Geometry of supergravity-matter coupling

NuclearPhysicsB (Proc. Suppl.)18B (1990)113-120 North-Holland

113

GEOMETRY OF SUPERGRAVITY-MATTER COUPLING Richard GRIMM Institut far Theoretische Physik, Universitr~t Karlsruhe, D-7500 Karlsruhe 1, West Germany A striking analogy between K~hler geometry and supersymmetric gauge theory allows for a unified description of supergravity and matter in a suitably generalized superspace geometry.

1. Supersymmetry, as arising in the framework of graded Lie algebras, allows to relate bosons and fermions in a way consistent with quantum field theory. The generators of supersymmetry transformations are odd operators defined as Majorana spinors with respect to Lorentz transformations and subject to anticommutation relations which connect them to space-time translations. The parameters of supersymmetry transformations are anticommuting variables, gauged supersymmetry transformations give rise to supergravity. Owing to these properties, supermultiplets can be described in terms of superfields f(x,0), i.e. functions of space-time coordinates and anticommutiug variables. The component field content of a superfield is determined from its power series expansion in the odd variables, which terminates after a finite number of steps. Differential geometric methods can be applied to this enlarged space, giving rise to superspace geometry [1,2]. Supersymmetry transformations are then related to generalized diffeomorphisms. Being rather well understood, N=I supersymmetry in four space-time dimensions enjoys a certain popularity because it is also believed to survive in some way or another the adjustment of superstring theories to the description of our real world. The basic structures to be studied in this context are the interactions between supersymmetric matter and supergravity. Supersymmetrie matter systems as well as supergravity matter couplings are closely related to K~hler geometry [3-6]. The geometric description of supergravity matter couplings which will be summarized here relies on an intriguing interplay between

complex geometry and the geometric formulation of supersymmetric gauge theories [7,8]. 2. In complex geometry the fundamental form fl = i d~~ dzk gk~(Z,i)

(1)

is defined in terms of the metric gk~ " A K~ller m~nifold is defined to have closed fundamental form, i.e. dfl = 0

(2)

This is a constraint equation on the metric which is solved in te-,~s of the K~hler potential,

gk~ = 0k~ K(z'z)

(3)

The metric is invarinnt under Kihler transformations of the K~hler potential:

K(z,i) --+ KCz,~-) + F(z) + F(~)

(4)

The fundamental two form may then be viewed as the invariant field strength of the one form gauge potential

F = dzka,_K - dzP0-K p K

(5)

which under Kihler transformations changes as:

0920-5632/91/$3.50 © Elsevier Science Publishers B.V. (North-Ilolland)

r -.

r +

d ImF

(6)

R. Grimm/Geometry of supergrav/ty-matter coupling

114 Cleady, dr =

(7)

ifl

is invariant. 3. Matter supermultiplets consisting of a complex scalar, a Majorana spinor, and an auxiliary field (which is also a complex scalar) are arranged in chiral superiields ~bk,~, complex conjugate to each other and subject to constraints

D&~k=0, Da~=0

model where the scalar fields take their values in a KfLhler manifold. This becomes most transparent once (10) is expanded in component fields. Assigning components, i.e. A ,Xa,,F k , ~ -

and performing the spinorial integrations gives rise to the component field action (indices kJ and p,q are Kfthler whereas m,n ate space-time)

(s) L - ~d4x [- gk~

Assuming absence of gauge and grav~*.ational inter~tions the chirality conditions axe stated in terms of the usual spinorial derivates, which in Weyl notation satisfy

aa Om

i xko.m~my~_ ~gk~mxkcrm~) +~gk~

(9)

The supersymmetric action

L = Jd4xd2~i~K(~,~)

TlmnOmAk@nAf~

This form of the action dearly exhibits the Kfthler structure of the theory with notations such that

(10) 02K(A, A) gk~(A,A) =

isinvariantunder superfieldtransformations K(~b,~) --~ K(~,~) + F(~b) + F(~)

(13) 0AkaA~

(11)

and ~mX k = amx k +

because the spinorial integrations over F(~) and ~'(~) give rise to space-time derivatives as a consequence of (8) and (9). K(~b,~) is interpreted as a Kftlder potential, the role of the coordinates of complex geometry is taken by the chiral superfields ~b,~. Equation (11) is a supezfield K~hler transformation. Supersymmetry and Kftlfler invariance of the action (10) are intimately related: the spinorial derivatives appearing in the volume element and which insure supersymmetry axe at the same time responsible for Kftlder invariance due to the chiral constraints (8). The action (10) describes the supersymmetric nonlinear

rkp~OmA p

(14)

