D=7 / D=6 Heterotic Supergravity with Gauged R-Symmetry
aa r X i v : . [ h e p - t h ] D ec MIFPA-10-30Imperial/TP/10/KSS/01AEI-2010-123 D = 7 /D = 6 Heterotic Supergravitywith Gauged R-Symmetry
T.G. Pugh, ∗ E. Sezgin † and K.S. Stelle ‡∗ ‡ The Blackett Laboratory, Imperial College, Prince Consort Road, London SW72BZ, UK † George and Cynthia Woods Mitchell Institute for Fundamental Physics andAstronomy,Texas A&M University, College Station, TX 77843, USA ‡ AEI, Max Planck Institut f¨ur Gravitationsphysik, Am M¨uhlenberg 1, D-14476Potsdam, Germany
ABSTRACT
We construct a family of chiral anomaly-free supergravity theories in D = 6 starting from D = 7 supergravity with a gauged noncompact R-symmetry, employing a Hoˇrava-Wittenbulk-plus-boundary construction. The gauged noncompact R-symmetry yields a positive (deSitter sign) D = 6 scalar field potential. Classical anomaly inflow which is needed to cancelboundary-field loop anomalies requires careful consideration of the gravitational, gauge, mixedand local supersymmetry anomalies. Coupling of boundary hypermultiplets requires carewith the Sp(1) gauge connection required to obtain quaternionic K¨ahler target manifolds in D = 6. This class of gauged R-symmetry models may be of use as starting points for furthercompactifications to D = 4 that take advantage of the positive scalar potential, such as thoseproposed in the scenario of supersymmetry in large extra dimensions. ∗ email: [email protected] † email: [email protected] ‡ email: [email protected] ontents D = 7 S / Z Orbifold 74 Dimensional Reduction and the Diagonalised Basis for Fields 95 Introduction of Boundary Yang-Mills Fields and the Modified BoundaryConditions 136 The Boundary Yang-Mills Action and Classical Anomalies 157 Coupling Boundary Localised Hypermultiplets 188 Extensions of the Model and Further Classical Anomalies 22 D = 7 Introduction
Anomaly-free chiral N = (1 ,
0) gauged supergravities in D = 6 [1, 2, 3, 4, 5, 6, 7] have intrigu-ing possible phenomenological applications, in particular for scenarios involving supersymme-try in large extra dimensions [8, 9]. A key challenge with such supergravity models has beento embed them in string or M-theory while also ensuring the absence of quantum gravitationalor gauge anomalies. One way to generate anomaly-free chiral models is the Hoˇrava-Wittenmechanism [10, 11], which involves compactification on a line interval while at the same timesupposing that matter fields appear on the end-walls of the interval in such a combinationas to cancel the quantum anomalies. The basic Hoˇrava-Witten scenario involves a stage ofKaluza-Klein reduction on S / Z followed by a search for anomaly-cancelling matter combi-nations with which to populate the bounding walls. In order to obtain an N = (1 , , D = 6theory with gauged U (1) R-symmetry in this way, one would need to begin this stage ofreduction with an appropriate D = 7 theory. For this purpose, we shall use the constructionof Ref. [12] which achieved D = 6 R-symmetry starting from N = 1 , D = 10 supergravityand reducing on the noncompact space H (2 , D = 7 supergravitycoupled to Super Yang-Mills with an SO (2 ,
2) noncompact gauge group. The noncompactnature of this D = 7 gauge group is essential for allowing subsequent truncation to a chiral D = 6 theory that retains an R-symmetry gauging of the sort found in Ref. [13].Reduction on a noncompact space obviously raises a number of important issues whichwould need to be addressed before such a construction could be considered physically rea-sonable. We will comment on this problem, but this issue will not be our main focus here.Rather, we will focus on another major problem arising with chiral D = 6 gauged supergravitymodels: ensuring the absence of mixed gravitational, supersymmetry and gauge anomalies.The anomaly analysis of Ref. [10, 11] for the reduction of D = 11 M-theory on S / Z yielded E Super-Yang-Mills matter multiplets on each of the two D = 10 bounding walls. A similaranalysis involving the reduction of the D = 7 theory obtained in [12] on S / Z down to D = 6will be our main focus in the present paper. SU (2) gauged half-maximal D = 7 supergravity,and its coupling to vector multiplets have been studied on a manifold with boundaries in Refs[14, 15]. There are important differences in the models considered in these papers and theones we study in this paper, the most important one being that, unlike in [14, 15], we heremaintain R-symmetry gauging on the boundary. As mentioned above, starting from a non-compact gauge theory in D = 7 is essential for this to work. Furthermore, we will study thecouplings of the scalar fields surviving the Z projection on the boundary, and will determinethe complete set of boundary conditions needed for closure under supersymmetry.In Section 2 we will review the N = 1 , D = 7 gauged supergravity which will describe ourbulk theory [16]. This can be obtained starting from N = 1 , D = 10 supergravity reducedon H (2 ,
2) as in [12]. Then, in Section 3 we will go on to consider this theory on an S / Z orbifold and we will demonstrate the necessity of appropriate Gibbons-Hawking-York terms.After this, we will continue on in Section 4 to consider a dimensional reduction of the D = 7bulk theory to D = 6 by taking a limit of vanishing orbifold width. This will be necessary toprepare the appropriate variables for subsequent bulk-boundary coupling.3he coupling of D = 6 supersymmetric boundary-localised matter to the D = 7 bulktheory involves some delicate steps. In Sections 5 and 6, we will concentrate on the couplingof boundary D = 6 vector multiplets to the D = 7 bulk fields. This involves, firstly, a carefulconsideration of how the boundary conditions for the bulk fields need to be modified in thepresence of the boundary fields, as discussed in Section 5.Since the raison d’ˆetre of the boundary fields is to provoke a “classical” anomalous gaugevariation which can be used to compensate for quantum anomalies occurring via quantumloops on the D = 6 boundaries, one expects the bulk-plus-boundary field construction toproduce a non-vanishing variation under gauge symmetries. However, since the closure of thesupersymmetry algebra generates gauge transformations, one finds that the classical gaugeanomalies are accompanied by classical supersymmetry anomalies as well. Accordingly, onecannot carry out the construction of the bulk-plus-boundary system while requiring exactsupersymmetry invariance. Instead, one must be guided by the necessity of ensuring thatthe total variation of the bulk-plus-boundary system satisfies the Wess-Zumino consistencyconditions, in order to have the structure necessary to cancel anomalies that will arise fromboundary-field quantum loops. This is discussed in Section 6.In Section 7, we will consider the coupling of boundary-localised hypermultiplets. Thisproceeds in a similar way to the coupling of the boundary vector multiplets. However, asthere is no bosonic anomaly associated to the hypermultiplets, there will be no correspondingsupersymmetry anomaly. The coupling of hypermultiplets is complicated by the fact thatthe scalars of the bulk and boundary sectors are required to combine to form a quaternionicK¨ahler manifold (QKM). We will demonstrate that this imposes a constraint on the Sp (1)connection of the boundary sector which sets it equal to the Sp (1) connection of the bulk.The models we construct in Sections 5 and 6 will be Wess-Zumino consistent, but willnot yet provide the full set of classical anomalies that are needed to cancel all the quantumanomalies. In Section 8, we will consider extensions of the present model that can give riseto the remaining cancellations. We will consider the supersymmetric extension of the thebulk model Chern-Simons terms, focusing particularly on a topological mass term. As well asexamining alternative boundary conditions, we finally will look at the coupling of boundary-localised tensor multiplets.In Section 9, we will consider an explicit example of an anomaly-cancelling system. Todo this, we will calculate the anomaly polynomial produced by one-loop quantum effects. Wewill then show how the Wess-Zumino consistent classical anomalies constructed so far can bearranged so as to cancel these quantum anomalies.In Appendix A, we examine the limit of coincident boundaries when the boundaries arepopulated with vector multiplets and will show the emergence of gauged supergravity in D =6. In Appendix B, we will provide the bulk-plus-boundary construction with a supersymmetricset of boundary conditions in an equivalent formulation in which the bulk 3-form potential isdualised to a 2-form potential. 4 D = 7 Seven dimensional N = 1 supergravity in the absence of boundaries has been well studied,and the action of the supergravity multiplet coupled to n vector multiplets is known [16]. Thefields in this action form a reducible multiplet with field content,(ˆ e NM , ˆ A MNR , ˆ σ, ˆ A ˆ IM , φ α , ˆ ψ AM , ˆ χ A , ˆ λ ˆ rA ) , (2.1)where M = 0 , . . . , , g MN and N = 0 , . . . , , η MN = diag( − + · · · +).The scalars φ α with α = 1 , , ..., n parametrise a coset, SO ( n, SO ( n ) × SO (3) , (2.2)for which we can form the representative elements L i ˆ I and L ˆ r ˆ I , where ˆ I = 1 , . . . , n + 3 isan SO ( n,
3) index, which is raised and lowered with the SO ( n,
3) invariant metric η ˆ I ˆ J =diag( − − − + . . . +). i = 1 , . . . , SO (3) index and ˆ r = 1 , . . . , n is an SO ( n ) index;these are raised and lowered with the Kronecker deltas δ ij and δ ˆ r ˆ s respectively. The cosetrepresentatives satisfy the relations − L i ˆ I L i ˆ J + L ˆ r ˆ I L ˆ r ˆ J = η ˆ I ˆ J , (2.3) L i ˆ I L ˆ Ij = − δ ij , L ˆ r ˆ I L ˆ I ˆ s = δ ˆ r ˆ s , L i ˆ I L ˆ I ˆ r = 0 . (2.4)The spinors are symplectic Majorana and carry an Sp (1) doublet index A = 1 , ǫ AB . The Sp (1) indices will often be suppressed, as in¯ χσ i ǫ = ¯ χ A σ iAB ǫ B .The action for these fields, up to terms quadratic in fermions, is given by S SG = 12 κ Z d x ˆ e (cid:26)
12 ˆ R − g e ˆ σ ˆ F iMN ˆ F MNi − g e ˆ σ ˆ F ˆ rMN ˆ F MN ˆ r − e − σ ˆ F MNRS ˆ F MNRS − √ g ˆ ε MNRST UV ˆ A MNR ˆ F ˆ rST ˆ F ˆ rUV − ∂ M ˆ σ∂ M ˆ σ −
12 ˆ P i ˆ rM ˆ P Mi ˆ r − g e − ˆ σ (cid:18) C i ˆ r C i ˆ r − C (cid:19) − i ψ M ˆ γ MNR ˆ D N ˆ ψ R − i χ ˆ γ M ˆ D M ˆ χ − i g ˆ¯ λ ˆ r ˆ γ M ˆ D M ˆ λ ˆ r − i χ ˆ γ M ˆ γ N ˆ ψ M ∂ N ˆ σ − g ˆ¯ λ ˆ r σ i ˆ γ M ˆ γ N ˆ ψ M ˆ P i ˆ rN (2.5) Our conventions are: ψ A = ǫ AB ψ B , ψ A = ψ B ǫ BA and ǫ AB ǫ BC = − δ CA . i √ e − ˆ σ ˆ F MNRS (cid:18) ˆ¯ ψ [ L ˆ γ L ˆ γ MNRS ˆ γ T ˆ ψ T ] + 4 ˆ¯ ψ L ˆ γ MNRS ˆ γ L ˆ χ − χ ˆ γ MNRS ˆ χ + 1 g ˆ¯ λ ˆ r ˆ γ MNRS ˆ λ ˆ r (cid:19) + 18 g e ˆ σ ˆ F iMN (cid:18) ˆ¯ ψ [ L σ i ˆ γ L ˆ γ MN ˆ γ T ˆ ψ T ] − ψ L σ i ˆ γ MN ˆ γ L ˆ χ + 3 ˆ¯ χσ i ˆ γ MN ˆ χ − g ˆ¯ λ ˆ r σ i ˆ γ MN ˆ λ ˆ r (cid:19) − i g e ˆ σ ˆ F ˆ rMN (cid:18) ˆ¯ ψ L ˆ γ MN ˆ γ L ˆ λ ˆ r + 2 ˆ¯ χ ˆ γ MN ˆ λ ˆ r (cid:19) − i √ ge − ˆ σ C (cid:18) ˆ¯ ψ M ˆ γ MN ˆ ψ N + 2 ˆ¯ ψ M ˆ γ M ˆ χ + 3 ˆ¯ χ ˆ χ − g ˆ¯ λ ˆ r ˆ λ ˆ r (cid:19) + 12 √ e − ˆ σ C i ˆ r (cid:18) ˆ¯ ψ M σ i ˆ γ M ˆ λ ˆ r − χσ i ˆ λ ˆ r (cid:19) + 12 g e − ˆ σ C ˆ r ˆ si ˆ¯ λ ˆ r σ i ˆ λ ˆ s (cid:27) , (2.6)where ˆ F MNRS = 4 ∂ [ M A NRS ] is the field strength, invariant under tensor gauge transforma-tions δ ˆ A MNR = 3 ∂ [ M ˆ λ NR ] ; ˆ ε = 1. Furthermore,ˆ F ˆ IMN = 2 ∂ [ M ˆ A ˆ IN ] + f ˆ J ˆ K ˆ I ˆ A ˆ JM ˆ A ˆ KN , ˆ F iMN = ˆ F ˆ IMN L i ˆ I , ˆ F ˆ rMN = ˆ F ˆ IMN L ˆ r ˆ I , ˆ ω MNR ( ˆ A ˆ I ) = ˆ F ˆ I [ MN ˆ A ˆ IR ] − f ˆ I ˆ J ˆ K ˆ A ˆ IM ˆ A ˆ JN ˆ A ˆ KR . (2.7)ˆ D M = ˆ ∂ M + 14 ˆ ω µµν γ µν + 12 √ Q iM σ i , ˆ Q iM = i √ ǫ ijk ˆ Q jkM , ˆ Q ijM = L ˆ Ij (cid:16) δ ˆ K ˆ I ∂ M + f ˆ I ˆ J ˆ K ˆ A ˆ JM (cid:17) L i ˆ K , ˆ Q ˆ r ˆ sM = L ˆ I ˆ r (cid:16) δ ˆ K ˆ I ∂ M + f ˆ I ˆ J ˆ K ˆ A ˆ JM (cid:17) L ˆ s ˆ K , ˆ P i ˆ rM = L ˆ I ˆ r (cid:16) δ ˆ K ˆ I ∂ M + f ˆ I ˆ J ˆ K ˆ A ˆ JM (cid:17) L i ˆ K , C = − √ f ˆ I ˆ J ˆ K L ˆ Ii L ˆ Jj L ˆ Kk ǫ ijk ,C i ˆ r = 1 √ f ˆ I ˆ J ˆ K L ˆ Ij L ˆ Jk L ˆ K ˆ r ǫ ijk , C ˆ r ˆ si = f ˆ I ˆ J ˆ K L ˆ I ˆ r L ˆ J ˆ s L ˆ Ki , (2.8)and ˆ R is the curvature defined with respect to the torsion-free Levi-Civita connection. Thevectors ˆ A ˆ JM gauge a group K ⊂ SO ( n,
