Anomalies in two-dimensional gravity

Volume 200, number 1,2

PHYSICS LETTERS B

7 January 1988

A N O M A L I E S IN T W O - D I M E N S I O N A L GRAVITY Takeshi FUKUYAMA ~ and Kiyoshi K A M I M U R A , a Department of Physics andAppliedMathematics. Osaka Institute of Technology. Asahiku Omiya. Osaka 535. Japan b Department of Physics. Toho University. Funabashi274. Japan

Received 20 July 1987

We discuss two-dimensionalgravity interacting with the chiral fermions. In the canonical quantization the general coordinate invariance is preserved in full orders when the local Lorentz anomaly is taken into account. The Lorentzgauge degree of freedom turns out to be dynamical and has negative metric.

Two-dimensional gravity seems to be an attractive exercise as the prototype of four- or higher-dimensional gravity. It is also keeping much attention from the side of two-dimensional field theories and string field theories/l la Polyakov [1]. ( 4 k + 2 ) - ( k = 0 , 1, 2 .... ) dimensional gravity coupled with chiral fermions seems to be especially important in connection with the unified scheme by means of supersymmetry [ 2 ]. In this paper we discuss two-dimensional gravity coupled with chiral fermions in the framework of canonical quantization. This system shows gravitational and/or local Lorentz anomalies and the study will reveal the fundamental feature of the theory in ( 4 k + 2 ) dimensions [3]. Then arises the question whether this anomalous theory is quantized consistently without loss of general covariance. If the answer is yes, then the current arguments of anomaly cancellation lose their significance. Indeed several authors have discussed such possibilities for the chiral Schwinger model and its non-abelian extensions [4] and for the Lorentz gauge theory [5]. We will show in this paper that the local Lorentz anomaly recovers the general covariance, but it results in the breakdown of unitarity. That is, the Lorentz gauge degree of freedom becomes a dynamical variable with negative metric and will appear in the external lines. We start with the lagrangian

massless chiral spinors in two dimensions. Hereafter we use r/oo=l, q l ~ = - I for the flat metric, dots (dashes) for the time (space) derivatives and f O g = f l O g ) - (Of)g, LPg~=-½~-g(R-2A)N,

(2)

5¢~, = ½ i x / ~ e~'~(g77~0~ ¢/).

(3)

They are based on the action invariant under local O(2,1) transformations and are obtained by fixing some of the gauge freedom [ 6 ]. The remaining symmetries of the classical theory are the local Lorentz (O(1,1)) and the general coordinate transformations. Both of them are gauge symmetries and we will formulate this system by using Dirac's constraints theory [ 7 ]. As usual we parameterize the zweibein by the Weyl variable Z, the local Lorentz variable F and the lapse and shift functions 1/o and r/l as -l}\shF

chFJ"

(4)

They are the four dynamical variables of pure gravitation in two dimensions. The spin connection oga~, and the scalar curvature R are defined using ea-- dxl%a by de~ = o9~ heb,

dogo~=½Reoe~,

(5)

or more explicitly, in terms of Z, F, r/o and q, by ~= ~o~ + ~,.

(1)

Lfgra~ is the pure gravity lagrangian and ~v, is that of 75

Volume 200, number 1,2

PHYSICS LETTERSB

H and T are the generators of general coordinate transformations and satisfy the classical Poisson bracket algebra

o)o,o, = (q,/*to)[2- (ql +,h z')] - F + O1;+Z'l/o) =O~o, o),.o, = ( l h l o ) [ 2 - ( ~ l ; + q , Z ' ) ] - F ' = - w , ,

(6)

R = 2 (e-2Z/t/o) (6) l - oJ6).

(7)

Although LPgravcontains terms of second-order derivatives they disappear by a partial integration, and is

(8)

The spinor lagrangian (3) becomes the sum of the lagrangians for different chiralities, L ~ = • 5a? + ~, LPT,

j=|

j=[

(9)

where 5¢f =iez-+F{v~+ q)f $ 01o + qt)~.U]+-~U'f

-'F ½[ T O)o + (~/o + nl )o.~, ] ~.vt-+~tf }.

{H+ (x), H_+(y)} = + 2[H+_ ( x ) O 6 / O x - H + (y)O6/Oy], {H+(x),H_(y)}=O,

(10)

# + are the numbers of fermions of the indicated chiralities. L (+) (L (-)) contains z + F ( x - F ) and qo+l/l (t/o-th) only. Furthermore z + F disappear from £f~, if we use the new variables ~1+- = e(X ±V)/2{ll+ ,

(11)

2~± =i[~*± ~t ± -T ½(no + t/,)(~-+ O, q/-)].

