Dimensional reduction of fermions in generalized gravity

Nuclear Physics B242 (1984) 473-502 © North-Holland Publishing C o m p a n y

D I M E N S I O N A L R E D U C T I O N OF F E R M I O N S IN GENERALIZED GRAVITY* C. W E T T E R I C H

Institute for Theoretical Pl~l~sics, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland Received 21 February 1984

We discuss possibilities of obtaining chiral four-dimensional fermions from dimensional reduction of pure higher dimensional gravity. We explore a modification of riemannian geometry where the Lorentz rotations are treated in close analogy to usual gauge theories. The metric is not the product of two vielbeins and the vielbein m a y not be invertible everywhere. The bundle structure of Lorentz transformations is distinguished from the bundle structure of tangent space rotations and the gravitational index theorems have to be modified for this case. We also investigate noncompact internal spaces with finite volume in the context of riemannian geometry. Chiral fermions are obtained in the latter case. As a byproduct of this work, we find that for the usual torsion theories the Dirac operator is not the relevant mass operator for dimensional reduction of fermions.

1. Introduction

Gravity in more than four space-time dimensions can give a unified description of gauge interactions and four-dimensional gravity if all spacelike dimensions except three are curled up to form an "internal" space with very small characteristic length scale. Typically, this length scale is roughly of the order of the Planck length. Isometries of the internal space correspond to gauge symmetries in the effective four-dimensional theory obtained by integrating the action over the internal space. This idea has been pursued by many authors [1]. There is, however, a major difficulty in constructing a realistic model along these lines: assuming riemannian geometry in higher dimensions and the internal space to be compact, this theory fails to predict chiral fermions in the effective four-dimensional theory. Starting with a spinor coupled to gravity in d(d> 4) dimensions, one always ends in the fourdimensional theory with fermions belonging to vectorlike representations of the gauge group. Typically, all the fermions have masses of the order of the Planck mass, although there are some known cases where for a special choice of parameters the four-dimensional fermions are massless. A natural explanation of the observed small fermion masses requires chiral fermions, where the left-handed and right-handed * Work in part supported by Schweizerischer Nationalfonds. 473

474

C. Wetterich / Reduction offermions

Weyl spinors belong to inequivalent representations of the low-energy gauge group SU(3) × SU(2) x U(1). Fermion chirality is a very fundamental feature of a theory, and the number of chiral fermions is independent of many details of the theory. In dimensionally reduced gravity theories with compact internal space the number of chiral fourdimensional fermions can be expressed [2] by an index related to the topology of internal space. This index has been discussed in detail by Witten [3], who finds, by applying a theorem due to Atiyah and Hirzebruch [4], that the number of chiral fermions obtained from dimensional reduction of a Weyl spinor coupled to riemannian geometry with d - 4 spacelike dimensions forming a compact oriented differentiable manifold without boundary is zero. In this case, no chiral fermions can be obtained with respect to any gauge group which corresponds to an isometry of internal space. A realistic dimensionally reduced gravity theory should circumvent this no-go theorem. One could of course speculate that the observed fermions and gauge bosons are composite rather than fundamental and that the observed gauge symmetries do not correspond to isometries. However, much of the beauty and unification of dimensionally reduced gravity is lost, and very little can actually be calculated beyond the pure speculation. A second alternative is the introduction of supplementary gauge symmetries in higher dimensions in addition to the symmetries of d-dimensional general coordinate and Lorentz transformations. Chiral fermions can be obtained from dimensional reduction in this case [5, 6, 3]. Again, one of the main motivations to study higher dimensional gravity, the unification of gravity and gauge symmetries, is lost. Remaining with general coordinate and Lorentz transformations in d dimensions (we denote this group by gena) as the only local symmetries of the theory, we must either abandon the assumption that the internal space is compact without boundary or we have to modify riemannian geometry. In this paper we first discuss modifications of riemannian geometry consistent with gen a being the local transformation group of the theory. More drastic modifications, where the group of d-dimensional Lorentz transformations is replaced by some other transformation group, have been investigated by Weinberg [7]. In the last section we discuss noncompact internal spaces. We believe that four-dimensional gravity is well described by riemannian geometry with an Einstein-Hilbert action linear in the curvature scalar. This is, however, no indication that d-dimensional gravity should be riemannian too. Indeed, gen a invariance together with a stable ground state, which is locally a direct product of four-dimensional Minkowski space 63]L4 and a D-dimensional internal space K D with small characteristic length scale, is in general sufficient to guarantee the effective four-dimensional gravity to be riemannian with Einstein-Hilbert action. A ground state 637C4x K D implies that the symmetry group governing four-dimensional gravitational interactions is gen 4. Furthermore, after dimensional reduction, all

C. Wetterich / Reduction offermions

475

particles appearing in the effective four-dimensional theory have generically masses of the order of the inverse characteristic length of internal space, except those for which there is a special reason why they have to be massless or very light. Gauge symmetry guarantees massless gauge bosons and a nonvanishing chirality index would guarantee massless fermions. There is a big problem of obtaining massless or very light scalars, the gauge hierarchy problem. (For some remarks on this problem in the context of dimensionally reduced gravity see ref. [8].) Finally, gen a invariance guarantees a massless graviton. No other gravitational object except the graviton is expected to be massless or light. Kinetic and interaction terms for the graviton will be dominated by the gen 4 invariant with lowest mass dimension. This is the curvature scalar. Thus the effective four-dimensional action for the graviton will be dominated by an Einstein-Hilbert term. Stability of 6YL4x K D requires Newton's constant to be positive and excludes ghosts or tachyons. We end in four dimensions with usual gravity, whereas all other "geometrical" objects correspond to different tensor fields whose contribution is suppressed by their huge mass of the order of the Planck mass. Since by these reasons we cannot find direct experimental hints on the structure of d-dimensional gravity, we should investigate if a generalized form of d-dimensional gravity can help to generate chiral fermions. In sect. 2 we discuss the geometrical objects of a generalized gravity theory with gen a transformation group. We give up the constraints of riemannian geometry that the metric is the square of the vielbein and that the vielbein is covariantly constant. The spin connection is considered to be a free dynamical field. This generalized gravity theory resembles a gauge theory for the Lorentz group. The curvature tensors constructed from the tangent space connection and from the spin connection are not identical any more. We keep the constraint that the tangent space connection is consistent with metricity. In sect. 3 we give an example for generalized gravity, spheres with vanishing spin connection in arbitrary dimensions D. For D even, the vielbein cannot be invertible everywhere. In our example it vanishes at the south pole. The isometry group is SO(D) (instead of SO(D + 1) for spheres in riemannian geometry). We find that the Dirac operator has a non-zero chirality index in this case. In sect. 4 we carry out the dimensional reduction of spinors for our generalized gravity. The usual Dirac operator is in general not the correct mass operator for fermions any more. It has to be replaced by a modified operator in order to guarantee hermiticity. (The modified mass operator is just the hermitian part of the Dirac operator.) These remarks also apply to dimensional reduction of the usual gravitational torsion theories. In sect. 5 we discuss in detail our example of spheres with vanishing spin connections for D = 2. We find that in contrast to the Dirac operator, there are no normalizable massless spinors with respect to the hermitian mass operator. In sect. 6 we apply index considerations to discuss the prospects of generalized gravity to generate chiral fermions. We find that an invertible vielbein excludes the

476

C. Wetterich / Reduction offermions

existence of chiral fermions for compact internal spaces without boundary. However, the no-go theorem described above does not apply if the determinant of the vielbein vanishes at at least one point of the internal manifold. Finally we discuss in sect. 7 a different alternative to circumvent the no-go theorem: we keep the structure of riemannian geometry, and assume that the internal space is a noncompact space with small finite volume. (Of course, this alternative could be combined with the first one and one may consider generalized gravity with noncompact internal space.) The main idea of unification of gauge interactions with gravity in the context of higher dimensional gravity only requires that the supplementary dimensions form an internal space with a small characteristic length scale and a compact group of isometries. Harmonic expansion and dimensional reduction can be carried out for noncompact internal spaces with finite volume. We give an explicit example for a noncompact space in D dimensions which has finite volume and admits chiral fermions with respect to its isometry group SO(D). For D = 10, we obtain after dimensional reduction two generations of left-handed fermions belonging to the 16-dimensional complex spinor representation of SO(10). However, many problems remain to be investigated in this approach.

