hep-th/0412190 December, 2004 On massive tensor multiplets Sergei M. Kuzenko School of Physics, The University of Western Australia, 5 35 Stirling Highway, Crawley W.A. 6009, Australia 0 [email protected] 0 2 n a J 2 Abstract 1 2 Massive tensor multiplets have recently been scrutinized in hep-th/0410051 and v hep-th/0410149, as they appear in orientifold compactifications of type IIB string 0 9 theory. Here we formulate several dually equivalent models for massive = 1, 2 N 1 tensor multiplets in four space-time dimensions. In the = 2 case, we employ 2 N harmonic and projective superspace techniques. 1 4 0 / h t - p e h : v i X r a 1 Introduction Recently, therehasbeen renewed interest in4D = 1 massive tensor multiplets andtheir N couplings to scalar and vector multiplets [1, 2]. Such interest is primarily motivated by the fact that massive two-forms naturally appear in four-dimensional = 2 supergravity N theories obtained from (or related to) compactifications oftype IIstring theory on Calabi- Yau threefolds in the presence of both electric and magnetic fluxes [3, 4]. This clearly provides enough ground for undertaking a more detailed study of massive = 1 and N = 2 tensor multiplets. N In = 1 supersymmetry, the massive tensor multiplet (as a dual version of the N massive vector multiplet) was introduced twenty five years ago [5], and since then this construction1 has been reviewed in two textbooks [8, 9]. In the original formulation [5], the mass parameter, m2, in the action 1 1 S = d8zG2 m2 d6zψαψ +c.c. (1.1) tensor α −2 Z − 2 (cid:26) Z (cid:27) was chosen to be real. Here 1 G = (Dαψ +D¯ ψ¯α˙) , D¯ ψ = 0 , (1.2) α α˙ α˙ α 2 where the dynamical variable ψ is an arbitrary chiral spinor superfield, and the (massless α gauge) field strength G is a real linear superfield, D2G = D¯2G = 0. The choice of a real mass parameter seemed to be natural in the sense that, for M = m, the above system is dual to the massive vector multiplet model2 1 1 1 S = d6zWαW + M2 d8zV2 , W = D¯2D V , V = V¯ , (1.3) vector α α α 4 Z 2 Z −4 which involves an intrinsically real mass parameter. The mass parameter is also intrinsi- cally real in the vector-tensor realization [5] (inspired by [11]) 1 1 S = d8zG2 + d6zWαW +M d8zGV v−t α −2 Z 4 Z Z 1 1 1 = d8zG2 + d6zWαW M d6zWαψ +c.c. , (1.4) α α −2 Z 4 Z − 2 (cid:26)Z (cid:27) 1The work of [5] is actually more famous for the massless tensor multiplet (as a dual version of the chiral scalar multiplet) introduced in it, see also [6, 7]. 2This duality is a supersymmetric version of the duality between massive one- and two-forms [10]. 1 which describes the same multiplet on the mass shell (massive superspin-1/2), and which possesses both the tensor multiplet gauge freedom i δψ = D¯2D K , K = K¯ (1.5) α α 4 and the vector multiplet gauge freedom δV = Λ+Λ¯ , D¯ Λ = 0 . (1.6) α˙ It was recently pointed out [1, 2], however, that giving the mass parameter in (1.1) an imaginary part,3 m2 m(m+ie) , (1.7) −→ leads to nontrivial physical implications, for the mass can now be interpreted to have both electric and magnetic contributions which are associated with the two possible mass terms B ∗B and B B for the component two-form. The dual vector multiplet is ∧ ∧ then characterized by the mass M = √m2 +e2. What makes the replacement (1.7) really interesting is that