Sigma Models and Complex Geometry

Home , Superspace

Ulf Lindström

Theoretical Physics Department of Physics and Astronomy University of Uppsala Sweden

Kerkyra, September 12, 2009

Ulf Lindström Superspace is smarter Outline

Sigma models Supersymmetry Superspace SUSY sigma models Complex geometry Quotients Hyperkähler quotient

Ulf Lindström Superspace is smarter Outline

Sigma models Supersymmetry Superspace SUSY sigma models Complex geometry Quotients Hyperkähler quotient

Ulf Lindström Superspace is smarter Outline

Sigma models Supersymmetry Superspace SUSY sigma models Complex geometry Quotients Hyperkähler quotient

Ulf Lindström Superspace is smarter Outline

Sigma models Supersymmetry Superspace SUSY sigma models Complex geometry Quotients Hyperkähler quotient

Ulf Lindström Superspace is smarter Outline

Sigma models Supersymmetry Superspace SUSY sigma models Complex geometry Quotients Hyperkähler quotient

Ulf Lindström Superspace is smarter Outline

Sigma models Supersymmetry Superspace SUSY sigma models Complex geometry Quotients Hyperkähler quotient

Ulf Lindström Superspace is smarter Outline

Sigma models Supersymmetry Superspace SUSY sigma models Complex geometry Quotients Hyperkähler quotient

Ulf Lindström Superspace is smarter Sigma models in d=2 Generalized complex geometry Generalized Kähler geometry (2,2) superspace Sigma model description of GKG The role of K New vector multiplets T-duality (4,4)

Ulf Lindström Superspace is smarter Collaborators

Tom Buscher, Anders Karlhede, Nigel Hitchin, Chris Hull, Martin Rocek,ˇ Rikard von Unge, Maxim Zabzine, Malin Göteman, Itai Ryb

Ulf Lindström Superspace is smarter References

L. Alvarez-Gaumé and D. Z. Freedman, “Geometrical Structure And Ultraviolet Finiteness In The Supersymmetric Sigma Model,” Commun. Math. Phys. 80, 443 (1981)

C. M. Hull, A. Karlhede, U. Lindstrom and M. Rocek, “Nonlinear Sigma Models And Their Gauging In And Out Of Superspace,” Nucl. Phys. B 266, 1 (1986)

N. J. Hitchin, A. Karlhede, U. Lindstrom and M. Rocek, “Hyperkahler Metrics And Supersymmetry,” Commun. Math. Phys. 108, 535 (1987).

S. J. . Gates, C. M. Hull and M. Rocek, “Twisted Multiplets And New Supersymmetric Nonlinear Sigma Models,” Nucl. Phys. B 248, 157 (1984).

M. Gualtieri, “Generalized complex geometry,” Oxford University DPhil thesis, arXiv:math.DG/0401221.

U. Lindstrom, M. Rocek, R. von Unge and M. Zabzine, “Generalized Kaehler manifolds and off-shell supersymmetry,” arXiv:hep-th/0512164.

Ulf Lindström Superspace is smarter Origin

IL NUOVO CIMENTO VOL. XVI, N. 4 16 Maggio 1960

The Axial Vector Current in Beta Decay (*).

M. GELL.-MA~-N (**) Coll~ge de Fra~ce and Ecole Normale Supdrieure - Paris (***)

M. LEVY FacuZte des ~%ie~ces, Orsay, aml Ecole Normale Sup~rieure - .Paris (***)

(ricevuto il 19 Febbraio 1960)

Summary. -- In order to derive in a convincing manner the formula of Goldberger and Treiman for the rate of charged pion decay, we consider the possibility that the divergence of the axial vector current in p-decay may be proportional to the pion field. Three models of the pion-nucleon interaction (and theUlf Lindströmweak current) Superspaceare presented is smarterthat have the required property. The first, using gradient coupling, has the advantage that it is easily generalized to strange particles, but the disadvantages of being unrenormalizable and of bringing in the vector and axial vector currents in an unsymmetrical way. The second model, using a strong interaction proposed by SCHWlNGER and a weak current proposed by POLKINGHORNE, is renormMizable and symmetrical bel,ween V and A, but it involves postulating a new particle and is hard to extend to strange particles. The third model resembles the second one except that it is not necessary to introduce a new particle. (Renormalizability in the usual sense is then lost, however). Further research along these lines is suggested, including consideration of the possibility that the pion decay rate may be plausibly obtained under less stringent conditions.

