The three form multiplet in N=2 superspace

Transcription

1 The three form multiplet in N=2 superspace arxiv:hep-th/ v1 13 Jun 1996 Maximilian F. Hasler Centre de Physique Théorique, C.N.R.S. Luminy, case 907, F Marseille Abstract We present an N = 2 multiplet including a three index antisymmetric tensor gauge potential, and describe it as a solution to the Bianchi identities for the associated fieldstrength superform, subject to some covariant constraints, in extended superspace. We find that this solution is given in terms of an hypermultiplet (in the antisymmetric tensor representation where the pseudoscalar auxiliary field is replaced by the fieldstrength of the three form potential. The analysis is performed in flat superspace, considering a supersymmetry algebra including a central charge, which is, however, trivially represented on the multiplet. We give the transformation laws for the multiplet as well as invariant superfield and component field lagrangians, and mention possible couplings to other multiplets. anonymous ftp or gopher: cpt.univ-mrs.fr Keywords: Extended supersymmetry, superspace gauge theories, supermultiplets. June 1996 CPT 96/P.3349 hep th/ allocataire M. E. S. R.; E mail: hasler@cpt.univ-mrs.fr

2 Introduction Although N = 2 supersymmetric theories are not yet directly related to the observable world (at least in what concerns particle physics, they started playing an important role in modern theoretical physics in recent years. This is in particular due to the fact that they yield the first exactly solvable models in four dimensional field theories, in spite of the rich physics they display, including also non perturbative phenomena [13]. In this paper, we discuss an N = 2 supermultiplet including a three index antisymmetric tensor gauge potential. We suppose this multiplet to play a crucial role in various scenarios related to supersymmetry breaking, somehow in analogy to the case in N = 1: For instance a gaugino condensate is associated, from a geometrical point of view, to the derivative dq = tr F 2 of the Yang Mills Chern Simons form Q = tr(af 1 3 A3, which is just a special case of a three form potential. Similarily, curvature squared terms, which seem to be involved in another recently discussed supersymmetry breaking mechanism [10], are associated in the same way to the gravitational Chern Simons forms. For N = 1, the three form multiplet has been known for some time [6], and recently a rather complete description of its couplings to the N = 1 supergravity matter system has been given [1]. However, to our knowledge, the corresponding N = 2 multiplet has not been studied yet. It is the purpose of the present paper to fill this gap: First, we discuss the field content and the supersymmetry transformations on the level of component fields, and give an invariant lagrangian density for the multiplet. Then, we turn to a geometric description of the multiplet as fieldstrength of a three form gauge potential in extended superspace. We find that this solution is given in terms of an hypermultiplet ( linear superfield subject to an additional constraint 1 similar to the case in N = 1. We also comment on the geometrical interpretation of the previously defined supersymmetry transformations. Next, we give an invariant kinetic lagrangian density, and comment on dynamics of the multiplet. Finally, we allude to possible couplings to other multiplets and N = 2 supergravity. Our spinor notations are those of Wess and Bagger [16], and concerning the internal structure group of N = 2 superspace, we adopted the conventions of [12]: In particular, we raise and lower internal SU(2 indices from the left by the antisymmetric tensors ǫ BA and ǫ BA with ǫ 12 = ǫ 21 = 1. 1 Another extra constrained hypermultiplet seems to be known, but it is apparently impossible to write an action for this multiplet [4]. 1

