the weyl anomaly for singular surfaces

Volume 203, number 3 PHYSICS LETTERS B 31 March 1988

THE WEYL ANOMALY FOR SINGULAR SURFACES

David LANCASTER Department of Theoretical Physics, University of Oxford, 1 Keble Road, Oxford OX I 3 NP, UK

Received 3 December 1987

The Weyl anomaly for both world sheet and background conical singularities contains a term in ¢ I apex- I analyse this using extended string states and by relating it to the system of a particle moving on a cone subject to a Bohm-Aharanov flux. Some orbifold results are reproduced.

1. Introduction

The fact that strings can propagate consistently on certain singular backgrounds corresponding to orbifolds has been known for several years [ ! ]. In fact, singular surfaces are relevant to string theory in several contexts: Witten uses a singular world sheet as a convenience in deriving the form of • and f in string field theory [ 2 ], compact lorentzian world sheets are generally liable to have singular points, and background orbifolds are important in discussing com- pactification. In this paper I consider a general two- dimensional conical singularity both in the background and world sheet. The Weyl anomaly, as used by Polyakov to fix 26 dimensions [3], is determined and I show that there is an additional term with a coefficient (dependent on cone opening angle) that must vanish for the amplitude to be physical.

The case of a flat background, but a world sheet with conical singularity introduces the techniques and is a useful preparation for the background singularity case. These techniques amount to relating the Weyl anomaly to determinants calculated for the physical situation of a particle moving on a cone, subject to a Bohm-Aharanov flux through the apex.

Next, the background is made conical in two directions thereby introducing a singularity. By means of a classical argument relating background and world sheet singularities I require that the world sheet metric should be gauge fixed to have a singularity of correlated strength. This amounts to a world sheet view of the naive classical picture that a smooth string

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moving over a cone will develop a kink unless the cone cerresponds to an orbifold [4]. In order to make this argument I am obliged to work with extended states of a novel kind. One can then proceed as before and calculate the Weyl anomaly with the help of an exact background field expansion.

The conclusion discusses the extended states in greater detail. Some are identified as lying in twisted sectors, their ground state energy is calculated to con- firm this.

2. World sheet singularity

The Polyakov functional integral consists of two pieces. There is a Faddeev-Popov determinant (det P*P in the notation of Alvarez [ 5]) from fixing the world sheet metric to have the form gap= exp(2~)~aa. Rap is the standard metric for a cone with opening angle,8 (the shorthand/~=p/2n will often be used), it is flat everywhere except at the apex. The second piece comes from integrating the string coordinates on a flat background and is simply the determinant of the covariant laplacian (D~, o anticipating later notation).

The Weyl anomaly W ( 8 ¢ ) picks up contributions from the variation of each of these terms. Using heat kernel regularisation they may be written as traces.

0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division)


W(80)=½dS( lnde t DLo ) 2 +½8(lndetP*P)

= d [ T r 80 exp( - eD~,o) ]

+ ½ [ - 4 Tr 60 e x p ( - e P * P )

+2 Tr 80 exp( - EPP*)] . (1)

General principles require that the anomaly be a local quantity. For a smooth world sheet without boundaries there is only one such term of the correct dimension (as usual there is also an infinite term of lower dimension). However, the cone has a distin- guished point and a second term is possible.

W(60) =A f , , /~g 80 d z a + g 801~p~x • (2)

Both terms are non-trivial because on integration they are non-local (e.g. the B term will be In det v/-gl ao~,). Furthermore, the presence of the cone will not disturb the value of the first coefficient from its smooth world sheet value ofA = ( d - 2 6 ) / 2 4 n .

To determine the/~-dependent coefficient B, consider a global Weyl scaling 8 0 = 8 0 o = const, about 0 = 0. In this case (1) becomes

I~(8 0o) =80o[d Tr exp( - el)~,o)

- 2 Tr exp( - eP*/5) + T r exp( - ePP *)] . (3)

The operators PfP, PP* may be evaluated on the pure cone using a basis of polar (wedge) coordinates, after a simple diagonalisation they become

PP*- 2 \ 0 , ' (4)

where the operators I)~,~ are gauge-covariant lapla- cians on a pure cone with opening angle r , and with a background Bohm-Aharanov gauge field corresponding to flux ~ through the apex. Quantities such as the determinant will be symmetric periodic functions of the flux.

