Gravitational anomalies: A soluble two-dimensional model

Volume 153B, number 1,2 PHYSICS LETTERS 21 March 1985

G R A V I T A T I O N A L A N O M A L I E S : A S O L U B L E T W O - D I M E N S I O N A L M O D E L *

H. L E U T W Y L E R

Institute for Theoretical Physics, University of Bern, Sidlerstrasse 5, CH- 3012 Bern, Switzerland

Received 3 January 1985

The determinant associated with the propagation of chiral fermions in a curved two-dimensional space is calculated explicitly. The short distance singularities of the propagator generate a Lorentz anomaly. Coordinate invariance remains intact.

As shown by Alvarez-Gaum6 and Witten [1 ], one- loop graphs describing the propagation of chiral fermions in an external gravitational field in a space- time of dimension 2 (mod 4) contain anomalies. Sev- eral authors have analyzed the general structure of these anomalies [2]. Furthermore, Langouche [3] has explicitly calculated the change produced in the Weyl determinant by infinitesimal frame rotations and coordinate transformations in a curved space-t ime of two dimensions, confirming the result of the general alge- braic analysis. The purpose of the present paper is to show that the two-dimensional case is soluble: in two dimensions, the Weyl determinant can explicitly be calculated in the presence both of an external gravitational field and an external U(1) gauge field.

We consider a space of euclidean signature, guy dxU dxv > 0. The Weyl operator is of the form

D = iaaeUa(~ u + i6o# + i A u ) . (1)

There are two inequivalent representations of the two- dimensional Weyl matrices, both of which are one-dimensional. We choose o 1 = 1, o 2 = i. The zweibein vec- tors e~(x) , e~(x) fix the metric guy(x) and its inverse g V(x): guy= e~ae a

as well as the spin connection _ l a _ - ~ a

6ou- - ~ e u e a~e# , (2)

* Work supported in part by Schweizerischer Nationalfonds.

0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

where e -a¢ differs from the numerical antisymmetric tensor e ~ = ea¢ , e 12= 1 by the factor (de tguv) 1/2=

= =

The curl of the spin connection is determined by the curvature

1-- a u co v - a ~,co u = -~e uvR . (3)

We assume that the geometry is asymptotically flat and that the gauge field A u ( x ) disappears sufficiently fast as Ix[ -+ oo (more specific conditions on the global properties are given below). The main result of the present paper is the following explicit formula for the determinant of the Weyl operator:

In det D = - l z g ( w ) + Zg(A) ,

1 Zg(O) =-- " ~ f dx dy eauuOlavv(x ) A g l (x, y )

X ap [(e °a - iV~g °°) va(Y)] , (4)

where A~-l(x, y ) is the inverse of the Laplace-Bel- trami operator

a llN/r-gg#Va uAg l (x, y ) = 6 (x -- y ) . (5)

Although, in principle, the logarithm of the determinant is fixed only up to a local polynomial in the fields e ~ X a a ( ) , e u ( x ) , A u ( x ) and their derivatives, there is no genuine ambiguity consistent with the symmetries of the problem. As specified above, the determinant is

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invariant under coordinate transformations. The modulus I det D I only depends on the geometry guy and on the field strength OuAv- OvA u. The contributions of the gauge field and of the curvature are of opposite sign: the gauge field strength decreases the modulus o f the determinant, whereas the curvature increases it. I f there is no gauge field, I det D I is given by

Idet DI

=exp(-1-~2~f dxN/ffR(x) Agl(x,y)R (Y)'v/ff dY ) •

The phase of the determinant is not fixed by the geometry and by the gauge field strength alone: it explicitly depends on the zweibein vector ("Lorentz) anomaly") and on the gauge potential ("U(1) anomaly"). Under an infinitesimal rotation of the zweibein, ~e~a = -p(x) Cab e~b(x ) and an infinitesimal gauge transformation, 6A u = bu a(x), the phase changes by

~ ln det D= ~)-6~f dx v~p(x)R(x)

- ~ f dx ,~(x) e"%,A.(x) . (6).