3) with n + 3 generators. Possible gauge groups arediscussed in [17]. Of special interest are certain non-compact gauge groups which allow anR-Symmetry gauging upon dimensional reduction to D = 6 followed by chiral truncation. Weshall make restrictions to such gaugings in Section 3, but for now we will leave the constructiongeneral. 6he action is invariant under the following local supersymmetry transformations, δ ˆ e MM = i ˆ¯ ǫγ M ˆ ψ M ,δ ˆ ψ M = 2 ˆ D M ˆ ǫ − √ e − ˆ σ ˆ F RSLT (cid:0) ˆ γ M ˆ γ RSLT + 5ˆ γ RSLT ˆ γ M (cid:1) ˆ ǫ − i g e ˆ σ ˆ F iRS σ i (cid:0) γ M ˆ γ RS − γ RS ˆ γ M (cid:1) ˆ ǫ − √ ge − ˆ σ C ˆ γ M ˆ ǫ ,δ ˆ χ = −
12 ˆ γ M ˆ ∂ M ˆ σ ˆ ǫ − √ e − ˆ σ ˆ F MNRS ˆ γ MNRS ˆ ǫ − i g e ˆ σ ˆ F iMN σ i ˆ γ MN ˆ ǫ + √ ge − ˆ σ C ˆ ǫ ,δ ˆ A MNR = 3 i √ e ˆ σ ˆ¯ ǫ ˆ γ [ MN ˆ ψ R ] − i √ e ˆ σ ˆ¯ ǫγ MNR ˆ χ ,δ ˆ A ˆ IM = − ge − ˆ σ (cid:16) ˆ¯ ǫσ i ˆ ψ M + ˆ¯ ǫσ i ˆ γ M ˆ χ (cid:17) L ˆ Ii + ie − ˆ σ ˆ¯ ǫ ˆ γ M ˆ λ ˆ r L ˆ I ˆ r ,δ ˆ σ = − i ˆ¯ ǫ ˆ χ ,δL i ˆ I = 1 g ˆ¯ ǫσ i ˆ λ ˆ r L ˆ r ˆ I ,δL ˆ r ˆ I = 1 g ˆ¯ ǫσ i ˆ λ ˆ r L i ˆ I ,δ ˆ λ ˆ r = − e ˆ σ ˆ F ˆ rMN ˆ γ MN ˆ ǫ + ig ˆ γ M ˆ P i ˆ rM σ i ˆ ǫ − i √ ge − ˆ σ C i ˆ r σ i ˆ ǫ . (2.9) S / Z Orbifold
The action has a Z parity symmetry under which x → − x , and the following fields haveeven parity: (ˆ e µν , ˆ e , ˆ A µν , ˆ σ, ˆ A I ′ µ , ˆ A I , φ ri , ˆ ψ µ + , ˆ ψ − , ˆ χ − , ˆ λ r − , ˆ λ r ′ + ) , (3.1)whilst the odd-parity fields are(ˆ e µ , ˆ e ν , ˆ A µνρ , ˆ A Iµ , A I ′ , φ r ′ i , ˆ ψ µ − , ˆ ψ , ˆ χ + , ˆ λ r + , ˆ λ r ′ − ) , (3.2)where the scalars ( φ ri , φ r ′ r ) parametrize the coset (2.2). The supersymmetry transformationrules are consistent with these parity assignments provided that ǫ + has even parity and ǫ − has odd parity. In the definitions (3.1) and (3.2), we have split up the index M into the 7direction and the directions normal to it, which are labelled by µ = 0 , . . . ,
5. We have alsodefined a chiral projection operator P ± = (cid:0) ± γ (cid:1) , which projects onto chiral spinors in thestandard way, i.e. χ ± = P ± χ . The ˆ r and ˆ I indices have also been split asˆ I = { I , I ′ } , I = 1 , ..., p + 3 , I ′ = p + 4 , ..., n + 3ˆ r = { r , r ′ } , r = 1 , ..., p , r ′ = p + 1 , ..., n (3.3)7here 0 ≤ p ≤ n . Next, we observe that the requirement that the Yang-Mills field strength(2.7) have a definite parity imposes the conditions f IJ K = f I ′ J ′ K = 0 . (3.4)The possible groups K which posses this property and which reduce to give a gauged super-gravity in 6 dimensions are SO (3 , SO (2 ,
1) and SO (2 ,
2) [17]. Since the action is invariantunder a Z symmetry, we can formulate the action integral on a manifold M × I , where M isan arbitrary D = 6 spacetime and I = S / Z is an interval with boundaries ( ∂M ) at x = 0and x = L . This will result in multiplication of the action by a factor of 2 since the interval I is half the size of the circle S . Assuming that all fields are continuous and smooth, theparity assignments then imply the following boundary conditions:(ˆ e µ , ˆ e ν , ˆ A µνρ , ˆ A Iµ , A I ′ , φ r ′ i , ˆ ψ µ − , ˆ ψ , ˆ χ + , ˆ λ r + , ˆ λ r ′ − ) (cid:12)(cid:12) ∂ M = 0 ,∂ (ˆ e µν , ˆ e , ˆ A µν , ˆ σ, ˆ A I ′ µ , ˆ A I , φ ri , ˆ ψ µ + , ˆ ψ − , ˆ χ − , ˆ λ r − , ˆ λ r ′ + ) (cid:12)(cid:12) ∂M = 0 . (3.5)The boundary conditions on φ –scalars imply that the even-parity coset representatives ( L iI , L rI )parametrize the coset SO ( p, /SO ( p ) × SO (3), and L r ′ I ′ = δ r ′ I ′ , whilst the odd-parity cosetrepresentatives ( L iI ′ , L rI ′ , L r ′ I ) vanish on the boundaries.The fields whose ∂ derivatives vanish at the boundaries are the parity even ones. In adiagonalised basis which will be spelled out in the next section (see eqn. (4.6)), they arrangethemselves into D = 6 supergravity plus a single tensor multiplet, ( n − p ) vector multipletsand ( p + 1) hypermultiplets.We also note that our parity assignments differ from those used in [14, 15] in two respects.Firstly, while the coupling constant g is declared to be parity odd in [14, 15], we take it hereto be parity even. Secondly, while all the vector fields are taken to be parity odd in [14, 15],here we split them into two sets, and we assign even parity to one of these sets. Both ofthese differences crucially depend on our working with a noncompact gauged supergravity in D = 7.In order that the Euler-Lagrange variational principle be consistent with these boundaryconditions, the action has to be supplemented by suitable additional terms defined on theboundary, known as Gibbons-Hawking-York terms. Then the total action takes the form S = Z M d x L SG + Z ∂M d x L GHY . (3.6)In the rest of this section, we will determine L GHY . We will consider explicitly the boundaryat x = 0. The boundary located at x = L can be treated similarly.To begin with, let us consider a general variation of the Einstein-Hilbert term. It containsa normal derivative of the metric variation, which must be avoided in order that the boundaryconditions implied by the variational principle are not over constrained. To achieve this, asis well known, one adds an extrinsic curvature term so that the total action becomes S EH + S GHY = 12 κ Z M d x ˆ e ˆ R + 1 κ Z ∂M d x q − ˆ h ˆ K , (3.7) We could alternatively have defined ˆ R with respect to the spin connection which would then contain fermi K is the extrinsic curvature, which is defined as follows. Let ˆ n N denote the unit vectornormal to the boundary pointing out of M . We construct the induced metric ˆ h MN asˆ g MN = ˆ h MN + ˆ n M ˆ n N ; ˆ n M ˆ h MN = 0 . (3.8)Consequently, contraction with ˆ h MN projects onto components of vectors in directions tangentto the boundary. The extrinsic curvature is defined asˆ K = ˆ h MN ˆ K MN , ˆ K MN = ˆ h PM ˆ h QN ˆ ∇ P ˆ n Q . (3.9)Then the general variations of (3.7) yields, modulo the Einstein field equation, (cid:0) δS EH + δS GHY (cid:1) | EOM = − κ Z ∂M dx q − ˆ h (cid:16) ˆ K MN − ˆ K ˆ h MN (cid:17) δ ˆ g MN . (3.10)This vanishes, however, upon imposing the boundary conditions (3.5), which in particularimply K µν | ∂M = 0 . (3.11)Turning to the general variation of the fermionic kinetic terms, they all involve fermion vari-ations of both chiralities. In order that the boundary conditions implied by the variationalprinciple are not over constrained, we add suitable Gibbons-Hawking terms such that S F + S GHY = 1 κ Z M d x ˆ e (cid:26) − i ψ M ˆ γ MNR ˆ D N ˆ ψ R − i χ ˆ γ M ˆ D M ˆ χ − i g ˆ¯ λ ˆ r ˆ γ M ˆ D M ˆ λ ˆ r (cid:27) + 1 κ Z ∂M d x q − ˆ h (cid:26) − i ψ µ ˆ γ µν ˆ ψ ν − i χ ˆ χ − i g ˆ¯ λ r ˆ λ r + i g ˆ¯ λ r ′ ˆ λ r ′ (cid:27) . (3.12)As a result, we obtain the total variation, modulo the fermion equations of motion, (cid:0) δS F + δS GHY (cid:1) | EOM = 1 κ Z ∂M d x q − ˆ h (cid:26) − i ˆ¯ ψ µ − ˆ γ µν δ ˆ ψ ν + − i ˆ¯ χ + δ ˆ χ − − ig ˆ¯ λ r + δ ˆ λ r − + ig ˆ¯ λ r ′ − δ ˆ λ r ′ + (cid:27) , (3.13)which is set to zero when the parity-odd fields vanish on the boundary.One can check that there is no need for any further Gibbons-Hawking terms, and we con-clude that the total action with a well-defined variational principle yielding the bulk equationsof motion and the boundary conditions (3.5) is given by S SG + S GHY + S GHY . In describing the coupling of matter fields to supergravity on the boundary, which we shall doin the next section, it is convenient to express the parity-even bulk fields in a diagonal basis squared terms. However that definition contributes a total derivative which is subsequently eliminated byadding appropriate Gibbons-Hawking-York terms, with no further effect in the bulk plus boundary theory thatwe will construct [18]. S SG on a circle and then will chirally truncate the theory such that we retainonly the even-parity fields. This amounts to taking a limit in which the boundaries are emptyand coincident, which results in a D = 6 , N = (1 ,
0) supergravity.We begin by making a Kaluza-Klein ansatz for the the metric,ˆ g MN = (cid:18) e αφ g µν + e βφ A µ A ν − e βφ A µ − e βφ A µ e βφ (cid:19) . (4.1)We chose values for α and β so as to obtain the standard Einstein-Hilbert gravitational actionin D = 6, α = − √ , β = − α . (4.2)We will chose our notation such that hatted fields have their indices raised and lowered withˆ g MN , while unhatted fields have their indices raised and lowered with g µν . We work with thecorresponding vielbein basis,ˆ e µµ = e αφ e µµ , ˆ e µµ = e − αφ e µµ , ˆ e µ = − e βφ A µ , ˆ e µ = e − αφ A µ , ˆ e µ = 0 , ˆ e µ = 0 , ˆ e = e βφ , ˆ e = e − βφ . (4.3)We note here that in order for the gauge choice (4.3) to be invariant under the supersymmetrytransformations (2.9), we must make a compensating Lorentz transformation with parameter λ µ = − i ¯ ǫ + γ µ ψ . As the veilbein is the only boson that transforms under Lorentz symmetry,the effect of this additional transformation on all other fields can be ignored, since it is higherorder in fermions.Working in a frame in which ˆ n M = (0 , , , , , , −
1) implies that ˆ n M = − ˆ e M . Substi-tuting this into (3.8) we see that,ˆ g MN = (cid:18) ˆ h µν + e βφ A µ A ν ˆ h µ − e βφ A µ ˆ h µ − e βφ A µ ˆ h + e βφ (cid:19) . (4.4)Comparing (4.1) and (4.4), we can read off the components of ˆ h asˆ h µν = e αφ g µν , ˆ h µ = ˆ h = 0 ; (4.5)this will be useful when determining the surface variations later on.10e can now diagonalise all kinetic terms by making the following redefinitions [17] σ = ˆ σ − αφ , ϕ = 12 ˆ σ + 4 αφ ,ψ r = 1 g √ e αφ ˆ λ r − , λ r ′ = 1 √ e αφ ˆ λ r ′ + ,χ = √ e αφ (cid:18) ˆ χ − + 14 ˆ ψ − (cid:19) , ψ = 1 √ e αφ (cid:16) ˆ ψ − − ˆ χ − (cid:17) ,ψ µ = 1 √ e αφ (cid:18) ˆ ψ µ + − γ µ ˆ ψ − (cid:19) , ˆ ǫ + = 1 √ e αφ ǫ , (4.6)Φ I = 1 g ˆ A I , A I ′ µ = ˆ A I ′ µ ,B µν = 1 √ A µν . With these definitions, the D = 6 supergravity action becomes S SG (6) = 2 Lκ Z d xe (cid:26) R − g e σ F r ′ µν F µνr ′ − e − σ G µνρ G µνρ − ∂ µ σ∂ µ σ − ∂ µ ϕ∂ µ ϕ − P irµ P µir − P rµ P µr − P iµ P µi − g e − σ (cid:16) C ir ′ C ir ′ + 2 S ir ′ S ir ′ (cid:17) + 124 g ε µνρσλτ G µνρ ω σλτ ( A r ′ ) − i ψ µ γ µνρ D ν ψ ρ − i χγ µ D µ χ − i g ¯ λ r ′ γ µ D µ λ r ′ − i ψγ µ D µ ψ − i ψ r γ µ D µ ψ r −
12 ¯ ψ r σ i γ µ γ ν ψ µ P irν −
