September, 2004 OCU-PHYS 213 hep-th/0409060 Supersymmetric U(N) Gauge Model and Partial Breaking of = 2 Supersymmetry N 5 0 0 2 n K. Fujiwaraa∗ , H. Itoyamaa† and M. Sakaguchib‡ a J 0 2 3 v 0 a Department of Mathematics and Physics, Graduate School of Science 6 0 Osaka City University 9 0 b Osaka City University Advanced Mathematical Institute (OCAMI) 4 0 / h 3-3-138, Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan t - p e h : v i X r Abstract a Guided by the gauging of U(N) isometry associated with the special Ka¨hler ge- ometry, and the discrete R symmetry, we construct the = 2 supersymmetric action N of a U(N) invariant nonabelian gauge model in which rigid = 2 supersymmetry N is spontaneously broken to = 1. This generalizes the abelian model considered by N Antoniadis, Partouche and Taylor. We shed light on complexity of the supercurrents of our model associated with a broken = 2 supermultiplet of currents, and discuss N thespontaneously brokensupersymmetryas an approximate fermionicshiftsymmetry. ∗e-mail: [email protected] †e-mail: [email protected] ‡e-mail: [email protected] I. Introduction Continuing investigations have been made for more than two decades on supersymmetric field theories, hoping to obtain realistic description of nature by broken = 1 supersym- ∗ N metry at an observable energy scale. On the other hand, it is most natural to view that physics beyond this energy scale is controlled by string theory, which, without nontoroidal backgrounds, produces extended supersymmetries in four dimensions. Breaking of extended supersymmetries in this vein provides a bridge between gauge field theory and string theory. String theory does not possess genuine coupling constants: instead, they are the vacuum expectation values of some supersymmetry preserving moduli fields. We are thus led to search for the possibility of spontaneous partial breaking of extended supersymmetries in four dimensions. In the context of = 2 supergravity [4], spontaneous breaking of local = 2 super- N N symmetry to its = 1 counterpart has been accomplished by the simultaneous realization N of the Higgs and the super Higgs mechanisms. Sizable amount of literature has been ac- cumulated till today along this direction [5, 6, 7]. There have been active researches car- ried out on nonlinear realization of extended supersymmetries in the partially broken phase [8, 9, 10, 11, 12, 13]. These are closely related to the effective description of string theory [14], brane dynamics [15, 16, 17, 18, 19, 20] and domain walls [21]. After [8, 9] and prior to the remainder of the works on nonlinear realization, there was a work within the linear realization done by Antoniadis, Partouche and Taylor [22] who constructed an = 2 supersymmetric, self-interacting U(1) model with one (or several) N abelian = 2 vector multiplet(s) [23] which breaks = 2 supersymmetry down to = 1 N N N spontaneously. See also [24, 25]. The partial breaking of supersymmetry is accomplished by the simultaneous presence of the electric and magnetic Fayet-Iliopoulos terms, which is a generalization of [26]. In the present paper, generalizing the work of [22], we construct the = 2 supersymmetric action of a U(N) invariant nonabelian gauge model in which rigid N = 2 supersymmetry is spontaneously broken to = 1. The gauging of U(N) isometry N N associated with the special K¨ahler geometry, and the discrete R symmetry are the primary ingredients of our construction. Let usrecallthat partialbreaking ofextended rigidsupersymmetries appearsnot possible on the basis of the positivity of the supersymmetry charge algebra: Q¯i ,Q = 2(1) δi H. (1.1) α jα˙ αα˙ j (cid:8) (cid:9) In fact, if Q 0 = 0, one concludes H 0 = 0 and Q 0 = 0 for all i. If Q 0 = 0, then 1 i 1 | i | i | i | i 6 H 0 = E 0 with E > 0 and Q 0 = 0 for all i. The loophole to this argument is that the i | i | i | i 6 ∗See [1, 2, 3] to review 2 use of the local version of the charge algebra is more appropriate in spontaneously broken symmetries and the most general supercurrent algebra is Q¯j, m(x) = 2(σn) δj Tm(x)+(σm) Cj, (1.2) α˙ Sαi αα˙ i n αα˙ i (cid:8) (cid:9) where m and Tm are the supercurrents and the energy momentum tensor respectively. We Sαi n have denoted by Cj a field independent constant matrix permitted by the constraints from i the Jacobi identity [27]. This last term does not modify the supersymmetry algebra acting on the fields. The abelian model of [22] and our nonabelian generalization provide a concrete example of this local algebra within linear realization from the point of view of the action principle. The Lagrangian of our model has noncanonical kinetic terms coming from the nontrivial K¨ahler potential and does not fall into the class of renormalizable Lagrangians. As a model with spontaneously broken = 2 supersymmetry, the prepotential is present from the N F beginning of our construction. This is in contrast with breaking = 2 to = 1 by the N N operator (superpotential) W(Φ), where appears aposteriori according to the recent devel- F opments beginning with Dijkgraaf and Vafa [28]. The model has a U(1) sector interacting with an SU(N) sector and the spontaneously broken supersymmetry acts as an approximate fermionic shift symmetry. Piecing through all these properties, we conclude that the action of the model should be regarded as a low energy effective action which applies to various processes and that the dynamical effects including those of (fractional) instantons are to be contained inthe prepotential as aninput. This input should besupplied by a separate means of calculation. The connection with the exact determination of the prepotential via [29, 30] and from integrable systems [31] [32] offers a new avenue of thoughts with this regard. In section II, we provide the construction of the = 2 supersymmetric action of the N U(N) invariant nonabelian gauge model which is equipped with the Fayet-Iliopoulos D term and a specific superpotential. Gauging of the noncanonical kinetic terms coming from the K¨ahler potential is a necessary step to complete the action. In section III, we provide the transformation law of the extended supersymmetries associated with the model. We note that the SU(2) automorphism of = 2 supersymmetry has been fixed in the parameter N space. In section IV, we fix the form of the prepotential and determine the vacuum with unbroken gaugesymmetry. Weexhibit partialbreaking of = 2supersymmetry anddiscuss N a mechanism which enables this. In section V, we examine a broken = 2 supermultiplet N of currents [33] associated with the model. The U(1) current is not conserved except R for the case where the prepotential has an R-weight two. Despite this, we show that the broken = 2 supermultiplet of currents provides a useful means to construct the extended N supercurrents. We shed light upon their complexity. In section VI, we discuss a role played by the spontaneously broken supersymmetry. We see that it acts as a approximate U(1) fermionic shift symmetry in the limit of letting the magnetic Fayet-Iliopoulos term large 3 relative to the electric one. Our discussion in section two and that in section three leading to = 2 supersymmetric Lagrangian exploit an algebraic operation denoted by R. This N operation is defined by including the sign flip of the