Niccolò Cribiori, Gianguido Dall'Agata
We revisit the construction of N=2 superconformal multiplets using rheonomic superspace techniques. We apply the result to the derivation of off-shell Poincaré supersymmetric models where a tensor multiplet couples to gravity and to an arbitrary number of vector multiplets. We also analyze gaugings involving the tensor field.
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We construct the dilaton Weyl multiplet for $N=2$ conformal supergravity in four dimensions. Beginning from an on-shell vector multiplet coupled to the standard Weyl multiplet, the equations of motion can be used to eliminate the supergravity auxiliary fields, following a similar pattern as in five and six dimensions. The resulting 24+24 component multiplet includes two gauge vectors and a gauge two-form and provides a variant formulation of $N=2$ conformal supergravity. We also show how this dilaton Weyl multiplet is contained in the minimal 32+32 Poincare supergravity multiplet introduced by…
We construct the complete coupling of $(2,0)$ supergravity in six dimensions to $n$ tensor multiplets, extending previous results to all orders in the fermi fields. The truncation to $(1,0)$ supergravity coupled to tensor multiplets exactly reproduces the complete couplings recently obtained.
We discuss the coupling of vector-tensor multiplets to N=2 supergravity.
Starting from the 48+48 component multiplet of supercurrents for a rigid N=2 tensor multiplet in four spacetime dimensions, we obtain the transformation of the linearized supergravity multiplet which couples to this supercurrent multiplet. At the linearized level, this 48+48 component supergravity multiplet decouples into the 24+24 component linearized standard Weyl multiplet and a 24+24 component irreducible matter multiplet containing a real scalar field. By a consistent application of the supersymmetry algebra with field dependent structure constants appropriate to N=2 conformal supergravit…
We review some results on the complete coupling between tensor and vector multiplets in six-dimensional $(1,0)$ supergravity.
We will use the covariant superform approach to develop a new density formula for $\mathcal{N}=2$ conformal supergravity which is based on a fermionic multiplet whose lowest component is a dimension-5/2 spinor. We will show that this density formula admits an embedding of the real scalar multiplet of [arXiv:1712.02309]. Upon using the embedding of the tensor multiplet into the real scalar multiplet, we will construct a new higher derivative action of the tensor multiplet in $\mathcal{N}=2$ conformal supergravity.
We present some multiplets of N=2 off-shell supergravity in five dimensions. One is the Super Yang-Mills multiplet, another one is the linear multiplet. The latter one is used to establish a general action formula from which we derive an action for the Super Yang-Mills multiplet. The Super Yang-Mills multiplet is used to construct the nonlinear multiplet with gauged SU(2). This nonlinear multiplet and the action formula for the Yang-Mills multiplet enable us to write down an SU(2) gauged supergravity which we finally truncate to arrive at gauged supergravity with gauge group SO(2).
Extending prior investigations, we study three of the the four distinct minimal (4,0) scalar multiplets coupled to (4,0) supergravity. It is found that the scalar multiplets manifest their differences at the component level by possessing totally different couplings to the supergravity fields. Only the SM-I multiplet possesses a conformal coupling. For the remaining multiplets, terms linear in the world sheet curvature and/or SU(2) gauge field strengths are required to appear in the action by local supersymmetry.
We construct the supercurrent multiplet that contains the energy-momentum tensor of the (2,0) tensor multiplet. By coupling this multiplet of currents to the fields of conformal supergravity, we first construct the linearized superconformal transformations rules of the (2,0) Weyl multiplet. Next, we construct the full non-linear transformation rules by gauging the superconformal algebra OSp(8^*|4). We then use this result to construct the full equations of motion of the tensor multiplet in a conformal supergravity background. Coupling N+5 copies of the tensor multiplet to conformal supergravit…
Some general features of locally supersymmetric theories (N=1 in four dimensions) involving Chern-Simons forms and antisymmetric tensors are sketched out. The relevance of the three-form multiplet both for the description of Chern-Simons forms and the supersymmetry properties of the gaugino condensate is pointed out.
We propose new interactions of a (massive) vector multiplet with chiral multiplets and (D-type) spontaneously broken supersymmetry in four-dimensional N=1 supergravity, due to the generalized Fayet-Iliopoulos (FI) terms. Our actions are invariant under linearly realized off-shell supersymmetry and Kaehler-Weyl transformations. We compute the scalar potentials and pinpoint some physical restrictions arising in this approach.