One might worry that the choice of polar coordinates is hiding some physics at the origin. This is borne out by an analysis in complex coordinates which indicates that there is a curvature contribution,/?, proportional to a delta function there. Alter- natively this may be interpreted as a term describing

the field strength of the Bohm-Aharanov flux. In cal- culating the heat kernel a transformation is made that removes this term, so that the entire effect of the Bohm-Aharanov flux appears as non-trivial period- icity of the wave functions.

The trace can therefore be written as

Tr exp( - ~ptp)

= Tr exp ( - ½ EI3~,_tr)+ Tr exp( l - ~ ~D~)'2

= 2 Tr exp ( - ½ ~l)J.#) . (5)

Doing the same for p/at the expression for I~(80o) (3) becomes

IYF(80o) = 8 0 o [ d T r exp( - d3~,o)

- 4 Tr exp( - ½ ~ , # ) +2 Tr exp( - ½ ~I3~,2#)] .

(6)

These traces can be determined [6,7] using an expression for the heat kernel on a cone with flux ~. This was originally written down at the beginning of the century and brought to modern attention by Dowker [ 8 ].

Tr exp ( - ~]~,~) = vol/4n e

+ (1/2/~)[~(6- 1 ) + ~ (1 -#21, for0~<~< 1. (7)

Notice that the general expression for the finite part of the anomaly (2), when evaluated for constant scaling about 0 = 0 becomes

I'V(80o)=60o( ~--426 4 n ( 1 - f l ) + B ) . (8)

A d-dependent form for B follows from equating this to the expression (6) and using the finite part of (7). In order for the Weyl anomaly to vanish each of the coefficients A and B must be separately equal to zero. As usual d=26 follows from A=0, and in this dimension,

B= (2//~)× 1, for 0</~< ½ ,

B= ( 2/~) × (1-f lz) + ( n - ~ ) 2 ,

for n-½ <~fl<~n+ ½, n~t~ . (9)

B only vanishes for/~= 1. In this derivation the significance of # = ½ arose from the restriction on ~ in

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(7), its geometrical significance is that for ]~< ½ all geodesics self-intersect. In the limit/7-00, the cone becomes a cylinder.

3. Background singularity

I only consider the simplest type of singularity, in which the background metric describes a cone in two dimensions, and is independent o f all other directions. This can easily be generalised to the case where several pairs of dimensions are treated indepen- dently, a situation that frequently occurs in simple orbifold models. It is also related to cosmic string models [ 9 ]. Concentrating on the two directions and using complex conformal coordinates for the cone (opening angle fib), the a model is

(io)

As usual, one would like to make a background field expansion. Because the space is flat the equation of mot ion for the background classical solutions zB (a) is 0dZB = 0. Here I have assumed that g~a is gauge fixed either fiat or conical, and I am using complex world sheet coordinates. Thus zB is analytic or antianalytic in a and will be characterised by a winding number

N = (1/2n)[arg zB]~ cxp(2~i)

for N positive za = a W, and for N negative z B ~--- ~ --N.

To fix N one must be more precise about what amplitude is being calculated. The simplest, spherical topology, vacuum to vacuum diagram is inappro- priate in this situation, za = const, with N = 0 is the only finite action solution. In other words the classical world sheet has degenerated to a world line, which is unable to probe stringy aspects o f the problem. It will also become apparent that it would lead to diffi- culties in the background field expansion.

In order to probe the singularity one needs a non- degenerate world sheet. This can be obtained by cal- culating the amplitude to go from the vacuum to an extended state. By appropriate choice of that state, the winding number can be fixed.

Introducing a boundary to the world sheet in this

manner could affect the analysis of the Weyl anomaly. However, there are no non-trivial local terms at the boundary with free coefficients that could be added to (2); for this reason the off-shell propagator [ 10] is generally Weyl invariant #~. It is also worth being clear about the issue of conformal Killing vectors. Usually in disc topologies, string tree diagram amplitudes vanish unless there are sufficient vertex operators to soak up the conformal Killing vectors. This is true for an open string disc; however the boundary conditions on metric fluctuations for a closed string disc with an extended state boundary are different [ 11 ]. The calculation here is done on an infinite disc, there are no normalisable conformal Killing vectors and in effect the modified boundary conditions are being used.