3 The three form multiplet The set of fields that we will call the three form multiplet Σ in the sequel consists of a three index gauge potential C lmn (x, another two index antisymmetric tensor S mn (x, an isotriplet of real scalar fields which we write as traceless, hermitean 2 2 matrix Z B A(x, as well as an isodoublet of Weyl spinors ζ A α (x, ζ α A(x and a real scalar auxiliary field H(x: The field tensors Σ ( C lmn, S mn, Z B A; ζ A α, ζ α A H. (1 Z m = 1 2 εmnkl n S kl, Σ = m C m (2 (where C klm = ε klmn C n etc. are invariant under the gauge transformations δs mn = m β n + n β m, δ C m = 1 2 εmnkl n γ kl. (3 We define the following supersymmetry transformations of spinorial parameters ξa α and ξ α Ȧ for the multiplet: δs mn = 2 ( ξ A σ mn ζ A + ξ A σ mn ζa δc lmn = 2 ε lmnk ( ξa σ k ζa + ξ A σ k ζ A δz BA = BA ( ξ B ζ A + ξ B ζa (4 δζ A α = ξ A α ( i 2 m C m H + ( i m Z A B δ A B Z m (σ m ξ B α δ ζ α A = ξ α A( i 2 m C m + H + ( i m Z B A + δ B A Z m ( σ m ξ α B δh = i ξ A ζ A i ξ A ζa One can verify that these transformations close on the multiplet: The commutator [δ 2, δ 1 ] of two supersymmetry transformations of parameters ξ i yields a spacetime translation of parameter ξ m = 2i ( ξ 2A σ m ξ A 1 + ξ A 2 σm ξ 1A (5 on all fields of the multiplet, plus gauge transformations with field dependent parameters for the potentials, explicitely and β m = ξ n S nm + 2i ( ξ 2A σ m ξb 1 + ξ B 2 σ mξ 1A Z A B (6 γ mn = ξ l C lmn + 4 ( ξ 2A ξ A 1 ξ A 2 ξ 1A Smn + 8 ( ξ 2A σ mn ξ B 1 ξ B 2 σ mn ξ1a Z A B. (7 The geometric interpretation of this will be discussed later on in this paper. 2

4 An invariant kinetic action for this multiplet is provided by the lagrangian density L kin = 1 2 H2 + 1 ( C 8 m m Z m Z 2 m (8 1 8 m Z A B m Z B A i 2 ζa ζ A. (Note that we did not introduce the normalization factors necessary to get the canonical kinetic terms for C m and Z B A in order to avoid many factors of 2. It should be noted here that C lmn does not propagate physical degrees of freedom. On shell, the above action describes the same 4+4 physical states as the usual hypermultiplet; the 4 physical spin 0 degrees of freedom given by the triplet of scalars and the antisymmetric tensor. Thus, in analogy to the case of the N = 1 three form multiplet [6], the pseudoscalar auxiliary field of the usual matter multiplet is replaced by the fieldstrength of the three form potential. Actually, the multiplet can be decomposed into one N = 1 three form multiplet and one N = 1 antisymmetric tensor multiplet [5]: This is done by writing ( L T Z B A, L = L (9 T L and splitting up the N = 2 multiplet into the two N = 1 multiplets Σ ( L, S mn ; Λ α, Λ α + ( T, T, C lmn ; η α, η α H. (10 We shall reconsider this issue on the level of superfields after the discussion of the superspace Bianchi identities in the following section. 3

5 Superspace geometry and Bianchi Identities We find that the previously introduced multiplet can be described as the components of the fieldstrength tensor Σ = dc of a three form potential in extended N = 2 superspace. We include an additional bosonic coordinate z, z which allows for the description of a supersymmetry algebra including a central charge [14]. Thus, we consider torsion T A = de A (A a, α A, Ȧ α, z, z with the following nonzero elements: and, as usual, T CB z γ β = 2i ǫ CB ǫ γβ, T γ β z CB = 2i ǫ CB ǫ γ β, (11 T C β a γ B = 2i δ C B (σa γ β. (12 By Poincaré s lemma dd = 0, they define the commutators ( C, B = T CB A A. (13 However, the superfields of the multiplet described here, as well as the gauge parameters, are taken independent of z, z, z C CBA = 0, z C CBA = 0, (14 i.e., the central charge is acting trivially on the multiplet. Furthermore, we impose the covariant constraints Σ δγβa = 0, Σ δγ z z = 0, ( δ δ, δ (15 and Σz c β B A α = 4 σ cβ α Z B A, Σ z c β B A α = 4 σ c β α Z B A (16 on the fieldstrength 2. Then the Bianchi identities dσ = 0, or E A E B E C E E E( E Σ CBA + 2 T E F Σ FCBA = 0, (17 give all other elements of Σ CBA in terms of Z B A and its derivatives: First, the identities with indices z z imply Σ z z BA = 0. (18 (These identities are just the Bianchi identities for a Yang Mills fieldstrength F BA in ordinary N = 2 superspace without the z coordinates [7], with the Yang Mills superfields occuring in F βα set to zero. 2 The last one should be considered as reality condition on Z B A. An antihermitean part of Z B A would just enlarge the field content of the multiplet by another 8+8 hypermultiplet, which would not be subject to the additional constraint we shall find below. 4