We present new off-shell formulations for the massive superspin-3/2 multiplet. In the massless limit, they reduce respectively to the old minimal (n=-1/3) and non-minimal ($n\neq -1/3, 0$) linearized formulations for 4D N=1 supergravity. Duality transformations, which relate the models constructed, are derived.
The lagrangian of the N=1, D=10 dual supergravity interacting with the Yang-Mills matter multiplet is constructed starting immediately from the equations of motion obtained from the Bianchi Identities in the superspace approach. The difference is established in comparison with the Gates-Nishino lagrangian at the fourth order level in fermionic fields.
The recent papers arXiv:1110.0971 and arXiv:1201.5431 have provided a superfield description for vector-tensor multiplets and their Chern-Simons couplings in 4D N = 2 conformal supergravity. Here we develop a superform formulation for these theories. Furthermore an alternative means of gauging the central charge is given, making use of a deformed vector multiplet, which may be thought of as a variant vector-tensor multiplet. Its Chern-Simons couplings to additional vector multiplets are also constructed. This multiplet together with its Chern-Simons couplings are new results not considered by …
We construct the Weyl multiplets of N=2 conformal supergravity in five dimensions. We show that there exist two different versions of the Weyl multiplet, which contain the same gauge fields but differ in the matter field content: the Standard Weyl multiplet and the Dilaton Weyl multiplet. At the linearized level we obtain the transformation rules for the Dilaton Weyl multiplet by coupling it to the multiplet of currents corresponding to an on-shell vector multiplet. We construct the full non-linear transformation rules for both multiplets by gauging the D=5 superconformal algebra F^2(4). We sh…
Recent advances in curved N=2 superspace methods have rendered the component reduction of superspace actions more feasible than in the past. In this paper, we consider models involving both vector and tensor multiplets coupled to supergravity and demonstrate explicitly how component actions may be efficiently obtained. In addition, tensor multiplets coupled to conformal supergravity are considered directly within projective superspace, where their formulation is most natural. We then demonstrate how the inverse procedure -- the lifting of component results to superspace -- can simplify the ana…
We describe in detail the derivation of a superconformal off-shell formulation of the alternative $\mathcal{N}=2$, $d=5$ ungauged supergravity of Nishino and Rajpoot, coupled to $n$ Abelian vector multiplets, using a general dilaton-Weyl multiplet. We generalize the vector multiplet coupling available in the literature and show under which assumptions that the scalar manifold reduces to the known case of SO(1,1) $\times$ SO(1,n)/SO(n). As an application of the formalism we propose generalized vector multiplet coupled higher curvature terms, whose construction we sketch briefly.
We extend the notion of central charge superspace to the case of local supersymmetry. Gauged central charge transformations are identified as diffeomorphisms at the same footing as space-time diffeomorphisms and local supersymmetry transformations. Given the general structure we then proceed to the description of a particular vector-tensor supergravity multiplet of 24+24 components, identified by means of rather radical constraints.
Variant dilaton Weyl Multiplet for N=3 conformal supergravity in four dimensions
本文通过耦合两个在壳矢量多重态并利用其场方程求解辅助场，构造了四维N=3共形超引力的新伸缩子Weyl多重态，其R对称性为SU(2)×U(1)×U(1)。
We couple in superspace a `dual' vector multiplet (C_{m_1... m_7}, ł^α) to the dual version of N=1 supergravity (e_m{}^a, ψ_m{}^α, M_{m_1... m_6}, χ_\a,Φ) in ten-dimensions. Our new 7-form field C has its 8-form field strength H dual to the 2-form field strength F of the conventional vector multiplet. We have found that the H-Bianchi identity must have the form N\wedge F, where N is the 7-form field strength in dual supergravity. We also see why only the dual version of supergravity couples to the dual vector multiplet consistently. The potential anomaly for the dual vector multiplet can be ca…
Supergravity with mimetic dark matter
本文提出了一种结合超对称引力与模仿暗物质的理论模型，通过约束chiral多重体和拉格朗日乘子多重体实现超对称性自发破缺，产生引力子、重引力子和两个标量场。
We consider the minimal off-shell formulation for four-dimensional N=2 supergravity with a cosmological term, in which the second compensator is an improved tensor multiplet. We use it to derive a linearized supergravity action (and its dual versions) around the anti-de Sitter (AdS) background in terms of three N=2 off-shell multiplets: an unconstrained scalar superfield, vector and tensor multiplets. This allows us to deduce the structure of the supercurrent multiplet associated with those supersymmetric theories which naturally couple to the supergravity formulation chosen, with or without a…
We couple a recently-established N=1 globally supersymmetric self-dual Yang-Mills multiplet in three dimensions to supergravity. This becomes possible due to our previous result on globally supersymmetric formulation based on a compensator multiplet. We further couple the self-dual vector to a supersymmetric sigma-model on the coset SO(8,n) / SO(8) X SO(n) via minimal couplings for an arbitrary gauged subgroup H_0 \subset SO(8) X SO(n). A corresponding superspace formulation is also presented.
