We discuss additional supersymmetries for N = (2, 2) supersymmetric
non-linear sigma models described by left and right semichiral superfields.Comment: 11 pages. Talk presented by U.L. at "30th Winter School on Geometry
  and Physics" Srni, Czech Republic January 2010

Abou Zeid

Ademollo

Banks

Barret

Buscher

Christopher M. Hull

Dine

Fröhlicher

Gates

Grisaru

Howe

Hull

Kiritsis

Kutasov

Mohab Abou Zeid

Ooguri

Sevrin

Nuclear Physics B

English

arXiv

We investigate the relation between supersymmetry and geometry for two dimensional sigma models with target spaces of arbitrary signature, and Lorentzian or Euclidean world-sheets. In particular, we consider twisted forms of the two-dimensional $(p,q)$ supersymmetry algebra. Superspace formulations of the $(p,q)$ heterotic sigma-models with twisted or untwisted supersymmetry are given. For the twisted (2,1) and the pseudo-K\"{a}hler sigma models, we give extended superspace formulations

Abou-Zeid, M.

Hull, C.

MPG.PuRe

The geometry of Sigma-models with twisted supersymmetry

summary:We discuss additional supersymmetries for $\mathcal{N}=(2,2)$ supersymmetric non-linear sigma models described by left and right semichiral superfields

Göteman, M.

Lindström, U.

Czech digital mathematics library

Sigma models with non-commuting complex structures and extended supersymmetry

Crossref

The geometry of sigma models with twisted supersymmetry

Institute of Mathematics AS CR, v. v. i.