6 Next, the identities with one index z correspond to the Bianchi identities dz = 0 for Z CBA i 2 Σ z CBA (19 such that Z = ds for some 2 form S, which is readily identified 3 as These identities imply and give with and finally S BA = i 2 C z BA. (20 Z A A = 0, (C γ ZBA = 0 = γ (C Z BA, (21 Z A cb α = 2 (σ cb ζ A α, Z α cba = 2 ( σ cb ζ α A, (22 ζ A α = 1 3 B α Z A B, ζ α A = 1 3 α BZ B A, (23 Z cba = 1 8 ε dcba (σ d α α ( A α ζ α A α A ζa α. (24 Note that the reduced identities (21 already imply that ε dcba d Z cba = 0. Then, the other components of dσ = 0 determine the components of Σ without z, z indices to be and Σ BA dc β α = 8 (σ dc βα Z BA, Σ β α dc BA = 8 ( σ dc β α Z BA, (25 Σdcb A α = 2 ε dcba (σ a ζ A α, Σdcb A α = 2 ε dcba ( σ a ζ A α, (26 where we chose to adopt the additional constraint Σdc B β A α = 0 which turns out to be conventional, i.e. just corresponds to a shift of the potential C lmn by the trace of this fieldstrength component. Finally, the vectorial fieldstrength is given by Σ dcba = ε dcba Σ, Σ = i 4 ( α A ζ A α + Ȧ α ζ α A. (27 On the other hand, one infers of course from the explicit definition of Σ = dc that Σ = 1 6 εdcba d C cba, (28 therefore the above result should be seen as an additional constraint on the superfield Z B A, requiring the imaginary part of its highest component to be a total derivative, ( α B α A + Ȧ α α ( B Z B A = d 2i ε dcba C cba. (29 3 to be precise, from the reality condition (16 follows that Σ z CBA = Σ z CBA, such that S BA = i 2 C z BA has the same fieldstrength Z and therefore describes the same physical object. (Roughly speaking, S BA and S BA just differ by a gauge transformation. 5

7 This concludes the analysis of the Bianchi identities; no other restrictions on Z B A are found. Note that we actually used here the same symbols for the superfields than for their lowest components which are precisely the fields presented in the preceding section. For convenience, we summarize their definitions here once again, denoting the projection on θ = 0 as usual by a vertical bar: Z B A(x = Z B A, S mn (x = i 2 C z mn, (30 ζ A α (x = 1 3 B α ZA B, ζ α A (x = 1 3 α B ZB A, (31 Z m (x = 1 2 εmnkl n S kl (x = 1 24 (σm α α ( α B A α A α α B Z B A, (32 Σ(x = 1 6 εklmn k C lmn (x = i ( α 12 B α A + Ȧ α α B Z B A, (33 and there remains just one real scalar auxiliary field, defined as H(x = 1 ( A 24 α α B α B A α Z B A. (34 Observe that the additional constraint (29 is analogous to the case in N = 1: There, the 3 form multiplet also corresponds to (anti chiral superfields satisfying the additional constraint α T = 0, α T = 0, (35 α α T α α T ε dcba d C cba. (36 Considering again the previously given decomposition of Z B A we find the chirality constraints (35 on T from the constraints (21, α (CZBA = 0, by setting all indices to one, while the linearity constraints on L and the additional constraint on T, T are most easily recovered from the relations B β ζ A α = ǫ BA ǫ βα (H + i 2 Σ, β B ζ α A = ǫ BA ǫ β α (H i 2 Σ, (37 again by setting the internal index B of the spinorial derivative to one. To conclude this section, we now turn back to the previously introduced supersymmetry transformations. We define them to be superspace diffeomorphisms of parameter ξ A such that the vielbeins E A remain invariant, i.e., L ξ E A (ι ξ + d 2 E A = 0 B ξ A = ξ C T CB A. (38 6