Fayet-Iliopoulos parameter ξ ξ → − into the standard discrete canonical transformation R. It is a legitimate algebraic process to use R to demonstrate the second supersymmetry and in section three we obtain = 2 N supersymmetry transformation by demanding the covariance under R. In Appendix A, we give a more pedagogical proof of = 2 supersymmetry of our action, using the canonical N R. The two approaches are thus shown to be equivalent. In Appendix B, we reexamine the = 1 current supermultiplet [34] in the Wess-Zumino model. N II. = 2 U(N) Gauge Model N Letusfirststateourstrategytoobtainthe = 2supersymmetric actionwithnonabelian N U(N) gauge symmetry. We adopt the = 1 superspace formalism to write down a U(N) N invariant action consisting of a set of = 1 U(N) chiral superfields and vector superfields N in the adjoint representation. The action at this level is equipped with the terms required for the gauging, the Fayet-Iliopoulos D term, and a generic superpotential. Imposing the discrete element of SU(2) automorphism of = 2 supersymmety algebra as symmetry of N our action [2, 22], we obtain the action mentioned in the introduction. What is meant by this last procedure is, however, a little more subtle than one might first thinkandwepause toexplainthishere inmoredetail. Inthepresence oftheFayet-Iliopoulos D term with its coefficient ξ, = 1 Lagrangian is in general not invariant under the discrete N R symmetry. (See (2.39)). Best one can do is therefore to consider simultaneously an inversion of the parameter ξ. (See (2.49)). Under this extended operation denoted by R, we will find R : , R : . (2.1) ′ ′ L → L L → L (See (2.26), (2.33).) Combining this with the algebra Rδ R 1 = δ , (2.2) 1 − 2 we conclude that our final actions (2.33) and (2.64) with (2.45) and (2.48) are invariant under = 2 supersymmetry. Here we denote by δ and δ , the transformation of the first 1 2 N supersymmetry and that of the second supersymmetry respectively. This definition R turns out to be consistent with an interpretation that full rigid SU(2) symmetry has been fixed in the parameter space. This is discussed in section III. 4 A. U(N) Gauge Model Let us introduce a set of = 1 chiral superfields N N2 1 − Φ(xm,θ) = Φat . (2.3) a a=0 X Here, t , a = 0,1,...,(N2 1), are N N hermitian matrices which generate u(N) algebra, a − × and t , aˆ = 1,...,(N2 1), generate su(N) algebra aˆ − [t ,t ] = ifcˆt . (2.4) aˆ ˆb aˆˆb cˆ The index 0 refers to the overall u(1) generator. The scalar fields A = Aat in Φ undergo a the adjoint action A UAU , (2.5) † → under U(N). The kinetic term for A is generated by = d2θd2θ¯K(Φa,Φ a), (2.6) K ∗ L Z where K(Aa,A a) is the K¨ahler potential. The K¨ahler potential we employ is given by ∗ i K(Aa,A a) = (Aa A a ), (2.7) ∗ 2 Fa∗ − ∗ Fa where = ∂ = d and is an analytic function of A.⋆ The K¨ahler potential can be Fa aF dAaF F written using a hermitian metric on the bundle compatible with the symplectic structure as I i 0 K = Ω Ω¯ , Ω Ω¯ = ΩT Ω . (2.8) −2 | | − I 0 ! ∗ − (cid:10) (cid:11) (cid:10) (cid:11) The K¨ahler metric gab∗ = ∂a∂b∗K = Im ab (2.9) F constructed this way always admits a U(N) isometry. The holomorphic Killing vectors k = k b∂ are generated by the Killing potential D , to be introduced shortly, as a a b a kab = −igbc∗∂c∗Da, ka∗b = igcb∗∂cDa. (2.10) ⋆ Aa The Ω = ( ) can be regarded as a section of a holomorphic symplectic bundle on a special K¨ahler Fb geometry (see [34] and references therein). We work in special coordinates in this paper. 