Lattice gravity and strings

Volume 150B, number 1,2,3 PHYSICS LETTERS 3 January 1985

LATrICE GRAVITY AND STRINGS ~

A. JEVICKI and M. NINOMIYA Department of Physics, Brown University, Providence, R! 02912, USA

Received 1 October 1984

We are concerned with applications of the simplicial discretization method (Regge calculus) to two-dimensional quantum gravity with emphasis on the physically relevant string model. Beginning with the discretization of gravity and matter we exhibit a discrete version of the conformal trace anomaly. Proceeding to the string problem we show how the direct approach of (finite difference) discretization based on Nambu action corresponds to unsatisfactory treatment of gravitational degrees. Based on the Regge approach we then propose a discretization corresponding to the Polyakov string. In this context we are led to a natural geometric version of the associated Liouville model and two-dimensional gravity.

Nonperturbative, discrete lattice methods might offer a fruitful approach to the problem of quantum gravity. One could initially think of a lattice ~i la Yang-Mills treating the gauge symmetry of gravity in a group theoretic manner. This, however, we do not think is satisfactory (due to noninvariance of loops and also we believe wrong universality class); rather one is led to the more natural geometric discretization (for discussion of these issues see ref. [1] ).

The simplicial discretization introduced by Regge offers a natural framework with variables being the geodesic lengths and curvature expressed through the deficit angles [2] (for more recent, systematic treat- ments see refs. [3,4] ). Concerning the continuum issue of coordinate invariance there are in this context inter- esting proposals [5,6].

We are interested in quantum applications of the Regge method and in this paper we concentrate on the problem in two dimensions. Our results and implications will therefore concern the theory of lattice surfaces (string model). This has already received a lot of attention lately [7 -9 ] . The problems of interest in two dimensions go as follows. The Regge version of the Einstein theory is simply

S E - H (l) = . ~ e i + ~k 0 ~ A (A), (1) t A

¢' Supported by the US Department of Energy under Contract DE-AC02-76ER03130.A013 - Task A.

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

where e i are the deficit angles at vertices and A (A) are areas of triangular simplices (both given in terms of bone lengths {la}); the total deficit is the Euler number implying trivial dynamics. Of real physical interest is the addition of matter fields

Sm = f d2x ,X/r~ [gUY(x) i) u c~ (x) a v qb (x) + m 2¢2 (x)], (2)

and this is essentially the problem we study in what follows. It encompasses the string problem and also the R 2 gravity problem (its discretization is related to the problem of discretizing the mass term, we shall comment later).

In the continuum (2) contains nontrivial dynamical effects given essentially through the trace anomaly [ 10]. To summarize: at the classical level the action is invariant under local Weyl rescalings

guy(x) -+ or(x) guy (x), (3)

but the regulators break this symmetry producing the anomaly at the quantum level. This for the string ((b(x) --+ X ( x ) ) according to the quantization of Polyakov [ 10] implies the nontrivial Liouville dynamics

S e f f = x f , / i - ~ ) + a ~ T f d x f d x ' V ~ R 2! . (4) a V~R

The correct discrete version o f surfaces should then contain these effects. For instance, for latter reference

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_N

\

Fig. 1.

let us consider a naive discretization of (2): one would triangulate the parameter space x (for example, in terms of a regular lattice in fig. 1) and simply represent the metric tensorguv(x ) ~guv(A) by 2 X 2 matrices on triangles. The scalar fields as nsual live on vertices with a finite difference representation for 0 u ~ av4,:

(4,1 -- 4,3) 2

(~1 - 4 '3) (4,2 - 4 ,3)

so the naive action is simply

S --- ~ g(A)l /2 g ( A ~ ' G~v(4,), A

(4,1 --4,3) (4,2 --4,3),~,|

(4,2 - 4 ,3)2 / (5)

(6)

and clearly this construction (0 has no attention to general coordinate invariance, (ii) has however still a version of local Weyl transformations, on each triangle guy(A) ~ a(A)guv(A) and consequently there is no dynamical effect of anomaly.

The way to discretize is through Regge calculus and geodesic lengths. For the metric tensor and the geometry of fig. 1 we defme ,a

( 1 2 ~(12-12-12)) Guy(t) = (7)

½(12_12_12) l 2

*1 The notation for lengths {li) on the triangle corresponds to sides opposite to the vertices {i), respectively.

and obtain for the action (2) the representation:

S = ~ 1 [ /2(4 ,1_4,2)(4 ,1_~3) AA--05

+/2 (4,2 --4,1) (4,2 --4,3) + 12 (4,3 --4,1) (t~3 --4,2)1.

