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PreprinttypesetinJHEPstyle-HYPERVERSION Poincaré Dual of D=4 N=2 Supergravity with Tensor 5 0 Multiplets 0 2 n a J 5 2 Luca Sommovigo 2 v Dipartimento di Fisica, Politecnico di Torino, 8 Corso Duca degli Abruzzi 24, I-10129 Torino, and 4 Istituto Nazionale di Fisica Nucleare (INFN) 0 Sezione di Torino, Italy 1 0 E-mail: [email protected] 5 0 / h Abstract: We study, in an arbitrary even number D of dimensions, the duality between t - p massive D/2 tensors coupled to vectors, with masses given by an arbitrary number of e “electric” and “magnetic” charges, and (D/2 1) massive tensors. We develop a formalism h − : to dualize the Lagrangian of D = 4, N = 2 supergravity coupled to tensor and vector v i multiplets, and show that, after the dualization, it is equivalent to a standard D = 4, X N = 2 gauged supergravity in which the Special Geometry quantities have been acted on r a by a suitable symplectic rotation. Contents 1. Introduction 1 2. General dualization procedure for the linearized theory 2 2.1 D = 4d 4 2.2 D = 4d+2 5 3. The D = 4 N = 2 Supergravity theory coupled to vector multiplets and scalar–tensor multiplets 6 4. Symplectic rotation 12 5. Conclusions 13 1. Introduction It is a well known fact that orientifold compactifications of IIB string theory with fluxes turned on, give rise to a theory in which tensor multiplets are present, and are endowed with a mass; this fact has recently given rise to a renewed interest in theories describing tensor multiplets coupled to scalar and vector multiplets [1, 2, 3]. This same topic has been considered also in [4] where a theory describing, in D even dimensions, a number of massive D/2–form tensors B with masses given by “magnetic” I and “electric” couplings has been dualized to a theory of massive D/2 1–dimensional − form fields, generalizing the well-known duality in 4 dimensions between massive 2–form tensors and massive vectors. The “electric” and “magnetic” couplings which give mass to the tensors arise naturally from Fayet-Iliopoulos terms in the context of the dualization of n massless scalars to 2–forms in D = 4, N = 2 supergravity [5, 6, 7]. In this light, the T theory of [4] can be thought of as a generalization to an arbitrary number of dimensions of the linearized (without coupling to scalars) bosonic kinetic and mass terms for the n T 2–forms B fromD = 4, N = 2supergravity coupledton tensormultiplets andn vector I T V multiplets [6]. Actually in [4] the proof of this generalized duality has been given only for two particular cases: that of a single tensor B, when D = 4d and hence B is a 2d–form field with coupling constants m and e, and that of a couple of tensors, B with I = 1,2 when D = 4d+2 and I hence the B ’s are (2d+1)–form fields, taking the coupling constants to be mIΛ = mδIΛ I and eI = eǫI, with Λ= 1,2. Λ Λ Inbothcases, thesquaredmassof the(D/2 1)–form after thedualization hasbeenshown − to depend on both “electric” and “magnetic” charges, via the relation: – 1 – µ = e2+m2, (1.1) so that no distinction appears between elepctric and magnetic charges. We will extend the analysis of [4] to the more general case in which an arbitrary number of “electric” and “magnetic” charges is present, in the linearized theory, showing that also in this case the Poincaré dual of a theory of