(*) Supported in part by the Alfred P. Sloan Foundation and by the United States Air Force through the European Office, Air Research and Development Command. (**) National Science Foundation Senior Postdoctoral Fellow. (***) Permanent address: California Institute of Technology, Pasadena, Cal. ('.*) Postal address: Laboratoire de Physique Th~orique et Hautes Energies, B.P. 12, Orsay (Seine et ()i,~e). THE AXIAL VECTOR CURICENT IN BETA DECAY 717

Yet we have evidenee that the weak interactions are symmetrical between V and A, particularly their apparent equality of strength and the fact that for the leptons, which have no strong couplings, the weak coupling is just r,,(l+y~).

5. - The a model.

We have another example of a theory in which eq. (5) holds, if we take a Lugrangian for the strong interactions that iv essontially one proposed by SeHW~NG~:r,, (~) and then for the axial vector current the form suggested by POLKINGI-IOI%NE(17). Again, for simplicity, we restrict ourselves to mtcleons and pions only, except that we introduce (following SCHW~GE~) ~ new scalar meson ~, with isotopic spin zero. It has strong interactions, and thus might easily have escaped observation if it is much heavier than ::, so that it would disintegrate immediately into two pions. It woald appear experimentally as a resonant state of two pions with J=0, [ :0. We tak(+ for our Lagrangian the following one, which leads to a renor- realizable theory of the strong int(~r~wtions:

/z;n- (36) ~?2 : ---N[)' ~ + nt,,--go(o 4- i'r'TrT,~)]N-- 2 2

+ a~) 2 _ _2 a(a ~ ÷ z2)] Ulf Lindström Superspace is smarter ]0 ] where fo = gol2mo. We have the usual pseudos(;alar theory of the pion, with the a added in a rather symmetric~fl way. The nature of the symmetry iv made much clearer if we perform a translation of the field v~ri-~bl.e a and re-e~press the Lagran- glair in terms of the variable:

(37) a' ~ a ----1 2],, " We have: (a,~,)o 'u°~ (3s) '2 2 (:~2 + ~,~) _

-- z,, [~-~ + ~r.. ~ 12 A OJ , [

(*G) j. SCI~WINGER: Ann. Ph.~s., 2, 407 (1957). (17) j. C. I)OLKINGI[OI{NE:-~'tto~'o Cimetdo, 8, 179, 781 (1958). Sigma models

φi :Σ Ñ T » i j S dφ Gij pφq dφ Σ 2 i B2 i B j i B k ∇ φ : φ φ Γjk φ 0

» ! ) » d¡2 µν i j S µ dξ η BµX Gij pXqBνX ...... ΣB BΣ

Ulf Lindström Superspace is smarter i) The mass-scale µ shows that the model typically will be non-renormalizable for d ¥ 3 but renormalizable and classically conformally invariant in d 2. ii) We have not included a potential for X and thus excluded Landau-Ginsburg models. iii) There is also the possibility to include a Wess-Zumino term. We shall return to this when discussing d 2 iv) From a quantum mechanical point of view it is useful to think of Gij pXq as an inﬁnite number of coupling constants:

p q 0 1 k Gij X Gij Gij,k X ... v) Classically, it is more rewarding to emphasize the geometry and think of Gij pXq as a metric on the target space T . This is the aspect we shall be mainly concerned with.

Ulf Lindström Superspace is smarter vi) The invariance of the action S under Diff pT q:

i i1 1 X Ñ X pXq , Gij pXq Ñ Gi1j1 pX q

(field-redefinitions from the point of view of the field theory on Σ), implies that the sigma model is defined by an equivalence class of metrics. N.B. This is not a symmetry of the model since the “coupling constants” also transform. It is an important property, however. Classically it means that the model is extendable beyond a single patch in T , and quantum mechanically it is needed for the effective action to be well-defined.

