Complex Geometry and Supersymmetry Pos(CORFU2011)070 , G = N (2.1) Ulf Lindström ] It Is Also Described 18 , There Is Always a Natural Equipped with a Metric ) ]

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PoS(CORFU2011)070 http://pos.sissa.it/ supersymmetry differs depending on the num- ) 2 , 2 ( ∗ ulf.lindstrom@physics.uu.se ber of manifest supersymmetries. The differences correspondGeneralized to Kähler different Geometry. aspects/formulations of Speaker. I stress how the form of sigma models with ∗ Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. c

Proceedings of the Corfu Summer Instituteand 2011 Gravity School and Workshops on ElementarySeptember Particle 4-18, Physics 2011 Corfu, Greece Ulf Lindström Division of Theoretical Physics Department of Physics and Astronomy Uppsala University SWEDEN E-mail: Complex Geometry and Supersymmetry PoS(CORFU2011)070 , g = N (2.1) Ulf Lindström ] it is also described 18 , there is always a natural equipped with a metric ) ]. In [ 0 1 , M 1 19 = ) ± . = ( ( ], which we now turn to. ] in his PhD thesis on Generalized J ) N 23 dB 18 ± ( or = ∇ ) 2 H complex structures and these, in turn, are , . The defining properties may be summa- , 1 H . g , , i.e., a manifold B = ( = ) H ) 1 + H N 2 ± , − g , ( ) Bihermitean 1 g ) ± 1 2 1 gJ = ( : ) , J 2 , ± ± t ( E g 0 J , -field, or NSNS two-form, is conveniently combined with Γ = ( B M Table 1: ] for related early discussions. , ( = N ) , 11 16 ) ± ( 2 − , Γ ], [ . This 2 = B 15 ) and a closed three-form ± = ( : 2 ) ( J E ± N ( J supplemented with (integrability)conditions acoording to ]. See also [ ) ) 14 ± ( J ]-[ , 1 g , M ( sigma models and complex geometry that I find remarkable: To each superspace formulation A reformulation of the data in Table.1 more adapted to Generalized Complex Geometry is as Generalized Kähler Geometry was defined by Gualtieri [ In this brief presentation I report on an aspects of the relation between twodimensional Bihermitean geometry is the set ) 2 , 2 the set In words, the metric iscovariantly constant hermitean with with respect respect to toand connections both which a are torsion the formed sumterms from of of the the a Levi-Civita closed potential connection two-form three-form.the metric Locally, into one the tensor three-form may be expressed in 2. Formulations of Generalized Kähler Geometry how GKG is a reformulation of the bihermitean geometry of [ rized as follows: Complex Geometry and Susy 1. Introduction ( of the sigma model, be it Complex Geometry. The latter subject was introduced by Hitchin in [ 2.1 Generalized Kähler Geometry I; Bihermitean Geometry. two complex structures corresponding formulation of the Generalized Kähler Geometry onthe the relevant target formulations space. of I Generalized first Kähler introduce are Geometry and collected then from the a sigma models.geometry: number The [ of results papers where we have used sigma models as tools to probe the PoS(CORFU2011)070 such (2.6) (2.7) (2.2) (2.3) (2.5) (2.4) is the c d : , Ulf Lindström ) G 2 reads ( ∗ , and second the J T ∗ ) 1 T ( ⊕ ⊕ T J T ∈ − , η ) = in , : ) ± ( + ) 0 ξ ν G y y . gJ ı . = ∗ dx = ) − 11 , = T , : ± µ ∗ Y ( I , ) − η ) ⊕ , ∂ x T ω − ± ( ı ξ ( ) = ( T = ( ! ⊕ ± ) ω ω 2 d c + ( 2 ) ∈ , T 2 1 x 1 − 0 1 1 0 c ( ( Y dd J , , d − =