n massive D/2–form fields is a theory describing n massive T T D/2 1–form fields. Moreover their square mass matrix will be given by the “sum of the − squares” of the electric and magnetic charge matrices: (µ2)IJ = e2+m2 IJ eIΛδΛΣeJΣ+mIΛδΛΣmJΣ (1.2) ≡ thus generalizing relation (1.1(cid:0)). (cid:1) (cid:0) (cid:1) The whole procedure requires the matrices eI, mIΛ and e2+m2 IJ to be invertible, Λ therefore from now on we will suppose that n = n = n, so that all these matrices are T V (cid:0) (cid:1) square, and that dete, detm, det e2+m2 = 0 in order to guarantee the existence of the 6 inverses. (cid:0) (cid:1) The paper is organized as follows: in section 2 we prepare the setting and write down the dual Lagrangian for a theory of n massive tensors in an arbitrary even number D of dimensions, distinguishing between D = 4d and D = 4d + 2 cases, generalizing the procedure of [4]. In section 3 we apply the whole formalism to the Lagrangian of D = 4 N = 2 supergravity coupled to n massive tensors and n vectors obtained in [6]. Finally, in section 4 we reinterpret the dual Lagrangian obtained with the dualization as a standard D = 4 N = 2 gauged supergravity, where the gauge is purely electric and the quantities of the special geometry can be obtained from the standard ones by means of a symplectic rotation. We note that the fact that the N = 2 tensor coupled theory can be related to a standard N = 2 supergravity by means of duality transformations does not mean that they are the same theory. As a matter of fact we can show that they give rise to different N = 1 theory when a suitable truncation to N =1 is performed (see reference [8]). 2. General dualization procedure for the linearized theory In this section we generalize the procedure of reference [4] to obtain the dual Lagrangian of a theory of massive tensors in an arbitrary even number of dimensions with an arbitrary number of charges. Thelinearized Lagrangian of a numbern of massless vector fieldsand a numbern of tensor fields, as coming from D = 4 N = 2 supergravity [5, 6], is1: 1 1 T,V = δIJ ∗ eI Λ+ mJΛB B I J Λ J I L −4 H ∧ H − F 2 ∧ (cid:18) (cid:19) 1Notethat,in orderto makecontact with [6]one hastoredefinethe“electric” and“magnetic” charges as 2eI →eI, 2mIΛ →mIΛ. Λ Λ – 2 – 1 + δΛΣ Λ +mIΛBI ∗ Σ +mIΣBJ (2.1) 2 F ∧ F (cid:0) (cid:1) (cid:0) (cid:1) where δ is the usual Kronecker delta, Λ, are the field-strengths of the gauge vectors I F H AΛ and the tensors B respectively. I Lagrangian (2.1) can be immediately generalized to an arbitrary (even) number D of dimensions by promoting B and Λ to be D/2–form fields2. As is well known, Lagrangian I F (2.1) and its generalization to higher dimensions are invariant under the following gauge transformations: δAΛ = mIΛΩ , δB = dΩ , (2.2) I I I − whereΩ isaD/2 1-formfield,ifandonlyifthegeneralizedtadpolecancellationcondition: I − eImJΛ eJmIΛ = 0 (2.3) Λ Λ ∓ holds. The minus sign holds in the D = 4d case, when the B are even forms (and hence I they commute with each other), while the plus sign holds in the D = 4d+2 case, when the B are odd forms (anticommuting with each other). I It will prove convenient to add to (2.1) a total derivative: 1 eI m−1 Λ Σ, (2.4) − 2 Λ IΣF ∧F in such a way that the topological term(cid:0) pro(cid:1)portional to the “electric” charge eI can be Λ written in a manifestly gauge