Ulf Lindström Superspace is smarter Supersymmetry

The algebra depends on the dimension d and the number N of supersymmetries. In d 4 we have

t a bu abp µ q ab p q ab Qα, Qβ 2δ γ C αβPµ CαβZ γ5C αβY

Qa are translation-invariant spinors that satisfy a Majorana reality condition and transform under some internal symmetry group G UpNq (corresponding to the index a).

Weyl-spinors and N 1: tQα, Q α9 u 2iBαα9

Ulf Lindström Superspace is smarter Superspace

Just like, e.g., translations are represented by differential operators acting on functions on Minkowski space

µ δP Φpxq irξ Pµ, Φpxqs ,

Pµ iBµ ,

supersymmetry transformations may be represented by differential operators acting on functions on superspace:

α α9 δQΦpx, θq ir Qα ¯ Q α9 , Φpx, θqs

where

1 α9 1 α Q iB θ¯ B 9 , Q 9 iB 9 θ B 9 . α α 2 αα α α 2 αα B B and α : Bθα .

Ulf Lindström Superspace is smarter Covariant derivatives

tDα, D α9 u 2iBαα9 s s tD, Qu tDs, Qu tD, Qu tDs, Qu 0

1 α9 1 α D B i θ¯ B 9 , D¯ 9 B 9 i θ B 9 , α α 2 αα α α 2 αα Using these we may impose covariant constraints. E.g., chirality: s Dα9 φ 0 Dαφ¯ .

Ulf Lindström Superspace is smarter SUSY sigma models

Ex. (d 2, N p2, 2q chiral ﬁelds)

¯ tDα, Dβu 2iBαβ

φpzq Ñ φpz, θq :

2 X φ| , Ψα Dαφ| , F D φ| » S Ñ dzdzD¯ 2D¯ 2 K pφ, φ¯q » pB p ¯ qB¯ ¯ q dzdz¯ XGXX¯ X, X X ...

Ulf Lindström Superspace is smarter where

p ¯ q B B p ¯ q GXX¯ X, X X X¯ K X, X

ðñ T carries Kähler Geometry

Susy σ models ðñ Geometry of T

d= 6 4 2 Geometry N= 1 2 4 Hyperkähler N= 1 2 Kähler N= 1 Riemannian

(Odd dimensions have the same structure as the even dimension lower.)

Ulf Lindström Superspace is smarter Complex Geometry

Complex structure: J : TM ý J2 ¡1

1 p ¨ q Projectors: π¨ : 2 11 iJ

Nijenhuis: N pJq 0 ðñ π©rπ¨u, π¨vs 0

Hermitean Metric: Jt GJ G

Kähler: ∇J 0, Gzz¯ Bz Bz¯K pz, z¯q

Hyperkähler: JA, A 1, 2, 3 JAJB ¡δAB ABCJC

Ulf Lindström Superspace is smarter Hyperkähler

» S d 4xD2D¯ 2K pφ, φ¯q

Extra, non-manifest SUSY: ¯ ¯ δφi D¯ 2pε¯Ω¯ i q , δφ¯i D2pεΩi q . Invariance of the action and closure of the algebra (on-shell) iff the following JpAq represent a Hyperkähler geometry:

£ £ 0 Ω¯i 0 iΩ¯i Jp1q j Jp2q j Ω¯ i 0 ¡iΩ¯ i 0 ¯j ¯j £ iδi 0 Jp3q j 0 ¡iδ¯i ¯j

Ulf Lindström Superspace is smarter Quotient

Back to bosonic: » i µ j S dxBµφ Gij pφqB φ .

i A i i δφ λ kApφq rλk, φs Lλk φ .

Under such a transformation the action varies as » i µ j δS dxBµφ Lλk Gij pφqB φ .

It is thus an invariance of the action if it is an isometry

Lλk Gij 0 .

