UUUITP-17/14
Extended supersymmetry of semichiral sigma model in D . Ulf Lindstr¨om a a Department of Physics and Astronomy, Division of Theoretical Physics, Uppsala University,Box 516, SE-751 20 Uppsala, Sweden
Abstract
Briefly : Using a novel (1 ,
1) superspace formulation of semichiral sigma models with 4 D target space, we investigate if an extended supersymmetry in terms of semichirals iscompatible with having a 4 D target space with torsion. In more detail : Semichiral sigma models have (2 ,
2) supersymmetry and GeneralizedK¨ahler target space geometry by construction. They can also support (4 , ,
1) superspace and construct the extra(on-shell) supersymmetries there. We then find the conditions for a lift to (2 ,
2) superspace and semichiral fields to exist. Those conditions are shown to hold for Hyperk¨ahlergeometries. The SU (2) ⊗ U (1) WZW model, which has (4 ,
4) supersymmetry and asemichiral description, is also investigated. The additional supersymmetries are found in(1 ,
1) superspace but shown not to be liftable to a (2 ,
2) semichiral formulation. a r X i v : . [ h e p - t h ] N ov ontents N = (4 , in N = (1 , superspace 4 N = (2 ,
2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 N = (4 ,
4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 S × S model 8 ,
1) . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.3.1 Deriving the transformations on the (1 ,
1) coordinate fields . . . . . 114.3.2 ( X L , X R ) coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 14 SU (2) ⊗ U (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 (1 ,
17B An alternative dual form 19
Generalized K¨ahler geometry is efficiently probed by (2 ,
2) supersymmetric sigma modelsin D = 2, [1]. Of particular interest for the present investigation is the symplectic case,i.e., sigma models that depend on semichiral superfields only. Additional supersymme-tries for these models were discussed in [2], and in [3]. In the latter article focus is onfour-dimensional target spaces and it is shown that a very general ansatz for additionalsupersymmetries leads to an on-shell extended supersymmetry and restricts the targetspace geometry to be hyperk¨ahler. 1n [3] this is seen as a shortcoming of the ansatz, since it is argued that the SU (2) ⊗ U (1)WZW model of [4] constitutes a counterexample. It has nonzero torsion and when coor-dinatized by chiral and twisted chiral superfields it has “manifest” (4 ,
4) supersymmetry.It further has a dual semichiral description [5] which is then expected to also display the(4 ,
4) supersymmetry .In this paper we investigate the possibility that the (2 ,
2) semichiral conditions areincompatible with “manifest” (4 ,
4) transformations. To study this problem, we descendto (1 ,
1) superspace and develop an on-shell formalism for the extra super symmetries, aformulation which retains the relation to (2 ,
2) semichirals. We test this (1 ,
1) formalismon the second supersymmetry (which is non-manifest in (1 , ,
2) semichiralrealisation, as expected from [3].We also derive the extra supersymmetries for the WZW model [4] in (1 ,
1) superspacein the relevant coordinates. When subjected to the same test they fail to satisfy someof the conditions. This leads to the surprising conclusion that (4 ,
4) supersymmetry ina (1 ,
1) formulation of a (2 ,
2) sigma model with on-shell supersymmetry is incompatiblewith the introduction of the (2 ,
2) auxiliary fields.
Consider a generalized K¨ahler potential [1] with one left- and one right semichiral fieldand their complex conjugates, K ( X L , X R ), where L = ( (cid:96), ¯ (cid:96) ) and R = ( r, ¯ r ). The action, S = (cid:90) d xd θd ¯ θK ( X L , X R ) (2.1)has manifest N = (2 ,
2) supersymmetry. The supersymmetry algebra is defined in termsof the anti-commutator of the covariant supersymmetry derivatives as { D ± , ¯ D ± } = i∂ ++= (2.2)and the semichiral fields are defined by their chirality constraints as [7]¯ D + X (cid:96) = 0 , ¯ D − X r = 0 . (2.3) By “manifest” we shall mean “as realised by transformations of (2 ,
