Gauge field corrections to 11-dimensional supergravity via dimensional reduction

Eur. Phys. J. C (2018) 78:686 https://doi.org/10.1140/epjc/s10052-018-6152-y Regular Article - Theoretical Physics Gauge field corrections to 11-dimensional supergravity via dimensional reduction Hamid R. Bakhtiarizadeha Department of Physics, Sirjan University of Technology, Sirjan, Iran Received: 28 December 2017 / Accepted: 10 August 2018 © The Author(s) 2018 Abstract Using the fact that eleven-dimensional supergravity yields type IIA supergravity under dimensional reduction on a circle, we determine higher-derivative terms of 11-dimensional supergravity including the R 4 , (∂ F4 )2 R 2 and (∂ F4 )4 terms. form [22,23]. The dimensional reduction of the metric gives rise to the ten-dimensional metric, a vector field, and a scalar (the dilaton). According to this, the metric of elevendimensional theory has to be expressed in terms of the tendimensional one as follows: 2 4 (11) = e− 3  gμν + e 3  C1μ C1ν , gμν 4 (11) gμz = e 3  C1μ 1 Introduction The low-energy effective action of M-theory is known as the 11-dimensional supergravity. This theory is described by massless modes of M-theory (the graviton, the three-form and the gravitino), which contains a membrane as a fundamental object. This theory also consists of the lowest-order supergravity action [1] plus an infinite number of higherderivative terms beyond the leading order. There exists a variety of methods which can be used to capture these higher-derivative terms. Let us briefly review some of them. The perturbative analyses of the scattering amplitudes is one of the important methods to determine the structure of the higher-derivative corrections to the 11d supergravity [2,3]. Besides the approaches based on the perturbation analyses, there are other methods to derive the higherderivative effective action of the M-theory. The famous methods are the analyses performed by computing the scattering amplitudes of superparticles [4–11], the superfield method [12–20] and by applying Noether’s method [21]. Among these approaches, we employ the straightforward dimensional reduction method to determine the higherderivative corrections to 11d supergravity. We assume that all fields are independent of the coordinate z = x 11 which (11) we choose to correspond to a spacelike direction (ηzz = 1) and then we rewrite the fields and action in a ten-dimensional form. Let us now consider the dimensional reduction of the bosonic fields of 11d supergravity, the metric and the three- and 0123456789().: V,-vol (1.1) whereas the dimensional reduction of the 3-form potential in D = 11 gives rise to a three-form and a two-form which are the fields of the 10d supergravity theory C3 (11) μνρ = C 3μνρ and C3 (11) μνz = Bμν , (1.2) with the corresponding field strengths F4 = dC3 and H = d B given by F4 (11) μνρλ = F4μνρλ and F4 (11) μνρz = Hμνρ . (1.3) The terms we would like to obtain consist of 4-form field strength and Riemann tensor. The dimensional reduction of 4-form field strength is given by Eq. (1.3), whereas the dimensional reduction of the Riemann tensor needs more considerations. For our intended purposes, it is sufficient to study the dimensional reduction of 11-dimensional supergravity which involves four massless fields. So we need the transformations (1.1) at the linear order. Assuming that the massless fields are small perturbations around the flat background, i.e. gμν = ημν +2κh μν ;  = φ0 +2κφ; C1μ = 2κc1μ . (1.4) The transformation of gμν , which is introduced in Eq. (1.1), takes the following linear form for the perturbations: h (11) μν = h μν . (1.5) On the other hand, the linearized Riemann curvature is defined as Rμν ρλ = κ∂[μ ∂ [ρ h ν] λ] . a e-mail: 4 (11) gzz = e 3 , (1.6) 123 686 Page 2 of 8 Eur. Phys. J. C (2018) 78:686 The Eq. (1.5) implies that the transformation of the linearized Riemann tensor, when carries no Killing index, is (11) Rμνρλ = Rμνρλ . (1.7) The requirement of the dimensional reduction is a powerful tool to restrict the form of an effective action. The procedure of the dimensional reduction method is well known and quite simple. First we prepare the ansatz for the higher derivative effective action in which each term has some unknown coefficients. Then we consider the dimensional reduction of the ansatz by splitting the eleven-dimensional indices into the ten-dimensional ones and the 11th index z. Some of the generated terms can be transformed to the known couplings in ten dimensions under dimensional reduction rules. The comparison of these terms gives rise simultaneous equations among the unknown coefficients in the ansatz. By solving these equations and substituting the solutions into the ansatz, one can determine the possible forms of the higher-derivative effective action. The content of our paper is as follows. In Sect. 2, we first construct an ansatz for R 4 terms with unknown coefficients in 11 dimensions and then derive them by forcing the ansatz to match with the known R 4 terms in ten dimensions. In Sect. 3, we follow the same procedure to determine the (∂ F4 )2 R 2 terms in 11 dimensions. Finally, in Sect. 4 we will obtain (∂ F4 )4 terms. Section 5 is devoted to discussion. check on our computations and results in the following relations between the unknown coefficients1 : {C2 → −16C1 , C3 → 2C1 , C4 → 16C1 , C5 → −32C1 , C6 → −32 + 16C1 , C7 → 128 − 32C1 }. Inserting these conditions into the ansatz leads to the following R 4 terms in 11 dimensions:  e−1 L R 4 = 32 4Rabce R a d c g R b f d h R e f gh  (2.2) −Rabce R ab d f R cd gh R e f gh , plus some other terms with unknown coefficients which implicitly are zero. The reason is that they vanishes when we write them in terms of independent variables in which all symmetries (including mono- and multi-term symmetries), mass-shell and on-shell conditions as well as conservation of momentum are applied. In the above equation, e denotes √ −g, where g is the determinant of the metric in 11 dimensions. 3 (∂ F4 )2 R2 terms Let us now consider the ansatz of the (∂ F4 )2 R 2 part. By imposing the linearised lowest-order equations of motion [11], one obtains 24 possible terms in the action C1 F agh i ,e F bd f i,c Rabcd Re f gh +C2 F acg i ,e F bd f i,h Rabcd Re f gh 2 R4 terms +C3 F acg i ,e F bdhi, f Rabcd Re f gh An ansatz for the higher derivative effective action, which includes quartic terms of the Riemann tensor [21], is parametrized by C1 Rabcd R abcd Re f gh R e f gh + C2 Rabcd R abc d e f gh e R f gh R +C3 Rabcd R e f R gh R + C4 Rabcd R a e c g R b f d h R e f gh +C5 Rabce R ab dg R c f d h R e f gh + C6 Rabce R ab d f R cd gh R e f gh +C7 Rabce R a d c g R b f d h R e f gh . (2.1) ab cd e f gh +C4 F cdgh ,i F iabe, f Rabcd Re f gh +C5 F bc hi ,a F f ghi,e Rabcd Re f g d +C6 F be hi ,a F f ghi,c Rabcd Re f g d +C7 F be hi ,a F c f hi,g Rabcd Re f g d +C8 F ce hi ,a F f ghi,b Rabcd Re f g d +C9 F bghi, f F ce hi ,a Rabcd Re f g d +C10 F ab f h,i F ceg h,i Rabcd Re f (...truncated)