Ulf Lindström Superspace is smarter Algebra B rk , k s c Ck , k k i . A B AB C A A Bφi We may gauge the isometry using minimal coupling B i ÑB i ¡ A i pB ¡ A q i i µφ µφ AµkA µ AµkA φ ∇µφ , » i µ j S Ñ dx∇µφ Gij pφq∇ φ .

Extremizing this action w.r.t. A yields a new sigma model on the space of orbits of the gauge group: » ¡ © B i ¡ ¡1AB Bµ j S dx µφ Gij H kiA kjB φ , where i j HAB kAGij kB .

Ulf Lindström Superspace is smarter SUSY quotient

Such a quotient yields a new sigma model with a new target space. But we need to preserve additional structure such as SUSY.

Ex. Flat space, (i=1,2) » » S d 4xD2Ds 2K pφφ¯q d 4xD2Ds 2φi φ¯i

Isometry:

δφi iλφi , δφ¯ ¡iλφ¯i

Gauged action: » S d 4xD2Ds 2pφi φ¯i eV ¡ cV q

Ulf Lindström Superspace is smarter Extremizing:

¢ φi φ¯i V ¡ln c

Quotient potential: ¢ ¢ φi φ¯i K˜ c ln 1 c lnp1 ζζ¯q .... , c where

ζ φ1{φ2 .

K˜ is the potential for the Fubini-Study metric on CP1.

Ulf Lindström Superspace is smarter HKQ

In the example the Kähler geometry of the original model was inherited by the quotient geometry. It is much more difﬁcult to preserve hyperkähler geometry. This requires gauging a tri-holomorphic isometry and performing a quotient which respect to the complexiﬁed action of this isometry. The latter point arises already in the Kähler quotient just illustrated: The general formula is » 1 p¡ 1 tJV q V K pφ, φ¯q Ñ Kˆ pφ, φ,¯ V q K pφ, φ¯q dte 2 µ 0

Ulf Lindström Superspace is smarter 548 N. J. Hitchin, A. Karlhede, U . Lindstrom, and M. Rocek

Consider, however, the orthogonal complement of H in TM. The complement of TN in TM is spanned at m e N by the normal vectors gradµ^, A = 1,..., dim G and the complement of H in TN is spanned by the vertical vectors kA, where the vector fields kA arise from a basis of g. But

x g(grad/ ! Y) = dµ (Y) = !{X9 Y) = g(lX, Y), (3.23) and so (3.24)

Thus the vector space spanned by gradµ^ and k A is a complex vector space, and so the complement of H and hence H itself is a complex vector bundle. The metric on M was Kahlerian, so I commutes with the covariant derivative. Since H is complex, I commutes with the orthogonal projection and so I commutes with the connection VH. Thus the induced metric on M is Kahlerian. This infinitesimal calculation disguises one essential aspect of the problem: the vector fields X, IX generate a complex Lie algebra of holomorphic vector fields and hence a local holomorphic action of the complex group Gc obtained from the complex Lie algebra g® C. Suppose this extends to a global action (equivalent to the completeness of the vector fields IX) then the symplectic quotient M, considered as a complex manifold, is simply the ordinary manifold quotient as in Sect. 3A in the holomorphic setting. However, since Gc is not compact, we would obtain non Hausdorff behaviour in this quotient, unless we restrict the action of G c to a suitable open set of points in M. This open set consists of those points whose Gc orbits meet N and are called stable points (see Fig. 3). I n many cases in algebraic geometry [13] this idea of stability coincides with a pre existing algebraic definition called Mumford stability.

(D) Hyperk"hler Quotients Suppose finally that M4n is a hyperk"hler manifold having a metric g and covariantly constant complex structures I, J, K which behave algebraically like quaternions:

IJ= JI = etc. (3.25)

Gc orbit

Fig. 3. The orbits of the group G and of its complexification Gc. G acts on µ 1(0) and M is the quotient space corresponding to this action. The same space is obtained if one considers the extension of µ 1(0) by exp(/ X) and takes the quotient by Gc

Ulf Lindström Superspace is smarter Summary and conclusions part I

Supersymmetric sigma models provide a powerful tool to probe complex geometry. The more supersymmetries, the more specialized geometry N=2 in d=4 has a hyperkähler target space. Gauging isometries and taking a quotient leads to new models. The hyperkähler reduction is suggested to us by superspace. Additional supersymmetries, when examined at the p2, 2q level, lead to interesting new structures on the target space.