Archivum MathematicumM. Göteman; Ulf LindströmSigma models with non-commuting complex structures and extendedsupersymmetryArchivum Mathematicum, Vol. 46 (2010), No. 5, 323--331Persistent URL: http://dml.cz/dmlcz/141386Terms of use:© Masaryk University, 2010Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czARCHIVUM MATHEMATICUM (BRNO)Tomus 46 (2010), 323–331SIGMA MODELS WITH NON-COMMUTINGCOMPLEX STRUCTURESAND EXTENDED SUPERSYMMETRYM. Göteman and U. LindströmAbstract. We discuss additional supersymmetries for N = (2, 2) super-symmetric non-linear sigma models described by left and right semichiralsuperfields.1. IntroductionIn this short review we summarize some recent development in understandingextended supersymmetry for two-dimensional supersymmetric sigma models descri-bed entirely in terms of semichiral superfields. In the special case of one left andone right semichiral field we find that off-shell supersymmetry is impossible, butthat the model gives an interesting description of neutral (pseudo) hyperkählergeometry if we require pseudo supersymmetry instead. In the general case, weencounter Magri-Morosi concomitants in the conditions for closure of the algebraand find that the geometry has a nice description in terms of Yano f -structures onthe sum of two copies of the tangent space.Part of this review is based on a paper written in collaboration with MartinRoček and Itai Ryb.2. PreliminariesMost of this section is a review of results from [3].Sigma models are collections of maps φi, i = 1, . . . , d, from a space Σ to a targetspace T :(2.1) φi : Σ→ Tsubject to equations that derive from an action. We shall be concerned with thecase when Σ is a two-dimensional (1, 1) or (2, 2) superspace and the φis are real orcomplex superfields. For (1, 1) supersymmetry with real superfields and D-algebra(2.2) D2± = i∂+=,2000 Mathematics Subject Classification: primary 81Q60; secondary 51P05.Key words and phrases: supersymmetry, complex geometry.324 M. GÖTEMAN AND U. LINDSTRÖMthe action is(2.3)∫Σd2ξd2θD+φi(Gij(φ) +Bij(φ))D−φj .Here D± are the covariant derivatives w.r.t. the anticommuting spinorial coordinatesθ± and ∂+=are the derivatives w.r.t. the commuting coordinates ξ+= . The fieldequations that define the sigma model follow from the action (2.3) and read(2.4) ∇(−)+ D−φi = D+D−φi +D+φjΓ(−) ijk D−φk = 0 ,where the connection is defined to be the Levi-Civita connection w.r.t. the metricGij , plus torsion(2.5) Γ(−) ijk := Γ(0) ijk −12GilHljk ,and H is the B-field field-strength. We see that a lot of geometry on T enter theseexpressions. In particular we find that the target space has to carry a torsionfulgeometry and that the defining equation (2.4) for the sigma model contains thepullback of the Laplacian in this geometry. In fact, in [3], analyzing the sigmamodel (2.3), the following possible geometries were found:N=(1,1) N=(1,1) N=(2,2) N=(2,2) N=(4,4) N=(4,4)G G,B G G,B G G,BRieman. Riem. w. Torsion Kähler Bi-Herm. Hyperkähler Bi-Hyperherm.Tab. 1: The relation between the number of supersymmetries andtype of target-space geometry.In this analysis the starting point is the (1, 1) action (2.3) with G or both Gand B non-zero. Additional supersymmetries are non-manifest and follow from theansatz(2.6) δφi = ε+D+φkJ i(+)k(φ) + ε−D−φkJ i(−)k(φ) .This ansatz is for one additional left and one additional right supersymmetry.There may be more. For (2, 2) supersymmetry one finds the following set ofconditions from closure of the algebra :(2.7) J2(±) = −11 , N (J(±)) = 0 , [J(+), J(−)] · (FE) = 0 ,where the first line says that J(±) are almost complex structures and N is theNijenhuis tensor whose vanishing signals their integrability. Finally, (FE) is anexpression that has the structure of a field equation. The upshot of these conditionsis that the algebra closes on the kernel of the commutator of the two complexstructures and/or on-shell.Invariance of the action gives additional constraints:(2.8) ∇(±)J(±) = 0 , J t(±)GJ(±) = GSIGMA MODELS AND SUPERSYMMETRY 325The first condition requires each complex structure to be covariantly constantwith respect to the corresponding torsionful connections. The second condition ishermiticity of the metric with respect to both complex structures (bi-hermiteangeometry).More than one extra left and one extra right supersymmetry introduces a set ofleft and right complex structures J (A)(±) that each obey (2.7), (2.8) as well as someadditional constraints both from the algebra and from invariance of the action.The relevant case here is (4, 4) supersymmetry where A = 1, 2, 3 and the additionalconstraint from the algebra is that they obey the algebra of quaternions. Togetherwith the first condition in (2.7), we write this as(2.9) J (A)(±)J(B)(±) = −δAB + εABCJ (C)(±) .In both the (2, 2) and the (4, 4) case, on-shell closure of the algebra signals thata manifest (2, 2) formulation, if it exists, will have to contain auxiliary fields [7].On ker[J(A)(+) , J(B)(−)], however, a formulation in terms of (2, 2) geometry certainlyexists [3].The (2, 2) D-algebra is(2.10) {D±, D̄±} = i∂+=,and the constrained (2, 2) superfields that we shall need are chiral φ, twisted chiralχ and left and right semichiral XL/R fields obeying(2.11)D̄±φ = 0 , D±φ̄ = 0 ;D̄+χ = D−χ = 0 , D+χ̄ = D̄−χ̄ = 0 ;D̄+XL = 0 , D+X̄L = 0 ;D̄−XR = 0 , D−X̄R = 0 .In terms of these fields, a sigma model on ker [J(+), J(−)] is described by the action(2.12)∫d2ξ d2θ d2θ̄ K(φ, φ̄, χ, χ̄) ,where the (2, 2) supersymmetry is now manifest. When K is independent of eitherφ or χ, B = 0 and the geometry is Kähler, with K being the Kähler potential forthe metric G. In general, K will be a (linear) potential both for the metric andthe B-field. More precisely, since B is a gauge-field, it is better to think of K as apotential for H.The (4, 4) models on ker[J (A)(+) , J(B)(−) ] are described by the same action (2.12), withan equal number of chiral and twisted chiral fields, plus the following conditionson K [3]:(2.13) Kφiφ̄j +Kχiχ̄j = 0 , Kφiφ̄j −Kφj φ̄i = 0 .326 M. GÖTEMAN AND U. LINDSTRÖM3. The complement of ker [J(+), J(−)]. A first lookAs mentioned in the previous section, a (2, 2) formulation of the sigma models onthe complement (ker [J(+), J(−)])⊥ requires auxiliary fields. One type of (2, 2)-fieldswhose (1, 1) components include (spinorial) auxiliary fields are the semichiral ones[2]. We may thus consider(3.14)∫d2ξ d2θ d2θ̄ K(φ, φ̄, χ, χ̄,XL, X̄L,XR, X̄R) ,where, to get a sensible theory, we need to include an equal number of left andright semichiral fields and the generalized Kähler potential K satisfies some regu-larity conditions [10]. It is shown in [10] that (away from irregular points) (3.14)indeed gives a complete description of bi-hermitean geometry, or, equivalently, ofGeneralized Kähler geometry [6], [9], [11]. In what follows we shall be interestedin the complement (ker [J(+), J(−)])⊥ only, so we set the chiral and twisted chiralfields to zero and study(3.15)∫d2ξ d2θ d2θ̄ K(XL, X̄L,XR, X̄R) .The question we pose is what the last entry in Table 1 looks like from this (2, 2)perspective. More precisely, we make an ansatz for additional supersymmetriesamong the semichiral fields and read off the consequences from closure of thealgebra and invariance of the action. Clearly we will find the bi-hypercomplexgeometry this way, but there may be additional structure involving the auxiliaryfields and we also expect to find conditions on K analogous to (2.13) in the kernel.Previously the question of (4, 4) supersymmetry has been partly addressed, indoubly projective superspace, for (4, 4) multiplets containing (2, 2) semichiral and(2, 2) auxiliary fields in [8]. This corresponds to on-shell closure of the fields in theaction (3.15) and complements the present analysis.In our first analysis [4], we restrict ourselves to one set of semichiral fields. Thecorresponding target-space is thus four (real) dimensional. From a general ansatz forthe additional supersymmetries we deduce that off-shell closure of supersymmetry,{Q, Q̄} = i∂, is impossible. However, interestingly, if we instead ask for pseudosupersymmetry1 , {Q, Q̄} = −i∂, off-shell closure is possible and we elaborate theconsequences for the case of linear transformations:(3.16)δXL = iε̄+D̄+(X̄L + XR +1κX̄R)+ iκε̄−D̄−XL −iκε−D−XL ,δXR = iε̄−D̄−(X̄R − (κκ̄− 1)XL +κκ̄− 1κ̄X̄L)− iκ̄ε̄+D̄+XR +iκ̄ε+D+XR .1This makes the full symmetry of the model a twisted supersymmetry [1].SIGMA MODELS AND SUPERSYMMETRY 327The only parameter here, modulo field redefinition and R-symmetry, is the complexparameter κ. The asymmetry between left and right fields is irrelevant and is anartefact of our choices of redefinitions. These transformations close off-shell to apseudo supersymmetry.Invariance of the action (3.15) under the transformations (3.16) requires thatthe following equations are fulfilled(3.17) K11̄ −K12 − κ̄K1̄2 = 0 , (κκ̄− 1)K22̄ +K12 − κK12̄ = 0 ,where the subscripts 1 and 2 denote derivatives w.r.t. the left and right semichiralfield, respectively. The system (3.17) may be solved by separating variables to givea two-parameter family of solutionsK = F (y) + F̄ (ȳ) , y = αXL + βX̄L + γXR + δX̄R ,(3.18)whereγ = αβα+ κ̄β , δ =αβκα+ β .