(12)

This is due to the classical invariance of 5% under local Lorentz and the conformal transformations. The hamiltonian takes the form

with

=

(Pz +--N')(Px 4-Z')-T (Pz 4-N')'

+ o - A N e 2 Z 4-~-01

(13)

~±

.

(17)

It guarantees the general covariance of the classical theory. The constraint PF= 0 appears since F does not appear in the lagrangian explicitly. The local Lorentz invariance is the trivial symmetry in our set of dynamical variables. Quantization is performed by constructing the operator expressions of the constraints (14) and (15) satisfying the quantum algebra corresponding to (16). It ensures the general coordinate and local Lorentz invariance in the quantum theory. We first follow the same line [8] as in the case of scalar fields [9]. It consists of two parts. One is the normal ordering and the other is the quantum corrections of the parameters. Constraints H+ defined in (17) is written by introducing a + and their conjugate momenta n + as

a+_=(z4-N)lx/2, n±=(Pz4-PN)Ix/2,

(18)

/'/+ = ½(7~+ 4-0"~_)2--"~(1/v/-2)(n + 4-a~_)' - ½(~ _ -T- a'_

0 ~ - f dx (rloH+rll T)

(16)

H+ =-H+_ T

~+(-(Pgrav= (1/?]0) [•-- (~;"~)~' ~]I)](]Q--?]l N') - (q;+Z'qo)N' +tlo eZXAN.

7 January 1988

)~ +__(1/,,/2) (~_ -z- o-- )'

- ( A / x / 2 ) ( a + - a _ ) exPIx/2 (e+ + a _ ) ] +~+01q/-.

(19)

with three constraints It must be noted that the signs of the bilinear terms of a+ and a_ modes are opposite and one of them behaves as the ghost, which is cancelled by the other. We assume that our space is closed and perform normal mode expansions:

H = P x P N + N' Z' - N " - A N C x

+tl~+g.-+ ~ ,~, - ~ - O ~ - ) = 0 , "

T = P z z ' - P i +PNN' + ½(~+ 0,t.u + + ~ - 0 , ~ - ) =0,

(14) a + ( x ) = n=~-~

and PF=0. 76

(15)

2~--LC,( a ± [ n ] + a ~ [ - n ] )

X exp[ in( 2n/L )x],

(20)

PHYSICS LETTERSB

Volume 200, number 1,2

A ? - ~c+

m~L(a±[nJ-a~[-n])

7t±(x)= ,=~-~ - i

×exp[in(2zt/L)x],

(20 cont'd) Af=~

with

o),,-~(2~/L)lnl

(n>> 1).

The canonical quantization means

[a±[n],a~[m]]=h6 ..... (others are zero).

(21)

We expand all other fields correspondingly. As the quantum expressions of H± we take H± = :{½(~r+ _+a+)2-T- (1/~/2fl~)(g+ _+a+)' - ½ ( ~ _ ~ a'_

)~ _+(1/,,/2 ~ , ) ( ~ _

~" )'

-v

-(A/x/2)(c+a + - c _ a ) ×exp[x/2(p+a+ + p _ a _ ) ] + n - O , q / - } : ,

(22)

where fl~, fl~, c± and p+ are parameters taking the value I in the classical case. The short-distance behavior of the commutation relations are determined completely and we can find the equal-time commutators

[,q+ (x), H+ (y)l = + 2ih [H± (x) Od/Ox- H+ (y) Od/Oy]

-4 2ifiA(A f - 1 )[ VN(x)Od/Ox- VN(y)Od/Oy] 1" -Y-~lhAA c+ [ Vo(x)Od/Ox- Vo(y)Oa/Oy]

~ifiA~ O6/Ox3,

7 January1988

(23)

and [H+ (x), H_ (y)] = -2ihA [(A J- - 1 ) VN(y)Od/Ox

+ (AY - 1 ) VN(X)O~/OX]

c_ ~h

l(p++p_ ~ fl~

(24)

where

h ) 2g(Pz+P2-) .

(27)

(28)

From the requirement that (23) and (24) take the same form as the corresponding classical Poisson brackets (16), the following constraints on the parameters appear: A~=0,

A~=0,

A+=I.