invariant way: 1 eI m−1 Λ+mJΛB Σ +mKΣB . (2.5) − 2 Λ IΣ F J ∧ F K By means of a gauge trans(cid:0)form(cid:1)atio(cid:0)n, the field-str(cid:1)engt(cid:0)hs of the vector(cid:1)s can be reabsorbed into the tensors: Ω = m−1 AΛ A′Λ AΛ+δAΛ = 0; B′ B + m−1 Λ (2.6) I IΛ ⇒ ≡ I ≡ I IΛF (cid:0) (cid:1) (cid:0) (cid:1) and the Lagrangian can be written in the simpler form: T,V = 1δIJ ∗ + 1 m2 IJ B ∗B 1(em)IJ B B (2.7) I J I J J I ±L −4 H ∧ H 2 ∧ − 2 ∧ the two signs referring respectively to the(cid:0)case(cid:1)D = 4d and D = 4d+2. We have defined the following matrices: m2 IJ mIΛδΛΣmJΣ (2.8) ≡ (cid:0)(em(cid:1))IJ eIΛmJΛ. (2.9) ≡ 2We adopt a mostly minus metric (+,−,...,−) so that, in order to have positive kinetic energy, when D=4d+2 an overall minussign in front of theLagrangian is needed. – 3 – It should be noted that both matrices have a well-defined symmetry in the indices I,J: m2 IJ is symmetric by construction while (em)IJ, due to the tadpole cancellation condition (2.3) is symmetric if D = 4d and antisymmetric if D = 4d+2. (cid:0) (cid:1) Starting from the Lagrangian of equation (2.7), the process of dualization requires that I H and B be considered as independent fields, enforcing the relation = dB by means of I I I H the equations of motion for a suitable Lagrange multiplier ρI that should be added to the Lagrangian, which becomes: T,V = 1δIJ ∗ + 1 m2 IJ B ∗B 1 (em)IJ B B + I J I J J I ±L −4 H ∧ H 2 ∧ − 2 ∧ +ρI ( dB ). (cid:0) (cid:1) (2.10) I I ∧ H − In order to get the dualized Lagrangian, at first one has to write down the equations of motion for the independent fields B and , then to solve these equations in terms of the I I H dualfieldρI andfinallyto substitutetheresultingexpressionsintotheoriginal Lagrangian. IJ Because of the (anti)-symmetry properties of the matrix (em) and the fact that the B I and fields are even or odd forms depending on the fact that D = 4d or D = 4d+2, it I H is more convenient to consider the two cases separately. 2.1 D =4d When D = 4d, the B ’s are 2d–forms, and hence they commute with all other p–form I fields; the equations of motion are: 1 δIJ∗ +ρI = 0 (2.11) J 2 H m2 IJ∗B (em)IJ B +dρI = 0 (2.12) J J − Equation (2.11) is easily inv(cid:0)erti(cid:1)ble in order to get in terms of ∗ρI, while from equation I H (2.12) and its Hodge dual it is possible to write B in terms of both dρI and ∗dρI as: I B = e2+m2 −1 ∗dρJ + e J dρK (2.13) I IJ m K (cid:18) (cid:19) (cid:16) (cid:17) ∗B = (cid:0) e2+m(cid:1)2 −1 dρJ e J ∗dρK (2.14) I − IJ − m K (cid:18) (cid:19) (cid:16) (cid:17) (cid:0) (cid:1) where the following matrices have been introduced: e I eIδΛΣ m−1 (2.15) m J ≡ Λ JΣ (cid:16)e2 (cid:17)IJ eIδΛΣe(cid:0)J = (cid:1)e2 JI. (2.16) Λ Σ ≡ As m2 IJ, also e2 IJ is sym(cid:0)met(cid:1)ric by constructio(cid:0)n. (cid:1)Using these symmetries, together with the following identities involving the product of two matrices: (cid:0) (cid:1) (cid:0) (cid:1) – 4 – IK e J 2 IJ (em) = e (2.17) m K e I (cid:16) 2 (cid:17)KJ (cid:0) e(cid:1) I KL e J e J 2 KI e = (em) = e (2.18) m K m K m L m K (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:0) (cid:1) (cid:0) (cid:1) which allow a number of simplifications, the dualized Lagrangian turns out to be: dual = 1 e2+m2 −1dρI ∗dρJ+1 e2+m2 −1 e KdρI dρJ δ ρI ∗ρJ. (2.19) L 2 IJ ∧ 2 IK m J ∧ − IJ ∧ (cid:16) (cid:17) (cid:0) (cid:1) (cid:0) (cid:1) As