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A Note on Embedding of M-Theory Corrections into Eleven-Dimensional Superspace

By analyzing eleven-dimensional superspace fourth-rank superfield strength F-Bianchi identities, we show that M-theory corrections to eleven-dimensional supergravity can not be embedded into the mass dimension zero constraints, such as the (\g^{a b})_{\a\b} X_{a b}{}^c or i (\g^{a_1... a_5})_{\a\b} X_{a_1... a_5}{}^c -terms in the supertorsion constraint T_{\a\b}{}^c. The only possible modification of superspace constraint at dimension zero is found to be the scaling of F_{\a\b c d} like F_{\a\b c d} = (1/2) \big(\g_{c d}\big)_{\a\b} e^\Phi for some real scalar superfield \Phi, which alone is further shown not enough to embed general M-theory corrections. This conclusion is based on the dimension zero F-Bianchi identity under the two assumptions: (i) There are no negative dimensional constraints on the F-superfield strength: F_{\a\b\g\d} = F_{\a\b\g d} =0; (ii) The supertorsion T-Bianchi identities and F-Bianchi identities are not modified by Chern-Simons terms. Our result can serve as a powerful tool for future exploration of M-theory corrections embedded into eleven-dimensional superspace supergravity.


Introduction
If M-theory [1] unifies superstring theories [2], such as type-I, heterotic and type-IIA superstring theories, and its low energy limit is described by eleven-dimensional (11D) supergravity [3], it is natural to expect that there must be high energy corrections to 11D supergravity compatible with supergravity formulation itself, just as superstring corrections can be embedded into 10D supergravity [2] [4]. Based on this principle, there have been attempts to embed such M-theory corrections into 11D supergravity, such as in component formulation [5], as well as in superspace formulation [6] [7][8] [9].
In superspace formulation [7] [8], it is so far commonly believed or expected that such Mtheory corrections are most likely embedded into the generalized symmetric matrix components of the mass dimension zero (d = 0) supertorsion component T αβ c , such as (γ ab ) αβ X ab c or i(γ a 1 ···a 5 ) αβ X a 1 ···a 5 c with some appropriate superfields X ab c or X a 1 ···a 5 c [7][8] [10]. However, it is not clear whether we need only the corrections of T αβ c or F αβcd at d = 0 alone for embedding M-theory corrections, or we also need any negative dimensional (d < 0) constraints, such as F αβγd or F αβγδ for such modifications for F -Bianchi identities (BIs).
In this Letter, we will present a 'no-go theorem' for embedding M-theory corrections into constraints for superspace BIs with modified constraints only at d = 0. We will show that the d = 0 corrections of constraints are not enough for embedding M-theory corrections into 11D superspace supergravity. Our conclusion is based on two assumptions: (i) All the F -superfield strength constraints at d < 0 vanish; (ii) The supertorsion T -BIs and F -BIs are not modified by Chern-Simons terms. In addition to these assumptions, our conclusion also relies on the so-called 'conventional constraints' that relate various superfields in the most general expansions of the superspace derivatives: E α and E a [9]. These conventional constraints are restrictive, e.g., the one-gamma term in T αβ c is only the standard one: i (γ c ) αβ , while the two-gamma term (γ de ) αβ X de c and the five-gamma term (γ d 1 ···d 5 ) αβ X d 1 ···d 5 c corrections are of a general form. Furthermore, the X's themselves are restricted e.g., X ab b = 0, etc., as will be shown later.
A statement for the necessity of the F -constraints at d < 0 has been given in [8], but without any proof. In the present paper, we provide explicit evidence for that claim. By studying the F -BI at d = 0, we show that, as long as the F -constraints at d < 0 are absent, there can be no such corrections as (γ de ) αβ X de c or (γ d 1 ···d 5 ) αβ X d 1 ···d 5 c possible in T αβ c other than the standard one-gamma term that can embed M-theory corrections.