5 These form an algebra [k ,k ] = fc k . The Aa and transform in the adjoint represen- a b − ab c Fa tation of U(N) δ Aa = faAc, δ = fc . (2.11) b − bc bFa − abFc One finds that the commutator of two of δ is given by [δ ,δ ] = fc δ . Comparing this with a a b ab c the commutator of two Killing vectors, we are able to identify δ with k . The equation a a − (2.11) is rewritten as k c∂ Aa = faAc, k c∂ = fc . (2.12) b c bc b cFa − baFc The isometry group can be embedded in the symplectic group, and the D is given by a 1 1 fb 0 D = Ω T Ω¯ = ( fb A c + fb Ac), T = ac . (2.13) a −2 | a −2 Fb ac ∗ Fb∗ ac a 0 fb ! − ac (cid:10) (cid:11) Note that D are completely determined by this formula while D is determined up to a aˆ 0 constant. In order to gauge the U(N) isometry, we introduce a set of = 1 vector superfields N N2 1 − V(xm,θ,θ¯) = Vat . (2.14) a a=0 X The U(N) gauging of is accomplished [35] by adding K L 1 Γ = d2θd2θ¯Γ, Γ = dαe2iαva(ka−ka∗)vcDc , (2.15) L Z (cid:20)Z0 (cid:21)va Va → where [ ] means the replacement of va by Va after evaluating . Combining va Va K ··· → ··· L with , we obtain Γ L i i LK +LΓ = −gab∗DmAaDmA∗b − 2gab∗ψaσmDm′ ψ¯b + 2gab∗Dm′ ψaσmψ¯b 1 1 +gab∗FaF∗b gab∗,c∗Faψ¯bψ¯c gbc∗,aF∗cψaψb − 2 − 2 1 1 +√2gab∗(λcψakc∗b +λ¯cψ¯bkca)+ 2DaDa , (2.16) where we have exploited 41gac∗,bd∗ψaψbψ¯cψ¯d = 0 as gac∗,bd∗ = 0 for the choice of K in (2.7). The covariant derivatives are defined as 1 Aa = ∂ Aa vb k a, (2.17) Dm m − 2 m b ψa = ψa +Γa Abψc, (2.18) Dm′ Dm bcDm 1 ψa = ∂ ψa vb ∂ k aψc , (2.19) Dm m − 2 m c b 6 where Γabc = gad∗gbd∗,c. The gauged kinetic action for the vector superfield V is given by i 1 = d2θτ a b +c.c. , = D¯D¯e VD eV = at , (2.20) LW2 −4 abW W Wα −4 − α Wα a Z where τ = (τ ) +i(τ ) is an analytic function of Φ, and will be determined by requiring ab 1 ab 2 ab = 2 supersymmetry. The is evaluated as 2 N LW 1 1 1 1 = τ λaσm λ¯b τ¯ λaσmλ¯b (τ ) va vbmn (τ ) ǫmnpqva vb LW2 −2 ab Dm − 2 abDm − 4 2 ab mn − 8 1 ab mn pq √2 −i 8 (∂cτabψcσnσ¯mλa −∂c∗τa∗bλ¯aσ¯mσnψ¯c)vmb n 1 √2 i i +2(τ2)abDaDb + 4 (∂cτabψcλa +∂c∗τa∗bψ¯cλ¯a)Db + 4∂cτabFcλaλb − 4∂c∗τa∗bF∗cλ¯aλ¯b i i −8∂c∂dτabψcψdλaλb + 8∂c∗∂d∗τa∗bψ¯cψ¯dλ¯aλ¯b, (2.21) where we have defined 1 va = ∂ va ∂ va favb vc, (2.22) mn m n − n m − 2 bc m n 1 λa = ∂ λa favb λc. (2.23) Dm m − 2 bc m In addition, we include the superpotential term = d2θW(Φ)+c.c. W L Z 1 1 = Fa∂aW ∂a∂bWψaψb +F∗a∂a∗W∗ ∂a∗∂b∗W∗ψ¯aψ¯b , (2.24) − 2 − 2 and the Fayet-Iliopoulos D-term [26] = ξ d2θd2θ¯V0 = √2ξD0. (2.25) D L Z The superpotential W will be determined by requiring = 2 supersymmetry. Finally, N putting all these together, the total action is given as = + + + + . (2.26) K Γ 2 W D L L L LW L L For the sake of our discussion in the next subsection, we present the on-shell action, eliminating the auxiliary fields by using the equations of motion 7 1 Da = Dˆa (τ 1)ab D +√2ξδ0 , (2.27) − 2− 2 b b (cid:18) (cid:19) Fa = Fˆa gab∗∂b∗W∗, (2.28) − F a = Fˆ a gba∗∂ W , (2.29) ∗ ∗ b − where √2 Dˆa = − 4 (τ2−1)ab ∂dτbcψdλc +∂d∗τb∗cψ¯dλ¯c , (2.30) (cid:0) (cid:1) i 1 Fˆa = −gab∗ −4∂b∗τc∗dλ¯cλ¯d − 2gcb∗,dψcψd , (2.31) (cid:18) (cid:19) i 1 Fˆ∗a = gba∗ ∂bτcdλcλd gbc∗,d∗ψ¯cψ¯d . (2.32) − 4 − 2 (cid:18) (cid:19) The action takes the following form; L = + + + + (2.33) ′ kin pot Pauli mass fermi4 L L L L L L where 1 1 Lkin = −gab∗DmAaDmA∗b − 4(τ2)abvmanvbmn − 8(τ1)abǫmnpqvmanvpbq (2.34) 1 1 i i −2τabλaσmDmλ¯b − 2τa∗bDmλaσmλ¯b − 2gab∗ψaσmDmψ¯b + 2gab∗Dmψaσmψ¯b, 1 1 1 Lpot = −2 τ2−1 ab 2Da +√2ξδa0 2Db +√2ξδb0 −gab∗∂aW∂b∗W∗, (2.35) (cid:18) (cid:19)(cid:18) (cid:19) (cid:0) (cid:1) √2 √2 LPauli = −i 8 ∂cτabψcσnσ¯mλavmb n +i 8 ∂c∗τa∗bλ¯aσ¯mσnψ¯cvmb n, (2.36) 1 i 1 Lmass = −2∂a∂bWψaψb −gab∗∂aW −4∂b∗τc∗dλ¯cλ¯d − 2gcb∗,dψcψd (cid:18) (cid:19) 1 i 1 ∂a∗∂b∗W∗ψ¯aψ¯b gab∗ ∂aτcdλcλd gac∗,d∗ψ¯cψ¯d ∂b∗W∗ −2 − 4 − 2 (cid:18) (cid:19) 1 +√2gab∗ λ¯cψ¯bkca +λcψakc∗b (cid:0) (cid:1) √2 1 − 4 τ2−1 ab 2Da +√2ξδa0 ∂dτbcψdλc +∂d∗τb∗cψ¯dλ¯c , (2.37) (cid:18) (cid:19) (cid:0) (cid:1) (cid:0) (cid:1) i i Lfermi4 = −8∂c∂dτabψcψdλaλb + 8∂c∗∂d∗τa∗bψ¯cψ¯dλ¯aλ¯b 1 −16 τ2−1 ab ∂dτacψdλc +∂d∗τa∗cψ¯dλ¯c ∂fτbeψfλe +∂f∗τb∗eψ¯fλ¯e (cid:0) (cid:1) (cid:0) (cid:1)(cid:0) (cid:1) i 1 i 1 −gab∗ 4∂aτcdλcλd − 2gac∗,d∗ψ¯cψ¯d −4∂b∗τe∗fλ¯eλ¯f − 2geb∗,fψeψf . (2.38) (cid:18) (cid:19)(cid:18) (cid:19) 8 B. Discrete R-symmetry We shall show that our Lagrangian (2.33) can be made invariant under the action ′ L λa ψa R : , (2.39) ψa ! → λa ! − which isa discrete element oftheSU(2)R-symmetry that actsasanautomorphismof = 2 N supersymmetry. First, we examine the invariance of , and under the action (2.39). The Pauli fermi4 kin L L L invariance of and that of under (2.39) require Pauli fermi4 L L ∂ τ = ∂ τ , (2.40) c ab a cb and ∂ ∂ τ = ∂ ∂ τ , ∂ τ = , (2.41) c d ab a b cd c ab abc F respectively. In addition, the invariance of the fermion kinetic terms in implies that kin L Im(τ ) = Im( ) (2.42) ab ab F and −2∂a∂b∗Dc = τadfcdb +τb∗dfcda, (2.43) as well as the last condition in (2.41) which comes from that the terms with a derivative of A vanish. The first condition (2.42) comes from the terms with a derivative of λ or ψ ∗ while the second one (2.43) from those including va . For the boson kinetic terms in , m Lkin the invariance is obvious because they do not contain fermionic fields. From the conditions (2.41) and (2.42), we conclude that τ = , (2.44) ab ab F so that gab∗ = (τ2)ab. It is easy to show that the Killing potential Da defined in (2.13) solves the condition (2.43). Secondly, we examine the invariance of the λλ and ψψ mass terms in under (2.39). mass L The key relation required for this invariance is i 1 1 gcd∗∂cτab∂d∗W∗ = gcd∗∂cWgad∗,b ∂a∂bW. (2.45) − 4 2 − 2 Writing the U(N) invariant function W as W = eA0 +mφ(A), where the e and m are real constants, it reduces to 1 abc( )cd(∂dφ ∂d∗φ∗) = ∂a∂bφ, (2.46) F − ∗ F −F 9 which can be solved by φ = +const. Thus we can choose 0 F W = eA0 +m , (2.47) 0 F up to an irrelevant constant. Thirdly, we examine the ψλ terms in . Because ψaλb is odd under the action (2.39), mass L the coefficient, √12gac∗kb∗c− √82(τ2−1)cd∂aτcb(Dd+2√2ξδd0), must be odd. This implies the key relation for the invariance 1 i∂ D +i∂ D (τ 1)cd∂ τ D = 0, (2.48) a b b a − 2 2− a cb d as well as R; ξ ξ. (2.49) → − The equation (2.48) can be proven as follows. First, we note that fc + fc = fc Ae, (2.50) Fac db Fbc da −Fabc de which is derived as a derivative of the second relation in (2.12). Using this relation and the definition (2.13), one finds that i i∂ D +i∂ D = fc A dAe. (2.51) a b b a −2Fabc de ∗ On the other hand, the Killing potential is shown to be rewritten as 1 D = fb A cAd( ) = ig fb A cAd (2.52) a 2 cd ∗ Fa∗b −Fab − ab cd ∗ by using the second relation in (2.12). The equations (2.51) and (2.52) are enough to see that the equation (2.48) is true. Lastly, we examine . The invariance of under (2.49) follows from the fact that pot pot L L the term linear in ξ in vanishes: pot L 1 √2 √2 (τ 1)abD √2ξδ0 = ξga0( ig fb A cAd) = i ξf0A cAd = 0 (2.53) − 2 2− a b − 2 − ab cd ∗ 2 cd ∗ where we have used (2.44) and (2.52). In summary, we have shown that our on-shell action (2.33) admits the discrete R- symmetry (2.39) and (2.49) if we choose τ as (2.44) and W as (2.47). ab We will show that the discrete R-symmetry can be realized in the off-shell action (2.26) with (2.44) and (2.47). In an ungauged theory without a superpotential, the discrete action on the auxiliary fields is Da Da and Fa F a. In our model, this is modified as is seen ∗ → − → 10