2. Geometrical objects in generalized gravity We assume that the theory can be formulated on a d-dimensional manifold with one timelike and d - 1 spacelike dimensions (we use signature ( + , - , - , ---)) and that this manifold admits an invertible metric g,, = g,,:

Under infinitesimal general coordinate transformations x ~ ~ x" + ( " ( x )

(2)

a covariant vector A, in tangent space transforms according to

A~ ~ A, + 8~A,, 8~A. = - O.~"A. - UO.A.,

(3)

with the usual generalizations for other tensors. The derivative O,A~ is in general not a tensor. In order to describe covariant derivatives transforming as tensors, we have to introduce a connection F,, x in tangent space. Under infinitesimal coordinate transformations a connection transforms according to

C. Wetterich / Reduction offermions

477

and the covariant derivative D~A~ = oua~ - F~,f'Ax

(5)

is a tensor. We always will assume that the connection is metric

Dog~ = O~g.~- r~)g~p- r~%~ = o.

(6)

Greek indices are raised with the inverse metric g~'" and lowered with g.; and these operations commute with covariant differentiation: D~A ~ = a . A ~ + r j A x

= D~(g~°Ao)= g~PD.A..

(7)

The commutator of two covariant derivatives defines again a tensor [D~, D~ ] A s = - R~?o A x - R~?D~ A~ ,

(8)

where the curvature tensor R..Xo and the antisymmetric part of the connection R , , ~ (which, by comparison with (4) is a tensor) are defined by

R , ? . = a , r . ) - a~r,) + r , ? r #

- r~? W ,

Ru~x = F,~x - irp~x .

(9) (10)

The Ricci tensor R.~ and the curvature scalar R are defined by Ru~ = R~,~x~,

R = g~'~R~.

(11)

Scalars under general coordinate transformations involving the metric and the connection can be constructed by contractions of the tensors guy, R,, x" and R~,~, like R, .,,~ox~,t~ ~ o x , R ~ X R ~ etc. We now want to introduce spinors. A spinor is a scalar with respect to general coordinate transformations 3,4, = - ( " 0 ~ p

(12)

belonging to a fundamental spinor representation of the Lorentz group Spin (1, d - 1). A Dirac spinor is a 2 [d/2] component representation transforming under infinitesimal local Lorentz transformations with parameters e,,,,(x)=-e,,m(X ) according to 8 e ~ = -- !82rnn2mnd'~1/ ,

(13)

478

where the d ( d obeying

C. Wetterich / Reduction of fermions

1 ) / 2 generators X"" can be constructed from d Dirac matrices ym { gm, yn } = 2~jmn ,

X'"=

- ¼[ym,3,"],

(14)

with ~ " " = ~m. = d i a g ( 1 , - 1 , - 1 . . . . 1). Except for d = 5, 7modS, the Dirac spinor is a reducible representation of the Lorentz group and can be decomposed into Majorana, Weyl or Majorana-Weyl spinors (compare ref. [9]). A Lorentz vector e ~ transforms according to (15)

6ee '~ = e r a : " ,

and ~'~" is an invariant tensor which will be used together with 7/m. to raise and lower Latin indices. The definition of a covariant spinor derivative requires a spin connection %ran = - % , , , related to the Lorentz group with '3~%m. = -- O f f % r e .

-- U O . % , . .

,

(16)

6 e % m , = e ~ P % p , + e l % r a p -- O u e m , .

Then the covariant derivatives (17) D~e rn = O~e m + %fnnen,

(18)

transform as a spinor and vector respectively under the Lorentz group. The covariant derivative of ~lm, vanishes. Again the commutator of two covariant derivatives is a tensor _ 1 ,2

(19)

~''",f, - R ~ , X D x ~ , ,

and F~. . . . the curvature tensor of the spin connection w, isgiven by ~vm~pn

•

(20)

The construction of a covariant kinetic term for a spinor needs a vielbein e~,m transforming as a vector under both general coordinate and Lorentz transformations 8~e~" = - O u ~ e~ m - ~ O~e Z ,

8e%m=eme. n.

(21)

C. Wetterich / Reduction of fermions

479

Multiplication with the vielbein converts a Lorentz vector e n into a covariant vector e~ = e~%,,

(22)

and we use the vielbein to convert Latin into Greek indices. If the vielbein has an inverse em~ em~ea n = 8 n ,

e umem ~ =

~,

(23)

(this will not always be the case in our approach), the inverse vielbein can be used to convert Greek into Latin indices (24)

e m = e,n~e~.

The covariant derivative of the vielbein is

D u e , ' = O~,e,m - Fu~,Xexm + O~nr"ne,,n .

(25)

With the use of the vielbein, a scalar Dirac operator can be constructed:

v~'Du~ = g~'*e..nymD~,~b,

(26)

and a possible kinetic term for the spinor is ~4~ = ½ i g l / 2 ~ y " D u ~ +

h.c.

(27)

with gi/2 = Idet g,~l 1/2. A general gen a invariant gravity theory can be constructed by forming scalars from the various tensors described above. The bosonic geometrical objects of such a general theory are the tensors g ~ and eft" and the connections F,~x and w~mn. Riemannian geometry has only one independent geometrical object, the vielbein e Z , and the other quantities are related to the vielbein by three constraints:

g ~ = egmeum,

(28)

Due~m = 0,

(29)

R ~ x = 0.

(30)

The first constraint (28) links the metric and the vielbein. It guarantees the existence of an inverse vielbein

eft' = g~'~e/'~l,,,,, = e~'m,

(31)

C Wetterich / Reduction offermions

480

and implies e~,m = e,~. The second constraint (29), together with (28), identifies the bundle structure of the Lorentz group with the principal SO(1, d - 1 ) bundle of tangent space rotations up to the double covering of SO(1, d - 1) by Spin (1, d - 1). From the commutator of two covariant derivatives of the vielbein one obtains

O:[D~,,Du]epm=Fj",,eon-Rj,

uXpexm.

(32)

The c u r v a t u r e s Ruurn n and F~,u.,. are identified and all characteristic classes constructed from the connections F~,ux and O)#m n a r e identical. According to (25), the spin connection can be expressed through F,ux and derivatives of the vielbein. As an example for the consequences of (28), (29), the spin connection cannot vanish identically on a sphere (with euclidean metric) except on S1, S 3 and S v, since a vanishing ~ requires a trivial bundle structure of the Lorentz group and therefore a trivial tangent bundle. Finally, the third constraint (30) excludes torsion and uniquely determines F~ x in terms of derivatives of the metric

=

r(L%?

=

{gx,.(a.gu. + aug..

a g.u).

