unchanged by quantum corrections to the effective superpotential, irrespective of the matter content of the theory.     

Conformal gravity and the bosonic string

We show that the Callan et al. effective equations of motion for the bosonic closed string in a non-trivial background may not be derived only from the bosonic part of theN=1 supergravity action of Manton and Chapline, but also from just a general conformally invariant theory of gravity. In our scheme the invariance under gauge transformations of the antisymmetric tensor field turns out to be a consequence of the metricity condition.     

On the origin of supergravity boundary terms in the AdS /CFT correspondence

The standard formulation of the AdS/CFT correspondence is incomplete since it requires adding to a supergravity action some a priori unknown boundary terms. We suggest a modification of the correspondence principle based on the Hamiltonian formulation of the supergravity action, which does not require any boundary terms. Then all the boundary terms of the standard formulation naturally appear by passing from the Hamiltonian version to the Lagrangian one. As examples the graviton part of the supergravity action on the product of AdS_d+1 with a compact Einstein manifold ϵ and fermions on AdS_d+1, are considered. We also discuss conformal transformations of gravity fields on the boundary of AdS and show that they are induced by the isometrics of AdS.     

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 $D_{IJK}$ .     

Einstein gravity emerging from quantum Weyl gravity

We advocate a conformal invariant world described by the sum of the Weyl, Dirac, and Yang-Mills action. Quantum fluctuations bring back Einstein gravity so that the long-distance phenomenology is as observed. Formulas for the induced Newton's constant and Eddington's constant are derived in quantized Weyl gravity. We show that the analogue of the trace anomaly for the Weyl action is structurally similar to that for the Yang-Mills action.     

Conformal off-mass-shell extension and elimination of conformal anomalies in quantum gravity

Radiative corrections in the Einstein quantum gravity are made manifestly conformally invariant without changing the S matrix. The conformally invariant form of the classical gravitational action is restored. It is shown, that conformal anomalies, discovered in gravitating systems, do not affect the S matrix. Off the mass shell these anomalies are eliminated by the appropriate choice of a regularization.     

An extension of Noether's theorem to transformations involving position-dependent parameters and their derivatives

Guided by the example of gauge transformations associated with classical Yang-Mills fields, a very general class of transformations is considered. The explicit representation of these transformations involves not only the independent and the dependent field variables, but also a set of position-dependent parameters together with their first derivatives. The stipulation that an action integral associated with the field variables be invariant under such transformations gives rise to a set of three conditions involving the Lagrangian and its derivatives, together with derivatives of the functions that define the transformations. These invariance identities constitute an extension of the classical theorem of Noether to general transformations of this kind. An application to the case of gauge fields demonstrates the existence of two distinct types of conservation laws for such fields.     

On the new definition of off-shell effective action

We analyze the recently proposed definition of the off-shell, gauge-invariant, gauge-independent, effective action $\text{Γ}$, utilizing an invariant metric on the field space. It is shown how to establish correspondence between $\text{Γ}$ and the standard effective action, calculated in some particular (Landau-type) gauge. Several examples are explicitly discussed, including Yang-Mills theory, the effective potential in scalar QED, and one-loop quantum gravity. Generalization to the case of super-invariant theories (e.g. super-Yang-Mills and supergravity) is also presented.     

Self-dual solutions to euclidean gravity

Recent work on Euclidean self-dual gravitational fields is reviewed. We discuss various solutions to the Einstein equations and treat asymptotically locally Euclidean self-dual metrics in detail. These latter solutions have vanishing classical action and nontrivial topological invariants, and so may play a role in quantum gravity resembling that of the Yang-Mills instantons.     

Double field formulation of Yang-Mills theory

Based on our previous work on the differential geometry for the closed string double field theory, we construct a Yang-Mills action which is covariant under O(D,D) T-duality rotation and invariant under three-types of gauge transformations: non-Abelian Yang-Mills, diffeomorphism and one-form gauge symmetries. In double field formulation, in a manifestly covariant manner our action couples a single O(D,D) vector potential to the closed string double field theory. In terms of undoubled component fields, it couples a usual Yang-Mills gauge field to an additional one-form field and also to the closed string background fields which consist of a dilaton, graviton and a two-form gauge field. Our resulting action resembles a twisted Yang-Mills action.     

N = 4 Yang-Mills theory on the light cone

The 10-dimensional supersymmetric Yang-Mills theory is constructed in the light-cone gauge. When the theory is dimensionally reduced to four dimensions it is shown that the corresponding N = 4 theory is conveniently described in terms of a scalar superfield. This formalism avoids the problem of auxiliary fields but is Lorentz invariant only on the mass shell. Similar formalisms in terms of scalar superfields are also sketched for the other supersymmetric Yang-Mills as well as for N = 8 supergravity.     

Comments on massive and massless Yang-Mills lagrangians with a quartic coupling of Faddeev-Popov ghosts

The complicated Lagrangians for massless as well as massive Yang-Mills fields and their BRS and anti-BRS transformations proposed by Curci and Ferrari are simplified by the help of an auxiliary field. In the massive non-Abelian case, the mechanism of the breakdown of physicalS-matrix unitarity is clarified and contrasted to the situation in the Abelian case.     