For weak gravitational fields, one can choose the renormalization procedure such that the effective ac- tion becomes independent of the Lorentz frame; it then breaks coordinate invariance [1 ]. Bardeen and Zumino [2] have shown that, even if the curvature is not treated as a perturbation, a local functional which transfers the disease from frame dependence to coordinate dependence can still be constructed. In the two- dimensional model, it suffices [3] to add the local term f dx euvwv~ ln(k~e~+i2)/247r to In det D, to arrive at a frame independent determinant, which then breaks coordinate invariance through the constant vector ks. Note however, that this term cannot be regarded as the finite remnant of one of the counter terms needed in the renormalization procedure, because it is not a polynomial in the zweibein matrix and its inverse. [Alternatively, one may observe that, according to the Gauss-Bonnet theorem, V~-R is the divergence ~uG u of a vector field G u which only depends on the metric and its first derivatives. A term proportional to f dx GUwu therefore also removes the Lorentz anomaly. The explicit expression (see, e.g., ref. [4]) for G tt, however, shows that this term also

contains illegal denominators.] The transfer from frame dependence to coordinate dependence cannot be achieved within the class of local polynomials which characterize the ambiguities of the renormalization procedure. The determinant can be renormalized in a coordinate invariant manner, but there is no renormalization in the proper sense which would make it frame independent. We consider it likely, that this statement also holds in higher dimensions.

The reason why it is possible to explicitly calculate the determinant of D is that two-dimensional rieman- nian spaces are conformally fiat: in a suitable local coordinate system ("isothermal" coordinates) we have

guy(X) = eaG(x)~uv. (7)

In this coordinate system the zweibein field is of the form

e~a = e -2G 6~ {6ab cos 2F - eab sin 2F}. (8)

In terms of the fields F(x), G(x), which specify the zweibein, the spin connection becomes

w u = OuF + euv~vG. (9)

The gauge field Au(x ) can be represented in an analogous manner

A u = OuA + euvOvB. (10)

With these substitutions, the Weyl operator takes the form

D = eMi(a l + i~2)e N ,

M = - i A - B + i F - 3 G , N = i A + B + i F + G . (11)

In this coordinate system the propagator is therefore known explicitly:

D-1 = e -U[ i (~ l + i~2)] -1 e -M

The propagator determines the change in the determinant under an infinitesimal deformation of the fields e~a, A~t

6 In det D = Tr (6DD-1) . (12)

The trace involves the value of D - 1 (x, y ) and of the derivative ~ D - l ( x , y ) at x = y. Since D-1 is singular there, the formula (11) does not make sense as it stands; the determinant of D requires renormalization. The singularities of D - l ( x , y ) at x = y are determined by the local properties of the fields e, A near x. The

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singular parts of D -1 and of OuD -1 are local polynomials in the fields e, e - l , A and their derivatives. Accordingly, the change in the determinant is uniquely specified by (12) up to (an integral over) a polynomial in the fields e, e - l , 6e, A, 6A and their derivatives.

To work out the change in the renormalized determinant, we use the regularization [5]

L e = f f dX Tr(6DD + e-aDD+). (13) e

Formally, L e reduces to Tr(6DD -1) as e ~ 0. As we will see below, the behaviour of Le at e ~ 0 is of the form

L e = (1/e) l 1 + L + O(e), (14)

where ll is a local polynomial. The renormalized determinant therefore obeys

6 lndet D = L + / , (15)

where l is a local polynomial in the fields e, e -1, 6e, A, 6A and their derivatives. It remains to work out the quantity L and to integrate the relation (15).

In the following we disregard problems connected with coordinate patching: we evaluate the determinant in the vicinity of a geometry for which there is a global isothermal coordinate system. Furthermore, we assume that the fields G(x), F(x), A (x) and B(x) all tend to zero as tx [ -~ oo. Consider an arbitrary deformation 6guv(x ) of this geometry. A suitable infinitesimal coordinate transformation, 6x u -- ~u, brings the deformed metric into isothermal form, i.e.

6gu~,= Vu~v+ Vv~u + 46G guv. (16)

Explicitly, ~u is given by

1 x v _ yV ~a.ag xCv)], ~U(x) = ~-£~fdy ~ [ag~.O ,) 1 P' h

6g~v = gUXagav, ~u = guv~ v . (17)

The change in the metric determines the change in the zweibein up to an infinitesimal angle of rotation, which we denote by 26F

6 ~ = ~c~ac~-e~aa~u -- 26Ge~a -- 26Feab ~ . (18)

The corresponding change in the spin connection is given by

6cou = ~'~ac~u + coc~au~ '~ + au6F + ~uva ~6G . (19)

Finally, we decompose the change in the gauge field according to

6A u = ~c '~A u + Ac~au~ a + ~ufA + ~uv~V6B . (20)

Note that the quantities ~U(x), 6G(x), 6F(x), 6A (x), 6B(x) are not determined by the properties o f ~ , 6eUa, A u, 6A u at the point x, but are nonlocal functions, given by integrals of the type (17).