-

(33)

The generalized gravity we discuss here abandons the three constraints (28), (29), (30) of riemannian geometry. Eqs. (28) and (29) are now replaced by

e~,meum = g~u + k~u,

(34)

D~,eum = U~um ,

(35)

and the constraint (30) may be dropped. The quantities e~m, ggu, /~vx and OkF.mn a r e then independent quantities and we have new tensors k,,, U,~ m, and R,~ x. New invariants can be constructed with these tensors. These include the quadratic terms _lxum__ k~,~k ~'~, H lP '~" and R~,uXR~'~x. For the effective four-dimensional gravity after dimensional reduction, we need the resulting coefficients of these quadratic terms for the four-dimensional tensors to be positive. The ground state is then characterized by

k,,~ ) = O,

(4)m - 0, U~u __

R.u - O. (4)~k

__

(36)

Deviations from four-dimensional riemannian geometry will be suppressed by inverse powers of the Planck mass. In contrast, the effective action for the d-dimensional theory may contain quadratic invariants with negative coefficients, leading to internal spaces with nonvanishing values of .~k (°), U~Dm) or __~,R (D)x and modifying the particle spectrum obtained from dimensional reduction. The index theorem leading to the no-go theorem for chiral fermions on compact internal spaces is only based on the bundle structure of the transformation groups. It

481

C Wetterich / Reduction offermions

assumes the identification of Lorentz transformations with tangent space rotations up to double covering. We want to circumvent this no-go theorem and are therefore mostly interested in the generalizations (34) and (35). As we will show in sect. 6, the vanishing of k ~ implies a zero chirality index on compact spaces. Our generalization allows us to decouple the bundle structure of Lorentz and tangent space rotations. The Lorentz bundle may be trivial even if the tangent bundle is not. The spin connection O,)p.mn is completely independent of the tangent space connection F,~~ and we can have vanishing spin connection on an internal manifold forming a sphere in arbitrary dimensions. From the commutator of two covariant derivatives of the vielbein we now obtain [Dt~,Dv]eom=f~mneo'~-R~Xoexm-R~XD~eo

m

= D~,U~o m - D.U~o m = V~o",

(37)

and for V~.p" nonvanishing the curvatures R and F are different. The characteristic classes constructed from these curvatures can now be different. The difference between two connections is a tensor. Since the Levi-Civita connection _F(LC)~vxdefined in (33) transforms as a connection in tangent space, we can always write q u a = F(LC).~x + L.~~ ,

(38)

with L.. x a tensor. Similarly, if an inverse vielbein exists, we can define a connection of the Lorentz group in terms of derivatives of the vielbein and the inverse vielbein: ,o(e)~,,p=e~"ko(e),,,,p,

09( e ),nnp

= -- ~'~mnp q- ~nprn - ~ p . . . .

(39)

~2mnp = -- 12 ( em~e~ ~" -- en~em" ) c3 e,,p .

One writes %mn = ~(e)~,~. + K~mn,

(40)

witfi K~,~. a tensor. The tensors L ~ x and K~,~. can be expressed in terms of R ~ x and U~,.: L.~x = ½(R,~ x + RX,~ + RX~u),

(41)

g m n p = gmt~gtznp 1

?t

= -- ~ { R m n e x p - Rnp;~ehrn -~- R p m X e x n -~ Cmn p -

Unto p -

Unp m -~" fpnrn-~- Upm n -

Umpn} .

(42)

482

C. Wetterich / Reduction offermions

We note that e,m = era, implies U,,p = standard theory of torsion.

U,p~. For U,~,~ = 0 one recovers the

3. An example for generalized gravity: spheres with vanishing spin connection In this section we want to give a concrete example for generalized gravity: spheres with vanishing spin connection in arbitrary dimensions. We will restrict the discussion to the D-dimensional internal space with euclidean signature O~b = -6ab. Once the kinematics of dimensional reduction is carried out [10], the problem of chiral fermions relies entirely on the internal space. For D = 2 m o d 4 the kinematics of dimensional reduction lead to chiral four-dimensional spinors provided the appropriate fermionic mass operator has zero eigenmodes if0 and that the D-dimensional Weyl spinors ~ = ½(1 + Fo)~0 and q~o = ½(1 - Fo)~0 belong to inequivalent representations of the isometry group. (FD is the D-dimensional generalization of 2/5.) Consider a space with topology and differentiable structure of a D-dimensional sphere S °. We use stereographic coordinates. Embedding the sphere with unit radius in a fiat (D + 1)-dimensional space with coordinates z i, ziz ~= 1, the coordinates y~ on S D can be induced by stereographic projection y~ = z~(1 + zD+I) -1,

a = 1...D.

(43)

This coordinate system can be used everywhere on S ° except on the south pole z o+1 = _ 1. Inclusion of the south pole, which requires a second coordinate patch, will be discussed below. To keep the discussion simple, we will assume that the tangent space connection has no torsion: L ~ v = R ~ v = 0. The ground state of our theory will be characterized by given values for e~", g ~ and ~0e~b which we denote by b~~, ~ and d ~ b. (We always assume for the ground state + = 0.) Isometries of the ground state are those combinations of D-dimensional coordinate and Lorentz transformations which leave b,~, ~ and &~b invariant. Such isometries are equivalent with gauge symmetries in the effective four-dimensional action. Let us assume a ground state characterized by

b~a=f(r)62, g~t~ = - g ( r ) 6 ~ , d~, b = 0,

(44)

with

r2=]F_,y~y ~.

(45)

C Wetterich /

483

Reduction offermions

(The standard metric on S D has g ( r ) = 4(1 + r2)-2.) Obviously, ~,~ is invariant under rotations among the y~ forming the group SO(D). Infinitesimal SO(D) rotations are described by D ( D - 1)/2 Killing vectors K~(y): (46)

y,~ ~ y , ~ + 6y'~ =y'~ + O~K~'~(y),

with OZKz~ - ½0ABKAB~,

KAB ~ =- --KBA ~,

0 A e = --0 8A ;

A, B = 1 . . . D ,

KAe" = ( 3 ~ B -- 6~6~A) Y~"

(47)

The invariance of the metric is easily checked:

The vielbein b," is also invariant under SO(D) transformations, provided the coordinate transformations (46) are combined with appropriate Lorentz transformations: 6 k , a = - ½0A"( O~KAsrOv ° + KABvOvO~~ ) -- b~bkh" = 0,

(49)

with o

1 AB

(50)

The isometry group is a linear combination of a special set of coordinate transformations and Lorentz transformations. We note that these Lorentz transformations are independent of y. The inhomogeneous part (see eq. (16)) of the SO(D) transformation of d ~ b therefore vanishes and d~,ab=0 is indeed invariant under SO(D) transformations. The SO(D) gauge transformations act on a spinor as follows [11]:

= --½0A%B

,

(51)

with SAe= KAB"O. -- NaB, ~"Ae = 6aA3bB 2"b"

(52)

484

C. Wetterich / Reduction of fermions

Since &,~b vanishes, one immediately finds that a constant spinor X is a zero eigenmode of the Dirac operator °~°

o~

(53)

C D.x = F O~X =0.

