Nonminimal string corrections and supergravity

Abstract

We reconsider the well-known issue of string corrections to Supergravity theory. Our treatment is carried out to second order in the string slope parameter. We establish a procedure for solving the Bianchi identities in the non minimal case, and we solve a long standing problem in the perturbative expansion of D=10, N=1 string corrected Supergravity, obtaining the H sector tensors, torsions and curvatures.PACS number: 04.65.+e 1 IntroductionIn view of the renewed interest in string corrected D=10, N=1 Supergravity [1], we revisit an outstanding problem concerning the case to second order in the perturbative expansion. String corrected D=10 and N =1 supergravity is believed to be the low energy limit of string theory, [1]- [4]. Some years ago a program was developed by Gates and collaborators to incorporate string corrections into the supergravity equations of motion [2]-[4]. This approach solved the problem of maintaining manifest supersymmetry. Recently the bosonic equations of motion for D=10, N=1 supergravity fields at superspace and component levels have been obtained and have been shown to be derivable from a lagrangian [1]. The authors have done this to first order in the string slope parameter perturbatively. A long standing problem has been that of obtaining the successful closure of the Bianchi identities to second order. It was stated in [5] that proceeding to higher than first order would yield interesting results. However in practice doing so was not so straightforward. 1In the present work we establish a procedure for solving the Bianchi identities to second order and we solve the long standing problem of closure in the H sector. It was suggested that a second order solution would require a modification of the torsion T αβ g to second order, [1], containing the so called X tensor. We propose an Ansatz for the X tensor and show that it allows for closure in the H and Torsion sectors. We also find the relevant torsions and show mutual consistency. In the curvature sector we identify R (2) αβde . Crucial to this solution are the results outlined in section (4). The perturbative approach by Gates and coworkers is well documented and discussed in the literature, and we will not recount it here. For a recent review and for an up to date commentary see [1], and references therein. Our starting point will be the Bianchi identities as listed in [5]. The sigma matrix identities and symmetries are recorded in [3].These geometrical methods nowadays are known as deformations [1], and the constraints have sometimes been referred to in the past as beta function favored (βF F ) constraints [7] (see for related subjects e.g. [8]). In the past, such methods allowed for the determination of the most general higher derivative Yang-Mills action to order γ 3 , which is globally supersymmetric and Lorentz covariant in D=10 spacetime (see e.g. [9]), a result which is important for topologically nontrivial gauge configurations of the vector field, e.g., for compactifield string theories on manifolds with topological...

String Corrections to the Riemann Curvature Tensor and the Third-Order Solution

“…This now simplifies the result first found in [7], and hence greatly simplifies the calculation at third order. We find that (18) and (20) become…”

“…Here we modify known results and find the string corrected Riemann curvature tensor. In particular we simplify a crucial result found in [7]. Our formalism is well discussed in [1] and [7].…”

“…In particular we simplify a crucial result found in [7]. Our formalism is well discussed in [1] and [7]. However the following brief outline will help to make this paper self contained.…”

“…It was required to find the so called X tensor discussed in [2], and then finding a consistent set of solutions to the Bianchi identities in superspace in the torsion, curvature and H sectors at dimension zero through 3 2 . This was achieved in [1] and [7] to second order. Here we modify known results and find the string corrected Riemann curvature tensor.…”

“…In the past we have studied ten dimensional supergravity in superspace, [1,7]. We followed the perturbative approach of Gates et al [2,3,5].…”