expected, (2.19) is the Lagrangian describing a theory of n massive (2d 1) vector fields, whose mass matrix is (µ2)IJ = e2+m2 IJ3. − (cid:0) (cid:1) 2.2 D =4d+2 In this case, the B fields are 2d+1–forms, hence anticommuting with all other odd forms; I taking into account these different properties, we can obtain the equations of motion: 1 δIJ∗ +ρI = 0 (2.20) J −2 H m2 IJ∗B (em)IJ B dρI = 0 (2.21) J J − − (cid:0) (cid:1) again, as in the previous case, the equations of motion for the fields are easily inverted, I H while after some algebraic manipulation on the equations for the B and their Hodge dual, I IJ remembering that (em) is now an antisymmetric matrix, and hence e I 2 KJ e I KL e J e J 2 KI e = (em) = e (2.22) m K m K m L − m K (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) we get the relations: (cid:0) (cid:1) (cid:0) (cid:1) B = e2+m2 −1 ∗dρJ + e J dρK (2.23) I IJ m K (cid:18) (cid:19) (cid:16) (cid:17) ∗B = (cid:0)e2+m2(cid:1)−1 dρJ + e J ∗dρK (2.24) I IJ m K (cid:18) (cid:19) (cid:16) (cid:17) (cid:0) (cid:1) Using (2.23) and (2.24) in (2.10), we get the following expression: dual = 1 e2+m2 −1dρI ∗dρJ 1 e2+m2 −1 e KdρI dρJ+δ ρI ∗ρJ. (2.25) L −2 IJ ∧ −2 IK m J ∧ IJ ∧ (cid:16) (cid:17) (cid:0) (cid:1) (cid:0) (cid:1) which is the Lagrangian describing n massive vectors in D = 4d+2 dimensions. We have therefore shown that, in an even number D of dimensions, a theory of n massive D/2–form tensors is dual to a theory of n massive D/2 1–form tensors. − 3Hereand in thenext case, thetopological term has been written for completeness, even if its presence does not modify the equations of motion. – 5 – 3. The D = 4 N = 2 Supergravity theory coupled to vector multiplets and scalar–tensor multiplets The procedure seen in section 2 can be applied to the Lagrangian of the D = 4 N = 2 supergravity theory coupled to n vector multiplets and n scalar-tensor multiplets [6]. V T The first step is to write the Lagrangian of reference [6] in first-order formalism, which is more practical to perform the dualization. Actually we do not need all the Lagrangian, but only the terms coupled either to I or BI, which are collected in bos and Pauli: H L L T,V LD=4 = Lbos+LPauli (3.1) 1 1 1 bos = Im ΛΣ ˆΛ ∗ ˆΣ + Re ΛΣ ˆΛ ˆΣ IJ I ∗ J + L −2 N F ∧ F 2 N F ∧F − 4M H ∧ H 1 + AI eI ˆΛ mJΛB B (3.2) I Λ J I H ∧ − F − 2 ∧ (cid:18) (cid:19) LPauli = −2i HI ∧∗ UIAαψA∧γ∧γζα−UIAαψA∧γ ∧γζα + i ∆I(cid:16)α ζ γζβ IAαψ ζ + ψA(cid:17)ζα + I β α I A α JAα − H ∧ −H ∧ U U 2iIm ΛΣ ˆΛ LΣ ψA (cid:16)ψBǫAB − LΣ ψA ψB(cid:17)ǫAB + + − N F ∧ ∧ − ∧ (cid:20) (cid:21) (cid:16) (cid:17) 2Im ΛΣ ˆΛ f¯Σı λ¯ıAγ ψBǫAB − +fiΣ (cid:0)λiAγ ψBǫA(cid:1)B + + − N F ∧ ∧ ∧ (cid:20) (cid:21) (cid:16) (cid:17) (cid:16) (cid:17) −4i ImNΛΣFˆΛ ∧ ∇ifjΣλiAγ ∧γλjBǫAB −∇¯ıfΣ¯λ¯ıAγ∧γλB¯ǫAB + +2i ImNΛΣFˆΛ ∧(cid:16)LΣζαγ∧γζβCαβ −LΣζαγ∧γζβCαβ (cid:17) (3.3) (cid:16) (cid:17) where we have defined the following quantities: ˆΛ Λ +mIΛB (3.4) I F ≡ F AI AIdqu (3.5) u ≡ γ γ dxµ (3.6) µ ≡ (...)± (...) i∗(...) (3.7) ≡ ± And the bilinear fermions (...)± enjoy the following property: ∗(...)± = i(...)± (3.8) ∓ the Hodge dual operator ∗ being defined as: 1 ∗dxµ1 ... dxµk √ gǫµ1...µk dxµk+1 ... dxµn. (3.9) ∧ ∧ ≡ (n k)! − µk+1...