(We denote F u the "internal Dirac matrices" { F ", F b } = 2~/uh, ]>" = ~"aF".) Under SO(D) transformations, X transforms (54)

6 X = ½0A~XaBX,

and for D even, the Weyl spinors

(55) belong to inequivalent spinor representations of SO(D). For D = 4 rood 4, X + and X belong to real or pseudoreal representations, whereas for D = 2 m o d 4 , X+ and X- are in complex representations conjugate to each other. For example, for D = 2 X+ and X have opposite abelian charges. For D = 10, X + belongs to a 16 representation and X transforms as 16. However, before asserting the existence of chiral four-dimensional spinors in this model, we have to discuss the kinematics of dimensional reduction for our generalized gravity (see next section) and to analyze a complete harmonic expansion of an arbitrary spinor function q~(y). In order to understand the bundle structure of our example, we have to investigate what happens in the coordinate patch covering the south pole. In particular, we will find that the vielbein ~ must vanish at the south pole. On the northern hemisphere (NH) we have introduced stereographic coordinates y~. They can be extended everywhere on S D except on the south pole S. To cover the whole southern hemisphere (SH), we need a different projection and define coordinates y'~=a"Bzt~(1-z°+l)-l,

a, f l = l . . . D ,

(56)

with a ~ to be chosen either a"~ = 6ff or a"~ = d i a g ( - 1 , 1 . . . 1) in order to have the standard orientation. In turn, the coordinates y ' cannot be extended to the north pole (N), z D+ 1 = + 1. The relation between the coordinates y and y ' in the overlap of N H and SH is characterized by a transition function ~,~: y,~ = a~yl~r

2,

r 2 = y'y~y. Ct

y~ = a~y'er '-2 ,

r '2 = E y ' a y

'a

= r -2 ,

(57)

(58)

C. Wetterich / Reduction offermions

485

On the equator z D+I = 0, r 2 = r '2 = 1 , the transition function describes a rotation. In two dimensions, we may introduce the azimuthal angle cp with cos cp = y'l / r" , sin cp = y'2/r'.

(59)

Using a/~ = diag( - 1, + 1), one has

ep.

/~ --!1/cos2qo, sin 2 ~ v ,

-sin2cp] cos2cp ] "

=r'2k

(60)

This function belongs to a nontrivial homotopy class of %(SO(2)). In general, the transition functions (58) on the equator are mappings from S D 1 on SO(D) and can be characterized by a homotopy class of rrD I(SO(D)). We next want to describe vector fields in the tangent space of the sphere. We may parametrize such a field by A,~(y) in N H and a~(y') in SH. If A and A' are to describe the same field, they must be related in the overlap region of N H and SH by the transition function:

a'(y') = ~JAe(y ) ,

(61)

for y ' = y ' ( y ) according to relation (57). The transition function characterizes the bundle structure related to the tangent space of the sphere. The vielbein 0~a is a set of covariant vectors, but its Lorentz index allows for a more complicated relation between the vielbein %a(y) defined in N H and e~(y') defined in SH. A local Lorentz transformation eba(y)

b~a(y) ---, O~b(y) ~b~(y) leaves the product 0 ~

(62)

invariant. Consistency requires 0 " ~ ( y ') = ~,/~O~b( y ) ¢b~ .

(63)

The transition function for the spinor ~b is governed by the same Lorentz transformation £h ~ as for the vielbein (up to double covering). The Lorentz transformation £b° defined in the overlap of N H and SH is again characterized by a homotopy class ~rD_ I(SO(D)). They specify the principal bundle £ (¢.)g) of Lorentz transformations. In the standard approach to gravity, the homotopy classes of ~-a and ~ must be the same. This links the bundle structure of E (°'.)g) to the structure of the tangent bundle T ( M ) . For example, the standard vielbein ~ induced on S D from the flat metric in D + 1 dimensions reads 2 e ~ ( Y ) = 1 + 1.2 3,,, u

e, a ( y , ) =

2

1 + r'2 6~.

(64)

486

C. Wenerich / Reduction of fermions

In the overlap of N H and SH one has O'aa = r20a a = ~aB@Bb~ba ,

(65)

implying that E and q,-1 belong to the same homotopy class since their combination must add to a transition function in the trivial homotopy class. For D = 2 one needs

~ha =

cos2qo, -sin2cp,

sin 209) cos2cp "

(66)

The nontrivial transition function Eb~ forbids then the spin connection &,~b to vanish everywhere on the sphere: the transition function for & involves an inhomogeneous term (compare (16)) Eo~,ab(y" ) = d&B~-l % c a ~ d + ~ - l j O ~ c b

,

(67)

and a vanishing ~ , b on N H implies a nonvanishing ~',b on SH. In our generalized approach, %~b is a free field and we can investigate the consequences of w~ b vanishing everywhere. In this case Eh" must be a constant Lorentz transformation, independent of y. This transformation belongs to the trivial homotopy class and we will simply take Eb" = 8~. Starting on N H with a vielbein ~(y)

=f(r)8~,

(68)

the vielbein on SH is given by (69) On the two sphere S 2 this vielbein is easily visualized: 1

1

sin2q0,

cos2ep ]

(70)

Going once around the south pole involves a rotation of 4~r among the two vectors ~1 and ~2. The vielbein ~ " is uniquely defined only if it vanishes at the south pole, requiring f ( r ) to vanish faster than 1 / r 2 for r--, oo(--" r ' ~ 0). For D even the vanishing of det ~," at some point on S D is an intrinsic feature of the trivial bundle structure of the Lorentz transformations. Let us demonstrate this for D = 2. For trivial Eh~ = 6~, the vielbein ~1 and ~ 2 play the role of normal vector fields with transition functions uniquely determined by q,. It is well known that S 2 does not admit vector fields defined everywhere and without zeros. Each e2 must vanish at at least one point on S 2, leading to det e~, = 0 at this point. A Lorentz transition

C Wetterich / Reduction offermions

487

function Eh~ belonging to the trivial homotopy class does not change this situation. The reason why in the usual approach to gravity an invertible positive definite metric on S 2 can be constructed from the product of two vielbeins (g~a = e~aeB~) is related to the nontrivial homotopy class of £b" in this case. Conversely, the possibility of chiral fermions in our generalized theory of gravity requires the vanishing of OJ at S. We will come back to this feature more systematically in sect. 6 and show that zeros of the vielbein at some points on the compact manifold are a necessary condition for the existence of chiral spinors. As an alternative, one may exclude the points where ~ vanishes from the manifold and consider dimensional reduction on open internal spaces with finite volume. This question is addressed in the last section.

4. Kinematics of dimensional reduction for spinors in generalized gravity Before proceeding further, we have to generalize the dimensional reduction of fermions [10] to the case when the constraints (28)-(30) of riemannian geometry do not hold. The important modification comes from the fact that the Dirac operator is not hermitian any more. As a consequence, the internal Dirac operator is not the relevant mass operator for the dimensional reduction of fermions. Instead, a modified hermitian operator ® has to be defined (@ is the hermitian part of the Dirac operator). The eigenvalues of this operator restricted to internal space dictate the mass spectrum of fermions after dimensional reduction. These remarks also apply to the usual torsion theories, where zero modes of the Dirac operator have been found for various cases [12]. Let us first discuss the d-dimensional field equations for spinors, following the analysis of ref. [9]. These field equations are equivalent to the complete set of field equations derived from the effective four-dimensional action obtained from dimensional reduction. The Dirac operator can be written in the form

= "y~tO~ -b lo)~ttpyP "b ~03rnn~pq:l

mnp ,

(71)

with .}tmnp =

~/[rn~n~/p]

(72)

the totally antisymmetrized product of three Dirac matrices and &,,np totally antisymmetric: ~

z

O)mnp = ~[mnp]

(Omnp = e~rnWgnp.