2) superfields.” Another case of supersymmetries being obstructed occurs when dualisation is along isometries thatdo not commute with the extra super symmetries. This leads to nonlocal realisations of the extra susysin the dual model [6]. Here, however, the extra susys commute with the isometry used in dualising. J (+) and J ( − ) that both preserve the metric G J ( ± ) t G J ( ± ) = G (2.4)as well as by an anti-symmetric B -field whose field strength H enters in the form of torsionin the covariant constancy conditions0 = ∇ ( ± ) J ( ± ) = (cid:0) ∂ + Γ (0) ± H G − (cid:1) J ( ± ) , (2.5)where Γ (0) is the Levi-Civita connection. These conditions identify the geometry as bi-hermitean [8], or generalized K¨ahler geometry (GKG) [9].The fact that our superfields are semichiral specifies the GKG as being of symplectictype where the metric g and the B -field take the form G = Ω[ J (+) , J ( − ) ] B = Ω { J (+) , J ( − ) } . (2.6)The matrix Ω is defined as Ω = (cid:32) K LR − K RL (cid:33) (2.7)and the submatrix K LR is the Hessian K LR = (cid:32) K (cid:96)r K (cid:96) ¯ r K ¯ (cid:96)r K ¯ (cid:96) ¯ r (cid:33) . (2.8)An additional condition results from the target space being four-dimensional and reads[7] { J (+) , J ( − ) } = 2 c, ⇒ B = 2 c Ω , (2.9)where c in general is a function of the coordinates. In [3] it is shown that a general ansatz for (4 ,
4) susy in a semichiral sigma model δ X (cid:96) = ¯ (cid:15) + ¯ D + f ( X L , X R ) + g ( X (cid:96) )¯ (cid:15) − ¯ D − X (cid:96) + h ( X (cid:96) ) (cid:15) − D − X (cid:96) ,δ X ¯ (cid:96) = (cid:15) + D + ¯ f ( X L , X R ) + ¯ g ( X ¯ (cid:96) ) (cid:15) − D − X ¯ (cid:96) + ¯ h ( X ¯ (cid:96) )¯ (cid:15) − ¯ D − X ¯ (cid:96) ,δ X r = ¯ (cid:15) − ¯ D − ˜ f ( X L , X R ) + ˜ g ( X r )¯ (cid:15) + ¯ D + X r + ˜ h ( X r ) (cid:15) + D + X r ,δ X ¯ r = (cid:15) − D − ¯˜ f ( X L , X R ) + ¯˜ g ( X ¯ r ) (cid:15) + D + X ¯ r + ¯˜ h ( X ¯ r )¯ (cid:15) + ¯ D + X ¯ r , (2.10) This gives the B field in a particular global gauge as B = B (2 , + B (0 , with respect to both complexstructures. δ , δ ] (cid:96) = − (cid:15) +[2 ¯ (cid:15) +1] | f ¯ (cid:96) | ∂ ++ .... (2.11)which has the wrong sign for supersymmetry. It is an interesting fact that on-shell closureof the algebra, together with conditions that come from invariance of the action, requiresthat the function c ( X L , X R ) defined by (2.11) is constant with absolute value less than one,which means that the geometry is hyperk¨ahler. This on-shell closure is different than theone which arises in the general (1 ,
1) discussion of extended susy [8] which locates the non-closure of the algebra to the (+ , − ) sector where the commutator [ J (+) , J ( − ) ] multipliesthe field equation. In the present case left or right susy alone require field equations.In [3] we argue that the ansatz for the additional supersymmetry is too restrictiveand should include central charge transformations. The reason for trying to find a moregeneral ansatz is that there is a known example of a BiLP with (4 ,
4) supersymmetry,mentioned in the introduction, that has a dual semichiral formulation which manifestlyviolates the hyperk¨ahler condition [5]. The duality and extra supersymmetry will bediscussed in detail in Sec.4 below. Since the isometry used in the dualisation commuteswith the extra supersymmetry, the dual model is expected to have the extra symmetry aswell. To investigate this, bearing in mind that the relevant algebra only closes on-shell, wenow develop a novel N = (1 ,
1) form of the semichiral model and its additional symmetries. N = (4 , in N = (1 , superspace We want to find out under what conditions a semichiral sigma model in 4 D supportsadditional complex structures forming an SU (2) algebra I ( a )(+) I ( b )(+) = − δ ab + (cid:15) abc I ( c )(+) (3.12)with a = 1 , , I (3)(+) := J (+) . To this end, we discuss the situation in(1 ,
1) superspace [8]. This discussion is general, only the later applications in the examplesection will be limited to 4 D . We replace spinor derivatives according to¯ D ± → D ± + iQ ± , D ± → D ± − iQ ± . (3.13)The general form of the (2 ,
2) sigma model reduced to (1 ,
1) has a Lagrangian that reads L = D + X A E AB ( X ) D − X C + Ψ R + K RL Ψ L − := L + L , (3.14) This acronym stands for “bihermitean local product” and refers to a 2 D sigma model with chiral andtwisted chiral superfields only. Corresponding to (+)-supersymmetries. The general case also involves ( − )-supersymmetries. E AB := G AB + B AB and we have completed the square for the spinor auxiliary fields ψ ± and defined Ψ R + := ψ R + − D + X A J R (+) A Ψ L − := ψ L − − J L ( − ) A D − X A . (3.15)Assume that we have found the additional transformations of the (1 ,
