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...

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We consistently incorporate Yang Mills matter fields into string corrected (deformed), D=10, N=1 Supergravity. We solve the Bianchi identities within the framework of the modified beta function favored constraints to second order in the string slope parameter γ also including the Yang Mills fields. In the torsion, curvature and H sectors we find that a consistent solution is readily obtained with a Yang Mills modified supercurrent A abc . We find a solution in the F sector following our previously developed method.

The string corrections to the Riemann Curvature tensor are found to first order in the string slope parameter, here proportional to γ. This is done for D=10 supergravity, the presumed low energy limit of string theory. We follow the perturbative approach. We also simplify a crucial result in our previous solution.