(73)

488

C Wenerich / Reductionoffermions

The hermitian kinetic term (27) for a Dirac or Weyl spinor is of the form

~ = ½ig1/Z~v~D~+ + h.c.

= gl/2{

+ h.c.) + i

o'op+
(74)

We note that the spinor does not couple directly to the trace of the spin connection ~0".p due to the identity i ~ 7 " + + h.c. = 0.

(75)

The derivative term can be written

½ig'/Zfy. a.+ + h.c.

=

ig'/Zffv. O.+ + ½iO.( g'/Ze" m) ~'~,"'+ -- ½i Op.( gl/2~/]t~t+ ).

(76)

Using the identities

O.(g~/Ze".,)=gl/:g"~( O.e..,-- F'LC).ffex m) = gl/2g.V( D~Gm_ ~o.fe~p + F~vXexm-- F{LC).vXexm) =gl/2g~v(U~vm+h¢v~'eXn,+o~¢vm)

(77)

= gila(GlUm q- L ~ . , q- ~OtZjzrn), we obtain the hermitian part of the Dirac operator up to a total divergence:

~4, = igl/2f@~/-- ½iO~( gl/2~V~ + ), "~

(78)

The d-dimensional field equations for a Dirac spinor then read 6) 4 = 0.

(79)

Since U~um and L'~., are tensors, @ commutes with gen a transformations as well as -/"D~. It can be written similar to eq. (71):

@=Y~O~ q-½WLC(e)"~pyP +aWmnpY -

m . ,

,

(80)

C Wetterich / Reductionoffermions

489

~0LC(e)",p = euup + U".r + L",p

(81)

where

has the same transformation properties as cO~,p. It obeys the relation

~LC(e)~e=g l/2O~(g'/2e~)

(82)

and can be expressed entirely in terms of the vielbein, the metric, and their derivatives. Consistent with the observation that the trace of the spin connection does not couple to spinors (75), ~o"p does not appear in the field equations. All spin connections with the same totally antisymmetric part lead to the same field equations. This is also relevant for the usual torsion theories (U~p = 0): for vanishing spin connection, constant spinors correspond to zero eigenmodes of y"D,, but in general not to zero eigenmodes of the operator @ which is relevant for the field equations. For e~m = era, one h a s ( O L C ( e ) ~ p = with ~o(e)mnp defined by eq. (39). This discussion is easily generalized to spinors obeying a Majorana constraint (Majorana spinors or Majorana-Weyl spinors; for details see ref. [9]). In this case the hermitian kinetic term for spinors is

COm,,p

to(e)~'#p

(s3) Due to Fermi statistics the bilinear ~,ym~ vanishes identically. Again, there is no coupling of the spinor to w~p. The field equations derived from (83) involve the combination 0, ( gl/2 e.m ) and read @~p = 0.

(84)

Again, the operator ® appears to be the relevant operator. For d = 2, 3, 4, 8, 9 rood 8 we may decompose a Dirac spinor into two Majorana spinors ~1 and t~2. If we use the hermitian kinetic term (74) for the Dirac spinor, it decomposes into the sum of two_independent kinetic terms for +1 and '~2 of the form (83). This would not be true for a kinetic term instead of In this case, there would be nontrivial mixings between ~Pl and ~2. The decoupling of ~1 and ~P2, however, is crucial for a consistent dimensional reduction [10]. Let us next discuss dimensional reduction for a spinor. The treatment will be completely parallel to ref. [10] where the reader is referred to for a more complete and detailed discussion. In view of the discussion of open internal spaces with finite volume in sect. 7 we will only assume that space-time is locally characterized by a direct product of four-dimensional Minkowski space and a D-dimensional internal space with finite volume, characterized by a small length scale. (" Locally" refers to a

igX/2~py~Du~p

igl/2~@~.

490

C. Wetterich / Reduction of fermions

local region in the usual four coordinates.) It is then useful to express the dependence of fields on the internal coordinates y by a harmonic expansion and to derive an effective four-dimensional action by integration of the action over the internal space. For spinors with a hermitian kinetic term, all information is obtained [10] by expanding a D-dimensional Dirac spinor over the internal space K °. We assume that K D is a D-dimensional space with finite volume (not necessarily compact) and signature ~,b = - 8 , b . We consider an arbitrary normalizable Dirac spinor + ( y ) on KD--this means that fdDyga~/2+*+ is a finite positive number. A normalizable spinor + ( y ) can be seen as a set of 2 L~/21 normalizable functions. (We remind that + ( y ) are commuting variables in this context.) Since the normalizable functions on K D form a vector space (of infinite dimension), we can express every normalizable spinor q~(y) in terms of a normalized basis of this vector space:

~pCy)=a?p,(y), f d°y gl/2fit+j = aij , go = Idet g,~#[,

(85)

with finite complex constant coefficients a i. We want to choose a basis { q~i(Y)} with q~i classified by representations of the isometry group and eigenfunctions of an appropriately defined mass operator. We also assume that K ° admits a group of isometries. For our purpose, this means that all geometrical quantities ~,¢, ~", i',~ v, &,be characterizing the ground state K D are invariant under some subgroup of genD transformations 6~,¢ = -0~S.~,¢ = 0

etc.

(86)

Here 0 z are infinitesimal transformation parameters independent of the internal coordinates y and the generators Sz are appropriate differential operators fulfilling the usual commutation relations for Lie groups

[Sx,Sy ] =f~yzS~

(87)

(compare the example in sect. 3). The action of the isometry group on a spinor ~b( y ) is given by aq~ = - o z s z + ,

(88) with kz.ab characterizing the Lorentz transformations needed to guarantee the

491

C Wetterich / Reduction offermions

invariance of the vielbein 0oa: O~e~, ~b = -

e,,b =

eh~,

6~o ~ = - O Z ( O , , K f ~ , ~ + Kfat)O,~) - b~bkh~= 0.

(89)

Everywhere where an inverse vielbein ~.~ exists, the generators S z acting on a spinor can be expressed S~q, = Kz°b,q, - ½b~,Kf~,,'¥tchZ"'~q., _ ~Kfifj~,,bO.2.b4,

+ i k." fia ,,a

b

rv,,b.,,

(90)

with covariant derivatives b . constructed from the connections &~h~, and I'~t~v: b.,/, = oo~-'

~09aabX" ° " ~ " ~~" ,

DoKf = OoKf + L f u : ' ,

R , J = e ~ r - e~,~~'.

(91)

For riemannian geometry (~o~ = 0,,,, ko/~r = 0, /.)~, = 0), the generators & are called the Lie derivatives of 4. Suppose S:4,~ can be expanded in terms of the basis functions %: S.~p, = ~,j ( T: )j,, [ T~, T,.] =f~,.:T~,

(92) (93)

with complex constant (~)~,. The (~)ji form an infinite dimensional reducible representation of the isometry group. Reducing to irreducible representations H with components n, we choose a basis 4',tf~ (k is a supplementary, so far unspecified index) with ~P(Y)= a,,,,k~P,,nk(Y), S~P,,Hk ( Y ) = ~',,'Hk ( Y ) ( T z ) , , ' . ,

f d DY g D1/2~n'tt'k" "l~ n l t k = 6n',fitl'H6k'k •

(94)

C. Wetterich / Reduction of ferrnions

492

°(x

Next_ we observe that S: commutes with the Dirac operator and also with U ~aF °a a. and L .~F .

[s: ,

I s .~ , Uo5, , F o ] = 0 ,

~/

~

[S: , oLo, a F ° ] = 0 ,

(95)

implying [S., 4 ] = O.