Ulf Lindström Superspace is smarter Sigma models in d=2

The (1,1)-D-algebra:

2 B D¨ i

» ¡ © 2 i p q j S d xD D¡ D ϕ Gij Bij D¡ϕ .

The (1,1) analysis by Gates Hull and Rocekˇ gives:

Susy (0,0) (1,1) (2,2) (2,2) (4,4) (4,4) Bgd G, B G G, B G G, B Geom Riem. Kähler biherm. hyperk. bihyperc.

Ulf Lindström Superspace is smarter The (1,1)-D-algebra:

2 B D¨ i

» ¡ © 2 i p q j S d xD D¡ D ϕ Gij Bij D¡ϕ .

The (1,1) analysis by Gates Hull and Rocekˇ gives:

Susy (0,0) (1,1) (2,2) (2,2) (4,4) (4,4) Bgd G, B G G, B G G, B Geom Riem. Kähler biherm. hyperk. bihyperc.

Ulf Lindström Superspace is smarter Ansatz for the extra supersymmetries:

i i j ¡ i j δϕ Jp qj D ϕ Jp¡qj D¡ϕ

Invariance of the action and closure of the algebra requires the geometry to be bi-hermitean:

2 Jp¨q ¡11

t Jp¨qGJp¨q G

NpJp¨qq 0

p¨q 2 p¨q 0 ¡1 ∇ Jp¨q 0 , Γ Γ ¨ G H , H : dB

H HJp¨qJp¨q

Ulf Lindström Superspace is smarter Generalized Complex Geometry

Complex structure: J : TM ` T ¦M ý J 2 ¡1 1 Π¨ : p11 ¨ J q 2 “Nijenhuis”:

NCpJ q 0 ðñ Π©rΠ¨u, Π¨vsC 0 where u pU, ξq , v pV , ρq

1 ru, vs rU, V s L ρ ¡ L ξ ¡ dpı ρ ¡ ı ξq C U V 2 U V The automorphisms of this courant bracket are diffeomorphisms and b-transforms: ebpU, ξq pU, ξ ibq , db 0 .

Ulf Lindström Superspace is smarter In a coordinate basis pBx , dxq a b-transform acts on J as follows:

¢ ¢ 1 0 1 0 J , b 1 ¡b 1

In such a basis, the natural pairing

pU, ξq, pV , ρq ¡ ıU ρ ıV ξ is represented by the matrix ¢ 1 0 I 0 1

A ﬁnal requirement ofn GCG is that

J t IJ I

Ulf Lindström Superspace is smarter Generalized Kähler geometry

Generalized Kähler:

D pJ1, J2q ; rJ1, J2s 0

2 G ¡J1J2 , G 1

Ex. Kähler (ω GJ): ¢ ¢ ¢ J 0 0 ¡ω¡1 0 G¡1 J J G 1 0 ¡Jt 2 ω 0 G 0

GKG Ø Bi-Hermitean (the G-map):

J p1,2q ¢ £ ¢ p q p¡q ¡1 ¡1 1 0 J ¨ J ¡pωp q © ωp¡qq 1 0 tp q tp¡q B 1 ωp q © ωp¡q ¡pJ ¨ J q ¡B 1

Ulf Lindström Superspace is smarter General description of GKG

So Bi-Hermitean and GKG data are equivalent. But what is the most general description? Again, superspace has the answer.

The description should be p2, 2q symmetric, as we know from GHR. They found the complete description of kerrJp q, Jp¡qs but its complement was not described.

The kernel corresponds to the target space geometry of a sigma model with chiral and twisted chiral p2, 2q-superﬁelds. The complement is coordinatized by semi-chiral ﬁelds.