The net effect of an arbitrary deformation of the zweibein and an arbitrary deformation of the gauge field is the following change in the Weyl operator:

6 D = P D - DP,

P = ~a3 a - i6A - 6B + i a F - 36G ,

= ~c'ac, - i6A - 6B - i6F - 6G . (21)

The functional Le can therefore be rewritten as

L e = Tr(P e -eDD+ - ff e-eD+D). (22)

To evaluate this expression, we need to study the heat kernel (x I exp( -M)D +) ly), defined by

(e-XDD+f)(x) = f ( x le-aDD+ly) dy X/~'f0' ) (23)

at small values of k. The behaviour is governed by the expansion

(xIe-aDD+Iy) = ~ e -a(x'y)/4a ~ XnHn(x,y) . (24) 47rX n=0

The function o ( x , y ) i s the square of the geodesic distance from y to x ; i t obeys the differential equation

gUVOuo BvO = 4a . (25)

The expansion coefficients Hn(x, y ) can be worked out recursively by solving the differential equations (n = 0, 1,2 .... ;H_ t = 0)

½a#o "duHn + (n + ¼Ao - 1 ) H n = -DD+Hn_I (26)

with the boundary condition Ho(x , x) = 1. In (26) we have made use of the following notation. The symbol A denotes the Laplace-Beltrami operator. The differential operators du, du are defined by

du = Vu + iPu, du = Vu + iPu, (27)

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I'u = c°u + A u , I'u = - c ° u + A u , (27 cont 'd)

where V u is the covariant derivative with respect to the metric gu,; On scalars like Hn, the covariant derivative V u reduces to the ordinary derivative. On ten- sors, the covariant derivative does not commute:

[V u, Vv] v a = Rafluvv ~ ,

a a a a 3' a 7 R my = auP~v - avP'~u + Pu~,P~v - Pv~rtsu

1 ~ - ( 2 8 ) = ~e 3 e u v R .

In terms of the operators intro.duced above we have

D = i e u d u , D + • u*~ = xe du, (29)

where e u denotes the complex zweibein vector

e u = oae" a = ~ + i e ~ , _ v_ . - v ev. - gt~v e - --lel~ve ,

e~e v* =gUy _ i ~ v , due v= eVdu. (30)

The representation (22) shows that l 1 and L are determined by the expansion coefficients H0, H 1 and by the analogous quantities H0, H I which occur in the expansion of the kernel (x lexp(-M)+D)lY). In particular,

L = ~ - ~ f dx x / g [ P H l ( X , y ) - P ~ I i ( x , Y l l x = y . (31)

We thus need to know the values of i l l , H1 and of their first derivatives at x = y. These quantities can be extracted from the recursion relation (26) as follows. The derivatives of the geodesic distance at x = y are given by

o = Vao = V~V~V~o = 0 , VaV~o = 2gog,

VaVtsV.rVa a = ~( -2g~gTa + gaTgfla + gaagfl'~) R . (32)

[To demonstrate these properties, let x ( s ) be a geodesic emanating from y, where s is the arc length, such that o[x (s), y] = s 2. The differential equation satisfied by the geodesic implies that the nth derivative of o with respect to s is given by 2 a l . . . :? an Val .-- Vana. Since the direction :~a of the geodesic at s = 0 is arbitrary, the totally symmetric part of Va 1 ... VanO van- ishes at x = y , i fn > 2. Finally, the commutation rela- tions of the covariant derivative allow one to express all components of this quantity in terms of totally symmetric derivatives.] With these properties of the

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geodesic distance, the recursion relation for H1 leads to

H1 = - D D + H 0 , daHl = -½daDD+H0, (33)

where it is understood that one first evaluates the derivatives and then sets x = y. We therefore need the first three derivatives ofH0(x, y ) at x = y. The differential equation satisfied by H 0 implies

daJ/0 = 0 , d~d~H 0 = - ~ V a V ~ A a + a i F ~ , (34)

dad3d.rH 0 = _ f i Va \73\7vAo + li(7aF3~,+ VOFav),

with Fo¢ = ~aPts - alsFa. Putting things together, we obtain

L = l f d x V~[-~i~c'V"coa¢ q-7/- - -

- l( i~COa + i 6 F - 6 G ) R + (~aO a - 2i~aAa

- 2i6A - 26B + 46G) ~°°3oAo ] . (35)