Bäcklund transformations and local conservation laws for self-dual Yang-Mills fields with arbitrary gauge group

A process is described for deriving infinite sets of local conservation laws for self-dual Yang-Mills fields with arbitrary gauge group. This process utilizes a set of recently constructed Bäcklund transformations which leave the self-duality equation invariant.     

Calculation of BRS cohomology with spectral sequences

A method for finding the general form of the BRS cohomology spaceH for the various gauge and supersymmetry theories is presented. The method is adapted for use in the space of integrated local polynomials of the gauge fields and ghosts with arbitrary numbers of fields and dervivatives. The technique uses the Hodge decomposition in a Fock space with a Euclidean inner product, and combines this with spectral sequences to generate simple and soluble equations whose solutions span a simple spaceE isomorphic to the complicated spaceH. The technique is illustrated for pedagogic purposes by the detailed calculation of the ghost charge zero and one sectors ofH for Yang-Mills theory with gauge groupSO (32) in ten dimensions. The method is appropriate for supersymmetric theories, gravity, supergravity and superstrings where higher order terms with many derivatives occur naturally in the effective action.Research supported in part by the Robert A. Welch Foundation and NSF Grants PHY 77-18762 and PHY 9009850     

Weyl group, <Emphasis Type="Italic">CP</Emphasis> and the kink-like field configurations in the effective <Emphasis Type="Italic">SU</Emphasis>(3) gauge theory

Effective Lagrangian for Yang-Mills gauge fields invariant under the standard space-time and local gauge SU(3) transformations is considered. It is demonstrated that a set of twelve degenerated minima exists as soon as a nonzero gluon condensate is postulated. The minima are connected to each other by the parity transformations and Weyl group transformations associated with the color su(3) algebra. The presence of degenerated discrete minima in the effective potential leads to the solutions of the effective Euclidean equations of motion in the form of the kink-like gauge field configurations interpolating between different minima. Spectrum of charged scalar field in the kink background is discussed.     

Soft gauge algebras

Gauge transformations whose algebra closes only modulo field dependent terms (soft gauge algebras) are studied in detail. The results are explicitly applied to a supersymmetric gauge theory, to gravity and to conformal gravity, all seen as gauge theories overx-space; the obvious applications to supergravity are pointed out. A consistency requirement for the gauge transformations of those fields which appear in the algebra is seen to rule out “local translations” as independent gauge transformations.     

Letter: Inflationary Dilatonic de Sitter Universe from Super Yang-Mills Theory Perturbed by Scalars and Spinors

In this paper a quantum N = 4 super Yang-Mills theory perturbed by dilaton-coupled scalars and spinor fields, is considered. The induced effective action for such a theory is calculated on a dilaton-gravitational background using the conformal anomaly found via the AdS/CFT correspondence. Considering such an effective action (using the large N method) as a quantum correction to the classical gravity action with cosmological constant, we study the effect from the dilaton on the scale factor (this corresponds to an inflationary universe without dilaton). It is shown that, depending on the initial conditions for the dilaton, the dilaton may slow down, or accelerate, the inflation process. At late times, the dilaton is decaying exponentially. Different possible cases corresponding to a dilatonic dS Universe are analyzed with respect to the equations of motion.     

Actions for non-abelian twisted self-duality

The dynamics of abelian vector and antisymmetric tensor gauge fields can be described in terms of twisted self-duality equations. These first-order equations relate the p-form fields to their dual forms by demanding that their respective field strengths are dual to each other. It is well known that such equations can be integrated to a local action that carries on equal footing the p-forms together with their duals and is manifestly duality invariant. Space-time covariance is no longer manifest but still present with a non-standard realization of space-time diffeomorphisms on the gauge fields. In this paper, we give a non-abelian generalization of this first-order action by gauging part of its global symmetries. The resulting field equations are non-abelian versions of the twisted self-duality equations. A key element in the construction is the introduction of proper couplings to higher-rank tensor fields. We discuss possible applications (to Yang-Mills and supergravity theories) and comment on the relation to previous no-go theorems.     

Einstein Supergravity and New Twistor String Theories

A family of new twistor string theories is constructed and shown to be free from world-sheet anomalies. The spectra in space-time are calculated and shown to give Einstein supergravities with second order field equations instead of the higher derivative conformal supergravities that arose from earlier twistor strings. The theories include one with the spectrum of N = 8 supergravity, another with the spectrum of N = 4 supergravity coupled to N = 4 super-Yang-Mills, and a family with N ≥ 0 supersymmetries with the spectra of self-dual supergravity coupled to self-dual super-Yang-Mills. The non-supersymmetric string with N = 0 gives self-dual gravity coupled to self-dual Yang-Mills and a scalar. A three-graviton amplitude is calculated for the N = 8 and N = 4 theories and shown to give a result consistent with the cubic interaction of Einstein supergravity.