Ulf Lindström Superspace is smarter (2,2) superspace

The (2,2)-D-algebra:

t ¨ ¯¨u B D , D 2i

Reduction to (1,1):

1 ¨ D¨ : ? D¨ D¯¨ 2 i ¨ Q¨ : ? D¨ ¡ D¯¨ 2 The (1,1)-D-algebra:

2 B D¨ i

Ulf Lindström Superspace is smarter (2,2) superﬁelds

Chiral ﬁelds φ:

D¯¨φ 0 ñ D¨φ¯ 0

Twisted chiral ﬁelds χ:

D¯ χ D¡χ 0 ñ D χ¯ D¯¡χ¯ 0

Left/Right semi-chiral ﬁelds XL{R: ¯ ñ ¯ D XL 0 D XL 0

¯ ñ ¯ D¡XR 0 D¡XR 0

These are all the ﬁelds needed.

Ulf Lindström Superspace is smarter Complex linear ﬁelds Σφ: ¯ ¯ D D¡Σφ 0 ñ D D¡Σ¯ φ 0

Dual to chiral ﬁelds

Complex twisted linear ﬁelds Σχ: ¯ ¯ D D¡Σχ 0 ñ D D¡Σ¯ χ 0

Dual to twisted chiral ﬁelds

Ulf Lindström Superspace is smarter N=(1,1) content

Deﬁne: ¢ i 0 J : 0 ¡i

Chiral ﬁelds: ¢ φ Φ: ñ ¨Φ J ¨Φ φ¯ Q D

Twisted chiral ﬁelds: ¢ χ χ : ñ ¨χ ¨J ¨χ χ¯ Q D

Read off the non-manifest second susy by projecting to the θ2 independent part.

Ulf Lindström Superspace is smarter Semi-chiral ﬁelds:

| | XL{R : XL{R , ψL¡{R : Q©XL{R ¢ ¢ XL{R ψL¡{R XL{R : ¯ , ΨL¡{R : ¯ XL{R ψL¡{R Q XL JD XL, Q¡XR JD¡XR and

¡ B Q ΨL¡ JD ΨL¡, Q¡ΨL¡ i XL

¡ B Q¡ΨR JD¡ΨR , Q ΨR i XR

The Ψ’s are auxiliary fermions.

Ulf Lindström Superspace is smarter Relation to GKG

In p1, 1q:

δΨS 0 : ñ pJp¨q, G, H dBq ,

2 Jp¨q ¡1, NpJq 0, rJp q, Jp¡qs 0 ,

t c c Jp¨qGJp¨q G, H dp qωp q ¡dp¡qωp¡q

A complete description of GKG.

Ulf Lindström Superspace is smarter The dependence on the generalized Kähler potential is non-linear (for simplicity consider semi-schiral ﬁelds only): E.g., ¢ ¢ LR RL j 0 K jKRL K CRR Jp q RL RL , Jp¡q K CLL K jKLR 0 j where ¢ i 0 j : , C : rj, K s , K RLK 11 . 0 ¡i LR

Only a symplectic form Ω depends linearly on the Hessian of K : ¢ 0 K Ω LR . ¡KRL 0

The metric and B-ﬁeld depend non-linearly:

G ΩrJp q, Jp¡qs , B ΩtJp q, Jp¡qu

Ulf Lindström Superspace is smarter Generating function

There are two special sets of Darboux coordinates for the p L q symplectic form Ω. One set, X , YL , is also canonical p R q coordinates for Jp q and the other set, X , YR is canonical coordinates for Jp¡q. The symplectomorphism that relates the two sets of coordinates has thus a generating function. This generating function is in fact the generalized Kähler-potential K pXL, XRq.

L Ð p L Rq Ñ R (X¢, YL) K X , X (X¢, YR) i 0 i 0 Jp q Jp¡q 0 ¡i 0 ¡i

This fact is a key ingredient in the proof that we have a complete description or GKG.