12 ¯ ψσ i γ µ γ ν ψ µ P iν − i ψ r γ µ γ ν ψ µ P rν − i χγ µ γ ν ψ µ ∂ ν σ − i ψγ µ γ ν ψ µ ∂ ν ϕ − i e − σ G µνρ (cid:18) ¯ ψ [ λ γ λ γ µνρ γ τ ψ τ ] − ψ λ γ µνρ γ λ χ − ¯ χγ µνρ χ + ¯ ψγ µνρ ψ + ¯ ψ r γ µνρ ψ r − g ¯ λ r ′ γ µνρ λ r ′ (cid:19) − P iµ (cid:18) ¯ ψ [ ρ σ i γ ρ γ µ γ τ ψ τ ] + ¯ χσ i γ µ χ + 1 g ¯ λ r ′ σ i γ µ λ r ′ − ¯ ψ r σ i γ µ ψ r − ¯ ψσ i γ µ ψ (cid:19) − i P rµ ¯ ψγ µ ψ r − i g e σ F r ′ µν (cid:18) ¯ ψ ρ γ µν γ ρ λ r ′ + ¯ χγ µν λ r ′ (cid:19) − e σ C irr ′ ¯ λ r ′ σ i ψ r + ie σ S rr ′ ¯ λ r ′ ψ r − e σ S ir ′ ¯ λ r ′ σ i ψ + 12 √ e − σ λ r ′ σ i γ µ ψ µ (cid:16) C ir ′ − √ S ir ′ (cid:17) + 12 √ e − σ λ r ′ σ i χ (cid:16) C ir ′ − √ S ir ′ (cid:17) (cid:27) , (4.7)where ε µνρσλτ = ˆ ε µνρσλτ , and we have used the following definitions: G µνρ = 3 ∂ [ µ B νρ ] , F r ′ µν = 2 ∂ [ µ A r ′ ν ] + f s ′ t ′ r ′ A s ′ µ A t ′ ν ,ω µνρ ( A r ′ ) = F r ′ [ µν A r ′ ρ ] − f r ′ s ′ t ′ A r ′ µ A s ′ ν A t ′ ρ ; (4.8)11he elements of the Maurer-Cartan forms are defined as P irµ = L Ir (cid:16) δ KI ∂ µ − f r ′ I K A r ′ µ (cid:17) L iK ,Q ijµ = L Ij (cid:16) δ KI ∂ µ − f r ′ I K A r ′ µ (cid:17) L iK ,Q rsµ = L Ir (cid:16) δ KI ∂ µ − f r ′ I K A r ′ µ (cid:17) L sK , (4.9)the axion field strengths are defined as P iµ = e ϕ (cid:16) ∂ µ Φ I + f I r ′ J A r ′ µ Φ J (cid:17) L iI , P rµ = e ϕ (cid:16) ∂ µ Φ I + f I r ′ J A r ′ µ Φ J (cid:17) L rI , (4.10)gauge functions are defined as C kr ′ = 1 √ ǫ kij f r ′ IJ L Ii L Jj , C irr ′ = f r ′ IJ L Ii L Jj ,S ir ′ = − e ϕ f r ′ IJ Φ J L Ii , S rr ′ = − e ϕ f r ′ IJ Φ J L Ir , (4.11)and the covariant derivative is defined as D µ ǫ = (cid:18) ∂ µ + 14 ω µµν γ µν + 12 √ Q iµ σ i (cid:19) ǫ , Q iµ = i √ ǫ ijk Q jkµ . (4.12)Truncating the supersymmetry transformations (2.9) and writing the result in terms of theredefined fields gives the transformations under which the action (4.7) is invariant: δe µµ = i ¯ ǫγ µ ψ µ ,δψ µ = D µ ǫ − i P iµ σ i ǫ + 124 e − σ G ρστ γ ρστ γ µ ǫ ,δχ = − γ µ ∂ µ σǫ − e − σ G µνρ γ µνρ ǫ ,δB µν = − ie σ ¯ ǫγ [ µ ψ ν ] + i e σ ¯ ǫγ µν χ ,δσ = − i ¯ ǫχ ,δA r ′ µ = ie − σ ¯ ǫγ µ λ r ′ ,δλ r ′ = − e σ γ µν F r ′ µν ǫ − i √ g e − σ (cid:16) C ir ′ − √ S ir ′ (cid:17) σ i ǫ ,δψ = i γ µ (cid:0) P iµ σ i − i∂ µ ϕ (cid:1) ǫ ,δψ r = i γ µ (cid:0) P irµ σ i + i P rµ (cid:1) ǫ ,δϕ = i ¯ ǫψ ,δL rI = ¯ ǫσ i ψ r L iI ,δL iI = ¯ ǫσ i ψ r L rI ,δ Φ I = − L Ii e − ϕ ¯ ǫσ i ψ − iL Ir e − ϕ ¯ ǫψ r . (4.13)12he fields appearing here can be written in terms of N = (1 ,
0) multiplets in D = 6. Theseconsist of the supergravity multiplet ( e µµ , ψ µ , B + µν ), a single tensor multiplet ( B − µν , χ, σ ), vectormultiplets ( A r ′ µ , λ r ′ ) and hypermultiplets ( L rI , L iI , Φ I , ϕ, ψ, ψ r ). By making suitable redefini-tions, it is possible to demonstrate that the scalars of the hypermultiplets form the enlargedcoset SO ( p + 1 , SO ( p + 1) × SO (4) (4.14)which is a quaternionic K¨ahler manifold [17]. However we will not make these redefinitionshere.These redefined fields and transformations represent the induced supergravity which ispresent on the boundary and it is to this supergravity that we will couple boundary-localisedmatter in the following sections. When the boundaries are populated by this localised mat-ter, the transformations (4.13) will be modified corresponding to non-zero odd × odd termsappearing in the variation of these even-parity fields. However these transformations will beof higher order in the boundary couplings and so will be ignored in this paper. We will now consider turning on a boundary action describing vector multiplets( C Xµ , η XA ) , (5.1)where η is an Sp (1) pseudo-Majorana spinor with a doublet index A as before and X labelsthe adjoint representation of some gauge group K ′ . The supersymmetry transformations ofthese boundary fields must be given by their known flat-space forms modified by appropriatebulk dressings. We therefore make the ansatz, δC Xµ = ie − aσ ¯ ǫγ µ η X ,δη X = − e aσ γ µν H Xµν ǫ , (5.2)where H Xµν = ∂ µ C Xν − ∂ ν C Xµ + f X Y Z C Yµ C Zν and a is a constant which is to be determined.From our analysis in Section 4, we recognise that the scalar ϕ forms part of the D = 6quaternionic K¨ahler coset, and as such it is does not arise in the above transformation rules.An immediate consequence of having introduced a boundary action is the modification ofthe boundary condition (3.11) such that K µν − g µν K will now be proportional to the stresstensor of the boundary action. This condition is known as the Israel junction condition [19].On the other hand, since the supersymmetry transformation of the odd-parity gravitino ψ µ − contains the extrinsic curvature K µν , it follows that we must modify its boundary conditiontoo. Supersymmetry will then require that we modify other boundary conditions as well.13o determine these modifications, we begin by recording the supersymmetry transformationrules of the parity-odd fields δψ µ − = − K µν γ ν ǫ − e − σ F ρσλτ ( γ µ γ ρσλτ + 5 γ ρσλτ γ µ ) ǫ ,δχ + = − e αφ ∂ ¯7 ˆ σǫ − e − σ F µνρσ γ µνρσ ǫ , (5.3)where we have used the bulk supersymmetry transformations (2.9) and have made the follow-ing redefinitions K µν = e αφ ˆ K µν , K = K µν g µν ,ψ µ − = 1 √ e αφ ˆ ψ µ − , χ + = 1 √ e αφ ˆ χ + , (5.4) F µνρσ = 1 √ F µνρσ . We have also used the identity P − (cid:16) ˆ D µ ˆ ǫ (cid:17) = − ˆ D µ ( P − ) ˆ ǫ = − √ K µν γ ν e − αφ ǫ . (5.5)Examining these transformations, it follows that we need also to specify the modified boundaryconditions for F µνρσ , ∂ ˆ σ and χ + in a manner consistent with (5.3). Carrying out this processyields the modified boundary conditions ψ µ − (cid:12)(cid:12) ∂M = − be ( c + a ) σ H Xµν γ ν η X + 340 be ( c + a ) σ H ρσX γ µρσ η X + (fermi) ,χ + (cid:12)(cid:12) ∂M = 120 be ( c + a ) σ H Xµν γ µν η X + (fermi) ,e αφ ∂ ˆ σ (cid:12)(cid:12) ∂M = − be ( c + a ) σ H Xµν H µνX + (fermi) , (5.6) F µνρσ (cid:12)(cid:12) ∂M = 32 be (1+ c + a ) σ H X [ µν H ρσ ] X + (fermi) ,K µν (cid:12)(cid:12) ∂M = 12 be ( c + a ) σ H Xµρ H ρν X − be ( c + a ) σ H Xρσ H ρσX g µν + (fermi) , where b and c are further constants, which will be determined in the next section by consideringthe cancellation of certain terms in the supersymmetry variation. Furthermore, the bulkBianchi identity ∂ [ µ ˆ F νρστ ] = 0 implies that 1 + a + c = 0. The boundary conditions on allother parity-odd bulk fields vanish at lowest order in fermions.We can rephrase the boundary condition on F µνρσ in terms of a condition on A µνρ . How-ever, in order to do this we must first modify the bulk supersymmetry transformation ofˆ A MNR to δ ˆ A MNR = 3 i √ e ˆ σ ˆ¯ ǫ ˆ γ [ MN ˆ ψ R ] − i √ e ˆ σ ˆ¯ ǫ ˆ γ MNR ˆ χ + ∂ [ M ˆ f NR ] . (5.7) For clarity, these have been truncated to include only parity-odd fields that receive nontrivial boundaryconditions in the following analysis. f NR is an arbitrary function, linear in ˆ ǫ . This does not effect the bulk supersymmetryas ˆ A MNR always appears through ˆ F MNRS or multiplies a total derivative in (2.6). Making anansatz for the boundary condition on A µνρ and then enforcing that its variations under (5.2)and (5.7) match, we find that A µνρ (cid:12)(cid:12) ∂M ≡ √ A µνρ (cid:12)(cid:12) ∂M = 34 bω µνρ ( C ) + i be − aσ ¯ η X γ µνρ η X , (5.8)and f µν (cid:12)(cid:12) ∂M ≡ √ f µν (cid:12)(cid:12) ∂M = 32 bδ ǫ C Xµ C νX . (5.9)Consistency with the boundary Yang-Mills gauge transformations then requires that we im-pose the following boundary condition on the tensor gauge transformation parameter λ µν (cid:12)(cid:12) ∂M ≡ √ λ µν (cid:12)(cid:12) ∂M = 12 b∂ [ µ C Xν ] Λ X . (5.10)As we shall see later, the boundary conditions (5.9) and (5.10) will play a crucial role inthe identifications of the supersymmetry and gauge anomalies, respectively. Note also thatin determining (5.9), we needed to include the term bilinear in fermions. While we did notneed to specify the bilinear fermion terms in (5.6) to the order to which we are working indetermining the boundary action, there is a need to do so in the case of A Iµ in studying thecoincident-boundary limit of the bulk-plus-boundary system, as we shall see in Appendix A.In that case, the appropriate boundary condition can be seen to be A Iµ (cid:12)(cid:12) ∂M = − κ λ e − ϕ ¯ η X σ i γ µ η X L iI + (fermi) . (5.11)Next, we shall construct the boundary Yang-Mills action, and we shall see that certain can-cellations between the boundary action and the surface terms will fix the coefficients a, b, c ,which are already subject to the condition a + c + 1 = 0, as we have seen above. The general variation of the bulk action supplemented by the Gibbons-Hawking-York termsdefined in (3.7) and (3.12) is given by δS SG + δS GHY + δS GHY = Z M d x ˆ eδ L (7) (6.1)+ 1 κ Z ∂M d x q − ˆ h (cid:26) − (cid:16) ˆ K MN − ˆ K ˆ h MN (cid:17) δ ˆ g MN − i ˆ¯ ψ µ − ˆ γ µν δ ˆ ψ ν + − i ˆ¯ χ + δ ˆ χ − + 54 ∂ ˆ σδ ˆ σ − g e ˆ σ δ ˆ A Iµ L iI ˆ F µ i − e − σ ˆ F µνρ δ ˆ A µνρ − √ g ˆ ε µνρσλτ ˆ A µνρ ˆ F r ′ σλ δA τr ′ (cid:27) , δ L (7) contains no derivatives of the variations. However, in consideringthe variation of the bulk action under supersymmetry, which we shall do next, there will beextra surface terms due to the fact that further integrations by parts will be needed in order toleave the supersymmetry parameter undifferentiated. These are due to derivatives of ǫ presentin the variation of the gravitino and the 3-from. Collecting the resulting surface terms, wefind Z M d xδ ǫ L (7) = 1 κ Z ∂M d x q − ˆ h ˆ n M (cid:26) − i ˆ¯ ǫ ˆ γ MNR ˆ D N ˆ ψ R − i
52 ˆ¯ χγ M γ N ˆ ǫ ˆ ∂ N ˆ σ + i √ e − ˆ σ ˆ F RST U (cid:18) ǫ ˆ γ [ M ˆ γ RST U ˆ γ N ] ˆ ψ N + 8ˆ¯ ǫ ˆ γ RST U ˆ γ M ˆ χ (cid:19) + 18 g e ˆ σ ˆ F RSi (cid:16) ǫσ i ˆ γ [ M ˆ γ RS ˆ γ T ] ˆ ψ T − ǫσ i ˆ γ RS ˆ γ M ˆ χ (cid:17) (6.2)+ 16 e − σ ∂ N ˆ f RS ˆ F MNRS − √ g ˆ ε RST UV W M ˆ f RS ˆ F r ′ T U F r ′ V W (cid:27) . Substituting this into (6.1) and imposing the boundary conditions gives, after some algebra, δ ǫ S SG + δ ǫ S GHY = 1 κ Z ∂M d xeb (cid:26) − e − (1+ a ) σ ¯ ǫγ ρσ γ µ σ i η X H ρσX P iµ (6.3)+ i e − (2+ a ) σ ¯ ǫγ ρστ γ µν η X H µνX G ρστ − i e − (2+ a ) σ ¯ ǫγ λ γ ρστ γ µν γ λ η X H µνX G ρστ + i e − σ ¯ ǫγ µνρστ ψ τ H Xµν H ρσX + i e − σ ¯ ǫγ µνρσ χH Xµν H ρσX − g ε µνρσλτ ω µνρ ( C ) F r ′ σλ δ ǫ A τr ′ + 116 g ε µνρσλτ δ ǫ C Xµ C νX F r ′ ρσ F r ′ λτ (cid:27) , where S GHY = S GHY + S GHY as defined in (3.7) and (3.12). Next we construct the boundaryaction such that, together with the bulk action and subject to the modified boundary condi-tions (5.6), the total action is invariant under supersymmetry except for the last two termsin (6.3), which will be interpreted as supersymmetry anomalies and will be discussed in moredetail below.After some algebra we find that the boundary action is given by S Y M = 1 λ Z ∂M d xe (cid:26) − e − σ H Xµν H µνX − i η X γ µ D µ η X − i e − σ H Xρσ ¯ η X γ µ γ ρσ ψ µ − i e − σ H Xµν ¯ η X γ µν χ (cid:27) , (6.4)where we have determined that a = − b = κ λ , (6.5)as required to ensure certain cancellations between the variations of the boundary action andthe surface term. 16ne might have expected a term of the form G µνρ ¯ η X γ µνρ η X to appear in the boundaryaction, as such a term is present in the D = 6 actions of [1, 2] and was claimed to be presentin [14]. However the Noether procedure does not require such a term and thus it is absentin the boundary action that we have derived. In Appendix A we will demonstrate thatthis term emerges in the coincident-boundaries limit by considering the boundary condition A µνρ (cid:12)(cid:12) ∂M ∼ ¯ h X γ µνρ η X . In this limit the 4-form kinetic term F MNRS F MNRS will then giverise to the required term in the reduced action (A.8). A similar process is also described in[18].With the parameters a, b fixed as in (6.5) the completely determined boundary conditionstake the form: ψ µ − (cid:12)(cid:12) ∂M = − κ λ e − σ H Xµν γ ν η X + 3 κ λ e − σ H ρσX γ µρσ η X + (fermi) ,χ + (cid:12)(cid:12) ∂M = κ λ e − σ H Xµν γ µν η X + (fermi) ,e αφ ∂ ˆ σ (cid:12)(cid:12) ∂M = − κ λ e − σ H Xµν H µνX + (fermi) ,A µνρ (cid:12)(cid:12) ∂M = 3 κ λ ω µνρ ( C ) + iκ λ e σ ¯ η X γ µνρ η X + (fermi) ,A Iµ (cid:12)(cid:12) ∂M = − κ λ e − ϕ ¯ η X σ i γ µ η X L iI + (fermi) ,K µν (cid:12)(cid:12) ∂M = κ λ e − σ H Xµρ H ρν X − κ λ e − σ H Xρσ H ρσX g µν + (fermi) . (6.6)The boundary conditions on all other parity-odd fields in (3.2)are set to zero at lowest orderin fermions. The vanishing boundary conditions on L