J g X : − ξ I , . X y ) = ] requires the existence of two commuting = ∗ 2 : 0 = , 11 , L ) 1 T t I 3 ( 18 ± = 0 = ( − Bihermitean 2 ⊕ (+) ) ], is formulated on the sum of the tangent and cotan- J 2 , = η − gJ ω T ( x ) 18 I G C ± ] ω , t c ( (+) L ) → = ], [ Y J d 0 − ∗ c ], which for ( ± ] + into a generalized complex structure is first that it preserves 19 Table 2: d T = π J y : , , 21 ] = I , + ⊕ ) t X x 11 2 H J ( T ± in local coordinates where the complex structure is diagonal, and − J (+) : π = [ [ ) J : ω ) and the last line defines an almost product structure = , ∂ ∓ ) C ) J ] 1 π ( c ∗ (+) − ± 2.4 2 Y ( d T , ¯ J ∂ J ( [ X i ⊕ [ T equipped with an endomorphism which is a (generalized) almost complex 11 ∗ − T = ⊕ ) 2 T , 1 2 ( are the generalizations of the Kähler forms for the two complex structures, J denotes the Courant bracket [ ± ω C Generalized Complex Geometry [ with both GCSs satisfying ( Here differential which reads we see that the three-form is defined in terms of the basic data. 2.2 Generalized Kähler Geometry II; Description on The further requirements that turn the natural pairing on Generalized Complex Structures, i.e.: gent bundles structure, i.e., a map Complex Geometry and Susy where the matrix expressionintegrability refers condition to the coordinate basis with the Lie bracket,right Lie hand derivative side. and Generalized contraction Kähler of Geometry forms [ with vectorfields appearing on the Here PoS(CORFU2011)070 ± F . In (2.9) (2.8) ) ) (2.10) ± ( . Unlike J 2 ] , ) 4 [ ± ( Ulf Lindström K F , M    ( 1 . B 1 0 or as 0 − . E.g., restricting attention ) ) K ) − t ,    ( ± ) = . ( E ) ) J )    to the data in Table 2:    − , ± − ( ) ( 1 t ( ) g ) J 1 ) , J LR F − − − ( ) 1 2 t ( M − ω J JK ( t ( − ) 1 J F 0 ∓ ± ± − under ( = ) − 1 , as ) ω B − t ) (+) (+) LR − ) J ( (+) ∓ ] K ω ( ± J ( ) ( F ) J − 12 ) 2 − , , ]( (+) ± 0 ( ( , H 4 ) LL F , ) B ( − K , there exists locally defined non-degenerate “sym- g − ( , Conditions on ) d , ( (+) − ) 0 and [ J J ω ]. J ) [ M ± 6 0 ) 1 J ( , ( ± ∓ = , ± 2 J − t (+) ( ( ) , ) 0 E B ] gives the precise relation ± (+) (+) ( M ( LR 1 2 J i > ( ω Table 3: 18 K F 1 2 ) − ( v d    ) . The bihermitean data is recovered from = = )    ± (    ) ± J ( ± = 1 , (+) ( J /0,the left complex structure is given by v , Kähler geometry satisfies these condition, and so does bihermitean ( is an arbitrary contravariant vector field and the condition says that F F ∗ ) 1 0 B (+) 6= ± T J ν such that ] ( )    ) ⊕ − F ± ( = ( T J ) , 2 F , refers to the holomorphic property of 1 ( (+) ) J ) 0 [ , J ± 2 ( ( B The derivation from sigma models is given in [ The description is complete away from irregular points of certain poisson structures In the first condition As we have seen, the geometric data representing Generalized Kähler Geometry may be pack- Bihermitean geometry emphasizes the complex aspect of generalized Kähler Geometry. There Given the bi-complex manifold 1 2 tames the complex structures the Kähler case, the expressions are non-linear in second derivatives of aged in various equivalent ways as, e.g., 2.4 Summary to the situation Complex Geometry and Susy When formulated in where, e.g., each case, there is a complete description in terms of a Generalized Kähler potential geometry. In fact the Gualtieri map [ is another formulation where the (local) symplectic structure is in focus. 