1) coordinates gener-ated by the SU (2) set of complex structures I ( a )(+) as in (3.12). These transformations leavethe L part of the action invariant. We would now like to extend them to symmetries ofthe full action and subsequently check if the full set can come from transformations of the(2 ,
2) semichiral fields.There are obvious symmetries we can write down, field equation symmetries (also calledZilch symmetries), but there are many possibilites. N = (2 , To get a guide to the correct form, we use the fact that we know that one of the symmetrieshas the correct properties; the one generated by I (3)(+) = J (+) .The full action is invariant under the following transformations: δX L = (cid:15) + J D + X L δX R = (cid:15) + ψ R + δψ L − = (cid:15) + J D + ψ L − δψ R + = (cid:15) + D X R , (3.16)where J is the canonical complex structure diag ( i, − i ). They give δ L = − δX A ∇ ( − )+ D − X B G AB + δ Ψ R + K RL Ψ L − + Ψ R + δK RL Ψ L − + Ψ R + K RL δ Ψ L − = − (cid:15) + J A (+) C D + X C ∇ ( − )+ D − X B G AB + Ψ R + (cid:16) K RL δ Ψ L − + 2 (cid:15) + ∇ ( − )+ D − X B G RB (cid:17) + δ Ψ R + K RL Ψ L − + Ψ R + δK RL Ψ L − , (3.17)where capital letters from the beginning of the alphabet takes on all values L and R .The first of the terms in the last line is the variation of L under the J (3)(+) symmetry andvanishes in the action. We evaluate the remaining terms using δ Ψ L − = (cid:15) + J D + Ψ L − + (cid:15) + J L ( − ) R (cid:48) D − Ψ R (cid:48) + − (cid:15) + J L ( − ) AR (cid:48) Ψ R (cid:48) + D − X A + (cid:15) + (cid:2) J (+) , J ( − ) (cid:3) LA (cid:16) D + D − X A + D + X B Γ ( − ) ABC D − X C (cid:17) δ Ψ R + = (cid:15) + J R (+) R (cid:48) D + Ψ R (cid:48) + − (cid:15) + J R (+) AR (cid:48) Ψ R (cid:48) + D + X A (3.18)5hich follows from (3.16). (Here J L ( − ) AR denotes the derivative ( J L ( − ) A ) ,R etc.). Since G AB = Ω AC (cid:2) J (+) , J ( − ) (cid:3) CB (3.19)the second line may be rewritten as (cid:15) + σ LC G CA ∇ ( − )+ D − X A , (3.20)where we use the notation σ LC Ω CD = δ CD . This part of the variation will cancel thecovariant derivative term multiplying Ψ R + in (3.17). We are left with δ L = − (cid:15) + D − (cid:16) Ψ R + C RR (cid:48) Ψ R (cid:48) + (cid:17) + (cid:15) + D + (cid:0) Ψ R + K RL J Ψ L − (cid:1) , which ensures invariance of the action. Here where C RR (cid:48) is the commutator with thecanonical complex structure C RR (cid:48) := [ J, K RR (cid:48) ].In deriving (3.21) heavy use is made of integrability and covariant constancy of J ( ± ) as well as their explicit expressions [1].We note that the transformations that leave the action invariant may be written δX A = (cid:15) + J A (+) B D + X B + δ AR (cid:15) + Ψ R + δ Ψ L − = (cid:15) + J D + Ψ L − + (cid:15) + J L ( − ) R (cid:48) D − Ψ R (cid:48) + − (cid:15) + J L ( − ) AR (cid:48) Ψ R (cid:48) + D − X A − (cid:15) + K LR G RA ∇ ( − )+ D − X A δ Ψ R + = (cid:15) + J R (+) R (cid:48) D + Ψ R (cid:48) + − (cid:15) + J R (+) AR (cid:48) Ψ R (cid:48) + D + X A (3.21) N = (4 , We now investigate if our additional super symmetries (4.56) can be written in terms oftransformations of semichiral fields in (2 ,
2) superspace. In our (1 ,
1) language, the relationto semichirals is given by (3.15) in ( X L , X R ) coordinates . We denote a generic complexstructure by I AB and write the X transformations as δX A = (cid:15) + (cid:2) I A (+) B D + X B + M AR Ψ R + (cid:3) , where L is assumed to be invariant (up to total derivatives) under the first transformationon the RHS. Note that M AL = 0. The formula (3.22) is of the form in (3.21) and the mostgeneral expression compatible with dimensions and symmetries. From (3.15) we have that The reduction of ( X L , X R ). Ψ ˙ R + = (cid:15) + (cid:104)(cid:16) I ˙ R (+) R − [ M, J (+) ] ˙ RR (cid:17) D + Ψ R + + (cid:16) I ˙ R + A,R + M ( M, J (+) ) ˙ RRA (cid:17) D + X A Ψ R + − M ˙ RR,R (cid:48) Ψ R (cid:48) + Ψ R + (cid:105) δ Ψ ˙ L − = (cid:15) + (cid:104) − [ M, J ( − ) ] ˙ LR D − Ψ R + + M ( M, J ( − ) ) ˙ LRA D − X A Ψ R + + ( I ˙ L (+) L − M ˙ LR J R (+) L ) D + Ψ L − + ( I ˙ L (+) A,L − M ˙ LR J R (+) A,L ) D + X A Ψ L − − M ˙ LR,L Ψ L − Ψ R + + (cid:16) [ I (+) , J ( − ) ] ˙ LA − M ˙ LR [ J (+) , J ( − ) ] RA (cid:17) ∇ ( − )+ D − X A (cid:105) (3.22)where the Magri-Morosi concomitant for two endomorphisms I and J reads M ( I, J ) ABD = I FB J AD,F − J FD I AB,F − I AF J FD,B + J AF I FB,D . (3.23)When M AR = δ AR and I + = J (+) these transformations reduce to (3.21).From invariance of (3.14), we find a number of relations. First, raising and loweringindices on M with K RL , M L [ R, ˙ R ] − M [ R ˙ R ] ,L = 0 M [ R ˙ R ] = − K ˙ R ˙ L [ I (+) , J ( − ) ] ˙ LA G AL K LR M R ˙ R = − K ˙ RL [ I (+) , J ( − ) ] LA B AR (3.24)Note that only the antisymmetric part of M R ˙ R is determined by this . The D − terms inthe variation of L areΨ ˙ R + K ˙ RL (cid:15) + (cid:0) − [ M, J ( − ) ] LR D − Ψ R + + M ( M, J ( − ) ) LRA D − X A Ψ R + (cid:1) For this to yield a total D − derivative, we shall need − K ( ˙ R | L | [ M, J ( − ) ] LR ) = [ J, M ] ( ˙ RR ) + C ( ˙ R | R | M RR ) = 0 (3.25)and K [ ˙ R | L | M ( M, J ( − ) ) LR ] A D − X A = − D − ( K [ ˙ R | L | [ M, J ( − ) ] LR ] ) (3.26) Originally defined for a Poisson structure P and a Nijenhuis tensor N when it reads [10] C kjm = P lj N km,l + P kl N km,l − N lm P kj,l + N jl P kl,m − P lj N kl,m and is only a tensor when [ P, N ] = 0. The RHS of the equation containing M [ R ˙ R ] is antisymmetric due to hermiticity conditions. D + terms to yield a total derivative, we need K LR I R ˙ R − K ˙ RL I LL + K LR [ ˜ J , M ] R ˙ R + C LL K LR M [ R, ˙ R ] = 0and (cid:16) I RA, ˙ R + M ( M, J (+) ) R ˙ RA (cid:17) K RL + K ˙ RLB I BA + K ˙ RL (cid:0) I LA,L − M LR J R (+) A,L (cid:1) = (cid:16) ( I ˙ LL − M ˙ LR J R (+) L ) K ˙ L ˙ R (cid:17) ,A (3.27)where we have used (3.24) and the explicit form of J (+) . S × S model In this section we briefly recapitulate the dualisation of the BiLP formulation of the SU (2) × U (1) WZW model [5], albeit in a different version.We start from the following BiLP potential which gives a sigma model with targetspace geometry S × S ; K = − ln ˆ χln ˆ¯ χ + (cid:90) ˆ φ ˆ¯ φ ˆ χ ˆ¯ χ dq ln (1 + q ) q , (4.28)where ˆ φ is chiral, ¯ D ± ˆ φ = 0, and ˆ χ is twisted chiral, ¯ D + ˆ χ = 0 = D − ˆ χ . The potentialsatisfies the Laplacian K ˆ φ ˆ¯ φ + K ˆ χ ˆ¯ χ = 0 , (4.29)and hence the model has (4 ,
4) supersymmetry [8]. Changing coordinates to new chiraland twisted chiral fields, φ = ln ˆ φ, χ = ln ˆ χ , results in K → K = − χ ¯ χ + (cid:90) φ + ¯ φ − χ − ¯ χ dq ln (1 + e q ) , (4.30)and makes it amenable to dualisation of the translation symmetry φ → φ + λ, χ → χ + λ . (4.31) This is equivalent to dualising the scaling symmetry of (4.28). These isometries both commute withthe extra supersymmetry [11], [12].
8o apply the gauging prescription of [13] we add a term α ( χ − ¯ χ )( φ − ¯ φ ) , which representsa constant B -term, to the Lagrangian and rewrite the potential, up to generalized K¨ahlergauge transformations, as ( χ − ¯ χ ) + α ( χ − ¯ χ )( φ − ¯ φ ) + (cid:90) φ + ¯ φ − χ − ¯ χ dq ln (1 + e q ) . (4.32)Following [13], we find the first order action (in a WZ gauge); − V χ − αV χ V φ − V (cid:48) X (cid:48) − V φ X φ − V χ X χ + (cid:90) V (cid:48) dq ln (1 + e q ) , (4.33)where V φ , V χ and V (cid:48) are the Large Vector Multiplet (LVM) fields [13] , and the Lagrangemultipliers are combinations of semichiral fields X φ = i ( (cid:96) − ¯ (cid:96) − r + ¯ r ) X χ = i ( − (cid:96) + ¯ (cid:96) − r + ¯ r ) X (cid:48) = ( (cid:96) + ¯ (cid:96) − r − ¯ r ) . (4.34)Eliminating the LVM and massaging the integral we find the dual semichiral action in theform − α X φ + 1 α X φ X χ − (cid:90) X (cid:48) dq ln ( e q − . (4.35)This is the potential that is expected to have additional supersymmetries due to those ofthe dual BiLP model. The reduction of a semichiral model to (1 ,
1) superspace may be expressed in several usefulcoordinate systems. E.g., the ( X L , X R ) coordinates directly obtained in the reduction isrelated to the ( X, Y ) system where J (+) is canonical via a coordinate transformation[14],[1]. In pure gauge they become V φ = i ( ¯ φ − φ ) V χ = i ( ¯ χ − χ ) V (cid:48) = φ + ¯ φ − χ − ¯ χ .