iI ′ , L rI ′ and L r ′ I imply that the parity-oddC-functions C , C ir , C irs and C ir ′ s ′ are also set to zero on the boundary. We also note that inRef. [14], only the boundary condition on A µνρ was considered, while our boundary conditionscorrespond to the completion of this to a full orbit.At this point, it is important to check that these boundary conditions are also consistentwith the variational principle following from the bulk + boundary action S = S SG + S GHY + S Y M . For example, the variation of the gravitino gives the boundary contribution Z ∂M d xe (cid:26) − iκ ¯ ψ µ − γ µν − i λ e − σ H Xρσ ¯ η X γ ν γ ρσ (cid:27) δψ ν , (6.7)which is set to zero by imposing the boundary condition on ψ µ − given above. Similarly, wehave checked that the surface terms that arise in the variations of all the other fields cancelupon use of the stated boundary conditions and boundary field equations.Next, we turn to the nonvanishing last two terms in (6.3), which we now identify asthe residual supersymmetry anomaly. We note that there is also an anomaly in the bound-ary Yang-Mills transformation, and, together with the supersymmetry anomalies, they musttogether satisfy the Wess-Zumino consistency conditions. To see this in more detail, it isconvenient to add the local counterterm S ′ Y M = 132 λ g Z ∂M d xeε µνρσλτ ω µνρ ( C ) ω σλτ ( A ) . (6.8)17his also produces a gauge anomaly in the bulk Yang-Mills gauge transformations and putsthe total gauge anomaly into a symmetric form known as the consistent anomaly [20]. Thenthe total variation of the action S ′ = S SG + S GHY + S ′ Y M under the Yang-Mills gauge trans-formations is given by δ Λ S ′ = 132 λ g Z ∂M d xe (cid:26) ε µνρσλτ H Xµν H ρσX ∂ λ A r ′ τ Λ r ′ + ε µνρσλτ F r ′ µν F r ′ ρσ ∂ λ C Xτ Λ X (cid:27) (6.9)and the last two terms in (6.3) together with the supersymmetry variation of (6.8) yield thecorresponding supersymmetry anomaly δ ǫ S ′ = 132 λ g Z ∂M d xe (cid:26) ε µνρσλτ H Xµν H ρσX δ ǫ A r ′ λ A r ′ τ − ε µνρσλτ ω µνρ ( C ) F r ′ σλ δ ǫ A r ′ τ + ε µνρσλτ F r ′ µν F r ′ ρσ δ ǫ C Xλ C τX − ε µνρσλτ ω µνρ ( A ) H Xσλ δ ǫ C τX (cid:27) . (6.10)Finally, one may verify that these two anomalies indeed do satisfy the complete set of Wess-Zumino consistency conditions δ Λ δ Λ S ′ − δ Λ δ Λ S ′ = δ [Λ , Λ ] S ′ , (6.11) δ ǫ δ Λ S ′ − δ Λ δ ǫ S ′ = 0 , (6.12) δ ǫ δ ǫ S ′ − δ ǫ δ ǫ S ′ = δ ˜Λ S ′ , (6.13)where ˜Λ is the gauge transformation produced by the commutator of two supersymmetrytransformations in the standard way. Next, let us consider the coupling of boundary-localised hypermultiplets. We will carry outthis coupling assuming no boundary-localised vector multiplets are present. These could bereintroduced later in order to gauge the hypermultiplet symmetries. The calculation will besimilar to that carried out for vector multiplets in the previous sections. First we will find asupersymmetric set of boundary conditions, then we will construct the surface term producedupon varying the bulk action, and finally we will construct a boundary-localised action whichvaries to cancel this surface term.We begin by considering m hypermultiplets consisting of 4 m real scalar fields φ α andsymplectic Majorana-Weyl spinors ζ a ( a = 1 , ..., m ). By global supersymmetry, it is knownthat the scalars must parametrize a hyperk¨ahler manifold M , which is characterised by havinga holonomy group H contained in Sp ( m ). The scalar target manifold M may or may not haveisometries. This will not play a role in our construction below. Let us denote the vielbeinson M by V aAα . By supersymmetry, they must be covariantly constant ∂ α V βaA − Γ γαβ V γaA + ω αab V βbA + ω αAB V βaB = 0 , (7.1)18here Γ γαβ is the Levi-Civita connection, ω abα is an H ⊆ Sp ( m ) valued connection and ω ABα is an Sp (1) R valued connection on M . These connections can be expressed in terms of thevielbein as usual. The holonomy condition means that the Sp (1) R curvature associated withthe connection ω αAB vanishes. The vielbeins must furthermore obey the relations [21] g αβ V αaA V βbB = ǫ ab ǫ AB , V αaA V βaB + α ↔ β = g αβ δ BA , (7.2)where ǫ ab and ǫ AB are Sp ( n ) and Sp (1) R invariant tensors. We use the conventions ζ a ǫ ab = ζ b , ǫ ab ζ b = ζ a , ǫ ab ǫ bc = − δ ac (7.3)for raising and lowering indices with ǫ ab and similar conventions for ǫ AB . It is useful to define P aAµ = ∂ µ φ α V aAα . (7.4)We can write the globally supersymmetric boundary action for the hyperscalars as S H = 1˜ λ Z d x (cid:20) − P aAµ P µaA − i ζ a γ µ D µ ζ a (cid:21) , (7.5)where D µ ζ a = ∇ µ ζ a + ∂ µ φ α ω abα ζ b , with ∇ µ containing the Lorentz spin connection, and we haveintroduced a coupling constant ˜ λ . This action is invariant under the global supersymmetrytransformations δφ α = i √ ǫ A ζ a V αaA ,δζ a = 1 √ γ µ ǫ A P aAµ . (7.6)We now consider the coupling of this boundary hypermultiplet action to our D = 7 bulksupergravity system. We begin the construction by modifying the field transformations as δφ α = i √ e − aϕ ¯ ǫ A ζ a V αaA ,δζ a = 1 √ e aϕ γ µ ǫ A P aAµ . (7.7)As before, we consider the boundary conditions that can be imposed on bulk fields such thatthese conditions form an orbit under supersymmetry. The bulk fermions on which we will at-tempt to impose non-zero boundary conditions transform under the projected supersymmetryas δψ Aµ − = − K µν γ ν ǫ A − i e ϕ F iρσ σ iAB (3 γ µ γ ρσ − γ ρσ γ µ ) ǫ B ,δχ A + = − e αφ ∂ ˆ σǫ A − i e ϕ F iµν σ iAB γ µν ǫ B . (7.8) As in (5.3), we have simplified the discussion by including only parity-odd fields which receive non-zeroboundary conditions in these transformations. ψ Aµ − (cid:12)(cid:12) ∂M = 910 √ be ( c − a ) ϕ ζ a P aAµ − √ be ( c − a ) ϕ γ µν ζ a P νaA + (fermi) ,χ + (cid:12)(cid:12) ∂M = 110 √ be ( c − a ) ϕγ µ ζ a P aAµ + (fermi) ,e αφ ∂ ˆ σ (cid:12)(cid:12) ∂M = 110 be cϕ P aAµ P µaA + (fermi) ,F iµν σ iAB (cid:12)(cid:12) ∂M = ibe ( c − ϕ P aA [ µ P Bν ] a + (fermi) ,K µν (cid:12)(cid:12) ∂M = 12 be cϕ P aAµ P νaA − be cϕ P aAρ P ρaA g µν + (fermi) , (7.9)where a , b and c are constants to be determined, and, as before, all other parity-odd fieldsin (3.2) are set to zero at lowest order in fermions. Calculating the surface term producedupon variation of the bulk action under (2.9) and then imposing these boundary conditions,we find the total non-invariance of the bulk supergravity action: δS SG + δS GHY + δS GHY = bκ Z ∂M d xe (cid:26) i √ e ( c − a ) ϕ − σ ¯ ǫ A γ µ γ ρστ γ ν γ µ ζ a P νaA G ρστ − i e cϕ ¯ ǫ A γ µνρ ψ ρB P µaA P aBν − i e cϕ ¯ ǫ A γ µν ψ B P µaA P aBν − √ e ( c − a ) ϕ ¯ ǫ A σ iAB ζ a P iµ P µaB (cid:27) . (7.10)Then, by the Noether procedure, we find the following boundary action S H = 1˜ λ Z ∂M d xe (cid:26) − e aϕ P aAµ P µaA − i ζ a γ µ D µ ζ a − i √ e aϕ ¯ ζ a γ µ γ ν ψ Aµ P νaA + i √ ae aϕ ¯ ζ a γ µ ψ A P µaA (cid:27) . (7.11)Here we have set c = 2 a which is required for invariance. With this condition, the actionvaries to give δS H = 1˜ λ Z ∂M d xe (cid:26) i √ e aϕ ¯ ǫ A γ ν γ µ ( D µ − D µ ) ( ζ a P νaA ) − i √ e aϕ − σ ¯ ǫ A γ µ γ ρστ γ ν γ µ ζ a P νaA G ρστ + i e aϕ ¯ ǫ A γ µνρ ψ ρB P µaA P aBν + iae aϕ ǫ A γ µν ψ B P µaA P aBν + 12 √ e aϕ ¯ ǫ A σ iAB (2 aγ µ γ ν + γ ν γ µ ) ζ a P iµ P νaB (cid:27) . (7.12)The D µ ( ζP ) term, with D µ defined in (4.12) and (4.9), arises from the variation of the ζψ µ P term. Furthermore, the D µ ( ζP ) term, with the covariant derivative defined with respect tothe pull-backed connection ∂ µ φ α ω αAB , comes from the variation of the P term in (7.11). The P G, P P and P P terms cancel the bulk surface term (7.10), as long as b = κ ˜ λ and a = , while20he term proportional to ( D µ − D µ )( ζP ) vanishes as long as the boundary Sp (1) R connectionis set equal that for the bulk at the boundary location, i.e. Q ABµ (cid:12)(cid:12) ∂M = ∂ µ φ α ω ABα , (7.13)where Q ABµ = i ǫ ijk Q jkµ σ ABi and Q jkµ is defined in (4.9).Owing to the order in fermions to which we have been working, this equation is valid only topurely bosonic order. We also note that the coupling of these boundary hypermultiplets doesnot produce any classical non-invariances such as those which arose for the vector multiplets.Substituting (7.13) into the field strength for Q ABµ and then using the boundary conditions C | ∂M = C ir | ∂M = 0, we find P aA [ µ P Bν ] a = − i ǫ ijk (cid:18) P ir [ µ P jrν ] + 12 √ ǫ ijl C lr ′ F r ′ µν (cid:19) σ kAB (cid:12)(cid:12) ∂M . (7.14)This implies that the Sp (1) R curvature of the boundary hypermultiplets is identified with the Sp (1) R curvature of the bulk scalars. The fact that this is nonzero is consistent with the factthat the full manifold parametrised by the 4 p + 4 scalars from the bulk and the 4 m scalarsfrom the boundary hypermultiplets parametrise a QKM in the limit of coincident boundaries.As before, we note that a term of the form ¯ ζ a γ µνρ ζ a G µνρ is not present in the boundaryaction, although it is present in the 6D hypermultiplet coupled action as given in Refs [1, 2]and in Ref. [14]. At the purely bosonic order, as required for the coupling process considered inthis section, the boundary condition simply sets A µνρ equal to zero on the boundary. However,at higher order in fermions the boundary condition will be of the form A µνρ | ∂M ∼ ¯ ζ a γ µνρ ζ a .This will then give rise to the required term in the coincident boundaries limit in an analogousway to that described in Section 6 and Appendix A.The scalar kinetic term in the boundary action (7.11) is multiplied by an unusual factor e ϕ , which also results in the unusual Noether coupling term e ϕ ¯ ζ a γ µ ψ A P µaA . This can beunderstood by bearing in mind that the hyperscalar ϕ as well as the newly-coupled boundaryscalars must together form a QKM in the limit of coincident boundaries.Note that the gauged U (1) R lies in the SO ( n,
3) isometry group of the bulk sigma model.Furthermore, the boundary hyperk¨ahler manifold M does not necessarily have any isometries.Consequently, the gauge field A r ′ µ does not arise in the definition of the covariant derivativegiven in (7.4). However, the local U (1) R symmetry is nonetheless realised as a result of thethe boundary condition (7.13). This condition is crucial for the quaternionic K¨ahler structureon the overall scalar manifold, N , which arises under local supersymmetry, as expected.The manifold N is a single irreducible QKM of dimension 4 m + 4 p + 4, with coordinates( φ α , φ ir ′ , Φ I , ϕ ), whose holonomy group is contained in Sp ( m + p + 1) × Sp (1). In the absenceof the m boundary hypermultiplets, and in the coincident boundaries limit, it is known that N can be described as the quaternionic K¨ahler coset SO ( p + 1 , /SO ( p + 1) × SO (4) [17].In the presence of m boundary hypermultiplets, however, the structure of the overall scalar An analogous condition has been found in [14] with all the bulk scalars set to zero. N arising in the coincident boundaries limit depends on the specific properties of M . It would be interesting to determine, for example, the conditions on M under which N becomes a symmetric or homogeneous QKM. In order to cancel the complete set of anomalies, it is necessary to consider various modifica-tions to the model described so far. One such modification is the addition of a bulk topologicalmass term for the 3-form potential [22, 17]. Another is the inclusion of further bulk Chern-Simons terms together with further modifications to the boundary conditions, while a thirdis the coupling of boundary-localised tensor multiplets. We will consider all three of theseextensions in the following section.
