On higher-derivative couplings in four-dimensional N= 2 supergravity

39500_2Butter.pdf On higher-derivative couplings in four-dimensionalN= 2supergravity

Daniel Butter and Sergei M. Kuzenko

School of Physics, University of Western Australia

XVII European Workshop on String Theory

Padua, 6 Sept 2011

Based on

JHEP 1103 (2011) 047 [arXiv: 1012.5153]

Daniel Butter and Sergei M. Kuzenko (UWA)Higher derivativesPadua 1 / 18 Component approaches (using superconformal tensor calculus) arepo werful. All major results inN= 2supergravity were obtained using only component approaches.

Just a few examples:Poincar

´e sugra = conformal sugra + compensators

[de Wit, Van Holten, Van Proeyen, et al.]quaternionic geometry of non-linear-models[Bagger & Witten (1983)]

What can superspace offer?More general (and manifestly supersymmetric) action principle Integrals over full superspace, chiral superspace, projective (or isotwistor) superspace [SMK, Lindstr

¨om, Rocek, & Tartaglino-Mazzucchelli (2008, 2009)]More general off-shell supermultiplets with finitely many or infinitely many

auxiliaries.A formalism to generate quaternionic K

¨ahler metrics.

[SMK, Lindstr

¨om & von Unge (2009)]Procedure to construct higher derivative actions[DB & SMK (2010)]Daniel Butter and Sergei M. Kuzenko (UWA)Higher derivativesPadua 2 / 18

Outline

1Projective-superspace approach toN= 2supergravity2Tensor multiplet actions in supergravity

3Higher derivative vector and tensor couplings

Daniel Butter and Sergei M. Kuzenko (UWA)Higher derivativesPadua 3 / 18

N= 2supergravity in superspace

N= 2superspaceM4j8with local coordinatezM= (xm;

i;i_)withi= 1;2Geometry described by covariant derivatives[Grimm (1980); Howe (1981)] D

A= (Da;Di;D_i) =EAM@M+12

bcAMbc+ jk AJjk fD i;Dj g= 4SijM+ 2"ij"Y M + 2"ij"W_ _M_ _(=Lorentz curvatures + 2""ijSklJkl+ 4YJij(=SU(2) curvatures fD i;D_ jg=2iij(c)_Dc+Lorentz and SU(2) curvatures Encodes "minimal multiplet" = conformal supergravity + vector compensator [de Wit, Van Holten & Van Proeyen (1980)] [Breitenlohner & Sohnius (1980)] W isN= 2superconformal Weyl tensor[Bergshoeff, de Roo & de Wit (1981)] W (ab)  ijTabij+ (icdi)Cabcd+


S

ij,YandG_involve components of vector compensatorDaniel Butter and Sergei M. Kuzenko (UWA)Higher derivativesPadua 4 / 18

Multiplets ofN= 2supergravity

Generic chiral multiplets(obeyD_i = 0) are too long forN= 2applications.

Vector multiplet(Am;; i; Dij)Reducedchiral multipletW:(Dij+ 4Sij)W= (Dij+ 4Sij)W[Grimm, Sohnius & Wess (1978); Howe (1982)]

Superfield:W +ii+ (iabi)Fab+ (ij)Dij+


Canonical chiral action:S=Z d

4xd4EW2Eis chiral measure

Tensor multiplet(gij; i; bmn; F)

[Wess (1975); de Wit & van Holten (1979); Breitenlohner & Sohnius (1980); Sohnius, Stelle & West (1981)]SuperfieldGijwith(Gij)=ikjlGkland constraintD(iGjk)=D(i

_Gjk)= 0Schematic expansionGijgij+(i j)+(iaj)~ha+ (ij)F+


Constructed in terms of a chiral prepotential via G ij=14 (Dij+ 4Sij) +14 (Dij+ 4Sij) ygauge invariance: = iwhereis reduced chiral, analogous tobmn=Fmn [Howe, Stelle & Townsend (1981); Gates & Siegel (1982); Siegel (1983); M ¨uller (1987)]Self-couplings of tensor multiplets are difficult in standardN= 2superspace. A more general superspace is required where tensor multiplets are natural. Daniel Butter and Sergei M. Kuzenko (UWA)Higher derivativesPadua 5 / 18

Supergravity in projective superspace

Introduce auxiliary bosonic dimensions=) M4j8CP1

[SMK, Lindstr

¨om, Rocek, & Tartaglino-Mazzucchelli (2008, 2009)]AuxiliaryCP1=S2parametrized by isotwistorvi2C2n f0gwithvicviRequire all superfields to have fixed homogeneity inviConsider new derivativesD+:=viDiandD+

_:=viDi_which obey algebra fD +;D+ g= 4S++M+ 4YJ++;fD+;D+ _g= 8G_J++ S ++:=Sijvivj; J++:=JijvivjLook for short multiplets annihilated byD+andD+ _=)

must be Lorentz scalars and annihilated byJ++=)holomorphic inviWeight-nisotwistorQ(n):Q(n)(z;cv) =cnQ(n)(z;v);D+Q(n)=D+