(8)

The above form for the metric tensor satisfies the following requirements: (i) it leads to a symmetric action (under permutations of vertices of a triangle); (ii) for the case of a fiat rectangular lattice (/1 = 1, l 2 = 1, l 3 --- x/~) it gives the Wilson action; (iii) when the 4,(i) are the surface coordinates X(i) the classical solution for the l is simply:

11=IX 2 - X 3 1 , l 2 = I X l - X 3 [ , 1 3 = I X l - X 2 1

and the action reduces to a sum over areas of triangles (total area o f the surface). Actually the above neces- sary requirements specify uniquely guy as a function of l ' s . The result (8) also agrees with the construction of Sorkin [1 I] who had developed a general exterior calculus on simplicial nets ,2. Concerning the mass term we mention the following. Its discretization is equivalent to the problem of discretizing N ~ R 2 action, this is seen by coupling x/fiR (x) 4,(x) (whose discretization is ~i ei4,i) and taking the m 2 ~ ,~ limit. However, we find that the discretization for this is not unique. Namely, we could approximate

mE f d2x "vl'gR 2~ m 2 ~. Si4, 2, (9) !

where S i is some area. The choice for S i to be the area of a dual polygon would correspond to the suggested R 2 form of ref. [14]. But we can also take forSi the 1]3 (sum of areas a t / ) which would give a different form (its advantage might be positive definiteness). The main point is that the construction is not unique.

We can now discuss the formulation of trace anomaly in the Regge set up. Obviously this regularization breaks the local Weyl symmetry: since triangles share edges rescaling lengths on one particular triangle would neces- sitate the same rescaling for the neighboring triangle and so on. What remains only is a global transformation

*2 This Regge lattice, even though different does have some similarities with recently introduced random lattices of ref. [4]. For a review see ref. [12]. Rigid simplicial lattices were also studied recently, see, e.g., ref. [13].

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(the anomaly is indeed invariant under this). We exhibit the anomaly in lattice perturbation theory: we consider the weak field expansion and integrate out the scalar field up to second order. Simply expanding

l l = l + h l , l 2 = l + h 2, l 3=x , /2 ( l+h3 ) , (10)

the action (8) becomes

1 n~ ~ [Au¢(n)] 2 + S I + O(h2 ) (11) S=~ u

where the O(h) tem~ can be put into the form:

1

SI ='2 ~n [huv(n) Tuv(¢) + hm'(n) Tuv(¢)] (12)

with huv and huv being certain linear combinations of h i at the sites n, n-/1, n-/11-/12. The notational analogs of the energy-momentum tensor are

_ 1 Tuv(n ) - ~ euL, a b A a (~(n) Ab(~(n),

Tuv(n) --¢ Eu,,,, b 8,, ¢(n) 6be(n), (13)

with A a and 8 a representing left and right derivatives and Eu~,ab = 6 uv6 ab --~ ua6 vb --~ ~b 6 va" The doubling occurs due to the fact that for each site we attach two triangles (fig. 1) otherwise things are actually totally paral- lel to the continuum treatment. One can integrate the scalar field concentrating on the second order

~ ~ ( ( h T + f t T ) n (hT+f tT)m) , (14) n m

which through lattice propagators in the continuum limit produces a nonlocal effect. A longer computation shows that what on f'mds corresponds (up to local counter terms) to the O(h 2) expansion of

4~Tr'n~m e(n) AOnm e(m), (is)

where AO m is the Wilson propagator, e(n) the deficit angles and to identify the result we have used the fact that to first order in the expansion (10) the deficit angles can be shown to equal

e(n) = - 4 (AuA vhuv - 6uvA2huv} (n +/11 -/12). (16)

So to this order we showed the presence of the anomaly; the basic reason it is detectable comes from the fact that it is a nonlocal term in comparison with local counter terms.

Let us now proceed to the considerations of the

string model and describe what its discretization should be. In the literature several different procedures are used [7-9] , the most direct being a finite difference discretization [8] of the Nambu action f d2x [det OuX 0vX ] 1/2 on a regular parameter space lattice:

S N = ~ [det GO()] 1/2 A

= ~ ((X 1 - X 3 ) 2 (X 2 - X 3 ) 2 A

- [(X 1 --X3)" (X 2 -X3)] 2) 1/2. (17)

However transforming this into the gravitational lan- guage we can show that it corresponds precisely to the naive discretization of gravity. Namely, consider the other way around the naive action (eq. (6)) with surface coordinates (~i ~ Xi); it is possible to integrate over the 2, × 2 matrix variablesgu~(A ) at each triangle separately (the transformations are due to Gervais and Neveu in connection with the continuum treatment where there is the issue of regulators, here the discrete- ness makes the statement complete). The identity is

f u n dguu (A) exp { -- [~0 V~( A ) + vrgguZ'(A) Gu,,(X)I } ~ v

= d(f~" exp ( - X0 Vrp)) exp ( - x / ~ ) / x / ~ . (18)

It can be proved by: diagonalizing Guy(X), changing to the eigenvalues of guY(A) so the left-hand side of(18) becomes

,fd~, 1 d~ 2 I~, 1 --~k2[

X exp { - [;ko/Gkl~2) 1/2 + (~'1 + ~'2) vCG-/(;kl;k2) 1/2 ]),

(19)

and now after a change of integration variables

P = (~kl~2)1/2 , t = X 1 + ~2 integration over t leads to eq. (18). The implications of the above correspondence are that in direct discretization of the Nambu action the gravitational degrees are mistreated. This comes from the fact that as explained the naive lattice gravity has problems with the anomaly.