Ulf Lindström Superspace is smarter The roles of K

p ¯ ¯ q K φ, φ, χ, χ,¯ XL{R, XL{R is the superspace Lagrangian for a p2, 2q sigma model with Generalized Kähler target space geometry.

p ¯ ¯ q K φ, φ, χ, χ,¯ XL{R, XL{R is generalized Kähler potential for the metric G and B-ﬁeld

For ﬁxed chiral and twisted chiral ﬁelds, p ¯ ¯ q K φ, φ, χ, χ,¯ XL{R, XL{R generates symplectomorphisms.

Ulf Lindström Superspace is smarter New vector multiplets

p ¯ ¯ q K φ, φ, χ, χ,¯ XL{R, XL{R

(Abelian) Isometries:

pB ¡B q kφ i φ φ¯

pB ¡B ¡B B q kφχ i φ φ¯ χ χ¯

pB ¡B ¡B B q kLR i L L¯ R R¯

Ulf Lindström Superspace is smarter The corresponding gauged Lagrangians:

φ Kφpφ φ¯ V , xq

φ χ 1 Kφχpφ φ¯ V , χ χ¯ V , ipφ ¡ φ¯ χ ¡ χ¯q V , xq

p ¯ L ¯ R p ¡ ¯ ¡ ¯ q 1 q KX XL XL V , XR XR V , i XL XL XR XR V , x with gauge transformations for the vectors;

δV φ ipΛ¯ ¡ Λq ¯ δV χ ipΛ˜ ¡ Λ˜q

1 ¯ δV Λ¯ Λ Λ˜ Λ˜

L{R p¯ ¡ q δV i ΛL{R ΛL{R

1 ¯ ¯ δV ΛL ΛL ΛR ΛR

Ulf Lindström Superspace is smarter The invariant ﬁeld strengths are the usual ones

φ φ W iD¡D¯ V , W¯ iD¯¡D V

χ ¯ χ W˜ iD¯¡D¯ V , W˜ iD¡D V and the new

1 1 L R D¯ D¯¡p ip qq F 2 V V V

1 1 L R ˜ D¯ D¡p ip ¡ qq F 2 V V V

1 1 φ 1 D¯ pV ipV V χqq : D¯ V G 2 2

1 1 φ χ 1 ¡ D¯¡pV ipV ¡ V qq : D¯¡V˜ G 2 2

Reduction to (1,1). Non-abelian extensions. Applied to T-duality.

Ulf Lindström Superspace is smarter T-duality

φ χ 1 Kφχpφ φ¯ V , χ χ¯ V , ipφ ¡ φ¯ χ ¡ χ¯q V q

1 1 1 1 ¯ ¡ V ¡ ¯ V¯ ¡ V˜ ¡ ¯ V˜ 2XL 2XL 2XR 2XR

δXL,R: ˜ ñ V and V pure gauge. Plug back to ﬁnd Kφχpφ, φ,¯ χ, χ¯q

δV , δV˜ : ñ B p ¯ q V Kφχ XL etc. Solve to give V XL,R, XL,R ,...... ˆ p ¯ q Plug back to ﬁnd K XL,R, XL,R A similar relation starting from the gauged semi-chiral action also displays the duality between (twisted) chiral and semi-chiral models. Ulf Lindström Superspace is smarter Additional supersymmetry

The chiral sector, same as described for d 4 above:

K Ñ K pφ, φ¯q

a α ¯ a a¯ α a¯ δφ ¯ DαΩ pφ, φ¯q, δφ¯ DαΩ¯ pφ, φ¯q

On-shell algebra. ¢ p q iδa 0 J 3 i b , j 0 ¡iδa¯ b¯ ¢ ¢ p q 0 Ωa p q 0 ¡iΩa 1 i b¯ 2 i b¯ J j ¯ a¯ , J j ¯ a¯ Ωb 0 iΩb 0

Ulf Lindström Superspace is smarter The semi-sector:

Ñ p ¯ ¯ q K K XL, XR, XL, XR

The general situation not known at the p2, 2q level.