µn ∧ ∧ − Moreover IJ and AIu are the remnants of the quaternionic metric huˆvˆ according to the M decomposition given in [9, 5]: – 6 – g + AIAJ AK huˆvˆ = uv MIAJKu v MIK u . (3.10) (cid:18) MIK v MIJ (cid:19) LΛ and fΛ are the upper components of the symplectic sections of standard N = 2 super- ¯ı gravity, while Im ΛΣ and Re ΛΣ are the imaginary and real parts of the period matrix N N ΛΣ of the special geometry; IAα and ∆Iαβ are the remnants of the quaternionic vielbein N U and symplectic connection in the dualized directions. (ζα) ζ are the (anti)-chiral hyper- α inos, (λ¯ı ) λiA are the (anti)-chiral gauginos and (ψA) ψ are the (anti)-chiral gravitinos. A A Finally, ǫAB is the totally antisymmetric tensor andCαβ is the charge conjugation matrix. In order for the theory to be supersymmetric, as a consequence of the Ward identity of supersymmetry, the electric and magnetic charges must satisfy the constraint eImJΛ eJmIΛ = 0, (3.11) Λ Λ − which coincides withequation (2.3) inthecaseD = 4d. Lagrangian (3.1) isinvariant under thegaugetransformations parametrized bya1–form fieldΩ already describedin(2.2). As I explained in the previous section, we can add to the Lagrangian a topological term which does not modify the equations of motion but provides a more compact expression for the terms of the type . F ∧F The invariance of the action is now explicit, due to the fact that the fields B and Λ I F appear only through either ˆΛ or , which are invariant under (2.2) by construction, I F H since: δ ˆΛ = dδAΛ +mIΛδB = mIΛΩ +mIΛΩ = 0 (3.12) I I I F − δ = dδB = ddΩ = 0 (3.13) I I I H The effect of the gauge fixing of equation (2.6) is the replacement of ˆΛ with mIΛB . I F Collecting all the fermionic bilinears, it is possible to rewrite the Lagrangian in a more compact expression: T,V = 1 IJ ∗ + AI + SI 1 m2 IJ B ∗B + LD=4 −4M HI ∧ HJ HI ∧ HI ∧ − 2 I ∧ J 1 (e˜m)IJ B B +B TI +ρI ( d(cid:0)B )(cid:1) (3.14) I J I I I −2 ∧ ∧ ∧ H − where m2 IJ is defined as in (2.8) with the matrix Im instead of δ. Notice that Im is N N negative definite, and hence the sign of the term proportional to m2 IJ is different from (cid:0) (cid:1) that in (2.10). We have also introduced the following quantity: (cid:0) (cid:1) e˜IΛ = eIΛ Re ΛΣmIΣ (3.15) − N and (e˜m)IJ is defined as in (2.9) replacing eI with e˜I. The TI are the Pauli terms arising Λ Λ from the coupling of the fermions to ˆ, while SI is the remnant of the quaternionic Pauli F terms coupled to the differential dqI of the axionic coordinates which have been dualized. – 7 – IJ Notice that (e˜m) is symmetric in the indices I,J because of the tadpole cancellation condition and of the symmetry of the real part of the period matrix ΛΣ. N From Lagrangian (3.14) it is easy to write down the equations of motion for the fields I H and B : I δ 1 L = 0 = IJ∗ = SI +AI ρI (3.16) J δ ⇒ 2M H − I H δL = 0 = m2 IJ ∗B +(e˜m)IJ B = TI dρI (3.17) J J δB ⇒ − I (cid:0) (cid:1) and their Hodge dual: 1 IJ = ∗ SI +AI ρI (3.18) J 2M H − m2 IJ B +((cid:0)e˜m)IJ ∗B = ∗(cid:1)dρI ∗TI (3.19) J J − equations (3.16) and (3.18(cid:0)) ca(cid:1)n be immediately inverted to obtain the expression for I H (respectively ∗ ) in terms of ∗ρI (ρI) while the usual linear combination of equations I H (3.17) and (3.19) gives the expression for B (∗B ) in terms of dρI and ∗dρI: I I = 2 ∗ SJ +AJ ρJ (3.20) I IJ H M − ∗ I = 2 IJ (cid:0)SJ +AJ ρJ (cid:1) (3.21) H M − J J B = e˜2+m(cid:0)2 −1 ∗TJ (cid:1) e˜ TK ∗dρJ e˜ dρK (3.22) I IJ − m − − m " (cid:18) (cid:19)K ! (cid:18) (cid:19)K !