We study the vacuum structures of four-dimensional $\mathcal{N} = 1$ old minimal supergravity with higher derivative corrections. We find that $\mathcal{N} = 1$ supergravity with Riemann curvature square corrections and higher derivative matter chiral multiplets induces a non-trivial de Sitter vacuum, even in the absence of superpotentials. This vacuum generically breaks supersymmetry. We show that the auxiliary fields in the gravity and the chiral multiples play important roles to generate a potential in supersymmetric higher derivative theories.
We construct the general four-dimensional N=2 supergravity theory coupled to vector and vector-tensor multiplets only. Consistency of the construction requires the introduction of the vector fields dual to those sitting in the same supermultiplets as the antisymmetric tensors, as well as the scalar fields dual to the tensors themselves. Gauge symmetries also involving these additional fields guarantee the correct counting of the physical degrees of freedom.
We consider consistent truncations of N=2 supergravites in the presence of tensor multiplets (dual to hypermultiplets) as they occur in type IIB compactifications on Calabi--Yau orientifolds. We analyze in detail the scalar potentials encompassing these reductions when fluxes are turned on and study vacua of the N=1 phases.
The recent paper arXiv:1205.6881 has developed superform formulations for two versions of the vector-tensor multiplet and their Chern-Simons couplings in four-dimensional N = 2 conformal supergravity. One of them is the standard vector-tensor multiplet with the central charge gauged by a vector multiplet. The other is the variant vector-tensor multiplet with the property that its own one-form gauges the central charge. Here a more general setup is presented in which the known versions reside as special cases. Analysis of the setup demonstrates that under certain assumptions there are two disti…
$SU(2)\times SU(2)$ dilaton Weyl multiplets for maximal conformal supergravity in four, five, and six dimensions
本文分别在四维、五维为$N=4$和$N=2$共形超引力构造了具$SU(2)\times SU(2)$ R对称的新dilaton Weyl多重态（旧为$USp(4)$），并首次在六维为$(2,0)$共形超引力构造同R对称的dilaton Weyl多重态。
We study, in an arbitrary even number D of dimensions, the duality between massive D/2 tensors coupled to vectors, with masses given by an arbitrary number of ``electric'' and ``magnetic'' charges, and (D/2-1) massive tensors. We develop a formalism to dualize the Lagrangian of D=4, N=2 supergravity coupled to tensor and vector multiplets, and show that, after the dualization, it is equivalent to a standard D=4, N=2 gauged supergravity in which the Special Geometry quantities have been acted on by a suitable symplectic rotation.
We present the action for a self-dual tensor in six dimensions, coupled to a (2,0) conformal supergravity background. This action gives rise to the expected equations of motion. An alternative look upon one of the gauge symmetries clarifies its role in the supersymmetry transformation rules and the realisation of the algebra.