8 This restricts the parameters to be x m and z, z independent, and the θ and θ components of ξ a and ξ z to be given in terms of ξa α and ξ α Ȧ. We combine these diffeomorphisms with a field dependent gauge transformation δ γ C = dγ with γ = ι ξ C (39 in order to obtain the very simple transformation laws δc = ι ξ Σ, δσ CBA = L ξ Σ CBA. (40 The relevant components of these formulae (or rather their values for θ = 0 were given in the previous section. (Note that we take the lowest component of ξ m and ξ z to vanish when we are interested in the supersymmetry transformations. One can show that the commutator [δ 2, δ 1 ] of two such transformations yields again a transformation of the same type, but of parameter plus a gauge transformation of parameter ξ A 21 = ι ξ 2 ι ξ1 T A ξ B 2 ξc 1 T CB A (41 γ 21 = ι ξ2 ι ξ1 Σ, (γ 21 BA = ξ C 2 ξ 1 Σ CBA. (42 isentangeling furtherly, we can describe this combined transformation also as a plain diffeomorphism of parameter ξ 21 plus a gauge transformation of parameter γ 21 + ι ξ21 C, which then yields the formulae given in the first section. A final remark concerns the piece in these transformations that stems from the z, z component of the parameter ξ: It actually has the form of something one could view as a kind of relic central charge transformation, acting nontrivially only on the three form potential, which becomes shifted by the antisymmetric tensor s fieldstrength, δ z C cba = Z cba, = 2i ( ξ z ξ z. (43 However, according to the definition of Z = ds, this also is nothing else than a gauge transformation, namely of parameter γ = S. 7

9 Superfield lagrangians and couplings The previously given kinetic action for the three form multiplet can actually be obtained as the θ 4 component of a chiral superfield lagrangian as follows: First, we consider an explicit solution to the constraints on Σ, namely C zba = i 2 S BA, C zba = i 2 S BA, with S BA β α = 4 ǫ BA ǫ βα S, S β α BA = 4 ǫ BA ǫ β α S, S B α β A = 0. (44 Here, S and S are (anti chiral superfields, in terms of which the fieldstrength multiplet is then given by Z B A = 1 4 ( ϕ A B ϕ S Ḃ ϕ ϕ AS. (45 (Note that S and S must derive from the same real prepotential, such that the imaginary part of their highest component is indeed a total derivative C m m. Then, the previously given kinetic action can be written as L kin = Re d 4 θ SZ, Z = 1 8 A B Z B A. (46 (Here the chiral volume element is normalized such that d 4 θ θ 4 = 1. ue to the constraints (21 on Z B A, this superfield Z is actually a vector superfield, and the above lagrangian is indeed invariant under the gauge transformations of S [11]. We also want to mention that, given this explicit solution, one can add a mass term L mass = m 2 d 4 θ S 2 + h.c. (47 which breaks the gauge invariance. This yields a theory for a massive multiplet (cf. also [15] with twice the number of physical fields: L mass includes an explicit mass term for a irac spinor made of ζ A α and m A αs, and for Z B A and the antisymmetric tensor S mn. Moreover, an additional triplet of bosons, X B A m( A B S + B A S, appears as auxiliary fields. Finally, the diagonalization of the fields H and Σ yields a mass term for ms. However, this is a slightly delicate point: In complete analogy with the case in N = 1, one cannot really eliminate Σ = m C m directly, as it is not really an auxiliary field. Moreover, its equation of motion only requires Σ to be a constant, but not to vanish. As one can see from the supersymmetry transformations, a nonvanishing constant would give rise to an inhomogeneous transformation law of the spinor field, which seems to indicate broken supersymmetry. For a more detailed discussion of these issues, see for example [9]. As the 3 form gauge multiplet is a special case of a hypermultiplet in the antisymmetric tensor representation, it allows for self couplings and couplings to other multiplets in the same way than they do. Notably one could consider the 8

10 analogue of the action for the improved tensor multiplet [3], which can be coupled to a general N = 2 supergravity background [2]. For this multiplet, one also can add a mass term involving only the invariant fieldstrength multiplet. This should be of particular interest in the supergravity coupled case. On the other hand, another subject of special interest related to the three form multiplet are of course couplings to other gauge multiplets that can be formulated on geometric grounds. In analogy to N = 1 [8, 9] we consider, for example, a modified field tensor Σ = dc + κ H A (48 where H = db is the fieldstrength of a two form potential and A is a U(1 gauge potential. This leads to the modified Bianchi Identity dσ = κ H F, F = da, (49 which yields in particular a modified fieldstrength Σ, including the spinorial superpartners of A m and B mn. This leads to dynamics similar to the case in N = 1, notably quartic potential terms for the spinor fields, that might be of interest in scenarios such as gaugino condensation and related issues. These topics, as well as the coupling of this multiplet to N = 2 supergravity, are still to be worked out in detail and will be discussed elsewhere. Acknowledgements I should like to thank R. Grimm for introducing me to supersymmetry and for useful suggestions during the course of this work. 9

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