(96)

If for all (n, H, k) the function 6~nlf k can be expanded in the basis ~,'tt'k' we write

@~nHk ( Y ) = +nHk'( Y ) Mk'k ,

(97)

with complex constant Mk, k. mk, k fulfils the r61e of a mass matrix for the four-dimensional spinors after dimensional reduction. This mass matrix is hermitian (Mk,k * = Mkk, ) provided the integral over the total divergence fdOyO,(glD/at~,ttk,f~'°~,rfk) vanishes for all +,/~k (exceptionally, there is no summation over n and H). We then can diagonalize M~, k by unitary transformations and choose a basis of eigenstates of d):

f d DYgD1/2 d/n'H'M'j' ~-~nHMj = 8n,n6H,HSM,~j,j "

(9s)

The real eigenvalues M correspond to the masses of four-dimensional fermions. Alternatively, we could also expand in eigenfunctions of F° ~°De. However, one now has °et°

~' Da~nH k -- ~nHk,mk,k,

(99)

where ink, k is not hermitian except for riemannian geometry. The mass matrix for four-dimensional fermions after dimensional reduction turns out [10] to be ½(mkk, + rn~,k). Thus, even if the Dirac operator has zero eigenmodes (mkk, has zero eigenvalues), there are not necessarily massless fermions in the reduced four-dimensional theory. Zero eigenvalues of mkk, do not imply that the hermitian part ½(mkk,+m'~,k) has zero eigenvalues. We are back to an investigation of the spectrum of the operator d~, which is the only mass operator relevant for dimensional reduction.

C. Wetterich / Reductionoffermions

493

5. Spinor harmonics on the sphere for generalized gravity

We now come back to our example of sect. 3 for D-dimensional spheres with vanishing spin connection. We have found zero eigenmodes of the Dirac operator ]'~])~ in this case. However, the relevant question is rather the spectrum of o~: does 0~ have zero eigenmodes for this geometry? For e~°"-f(r)8~,- ~ = -g(r)6~/~, dGhc = O, R ~ f = 0 (eq. (44)) one finds: o

_ l yo-YTg ° _ f 8oo ( D - 2 )

+2

L % = o,

(100)

@~/= f F a ( 8 2 0 ~ + - ~1 ~

8~a ( ( D - 2 )

~ + 2 f f ] 4' ) ,

(101)

with f ' = d f ( r ) / d r , g'=dg(r)/dr. We remember that any ground state ° h with vanishing antisymmetric part %bc : leads to the same spin connection t%~ operator 0~. We first will discuss the harmonic expansion of spinors for the case D = 2 in some detail. This gives a simple illustration for the general procedure outlined in the last section, where many of the characteristic features can be seen explicitly. The isometry group is U(1), and we parametrize the orbit of U(1) by a new coordinate ep: yl = rcos

y2 = r sin q).

¢p,

(102)

In these coordinates the derivatives read O = yt~ 8~B Oy a with e12 = - e21 = l, en = E22

=

Or

%[3r2

0. For the Killing KlzaOa

=

Oq~

vector

' K 1 2 '~

(eq. 47) one has (104)

- 09).

The Dirac spinor in two dimensions has two components, which can be decomposed into one-component Weyl spinors ~+ and ~b-: q'= ( ~++ ) '

~ - + = ½ ( I + F D ) ~.

(105)

We use the following representation for the Dirac matrices F": 1

494

C. Wetterich / Reduction of fermions

The gauge transformation of a spinor is 6• = - ½0A~$AB¢ = 0( O~ + ½i~-3)~p =

iOQ~p,

(107)

with 0 = 0t2 = -0~1. We expand an arbitrary normalizable spinor in eigenfunctions of the charge operator Q:

~b( r, cg) = a.Mj}.Mj( r, cp) , 1

exp(in~)x+"MJ(r)

+~Mj(r'~)=v~.,,

),

(108)

exp(i(n+l)~)X~Mj(r) 4

Q~P,~tj = (n + ½) tp,Mj, with n an arbitrary integer*. The functions

(109)

XyMflr) + are normalized

dr g(r)[(X.M,j, ) X,,Mj+(X,.7,)X.Mj] f

+

+

-

g"

=6~,M6jT,

-

,

(110)

(without summation over n), implying f d2yg~/2 (LP.,M,j,)+ ~P. Mj =

6.,,,6M,M6/7,

(111)

The X+~Mj(r) are a complete basis for normalizable r dependent spinor functions, so that every normalizable spinor depending only on r can be written ¢.(r)

=

(x+(r))=EanMj(X+Mj(r)) x,,(,)

.j

(112) '

(We note that one may choose a different basis for each n.) Let us next look for eigenmodes of the mass operator @: o

=

(113)

One obtains

.~. {

exp(inep)X+,,Mj(r)

~'t) e x p ( i ( n + 1)qp)×XMj(r )

I

i i

Or + ~ --/- + Or + 2 f

exp(in~))X~tj(r)

r (114)

* All representations of U(1) are one-dimensional and labelled by the index n, which corresponds to the index H in the general case of sect. 4. We also note that in contrast to Kaluza's original five-dimensional example there are no neutral fermions in this case, since all fermions car~ half-integer charge.

C. Wetterich / Reduction offermions

495

and therefore _(f 1 f' n+l) i g Or + ~ ~ + - - r

g

2 f

X,,MI= Mx,,,~tl,

r X,,Mj= MX,,Mj.

(115)

We want to know if there are solutions of (115) with M = 0. If we could find a normalizable solution of l fOr+2 f

n)+ r Xn°=0'

(116)

we also would have a zero mode X,,'0 with n ' = - ( n + 1). The charges of X,,+0 and X,,'0 are n + ½ and - ( n + ½) respectively and therefore opposite to each other as required by charge conjugation [10]. A normalizable solution of (116) would indeed induce chiral fermions after dimensional reduction. Eq. (116) is easily integrated and the solution is +

+

n

X,,o = a,,or f ( r )

l/2

(117 t

(with a,,+0 an integration constant). Normalizability of this solution requires the integral (110)

I,,=~

drrg(r)[x+,,o] ~

(118)

to be finite. We have supposed that both f ( r = O ) and g(r=O) are positive constants and that f ( r ) and g(r) have no zero for finite r. Taking the limit r--+ 0, the integral I,, can only be finite if n > - 1.

(119)

Thus n is positive or zero. For r ~ m, consistency requires that f ( r ) vanishes faster than r-2 (compare sect. 3) and for compact spaces one needs lim g ( r ) ---, r 4.

(120)

r ~

As a consequence, I,, diverges for r --+ m for all n > 0. There are no normalizable zero modes of @! Coming back to the eigenvalues of the Dirac operator F° ~°D,~, they are obtained by omitting the term ½ f ' / f in eqs. (114) (116). The solution "+ --- + a,,or " X,,o

(121)

496

C, Wetterich / Reduction of ferrmons

is then normalizable for n = 0. The zero modes discussed in sect. 3 are therefore the only zero modes of the Dirac operator. We observe that in our example of generalized gravity the index [3] is different for the operators d0 and P"b~. This is due to the fact that both ~ and ~P~b~ are not elliptic operators. The coefficient b~a multiplying the derivative term vanishes at the south pole. We conclude that the choice of the correct mass operator is crucial to establish the spectrum of zero modes after dimensional reduction. This also applies to the usual torsion theories, where often zero modes of the Dirac operator are found, whereas we do not expect that the hermitian operator d0 = F° ~°D~ + ~L 1 ° ~ has any zero modes in most of these cases!