The Levi-Civita connection of the K~ler manifold and its curvature tensor are defined in the usual way. 4. The action (10) can also be understood as a special case of a D-term which occurs in supersymmetric gauge theories, the Kiihler transformations (11) corresponding to superfield gauge transformations. To see this more dearly, consider supersymmetric abelian gauge theory in superspace: there, a one form gauge potential

R. Grimm~Geometry of supergravity-matter coup//ug

.4-- EA-4A = ZaA, + EaAa + Eb-4&

(16)

115

The new parameters A,/t appemiug here do not siu~ up in the gauge traa~zmatio~ of - 4 a ~ becamm t ~

is defined with gauge transformations -4 --, .4 - idX

are chiral supe~elds: (17)

X is a real superfield, E A the frame of rigid supenpace with torsion defined by (9). The invariant field stzength

d.4---~r-- ~ EAEBJrBA

(18)

~rBA =- DB-4A - (-)abDAJB + TBAF-4F

(19)

Da/~ = O,

D&A=0

(26)

They are called pngauge trLns/mma~as. It is coa~nieat to decompose U,O into zeal and ~ parts: U+O corresponds to a gauge t ~ X. Oae then arrives at (zT)

is subjectto covariantconstraints

~

= o, ~

= 0

(20)

.4&= - DbV + pure gauge

(29)

v = ~ (u-o)

(29)

By definition,

o

•~=0

(21)

The solutions to (20) allow to express the spinorial components -4~,-4& of the gauge potential in terms of

is inert under X gauge transformations and dmages under pxegauge transformations as

V-~V+A+~

(3O)

spinorial derivatives of the prepotentials U,O: -4a -- iDaU

(22)

-4&= iD&O

(23)

Of course, the gauge transformationsof U,U must be

Finally, the constraint (21) serves to expzess the vectorial component .4a in terms of V such that

-4==~,~[V~V.JV+p~egauge

(31)

Replacing -4by the combination

t

adjusted to correctlyreproduce those of -4a,-4b, i.e. u -. u - x + ½ ~ 0 .-. 0 -

x

- ½A

-4 - ~ d(U+U) (24)

(32)

makes the X gauge transformations drop out, yieJdinE a superspace gauge potential of transformation law

(2s) .4 -~ ,4 + 2i d ImA

(33)

116

R. Grimm / Geometry of supergravity-matter coupling

5. Comparing sections 2 and 4 reveals a close analogy between Kftlflergeometry and supersymmetric abelian gauge theory. The fundamental two form f/ of complex geometry, which is invariant under reparametrization of complex coordinates corresponds to the field strength ~r=d.4 in supersymmetric gauge theory. Both 0 and 3r are subject to constraints - - eqs. (2) and (20,21) respectively. In both cases the solutions to the constraints allow for the appearance of new types of restricted gauge transformations: the complex metric is invariant under Kfthler transformations of the Kfflfler potential. In supersymmetric gauge theory the constraints are solved in terms of the prepotential superfield subject to pregauge transformations of chiral superfield parameters.

Complex geometry

n = id~E~dzkgkr~

Finally, the invariant two form fieldstrengths f / a n d ~"are obtained as exteriorderivativesof one form potentials r and .4 subject to completely analogous gauge transformations --eqs. (6) and (33) respectively. As shown by Zumino, and reviewed in Section 3, these analogies allow to interpret the general action for supersymmetric matter fields as a D-Term of supersymmetric gauge theory with the prepotential V replaced by the KaMer potential K and the role of complex coordinatestaken by chiralmatter superfields. The similaritiesin the formal structures of K~dfler geometry and supersymmetric gauge theory are visualized in the following table.

Supersymmetric gauge theory

~r= d.4 -- ½ EAF.BjrBA

df/=O

.4a= D ~ V , A & = -DaV K-~K+F+F

a-F = 0, 0 ~ ' = 0 P

P

V~V+A+A

D & A = 0, Da~. - 0

P = (dzkOk - d~.P0~)K

r ~ r + 2i d ImF

,4--*.4+ 2i d ImA

R. Grimm~Geometry of supergravity-matter coupling

6. It remains to couple supersymmetric matter to supergravity. In the pioneering work of C r o m e r et al. the most general coupling was performed in terms of component fields. In their approach, field dependent rescalings axe necessary in order to obtain a properly normalized Einstein term. The final component field action, after elimination of the auxiliary fields, is found to be K~lfler invariant prodded the fermionic fields are subject to certain chiral phase transformations. On the other hand, it turned out later that the structures described in the previous sections can be generalized such that supergravity and supersymmetzic matter appear in a unified geometric structure in superspace which directly yields correctly normalized actions and contains supezfield K ~ l e r transformations of all fields from the outset, thus providing a concise interpretation of the results of Cranmer et al. The relevant geometric structure is a generalization of WessZnmino superspace used to describe N--1 supergravity such that the new superspace torsion is defined as T A = dE A + EB~BA -'I-w(EA)EA-4

