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A New Look at Anomaly Cancellation in Heterotic M-Theory

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A New Look at Anomaly Cancellation in Heterotic M-Theory Physics Letters B 637 (2006) 93–96 www.elsevier.com/locate/physletb A new look at anomaly cancellation in heterotic M-theory Ian G. Moss School of Mathematics and Statistics, University of Newcastle upon Tyne, NE1 7RU, UK Received 2 September 2005; received in revised form 13 April 2006; accepted 13 April 2006 Available online 25 April 2006 Editor: N. Glover Abstract This Letter considers anomaly cancellation for eleven-dimensional supergravity on a manifold with boundary and theories related to heterotic M-theory. The Green–Schwartz mechanism is implemented without introducing distributions. The importance of the supersymmetry anomaly in constructing the low energy action is discussed and it is argued that a recently proposed action for low-energy heterotic M-theory gives a supersymmetric theory to all orders in the gravitational coupling κ. © 2006 Elsevier B.V. All rights reserved. 1. Introduction itino and the supergravity 3-form, so that now an action can be constructed which is non-singular and supersymmetric to Horava and Witten have argued that the strong coupling limit higher orders [8]. The theory reduces to the correct Yang–Mills of the ten-dimensional heterotic string is eleven-dimensional supergravity theory in 10 dimensions for the low-energy het- supergravity with gauge multiplets confined to two ten-dimen- erotic string by dimensional reduction, with the boundary con- sional hypersurfaces forming the boundary of the eleven- ditions playing a role in obtaining the correct 10-dimensional dimensional spacetime manifold [1,2]. This theory is a very gaugino terms [9]. promising starting point for phenomenological models based on The main issue to be addressed in this Letter is the effect of compactifications to four dimensions (see, for example, [3–7]). the new boundary conditions on anomaly cancellation. As we In applications such as these, it is important to know the action extend the theory to higher orders in the gravitational coupling, in as much detail as possible. we have to take account of the supersymmetry anomaly which 2 2 The form of the low-energy action originally put forward appears at order κ (i.e. κ times the gravitational action). The was found by relying partly on anomaly cancellation and super- most important point is that the existence of a supersymmetry symmetry. Gauge and gravitational anomalies in the theory can- anomaly implies that the classical action should not be super- cel via a novel modification of the Green–Schwartz mechanism symmetric at this order. However, it is reasonable to suppose involving the supergravity 3-form. The cancellation, which in- that the supersymmetry anomaly, like the gauge anomaly, is volves some remarkable algebraic coincidences, requires that cancelled by the Green–Schwartz mechanism, and the action the matter action contains a factor of order κ2/3 compared to the should therefore be supersymmetric up to the variation of the supergravity action, where κ is the eleven-dimensional gravita- Green–Schwartz terms [10,11]. This was not appreciated in [9], tional coupling strength. where it was shown that the supersymmetric variation of the Imposing local supersymmetry on the action fixes all of the new action for heterotic M-theory reduced to just the varia- terms at order κ2/3. However, when the same procedure is ap- tion of the Green–Schwartz terms. Now, taking into account pliedtoorderκ4/3, singular terms depending on the square of the supersymmetry anomaly, this theory is supersymmetric to the delta-function start to arise. This problem has recently been all orders in κ, at least when truncated to terms up to first order overcome by modifying the boundary conditions on the grav- in the Riemann tensor. A heuristic argument can be made for the cancellation of the supersymmetry anomaly by the Green–Schwartz terms. The E-mail address: [email protected] (I.G. Moss). Wess–Zumino consistency conditions relate the supersymme- 0370-2693/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2006.04.023 94 I.G. Moss / Physics Letters B 637 (2006) 93–96 S try anomaly to the gauge anomaly [10,11]. When the Green– boundary, then part of the supersymmetry anomaly I10 is can- Schwartz terms are added to the effective action, the total has celled [9]. (The rest of the anomaly is cancelled by the usual vanishing gauge variation. If the consistency conditions have