A topological mass term can be added to the bulk action described in Section 2, therebyarriving at a one-parameter extension. However, a mass term of the form hA ∧ F with aconstant mass parameter h violates the Z symmetry of the boundary. In order to respectthis Z symmetry, we need to allow the mass parameter h to undergo a jump at the boundarylocation when viewed from an upstairs perspective. To accomplish this, we dualise h to a6-form potential A such that the field equation for h , now treated as a scalar field, equates h to the dual of the A field strength, while the field equation for A implies that h is at leastpiecewise constant. In this formulation, we can now assign odd parity to h so as to renderthe term hA ∧ F parity-even. The resulting new terms in the bulk action are S h = 1 κ Z M d x ˆ e (cid:26) − ih e σ + h ˆ ε MNRST UV ˆ G MNRST UV (cid:27) (8.1)where ˆ G MNRST UV = 7 ∂ [ M ˆ A NRST UV ] + 136 ˆ F [ MNRS ˆ A T UV ] − √ ε MNRST UV e ˆ σ C − i e σ ˆ¯ ψ [ M ˆ γ NRST U ˆ ψ V ] + 8 i e σ ˆ¯ ψ [ M ˆ γ NRST UV ] ˆ χ + 27 i e σ ˆ¯ χ ˆ γ MNRST UV ˆ χ − i e σ ˆ¯ λ ˆ r ˆ γ MNRST UV ˆ λ (8.2)and the new terms in the supersymmetry transformation rules are δ ˆ ψ M = − he σ ˆ γ M ˆ ǫ ,δ ˆ χ = − he σ ˆ ǫ ,δ ˆ A MNRST U = − δ ˆ A [ MNR ˆ A ST U ] + 24 i e σ ˆ¯ ǫ ˆ γ [ MNRST ˆ ψ U ] − i e σ ˆ¯ ǫ ˆ γ MNRST U ˆ χ ,δh = 0 . (8.3)22he 6-form potential ˆ A µνρσλτ is parity even and ˆ A µνρσλ is parity odd. The action is nowinvariant under a modified tensor gauge transformation under which A must transform as δ ˆ A MNRST U = −
121 ˆ A [ MNR ∂ S ˆ λ T U ] . (8.4)In the presence of the boundaries, the supersymmetry of the bulk-plus-boundary actionis unaffected by this construction and the variational principle remains consistent, providedthat we impose the boundary condition h (cid:12)(cid:12) ∂M = 0 , A µ ...µ (cid:12)(cid:12) ∂M = 0 . (8.5)However, we may also consider the boundary value of h to be a constant h (cid:12)(cid:12) ∂M = h . (8.6)This will lead to the introduction of a new boundary term and modified boundary conditionsthat will produce further classical anomalies in the boundary Yang-Mills gauge symmetry.We now seek an orbit of boundary conditions which contains (8.6). As we are interestedin the effects of the topological mass term on classical anomalies, we consider boundaryconditions involving boundary vector multiplets as well as the constant h . However, becausethe hypermultiplets do not effect the classical non-invariances, we will not further considertheir simultaneous coupling here. Carrying out this process, we find an orbit of boundaryconditions given by (6.6) with the following modifications (up to quartic fermion terms): e αφ ∂ ˆ σ (cid:12)(cid:12) ∂M = − κ λ e − σ H Xµν H µνX − γ ) e σ +2 ϕ h ,K µν (cid:12)(cid:12) ∂M = κ λ e − σ H Xµρ H ρν X − κ λ e − σ H Xρσ H ρσX g µν + γe σ +2 ϕ h g µν ,C (cid:12)(cid:12) ∂M = − √ (cid:18)
45 + γ (cid:19) h g e ϕ +2 σ , (8.7)where γ is a parameter shortly to be determined. To find the total supersymmetric action upto a supersymmetry anomaly, we need to give the total boundary action S tot .B = Z ∂M d xe (cid:26) − λ e − σ H Xµν H µνX − i λ ¯ η X γ µ D µ η X − i λ e − σ H Xρσ ¯ η X γ µ γ ρσ ψ µ − i λ e − σ H Xµν ¯ η X γ µν χ + 132 λ g ε µνρσλτ ω µνρ ( C ) ω σλτ ( A )+ 4 h κ e σ +2 ϕ + 7 h κ ε µνρσλτ A µνρσλτ + ih κ λ e σ ω µνρ ( C )¯ η X γ µνρ η X (cid:27) . (8.8)23equiring supersymmetry up to a Wess-Zumino consistent anomaly determines the value of γ : γ = − . (8.9)It is interesting that this implies the boundary condition C (cid:12)(cid:12) ∂M = 0. One can further checkthat the above boundary conditions are consistent with the variational principle. The variationof the action (8.8) under tensor gauge transformations subject to the boundary conditions(6.6) gives the additional gauge anomaly contribution δ Λ S tot B = − h κ λ Z ∂M d x eε µνρσλτ H Xµν H ρσX ∂ λ C Yτ Λ Y . (8.10)Correspondingly, there is an additional contribution to the supersymmetry anomaly givenby − h κ λ Z ∂M d x eε µνρσλτ (cid:26) H Xµν H ρσX δ ǫ C Yλ C τY − ω µνρ ( C ) H Xσλ δ ǫ C τX (cid:27) . (8.11)As before, one may check that the inclusion of these anomalies continues to give a Wess-Zumino consistent system. Before evaluating the gauge/Lorentz anomalies that result from the variation of the bulk plusboundary action subject to the chosen boundary conditions, we need to discuss possible addi-tional extensions of the bulk model. Terms of types that may produce anomalous variationsare of the forms A ∧ tr R ∧ R, ω L , ω ( A ) , ω ( A ) ∧ tr R ∧ R where ω L and ω ( A ) are theLorentz and Yang-Mills Chern-Simons forms, respectively. The ω L and A (3) tr ∧ R ∧ R terms are known to arise in the K D = 11 supergravity supplementedwith the Duff-Minasian term A (3) tr R ∧ R ∧ R ∧ R . These have been used in a Hoˇrava-Wittenformulation of ungauged pure D = 7 supergravity [14]. However, in the non-compact D = 7model we are considering here, derivation from higher dimensions involves a noncompact in-ternal space of infinite volume. Indeed, as we saw in the Introduction, a 3-manifold of thiskind, known as H (2 , N = 1, D = 10 supergravity to the SO (2 ,
2) gauged supergravity in D = 7 [12], yielding a consistent Kaluza-Klein truncation.The same model can also be obtained from D = 11 supergravity by reducing on H (2 , × S ,again yielding a consistent Kaluza-Klein truncation. However, in the presence of the term D = 11 A (3) tr R ∧ R ∧ R ∧ R and even in the presence of the Yang-Mills sector in D = 10,a consistent Kaluza-Klein ansatz is not at present known. A preliminary investigation ofthe infinite volume problem suggests that the appropriate Weyl rescaling of fields needed to While a term of the type ω ( A ) does arise in the SO (5) gauged maximal D = 7 supergravity, it doesnot appear in any gauged half-maximal D = 7 supergravity. The half-maximal truncation of the maximaltheory studied in [12] might seem to indicate the presence of ω ( A ) but, in fact, such a term is not allowed bysupersymmetry in this system. We would like to acknowledge detailed discussions with Chris Pope on this point. D = 7 leads to vanishing coefficients in front of the ω ( A ) termand we expect this to be the case for the ω ( A )tr R ∧ R term as well. With this in mind,we shall not consider further the inclusion of higher-derivative terms in the bulk Lagrangianas given in Section 2, but supplemented by the topological mass term added in Section 8.1.However, we shall consider modifications of the boundary condition on A (3) occasioned by theinclusion of Chern-Simons terms for the bulk gauge fields and Lorentz connection such that A extra(3) (cid:12)(cid:12) ∂M = c A ω ( A ) + c L ω L + (fermi) , (8.12)where c A and c L are arbitrary constant coefficients. Extending the full set of supersymmet-ric boundary conditions (6.6) to incorporate this modification will, in particular, alter theboundary condition on the extrinsic curvature K µν which will now must include terms takingthe form K µν (cid:12)(cid:12) ∂M ∼ e − σ F µρr ′ F r ′ νρ + e − σ R µρ µν R νρ µν + · · · . (8.13)Since K µν picks up contributions for the boundary stress tensor, it follows that modificationsproportional to this, in turn, imply that the full boundary action must contain terms givenby S ext .B ∼ Z ∂M d xe (cid:26) e − σ F r ′ µν F µνr ′ + e − σ R µν µν R µν µν + · · · (cid:27) . (8.14)An R term of this type has been encountered in the Hoˇrava-Witten formulation of D = 11supergravity compactified on S / Z [23]. We note that the dilaton factors in (8.14) areequivalent to the dilaton factor multiplying the kinetic term in (6.4). In standard D = 6calculations, higher-derivative invariants with either e σ or e − σ factors multiplying the R termare possible [24, 25]. Supersymmetrizing the e σ variant would imply the presence of a term ofthe form B ∧ R ∧ R , whilst supersymmetrizing the e − σ variant implies that the 3-form fieldstrength appearing in the action is Chern-Simons modified such that G = dB + ω L . Sincethe boundary condition (8.12) implies that the field strength becomes Chern-Simons modifiedin the coincident boundaries limit (see Appendix A) we deduce that the necessary factor heremust be e − σ multiplying the R term present in this boundary action. A similar argumentalso applies to vector couplings, which is consistent with the fact that Noether coupling forcedus to determine the coefficient a = − − √ κ g Z M d x ˆ A (3) ∧ ˆ F ˆ r ∧ ˆ F ˆ r + 7! Z ∂M d xh A (6) . (8.15)Using the modified boundary conditions (8.12), the variations of these terms give the newtotal bosonic anomalyΩ = Z ∂M (cid:26) h κ (cid:16)(cid:0) c A − g h (cid:1) ω ( A ) + 2 c L ω L + κ λ ω ( C ) (cid:17) ∧ (cid:16)(cid:0) c A − g h (cid:1) tr F ∧ F + 2 c L R ∧ R + κ λ tr H ∧ H (cid:17) − κ g h ω ( A ) ∧ tr F ∧ F (cid:27) , (8.16)25here ω is defined by δω = dω . If we consider the gauge group for the boundary vectormultiplets K ′ to be the tensor product of simple groups K ⊗ . . . ⊗ K n g , we can define the4-forms G a , where a = 0 , . . . , n g + 1, as G = tr F ∧ F, G = tr R ∧ R, G = tr H (1)2 ∧ H (1)2 , . . . , G n g +1 = tr H ( n )2 ∧ H ( n )2 , (8.17)where dω ( A ) = tr F ∧ F = F r ′ ∧ F r ′ and dω ( C ) = tr H ∧ H = H X ∧ H X . Then theanomaly (8.16) is related to the following 8-form polynomialΩ clas = 8 h κ (cid:20) ( 13 c A G + 13 c L G + n g +1 X a =2 κ λ a ) G a (cid:21) ∧ (8.18) (cid:20) (cid:18) c A − h g (cid:19) G + 13 c L G + n g +1 X a =2 κ λ a ) G a (cid:21) , by the descent equations ω clas = d Ω and δ Ω = d Ω . The classical non-invariance produced so far obeys the Wess-Zumino consistency conditionsand produces terms of the correct forms to cancel the quantum anomalies. However theclassical anomaly produced is still not sufficiently general to completely cancel the anomaliesproduced by quantum effects and so to yield an overall invariant system. We therefore considera further extension of the model by adding n T boundary-localised tensor multiplets to theaction. These multiplets have the form ( B xµν , χ Ax − , φ x ), where x = 2 , . . . , n T + 1, which playa crucial role in the implementation of a generalized Green-Schwarz anomaly cancellationmechanism introduced in [26].Tensor multiplets of this form are known to exist in rigid D = 6 supersymmetry andaccordingly a coupling process similar to that shown in Sections 5 and 6 will be possible.However this process is complicated by the fact that the 3-form field strength H x = dB x isrequired, by closure of the supersymmetry algebra, to be self-dual: H = ⋆H . This has theconsequence that the na¨ıve kinetic term that one would write for B x vanishes. This problemmay be addressed by use of a non-manifestly Lorentz invariant action [27], or by reformulatingthe