9e now derive the metric in (
X, Y ) coordinates for (4.35) . To this end we firstcalculate the various ingredient matrices according to the formulae in [1].Without loss of generality, we set α to − X (cid:48) . The Y coordinates are defined to be K L =: Y . We find − y = K (cid:96) = i X χ + ln ( e X −
1) = ( (cid:96) − ¯ (cid:96) ) + ( r − ¯ r ) + ln ( e X − ⇒ X = ln (1 + e − ( y +¯ y ) ) r + ¯ r = (cid:96) + ¯ (cid:96) − ln (1 + e − ( y +¯ y ) ) ,r − ¯ r = − y − ¯ y ) − ( (cid:96) − ¯ (cid:96) ) (4.36)The relevant matrices of derivatives of K are K LL = − N (cid:32) E E (cid:33) , − K RR = 14 N (cid:32) − E M N + 1 2 − E (cid:33) − K LR = − N (cid:32) EE (cid:33) , − K RL = 1 e X (cid:32) − E − E (cid:33) − C LL = 2 i N (cid:32) − (cid:33) , − C RR = 2 iM N (cid:32) − (cid:33) (4.37)where we have introduced the notation N := e − ( y +¯ y ) = e X − , E := 2 e X − , M := 4 N + 1 , (4.38)for combinations that will occur frequently in our formulae. The metric and B -field in( X, Y ) coordinates can be calculated from the formulae in [1]: − E LL = J ( K LL K LR J K RL − K LR J K RL K LL − K LL K LR C RR K RL K LL ) = − σ E LY = J ( K LR J K RL + K LL K LR C RR K RL ) = − e X (cid:32) − e X − E − E − e X (cid:33) E Y L = J ( − K LR J K RL + K LR C RR K RL K LL ) = 1 e X (cid:32) − e X E E − e X (cid:33) − E Y Y = − J K LR C RR K RL = − Me X σ . Note that the tensor E depends on y + ¯ y only. From the formulae for E = G + B , it follows10hat the metric is G = 2 e X (cid:32) e − X EE M (cid:33) ⊗ σ , (4.39)with inverse G − = − e X N (cid:32) − M EE − e X (cid:33) ⊗ σ =: − e X N h ⊗ σ . Also B Y L = (2 e − X − . (4.40)In these coordinates J (+) is canonical while J ( − ) = e − X (cid:32) (2 − e X ) J − Eσ − M σ e X σ (2 − e X ) J + 2 Eσ (cid:33) =: e − X (cid:0) j ⊗ σ + (2 − e X ) ⊗ iσ (cid:1) . (4.41)where we use that J = iσ . (1 , (1 , coordinate fields In the original BiLP, (4.28) the additional super symmetries read [8] δ ˆ φ = ¯ (cid:15) + ¯ D + ˆ¯ χ + ¯ (cid:15) − ¯ D − ˆ χδ ˆ¯ φ = (cid:15) + D + ˆ χ + (cid:15) − D − ˆ¯ χδ ˆ χ = − ¯ (cid:15) + ¯ D + ˆ¯ φ − (cid:15) − D − ˆ φδ ˆ¯ χ = − (cid:15) + D + ˆ φ − ¯ (cid:15) − ¯ D − ˆ¯ φ , (4.42)and in the transformed version (4.30) they become δφ = e ¯ χ − φ ¯ (cid:15) + ¯ D + ¯ χ + e χ − φ ¯ (cid:15) − ¯ D − χδ ¯ φ = e χ − ¯ φ (cid:15) + D + χ + e ¯ χ − ¯ φ (cid:15) − D − ¯ χδχ = − e ¯ φ − χ ¯ (cid:15) + ¯ D + ¯ φ − e φ − χ (cid:15) − D − φδ ¯ χ = − e φ − ¯ χ (cid:15) + D + φ − e ¯ φ − ¯ χ ¯ (cid:15) − ¯ D − ¯ φ . (4.43) Since a lot of the objects have 2 D complex submatrices, it is convenient to introduce the Pauli matrices σ i and write matrices as direct products. ,
1) reduction with D ± → D ± . From (4.43) we then readoff the additional complex structures according to δϕ = (cid:104)(cid:16) J (1)( ± ) + iJ (2)( ± ) (cid:17) (cid:15) ± D ± ϕ + (cid:16) J (1)( ± ) − iJ (2)( ± ) (cid:17) ¯ (cid:15) ± D ± ϕ (cid:105) (4.44)For the J ( a )(+) we find J (1)(+) = e ¯ χ − φ e χ − ¯ φ − e ¯ φ − χ − e φ − ¯ χ (4.45) J (2)(+) = ie ¯ χ − φ − ie χ − ¯ φ − ie ¯ φ − χ ie φ − ¯ χ , (4.46)with J (3)(+) = J , the canonical complex structure.We would like to see what these complex structures look like in (1 ,
1) coordinates relatedto the semichiral description. While T-dual formulations are not in general related bycoordinate transformations, they are in this case due to the special choice of the isometrydirection; we have dualised along the common U (1). We now need to find the coordinatetransformation. To this end, we note that the relations (A.74), derived in the appendix, V φ = X χ + X φ V χ = X φ V = ln ( e X − . (4.47)do not completely determine the transformations. Identifying the LHS with BiLP fields(WZ-gauge) and writing out the RHS we have i ( ¯ φ − φ ) = − i ( r − ¯ r ) i ( ¯ χ − χ ) = i ( (cid:96) − ¯ (cid:96) − r + ¯ r ) φ + ¯ φ − χ − ¯ χ = ln ( e
12 ( (cid:96) +¯ (cid:96) − r − ¯ r ) − . (4.48)It turns out to be most convenient to identify the coordinate transformation to ( X, Y )coordinates where the complex structure derived from the semi side, J (+) , is canonical . The map of J ( ± ) under duality is discussed in [15] where the dual model appears in some preferredcoordinates. We have not investigated the relation to the present coordinates.