6. Chirality index in generalized gravity For compact internal spaces the number of chiral fermions is characterized by an index [4] which has been discussed extensively in the context of dimensional reduction by Witten [3]. Let us suppose we have an operator D acting on spinors containing first derivatives of the spinor and non-derivative terms and that this operator commutes with the gauge transformations and anticommutes with FD" (D is always even for this section.)

[Sz,L) ] = 0 ,

{FD,D} = 0 .

(122)

Examples for such operators are the Dirac operator F° ~°D~ or the hermitian operator 0~. We define [2] & ( > )

=

-

-

+

(123)

where n~- counts the number of zero modes of D in the Weyl spinor ~p+ belonging + to the complex representation C and similar for n c, n c etc. (C is the complex conjugate representation of C.) If D is the mass operator relevant for dimensional reduction (in our case £)), the number figc counts the number of chiral four-dimensional left-handed fermion generations belonging to the representation C, up to a numerical factor to account for eventual Majorana constraints in the (D + 4)dimensional theory [2]. If the vielbein e~°a is invertible everywhere, the operators F°~°D~ and @ are elliptic operators. For elliptic operators D on compact spaces, the index N c ( D ) remains invariant under continuous changes of the metric, the vielbein and the connections which preserve the isometry group. Although in our examples of sects. 3, 5 the operators J'~b~ and 6~ were not elliptic, the index of F°~°D~ was nonzero and the index of 6~ was zero, independently of the specific r dependence of the functions f ( r ) and g(r).

C Wetterich / Reductionoffermions

497

For elliptic operators D on compact spaces one also has [3]

&(5)=&(v +tA)

(124)

for arbitrary t, if A is a non-derivative operator obeying [&,A] =0,

{FD, A } = 0 .

(125)

Examples for operators obeying (125) are U . . F or L"..F". Thus N"c ( F°'~° D . ) -fitc(6~ ) for 0,fl invertible everywhere. In our example, however, the Dirac operator was not elliptic, and this is why we cannot conclude from the nonvanishing index of the Dirac operator F"L). on the existence of chiral massless fermions, which are rather given by the index of 6~. We can use the index to give for compact internal spaces a necessary condition for the existence of chiral fermions in our generalized gravity: The vielbein must not be invertible everywhere. Let us show by steps that an invertible vielbein implies Arc(D) = 0: we know [3, 4] that for riemannian geometry the chirality index vanishes for the Dirac operator for arbitrary compact spaces with arbitrary isometry group. For ordinary torsion theories 0 . " ~ . = ~',~, U.~. = 0, with nonvanishing ground state torsion R . ~ preserving the gauge symmetry ([R,~Y, &] = 0), one finds from (41) and (71): k r.b. ' F" '~° D, = ( r° "°n . ) R - *~' :*.. '~~r-- _ ±8-.[,:,~.1-

(126)

where (F~b~)R denotes the Dirac operator of riemannian geometry. By virtue of eq. (125), the Dirac operator has zero index also for nonvanishing torsion. We next release the constraint of vanishing U~B-, while still keeping b~"b~, = ~ . In this case we can again combine (41), (71) and (125) to derive that the Dirac operator has vanishing index. As long as b," is invertible, the Dirac operator still has vanishing index even if we release the constraint ~ " ~ , = g~t~" Indeed, we can define a new metric ~ , ~ ( e ) = ~ " b ~ and the Dirac operator has zero index with respect to the metric ~ B ( e ) . Now ~,~(e) can continuously be deformed into the metric ~ without breaking the gauge group. Therefore the Dirac operator has vanishing index with respect to ~ ¢ as well. In conclusion, we find for our generalized gravity that an invertible vielbein on a compact internal space excludes the existence of chiral fermions! It is certainly possible to work out systematically the properties of the index of various operators for the case of vielbeins not invertible everywhere. We do not know if this has been done already. Care has to be taken if the operator is not elliptic. We know no theorem excluding the existence of chiral fernfions in this case. 7. Chiral s p i n o r s o n o p e n internal s p a c e s w i t h f i n i t e v o l u m e ?

So far we have discussed a generalized version of gravity in order to circumvent Witten's no-go theorem for chiral fermions. We considered the internal space to be

498

C. Wetterich / Reduction of fermions

compact. Compactness of internal space is a basic assumption for the theorem of the vanishing of the chirality index in riemannian geometry. Instead of modifying riemannian geometry, we may investigate noncompact rather than compact spaces. Indeed, the basic idea for the unification of gauge interactions with gravity in the context of higher-dimensional gravity only assumes that the internal dimensions form a space with a very small characteristic length scale, or equivalently with very strong curvature. We could imagine an open space with finite volume of the order ( M p ) - e . The internal space also should admit a group of isometries which becomes the gauge group after dimensional reduction. Since we observe compact gauge groups, we would like the orbit of the gauge group to be compact. However, the gauge group needs not to act transitively and a compact orbit of the gauge group does not imply that the internal space is compact. As an example, consider our example of sect. 3, where we now take out the south pole of the manifold. The manifold has then the topology of N D and we only need one coordinate patch (eq. (43)). We again assume the ground state to be characterized by a vielbein 6 a =f(r)3~ and a metric ~ = - g ( r ) 3 ~ , (compare (44)), but we now consider riemannian geometry with g ( r ) = f ( r ) 2 . The function f ( r ) can be an arbitrary positive definite function. We consider the case where f vanishes faster than r -2 for r --* m. The volume of this space

VD=f dDyglD/2=f dDyf(r)D

(127)

turns out to be finite. This gives an example for an open space with finite volume and isometry group SO(D). Assuming the volume to be of the order ( M e ) D, the characteristic length scale of the internal space is the Planck length and the internal dimensions are therefore not directly observable. Dimensional reduction in the bosonic sector can be done completely parallel to the case of a compact internal space. The general treatment of the gauge interactions in ref. [11] does not use compactness of the internal space, but rather a finite volume. The gauge coupling is finite provided quantities like

f dDyg~/2~,~BK~K,t~

(128)

are finite. This is the case in our example. Consistency also requires

f dDyg~/2Ri

(129)

to be finite, with R~ the internal curvature scalar. For more complicated effective actions including terms like R~¢vaR~l~v~[13], the integral over these terms should be finite. We then expect finite gauge coupling and Newton's constant, normalizable

C Wetterich /

499

Reduction offermions

kinetic terms (up to a wrong sign if the "ground state" is unstable) and a well-defined scalar potential. We will not address in this paper the question if such a noncompact internal space could be an acceptable ground state, which means that it is at least a classically stable solution of the field equations derived from some appropriate effective action for d-dimensional gravity. We are first interested in the question if such an open internal space with finite volume could admit chiral fermions. Let us pursue our example with SO(D) symmetry: the relevant geometrical quantities are

O~a = f ( r ) 3 f f , °

OOabc

-f2(r)6~l~,

~',~ =

= Y--~-~ f--~'(3rh3~c- 3w3.b). r f2

(130)

One finds a vanishing totally antisymmetric part of the spin connection

(131)

02ab c = td[abc ] = O ,

and the Dirac operator is given by FDa°a~

~" ~ = F a f

1

~t~Oot+ ½(D -

1)

-]6,~,,f'

.