problem at the equation-of-motion level. We shall not attempt here a full analysis of thesecouplings. Although a full coupling would be necessary for detailed analysis of the classicalsupersymmetry anomalies, it is not necessary for analysis of the purely bosonic anomalies.This is due to the fact that bosonic anomaly contributions arising from boundary tensors canonly be generated by the variation of one type of term in the boundary action. This crucialanomaly-generating term type is analogous to the bulk Chern-Simons term g A ∧ F ˆ r ∧ F ˆ r ,and is of the same form as the standard anomaly counterterm that is seen in purely D = 6theories [20]. In our boundary action, it appears as Z ∂M v xa B x ∧ G a , (8.19) Note that we are using the Chern-Simons 3-form normalisation given in Equation (2.7), as in Reference[17], for both gauge and Lorentz symmetries. This gives rise to the factors of in the descent relations. v xa is a numerical coupling matrix analogous to the g which appears in the in the bulkaction, and where summation over the index x = 2 , ..., n T + 1 is understood. If B x is requiredto transform under the bosonic symmetries of the theory according to δB x = v ′ xa ω a , (8.20)then the variation of (8.19) will produce a non-invariance of the form Z ∂M v xa v ′ xb ω a G b . (8.21)Adding this to the classical anomaly generated so far, we can write the total anomaly asΩ tot = Z ∂M v Ia v ′ Jb η IJ ω a ∧ G b , (8.22)where the index x has been extended to a new index I = 0 , . . . , n T + 1. In general, the index a = 0 , ..., n g + 1. However, if n g < n T , then the matrix v Ia v ′ Jb η IJ has non-maximal rank,which turns out to put a severe restriction on the quantum anomaly polynomial [26, 14]. Thisrestriction is lifted for n T ≥ n g . For simplicity, we shall assume that n T = n g from here on.Then, we find that the vector v Ia is given by v a = v ′ a = (cid:16) c A √− h κ − g √− h , c L √− h κ , κ √− h λ ) , . . . , κ √− h λ n g +1 ) (cid:17) ,v a = v ′ a = (cid:16) κg √− h , , , . . . , (cid:17) , v Ia = v xa , v ′ Ia = v ′ xa , for I = 2 , . . . , n T + 1 , (8.23) η IJ = diag( − , + , . . . , +) and we have assumed h < We shall now construct an example of an anomaly-free model in the D = 7 /D = 6 Hoˇrava-Witten setting that we have been constructing in this paper. As we wish to end up withan R -symmetry gauged model, we need to start with a matter-coupled noncompact gauged D = 7 theory. The possible non-compact gauge groups and the surviving even-parity bulkfields have been listed in [17]. Here, we shall consider the SO (2 ,
1) gauged D = 7 modelwhich consists of minimal supergravity coupled to one vector multiplet. The bulk scalarsparametrize the coset SO (1 , /SO (3) and the SO (1 ,
2) subgroup of SO (1 ,
3) is gauged. Thestructure constants are given by [17]ˆ f ˆ I ˆ J ˆ K = ǫ ijk , i = 1 , , , (9.1)27here ǫ ijk are the SO (1 ,
2) structure constants. In (3.3), we now have p = 0 , n = 1, and theresulting even-parity fields form the multiplets( e µµ , ψ µ + , B − µν ) , ( B + µν , χ − , σ ) , ( ψ − , ϕ, Φ I ) , ( A µ , λ ) , (9.2)with supersymmetry transformations as given in (4.13). The vector field A µ gauges the R-symmetry group U (1) R . We have denoted the D = 6 chiralities of the fermions explicitlyfor convenience, and we have split the 2-form potential into parts that have self-dual andanti-self-dual field strengths.The chiral fermions ( ψ µ + , χ − , λ , ψ − ) give rise to gravitational, U (1) R and mixed gravitational- U (1) R anomalies on the boundaries. The anomalies are encoded in an 8-form polynomial madeup of the Riemann and Yang-Mills curvature forms, via the descent equations. The standardanomaly formulae giveΩ( ψ µ + ) = 524 F − F trR + 15760 (cid:20)
245 tr R − ×
434 (tr R ) (cid:21) , Ω( χ − ) = − F − F trR − (cid:20) tr R + 54 (tr R ) (cid:21) , Ω( λ ) = 124 F + 196 F trR + 15760 (cid:20) tr R + 54 (tr R ) (cid:21) , Ω( ψ − ) = − (cid:20) tr R + 54 (tr R ) (cid:21) , Ω( B µν + ) = 15760 (cid:2) −
28 tr R + 10 (tr R ) (cid:3) , (9.3)where F is the U (1) R field strength, and we have suppressed the wedge symbol, so that, forexample F tr R = F ∧ F ∧ tr R ∧ R .The total anomaly coming from the bulk fields on each boundary is half of the total bulkanomaly. Thus on a given boundary we haveΩ bulkgrav/U (1) R | ∂M = 548 F − F trR + 15760 (cid:20)
122 tr R −
552 (tr R ) (cid:21) . (9.4)Next, we need to compute the quantum anomalies that result from the introduction of n V gauge, n H hyper and n T tensor multiplets on a given boundary. It is useful first to computethe total gravitational anomaly. Summing up the bulk contributions given in (9.4) and thoseof the boundary multiplets, the total gravitational anomaly on ∂M is given byΩ t ot.grav. | ∂M = h ( n V − n H − n T + 122)tr R + ( n V − n H + 7 n T −
22) (tr R ) i . (9.5)28he tr R term must necessarily vanish for anomaly freedom. As we have assumed thatthere is no bulk Lorentz Chern-Simons term, the vanishing of the tr R anomaly imposes theconstraint n H − n V + 29 n T = 122 . (9.6)Using this condition in (9.5), and including the contributions to the U (1) R and mixed gravitational- U (1) R anomalies ( i.e. the F and F tr R terms in (9.4), together with similar contributionsfrom all the boundary matter multiplets that have been introduced), we findΩ t ot.grav/U (1) R | ∂M = ( n T − R ) + [2( n V − n T ) + 5] F + [2( n V − n T ) − F tr R . (9.7)At this point, we need to specify n V , n H and n T such that the condition (9.6) is satisfied,where the boundary Yang-Mills gauge group has total dimension n V , and such that the n H hyperfermions form a set of representations of this group. A complete analysis of all thepossibilities is beyond the scope of the present paper. Instead, we shall give one example toillustrate how anomaly freedom can be achieved in the bulk-plus-boundary system that wehave constructed. We shall take the gauge group on a given boundary to be K ′ = E × E , (9.8)so that n V = 78 + 133. Furthermore, we shall introduce two tensor multiplets, and fivehypermultiplets in fundamental representations of E and five fundamental representations of E . Thus, all in all, we have n T = 2 ,n V = 78 + 133 ,n H = 5 × (27 ,
1) + 5 × (1 , . (9.9)Using this data and employing the relationsTr H = 4tr H , Tr H = (tr H ) , tr H = (tr H ) , Tr H = 3tr H , Tr H = (tr H ) , tr H = (tr H ) , (9.10)where Tr(tr) denote the trace in the adjoint (fundamental) representation, we find that thetotal one-loop anomaly polynomial is encoded byΩ − loop = − (cid:0) tr R (cid:1) + 14116 F + 13364 F tr R + F (cid:18) tr H + 34 tr H (cid:19) −
196 tr R (cid:0) tr H + 2tr H (cid:1) + 1576 h (cid:0) tr H (cid:1) − (cid:0) tr H (cid:1) i . (9.11) In the standard N = 1, D = 6 anomaly cancellation, the equivalent relation is given by n H − n V + 29 n T =273. The difference here is due to two factors. Firstly, our n T counts the number of boundary-localised tensormultiplets whilst the n T in the standard equation counts the total number of tensor multiplets. As one tensormultiplet comes from the reduction of the bulk supergravity multiplet, our n T differs from the standard setupby 1. Secondly, the quantum anomaly in our case is split across two boundaries and so differs from the standardresult by a factor of 2. Therefore in our case we have a different gravitational-anomaly cancellation conditionfrom the standard condition: n H − n V + 29 n T = (273 − / n T = n g = 2. We begin by making the following redefinitions˜ λ = λ (cid:18) − h κ (cid:19) ˜ λ = λ (cid:18) − h κ (cid:19) ˜ g = g (cid:0) − h κ (cid:1) ˜ c A = c A (cid:18) − h κ (cid:19) ˜ c L = c L (cid:18) − h κ (cid:19) , (9.12)where all the new parameters are dimensionless. This allows us to rewrite the anomalypolynomial (8.19) asΩ clas = − (cid:18)
13 ˜ c A G + 13 ˜ c L G + 14(˜ λ ) G + 14(˜ λ ) G (cid:19) ∧ (cid:18)(cid:18)
13 ˜ c A + 18˜ g (cid:19) G + 13 ˜ c L G + 14(˜ λ ) G + 14(˜ λ ) G (cid:19) + v a v ′ b G a ∧ G b + v a v ′ b G a ∧ G b . (9.13)In order for the system to be anomaly free, (9.13) must cancel the quantum anomaly polyno-mial Ω − loop = 14116 ( G ) −
164 ( G ) + 13364 G G + G (cid:18) G + 34 G (cid:19) − G (cid:0) G + 2 G (cid:1) + 1576 (cid:2) G ) − ( G ) (cid:3) . (9.14)This requirement places 10 constraints on the 21 parameters in (9.13) which leaves an 11dimensional space of solutions. In order to demonstrate that a solution exists in which allparameters are real, we give an example solution ,˜ c A = 0 . g = 0 . c L = 0 . λ = 3 . λ = 4 . v = 0 . , v = − . v = 1 . v = − . v ′ = 0 . v ′ = − . v ′ = 0 . v ′ = − . v = − . v = − . v = 0 . v = 0 . v ′ = 8 . v ′ = 0 . v ′ = 0 . v ′ = 0 . , (9.15)where we have dropped the underlines in v Ia for notational simplicity. This demonstrates thatanomaly-free bulk-plus-boundary models can indeed be constructed as we have described. Finding solutions to a large number of simultaneous equations such as these is greatly simplified by findingthe Groebner basis for the equations. This is most easily done using the program Singular or the Mathematicapackage STRINGVACUA. We may view the construction in this paper as a worked example of an anomaly-free modelwith gauged R-symmetry and a positive cosmological potential. A variety of approaches hasbeen followed in the search for realistic reductions of string/M-theory to candidate effective D = 4 theories. The standard compactifications and brane constructions limit to effectivesupergravity theories which populate only a sub-class of the available models that one mightwant to explore, however. In particular, the class of non-compact gaugings of supergravity hasbeen rather under-exploited to date. Such models depart from models with compact gaugedR-symmetries, such as the original D = 4 gauged N = 8 supergravity [28]. The discoveryof models with gauged R-symmetries then led on to searches for models with gauged non-compact symmetry groups [29, 30]. These were in turn obtained by reduction from higherdimensions on non-compact manifolds [31].The physical interest of models with non-compact gaugings is illustrated by cosmologicalapproaches such as the SLED program of supersymmetry in large extra dimensions [9], whichtakes as a starting-point example the D = 6 Salam-Sezgin model [13]. But non-compactgaugings have not yet figured prominently in the search for realistic string or M-theory particlephysics vacua. One reason for this has been the lack of a perceived link to the “ur-theories”in D = 10 and D = 11. A path towards such links has now been opened up, however, bythe reduction in Ref. [12], involving precisely the sort of non-compact manifold reductionenvisaged in [31]. So, it seems that a relevant chapter in the encyclopedia of string/M-theoryreductions