12 coordinate transformation to ( X L , X R ) coordinates will then give a non-canonical J (+) .Comparing this to (4.36), we identify y = χ − φ(cid:96) − ¯ (cid:96) = 2( ¯ χ − χ ) − ( ¯ φ − φ ) , (4.49)but (cid:96) + ¯ (cid:96) is left undetermined. In both coordinate systems we have J (3)(+) = J . Requiring thatthe coordinate transformation takes the canonical complex structure into itself determines (cid:96) + ¯ (cid:96) = φ + ¯ φ −
2( ¯ χ − χ ), which results in (cid:96) = φ − χ . (4.50)This gives the transformation Jacobian Λ = (cid:32) ∂L∂φ ∂L∂χ∂Y∂φ ∂Y∂χ (cid:33) = (cid:32) − − (cid:33) , (4.51)with inverse Λ − = − (cid:32)
11 1 (cid:33) , (4.52)These transformations correctly relates the BiLP metric derived from (4.30) to the semimetric (4.39). We now write the extra complex structures J ( a )(+) as J ( a ))(+) = (cid:32) A ( a ) − ( A ( a ) ) − (cid:33) (4.53)for a = 1 ,
2, with A (1) = (cid:32) e ¯ χ − φ e χ − ¯ φ (cid:33) = (cid:32) e ϕ + y e − ϕ +¯ y (cid:33) A (2) = (cid:32) ie ¯ χ − φ − ie χ − ¯ φ (cid:33) = (cid:32) ie ϕ + y − ie − ϕ +¯ y (cid:33) , (4.54)where ϕ := ( (cid:96) − ¯ (cid:96) ) + ( y − ¯ y ) . (4.55)The expressions for J ( a )(+) in ( X, Y ) coordinates then become J ( a )(+) = (cid:32) − E − Me X E (cid:33) ⊗ A ( a ) =: j ⊗ A ( a ) , j = − N , ( A ( a ) ) = N − , (4.56)where we again use the notation in (4.38). The complex structures J ( a )(+) preserve the metric(4.39), as confirmed by an explicit calculation.13 .3.2 ( X L , X R ) coordinates Using (4.37) in the Jacobian Λ = (cid:32) K LL K LR (cid:33) = 14 N (cid:32) N − ( E + σ ) + Eσ (cid:33) , Λ − = (cid:32) − K RL K LL K RL (cid:33) = (cid:32) σ − e − X ( − Eσ ) (cid:33) , (4.57)we find the expressions for J ( a )(+) in left right coordinates ( X L , X R ): J ( a )(+) = 14 N (cid:40)(cid:32) E − Me − X − e − X E (cid:33) ⊗ A ( A ) + (cid:32) M − M Ee − X E − e − X E (cid:33) ⊗ A ( A ) σ + Ne X (cid:34)(cid:32) − E (cid:33) ⊗ ¯ A ( A ) + (cid:32) E − (cid:33) ⊗ ¯ A ( A ) σ (cid:35)(cid:41) , (4.58)where Y ( X L , X R ) is given by the relations in (4.36). We want to show that there are hyperk¨ahler solutions to our problem in 4 D . To this endwe note that, when c is constant we have available the following hyperk¨ahler structure[16]; I := J (+) , J := 1 √ − c ( J ( − ) + cJ (+) ) , K := 12 √ − c [ J (+) , J ( − ) ] . (5.59)The relations (3.24) determine M in the three cases according to I : M R ˙ R = δ R ˙ R , M [ ˙ RR ] = 0 J : M R ˙ R = c δ R ˙ R √ − c , K : M R ˙ R K LR = − √ − c K ˙ RL J L ( − ) L = − √ − c J K ˙ RL M [ ˙ RR ] = − √ − c K ˙ RL J L ( − ) R = − √ − c C ˙ RR (5.60)14ach case satisfies the first relation in (3.24) (provided that c is constant) .The conditions (3.25) is satisfied by the hyperk¨ahler structure (5.60). The relation (3.26)is satisfied for I and J by direct insertion. For K we determine the full M L ˙ R = − √ − c K LR J K R ˙ R (5.61)and find that K [ ˙ R | L | [ M, J ( − ) ] LR ] = 0 , (5.62)and the issue becomes the vanishing of M . This is again confirmed by direct insertion ofthe K expressions from (5.60).As a final check we also find that the relations (3.27) and (3.27) are indeed satisfiedfor I , J and K . SU (2) ⊗ U (1) Using (4.56) and (4.41) we find that in (
X, Y ) coordinates[ J ( a )+ , J − ] = − e − X (cid:0) (2 − e X ) j ⊗ ( b ( a ) σ + a ( a ) σ ) + 2 N ⊗ a ( a ) iσ (cid:1) , (5.63)where j is defined in (4.41). Using (4.54) we have defined A (1) =: a (1) σ − b (1) σ =: 1 √ N ( cosψ σ − sinψ σ ) A (2) =: a (2) σ − b (2) σ =: 1 √ N ( − sinψ σ − cosψ σ ) , (5.64)and iψ := ( (cid:96) − ¯ (cid:96) ) + 32 ( y − ¯ y ) . (5.65)As is clear from (3.24), we shall need [ J ( a )+ , J − ] G − . We find[ J ( a )+ , J − ] G − = 2 a ( A ) h ⊗ σ + (2 − e X ) (cid:32) − (cid:33) ⊗ (cid:0) b ( a ) − a ( a ) iσ (cid:1) , (5.66)where h is defined in (4.40). In ( X L , X R ) coordinates this becomes(2 e − X − (cid:32) − (cid:33) ⊗ b ( a ) ( − Eσ ) + (cid:32) M EE (cid:33) ⊗ a ( a ) σ − (cid:32) − (cid:33) ⊗ a ( a ) iσ . (5.67)15e read off the matrices relevant to (3.24) M [ R ˙ R ] = K ˙ R ˙ L (cid:16) [ J ( a )+ , J − ] G − (cid:17) ˙ LL K LR = − M e X a ( A ) N σ M LR = K R ˙ L (cid:16) [ J ( a )+ , J − ] G − (cid:17) ˙ LR K RL = − e X N (cid:2) (2 e − X − b ( a ) (1 + Eσ ) + a ( a ) ( Eσ − iσ ) (cid:3) (5.68)We find that the quantities in (5.68) indeed satisfy the first relation in (3.24). Proceedingto (3.25) and (3.26), we find that (3.25) is also satisfied using (5.68), and that the b ( a ) terms in (3.26) cancel. However, the remaining terms in (3.26) must satisfy (cid:0) M F [ r K ¯ r ] F (cid:1) R = 0 M rr [ ,r K ¯ r ] L = 0 , (5.69)where knowledge of the form of J ( − ) along with partial information from (5.68) has beenused. While the first of these equations determines the remaining parts of M LR , the secondequations must be identically satisfied by M RR in (5.68). This is not the case. We have extended the (1 ,