2.3 Generalized Kähler Geometry III; Local Symplectic Description plectic” two-forms PoS(CORFU2011)070 ), di- and, (3.3) (3.2) (3.1) K , 2.10 (2.11) (2.12) (2.13) 0 ¯ a , 0 a ], as a gen- 4 = Ulf Lindström : t is shorthand for , ¯ ]. a LR , K 12 a and coordinates where = : c , and ¯ (+) r J , r ,... = ` : ) L ), and, as shown in [ . E.g., R R ) K 1 X ± . ( , ¯ − ` , 2.13 L ) λ `,    X ,( LR , , , , , ¯ ¯ r r t = K X X ), as a potential for the geometry, ( F 0 0 0 0 , : ( χ K K ∂ ∂ = ¯ 2 2 ] ++ , ` ` J L ) c ∂ ∂ = = = = ∂ X X R 3.3 i − 0 r ` ∂ ∂ φ a , ( a ( iK J φ X X ) = χ , K r r R ± − + } = − 5 − X X ¯ ¯ ¯ X D ` D D K K ± (+) ∂ ∂ D ¯ , ¯ 2 2 D ` ` J ¯ L D [ ∂ ∂ X X − (+) , = ∂ ∂ X ]: Ω 0 λ D ± ( a + 17 D =    sigma model uses the generalized Kähler Potential χ ¯ { , D g ) + = + 2 ¯ : D , D (+) 2 LR ( λ Z d K = = S . ¯ supersymmetry algebra of covariant derivatives is r (+) , ) r 2 F , = 2 : R = ( ¯ `, N `, , formulation of the ) 2 = has many roles: as a Lagrangian as in ( 2 : , = L 2 K ( d ), as a “prepotential” for the local symplectic form The The relations may be extended to the whole manifold in terms of gerbes [ The are canonical. ) − ( 2.11 rectly: Note that 3.1 Superspace I as before, and their complex conjugate. The collective indexnotation is taken to be; The covariant derivatives can be usedand to left constrain and superfields. right We semichiral shall superfields need [ chiral, twisted chiral erating function for symplectomorphisms betweenJ coordinates where the matrix ( The metric is and the local symplectic structures have potential one-forms 3. Sigma Models Complex Geometry and Susy where we we introduced local coordinates PoS(CORFU2011)070 ], 4 [ (3.6) (3.7) (3.5) (3.4) (3.8) (3.9) ) ), the (3.10) L . compo- Y R 3.6 , ) ] but with L 1 → , ]. X 22 2 ( Ulf Lindström ). Of the two L ( 23 derivative and a (+) formulation. The ) λ instead appears in 1 ) , and d , ) 1 2 , − = ( ( 2 ) as geometric objects: J ( formulations, but the role of → − ) (+) coordinates 2.1 is the remaining manifest 2 + sigma model [ , ) ) F 1 2 ) ( − } ], we restrict the potential to , ( 1 ¯ α ) ), ( 2 J , . . ¯ ( 14 2 ϕ c and , ( . , ) c R . Both complex structures are now α 1 2.13 R + 3 , X ϕ + , 2 Ψ superspace and the action ( { ( − . α : in ( ) ) ϕ = L − = X 1 JD , − | R − } 0 and its c.c., which are the Ψ , (+) i 2 R D : K λ ( φ − = superspace [ X α = { + ` + ) | iQ XED 1 K (+) L − L Q , , + λ + − X Ψ 2 right derivative as a sum of ) ( 6 ¯ i D ( L D − − formulation ) , − ( φ ) K 2 superfield as D Q -fields in the combination ( D model by the replacement − 1 + − superspace directly or via the , : , B ) + 2 ) ) to , D ) 1 D − 2 yields a model in which ( i = iQ ( ¯ 1 1 have been eliminated , D + ) , D , ) X 2 α − − 3.3 + − 2 D R + 1 2 : + ( ( ( ( , J D D ¯ + D Ψ = 1 Z ε ( | D + ( Z : i = ) to + X D is now manifest. The complex structure and ¯ = S D 0. This is the standard form of a = ) to Z 3.3 L − + S = (+) = Ψ equations. We solve this by going to = D J 3.3 α ¯ α ` S ϕ X δϕ + only ¯ D ) components of a and ± ) ` ( 1 J and X , components will now both be chiral. The action then reads 2 ) ) ) ( ` 1 2 Y ) then reduces as , , , ` 2 2 ( ( 3.3 X to simplify the expressions. ( ) ∈ R is a Lagrange multiplier field enforcing α X , Ψ ϕ Note that these spinors have the role of Lagrange multipliers in the We may reduce the action ( To discuss reduction of the action ( The reduction entails representing the Similarily, reduction of ( L 3 ], whose X ( 14 Here with the vector potential now identified (up to factors) with complex structure. It is found from the resulting action now involves the metric and and both the auxiliary spinors non-manifest and arise in the extra supersymmetry transformations as explained in [ auxiliary fields with algebraic field equations in the The action ( nents of the [ complex structures the non-manifest supersymmetry 3.3 Superspace III reduction goes via Complex Geometry and Susy 