has only just been opened.In the present paper, we have focused primarily on a process for generating a chiral,anomaly-free model starting from a gauged R-symmetry In order to provide a richer andmore fully worked-out scheme for D = 6 models such as those needed for the SLED program,we began with a gauged R-symmetry model in D = 7. To generate a chiral theory in D = 6, weused a Hoˇrava-Witten construction based on a slice of D = 7 bulk spacetime bounded by two D = 6 spaces which can then be populated with D = 6 supermatter as needed to construct ananomaly-free model. Hoˇrava-Witten type constructions, generalising the original D = 11/ D =10 construction of the heterotic string from M-theory [10, 11], can also be seen as domain-wall brane-solution constructions such as the D = 5/ D = 4 “heterotic M-theory” construction[32, 33]. These naturally produce chiral theories in the lower even dimension. But this thenraises the issue of potential quantum anomalies in the reduced theory. The mechanism ofanomaly cancellation involves anomaly inflow from the bulk higher-dimensional space togetherwith a careful choice of “matter” fields to populate the boundary brane spaces. In the D =11/ D = 10 construction, this uniquely yields the original E gauge multiplet on each boundingbrane [10, 11, 18, 34, 23]. As one goes down in dimensionality, the anomaly-cancellationrequirements become less stringent, so that in a direct D = 5/ D = 4 analysis [35], the onlyanomalies requiring cancellation are gauge and mixed gravitational-gauge anomalies, witha wide resulting set of anomaly-free constructions. The present D = 7/ D = 6 constructionpresents an intermediate scenario, with a detailed set of cancellation requirements as presentedin Section 9. These do not uniquely specify the boundary gauge groups and fields, but theydo impose a stringent set of anomaly-cancellation conditions on them. In the present paper,31e have not attempted a comprehensive study of the solutions to these conditions, but it maybe hoped that such a study might reveal classes of phenomenologically interesting scenarios.The main challenges to be met in carrying out the D = 7/ D = 6 construction revolvedaround the details of coupling 8-supercharge boundary matter to the 16-supercharge bulktheory. One needs to take care to provide necessary Gibbons-Hawking-York terms so asto ensure consistency between the bulk-plus-boundary variational equations and the chosenboundary conditions for the bulk fields. The halving of the supersymmetry at a boundaryis a natural consequence of any Hoˇrava-Witten type orbifold construction. But one needsto take great care here in handling the supersymmetric couplings, since in the absence of afully off-shell formalism, the classical boundary non-gauge-invariances of the bulk theory, asneeded for anomaly inflow, engender also supersymmetry anomalies.The occurrence of supersymmetry anomalies in Hoˇrava-Witten type constructions is al-ready familiar from the work of Refs [18, 34], but what is different about the constructionsmade in the present paper is the order at which these occur. In [18, 34], an iterative construc-tion to suppress the anomalies was carried out in powers of the boundary coupling constantfor the original D = 11 /D = 10 heterotic construction. In that case, the D = 10 boundaryaction and the corresponding boundary conditions for D = 11 bulk fields occurred at firstorder in the boundary coupling λ S boundary ∼ λ Z ∂M ∗ F (2) ∧ F (2) C (3) (cid:12)(cid:12) ∂M ∼ κ λ ω (3) (10.1)The bosonic anomaly, however, comes from substituting the boundary condition for C (3) intothe variation of the Chern-Simons term, δS ∼ κ Z ∂M δC (3) ∧ C (3) ∧ F (4) ∼ κ (cid:18) k λ (cid:19) Z ∂M δω (3) ∧ ω (3) ∧ F (2) ∧ F (2) (10.2)which gives an anomaly at third order in λ . This means that supersymmetric Noethercoupling can be caried out to second order in λ [18] without interference from anomalycomplications, whose discussion can be postponed until later on at third order in λ [34]. Inthe construction of the present paper, however, the discussion of anomalies cannot similarlybe postponed. This is because the bosonic anomaly in this case comes from a variation δS ∼ κ Z ∂M δA (3) ∧ ω (3) ( A ) ∼ κ k λ Z ∂M δω (3) ( C ) ∧ ω (3) ( A ) (10.3)which occurs already at first order in λ , i.e. it is of the same order as the boundary actionthat we are constructing.Thus, the best that one can arrange for in the present bulk-plus-boundary coupling isagreement with the Wess-Zumino consistency conditions, as discussed in Section 6. Reduc-tion of the D = 7/ D = 6 construction to a purely D = 6 theory by taking a coincidentboundary limit, as explained in Appendix A, confirms the correctness of this construction byyielding precisely the D = 6 Wess-Zumino consistent system that was found in Ref. [20]. Itis interesting to note that the construction of supersymmetric bulk-plus-boundary systems,32imilar to those considered here, is greatly simplified by the use of the ‘susy without b.c.’ formalism considered in [36]. This formalism, as currently constructed, requires an off-shellsupersymmetry realisation and so works only in cases with lesser degrees of supersymmetry.However, in the future this may provide a deeper understanding of complicated constructionssuch as those made in this paper.Another challenge encountered in the present construction is the coupling of boundaryhypermultiplets. These are in general necessary in order to arrange for gravitational anomalycancellation, but they do not affect the classical gauge or supersymmetry anomalies. However,the bulk-plus-boundary couplings in this sector lead to novel problems. Eight-supercharge( N = 2, D = 4 or N = 1, D = 6 supersymmetry) hypermultiplets coupled to supergravityrequire an overall quaternionic K¨ahler target-space manifold [21]. Indeed, the bulk D = 7theory dimensionally reduced to D = 6 and truncated to N = 1, D = 6 local supersymmetrygenerates precisely this kind of scalar target-space manifold [17]. However, when one includesadditional hypermultiplets on the D = 6 boundaries of the Hoˇrava-Witten construction, oneruns into the problem that one cannot simply add quaternionic K¨ahler manifolds to producean overall quaternionic K¨ahler manifold. The resolution of this problem led to the connectioncondition (7.13).A number of aspects of the constructions discussed in this paper call for further devel-opment. A fuller treatment of the hypermultiplet couplings will be given in a separate pub-lication, and a full analysis of the solutions to the anomaly-cancellation conditions is calledfor. Another open question deals with a very special class of remarkably anomaly-free D = 6theories with gauged U (1) R symmetries. These are: • the E × E × U (1) R invariant model in which the hyperfermions are in the (912 , , • the E × G × U (1) R invariant model with hyperfermions in the (56 , ,
1) representationof the gauge group [6], and • the F × Sp (9) × U (1) R invariant model with hyperfermions in the (52 , ,
1) represen-tation of the gauge group [7].We have determined that the construction of this paper cannot yield any of these models in acoincident brane limit. Thus, finding the higher-dimensional origins of these theories, if any,remains an outstanding open problem.More generally, the rˆole of noncompact gaugings and their higher-dimensional originsthrough reduction on noncompact spaces needs further consideration. Noncompact reductionsmay, as in the H (2 ,
2) reduction considered in [12], yield classically consistent Kaluza-Kleinreductions. But at the quantum level, this classical Kaluza-Klein consistency is surely bro-ken. Moreover, noncompact reductions from higher-dimensional theories would be expectedto lead to a continuous Laplace eigenvalue spectrum without a mass gap between the re-tained lower-dimensional and the higher truncated Kaluza-Klein states. One can imagine anumber of possible responses to this situation. One would be to consider a compactifica-tion of the reduction space, perhaps by modding out by discrete symmetries, but this would33lso likely be at the cost of introducing supersymmetry breaking at some new scale in theproblem. Another might be to look for discrete Laplace eigenfunctions in the midst of acontinuous-eigenvalue spectrum. Such situations are not unusual in other contexts, such ascondensed-matter physics. It remains to be seen whether they have a relevance in the contextof noncompact gauged R-symmetries.
Acknowledgements
We would like to acknowledge collaboration with Chris Pope and Eric Bergshoeff in earlystages of this project and for many subsequent discussions. We also thank Alex Kehagiasfor useful discussions. For hospitality during the course of the work, ES would like to thankthe Theoretical Physics Group at Imperial College London and National Technical Universityof Athens; KSS and TGP would like to thank the George P. and Cynthia Woods MitchellInstitute for Fundamental Physics and Astronomy at Texas A&M University; KSS would alsolike to thank the TH Unit at CERN and the Albert Einstein Institute, Potsdam. KSS wouldlike to thank as well the Mitchell Family for hospitality and a beautiful and quiet place towork during the Cooks Branch workshop in April 2010. TGP would like to thank NoppadolMekareeya for many helpful discussions. The research of E.S. was supported in part by NSFgrants PHY-0555575 and PHY-0906222. The work of K.S.S. was supported in part by by theSTFC under rolling grant PP/D0744X/1.
Appendices
A The Coincident Boundary Limit