(29)

The first one is the condition that the central charges from spinor fields must be cancelled with those from the gravitation. The last one appeared by the presence of the exponential operators. We have also shifted the constants in H+ in order to cancel out cnumber 3' terms from the RHS of (23). Our problem is thus to find a set of solutions satisfying (29). First of all, the parameters with their classical values (all parameters = 1) do not satisfy (29) due to the quantum corrections. Next we see that there is no solution for # + ~ # - (chiral fermion). This is because A ( = 0 and A f =1 give fl{- =ilL- and f l + = tiff then A ff = 0 is satisfied only for the case # + = # (Dirac fermion). This result corresponds to the one given in ref. [ 3 ] that in two dimensions ( ( 4 k + 2)-dimensional) general coordinate invariance is broken. We must remember that the local Lorentz invariance is kept on the quantum level manifestly. That is, PF= 0 is valid as a quantum constraint. It was argued that the general covariance is recovered by abandoning the local Lorentz invariance [ 10,5]. They studied the effective action 1

-- ½ihA[A + Vo(y)&~/Ox+AF Vo(x)OO/Ox],

h (c+p+ - c p_),

1

L~cfr= 192~ x//-g R O(xfl~ g.O)

X (ax/~

R + flO~,x/~ g~ ~ov/2 )

Vo(x) = : exp[x/2 (p+ a+ +p_ a_ )]:,

v~(x) ="

+½7 xfl~g g"Vo), a), + / z x / ~ ,

(c+a+ -c_ ~_)

×exp[ x/2(p+ a+ +p_ a _ ) ] :, Ag =

+

-

+

(25) + 7 - - # +, (26)

(30)

which is obtained by functional integration of (1) over ~,. The first term of eq. (30) comes from the fermion determinant, a and fl are related to # + by a = # + + # -, f l = # + - # - . The second and third terms are possible general covariant counterterms. 77

Volume 200, number 1,2

PHYSICS LETTERS B

LP~n-breaks the local Lorentz invariance explicitly. Li examined this action in detail [ 5 ] and concluded that the sign of y is crucial for unitarity. If ?, is positive the local Lorentz variable F appears in the dynamics as the physical one, whereas if y is negative F turns out to be a ghost variable. In the present canonical formalism F comes to couple through the local Lorentz anomaly when #+ ~ # - . This situation is the same as the Polyakov string [ 1 ], in that the Weyl freedom turns out to be the physical one through the anomaly when d e 26. In the path integral approach the local Lorentz anomaly may be found as the jacobian factor of the transformation from the general covariant bases to those of ~ ± in (11 ) [ 11 ]. Here we consult the result (30) and we find the additional contributions to H+. They are the compensation for using the variables

&+.

~ r r is, however, non-local and is not suitable for canonical quantization. We introduce an auxiliary scalar and write LPioc= l x / ~

7 January 1988

of the parameters a, b, c will be determined later. We follow the same procedure as before. We find H and T corresponding to (14) as

H = p z ( p x , + F , ) + p F Z , _ p , r _ e 2 Z A ( ~ + 2 a ¢ ) + ~p~ 2 +l(q~, _bp~.)2 + ~yp~?l 2

+(1/2y)(PF-bP~o_

]~,) 2

T = P z z ' - P ~ + PxN' + PFF' + P~q~'

01 7-),

where N = N - 2 a and ) , = c - b 2. Since the local Lorentz invariance is not maintained any more, the constraint corresponding to (15) does not appear. To clarify the structure of (35), we perform a canonical transformation generated by

W = P z z +/~x[N+ 2abF+ 2aq~- (2a 2 + ½c)x]

+ PFF+P~(~ + b F - 2 a z ) - N F ' - bz' ~o- cFx'.

g " O,,~oO,(o

(36)

+~o(ax/-----g R + b O , , ~ - g g," o),)

+ ½c,~---g g,'"o~,,~o,.

(35)

(31)

When we rewrite ~o by using the classical solution ~o~ as

In terms of the new variables (tilted in W), by omitting tildes, H+ = H g + H ~ + H F + H ~ ,

(37)

where =~o~ + 0 ,

H F = (1/2y)(PF++_yF') 2 - (pF+yF') '

o,,(,,~-gg,,'o,)~o ~' =axf---g R + bOvxf---g g~" o),,

+2abA eZXF, (32)

5~oc becomes

(38)

H~ =½(P~,+~o')z+(+Za-b)(P~,+_~o') '.

(39)

Hg++H~

with

takes the same form as (22)

oc= l ( - - g g.. O.Coo.Co

parameters

+ ½( c - b 2 ) x / - - g g~" o ) u ~

1/fl~_ = 1 - (2a 2 + ½c-T-2ab),

+ ½ ~ - g R[ l/O( ~ - g g.O)l

1/fl~ = 1 + (2a 2 + ½c++_2ab),

× (a 2x f ~

R + 2abO,,x/~g~"o)~ ) .

(33)

The equivalence of ~ r and ~oc may be recovered by killing the freedom of (? and some identification of the parameters. Now our lagrangian is ~9 ~---~/9grav "t- o ~ "t- ~(~loc,

(34)

with ~ , given in terms o f ~ +- as in (12). The values 78

c+ = 1 + (2a2 +½c).