2. Solving F -BI at d = 0 We first give the most important part of our results here, namely we analyze whether the d = 0 F -BI of the type (αβγδe): allows any non-trivial solution possible for embedding M-theory corrections [8]. For simplicity (as the assumption of our 'no-go theorem'), we put the d < 0 F -constraints to be zero: Accordingly, there remains only one term in (2.1) at d = 0: It is now a purely algebraic question whether there can be any non-trivial solution to (2.3), when we postulate for the d = 0 supertorsion constraint for T αβ c [8]. Here we have no corrections for the first one-gamma term, due to the 'conventional constraints' relating E α and E a as in eqs. (24) and (25) in [9]. On the other hand, X ab c and X a 1 ···a 5 c are some appropriate superfields that can possibly embed M-theory corrections [7] [8]. Accordingly, we need to put the most general corrections also into the d = 0 F -constraint 4 In particular, the lowest order on-shell physical superfield constraint in [11] corresponds to the special case of with U abc = U a 1 ···a 5 bc = 0. Since we are considering M-theory corrections, these X and U -superfields can be dealt as perturbation, namely we can first consider the satisfaction of the BI (2.3) at the linear order, temporarily ignoring the bilinear order (cross terms). For this reason, we concentrate on the analysis at the linear order in terms of X's and U's.
Based on the fluctuation analysis [9], we can impose on the X's the following 'conventional constraints' Under these constraints, X ab c has only 429 degrees of freedom, while X ⌊ ⌈5⌋ ⌉ c has 4,290 degrees of freedom. These constraints are analogous to the familiar torsion constraint T ab c = 0 commonly used in superspace, which does not delete any degrees of freedom. To be more specific, we saw in the fluctuation analysis in [9] that some components of superfield ∆ α β [9] : As seen from (2.8c) and (2.8d), the constraints (2.7) are just equivalent to determining Ψ ⌊ ⌈1⌋ ⌉ , Ψ ⌊ ⌈3⌋ ⌉ , Ψ ⌊ ⌈4⌋ ⌉ and Ψ ⌊ ⌈5⌋ ⌉ in terms only of H α b [9]. This situation is similar to the constraint T ab c = 0 in order to express the Lorentz connection φ ma b in terms of the vielbein e a m . Therefore we stress that there is no loss of degrees of freedom under the covariant constraints (2.7).
The original tensor superfields X's or U's are reduced into more fundamental irreducible components. For example, the original U abc has 605 components considering their symmetries, which can be decomposed into Substituting each irreducible component back into (2.3), we get the set of algebraic conditions to be satisfied for each irreducible components: The conceptually important ingredient here is that different irreducible components in (2.10), such as 165 vs. From now on, we use heavily the Fierz-type identities (5.1) -(5.4) and Lemmas (5.5) -(5.8) that will be given separately in section 6. We start with eq. (2.10a). This condition is identically satisfied for arbitrary A {1} due to the well-known Fierz identity in 11D (2.11) Next, due to Lemma 2 in (5.5), eq. (2.10b) implies that Eqs. (2.10c) and (2.10d) satisfy the assumption of Lemma 2, when B a,bc is identified with both for n = 55 and n = 66. Here we use the 'arrow' symbol instead of 'equality', due to the free index e, while B a,bc on the l.h.s. can be arbitrary including any such 'free' index. Therefore corresponding to (5.6a), we get the condition both for n = 55 and n = 66. Now the ae -and ce -contractions respectively yield Eqs. (2.10e), (2.10h), (2.10j) and (2.10ℓ) are solved based on Lemma 3 in (5.8), namely they imply nothing other than the vanishing of Eq. (2.10f) is understood as a sum of (5.2) + (5.4), when B a,bc and B a,⌊ ⌈5⌋ ⌉ are identified with (2.17) The like terms of the types (γ a ) αβ (γ ⌊ ⌈5⌋ ⌉ ) γδ and (γ ⌊ ⌈2⌋ ⌉ ) αβ (γ ⌊ ⌈2⌋ ⌉ ′ ) γδ in the sum (5.2) + (5.4) yield respectively the conditions Now if we look into only the ⌊ ⌈abcd⌋ ⌉ -component of this equation, and multiply it by ǫ cd abghk 1 ···k 5 , we get the condition so that the assumption of (5.7) in Lemma 2 is satisfied, and therefore B a,bc = 0, i.e., At this stage, eqs. (2.10g) and (2.10k) are the only remaining conditions to be solved. Eq. (2.10g) is regarded as the sum (5.1) + (5.3), when A a,b for the former, and A ab,cd for the latter are respectively identified with Accordingly, it is convenient to rewrite this (5.1) + (5.3) in terms of A a,b and A ab,cd instead of A's: Since each of the different γ -matrix structure is independent, we have the following three conditions Obviously, (2.29) and (2.28) lead to A ab = 0, A ab cb = 0, which via (2.27b) and (2.27c) implies also that A abcd = 0. Therefore we get We are now left with the condition (2.10k). As mentioned before, we can regard A for an arbitrary X ⌊ ⌈5⌋ ⌉ c = 0. The simplest way to get more explicit conditions from (2.31) is to multiply it by (γ ab ) αβ , and contract the indices αβ: We next multiply (2.32) by (γ b ) ǫ γ , to get the only solutions a = b = 0 as This is because the γ [4⌋ ⌉ -term yields 2a + b = 0, while the multiplication of the γ ⌊ ⌈6⌋ ⌉ -terms by iγ a yields 25a + 11b = 0. Therefore the only solutions to the condition (2.10k) are ( which is intact under the shift (2.36). The same is also true for F αβγd , which we skip here.