(132)

We note that this is the same operator as in sect. 5, eq. (10) for g = f 2 . This is an example how two different spin connections with the same totally antisymmetric part &~hc lead to the same mass operator. We consider a harmonic expansion of a Dirac spinor

+(y) = an.Mj~,,.M/(y), o (~) ~n HMj =

M~,, ttMI

(133)

,

where we restrict our basis functions to functions with vanishing divergence

f d";Oo(g,1/2

"t

/"%.M,)

= o.

(134)

The functions ~,,ngj are obtained as solutions of the equation

= m%n,,M,,

(135)

with positive definite or vanishing M 2. We expect the spectrum of ~)2 to be discrete.

500

c. Wetterich / Reduction of fermions

We first specialize to the case D = 2 (with notation as in sect. [5]). Zero modes of 6~ obey again eq. (116) with solution (117). However, the question of normalizability is now different, since g ( r ) vanishes faster than r 4 for r ~ ~ . The relevant integral (118) is given by I.-£~

drr2n+lf(r).

(136)

A finite value of f at r = 0 implies again that normalizable zero modes exist only for n >/0. Assuming lim f ( r ) -~ ar -N

(137)

r~(x?

we find a normalizable zero mode in X + for n = 0 provided 2 < N~< 4. This corresponds to a chiral spinor in ++ with charge + ½. The charge conjugate of this spinor is also massless. It is in ~ - and has charge - ½. For higher values of N we find even more chiral spinors. For 4 < N ~< 6 there are normalizable spinors for n = 0 and n - - 1 , and in general normalizable zero modes for 0 ~< n ~< M exist for 2 ( M + 1 ) < N ~< 2 ( M + 2). We have found an example of a nonvanishing chirality index of the mass operator 67. The n = 0 zero mode is easily generalized to more than two internal dimensions. We make an ansatz for a zero-mass solution: ~o = q ) ( r ) x ,

(138)

with X a constant spinor and ¢p(r) a real function of r. Since KAB'~C)acp(r) = O, the spinor ~0 has the same transformation properties as the constant spinor X in sect. 3: &Po = ½0AB~aB~O •

(139)

We remind ourselves that for D = 2 m o d 4 +~ and +o belong to complex spinor representations conjugate to each other. Solutions of the equation

4% = 0

(140)

obey qcpV + ½ ( D -

1 ) f7'- = 0

(141)

and are given by ~p(r) = cpof(r) -(D ,)/z

(142)

C. Wetterich / Reductionoffermions

501

The zero mode q)(r) is normalizable if the integral

In- fo~drr D

if(r)

(143)

is finite. Assuming again that f(r) vanishes like r - s for large r (eq. 137), we find a normalizable zero mode provided N > D. If there are no other normalizable zero modes, we again have found an example of chiral fermions. (By analogy with the two-dimensional example, we do not expect supplementary zero modes if f does not vanish much faster than r-(D+ 1).) Starting from d = 14 riemannian gravity, a small noncompact 10-dimensional internal space would lead to two identical generations of left-handed fermions belonging to the 16-dimensional spinor representation of SO(10)! The existence of two identical fermion generations is related to the fact that there exists no Majorana spinor in 14-dimensional gravity [10]. To make the picture more realistic, one would be tempted to start with a Majorana-Weyl spinor in d = 18 and use the supplementary four internal dimensions to provide a generation group for possibly four generations of chiral fermions. We do not know yet if the approach of a noncompact internal space will lead to a consistent theory. Many questions remain open. For example, the curvature tensor in our example is given by

o

= ( 2/" + /,2 ) ( 8,~6~ h _ f, +r2lf

rf

8~8,~ b)

f,2 \ J (144)

with curvature scalar

(145) For f(r) vanishing for large r like r---, 0o like

ar

limR--*(D-1)Nr2(U r-~oo

N,

N > 2, the curvature scalar R diverges for

2){(D-2)(N-2)r2-2D}

(146)

a 2

However, the integral

f dDyglD/2(R)M f drr(2M_D)(N_

1)-1

(147)

502

c Wetterich / Reduction offermions

is finite for all M < ½D. We do not know if an acceptable g r o u n d state a d m i t t i n g chiral fermions can be found. (Questions of classical stability of a possible g r o u n d state have b e e n discussed extensively in ref. [6].) Also a more systematic study of the h a r m o n i c e x p a n s i o n is needed. W e should n o t be too afraid of the divergences in geometrical quantities or in the zero mode solutions for spinors as r goes to infinity. F o r large r the characteristic length scale becomes extremely tiny since the curvature scalar diverges for r ~ oo. After all, our knowledge a b o u t the physics at very short distances is very poor. Problems occurring at extremely short characteristic length scales may be related to the need of a more complete physical theory at these distances. W e have neglected so far q u a n t u m fluctuations in our completely classical t r e a t m e n t of the theory. The c o n c l u s i o n of this paper can be made rather short: there are still interesting possibilities left open, how pure higher-dimensional gravity could lead to chiral spinors in four dimensions. They certainly merit further investigation.

References [1] Th. Kaluza, Sitzungsber. Preuss. Akad. Wiss. Berlin, Math. Phys. K1 (1921) 966; O. Klein, Z. Phys. 37 (1926) 895~ A. Einstein and P. Bergmann, Ann. Math. 39 (1938) 683; B. De Witt, in Relativity, groups and topology (Gordon and Breach, New York, 1964); J. Rayski, Acta Phys. Pol. 27 (1965) 89: R. Kerner, Ann. Inst. Henri Poincard 9 (1968) 143; A. Trautmann, Reports Math. Phys. 1 (1970) 29: Y.M. Cho and P.G.O. Freund, Phys. Rev. D12 (1975) 1711: L.N. Chang, K.I. Macrae and F. Mansouri, Phys. Rev. D13 (i976) 235; J. Scherk and J.H. Schwarz, Phys. Lett. 57B (1975) 463; Nucl. Phys. B153 (1979) 61; E. Cremmer and J. Scherk, Nucl. Phys. B103 (1976) 393; BI08 (1976) 409; J.F. LuciaN, Nucl. Phys. B135 (i978) 111; E. Witten, Nucl. Phys. B186 (1981) 412; A. Salam and J. Strathdee, Ann. Phys. i41 (1982) 316 [2] C. Wetterich, Nucl. Phys. B223 (1983) 109 [3] E. Witten, Princeton preprint (1983) [4] M.F. Atiyah and F. Hirzebruch, in Essays on topology and related topics, ed. A. Haefliger and R. Narasimhan (Springer, Berlin, 1970) p. 18 [5] N.S. Manton, Nucl. Phys. B193 (1981) 391; G. Chapline and N. Manton, Nucl. Phys. B184 (1981) 391; G. Chapline and R. Slansky, Nucl. Phys. B209 (1982) 461 [6] S. Randjbar-Daemi, A. Salam and J. Strathdee, Nucl. Phys. B214 (1983) 491; Phys. Lett. 124B (1983) 349; Trieste preprint (1983) [7] S. Weinberg, Phys. Lett. 138B (I984) 47 [8] C. Wetterich, CERN preprint TH3528, revised version (1983), to appear in Phys. Lett. B [9] C. Wetterich, Nucl. Phys. B211 (1983) i77 [10] C. Wetterich, Nucl. Phys. B222 (1983) 20 [11] C. Wetterich, Phys. Lett. ll0B (1982) 379 [12] C. Destri, C. Orzalesi and P. Rossi, Ann. Phys. 147 (1983) 321: Y.S. Wu and A. Zee, Univ. of Washington preprint 400 48-25P3 (1983) [13] C. Wetterich, Phys. Lett. lI3B (1982) 377