We consider N=4 SU(2)xU(1) gauged supergravity on asymptotically-AdS_5 backgrounds. By a near-boundary analysis we determine the boundary-dominant components of the bulk fields from their partially gauge-fixed field equations. Subdominant components are projected out in the boundary limit and we find a reduced set of boundary fields, constituting the N=2 Weyl multiplet. The residual bulk symmetries are found to act on the boundary fields as four-dimensional diffeomorphisms, N=2 supersymmetry and (super-)Weyl transformations. This shows that the on-shell N=4 supergravity multiplet yields the N=…
We present the description in central charge superspace of N=4 supergravity with antisymmetric tensor coupled to an arbitrary number of abelian vector multiplets. All the gauge vectors of the coupled system are treated on the same footing as gauge fields corresponding to translations along additional bosonic coordinates. It is the geometry of the antisymmetric tensor which singles out which combinations of these vectors belong to the supergravity multiplet and which are the additional coupled ones. Moreover, basic properties of Chapline-Manton coupling mechanism, as well as the SO(6,n)/SO(6)*S…
We derive the couplings of the 3-form supermultiplet to the general supergravity-matter-Yang-Mills system. Based on the methods of superspace geometry, we identify component fields, establish their supergravity transformations and construct invariant component field actions. Two specific applications are adressed: the appearance of fundamental 3-forms in the context of strong-weak duality and the use of the 3-form supermultiplets to describe effective degrees of freedom relevant to the mechanism of gaugino condensation.
We construct the complete coupling of $(1,0)$ supergravity in six dimensions to $n$ tensor multiplets, extending previous results to all orders in the fermi fields. We then add couplings to vector multiplets, as dictated by the generalized Green-Schwarz mechanism. The resulting theory embodies factorized gauge and supersymmetry anomalies, to be disposed of by fermion loops, and is determined by corresponding Wess-Zumino consistency conditions, aside from a quartic coupling for the gaugini. The supersymmetry algebra contains a corresponding extension that plays a crucial role for the consistenc…
Dilaton Weyl multiplets for $N = 3$ conformal supergravity in four dimensions
本文构造了四维 N=3 共形超引力的胀子 Weyl 多重态，通过耦合在壳矢量多重态并利用其场方程替换标准 Weyl 多重态的部分辅助场，最终得到 SU(2)×U(1) R 对称性的结果。
We propose a four-dimensional N = 1 supergravity-based Starobinsky-type inflationary model in terms of a single massive vector multiplet, whose action includes the Dirac-Born-Infeld-type kinetic terms and a generalized (new) Fayet-Iliopolulos-type term without gauging the R-symmetry. The bosonic action and the scalar potential are computed. Inflaton is the superpartner of goldstino in our model, and supersymmetry is spontaneously broken after inflation by the D-type mechanism, whose scale is related to the value of the cosmological constant.
We construct a 5D, N = 2 Euclidean theory of supergravity coupled to vector multiplets. Upon reducing this theory over a circle we recover the action of 4D, N = 2 Euclidean supergravity coupled to vector multiplets.
Higher derivative couplings with multi-tensor multiplets in 6D supergravity, action and anomalies
针对 nT>1 时无协变作用量的六维 (1,0) 超引力多张量多重态与 Yang-Mills 耦合系统，该研究用 Henneaux-Teitelboim 方法构造含规范反常的作用量，并给出适用于任意物质内容的超对称 Green-Schwarz 引力反常抵消项。
Projective superspace provides a natural framework for the construction of actions coupling hypermultiplets to conformal supergravity. We review how the off-shell actions are formulated in superspace and then discuss how to eliminate the infinite number of auxiliary fields to produce an on-shell N=2 supersymmetric sigma model, with the target space corresponding to a generic 4n-dimensional hyperkahler cone. We show how the component action coupling the hypermultiplets to conformal supergravity may be constructed starting from curved superspace. The superspace origin of the geometric data -- th…
We address the problem of classifying all N=2 supercurrent multiplets in four space-time dimensions. For this purpose we consider the minimal formulation of N=2 Poincare supergravity with a tensor compensator, and derive its linearized action in terms of three N=2 off-shell multiplets: an unconstrained scalar superfield, a vector multiplet, and a tensor multiplet. Such an action was ruled out to exist in the past. Using the action constructed, one can derive other models for linearized N=2 supergravity by applying N=2 superfield duality transformations. The action depends parametrically on a c…
Using the duality between color and kinematics, we construct the two- and three-loop amplitudes of half-maximal supergravity with matter multiplets and show that new divergences occur in D=4 and D=5. Bossard, Howe and Stelle have recently conjectured the existence of 16-supercharge off-shell harmonic superspaces in order to explain the ultraviolet finiteness of pure half-maximal supergravity with no matter multiplets in D=4 at three loops and in D=5 at two loops. By assuming the required superspace exists in D=5, they argued that no new divergences should occur at two loops even with the addit…
