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Noncovariant Gauges in Simple Supergravity

A gauge-averaging functional of the axial type is studied for simple supergravity at one loop about flat Euclidean four-space bounded by a three-sphere, or two concentric three-spheres. This is a generalization of recent work on the axial gauge in quantum supergravity on manifolds with boundary. Ghost modes obey nonlocal boundary conditions of the spectral type, in that half of them obey Dirichlet or Neumann conditions at the boundary. In both cases, they give a vanishing contribution to the one-loop divergence. The admissibility of noncovariant gauges at the classical level is also proved.


Introduction
The recent investigations of boundary conditions and heat-kernel asymptotics in Euclidean quantum gravity have led to new work on the application of noncovariant gauges in the quantization of Einstein's gravity and simple supergravity. 1−3 The motivations of such an analysis are as follows.
(i) To improve the understanding of the one-loop semiclassical approximation. If quantum theory is viewed as a theory of small disturbances of the underlying classical theory, one should be able to compute at least the first quantum corrections in powers ofh, if such a perturbative scheme can be of any help (despite the well known lack of perturbative renormalizability).
(ii) To obtain a complete picture of admissible boundary conditions for the quantization of gauge fields and gravitation. Boundary conditions are here viewed as an essential element of any quantization scheme, as is suggested by recent progress in Euclidean quantum gravity. 3,4 (iii) To ensure self-adjointness of the elliptic operators acting on graviton and gravitino perturbations, when a problem with boundary is studied. 1−3 (iv) To understand whether supergravity theories are at least one-loop finite in the presence of boundaries. 2,3 In simple supergravity, which is the object of our investigation, it is by now well known that one has a choice of local or nonlocal boundary conditions on gravitino perturbations.
The former may involve complementary projection operators at the boundary, 5 or may fix on the initial surface the whole primed part of tangential components of gravitino perturbations, and on the final surface the whole unprimed part of tangential components of gravitino perturbations. 6,7 The latter rely instead on the following scheme. 2,3 The massless Rarita-Schwinger potential subject to gauge conditions and linearized supersymmetry constraints is split into a regular part and a singular part. The regular part may be written as an infinite sum of modes multiplying harmonics having positive eigenvalues of the intrinsic three-dimensional Dirac operator of the boundary. The singular part is instead an infinite sum of modes multiplying harmonics having negative eigenvalues of the intrinsic three-dimensional Dirac operator of the boundary. One thus performs a nonlocal operation, i.e. the separation of the spectrum of a first-order elliptic operator into its positive and negative part. 2,8 This is closely related to a positive-and negative-frequency split, which is suitable both for scattering theory and one-loop quantum cosmology. 6,7 These nonlocal boundary conditions of the spectral type for gravitino perturbations can be written in the form 2 where the label (+) denotes the part of the perturbation potential corresponding to the regular part of the underlying classical theory. In Eqs. (1.1) and (1.2) one deals with the tangential components of the perturbation potential. Their expression in terms of spinor and tetrad fields is To ensure invariance of the boundary conditions (1.1) and (1. 2) under such gauge transformations, one begins by considering the conditions 2 In the (one-loop) quantum theory, the spinor fields ν C and µ C ′ should be regarded as the which, of course, act linearly on ν C and µ C ′ , respectively. On the other hand, contraction of Eq. (1.7a) with e n C A ′ leads to An analogous procedure for Eq. (1.8a) leads to (1.14)

Noncovariant Gauges in Simple Supergravity
At a deeper level, Eq. (1.13) is obtained by requiring that the boundary conditions should be invariant under the infinitesimal gauge transformations (1.5): Similarly, Eq. (1.14) is obtained if one requires that the boundary conditions should be preserved by the infinitesimal gauge transformations (1.6): In our paper, we consider a more general class of gauge-averaging functionals of the axial type. They are defined as where round brackets denote symmetrization over spinor indices of the same kind. In Concluding remarks are presented in Sec. 5.

Ghost Operators and Eigenvalue Equations
In the Faddeev-Popov path integral for simple supergravity in the axial gauge, one has to perform a gaussian average involving the left-hand sides of (1.17) and (1.18), jointly with their corresponding ghost terms.
In the case of massless gravitino perturbations, as we said in the introduction, the ghost operators are obtained by studying the behaviour of (1.17) and (1.18) under the infinitesimal gauge transformations (1.5) and (1.6), respectively. Thus, the ghost operators acting on the spinor fields ν C and µ C ′ read and One can write the eigenfunction expansions for the spinor fields ν C and µ C ′ as 2 and then analyze the following eigenvalue equations: We here study a flat Euclidean background bounded by a three-sphere. Thus, bearing in mind that 2 and respectively. Further details about the notation can be found in Ref. and In this section we study the one-loop properties of our ghost fields in flat Euclidean background bounded by a three-sphere. For this purpose, we expand the ghost fields in harmonics on a family of three-spheres centred on the origin 2 where the α n are block-diagonal matrices with blocks 1 1 1 −1 , and the harmonics obey  Setting for convenience a = 1, one can define the following ζ-function for a first-order, elliptic and positive-definite operator A (cf. (2.8)) with spectrum (3.13): Moreover, a first-order, elliptic and nonnegative operator B exists (cf. (2.9)), with spectrum (3.14) and finite-dimensional null-space, and its ζ-function can be defined as (n + 1)(n + 2)n −s , (3.16) where the dimension of the null-space has been included in the definition of the ζ-function, Following Refs. 1,2, the one-loop divergence for simple supergravity in the axial gauge reduces therefore to the contribution resulting from three-dimensional transverse-traceless perturbations for gravitons and gravitinos: when only one bounding three-sphere exists, and when two concentric three-sphere boundaries occur.

Classical Admissibility of Axial Gauges
The axial gauge-averaging functionals play an important role in the path-integral approach to quantum gravity, since they lead to self-adjoint operators on metric and gravitino per- The preservation of the gauge conditions (4.1) and (4.2) under the infinitesimal gauge transformations (1.5) and (1.6) leads to the following differential equations for the spinor fields ν A and µ A ′ : Hence the spinor fields ν A and µ A ′ are no longer freely specifiable, but they have to satisfy Eqs. The restriction on gauge fields resulting from the preservation of gauge conditions should not be surprising, and is indeed a familiar property in field theory. For example, it is well known that (Euclidean) Maxwell theory is invariant under the following infinitesimal gauge transformations: where A b is the electromagnetic potential and the function f is freely specifiable. However, after imposing a gauge condition, the function f has to obey a differential equation, instead of being freely specifiable. On choosing the axial gauge: it is possible to ensure that the gauge-transformed potential (4.5) does actually obey the gauge (4.6), provided that n b ∇ b f = 0. In the case of the Lorentz gauge:  This, however, does not fix r np and m np .

Concluding Remarks
The contributions of our paper are as follows. is increasing evidence that this programme is going to shed new light on the various approaches to the quantization of gauge theories, and on the fertile interplay between spectral geometry and quantum field theory.
As far as supergravity theories are concerned, a further line of development lies in the application to supersymmetric quantum cosmology. So far, the main emphasis has been on the application of Hamiltonian methods in such a branch of modern quantum cosmology (see Refs. 6,21 and references therein). It now remains to be seen whether the techniques applied in Refs. 1-3 and in our paper can improve the current understanding of the quantum state of the universe within a supersymmetric framework.