Based on these results and considerations, we conclude that the only degree of freedom possibly embedding M-theory corrections for the superspace constraints at d = 0 is the singlet component A {1} in (2.35), which implies that F αβcd can be only of the form proportional to the on-shell physical superfield constraint [11], scaled by some real scalar superfield Φ.
We mention here the importance of the conventional constraints (2.7). If these constraints were not imposed, we would have such corrections as for an arbitrary 32 × 32 matrix M α β satisfying (2.3) at the linear order in M. However, these corrections can not embed M-theory corrections, because they contribute as redundant degrees of freedom. We can also show that the gravitational superfield equation will not be modified at the linear order in M, because the (αβc, δ) -type BI is not modified by M. Therefore, such a matrix M is not enough for embedding M-theory corrections. We skip the details here, leaving them for a future publication.
Our analysis so far is concerned only with the linear order terms in (2.3) in the fluctuations in F αβcd and T αβ c for M-theory corrections. However, even if we include the bilinearorder terms in (2.3), our conclusion above remains intact. In other words, the linear-order satisfaction of the F -BI at d = 0 only by the limited form (2.38) will not be affected by the inclusion of next bilinear order terms. To put it differently, once the most general correction of the F αβcd has determined as in (2.3) at the linear order, in particular with no corrections for T αβ c , then it is straightforward to confirm that (2.38) satisfies also the F -BI at d = 0 to 'all orders' in the expansion in terms of Φ, because the only correction is just a scaling of F αβcd , while there is no derivative involved in the d = 0 BI (2.3).
This 'no-go theorem' established here is not so surprising from the viewpoint that (2.11) is the only available Fierz identity of the type (γ ⌊ ⌈m⌋ ⌉ ) (αβ| (γ ⌊ ⌈n⌋ ⌉ ) |γδ) = 0 allowing arbitrary integers m and n. In other words, such conditions as (2.3) has too many free indices to allow more degrees of freedom other than the singlet component A {1} as in (2.10a).

Effect of Scalar Superfield Φ
We have seen that the only possible correction of superspace constraints at d = 0 is the scaling of F αβcd in (2.38) by a real scalar superfield Φ. The next question is what sort of M-theory corrections can be embedded into this scalar superfield Φ.
The answer can be easily deduced from dimensional considerations. First, since the scalar superfield Φ is at d = 0, its spinorial derivative enters into the d = 1/2 constraints T αβ γ , T αb c and F αbcd . Therefore we have again the spinorial superfield J α [6] related to Φ by 6 with an appropriate constant ξ like in refs. [9][6]. The most general forms for these constraints are now with the unknown coefficients α 1 , α 2 , η, β 1 , β 2 , β 3 . The satisfaction of d = 1/2 BIs yields the following relationships among the coefficients Note the important fact that the exponential function e Φ is needed in F αβγd , while such a factor is absent in T αb c and T αβ γ in order to satisfy the F -BI at d = 1/2. For example, the (αβcde) -type F -BI tells us that all the F abcd -linear terms in T αb γ stay the same with no exponential function e Φ : where the symbol | F denotes the F abcd -linear part of T αb γ . As is well-known, the (αbcde) -type F -BI at d = 3/2 gives the expression for ∇ α F bcde in terms of T ab γ and F αbcd , where the latter contains the linear J's as

5)
Here the factor e Φ is involved via (3.2b). On the other hand, the (αbcd) -type T -BI gives the relationship 6) with no exponential factor e Φ with the J's. When (3.5) and (3.6) are used in the (aβγ, δ) -type BI, the former produces an exponential factor e Φ , while the latter does not.