the complex mass parameter can be interpreted as a vacuum expectation value for chiral scalars [12], m(m+ie) d6zψαψ d6zF(φ)ψαψ , (1.8) α α Z ←− Z with φ some chiral scalars, D¯ φ = 0. α˙ In the massive case, the gauge freedom (1.5) is broken if one does not use the vector- tensor formulation (1.4). It can be restored, however, if one implements the standard Stueckelberg formalism, as was done in [1, 2], and replaces the naked chiral prepotential ψ everywhere by α i ψ ψ + W , (1.9) α α α −→ m where the compensating vector multiplet has to transform as δV = mK under (1.5). The gauge symmetry thus obtained can be treated as a deformation of the transformations (1.5) and (1.6). In this note we continue the research started in [1, 2] and provide further insight into the structure of massive tensor multiplets. In section 2 we consider aspects of = 1 N tensor multiplets in curved superspace and introduce a model for the massive improved 3Unlikethescalarmultiplet,thisimaginarypartcannotbeeliminatedbyarigidphasetransformation of ψ as long as the explicit form of the linear scalar G, in terms of ψ and its conjugate, is fixed. α α 2 tensor multiplet. Unlike the ordinary tensor multiplet [5], the (massless) improved ten- sor multipet [13] is superconformal in global supersymmetry and super Weyl invariant in curved superspace. There are at least two reasons why the improved tensor multiplet is interesting: (i) it describes the superconformal compensator in the new minimal formu- lation of = 1 supergravity, see [8, 9] for reviews; (ii) it corresponds to the Goldstone N multiplet for partial breaking of = 1 superconformal symmetry associated with the N coset SU(2,2 1)/(SO(4,1) U(1)) which has AdS as the bosonic subspace [14, 15]. As 5 | × we demonstrate below, a remarkable feature of the improved tensor multiplet is that its super Weyl invariance remains intact in the massive case. In section 3 we introduce several realizations for the massive = 2 tensor multiplet, N describe its duality to the massive = 2 vector multiplet, and also sketch possible self- N couplings and couplings to vector multiplets. Section 4 is devoted to the description of the reduction of manifestly = 2 supersymmetric actions to = 1 superspace. The list N N of = 2 superspace integrations measures is given in the appendix. N Our = 1 supergravity conventions correspond to [9]. They are very similar to those N adopted in [16]. The conversion from [9] to [16] is as follows: E−1 E and R 2R. → → = 1 2 tensor multiplets N It is known that 4D = 1 new minimal supergravity can be treated as a super Weyl N invariant dynamical system describing the coupling of old minimal supergravity to a real covariantly linear scalar superfield L constrained by (¯2 4R)L = ( 2 4R¯)L = 0, see D − D − [8, 9] for reviews.4 The new minimal supergravity action 3 S = d8zE−1LlnL , E = Ber(E M) (2.1) SG,new κ2 Z A is invariant with respect to the super Weyl transformation5 [17] eσ/2−σ¯ ( βσ)M , ¯ eσ¯/2−σ ¯ (¯β˙σ¯)M¯ (2.2) Dα → Dα − D αβ Dα˙ → Dα˙ − D β˙α˙ (cid:16) (cid:17) (cid:16) (cid:17) 4In old minimal supergravity, the superspace covariant derivatives are = ( , , ¯α˙) = A a α D D D D E M(z)∂ + Ω βγ(z)M +Ω β˙γ˙(z)M¯ , with M and M¯ the Lorentz generators. They obey A M A βγ A β˙γ˙ βγ β˙γ˙ the (modified) Wess-Zumino constraints, and the latter imply that the torsion and the curvature are expressed in terms of a vector G = G¯ and covariantly chiral objects R and W subject to some a a αβγ additional Bianchi identities. 