We use the superconformal method to construct a new formulation for pure off-shell D=5, N=2 Poincaré supergravity and present its internal gauging. The main difference between the traditional formulation and our new formulation is the choice of the Dilaton Weyl Multiplet as the background Weyl Multiplet and the choice of a Linear compensating Multiplet. We do not introduce an external vector multiplet to gauge the theory, but instead use the internal vector of the Dilaton Weyl Multiplet. We show that the corresponding on-shell theory is Einstein-Maxwell supergravity. We believe that this gaugi…
Using the off-shell formulation for general N = 2 supergravity-matter systems developed in arXiv:0905.0063, we propose a construction to generate a restricted chiral superfield from any real weight-zero projective multiplet L. One can choose L to be composed of tensor multiplets, O(2n) multiplets, with n=2,3,..., and polar hypermultiplets. In conjunction with the standard procedure to induce a N = 2 linear multiplet from any chiral weight-one scalar, we obtain a powerful mechanism to generate higher-derivative couplings in supergravity. In the case that L is a homogeneous function of n tensor …
We analyze various supersymmetry multiplets containing the supercurrent and the energy-momentum tensor. The most widely known such multiplet, the Ferrara-Zumino (FZ) multiplet, is not always well-defined. This can happen once Fayet-Iliopoulos (FI) terms are present or when the Kahler form of the target space is not exact. We present a new multiplet S which always exists. This understanding of the supersymmetry current allows us to obtain new results about the possible IR behavior of supersymmetric theories. Next, we discuss the coupling of rigid supersymmetric theories to supergravity. When th…
We propose a nilpotent ${\cal N}=1$ tensor multiplet describing two fields, which are the Goldstino and the axion, the latter being realised in terms of the field strength of a gauge two-form. This supersymmetric multiplet is formulated in terms of a deformed real linear superfield, subject to a cubic nilpotency condition. Its couplings to a super Yang-Mills multiplet and supergravity are presented. To define a nilpotent tensor multiplet in the locally supersymmetric case, one has to make use of either real or complex three-form supergravity theories, which are variant realisations of the old …
We describe the couplings of six-dimensional supergravity, which contain a self-dual tensor multiplet, to $n_T$ anti-self-dual tensor matter multiplets, $n_V$ vector multiplets and $n_H$ hypermultiplets. The scalar fields of the tensor multiplets form a coset $SO(n_T,1)/SO(n_T)$, while the scalars in the hypermultiplets form quaternionic Kähler symmetric spaces, the generic example being $Sp(n_H,1)/Sp(n_H)\otimes Sp(1)$. The gauging of the compact subgroup $Sp(n_H) \times Sp(1)$ is also described. These results generalize previous ones in the literature on matter couplings of $N=1$ supergravit…
Within the framework of six-dimensional ${\cal N}=(1,0)$ conformal supergravity, we introduce new off-shell multiplets ${\cal O}{}^{*}(n)$, where $n=3,4,\dots,$ and use them to construct higher-rank extensions of the linear multiplet action. The ${\cal O}{}^{*}(n)$ multiplets may be viewed as being dual to well-known superconformal ${\cal O}(n)$ multiplets. We provide prepotential formulations for the ${\cal O}(n)$ and ${\cal O}{}^{*}(n)$ multiplets coupled to conformal supergravity. For every ${\cal O}{}^{*}(n)$ multiplet, we construct a higher derivative invariant which is superconformal on …
We systematically investigate the finite set of possible gauge groups and matter content for N = 1 supergravity theories in six dimensions with no tensor multiplets, focusing on nonabelian gauge groups which are a product of SU(N) factors. We identify a number of models which obey all known low-energy consistency conditions, but which have no known string theory realization. Many of these models contain novel matter representations, suggesting possible new string theory constructions. Many of the most exotic matter structures arise in models which precisely saturate the gravitational anomaly b…
Euclidean special geometry has recently been investigated in the context of Euclidean supersymmetric theories with vector multiplets. In the rigid case, the scalar manifold is described by affine special para-Kahler geometry while the target geometries of Euclidean vector multiplets coupled to supergravity are given by projective special para-Kahler manifolds. In this letter, we derive the Killing spinor equations of Euclidean N=2 supergravity theories coupled to vector multiplets. These equations provide the starting point for finding general supersymmetric instanton solutions.