variation of C, as we shall see shortly.) a unique solution, then the effective action with the Green– The gauge and supersymmetry variations of C are precisely Schwartz terms should have vanishing supersymmetry variation those which are required to maintain the gauge and supersym- as well. Below, we shall see how this works out explicitly. metry invariance of the 3-form boundary condition given in [9], We shall consider the gauge and supersymmetry anomaly namely cancellation now in more detail. We work throughout on the √ ‘downstairs picture’ of a manifold with boundary, rather than = 2 + 10 C (ωY ωχ ) (7) lifting to the covering space R × S1. For the present, we trun- 12 cate the action to first order in the Riemann tensor. The gauge 2 on the boundary, where ωY is the Chern–Simons form T tr F anomaly from the chiral fermion on one of the boundary com- and ponents can be described by a formal 12-form I12(F ) [12].To generate the anomaly, we introduce the notation T , such that lo- 1 a a ωχ = χ¯ ΓABCχ . (8) cally dTω = ω for a closed form ω. The anomalous variation of 4 the chiral fermion effective action under gauge transformations The constant in Eq. (7) is related to the gauge coupling con- A 2 2 δα is given by integrating a 10-form I10, defined by stant λ by = κ /2λ . This boundary condition is responsible for obtaining the correct combination of two-form and gaugino A = I10 TδαTI12. (1) fields in the action when the theory is reduced to 10 dimensions The anomalous variation of the chiral fermion effective action [9]. under supersymmetry variations has been discussed by Itoyama Since the Chern–Simons form has a gauge transformation = et al. [10,11]. It is given by the sum of two different 10-forms, δαωY d(tr αF), the boundary condition remains valid if the I S + I S , where variation of C is given by 10 10 √ I S = l TI (2) 2 10 η 12 δαC = da (9) 12 and I S is a gauge invariant expression which we leave for later. 10 where a = tr(αF ) on the boundary. (Some details of the use The operator l is an anti-derivative defined by l A = 0 and η η of p-form boundary conditions in quantum field theory can be l F = δ A. η η found in [15,16]. A more careful treatment would consider the In the case of the gauge group E , 8 Abelian BRST variations of the boundary condition, but these 1 are similar in form to the Abelian gauge variations with the I = tr F 2 3 (3) 12 12(4π)5 gauge parameter replaced by the ghost field, and the argument remains essentially unchanged.) and we have The fermion term (8) in the boundary condition is required 1 to make the boundary condition supersymmetric. It also plays I A = tr(αF ) tr F 2 2 (4) 10 12(4π)5 an important role in obtaining the correct ten-dimensional re- duction. Unfortunately, the gaugino enters into the variation of S 1 2 2 I = tr(δηAA) tr F the CGG term through the value of G = 6dC on the boundary, 10 12(4π)5 1 2 2 = √ 2 + + tr(δηAF )T tr F . (5) G tr F dωχ . (10) 3(4π)5 2 A The gauge anomaly has no fermion terms, therefore in order to Now the key observation of Horava and Witten was that I10 can be cancelled by a variation of the CGG term in the supergrav- avoid spoiling the anomaly cancellation, we have to add bound- ity action [13]. This can be done by requiring G ∼ tr F 2 on the ary corrections to the Green–Schwartz terms. The CGG term 11 boundary and δαC ∼ δ(x ) tr(αF ). If we follow this route fur- is taken from the usual supergravity action (with gravitational 2 ther, we are eventually lead to the theory with δ(x11)2 terms [2]. coupling κ /2 [17]), An alternative way to arrange the Green–Schwartz cancella- √ ∼ 2 2 tion was first described in [14].IfweletδαC da, where a is SC =− CGG. (11) = κ2 any 2-form which satisfies a tr(αF ) on the boundary, and re- M quire that G ∼ tr F 2 on the boundary, then the variation of the Green–Schwartz term is a total derivative, The boundary terms can only involve ωY , ωχ and F and they must vanish when ωχ = 0. The unique combination which has δCGG ∼ d(aGG). (6) the desired effect is 3 This integrates to give a boundary term which can cancel the 2 S =− ωY ωχ 2trF + dωχ . (12) anomaly (4). Similarly, if we add an extra supersymmetry vari- 3 6κ2 M ation δηC ∼ df to the 3-form, where f = tr(δηAA) on the ∂ I.G. Moss / Physics Letters B 637 (2006) 93–96 95 ∗ ˆ The variation of SC and S3 using (9) is then where F = (F + F)/2, Ω is the supergravity spin connection and Ω∗∗ = Ω + 1 ψDΓ ψE.Wehavenowdis- 3 ABC ABC 24 ABCDE 2 2 covered a new result that we must also add the term S for the δαSC + δαS3 =− tr(αF ) tr F .
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