5Under (2.2),the full superspacemeasure changesasd8zE−1 d8zE−1 exp(σ+σ¯), while the chiral → superspace measure transforms as d8zE−1/R d8z(E−1/R) exp(3σ). → 3 accompanied with L e−σ−σ¯ L . (2.3) → where σ(z) is an arbitrary covariantly chiral scalar parameter, ¯ σ = 0. The super Weyl α˙ D transformation of L is uniquely fixed by the constraint (2.6). The dynamical system (2.1) is classically equivalent to old minimal supergravity described by the action 3 S = d8zE−1 . (2.4) SG,old −κ2 Z Modulo sign, the functional (2.1) coincides with the action for the so-called improved tensor multiplet [13] S = µ d8zE−1Gln(G/µ) , (2.5) − Z with G obeying the same constraint as L above, (¯2 4R)G = ( 2 4R¯)G = 0 . (2.6) D − D − In the family of tensor multiplet models [5] of the form S = µ2 d8zE−1 (G/µ) , (2.7) Z F the action (2.5) is singled out by the requirement of super Weyl invariance. In particular, the free massless tensor multiplet action 1 S = d8zE−1G2 (2.8) −2 Z is not super Weyl invariant. Upon reduction to flat superspace, the action (2.5) becomes superconformal. As is well known, the general solution of (2.6) is 1 G = ( αψ + ¯ ψ¯α˙) , ¯ ψ = 0 , (2.9) α α˙ α˙ α 2 D D D with an arbitrary covariantly chiral spinor superfield ψ . The super Weyl transformation α of the prepotential ψ turns out to be uniquely fixed [9]: α G e−σ−σ¯G = ψ e−3σ/2ψ . (2.10) α α → ⇒ → Asaresult, addingamasstermtotheaction(2.5)doesnotspoilitssuperWeylinvariance. That is, the action 1 E−1 S[ψ,ψ¯] = µ d8zE−1Gln(G/µ) m (m+ie) d8z ψ2 +c.c. (2.11) − Z − 2 (cid:26) Z R (cid:27) 4 is invariant under arbitrary super Weyl transformations. The latter property uniquely singles out this model in the family of actions 1 E−1 S = µ2 d8zE−1 (G/µ) m (m+ie) d8z ψ2 +c.c. . (2.12) Z F − 2 (cid:26) Z R (cid:27) Therefore, the action (2.11) defines the massive improved tensor multiplet. This is a nontrivial theory, unlike the massless improved tensor multiplet that is known to be free. Upon reduction to flat superspace, the action turns into a superconformal model. Let us consider a dual formulation for the theory introduced in (2.11). We follow the procedure given in [5, 8, 9] and first relax the linear constraints (¯2 4R)G = D − ( 2 4R¯)G = 0 by introducing the “first-order” model D − 1 S = µ d8zE−1G ln(G/µ) 1 +M d8zE−1V G ( αψ + ¯ ψ¯α˙) auxiliary α α˙ − Z (cid:16) − (cid:17) Z (cid:16) − 2 D D (cid:17) 1 E−1 m (m+ie) d8z ψ2 +c.c. , (2.13) −2 (cid:26) Z R (cid:27) with M2 = m2 +e2 . (2.14) Here both scalars G and V are real unconstrained, and G is not related to ψ and its α conjugate. Topreserve thesuper Weylinvariance, however, V shouldtransformasfollows: µ V V (σ +σ¯) . (2.15) → − M Varying S with respect to V brings us back to (2.11). On the other hand, varying auxiliary S with respect to G and ψ allows one to express these variables in terms of V and auxiliary α the vector multiplet strength 1 W = (¯2 4R) V , ¯ W = 0 , αW = ¯ Wα˙ . (2.16) α α α˙ α α α˙ −4 D − D D D D One ends up with 1 E−1 M S[V] = d8z W2 +µ2 d8zE−1 exp V . (2.17) 4 Z R Z (cid:16) µ (cid:17) This actionis invariant under the super Weyl transformations (2.2) and (2.15). It