(40)

In other words, the parameter correction is understood as the introduction of counter terms in the lagrangian. However, the effect of p± is not involved in £P~o~.F and PF couple to this system in the form of H F . The sign of ~' is significant since F must be resealed depending on it,

PF/x/~--*PF,

x/~

F~F.

(41)

Volume 200, number 1,2

PHYSICS LETTERS B

For 7>0 H'~ = ½( PF + F' ) 2 - (1/X/2 flF)( PF + F ' ) ' -AFe2ZF,

(42)

7 January 1988

and is consistent with the a s s u m p t i o n y < 0. Eq. (46) is the same equation as in the case o f pure gravity and scalar coupling [ 8 ]. F o r 7 > 0 the first ofeqs. (45) is replaced by

and for y < 0

A(+°) + (llfl2 +fil6zO = 0 .

H F = _ [½ ( P F T - F ' ) 2 _ ( 1 / X / 2 f l F ) ( P F ~ F ' ) '

In this case, however, there is no solution consistent with X> 0. Thus we arrive at the conclusion that the gravitation coupled with chiral fermions cannot be consistent. Namely, F appears as the ghost freedom. It m a y be cured by adding a Weyl spin 3/2 field and an a n t i s y m m e t r i c tensor to cancel out the y term [2,3]. Lastly, we make a remark. In the canonical formalism we used (30) as the possible form of the counterterms o b t a i n e d by the one-loop calculation [4]. Yet our result is not restricted in the one-loop order. If the effective action changes its form by higher-order corrections, (37) will be modified. W i t h respect to this p o i n t the A d l e r - B a r d e e n t h e o r e m [12] is recalled, which insists that the form o f the chiral breaking term is not modified, even in higher-order loop corrections. We do not know the corresponding t h e o r e m for the local Lorentz anomaly. Therefore, at this stage, our result is not completely affirmative, though our result strongly suggests that all higher-loop effects are taken only into p± as in (46).

-AFe2XF],

(43)

with

1//L-= ---,f21 I, AF=-T-2abA&/17

(y~0).

(44)

~o a n d P~ decouple from other variables as is seen in the form o f H~+. In this canonical c o o r d i n a t e we set H~+ = 0 in H+_. It m a y ensure the equivalence between local and non-local lagrangians. Thus the q u a n t u m operators H+ are the sum o f (22) and the n o r m a l - o r d e r e d form o f ( 4 2 ) or (43) with the parameter given in (40) and (44). The closure o f the algebra is e x a m i n e d for each sign o f y. The c o n d i t i o n that the H+ algebra closes is, for y<0, A (+o)_ ( l l f l 2 + h / 6 n ) = 0 , A~ +,,f2AFIflFA=O,

Af =l.

(45)

(50)

These conditions result in the following relations: p+ = p ,

p - (h/21t)p 2 - 1 = 0 ,

b 2 +4a 2 = (h/24z0(# + +#-

References -2),

(46)

4ab= - (h/24=)(# + -#-), c = - 4 a 2.

(47)

It is necessary that # + > 0 for real a a n d b. The values o f a and b coincide with those o f the one-loop result used in ref. [5] (there are some differences o f factors that come from the different definitions), a = ~ [ x / h ( # - - I )/3re + x / h ( # + - I )/37c 1, b=~t~/h(#--1)/3r~-

hx/~-l)/3~].

(48)

F u r t h e r m o r e , in contrast to ref. [ 10], the value o f 7 is d e t e r m i n e d to be y= -h(#

+ +#-

-2)/24~ <0,

(49)

[ 1] A.M. Polyakov, Phys. Lett. B 103 (1981) 207. [2] M.B. Green andJ. Schwarz, Nucl. Phys. B 218 (1983) 43. [ 3 ] L. Alvarez-Gaum6 and E. Witten, Nucl. Phys. B 234 (1983) 269. [4] R. Jackiw and R. Rajaraman, Phys. Rev. Lett. 54 (1985) 1219; R. Rajaraman, Phys. Lett. B 154 (1985) 305. [5] K. Li, Phys. Rev. D 34 (1986) 2292. [6] T. Fukuyama and K. Kamimura, Phys. Lett. B 160 (1985) 259. [7] P.A.M. Dirac, Can. J. Math. 2 (1950) 189. [8] T. Yoneya, Report No. UT-Komaba 85-3, unpublished. [9] T. Fukuyama and K. Kamimura, Phys. Rev. D 35 (1987) 3768. [ 10] H. Leutwyler, Phys. Lett. B 153 (1985) 65. [ 11 ] K. Fujikawa, Proc. Kyoto Summer Institute (1985) (World Scientific, Singapore). [12] S.L. Adler and W.A. Bardeen, Phys. Rev. 182 (1969) 1517.

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