(34)

The first two right-hand terms axe those of traditional superspace, E A is the frame in superspace, and ~Bk the gauge potential for Lorentz transformations. The third term contains the one form gauge potential -4 defined as

The chiral weights of the frame are given as

~(z')=o. ~(z~) = I. ~(z~=-1

K(~b,~) ~ K(~b,~) + F(~b) + F(~)

(36)

.4 --~ -4 + ½ d ImP

(37)

(~,)

Thus the chiral weights of all q m t i t i e s a p ~ im the geometry (34) are determined, for instance those of the coefficients of the torsion two forms ate

~(TcsA ) = ~(~.A)_ ~(Ec) _ ~(zs)

(4o)

The structure (34) with definition (35 ! can be viewed as a special case of u(1) superspa~ geom~,7 ~J] (with identical torsion constraints). Observe that ~ suggestive way to display -4 and its curl ~rfdM is: .4 ----~ Kkd~bk - ~ K ~ d ~ + ~ E ~ Z D a ~ # ~ , ~

(41)

~r----½ gk~d~kd~ + ~ d ( E a ' ~ a ~ & ~ ) ( 4 2 1 Finally, given some genetic superfield S of choral weight w(S), its covariant derivatives with respect to Kftlfler transformations are defined as

~As = EA=aM s + ~(s)-4As

K(~,~) depends on chiral superfields ~b,~ and the torsion (341 is covariant with respect to superfield Kftlfler transformations

117

(43)

Clearly, the covariant derivatives of S Imve chiral weights

.(~AS) = ~(S) - ~(E A)

(44)

In this way, K ~ l e r transformations are umambiguously determined at the full superfield level as a consequence of our geometric construction.

118

R. Grimm / Geometry of supergravity-matter coupling

7. Component fields are defined as lowest components of superfiekls~ Therefore, having identified Kiihler transformations in the structure group of superspace geometry allows to determine K~ihler transformations of all the component fields in a constructive way ab initio. To be more specific we derive the K~hler transformation laws of gravitino and matter fields. As a consequence of the definitions

~m&(X) - 2Era&[

(46)

the gravitino carries chiral weights w(~bma ) = 1, W(~m& ) ---1

(47)

and is therefore subject to Kiilder transformations C m a---, #2ma exp(- ~ ImF)

(48)

~ m & "' ~m& exp(~ ImF)

(49)

For matter superfields of vanishing chiral weights, component fields and their chiral weights are defined as follows:

Ak(x) -" ~bk[

w(A k) = 0

8. The superfield action for the supergravity-matter system has a remarkably simple form, it is given as the superspace integral over the superdeterminant of the frame corresponding to the superspace geometry described in the previous section: L = -3J'E

By construction, this action is invariant under superspace diffeomorphisms (and thereby under general coordinate and local supersymmetry transformations) and under KRhler transformations. Less obvious is whether this action indeed describes supergravity and matter. But this can be verified by working out the superfield equations of motion, obtained from variations of the vielbein in the presence of constraints. Likewise, one can expand L in terms of component fields using standard superspace techniques. The resulting action density reads

e M1VI+ e baba +eDj/

w(Fk) - -2

(50)

(52)

As a consequence, the component field Kiihler transformations [F=F(¢) I =F(A)] are:

X ka ---,X ka expA(~ImF)

(53)

F k ~ F k exp(i ImF)

(54)

(56)

Except for the last term, this is just the supergravity action density in the usual notations. The last term, defined as Dj/= - ½/~aXal +~

Fk(x) = - ~ b k l

(55)

a

m

-&

(57)

contains contributionsof the matter sectorvia the lowest components of the superfields X a , X ~ , ~aX a .

But these superfieldsplay a specialrolein the geometry defined in (34). They are identifiedin the two form fieldstrengthY=dd such that

g. Grimm/Geometry of supergr~vity-mattercoupling

F~cz = - ~ @y~X~

(59)

As a consequence of the constraints on ~r and its superspace Bianchi identities they satisfy o,

= o

(0o)

and #ax a =

(01)

Since on the other hand ~r is given in terms of matter fields by means of (42), it is straightforward to read off the explicit form of X to be

i X~ =- - 12-gk~

(62)