We suggest a discretization based on Regge calculus. It is based on the analogy with Polyakov's action and

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uses the Regge construction (7):

S = ~ { [g ( I ,A ) ] I / 2g tW( I )G .v (X)+ XOV~) (20) A

with lengths at the bones (l} and coordinates of the surface (X) being independent variables. As noted ear- lier (in construction of guy(l)) at the classical level the equations can be solved for the l and the action (20) is indeed sum over areas of triangles and reduces to Nambu action (17), at the quantum level one is to integrate over the l and X independently and the gravitational degrees receive nontrivial dynamics. Concerning this we would like to stress that for a correct approach to the continuum gravitational counter terms need to be subtracted from (20). Namely integration over X would give

exp 1) C[l] - - ~ nm '

where C are terms in addition to the nonlocal anomaly term. This is similar to the situation with Wilson fermions. Presence of this effect can be easily exhibited by taking a translationally invariant configuration (for the l) , the space is flat, all the deficits are equal to zero and the integration o f X gives entirely the C effect:

D C [ I ] -~D- ' log[( / ] +12+12)/A(l)1. (22)

I f kept, this term would immediately lead to the infi- nite Haussdorf dimension.

A subtraction leads us to effective gravitational theory of the type

S='Y ~ e n ( 1 ) A n m ( l ) e m q ) + Xo ~ A ( 1 ) , (23) nm .~

which represents a simplicial discretization of the Liouville action. Anm denotes the full scalar propagator on the gravitational net (as such it will need renor- malization); the basic form is the inverse o f the qua- dratic form in (8):

= m o S ) n m . (24) Anm (M + 2 - 1

= ~ l ~ / A ~ (the sum is over all Explicitly: M n =m 2 triangles meeting at n), M(n , m are neighbors) = Z(l 2 - l~ - l~)/A (the sum over the two triangles having the link nm as boundary)Mnm = 0 otherwise.

This essentially geometric formulation of Liouville theory is suited for numerical studies [ 15]. These can

be used to simulate the continuum problems of quan- tized Liouville theory, two-dimensional gravity and the Polyakov string [ 1 6 - 2 0 ] .

We thank the Ecole Normale Sup6rieure, CERN and the Niels Bohr Institute for their hospitali ty. We ac- knowledge fruitful discussions with many colleagues, especially J.L. Gervais, M. Virasoro, H. Kawai, H.B. Nielsen and D. Weingarten. We have also learned from C. I tzykson that he and M. Bander considered a similar problem [21].

References

[ 1 ] A. Jevicki, Lectures IV Adriatic Meeting on Particle physics, ed. Dj ~ijaSki, N. Bilid, B. Dragovi~ and D. Popovi~ (World Sdentific, Singapore, 1984).

[2] T. Regge, Nuovo Cimento 19 (1961) 558. [3] M. RolSek and R.M. Williams, Phys. Lett. 104B (1981) 31;

B. Hasslacher and M. Perry, Phys. Lett. 103B (1981) 21. [4] R. Friedberg and T.D. Lee, Nucl. Phys. B242 (1984) 145. [5 ] M. Lehto, H.B. Nielsen and M. Ninomiya, Geometry from

pregeometrie quantum lattice, Nordita Report (1983); Pregeometrie quantum lattice and general relativity, Brown University preprint HET-543 (1984).

[6] M. Kaku, Generally covariant lattices, The random calculus, and .... City College preptint (1984).

[7] D. Weingarten, Phys. Lett. 90B (1980) 280; D. Foerster, Nucl. Phys. B170 (1980) 107; H. Kawai and Y. Okamoto, Phys. Lett. 130B (1983) 415; B. Durhuus, J. Frolich and T. Jonsson, Phys. Lett. 137B (1984) 93.

[8] A. Billoire, D. Gross and E. Marinari, Phys. Lett. 139B (1984) 75. D. Gross, Phys. Lett. 138B (1984) 185.

[9] T. Eguchi and M. Fukugita, Phys. Lett. l17B (1982) 223. [10] A.M. Polyakov, Phys. Lett. 103B (1984) 207;

L.S. Brown, Phys. Rev. D15 (1977) 1469; K. Fujikawa, Phys. Rev. D23 (1981) 2262.

[11] R. Sorkin, J. Math. Phys. 16 (1975) 2432. [12] C. Itzykson, Cargese Lectures (1983). [13] W. Celmaster, Phys. Rev. D26 (1982) 2955. [14] H. Hamber and R.M. Williams, IAS preprint (1984). [15 ] A. Jevieki and Lj. Huntt, work in progress. [16] J.L. Gervais and A. Neveu, CERN reports TH 3954/84;

TH 3955[84. [17] E. Onofri and M. Virasoro, Nucl. Phys. B201 (1982) 159. [18] E. D'Hoker and R. Jackiw, Phys. Rev. Lett. 50 (1983) 1719. [19] T. Banks and L. Susskind, IAS report (1984). [20] A.B. Zarnolodchikov, Phys. Lett. l17B (1982) 87;

J. Jurkiewicz and A. Krzywicki, Orsay preprint LPTHE 84/22.

[21] C. Itzykson and M. Bander, to be published.

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Lattice gravity and strings