In order to avoid this mismatch, we are forced to put η = 0, which in turn via (3.3) implies that The gravitino superfield equation can be obtained from the (aβγ, δ) -type BI, by contracting spinorial indices in several different ways, which should be consistent with each other. One way is to multiply this BI by i(γ a ) α β δ δ γ to get the trace part of the gravitino superfield equation while another way is to multiply the same BI by i(γ a ) δ γ to get up to terms, such as J 2 or the J's with fundamental physical superfields ignored as higher orders. Note that all the α 2 -dependent terms cancel each other in these equations. obviously, (3.8) and (3.9) lead to the conclusion that ξ = 0 as the only possible solution.
Unfortunately, this is a trivial solution, because this implies that J α = 0 in (3.1), so that Φ = const. Even though we did not mention, there are also other additional conditions on the independent parameters in (3.7), that will not change the conclusion here. This is because they provide more stronger conditions on the unknown parameters, but they never avoid the conclusion ξ = 0 above. Note also that our result here is in agreement with the argument about the absence of the off-shell J -superfield in [7].
From these considerations, we conclude that the scalar superfield Φ embedded into F αβcd as the exponent above is not enough to embed M-theory corrections, as long as the F -BIs are not modified by Chern-Simons terms.

Fermionic κ -Symmetry and Chern-Simons Modification
We next consider the fermionic κ -symmetry of supermembrane action [12], which justifies our assumption (i) about the vanishing F -constraints at d < 0. The standard supermembrane action is [12] with the pull-backs Π i A ≡ (∂ i Z M )E M A , for the superspace coordinates Z M and the inverse vielbein E M A in the 11D superspace we are dealing with. The fermionic κ -symmetry is dictated by [12] δ where δ κ E A ≡ (δ κ Z M )E M A . The general variation formula under δ κ E a = 0 is 3) The first term is from the variation of the kinetic term, while the second one is from the Wess-Zumino-Novikov-Witten (WZNW) term in (4.1).
We now study whether the non-zero F -constraints F αβγδ = 0, F αβγd = 0 at d < 0 are compatible with this fermionic symmetry (4.2). Using (4.3), we easily see that if there are such non-trivial F -constraints at d < 0, they will contribute only to the variation of the WZNW-term in (4.3), that are not simply cancelled by the variation of the kinetic term: It is unlikely that new corrections due to other radical and non-conventional corrections such as δ κ E a = 0 itself, or due to the addition of some other terms to the action I itself can lead to the fermionic κ -invariance of the conventional supermembrane action (4.1), because such corrections occur only at d < 0, which do not seem to communicate with other d ≥ 0 constraints. 7

Useful Algebraic Lemmas
In what follows, we list up some useful algebraic lemmas and relationships that play decisive roles for our conclusions in this paper. We start with the Fierz identity where A a,b are any arbitrary symmetric tensor superfield. In a similar fashion for an arbitrary tensor superfield B a,bc with the property B a,bc = −B a,cb , we have Similarly, for any arbitrary tensor superfields with the properties A ab,cd = +A cd,ab = (1/4)A ⌊ ⌈ab⌋ ⌉,⌊ ⌈cd⌋ ⌉ , A ⌊ ⌈ab,cd⌋ ⌉ = 0, A ab, ab = 0, we have where we have omitted the terms with the structure of (γ ⌊ ⌈5⌋ ⌉ ) αβ , because they are independent from the terms explicitly given here, and moreover, they are too messy whose structures are not decisive for our lemma below.