The induced metric for the solution with winding number N is

Yc~ = ( I zB 12)gb-t 0,~ZB0,8-YD = ~ap( I tr 12) INIgb-' ,

it suffers f rom a conical singularity with opening angle {Nlflb. The free world sheet metric g-a has its classical value given, up to a conformal scaling, by the induced metric ~',~a. In fixing the gauge it is inap- propriate to imagine the conformal factor to be so singular as to remove the cone altogether. It is therefore sensible to correlate the singularities in the gauge fixed metric g-a exp(20) to those in Y,~p- Specifically, g~a is chosen to have a cone singularity with opening angle r = INIPb. It is in this sense that the Faddeev- Popov determinant, usually background independent, does contribute to the anomaly. This kind of relation is also used in treating the scattering o f twisted states on an orbifold [ 12,13 ].

As mentioned earlier, this argument is a world sheet view of the classical picture that strings develop kinks unless the cone corresponds to an orbifold [4 ]. In fact it is a classical argument along these lines that lies at the heart of showing why orbifolds are preferred backgrounds, rather than a quantum mechanical re- quirement that the r - funct ion of the sigma-model on a cone should vanish. Orbifolds have/~b = 1/p so if I NI =P the world sheet can be fiat, and as I shall show, the anomaly vanishes.

~' It is true however that a proper treatment of boundaries would change the 4n(1-/~) in the first term of eq. (8) to the Euler characteristic of the world sheet in question. In the critical dimension this has no effect on the coefficient.

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Volume 203, number 3 PHYSICS LETTERS B 31March 1988

The previous section on world sheet singularities actually calculates the amplitude from vacuum to a "punctual" [ 10] state. In that case there is a fiat background and zB = const, is a reasonable solution. Classically, T,~p vanishes for any choice of 7up and there is no restriction on ~ . One can alternatively calculate amplitudes to extended states in a fiat background by setting/Tb = 1 in the following.

Proceeding to the quantum theory, the action (10) is used in the functional integral with the restriction that z(a) maps between the apexes of the correlated cones. The covariant background field expansion [ 14] can be performed exactly.

z=zB(1 +~gzB) ''~ (l 1)

The quantum field integration is actually done using tangent variables ~= ( I zB [ 2) (&- ~)/2 ~, with boundary condition ~ I apex = 0.

; ~ ' ~ ' e x p ( - - S ~ ) ,

1 Sc = ~-~n f v/g I D ~ I 2 d e a ,

with

As =i½ (/~b -- 1 )0~ ln(ZB/Zn). (12)

The U(1 ) gauge field A~ is apparently pure gauge, but does in fact correspond to a Bohm-Aharanov situation with flux ~,

1 As d~ ~

~ = 2re about apex

= ( - 1/2to)(1 --/~b)( Imln'-Bj.~ ~ exp(2.i)

= (/1b-- 1)N, (13)

I f the background is an orbifold and [Nl/Tb = 1 the flux is integer, and can be gauged away. The determinant about g~a arising from (12) is

det I)},<&_ l)N=det I~ ,Nf f b = d e t I )}d . (14)

In the other ( d - 2) directions the result is the same as in the previous section. Similarly the gauge fixing determinant is unchanged by the background except through the correlation of singularities. The coefficient B of the Weyl anomaly at d = 26 is

B = (1//1) × (/~2 - / ~ + 2),

for 0 </7~< ½,

B = (1//~) × [/~2 _ ( 6 n - 1 )/~+ 3n 2 - n + 2 l ,

for n - ½ ~</~ n ,

B = (1//1) X [/~2 _ (6n+ 1 )/1+ 3n 2 + n + 2 ] ,

for n~</3~< n+ -~.

Again this only vanishes for/~= 1.