We study off-shell N-extended Yang-Mills multiplets coupled to conformal supergravity in three spacetime dimensions. Superform formulations are presented for the non-Abelian Chern-Simons actions in the cases N=1, 2, 3, and the corresponding component actions are explicitly worked out. Such a Chern-Simons action does not exist for N=4. In the latter case, a superform formulation is given for the BF term that describes the coupling of two Abelian vector multiplets with self-dual and anti-self-dual superfield strengths respectively. The superform results obtained are used to construct linear mult…
${\mathcal N}=3$ nonlinear multiplet and supergravity
本文提出 ${\mathcal N}=3$ 非线性多重态并构造 $\mathcal N=3$ Einstein 超空间，由此给出庞加莱超引力运动方程，证明其解均为共形超引力解。
In N = 2 Poincare supersymmetry in four space-time dimensions, there exist off-shell supermultiplets with intrinsic central charge, including the important examples of the Fayet-Sohnius hypermultiplet, the linear and the nonlinear vector-tensor (VT) multiplets. One can also define similar supermultiplets in the context of N = 2 anti-de Sitter (AdS) supersymmetry, although the origin of the central charge becomes somewhat obscure. In this paper we develop a general setting for N = 2 AdS supersymmetric theories with central charge. We formulate a supersymmetric action principle in N = 2 AdS supe…
We propose the Starobinsky-type inflationary model in the matter-coupled $N=1$ four-dimensional supergravity with the massive vector multiplet that has inflaton (scalaron) and goldstino amongst its field components, whose action includes the Dirac-Born-Infeld-type kinetic term and the generalized (new) Fayet-Iliopoulos-type term, without gauging the R-symmetry. The $N=1$ chiral matter ("hidden sector") is described by the modified Polonyi model needed for spontaneous supersymmetry breaking after inflation. We compute the bosonic action and the scalar potential of the model, and show that it ca…
In a classical conformal invariant supersymmetric gauge theory, the chiral R-symmetry current, the supersymmetry current and the energy-momentum tensor constitute a supercurrent multiplet. There are two different superconformal anoamly multiplets in four-dimensional supersymmetric gauge theories, one originating from the supersymmetric gauge dynamics and the consequent nonvanishing β-function, and the other one coming from the coupling of the supercurrent multiplet to the external supergravity multiplet with non-trivial topology. We emphasize that in the gauge/gravity dual correspondence these…
Supersymmetric truncation of N=3 dilaton Weyl multiplet
本文研究了N=3稀尔顿韦伊尔多重体的超对称截断，发现通过不同方法可得到N=2共形超引力的32+32外现多重体，并证明其与庞加莱超引力多重体的等价性。
We derive the full N=2 supergravity Lagrangian which contains a symplectic invariant scalar potential in terms of electric and magnetic charges. As shown in reference [1], the appearance of magnetic charges is allowed only if tensor multiplets are present and a suitable Fayet-Iliopoulos term is included in the fermion transformation laws. We generalize the procedure in the quoted reference by adding further a Fayet-Iliopoulos term which allows the introduction of electric charges in such a way that the potential and the equations of motion of the theory are symplectic invariant. The theory is …
Torus reduction of maximal conformal supergravity
该研究将六维N=(2,0)共形超引力经二维环面维数约化得到四维N=4共形超引力，首次在四维极大超对称情形下推导出dilaton Weyl多重态，并给出六维与四维场之间的非线性关系。
As an extension of the recent construction of generalised Fayet-Iliopoulos terms in ${\cal N}=1$ supergravity given in [1], we present self-interactions for a vector multiplet coupled to conformal supergravity. They are used to construct new models for spontaneously broken local supersymmetry.
In this paper we derive the most general curvature squared action coupled to an arbitrary number of vector multiplets in four dimensional $N=2$ supergravity using the dilaton Weyl multiplet. The action that we derive is encoded in a single holomorphic prepotential.
We present the {\it canonical} set of superspace constraints for self-dual supergravity, a ``self-dual'' tensor multiplet and a self-dual Yang-Mills multiplet with $~N=1~$ supersymmetry in the space-time with signature $(+,+,-,-)$. For this set of constraints, the consistency of the self-duality conditions on these multiplets with supersymmetry is manifest. The energy-momentum tensors of all the self-dual ``matter'' multiplets vanish, to be consistent with the self-duality of the Riemann tensor. In particular, the special significance of the ``self-dual'' tensor multiplet is noted. This result…
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