_Q(n)= 0New action principle:S=12I C vidviZ d

4xd4d4E(S++)2L++L

++is an arbitrary real weight-2 isotwistor superfield; contourCaround poles

Generalization ofSrigid=12I

C vidviZ d

4xd2+d2+L++withL++(x;+;+)Daniel Butter and Sergei M. Kuzenko (UWA)Higher derivativesPadua 6 / 18

Examples of isotwistor superfields

Tensor multiplet is real weight-2 isotwistor superfield defined everywhere onCP1 G ++=Gijvivj;D+G ++= 0() D(iG jk)= 0; = (;_)RealO(2n)multipletsQ(2n)are generalization of tensor multiplets Q (2n)=Qi1


i2nvi1


vi2n;D+Q (2n)= 0() D(iQ j1


j2n)= 0

Also may consider multiplets which are non-polynomial inviArctic multiplet(n)is well defined nearvi= (1;0)("north pole")Antarctic multiplet

(n)is well defined nearvi= (0;1)("south pole")Real tropical multipletV(2n)is well defined near "equator"These haveinfinitenumbers of auxiliaries. On-shell, the arctic multiplet describes

the generalN= 2hypermultiplet.Daniel Butter and Sergei M. Kuzenko (UWA)Higher derivativesPadua 7 / 18

Vector multiplet in projective superspace

Introduce a real weight-zero tropical multipletVwith gauge transformation

V ! V+  +

whereis arctic multiplet andis antarctic.[Lindstr¨om & Rocek (1990)]Construct gauge invariant field strengthW

W=18I

C vidvi (D)2+ 4S V D _:=uiv kukDi_; S:=uiuj(vkuk)2Sij u iisfixedisotwistor required only to obeyvkuk6= 0along the contourC Construction isindependentof choice ofuiWautomatically obeys constraints ofN= 2vector multiplet

=) Vis vector multiplet prepotentialDaniel Butter and Sergei M. Kuzenko (UWA)Higher derivativesPadua 8 / 18

Outline

1Projective-superspace approach toN= 2supergravity2Tensor multiplet actions in supergravity

3Higher derivative vector and tensor couplings

Daniel Butter and Sergei M. Kuzenko (UWA)Higher derivativesPadua 9 / 18

A prototype action: Vector-tensorBFcoupling

Look for coupling between tensor multipletGijand a vector multipletW Prototype for more complicated actions (including tensor self-interactions). Two equivalent representations1pure projective action:L++=G++V

S=12I

C vidviZ d

4xd4d4E(S++)2G++V

Z d

4x"mnpqhmnpAq2pure chiral action:Lc= W

S=Z d

4xd4E W+c.c.

Z d

4x"mnpqbmnFpq

Advantage: OnceWis constructed, purely chiral action can be evaluated with conventional superspace techniques.

Let"s put these ideas to use...

Daniel Butter and Sergei M. Kuzenko (UWA)Higher derivativesPadua 10 / 18

Self-interacting tensor multiplets

Let"s start with a single tensor multiplet.Improved (superconformal) tensor multiplet actionL++=G++lnG++

Supergravity generalization of[Karlhede, Lindstr¨om & Rocek (1984)]Rewrite as chiral LagrangianLc= Wwith

W:=18I

C vidvi (D)2+ 4S lnG++=18I C vidvi MG ++++(G++)2  M:=13 (Djk+ 12Sjk)Gjk;_i=23

D_jGjiauxiliary isotwistoruidrops out=)SU(2)covariant integration possibleTaking contour around zero ofG++gives

W=18GM+116G3ijGij=G8

(Dij+ 4Sij)GijG 2 ;G:=r1 2

GijGijWwas constructed long ago in flat projective superspace[Siegel (1985)]Component form appeared in supergravity even earlier

[de Wit, Philippe, & Van Proeyen (1983)] Daniel Butter and Sergei M. Kuzenko (UWA)Higher derivativesPadua 11 / 18

Self-interacting tensor multiplets

Models of several tensor multiplets haveL++(cG++) =cL++(G++) =) L++=G++

AFA(G++)Rewrite as chiral integralLc= AWAwith

W

A:=18I

C vidvi (D)2+ 4S FAWithout specifyingFA, only general structure can be given: W

A=FA;BMB+FA;B;CijiBj

C; 

MA:=13

(Djk+ 12Sjk)Gjk

A;_iA=23

D_jGji AThe properties ofFA;BandFA;B;Cijwere first worked out in components using

superconformal tensor calculus.[de Wit & Saueressig (2006)]Daniel Butter and Sergei M. Kuzenko (UWA)Higher derivativesPadua 12 / 18

A more complicated example: GeneralO(2n)multiplet

A more complicated example with a singleO(2n)and a tensor multiplet L ++=Q(2n)(G++)n1=G++Q(2n)(G++)n=)S=Z d

4xd4E Wn+c.c.

