Anomaly Free Supergravity in D=10: I) the Bianchi Identities and the Bosonic Lagrangian

Abstract

Using a theorem by Bonora-Pasti and Tonin on the existence of a solution for D=10N=1 Bianchi identities in the presence of a Lorentz Chern Simons term, we find an explicit parametrization of the superspace curvatures. Our solution depends only on one free parameter which can be reabsorbed in a field redefinition of the dilaton and of the gravitello. We emphasize that the essential point which enables us to obtain a closed form for the curvature parametrizations and hence for the supersymmetry transformation rules is the use of first order formalism. The spin connection is known once the torsion is known. This latter, rather than being identified with Hµνρ as it is usually done in the literature, is related to it by a differential equation which reduces to the algebraic relation Hµνρ = - 3Tµνρ e4/3σ only at γ1=0 (γ1 being proportional to κ/g2). The solution of the Bianchi identities exhibited in this paper corresponds to a D=10 anomaly free supergravity (AFS). This theory is unique in first order formalism but corresponds to various theories in second order formalism. Indeed the torsion equation is a differential equation which, in order to be solved must be supplemented with boundary conditions. One wonders whether supplemented with a judicious choice of boundary conditions for the torsion equation, AFS yields all the interaction terms found in the effective theory of the heterotic string (ETHS). In this respect two remarks are in order. Firstly it appears that solving the torsion equation iteratively with Tµνρ = -1/3Hµνρ e-4/3σ as starting point all the terms of ETHS except those with a ζ(3) coefficient show up. (Whether the coefficient agree is still to be checked.) Secondly, as shown in this paper the rheonomic solution of the super Poincaré Bianchi identities is unique. Hence additional interaction terms can be added to the Lagrangian only by modifying the rheonomic parametrization of the [Formula: see text]-curvature. The only assumption made in our paper is that [Formula: see text] has at most ψ∧ψ∧V components (sector (1,2)). Correspondingly the only room left for a modification of the present theory is the addition of a (0, 3) part in the rheonomic parametrization of [Formula: see text]. When this work was already finished a conjecture was published by Lechner Pasti and Tonin that such a generalization of AFS might exist and be responsible for the ζ(3) missing term. Indeed if we were able to solve the [Formula: see text]-Bianchi with this new (0, 3)-part then the torsion equation would be modified via new terms which, in second order formalism, lead to additional gravitational interactions. The equation of motion of Anomaly Free Supergravity can be worked out from the Bianchi identities: we indicate through which steps. The corresponding Lagrangian could be constructed with the standard procedures of the rheonomy approach. In this paper we limit ourselves to the bosonic sector of such a Lagrangian and we show that it can indeed be constructed in such a way as to produce the relation between Hµνρ and Tµνρ as a variational equation.

Second order Hamiltonian formalism and constraints algebra for non‐polynomial supergravities

“…Furthermore, as it was shown in Ref. [3], the Bianchi identities can be solved in closed form also in the presence of the LorentzChern-Simons 3-form L2 (0) and this solution is guaranteed by the theroem of Ref. [ …”

On anomaly-free supergravity as an effective string theory

“…It is also, at least in principle, possible to add non-minimal terms by keeping more SO(1, 9) representations non-zero in the constraints imposed. Both these mechanisms have been suggested [11,13] to account for the ζ(3)R 4 corrections [14], and string loop terms should somehow turn up this way too. Strong restrictions on possible non-minimal terms must be provided by the fact that AFS and the heterotic sigma model have in common a non-trivial instanton solution [15].…”

“…It is obtained by imposing constraints on the superspace Bianchi identities, with the Lorentz Chern-Simons term added in the way required for gauge and gravitational anomaly cancellations [8], and then solving for the physical fields. Thanks to the observation made by Bonora, Pasti and Tonin [9] that the new O(α ′ ) terms do not add any new irreducible representations of SO (1,9) to the equations, these can indeed be solved explicitly, albeit after very cumbersome calculations [10,11,12]. The solution consists of a set of equations of motion, supersymmetry transformations, and x-space Bianchi identities.…”

“…We employ the shorthand notation (T µ T ν ) = T µρ 1 ρ 2 T νρ 2 ρ 1 and similarly for the higher "traces" and T 2 = T µ 1 .µ 3 T µ 1 .µ 3 . The coefficient γ 1 introduced in [11] is proportional to α ′ once AFS is interpreted as an effective string theory, and h 1 is an arbitrary constant, which can be removed by a redefinition of φ in terms of T 2 . We also have the trace of (3)…”

String κ-anomalies and D = 10 supergravity constraints: the solution of a puzzle

“…Moreover, the so called "poltergeister", appearing in N=1, D=10 supergravity theory when the Lorentz-Chern-Simons term is present, which in previous formulations [4,5,6] show up at order α ′ , in the formulation corresponding to (32) are The κ-anomaly cancellation mechanism in the heterotic superstring determines the superspace constraints for N=1, D=10 Supergravity-Super-Yang-Mills theory.…”