(5.5) Lemma 2: The vanishing of the l.h.s. of (5.2) implies that the following two conditions hold: Some remarks are in order for these lemmas: First, Lemma 1 is based on the fact that each sector of different structure of γ -matrices in (5.1) for the two pair of indices αβ and γδ is to be independently zero. This leads to the condition (3/8)A a,b − (1/48)η ab A c, c = 0, whose trace gives A c, c = 0, which in turn yields A a,b = 0, when re-substituted into this original equation. Second, Lemma 2 is also easy under (5.2), because we can require each of the two sectors with the γ -matrix structures (γ a ) αβ (γ bc ) γδ and (γ ⌊ ⌈2⌋ ⌉ ) αβ (γ ⌊ ⌈2⌋ ⌉ abc ) γδ should vanish independently. Note here that a simple contraction of two indices in (5.6a) does not lead to B a, a b = 0, due to the vanishing trace component. We did not write the condition of vanishing of the second line on the r.h.s. of (5.2), because it is just a necessary condition of the conditions (5.6a) and (5.6b). We also mention that the well-known identity (2.11) is nothing else than a special case of B a,bc = η a⌊ ⌈b v c⌋ ⌉ satisfying both (5.6a) and (5.6b). Third, Lemma 3 is straightforward, because each sector in the r.h.s. of (5.4) is to vanish independently. The last sector with (γ which combined with the vanishing of the first and second lines of the r.h.s. of (5.4) implies immediately B a,b 1 ···b 5 = 0. This is due to the difference in the coefficient in the first terms in these two sectors.

Concluding Remarks
In this Letter, we have shown that the d = 0 F -BI can not be satisfied by any correction of the type (γ ab ) αβ X ab c or i(γ a 1 ···a 5 ) αβ X a 1 ···a 5 c in dimension zero supertorsion constraint T αβ c , based on the two assumptions: (i) M-theory corrections to F -superfield strength at d < 0 are absent; (ii) The T and F -BIs are not modified by Chern-Simons terms. Additionally, we relied upon the 'conventional constraints' that relate and delete unnecessary freedom in the expansion of E α and E a . These conventional constraints restrict the structure of corrections in T αβ c , such that the one-gamma term i(γ c ) αβ in T αβ c receives no corrections, while the γ ⌊ ⌈2⌋ ⌉ or γ ⌊ ⌈5⌋ ⌉ terms can, satisfying the conditions in (2.7). We have seen that this result is valid for F -BI without Chern-Simons modifications.
Subsequently, we have also analyzed the BIs at d ≥ 1, and obtained some conditions on the constraints of T αβ γ , T αb c and F αbcd , at least for the case that the F -BIs are not modified by Chern-Simons terms. In particular, we have found that F αbcd = 0 in order to satisfy the matching exponential functions e Φ in the (aβγ, δ) -type BI at d = 3/2. On the other hand, the consistency of gravitino superfield equation out of the same (aβγ, δ) -type BI leads to the condition of vanishing of ∇ α Φ = 0, leading to the trivial solution Φ = const. This validates our conclusion, because this scalar superfield Φ was the only possible modification at d = 0. In other words, the modification of the constraint F αβcd at d = 0 is not enough for embedding M-theory corrections.
As has been mentioned, our result is not so surprising, but reasonable enough from the following viewpoints. Namely, the only Fierz identity of the form (γ ⌊ ⌈m⌋ ⌉ ) (αβ (γ ⌊ ⌈n⌋ ⌉ ) γδ) is nothing other than (2.11). This is also understandable from the fact that the F -BI (2.3) at d = 0 has four spinorial indices and one vectorial index as free indices, and therefore the vanishing of (2.3) gives such a strong condition as all the components in U's and X's are zero, except for the singlet A {1} in U abcd .
Our main conclusion in this paper can be bypassed by avoiding at least one of the two assumptions (i) and (ii) above. The assumption (i) seems very difficult to avoid, due to the fermionic symmetry of supermembrane action that seems to prevent the introduction of any F -constraints at d < 0. On the other hand, the assumption (ii) is also difficult, because there has been no other known example of such supertorsion T -BIs modified by Chern-Simons terms. We mention also that our result here does not contradict the works in [8], because the F -BIs we dealt with in our paper has not been analyzed explicitly in [8]. In fact, a statement about the necessity of the F -constraints at d < 0 was given in [8] without explicit proof. Our result in this paper provides a supporting proof with evidence for this statement, in terms of explicit computation of the F -BI at d = 0.
We believe that not only the conclusion presented in this paper, but also the technical ingredient of Fierz identities will be of great importance in the future, for exploring any possible M-theory corrections into 11D superspace supergravity.