is worth pointing out that the inhomogeneous piece on the right of (2.15) does not show up in the transformation of W : α W e−3σ/2W . (2.18) α α → The super Weyl transformations of the chiral spinors ψ and W are clearly identical. α α 5 Employing the Stueckelberg formalism, the action (2.17) can be replaced by the classically-equivalent one 1 E−1 S[V,Φ,Φ¯] = d8z W2 +µ2 d8zE−1Φ¯ e(M/µ)VΦ , (2.19) 4 Z R Z with Φ a compensating chiral scalar possessing a non-vanishing v.e.v. This action is invariant under the U(1) gauge transformation µ ¯ ¯ δV = Λ+Λ , δΦ = ΛΦ , Λ = 0 . (2.20) α˙ −M D The super Weyl transformation (2.15) turns into V V , Φ e−σΦ . (2.21) → → The model (2.17), or its equivalent realization (2.19), describes the dynamics of a massive improved vector multiplet in curved superspace. The mass term in (2.11) breaks the massless gauge symmetry [5] i δψ = (¯2 4R) K , K = K¯ (2.22) α α 4 D − D that leaves the field strength (2.9) invariant. Nevertheless, inspired by [11], one can preserve the gauge symmetry in the massive case by considering the following vector- tensor model 1 E−1 S[ψ,ψ¯,V] = µ d8zE−1Gln(G/µ)+ d8z W2 +M d8zE−1GV . (2.23) − Z 4 Z R Z This action possesses both the tensor multiplet (2.22) and vector multiplet (2.20) gauge symmetries. This action can also be seen to be super Weyl invariant provided V is chosen, say, to be inert under such transformations. By inspecting the equations of motion, one can check that the theory (2.23) is classically equivalent to (2.11) if M is chosen as in (2.14). One can also establish the duality of (2.23) to the improved vector multiplet (2.19) by dualizing the linear superfield G into a chiral scalar and its conjugate according to [5, 7]. Inthemassive case, followingStueckelberg, thegaugeinvariance(2.22)canberestored by introducing a compensating Abelian vector multiplet (with the gauge field and the V chiral field strength ) and replacing ψ in (2.12) by α α W i 1 ψ ψ + , = (¯2 4R) , = ¯ . (2.24) α α α α α −→ mW W −4 D − D V V V 6 Here transforms as δ = mK under (2.22) such that the combination mψ +i is α α V V W gauge invariant. The modified mass term remains to be super Weyl invariant. Since E−1 Im d8z 2 = 0 , Z R W we then obtain E−1 i 2 E−1 im d8z ψ + +c.c. = i d8z mψ2 +2iψ +c.c. (2.25) Z R (cid:16) mW(cid:17) Z R (cid:16) W(cid:17) = 2 3 tensor multiplets N To generalize the previous consideration to the case of = 2 supersymmetry, it is ad- N vantageous (in some respect, necessary) to make use of the = 2 harmonic superspace N R4|8 S2 [18, 19]. It extends conventional = 2 superspace R4|8 (paramerized by coor- × N dinates Z = (xa,θα,θ¯i), with i = ˆ1,ˆ2) by the two-sphere S2 = SU(2)/U(1) parametrized i α˙ by harmonics, i.e., group elements (u −, u +) SU(2) , u+ = ε u+j , u+i = u− , u+iu− = 1 . (3.1) i i ∈ i ij i i Forsimplicity, our consideration will berestricted to the study of globallysupersymmetric theories only. Let us start by recalling the model for a free massive = 2 vector multiplet [18, 19]. N Its dynamical variable V++(Z,u) is a real analytic superfield, D+V++ = D¯+V++ = 0, α α˙ where the harmonic-dependent spinor covariant derivatives D± and D¯± are defined in eq. α α˙ (A.5). The action6 is 1 1 S = d8ZW2 M2 dζ(−4)(V++)2 vector 2 Z − 2 Z 1 V++(u)V++(u′) 1 = d12Zdudu′ M2 dζ(−4)(V++)2 , (3.2) 2 Z (u+u′+)2 − 2 Z see [19] for the definition of harmonic distributions of the form(u+u+)−n, where (u+u+) = 1 2 1 2 u+iu+ . HereW(Z)isthe(harmonicindependent)chiralfieldstrengthofthe = 2vector 1 2i N multiplet [20], D¯iW = 0 , DαiDjW = D¯iD¯jα˙W¯ , (3.3) α˙ α α˙ 6The various =2 superspace integration measures are defined in the Appendix. N 7 which is expressed via the analytic prepotential V++(Z,u) as follows [19, 21]: 1 1 W(Z) = du(D¯−)2V++(Z,u) = (D¯+)2V−−(Z,u) , (3.4) 4 Z 4 V++(Z,u′) V−−(Z,u) = du′ . Z (u+u′+)2 The equation of motion is 1 (D+)2W M2V++ = 0 , (3.5) 4 − where one should keep in mind that the Bianchi identity is equivalent to (D+)2W = (D¯+)2W¯ . This equation implies that V++ is an = 2 linear superfield: N D++V++ = 0 V++(Z,u) = V(ij)(Z)u+u+ , (3.6) −→ i j where Vij obeys the constaints D(iVjk) = D¯(iVjk) = 0 D+V++ = D¯+V++ = 0 , (3.7) α α˙ ←− α α˙ as a consequence of the analyticity of the dynamical variable. It is now easy to arrive at (2 M2)V++ = 0 (2 M2)W = 0 . (3.8) − −→ − In the massless case, M = 0, the action (3.2) is invariant under the gauge transforma- tion [18, 19] δV++ = D++λ , (3.9) with the gauge parameter λ(Z,u) a real analytic superfield, D+λ = D¯+λ = 0. This α α˙ transformation leaves the field strength (3.4) invariant. Let us now turn to the massless = 2 tensor multiplet [22] formulated in harmonic N superspace in [23, 19]. The free action is 1 S = dζ(−4)(G++)2 , (3.10) 2 Z where G++(Z,u) is a restricted real analytic superfield under the constraints (3.6) and (3.7). One can can express G++(Z,u) = Gij(Z)u+u+ in terms of an unconstrained chiral i j superfield Ψ(Z) and its conjugate: 1 1 G++(Z,u) = (D+)2Ψ(Z)+ (D¯+)2Ψ¯(Z) , D¯iΨ = 0 . (3.11) 8 8 α˙ This superfield remains invariant under the gauge transformation δΨ = iΛ , D¯iΛ = 0 , DαiDjΛ = D¯iD¯jα˙Λ¯ . (3.12) α˙ α α˙ 8 As is seen, the chiral gauge parameter Λ satisfies the same constraints as the vector multiplet field strength. Recalling the construction of [11], the massive tensor (or vector) multiplet can be described by the action 1 1 S = dζ(−4)(G++)2 + d8ZW2 +M dζ(−4)G++V++ (3.13) v−t 2 Z 2 Z Z 1 1 1 = dζ(−4)(G++)2 + d8ZW2 + M d8ZWΨ+c.c. , (3.14) 2 Z 2 Z 2 (cid:26)Z (cid:27) which is invariant under the gauge transformations (3.9) and (3.12). The corresponding equations of motion are 1 1 (D¯−)2G++ +M W = 0 , (D+)2W +MG++ = 0 , (3.15) 4 4 as well as the complex conjugate of the first equation. Of primary importance for us will be another massive extension of (3.10) 1 1 S = dζ(−4)(G++)2 m (m+ie) d8ZΨ2 +c.c. . (3.16) tensor 2 Z − 4 n Z o This action generates the following equations of motion 1 1 (D¯−)2G++ m(m+ie)Ψ = 0 , (D−)2G++ m(m ie)Ψ¯ = 0 , (3.17) 4 − 4 − − which imply (2 M2)G++ = 0 , M = √m2 +e2 . (3.18) − The dynamical systems (3.2) and (3.16) turn out to be dual to each other provided M is chosen as above. The corresponding “first-order” action, which establishes the duality between these theories, is 1 1 S = dζ(−4)(G++)2 + M dζ(−4)V++ 8G++ (D+)2Ψ (D¯+)2Ψ¯ auxiliary 2 Z 8 Z (cid:16) − − (cid:17) 1 m (m+ie) d8ZΨ2 +c.c. , (3.19) − 4 n Z o where bothrealanalyticsuperfields V++ andG++ areunconstrained. VaryingV++ brings us back to (3.16). On the other hand, varying S with respect to G++ and Ψ and auxiliary using the equations obtained to eliminate these superfields, we end up with (3.2). 9