Linear tf:

¯ p¯ 1 ¯ q ¡ ¯ ¡1 ¡ δXL i¯ D XL XR κ XR iκ¯ D¡XL iκ D¡XL,

¡ ¯ p¯ ¡ p| |2 ¡ q |κ|2¡1 ¯ q ¡ ¯ δXR i¯ D¡ XR κ 1 XL κ¯ XL iκ¯¯ D XR ¡ ¡1 iκ¯ D XR,

Invariance:

¡ ¡ K11¯ K12 κK12¯ 0, p| |2 ¡ q ¡ κ 1 K22¯ K12 κ¯K12¯ 0.

Ulf Lindström Superspace is smarter tJ , J¡u 2c , ðñ p q| |2 p ¡ q| |2 1 c K12 1 c K12¯ 2K11¯K22¯.

Using the invariance condition we ﬁnd:

¡ |κ|2 1 c |κ|2¡1

Since c2 ¡ 1 is a constant we can form the following two local product structures:

1 2 S : ? pJ¡ cJ q, S 1 , c2 ¡ 1 1 2 T : ? rJ , J¡s, T 1 , 2 c2 ¡ 1 such that the commutator algebra of pJ , S, T q is SLp2, R).

Ulf Lindström Superspace is smarter The structures pJ , S, T q preserve a metric of signature p2, 2q and this geometry of the target space is called neutral hypercomplex.

When c2 1, the corresponding construction yields a triple of complex structures, the metric is positive deﬁnite and the geometry hyperkähler.

The general case is presently under investigation, i.e., 2d-dimensions and non-linear transformations. We do not expect that it will give a constant c, but it seems to have other interesting geometric properties related to Yano f-structures f 3 f 0.

Ulf Lindström Superspace is smarter Summary and conclusions part II

A complete description of GKG uses chiral, twisted chiral and semi-chiral superﬁelds. The generalized Kähler potential doubles as a (non-linear) potential for the metric and B-ﬁeld and as a generating function of symplectomorphisms. New vector-multiplets are available for gauging an important class of isometries. T-duality and quotients may be discussed in terms of these multiplets. Global issues (bi-holomorphic gerbes...) can be addressed. Additional supersymmetries, when examined at the p2, 2q level, lead to interesting new structures on the target space.

Ulf Lindström Superspace is smarter Telos

Ulf Lindström Superspace is smarter References

L. Alvarez-Gaumé and D. Z. Freedman, “Geometrical Structure And Ultraviolet Finiteness In The Supersymmetric Sigma Model,” Commun. Math. Phys. 80, 443 (1981)

C. M. Hull, A. Karlhede, U. Lindstrom and M. Rocek, “Nonlinear Sigma Models And Their Gauging In And Out Of Superspace,” Nucl. Phys. B 266, 1 (1986)

N. J. Hitchin, A. Karlhede, U. Lindstrom and M. Rocek, “Hyperkahler Metrics And Supersymmetry,” Commun. Math. Phys. 108, 535 (1987).

S. J. . Gates, C. M. Hull and M. Rocek, “Twisted Multiplets And New Supersymmetric Nonlinear Sigma Models,” Nucl. Phys. B 248, 157 (1984).

M. Gualtieri, “Generalized complex geometry,” Oxford University DPhil thesis, arXiv:math.DG/0401221.

U. Lindstrom, M. Rocek, R. von Unge and M. Zabzine, “Generalized Kaehler manifolds and off-shell supersymmetry,” arXiv:hep-th/0512164.

Ulf Lindström Superspace is smarter Duality

Back to the chiral complex linear duality:

» S d 2xD2D¯ 2K pΣ, Σ¯q » ¨ Ñ S˜ d 2xD2D¯ 2 K pS, S¯q ¡ φS ¡ φ¯S¯

˜ ˜ ñ ¯ ¯ δφS δφ¯S 0 D D¡S 0, D D¡S 0 ñ S Σ, S¯ Σ¯, S˜ Ñ S

˜ ˜ ðñ ¯ δSS δS¯ S 0 KS φ, KS¯ φ ñ K ¡ φS ¡ φ¯S¯ Kˆ pφ, φ¯q

Ulf Lindström Superspace is smarter