(3.19)Due to the linearity of the conditions (3.17), the solution integrated over the freeparameters is again a solution. The general K is thus(3.20) K(XL, X̄L,XR, X̄R) =∫dα dβ K(α, β;αXL + βX̄L + γXR + δX̄R) ,where K is a particular solution of the type (3.18).It is an interesting fact that the complex structures that follow from the solution(3.20) fulfill2(3.21) {J(+), J(−)} = −6 · 11 .This means that the H-field is trivial [10] and also that there are two local productstructures(3.22) S := 12√2(J(+) − 3J(−)), T := 14√2[J(+), J(−)] ,that preserve the metric G of signature (2, 2), and together with J(+) generateSL(2,C). The corresponding geometry is called neutral (pseudo) hyperkähler. Wehave thus reached an interesting conclusion; starting from a (2, 2) sigma-modeldescribed by a generalized Kähler potential K, the requirement that it in additioncarries non-manifest pseudo supersymmetry leads to neutral hyperkähler geometryon the target-space. Since K is a potential for all geometric objects in the (2, 2)model, including metric and complex structures, (albeit entering in a non-linearway), we see that it is also a potential for all the geometric objects in the neutralhyperkähler geometry. Furthermore, our approach provides a recipe for constructingsuch geometries.2Here we have chosen κ =√2 for definiteness, the argument goes through for any κ and onlydepends on the fact that |κκ̄+1κκ̄−1 | > 1.328 M. GÖTEMAN AND U. LINDSTRÖM4. The general caseIn this section we give a brief summary of the results of [5]. When there aremore then one set of semichiral fields, there is no problem with off-shell (ordinary)supersymmetry. The general ansatz isδ̄(±)X = U (±)ε̄±D̄±X , δ(±)X = V (±)ε±D±X ,(4.23)whereX :=XLX̄LXRX̄R =:(Xi),(4.24)and where we have assumed that there are an equal number d of left and rightfields but have suppressed the corresponding index, displaying only the generalindex i = 1, . . . , 4d. The matrices U and V are related via complex conjugationand rearrangement of the rows and columns. Note that the chirality constraints(2.11) imply that not all the entries in U and V are determined, in particular theymay be degenerate.Closure of the algebra gives the following equations:(4.25)[δ̄±, δ̄∓]Xi = 0⇐⇒M(U (±), U (∓))ijkD̄±XjD̄∓Xk= [U (±), U (∓)]imD̄±D̄∓Xm ,[δ̄±, δ∓]Xi = 0⇐⇒M(U (±), V (∓))ijkD̄±XjD∓Xk= [U (±), V (∓)]imD̄±D∓Xm ,[δ̄±, δ̄±]Xi = 0⇐⇒ N (U (±))ijkD̄±XjD̄±Xk = 0 ,and[δ±, δ̄±]Xi = iε̄±ε±∂+=Xi ⇐⇒M(U (±), V (±))ijkD̄±XjD±Xk=[(UV )(±)ij + δij]D̄±D±Xj+[(V U)(±)ij + δij]D±D̄±Xj .(4.26)Here N is the Nijenhuis tensor previously mentioned, and M is the Magri-Morosiconcomitant. Whereas the vanishing of N (J) for an almost complex structureJ implies that it is integrable, vanishing of M(J, L) for two commuting almostcomplex structures is related to their simultaneous integrability. In fact, M(J, L)is defined for two arbitrary endomorphisms J and L, but is only a tensor whenthey commute.Off-shell we have no relations between second derivatives and the product of twoderivatives. The relations (4.25)–(4.26) can thus only be satisfied if the left andright sides vanish independently. This sets the Nijenhuis tensors, the concomitantsand the commutators to zero separately (in the non-vanishing directions). ThereSIGMA MODELS AND SUPERSYMMETRY 329are many interesting aspects of this off-shell geometry. Here we only mention one:the existence of a Yano f -structure [12] on TM ⊕ TM .Consider the following 4d× 4d matrix defined on the sum of two copies of thetangent space:(4.27) F(±) := 0 U (±)V (±) 0 .The entries in U and V that are left undetermined in their definitions are now setto zero. Due to some of the conditions in (4.26), F(±) satisfies the condition for aYano f -structure(4.28) F3(±) + F(±) = 0 ,which is slightly weaker that the condition for an almost complex structure in thatit allows for degenerate matrices.Further, the two distributions that correspond to −F2(±) and 1 + F2(±) areintegrable in the sense of Yano, due to the vanishing of the