W n=18I C vidvi (D)2+ 4SQ(2n)(G++)n Let us assume the contour encloses a root ofG++. Then evaluate to find (suppressing combinatoric factor) W n/ G(Dij+ 4Sij)Rijn;G=r1 2

GijGij

R ijn=1G 2n  ij kl12G2GijGkl Q kl i1


i2n2Gi1i2


Gi2n3i2n2 Qmay be composite (e.g. product of smallerO(2j)multiplets)

A simple example:L++=(H++)2G

++=G++(H++)2(G++)2 W n=116

G(Dij+ 4Sij)Rij

2;Rij 2=1G 4  ij kl12G2GijGkl H (klHmn)GmnDaniel Butter and Sergei M. Kuzenko (UWA)Higher derivativesPadua 13 / 18

Outline

1Projective-superspace approach toN= 2supergravity2Tensor multiplet actions in supergravity

3Higher derivative vector and tensor couplings

Daniel Butter and Sergei M. Kuzenko (UWA)Higher derivativesPadua 14 / 18

A powerful observation!

Vector multiplet prepotentialV(z;vi)

weight-zero tropical multipletVector multiplet (field strength)

W=18I

C vidvi (D)2+ 4S

VTensor multiplet prepotential (z)

chiral multipletTensor multiplet G ++=14  (D+)2+ 4S++ +c.c.Daniel Butter and Sergei M. Kuzenko (UWA)Higher derivativesPadua 15 / 18

A powerful observation!

Vector multiplet prepotentialV(z;vi)

weight-zero tropical multipletVector multiplet (field strength)

W=18I

C vidvi (D)2+ 4S

VTensor multiplet prepotential (z)

chiral multipletTensor multiplet G ++=14  (D+)2+ 4S++ +c.c.V=V(G++)Daniel Butter and Sergei M. Kuzenko (UWA)Higher derivativesPadua 15 / 18

A powerful observation!

Vector multiplet prepotentialV(z;vi)

weight-zero tropical multipletVector multiplet (field strength)

W=18I

C vidvi (D)2+ 4S

VTensor multiplet prepotential (z)

chiral multipletTensor multiplet G ++=14  (D+)2+ 4S++ +c.c.V=V(G++) =F(W)Daniel Butter and Sergei M. Kuzenko (UWA)Higher derivativesPadua 15 / 18

Higher derivative vector and tensor couplings

System ofnVAbelian vector multipletsWIGiven holomorphic functionsF^A(WI)of degree one, construct composite tensor

multiplets=)G^Aij:=14 

Dij+ 4Sij

F ^A(WI) +c.c.

System ofnTtensor multipletsGij

AGiven real functionV^I(G++

A)of degree zero, construct composite vector multiplet =)W^I:=18I C vidvi (D)2+ 4S V ^I(G++ A) We may iterate the procedure by usingF^A(WI;W^J)andV^I(G++ A;G++ ^A). Each successive iteration adds two spinor derivatives (or one vector derivative). Distinct from more traditional way of generating higher derivative structures using chiral projection operator 
=196  (Dij+ 16Sij)Dij(D__16Y__)D__ .e.g. Given some scalar superfieldUand vector multipletsWZ d

4xd4E W
(W2
U)is a higher derivative chiral action. But can be

rewritten over full superspaceZ d

4xd4d4EWW2
U.Daniel Butter and Sergei M. Kuzenko (UWA)Higher derivativesPadua 16 / 18

Higher derivative vector and tensor couplings

Higher derivative actions we consider fall into the general class

I=12I

C vidviZ d

4xd4d4ES

++ (WI;W^I;G++ A;G++ ^A) +c.c.  D+ _ = 0howeverD+ ;D+ _ ;D _

6= 0Locally supersymmetric version of

I rigid=18I C vidviZ d

4x(D)2D4

+c.c.; D4:=148 DijDijIntegral naturally understood as over 3/4 of superspace.

Simplest example:

=WlogG++=WVwhereV= logG++ I=Z d

4xd4E WWZ

d

4xeFpqFpqZ

d

4xeFpq(DpGikDqGj`Gijkl)G

3The actions we generate include terms whichcannotbe lifted to full superspace

(without introducing non-covariant prepotentials). This represents a ne wclass of higher derivative terms in supergravity. Daniel Butter and Sergei M. Kuzenko (UWA)Higher derivativesPadua 17 / 18

Summary

Superspace techniques offer useful insight into (and generating functions for) the

structure of tensor multiplet actions inN= 2supergravity.Higher derivative terms involving vectors and tensors can be worked out explicitly.

It is easy to extend the contruction and consider composite vector multiplets built out of arctic multiplets andO(2n)multiplets, in addition to tensor multiplets. This

gives a very broad class of higher derivative terms.Some of these terms are naturally integrated over 3/4 of superspace andcannot

be lifted to full superspace. Daniel Butter and Sergei M. Kuzenko (UWA)Higher derivativesPadua 18 / 18