Working out ~aX a, projecting to component field formalism and putting together all the terms yields Dj/---

gk~(gmn@mAk @nA fJ + ~ x°k~rmct&~m~~

+ ~ Rk~l~(XkXI)(~p~q) - ½ gk~(Xk~°'~)ba Ak+ - ~

gk~

- gk~ (~m~mnxP)(~ Xk) - gk~ (~m #

mnk

- -~

and ~ m ~ the usual spin connection which appears in supergravity (expressed in terms of vierhein and gravitino). Combination of (56) and (63) shows that the superfield action (55) indeed describes all the kinetic terms of the supergravity-matter system. 9. In snmmary, the close a~logy between K~hler geometry and supersymmetric (abelian) gauge theory, already observed in the description of pure supersymmetric matter, allows for a concise description of the snpergravity-matter system as well. The crucial point is the identification of K~dder transformations in the structure group of superspace geometry, on a footing as Lorentz transformations. Kftlder transformations appearing geometrically st the full superfield level allow to derive the correct K~hlet transformations of all the component fields. Invaxiant actions can he constructed in terms of snperfields and of component fields without any need of field dependent rescalings. The corresponding component field supersymmetry transformations (mixing supergravity and matter fields) are directly obtained from our supsspace geometry by means of the usual methods. Of course, these conceptual and technical advantages are not restricted to the description of the kinetic te~ms alone. The correct superpotential term, for instance, reads in our formulation

Z )(~nX ) Lsuperpotential ~- ½J~-eK/2w(~) + h.c.

- ½ gk~ 9mn(~mxk)(~nXf~) with

119

(65)

(03) Here, the superfield R as appearing in superspace torsion has chiral weight w(R)=2 and K~ler transformstion

R ~ R e -½(F(~b)-~(~))

(06)

120

R. Grimm/Geometry of supergrsvity-matter coupling

Taking into account the transformation law of K as well, the superpotential term is invariant provided the superpotential itself transforms holomorphically such that W(~) ---, e-F(~)W(~b)

REFE11ENCES B. Z-mino, in: Recent Developments in Gravitation, Carg~se 1978, eds. M. L~vy, S. Deser, NATO ASI Series B44 (Plenum Press, New York 1979 2.

J. We s s , J. Bagger:

3.

B. Zumino: Phys. Lett. B S7(1979)203

(67)

As another example, supersymmetric Yang-Mills theory is formulated in terms of covariant superfiels

E. Cremmer, B. Julia, J. Scherk, S. Ferrara, L. Girardello, P. van Nieuwenhnizen: Phys. Lett. B 79(1978)231; Nucl. Phys. B 147(1979)105 E. Cremmer, S. Ferrara, L. Girardello, A. van Proeyen: Phys. Lett. B 116(1982)231; Nucl. Phys. B 212(1983)413

W (r) , W (r)&

of well-known geometrical origin and of chiral weights

5. .

a~W (r)) - 1, w(W(r)&) - - I

Supersymmetry and Supergravity (Princeton Univ. Press, Princeton 19S3)

(68)

The invariant kinetic term is then given as

7.

"...ro¢~jc~ abcc ~u~dj ~c ®comct~ia scin ~n~j bcroc~t, un~ ~ie gc~n~[ic~cn roa~¢~ an~e~gt, ~crn sot aide rock ~[aubcn..." [Albrecht Diirer: Hierinn sind begriffen vier b~cher..., N0rnberg 1528].

J. Bagger, in:

Supersymmetry, Bonn 1984, eds. K. Dietz, 11. Flume, G. Gehlen, V. Rittenberg, NATO ASI Series B125 (Plenum, New York 1985) P. Bin~truy, G. Girardi, 11. Grimm, M. Mtiller: Phys. Lett. B 189(1987)83; B 195(1987)389 P. Bin~truy, G. Girardi, 11. Grimm: Supergravity and Matter: A Geometric Formulation, preprint LAPP-TH-275/90, Annecy May 1990

1 [E,xr(r)~ur (s)~ t.L~ Lyang-Mills -- ~Jl~'" ,, ~ ~Cr)(s)~J -l- h.c. (69) with f(r)(s)-f(r)(s) some arbitrary holomorphic function of the (covariantly) chiral matter superfields, inert under K~hler transformations. In addition to these standard examples, the structure of K~ler superspace allows to accommodate gauged isometries of the K~ler metric, couplings of the linear multiplet (describing the antisymmetric tensor gauge field and its supersymmetric partners), and the description of supersymmetric Chern-Simons forms. In view of all this, K~ler superspace provides a powerful and elegant instrument for investigations of N=I supersymmetfic theories in four space-time dimensions. As is well known, geometric methods lead to coherent and convincing results, since:

J. Bagger: Nucl. Phys. B 211(1983)302

9.

M. Mttller: Nucl. Phys. B 264(1986)292