Nijenhuis-tensors in(4.25).Now, invariance of the action (3.15) gives the following set of differential equa-tions(4.29)(KiU(+)i[j)k] = 0 , j, k 6= a ,where a represents the left chiral directions. Similar relations hold for U (−) andV (±). These relations are the semichiral counterpart of (2.13) for commutingcomplex structures and corresponds to (3.17) in the pseudo supersymmetric model.It is interesting that (4.29) has a geometric formulation related to the f -structure,at least when U and V are curl-free in the lower indices. It may then be written asa suitable projection of(4.30) F t(±)BF(±) = B ,where we have combined the Hessian Kij of the generalized Kähler potential intoan antisymmetric tensor B on TM ⊕ TM as(4.31) B = 0 K−Kt 0 .5. On-shellWe know from [3] that the underlying geometry for the (4, 4) theory is bi-hyper-complex, also on (ker [J(+), J(−)])⊥. We may compare our results to this by imposingthe field equations. In a real basis for the covariant derivatives,D± =: 12 (D± − iQ±) ,(5.32)330 M. GÖTEMAN AND U. LINDSTRÖMgoing on-shell amounts to settingD±Xi = (π̄(±)D±X)i ,(5.33)where the projection operator isπ(±) : = 12(1 + iJ(±)).(5.34)Identifying J(±) with J(3)(±) of the quaternion worth of complex structures in thebi-hypercomplex geometry, and identifying the transformations (4.23) with theknown transformations in terms of J (1)(±) and J(2)(±) on shell, we identify(5.35) 12(J(1)(±) − iJ(2)(±))= U (±)π , 12(J(1)(±) + iJ(2)(±))= V (±)π̄on shell.Since the on-shell conditions gives a relation between second derivatives andthe product of two derivatives, the relations in (4.25) have more solutions. Clearlythe off-shell transformations form a subset of the on-shell ones. In fact, with theidentification (5.35) all the relations in (4.25) should be satisfied, possibly up toconstraints from invariance of the action, since the bi-hypercomplex geometry givesthe full answer. We have checked that this is indeed the case.Acknowledgement. We are grateful to our collaborators on one of the papersreviewed, Martin Roček and Itai Ryb. The research of U.L. was supported by VRgrant 621-2009-4066.References[1] Abou-Zeid, M., Hull, C. M., The geometry of sigma-models with twisted supersymmetry,Nucl. Phys. B 561 (1999), 293–315, [arXiv:hep-th/9907046].[2] Buscher, T., Lindström, U., Roček, M., New supersymmetric sigma models with Wess-Zuminoterms, Phys. Lett. B 202 (1988), 94–98.[3] Gates, S. J., Hull, C. M., Ročcek, M., Twisted multiplets and new supersymmetric nonlinearsigma models, Nuclear Phys. B 248 (1) (1984), 157–186.[4] Göteman, M., Lindström, U., Pseudo-hyperkahler Geometry and Generalized Kahler Geome-try, to be published in Lett. Math. Phys., arXiv:0903.2376[hep-th].[5] Göteman, M., Lindström, U., Roček, M., Ryb, I., Sigma models with off–shell N = (4, 4)supersymmetry and noncommuting complex structures, arXiv:0912.4724[hep-th].[6] Gualtieri, M., Generalized complex geometry, Ph.D. thesis, Oxford University, 2004, [math/0401221[math-dg]].[7] Lindström, U., Generalized N = (2, 2) supersymmetric non-linear sigma models, Phys. Lett.B 587 (2004), 216–224, [arXiv:hep-th/0401100].[8] Lindström, U., Ivanov, I. T., Roček, M., New N=4 superfields and sigma models, Phys. Lett.B 328 (1994), 49–54, [arXiv:hep-th/9401091].[9] Lindström, U., Minasian, R., Tomasiello, A., Zabzine, M., Generalized complex manifoldsand supersymmetry, Commun. Math. Phys. 257 (2005), 235–256.SIGMA MODELS AND SUPERSYMMETRY 331[10] Lindström, U., Roček, M., Unge, R. von, Zabzine, M., Generalized Kaehler manifolds andoff-shell supersymmetry, Commun. Math. Phys. 269 (2007), 833–849.[11] Lindström, U., Roček, M., Unge, R. von, Zabzine, M., Linearizing generalized Kählergeometry, JHEP 0704 (2007), 28pp., [arXiv:hep-th/0702126].[12] Yano, K., On a structure f satisfying f3 + f = 0, Tech. Rep. Univ. of Washington 12 (1961).[13] Yano, K., On a structure defined by a tensor field of type (1, 1) satisfying f3 + f = 0, TensorN. S. 14 (1963), 9.Theoretical Physics, Department of Physics and Astronomy,Uppsala University, Box 516, SE-751 20 Uppsala, SwedenE-mail: lindoulf@gmail.com

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Sigma models with non-commuting complex structures and extended
  supersymmetry

Sigma models with non-commuting complex structures and extended supersymmetry

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Institute of Mathematics AS CR, v. v. i.