We now consider taking the coincident boundaries limit when the boundaries are populatedwith vector multiplets as described in Section 6. This gives a six-dimensional gauged super-gravity theory similar to that described in [17].The orbit of boundary conditions in this D = 7 system involves both Neumann andDirichlet types, which have different effects on the reduced system. Let us first consider theNeumann boundary conditions with the example of the form field A µνρ . This is subject totwo boundary conditions: one on the x = 0 boundary and the other on the x = L boundary(where L is the interval length ). We can follow the work of [18, 37, 38] and use the fact that,in the limit of small interval length, it is sufficient to approximate the value of A µνρ in thebulk by a linear interpolation between the two boundary conditions: A µνρ = A µνρ (cid:12)(cid:12)(cid:12)(cid:12) x =0 (cid:18) − x L (cid:19) + A µνρ (cid:12)(cid:12)(cid:12)(cid:12) x = L x L . (A.1)We consider the simplified case in which the boundary at x = 0 is populated by vector34ultiplets in the way we have described and the boundary at x = L is empty. This meansthat the bulk field A µνρ becomes A µνρ = (cid:18) κ λ ω µνρ ( C ) + iκ λ e σ ¯ η X γ µνρ η X (cid:19) (cid:18) − x L (cid:19) . (A.2)This causes the six-dimensional 3-form field strength to become Chern-Simons modified:ˆ F µνρ = 3 ∂ [ µ ˆ A νρ ]7 − ∂ ˆ A µνρ = √ (cid:18) ∂ [ µ B νρ ] + 32 g ′ ω µνρ ( C ) + i g ′ ¯ η X γ µνρ η X (cid:19) , (A.3)where we have defined g ′ = Lλ κ in order to match the conventional result. If we now redefine G µνρ as the appropriately normalised bosonic part in the above equation i.e. G µνρ = 3 ∂ [ µ B νρ ] + 32 g ′ ω µνρ ( C ) , (A.4)then we find that G µνρ is invariant under the Yang-Mills gauge symmetry since B µν developsa gauge transformation due to the boundary condition (5.10): δ Λ B µν = − g ′ ∂ [ µ C Xν ] Λ X . (A.5)On the other hand, the field ˆ σ receives a Dirichlet boundary condition. In the small intervallimit, we can again interpolate between its two boundary values such that ∂ ˆ σ = ∂ ˆ σ (cid:12)(cid:12)(cid:12)(cid:12) x =0 (cid:18) − x L (cid:19) + ∂ ˆ σ (cid:12)(cid:12)(cid:12)(cid:12) x = L x L . (A.6)If we integrate this equation and impose the requirement that the average value of ˆ σ is thesame as in the empty boundaries case, then we obtainˆ σ = − ∂ ˆ σ (cid:12)(cid:12)(cid:12)(cid:12) x =0 (cid:18) ( x ) L − x + L (cid:19) + ∂ ˆ σ (cid:12)(cid:12)(cid:12)(cid:12) x = L (cid:18) ( x ) L − L (cid:19) + 45 σ + 25 ϕ . (A.7)Performing similar steps for all fields that receive non-trivial boundary conditions and thenincorporating these into the D = 7 bulk action together with the Gibbons-Hawking-Yorkterms and the boundary action, and ignoring any higher-order terms in λ or L , we obtain35he D = 6 action S SG (6) = 2 Lκ Z dx e (cid:26) R − g e σ F r ′ µν F µνr ′ − g ′ e − σ H Xµν H µνX − e − σ G µνρ G µνρ − ∂ µ σ∂ µ σ − ∂ µ ϕ∂ µ ϕ − P irµ P µir − P rµ P µr − P iµ P µi − g e − σ (cid:16) C ir ′ C ir ′ + 2 S ir ′ S ir ′ (cid:17) + 116 g ε µνρσλτ B µν F r ′ ρσ F r ′ λτ + 132 g g ′ ε µνρσλτ ω µνρ ( C ) ω σλτ ( A ) − i ψ µ γ µνρ D ν ψ ρ − i χγ µ D µ χ − i g ¯ λ r ′ γ µ D µ λ r ′ − i ψγ µ D µ ψ − i ψ r γ µ D µ ψ r − i g ′ ¯ η X γ µ D µ η X −
12 ¯ ψ r σ i γ µ γ ν ψ µ P irν −
12 ¯ ψσ i γ µ γ ν ψ µ P iν − i ψ r γ µ γ ν ψ µ P rν − i χγ µ γ ν ψ µ ∂ ν σ − i ψγ µ γ ν ψ µ ∂ ν ϕ − i e − σ G µνρ (cid:18) ¯ ψ [ λ γ λ γ µνρ γ τ ψ τ ] − ψ λ γ µνρ γ λ χ − ¯ χγ µνρ χ + ¯ ψγ µνρ ψ + ¯ ψ r γ µνρ ψ r − g ¯ λ r ′ γ µνρ λ r ′ − g ′ η X γ µνρ η X (cid:19) − P iµ (cid:18) ¯ ψ [ ρ σ i γ ρ γ µ γ τ ψ τ ] + ¯ χσ i γ µ χ + 1 g ¯ λ r ′ σ i γ µ λ r ′ + 1 g ′ ¯ η X σ i γ µ η X − ¯ ψ r σ i γ µ ψ r − ¯ ψσ i γ µ ψ (cid:19) − i g e σ F r ′ µν (cid:18) ¯ ψ ρ γ µν γ ρ λ r ′ + ¯ χγ µν λ r ′ (cid:19) − i g ′ e − σ H Xµν (cid:18) ¯ ψ ρ γ µν γ ρ η X − ¯ χγ µν η X (cid:19) − i P rµ ¯ ψγ µ ψ r − e σ C irr ′ ¯ λ r ′ σ i ψ r + ie σ S rr ′ ¯ λ r ′ ψ r − e σ S ir ′ ¯ λ r ′ σ i ψ + 12 √ e − σ λ r ′ σ i γ µ ψ µ (cid:16) C ir ′ − √ S ir ′ (cid:17) + 12 √ e − σ λ r ′ σ i χ (cid:16) C ir ′ − √ S ir ′ (cid:17) (cid:27) . (A.8)36arrying out the reduction of the supersymmetry transformations and averaging over x gives δe µµ = i ¯ ǫγ µ ψ µ ,δψ µ = D µ ǫ + 124 e − σ G ρστ γ ρστ γ µ ǫ − i P iµ σ i ǫ ,δχ = − γ µ ∂ µ σǫ − e − σ G µνρ γ µνρ ǫ ,δB µν = − ie σ ¯ ǫγ [ µ ψ ν ] + i e σ ¯ ǫγ µν χ + 1 g ′ δ ǫ C X [ µ C ν ] X ,δσ = − i ¯ ǫχ ,δA r ′ µ = ie − σ ¯ ǫγ µ λ r ′ ,δλ r ′ = − e σ γ µν F r ′ µν ǫ − i √ g e − σ (cid:16) C ir ′ − √ S ir ′ (cid:17) σ i ǫ ,δψ = i γ µ (cid:0) P iµ σ i − i∂ µ ϕ (cid:1) ǫ ,δψ r = i γ µ (cid:0) P irµ σ i + i P rµ (cid:1) ǫ ,δϕ = i ¯ ǫψ ,δL rI = ¯ ǫσ i ψ r L iI ,δL iI = ¯ ǫσ i ψ r L rI ,δ Φ I = − L Ii e − ϕ ¯ ǫσ i ψ − iL Ir e − ϕ ¯ ǫψ r ,δC Xµ = ie σ ¯ ǫγ µ η X ,δη X = − e − σ γ µν H Xµν ǫ . (A.9)Under these supersymmetry transformations, the action varies into the supersymmetry anomaly δ ǫ S = 2 L κ g g ′ Z ∂M d xe (cid:26) ǫ µνρσλτ H Xµν H Xρσ δ ǫ A r ′ λ A r ′ τ − ǫ µνρσλτ ω µνρ ( C ) F r ′ σλ δ ǫ A r ′ τ + ǫ µνρσλτ F r ′ µν F r ′ ρσ δ ǫ C Xλ C Xτ − ǫ µνρσλτ ω µνρ ( A ) H Xσλ δ ǫ C Xτ (cid:27) , (A.10)which is Wess-Zumino consistent with its gauge variation, δ Λ S = 2 L κ g g ′ Z ∂M d xe (cid:26) ε µνρσλτ H Xµν H Xρσ ∂ λ A r ′ τ Λ r ′ + ε µνρσλτ F r ′ µν F r ′ ρσ ∂ λ C Xτ Λ X (cid:27) . (A.11)We note that the action and variations obtained here are consistent with the general mattercoupled D = 6 supergravity described in [2, 20] for the case of a single tensor multiplet.We note also that that if one were to consider the boundary matter coupling startingfrom the boundary condition A µνρ ∼ c A ω µνρ ( A ) as described in Section 8.2, then the reducedaction would appear to contain kinetic terms of the form S ∼ Z d xe (cid:18) − g e σ − c A e − σ (cid:19) F r ′ µν F µνr ′ (A.12)which is known to exhibit interesting phase transition behaviour [39, 40]. The dilaton depen-dence arises from supersymmetry considerations as described in Section 6.37 D = 7 We now consider the equivalent construction for the theory in which the 3-form ˆ A MNR hasbeen dualised into a 2-form ˆ B MN . This has the D = 7 bulk action S SG = 1 κ Z d x ˆ e (cid:26)
12 ˆ R (ˆΓ) −
14 1 g e ˆ σ ˆ F iMN ˆ F MNi − g e ˆ σ ˆ F ˆ rMN ˆ F MN ˆ r − e σ ˆ G MNR ˆ G MNR −
58 ˆ ∂ M ˆ σ ˆ ∂ M ˆ σ −
12 ˆ P i ˆ rM ˆ P Mi ˆ r − g e − σ (cid:18) C i ˆ r C i ˆ r − C (cid:19) − i ψ M ˆ γ MNR ˆ D N ˆ ψ R − i χ ˆ γ M ˆ D M ˆ χ − i g ˆ¯ λ ˆ r ˆ γ M ˆ D M ˆ λ ˆ r − i χ ˆ γ M ˆ γ N ˆ ψ M ˆ ∂ N ˆ σ − g ˆ¯ λ ˆ r σ i ˆ γ M ˆ γ N ˆ ψ M P i ˆ rN + i √ e ˆ σ ˆ G MNR (cid:18) ˆ¯ ψ [ L ˆ γ L ˆ γ MNR ˆ γ T ˆ ψ T ] + 4 ˆ¯ ψ L ˆ γ MNR ˆ γ L ˆ χ − χ ˆ γ MNR ˆ χ + 1 g ˆ¯ λ ˆ r ˆ γ MNR ˆ λ ˆ r (cid:19) + 18 g e ˆ σ ˆ F iMN (cid:18) ˆ¯ ψ [ L σ i ˆ γ L ˆ γ MN ˆ γ T ˆ ψ T ] − ψ L σ i ˆ γ MN ˆ γ L ˆ χ + 3 ˆ¯ χσ i ˆ γ MN ˆ χ − g ˆ¯ λ ˆ r σ i ˆ γ MN ˆ λ ˆ r (cid:19) − i g e ˆ σ ˆ F ˆ rMN (cid:18) ˆ¯ ψ L ˆ γ MN ˆ γ L ˆ λ ˆ r + 2 ˆ¯ χ ˆ γ MN ˆ λ ˆ r (cid:19) + 12 √ e − ˆ σ C i ˆ r (cid:18) ˆ¯ ψ M σ i ˆ γ M ˆ λ ˆ r − χσ i ˆ λ ˆ r (cid:19) − i √ ge − ˆ σ C (cid:18) ˆ¯ ψ M ˆ γ MN ˆ ψ N + 2 ˆ¯ ψ M ˆ γ M ˆ χ + 3 ˆ¯ χ ˆ χ − g ˆ¯ λ ˆ r ˆ λ ˆ r (cid:19) + 12 g e − ˆ σ C ˆ r ˆ si ˆ¯ λ ˆ r σ i ˆ λ ˆ r (cid:27) (B.1)where ˆ G MNR = 3 ∂ [ M ˆ B NR ] − √ g ˆ ω MNR ( ˆ A ) (B.2)and all other definitions remain the same as before. This action has no Chern-Simons term,so we might expect no anomaly to occur. However, as we now see, this is not the case.38he action is invariant under the following local supersymmetry transformations: δ ˆ e MM = i ˆ¯ ǫγ M ˆ ψ M ,δ ˆ ψ M = 2 ˆ D M ˆ ǫ − √ e ˆ σ ˆ G RST (cid:0) ˆ γ M ˆ γ RST + 5ˆ γ RST ˆ γ M (cid:1) ˆ ǫ − i g e ˆ σ ˆ F iRS σ i (cid:0) γ M ˆ γ RS − γ RS ˆ γ M (cid:1) ˆ ǫ − √ ge − ˆ σ C ˆ γ M ˆ ǫ ,δ ˆ χ = −
12 ˆ γ M ˆ ∇ M ˆ σ ˆ ǫ − i e ˆ σ ˆ F iMN σ i ˆ γ MN ˆ ǫ − √ e ˆ σ ˆ G MNR ˆ γ MNR ˆ ǫ + √ e − ˆ σ C ˆ ǫ ,δ ˆ B MN = i √ e − ˆ σ (cid:16) ˆ¯ ǫ ˆ γ [ M ˆ ψ N ] + ˆ¯ ǫγ MN ˆ χ (cid:17) − √ g ˆ A ˆ I [ M δ ˆ A N ] ˆ I ,δ ˆ A ˆ IM = − ge ˆ σ (cid:16) ˆ¯ ǫσ i ˆ ψ M + ˆ¯ ǫ ˆ γ MN ˆ χ (cid:17) L ˆ Ii + ie − ˆ σ ˆ¯ ǫ ˆ γ M ˆ λ ˆ r L ˆ I ˆ r ,δ ˆ σ = − i ˆ¯ ǫ ˆ χ ,δL i ˆ I = 1 g ˆ¯ ǫσ i ˆ λ ˆ r L ˆ r ˆ I ,δL ˆ r ˆ I = 1 g ˆ¯ ǫσ i ˆ λ ˆ r L i ˆ I ,δ ˆ λ ˆ r = − e ˆ σ ˆ F ˆ rMN ˆ γ MN ˆ ǫ + ig ˆ γ M ˆ P i ˆ rM σ i ˆ ǫ − i √ ge − ˆ σ C i ˆ r σ i ˆ ǫ , as well as having a Z symmetry which acts as before but now with ˆ B µν assigned even parityand ˆ B µ odd parity. The action possesses a gauge symmetry under which ˆ B MN transforms as δ Λ ˆ B MN = √ g ˆ ∂ [ M ˆ A ˆ IN ] Λ ˆ I . (B.3)Once again, we begin our construction on a manifold with boundary by adding Gibbons-Hawking-York terms S GHY = Z ∂M d x q − ˆ h (cid:26) ˆ K − i ψ µ ˆ γ µν ˆ ψ ν − i χ ˆ χ (cid:27) . (B.4)Redefining exactly as before but now with B µν = √ ˆ B µν , G µν = √ ˆ G µν gives the D = 639upergravity transformations [17] δe µµ = i ¯ ǫγ µ ψ µ ,δψ µ = D µ ǫ − e σ G ρστ γ ρστ γ µ ǫ − i P iµ σ i ǫ ,δχ = − γ µ ∇ µ σǫ − e σ G µνρ γ µνρ ǫ ,δB µν = ie − σ (cid:18) ¯ ǫγ [ µ ψ ν ] + 12 ¯ ǫγ µν χ (cid:19) − g A r ′ [ µ δA ν ] r ′ ,δσ = − i ¯ ǫχ ,δA r ′ µ = ie − σ ¯ ǫγ µ λ r ′ ,δλ r ′ = − e σ γ µν F r ′ µν ǫ − i √ g e − σ (cid:16) C ir ′ − √ S ir ′ (cid:17) σ i ǫ ,δψ = i γ µ (cid:0) P iµ σ i − i ∇ µ ϕ (cid:1) ǫ ,δψ r = i γ µ (cid:0) P irµ σ i + iP rµ (cid:1) ǫ ,δϕ = i ¯ ǫψ ,δL rI = ¯ ǫσ i ψ r L iI ,δL iI = ¯ ǫσ i ψ r L rI ,δ Φ I = − L Ii e − ϕ ¯ ǫσ i ψ − iL Ir e − ϕ ¯ ǫψ r , (B.5)where now G µνρ = 3 ∂ [ µ B νρ ] − g ω µνρ ( A ) and B µν transforms as δ Λ B µν = 1 g ∂ [ µ A r ′ ν ] Λ r ′ . (B.6)Again, we can construct a consistent set of boundary conditions and in this we case find, ψ µ − = − κ λ e − σ H Xµν γ ν η X + 3 κ λ e − σ H ρσX γ µρσ η X + (fermi) ,χ = κ λ e − σ H Xµν γ µν η X + (fermi) ,e αφ ∂ ˆ σ = − κ λ e − σ H Xρσ H ρσX + (fermi) ,G µν = κ λ e − σ ǫ µνρσλτ H ρσX H λτX + (fermi) ,K µν = κ λ e − σ H Xµρ H νρX − κ λ e − σ H Xρσ H ρσX g µν + (fermi) . (B.7)Then, upon substituting this into the surface terms, obtained as before, a great deal ofcancellation occurs and we are left with δ ǫ S SG + δ ǫ S GHY = 1 λ Z ∂M d xe (cid:26) − e − σ ¯ ǫγ ρσ γ µ σ i η X H Xρσ P iµ (cid:27) . (B.8) Here we have, as in the previous case, set all the free parameters that can occur equal to values that willbe required by the variational principle, anticipating the final constructed boundary action. S B = 1 λ Z ∂M d x (cid:26) − e − σ H Xµν H µνX − i η X γ µ D µ η X − i e − σ H Xµν ¯ η X γ ρ γ µν ψ ρ − i e − σ H Xµν ¯ η X γ µν χ − i e σ G µνρ ¯ η X γ µνρ η X − ǫ µνρσλτ B µν H Xρσ H Xλτ + 132 g ǫ µνρσλτ ω µνρ ( C ) ω σλτ ( A ) (cid:27) , (B.10)gives the classical supersymmetry anomaly δ ǫ S = − λ g Z ∂M d xe (cid:26) ǫ µνρσλτ H Xµν H Xρσ δ ǫ A r ′ λ A r ′ τ − ǫ µνρσλτ ω µνρ ( C ) F r ′ σλ δ ǫ A r ′ τ + ǫ µνρσλτ F r ′ µν F r ′ ρσ δ ǫ C Xλ C Xτ − ǫ µνρσλτ ω µνρ ( A ) H Xσλ δ ǫ C Xτ (cid:27) , (B.11)whilst the classical gauge anomaly is δ Λ S = − λ g Z ∂M d xe (cid:26) ε µνρσλτ H Xµν H Xρσ ∂ λ A r ′ τ Λ r ′ + ε µνρσλτ F r ′ µν F r ′ ρσ ∂ λ C Xτ Λ X (cid:27) . (B.12)Once again these are Wess-Zumino consistent.It is interesting to note that these classical anomalies exist, in spite of the fact thatthere is no Chern-Simons term to provide anomaly inflow, because the inherited supergravitytransformation rules have forced a Green-Schwarz type of anomaly production upon us. Thisis very different mechanism from the 3-form case considered in Section 6, but gives rise toanomalies of exactly the same form. The bulk contribution (B.8) can also be produced by adding a term of the form S = 1 λ Z ∂M d xe (cid:26) ¯ η X γ µ σ i η X P iµ (cid:27) (B.9)to the boundary action and multiplying the R.H.S. of (B.7) by a corresponding factor. However, if this weredone, the action and boundary conditions would then no longer be consistent with the variational principle. eferences [1] H. Nishino and E. Sezgin, “The Complete N=2, d=6 Supervravity with Matter andYang-Mills Couplings,” Nucl. Phys.
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