# (cid:0) (cid:1) J J ∗B = e˜2+m2 −1 dρJ + e˜ ∗dρK TJ + e˜ ∗TK (3.23) I IJ m − m " (cid:18) (cid:19)K ! (cid:18) (cid:19)K !# (cid:0) (cid:1) Notice that there are again some sign differences with equations (2.13), (2.14) due to the new definition of m2 IJ, e˜2 IJ and e˜ I with Im instead of δ. By substituting (3.22– m J N 3.21) in (3.14), one gets the dualized Lagrangian: (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) = ∗AI AJ + ∗ρI ρJ 2 ∗AI ρJ + Dual IJ IJ IJ L M ∧ M ∧ − M ∧ +2 (cid:0)IJ ∗AI +∗ρI SJ + IJ∗SI SJ + (cid:1) M ∧ M ∧ K 1 e˜2+(cid:0)m2 −1d∗ρ(cid:1)I dρJ + 1 e˜2+m2 −1 e˜ dρI dρJ + −2 IJ ∧ 2 IK m ∧ (cid:18) (cid:19)J (cid:0) (cid:1) (cid:0) (cid:1) K + e˜2+m2 −1∗TI dρJ e˜2+m2 −1 e˜ TI dρJ + IJ ∧ − IK m ∧ (cid:18) (cid:19)J (cid:0) (cid:1) (cid:0) (cid:1) K 1 e˜2+m2 −1∗TI TJ + 1 e˜2+m2 −1 e˜ TI TJ (3.24) −2 IJ ∧ 2 IK m ∧ (cid:18) (cid:19)J (cid:0) (cid:1) (cid:0) (cid:1) It should be noticed that, in the absence of Fayet-Iliopoulos terms (eI = mIΛ = 0), the Λ equations of motion for would have given the standard N = 2 Lagrangian if dqI = ρI, I H − – 8 – where the minus sign is due to the opposite sign of the Lagrange multiplier term in the standard Lagrangian, when dualizing qI (see reference [5]) with respect to that in (3.14). ∇ This Lagrangian describes a system made up by n massive vectors, coupled to scalars (AI) and fermions (TI). Together with all the undualized terms of N = 2 D = 4 Lagrangian of [6], it describes a supergravity theory with massive vectors, which can not be identified with a standard N = 2 D = 4 supergravity, due both to the presence of a number of massive vectors, and to the fact that the vector- and hyper- multiplets are not written in the standard way. In order to recover a more usual Lagrangian, we make use of the Stu¨ckelberg mechanism, which allows oneto rewritethemass terms as kinetic terms for anumberof scalars coupled tovectors,withagaugeinvarianceinordertokeepthecorrectnumberofdegreesoffreedom fixed. The Stu¨ckelberg mechanism works as follows: at first it has to be noted that the kinetic term for the vectors is invariant under gauge transformations: δρI = dΩI (3.25) where ΩI is a 0–form gauge parameter. On the contrary the mass terms, in which the bare field ρI is present, are not invariant under (3.25). In order to extend this invariance to the whole Lagrangian, we can think of the mass terms as beingalready gauge-fixed, as in equation (2.6), sothat it is possibleto recover the gauge invariance introducing a new 0–form field φI together with a massless vector Λ, whose A gauge transformations leave ρI invariant. Summarizing we can write ρI as: ρI kI Λ+dφI (3.26) Λ − ≡ A where now all is invariant under the following gauge transformations: Λ Λ δ = dΩ (3.27) A δφI = kIΩΛ (3.28) Λ − since the kI are constant. The r.h.s. of equation (3.26) can also be interpreted as the Λ covariant derivative of the scalar field φI, where the gauge fields are the vectors Λ, and A kI have the role of Killing vectors. Λ After this redefinition, the Lagrangian reads: = kin + kin + Pauli+ Pauli+ 4f + 4f (3.29) L Lscal Lvect Lscal Lvect Linv Lnoninv where: kin = ∗AI AJ +∗ φI φJ +2∗AI φJ (3.30) scal IJ L M ∧ ∇ ∧∇ ∧∇ Pscaaulli = 2 IJ(cid:0)∗AI SJ +2 IJ∗ φI SJ (cid:1) (3.31) L M ∧ M ∇ ∧ 4f = ∗SI SJ (3.32) Linv MIJ ∧ – 9 –