1) formulation of semichiral sigma models to allow for a treatmentof extra super symmetries with on-shell closure. To exemplify the general method we haveshown that a set of hyperk¨ahler geometries arise as solutions of the conditions for extrasupersymmetry. We have further constructed the extra super symmetries in a semichiralmodels dual to a BiLP model with “manifest” (4 ,
4) susy on the BiLP side. This model failsthe criteria for the additional supersymmetry to be manifest as transformations of (2 , ,
4) supersymmetry is incompatiblewith the introduction of the (2 ,
2) auxiliary spinor fields. The key ingredient in the analysisis to show that invariance of the action fails (on-shell closure of the algebra is ensured byconstruction). Note that the analysis shows that not even an extra supersymmetry of onehandedness only is possible.Our analysis is carried out at the (1 ,
1) level, where conditions for additional super-symmetries are well established since thirty years [8].An analysis at the (2 ,
2) level already indicated that the remedy suggested in [3] willnot work; a formulation including central charge transformations will typically display theoriginal obstructions when we go on-shell. 16 further indication of problems with an extra supersymmetry comes from dualisationprocedure itself. One would expect the parent action, where the chirality constraints on thechiral and twisted chiral superfields have been relaxed, to have the extra supersymmetry.This would mean the the LVM gauge multiplet could carry extra supersymmetry. Thiswas concluded to be impossible under fairly general assumptions in [2].In view of this result, it is reasonable to conjecture that manifest extra supersymmetriesinvolving semichiral fields together with a 4 D target space is only possible in modelsincluding auxiliary fields such as in the (4 ,
4) superspace setting of [17].Acknowledgement: Discussions with M. Roˇcek at various stages of this work aregratefully acknowledged. Supported in part by VR grant 621-2013-4245
A Duality in (1 , In this section we reduce the action (4.33) (with α = −
1) to (1 ,
1) and eliminate the LVMthere instead. This makes clear the issue of coordinate transformations at the (1 ,
1) level.We replace covariant derivatives according to¯ D ± → D ± + iQ ± , D ± → D ± − iQ ± . (A.70)To facilitate the calculation we introduce the following notation: Y A ± := Q ± X A ,Z A := Q + Q ± X A , A = φ, χ, Xs (cid:96) := (cid:96) + ¯ (cid:96) , d (cid:96) := (cid:96) − ¯ (cid:96)s r := r + ¯ r , d r := r − ¯ r Σ := ψ + ¯ ψ Λ := ψ − ¯ ψ (A.71)and define the (1 ,
1) components of the LVM (in WZ gauge) as V χ | =: V χ , Q ± V χ | =: ( A + B ) ± , Q + Q ± V χ | =: FV φ | =: V φ , Q ± V φ | =: B ± , Q + Q ± V φ | =: GV X | =: V X , Q ± V X | =: C ± , Q + Q ± V X | =: H . (A.72)17he Lagrangian becomes F ( V φ − V χ − X χ ) + G ( V χ − X φ ) + H ( ln (1 + e V ) − X ) − V φ Z φ − V χ Z χ − V (cid:48) Z X − ( A + + Y χ + )( A − + Y χ − ) + Y χ + Y χ − +( B + + Y φ + + Y χ + )( B − + Y φ − + Y χ − ) − ( Y φ + + Y χ + )( Y φ − + Y χ − )+ (cid:18) C + − Y χ + (cid:18) e V e V (cid:19)(cid:19) (cid:18) e V e V (cid:19) (cid:18) C − − (cid:18) e V e V (cid:19) Y χ − (cid:19) − Y X + Y X − (cid:18) e V e V (cid:19) . (A.73)Integrating out F, G, H gives the coordinate transformation V φ = X χ + X φ V χ = X φ V = ln ( e X − . (A.74)Integrating V φ = , V χ , V determines F, G, H in terms of the components of the semis: F = Z φ G = Z φ + Z χ H = (cid:18) e V e V (cid:19) (cid:20)(cid:18) − e V e V (cid:19) Y X + Y X − + Z X (cid:21) . (A.75)(No contribution from C ± terms et.c.. 1.5 formalism). Finally, integrating A, B, C againdetermines these fields in terms of the semi components. This leaves us with a purely semiLagrangian; Y χ + Y χ − − ( Y φ + + Y χ + )( Y φ − + Y χ − ) − Y X + Y X − (cid:18) e X e X − (cid:19) − ( X χ + X φ ) Z φ − X φ Z χ − ln ( e X − Z X (A.76)As a check that this agrees with the reduced semi action, we integrate out the auxiliaryspinors Ψ ± and reconstruct the complex structures J ( ± ) . We shall need Y φ + = i (cid:2) iD + s (cid:96) − Λ r + (cid:3) Y φ − = i (cid:2) Λ (cid:96) − − iD − s r (cid:3) Y χ + = − i (cid:2) iD + s (cid:96) + Λ r + (cid:3) Y χ − = − i (cid:2) Λ (cid:96) − + iD − s r (cid:3) X + = (cid:2) iD + d (cid:96) − Σ r + (cid:3) Y X − = (cid:2) Σ (cid:96) − − iD − d r (cid:3) Z φ = − (cid:2) D + Σ (cid:96) − + D − Σ r + (cid:3) Z χ = (cid:2) D + Σ (cid:96) − − D − Σ r + (cid:3) Z X = i (cid:2) D + Λ (cid:96) − + D − Λ r + (cid:3) (A.77)From the variations we find: δ Λ r + : Λ (cid:96) − = 3 iD − s r − i (cid:18) e X e X − (cid:19) D − Xδ Σ r + : Σ (cid:96) − = iD − d r − (cid:18) e X − e X (cid:19) D − ( X χ + 2 X φ ) δ Λ (cid:96) − : Λ r + = − iD + s (cid:96) + 2 i (cid:18) e X e X − (cid:19) D + Xδ Σ (cid:96) − : Σ r + = iD + d (cid:96) + 2 (cid:18) e X − e X (cid:19) D + X χ . This implies that (in the notation of (4.38))Ψ (cid:96) − = i e X N (cid:2) (1 + E ) D − (cid:96) + 2 ED − ¯ (cid:96) − N + 1) ED − r − N + 1) D − ¯ r (cid:3) ]Ψ r + = i e X N (cid:2) ED + (cid:96) + 2 D + ¯ (cid:96) − (1 + E ) D + r − ED + ¯ r (cid:3) (A.78)These are the correct expressions for these auxiliary fermions, as may be checked usingthe matrices (4.37) in the formulae for J ( ± ) in [1]. B An alternative dual form
We have been studying the action (4.35). It can be cast into a different form which connectsto the results in [18]. We perform a Legendre transformation of the right semichiralsuperfields (for α = 1) ˜ K = K ( L, x, ¯ x ) − xr − ¯ x ¯ r (B.79)which together with the change L → − L (and some manipulations of the integral) bringsthe potential to the form found in [18]: − ( (cid:96) − ¯ r )(¯ (cid:96) − r ) + (cid:90) r +¯ r dq ln (1 + e q ) . (B.80)19e already know the metric in these ( X L , X R ) coordinates from [18] G = (cid:32) σ − Z − Z Zσ (cid:33) , (B.81)where Z := 11 + e r +¯ r = 11 + e (cid:96) +¯ (cid:96) − y − ¯ y , (B.82)and transformation to new ( X, Y ) coordinates reads y = K (cid:96) = ¯ (cid:96) − rr = ¯ (cid:96) − y . (B.83)The corresponding Jacobian is [1] J = (cid:32) − K RL K LL K RL (cid:33) . (B.84)We have K LL = − σ , K RR = − Zσ K LR = , K RL = (cid:48) (B.85)which implies G → (cid:32) − σ Zσ (cid:33) (B.86) References [1] U. Lindstrom, M. Rocek, R. von Unge and M. Zabzine, “Generalized Kahler man-ifolds and off-shell supersymmetry,” Commun. Math. Phys. , 833 (2007) [hep-th/0512164].[2] M. Goteman, U. Lindstrom, M. Rocek and I. Ryb, “Off-shell N=(4,4) supersym-metry for new (2,2) vector multiplets,” JHEP , 088 (2011) [arXiv:1008.3186[hep-th]].[3] M. Goteman, U. Lindstrom and M. Rocek, “Semichiral Sigma Models with 4DHyperkaehler Geometry,” JHEP , 073 (2013) [arXiv:1207.4753 [hep-th]].204] M. Rocek, K. Schoutens and A. Sevrin, “Off-shell WZW models in extended su-perspace,” Phys. Lett. B , 303 (1991).[5] U. Lindstr¨om, I. Ryb, M. Roˇcek, R.von Unge, M. Zabzine, “T-duality for the S piece in the S × S model,” October 2009, unpublished.[6] I. Bakas and K. Sfetsos, “T duality and world sheet supersymmetry,” Phys. Lett.B , 448 (1995) [hep-th/9502065].[7] T. Buscher, U. Lindstr¨om and M. Roˇcek, “New supersymmetric sigma models withWess Zumino terms,” Phys. Lett. B , 94 (1988).[8] S. J. Gates, Jr., C. M. Hull and M. Rocek, “Twisted Multiplets and New Super-symmetric Nonlinear Sigma Models,” Nucl. Phys. B , 157 (1984).[9] M. Gualtieri, “Generalized complex geometry,” Oxford University DPhil thesis,[arXiv:math/0401221].[10] F. Magri and C. Morosi, “A geometrical characterization of integrable Hamiltoniansystems through the theory of Poisson-Nijenhuis manifolds”, Quadrini del Diparti-mento di Matematica, Universita di Milano, 1984.[11] P. M. Crichigno, “The Semi-Chiral Quotient, Hyperkahler Manifolds and T-Duality,” JHEP , 046 (2012) [arXiv:1112.1952 [hep-th]].[12] M. Goteman, “N=(4,4) supersymmetry and T-duality,” Symmetry , 603 (2012)[arXiv:1208.2166 [hep-th]].[13] U. Lindstrom, M. Rocek, I. Ryb, R. von Unge and M. Zabzine, “T-duality andGeneralized Kahler Geometry,” JHEP , 056 (2008) [arXiv:0707.1696 [hep-th]].[14] J. Bogaerts, A. Sevrin, S. van der Loo and S. Van Gils, “Properties of semichiralsuperfields,” Nucl. Phys. B , 277 (1999) [hep-th/9905141].[15] I. T. Ivanov, B. b. Kim and M. Rocek, “Complex structures, duality and WZWmodels in extended superspace,” Phys. Lett. B , 133 (1995) [hep-th/9406063].[16] M. Goteman and U. Lindstrom, “Pseudo-hyperkahler Geometry and GeneralizedKahler Geometry,” Lett. Math. Phys. , 211 (2011) [arXiv:0903.2376 [hep-th]].[17] U. Lindstrom, I. T. Ivanov and M. Rocek, “New N=4 superfields and sigma mod-els,” Phys. Lett. B , 49 (1994) [hep-th/9401091].2118] A. Sevrin, W. Staessens and D. Terryn, “The Generalized Kahler geometry ofN=(2,2) WZW-models,” JHEP1112