3.2 Superspace II K and defining the where we have supressed the indices. Starting from generator of supersymmetry: PoS(CORFU2011)070 ˇ cek, , 216 JHEP 587 Ulf Lindström Phys. Lett. B Generalized Kahler geometry and T-duality and Generalized Kahler Nonabelian Generalized Gauge New N = (2,2) vector multiplets, respectively, which connects it to the , 067 (2005) [hep-th/0411186]. ) Generalized complex manifolds and − Generalized Kahler geometry from ( λ Linearizing Generalized Kahler Geometry, Generalized Kahler manifolds and off-shell Generalized Kahler geometry and manifest N A potential for Generalized Kahler Geometry, 0507 or , 291 (2006) [hep-th/0603130]. JHEP 77 (+) λ 7 First-order supersymmetric sigma models and target , 833 (2007) [hep-th/0512164]. , 235 (2005) [hep-th/0405085]. 269 257 sigma model, finally, is expressed directly in terms of the ) 1 , 263 (2010) [hep-th/0703111]. , 1 ( 16 Lett. Math. Phys. , 144 (2006) [hep-th/0508228]. 0601 , 020 (2009) [arXiv:0808.1535 [hep-th]]. , 056 (2008) [arXiv:0707.1696 [hep-th]]. , 062 (2009) [arXiv:0811.3615 [hep-th]]. sigma model is written directly in terms of the generalized Kähler potential. 0902 0802 formulation, i.e., the component formulation of the sigma model. JHEP Commun. Math. Phys. Commun. Math. Phys. ) model involves the one form ) 2 Generalized N = (2,2) supersymmetric nonlinear sigma models, 0910 ) , 0 , 2 2 , JHEP ( , 061 (2007) [hep-th/0702126]. 0 JHEP ( 1 ( -field, making that aspect of the geometry manifest. These are also the objets that JHEP B 0704 or , 008 (2007) [arXiv:0705.3201 [hep-th]]. ) 1 , 2 space geometry, JHEP IRMA Lect. Math. Theor. Phys. supersymmetric sigma models, 0708 Geometry, Multiplets, gerbes, supersymmetry, = (2,2) supersymmetric nonlinear sigma-models, supersymmetry, (2004) [hep-th/0401100]. The various sigma models have different formulations of Generalized Kähler Geometry man- ( [2] U. Lindstrom, R. Minasian, A. Tomasiello and M. Zabzine, [1] U. Lindstrom, [8] U. Lindstrom, M. Rocek, R. von Unge and M. Zabzine, [6] A. Bredthauer, U. Lindstrom, J. Persson and M. Zabzine, [7] U. Lindstrom, M. Rocek, R. von Unge and M. Zabzine, [9] U. Lindstrom, M. Rocek, I. Ryb, R. von Unge and M. Zabzine, [5] A. Bredthauer, U. Lindstrom and J. Persson, [4] U. Lindstrom, M. Rocek, R. von Unge and M. Zabzine, [3] U. Lindstrom, M. Rocek, R. von Unge and M. Zabzine, [12] C. M. Hull, U. Lindstrom, M. Rocek, R. von Unge and M. Zabzine, [10] U. Lindstrom, M. Rocek, I. Ryb, R. von Unge and M. Zabzine, [11] U. Lindstrom, M. Rocek, I. Ryb, R. von Unge and M. Zabzine, The ifest. Thus the Acknowledgement This short presentation highlights some aspects ofized the Kähler relation Geometry between derived sigma in models a and long General- and ongoing collaboration with Chris Hull, Martin Ro Complex Geometry and Susy 3.4 Summary metric and Rikard von Unge, Maxim Zabzineinsights and and contributions. others. I Iand am am Strings. also grateful Work supported grateful to by to all VR-grant the 621-2009-4066 of organizers my of collaborators the for Corfu their References workshop on Fields determine the local symplectic formulation. The PoS(CORFU2011)070 , 623 492 Nucl. Phys. Lett. , 490 Ulf Lindström 513 Nucl. Phys. B , 281-308 (2003). 54 Nucl. Phys. B Generalized Calabi-Yau metric Generalized Kähler Geometry in Properties of semichiral superfields, , 060 (2010) [arXiv:1005.5658 [hep-th]]. 1008 Quart. J. Math. Oxford Ser. 8 Oxford University DPhil thesis, [arXiv:math/0401221]. 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