(15)

4. Conclusion

To interpret these results it is necessary to examine more closely the nature of the states IN) that induce winding number N in the classical solutions. For N # + 1 they correspond to self intersecting curves in the background. I f the background is fiat,/lb = 1, the only amplitudes of the type studied which avoid an anomaly are (01 + 1 ) , the others are unphysical so one might be tempted to regard the states I N ¢ + 1 ) as illegal.

However, if one now considers an orbifold, fib = 1/p, it is precisely states of this kind that give rise to a physical amplitude; (01 + p ) has /~=1 and is anomaly free. In fact the states I + 1 ) on a fiat background and I + P ) on a/~b = 1/p orbifold are simply related. A non-intersecting curve on the flat space away from the singularity will develop p - 1 self in- tersections when it is moved continuously over the apex. There are certain states that cannot be de- formed away from the apex in this manner, these are identified as lying in the twisted sector. Specifically, the twist is given by the winding number modulo p.

To substantiate this identification, consider the ground state of the twisted sectors. Energy-momen- tum can be inserted, and the Weyl anomaly can be cancelled by placing a vertex operator at the singularity of the background space. This point is the image of the world sheet cone, so the vertex operator is not to be integrated over the world sheet. For example a term exp (ikx) l apex in the functional integral leads to a contribution to the anomaly of ( - k2/4fl) ~1 apex- TO see this, one shifts variables which results in a term in the action - ½ n k K ( a , tr)k, where K is the Green function on the cone: K = (I/42q7) In I trl 2. By regu- lating in a reparameterisation invariant manner

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(most simply, a p roper distance cut-off) the form above is found. The Weyl anomaly is p ropor t iona l to ~lapex, which is of the same form as discovered ear- l ie r (2) . In order to cancel it one must require

-k2 /4~+B=O. (16)

Using the form for B der ived in (15 ) and applying this to twisted states with the smallest winding number, - p/2 <~ N<~ p/2 (N¢ 0), one finds ( ~ ' = 1 )

M z = - 4 ( / ~ z - / ~ + 2) . (17)

This is the same answer for the twisted ground state energy as found using other methods [ 1,12,13 ]. By using more compl ica ted vertex operators one can probe the physical state content o f the states I N ) in more detail .

For non-orbifold backgrounds, there are an infinite number of twisted sectors. The Weyl anomaly never vanishes unless a vertex opera tor is inserted, so it is never possible to relate high winding number states to states in the untwisted sector as was done above. This si tuation, i f not a l ready pathological at this stage, would almost certainly become so with a compact background having more singularities.

Acknowledgement

aided my unders tanding of this subject, most of all I have benef i t ted f rom Paul Mansf ie ld ' s insight.

References

[ 1 ] L. Dixon, J. Harvey, C. Vafa and E. Witten, Nucl. Phys. B 261 (1985) 678;B 274 (1986) 285.

[2] E. Witten, Nucl. Phys. B 268 (1986) 253. [3] A.M. Polyakov, Phys. Left. B 103 (1981) 207. [4] J.A. Bagger, Proc. XXIII Intern. Conf. on High energy

physics (Berkeley, CA). [5] O. Alvarez, Nucl. Phys. B 216 (1982) 125. [ 6 ] D. Lancaster, unpublished. [7] J.S, Dowker, Phys. Rev. D 36 (1987) 620. [8] J.S, Dowker, J. Phys. A 10 (1977) 115. [9] J.A. Bagger, C.G. Callan and J.A. Harvey, Nucl. Phys. B

278 (1986) 550. [ 10] A. Cohen, G. Moore, P. Nelson and J. Polchinski, Nucl.

Phys. B 267 (1986) 143; in: Unified string theories, eds. M. Green and D. Gross (1986).

[ 11 ] D. Birmingham and C.G. Torre, Phys. Lett. B 194 (1987) 49.

[ 12] L. Dixon, D. Friedan, E. Martinec and S. Shenker, Nucl. Phys. B 282 (1987) 13.

[ 13] S. Hamidi and C. Vafa, Nucl. Phys. B 279 (1987) 465. [ 14] L. Alvarez-Gaum6, D.Z. Freedman and S. Mukhi, Ann.

Phys. 134 (1981) 85.

I would like to thank the many people who have

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the weyl anomaly for singular surfaces

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