Covariant path integral for chiral p-forms.

Chiral p-forms play a central role in supergravity and in string theory [1,2]. In particular, they contribute to the `miraculous" cancellation of the gravitational anomaly in type-IIB supergravity or superstring theory, making these theories quantum-mechanically consistent. The calculation of the gravitational anomaly for chiral p-forms was performed rst in [3] without using a Lagrangian but by guessing suitable Feynman rules that incorporate the chirality condition. A Lagrangian that leads to the correct equations of motion for chiral p-forms was given later in [4,5] both in at and in curved spacetimes. This Lagrangian generalizes to chiral p-forms the one constructed in [6] for chiral bosons in two dimensions, a model that has been extensivelly analysed during the last years [7]. Using the Lagrangian of [4] the authors of [8] recalculated the gravitational anomaly for chiral p-forms and found agreement with the work of [3] even though their Feynman rules turned out to be di erent. One feature of the Lagrangian given in [6] for chiral bosons, and of its generalization given in [4] for chiral pforms, is that it is not manifestly covariant. Furthermore, it leads to second class constraints in the hamiltonian formalism, which imply non usual commutation relations between the eld variables. In order to cure these di culties, an in nite number of auxiliary elds were introduced in [9]. These auxiliary variables do not carry physical degrees of freedom of their own and enable one to replace the second class constraints enforcing the chirality condition by an in nite number of rst class ones, along the lines of [10] and [11]. These rst class constraints generate a new gauge freedom and by xing the gauge through canonical methods one falls back on the original description of the chiral boson. Using this new formulation, the authors of [9] were able to derive a covariant path integral for chiral bosons and to show that it reproduces the correct physical amplitudes and anomalies. The purpose of this paper is to generalize the path integral derivation of [9] for chiral p-forms. The procedure is not entirely trivial because chiral p-forms have already a gauge invariance of their own contrary to chiral bosons -, which is furthermore reducible. Moreover, it turns out that the new gauge invariance associated with the in nite number of auxiliary elds added to achieve covariance, mixes in a non-trivial way with the original invariance of the p-forms, leading to even more reducibility. This requires the presence of further ghosts of ghosts. The more convenient way to handle this problem is to follow the lines of the eld-anti eld formalism [12{14] as we do here. Our paper is organized as follows. First we reproduce the results of [9] through the anti eld approach. We then go to the p-form case. We derive the explicit form of the pure rst class action, introducing an in nite number of auxiliary elds. This rst class description is veri ed to be physically equivalent to the second class description of [4,5], as in the chiral boson case [9]. We then derive the solution of the master equation and provide a gauge xing fermion leading to the desired manifestly covariant path integral. We close our paper with some comments on the applications of this work.