This expression appears to indicate that the determinant is not invariant under the infinitesimal coordinate transformation x -+ x + ~. One can however remove all terms involving ~ by adding the local polynomial

I f i (6eUe~_ SeU*eu) l = dx e°~ [ - li(.o~ 8w# + gaaw~

+ i A . 6 A o + ½0aA~g~6gu~] . (36)

Using the explicit representation (9), (10) for the fields w(~ and A a which implies, in particular,

x /gR = -4 i luOuG ,

one finds that the relation (15) is integrable, with

in det D = dx [ - ½ 8 , G ( O u G + iauF )

+ DuB(DuB + iauA)] . (37)

In (4) we have rewritten this result in a coordinate independent manner, a legitimate procedure, since 6 In det D is independent of ~.

Finally, we quote the explicit expressions for the contributions of the fermion loop to the energy momentum tensor and to the current. According to (4) an infinitesimal change in the zweibein matrix e~ and in the gauge field A u generates the following change in the determinant:


6 In det D = f d x x/g(½8g#vT uv + 6AufU ) + [ ,

1 T u v = ~ - ( - D U B D V B + 1 G v +1 ~D u DG zDUD~G+~gV~R),

]u = _ 1 i DUB D u = V u - i~uvV v = - i~uv O v 2rr '

Bfx) = - f zx{l(x,y) dy eU%,A~Cv) ,

o(x) -- - f A; l (x ,y ) dy ' -VTRCe),

~--~f dx e taV(AufAv - ½ w u S w v - ~ R e g 6 e ~ . (38) T=

Note that the energy momentum tensor T u" and the current ju are defined only up to local polynomials. We have made use of this freedom and have split off a purely imaginary local polynomial Tin such a manner that T uv and j u are frame independent and gauge invariant, i.e., only depend on the geometry and on the field strength A uv = O u A v - OvA u. (In Minkowski space, B is imaginary, T u~' and/ 'u are real.) The sources obey the following conservation laws

g u T uv = -AUVfu + ( i /96n)UUBuR ,

TUu= O, V u j u = 0 / 2 7 0 U v 0 u A v . (39)

The energy momentum tensor is not covariantly conserved, because the metric is not the only external field the system interacts with. The term -AUP/u de- scribes the energy and momentum generated or ab- sorbed by the gauge field, whereas the term proportional to euVOuR stems from the degree of freedom which specifies the direction of the zweibein. To dis- cuss the significance of this term, we switch the gauge

field off. In this case, V u T uv is a local polynomial (frame independence is spoiled only by the short distance singularities of the theory). It is, however, im- possible to construct a covariantly conserved tensor by merely adding a local polynomial to TUV: although T uv can be chosen in a frame independent manner, the presence of an additional degree of freedom independent of the geometry cannot be hidden. To arrive at a covariantly conserved energy momentum tensor, we need to pin down all external fields in terms of the metric. If the gauge field is switched off, this can be achieved, e.g. by imposing the gauge condit ion Vuw u = 0 on the spin connection (this condit ion does not break coordinate invariance and uniquely fixes the zweibein in terms of the metric; in isothermal coordinates, it implies F = 0). Formula (4) shows that in this gauge the determinant is real. The response of in det D to a deformation o f guy is given by the real part of the tensor T u" defined in (38), which is indeed conserved.

I am indebted to Hans Debrunner for useful infor- mation concerning global properties of two-dimensional differential geometries.

References

[1] L. Alvarez-Gaum6 and E. Witten, Nucl. Phys. B234 (1984) 269.

[2] W.A. Bardeen and B. Zumino, Lawrence Berkeley Lab. Preprint LBL-17639 (1984); F. Langouche, T. Sch/icker and R. Stora, Phys. Lett. 145B (1984) 342; L. Alvarez and P. Ginsparg, Harvard University Preprint HUTP-84/A016 (1984).

[3] F. Langouche, Phys. Lett. 148B (1984) 93. [4] L. Bieberbach, Differentialgeometrie, (Teubner, Leipzig,

1932). [5] H. Leutwyler, Phys. Lett. 152B (1985) 78.

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Gravitational anomalies: A soluble two-dimensional model