O N Sigma Model As a Three Dimensional Conformal Field

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June 1994

UR-1357

ER-40685-807

hep-th/9406010

ON Sigma Mo del as a Three Dimensional Conformal Field Theory

yS. Guruswamy, y S. G. Ra jeev and yyP. Vitale

yDepartmentofPhysics, UniversityofRochester

Ro chester, NY 14627 USA

e-mail: guruswamy, ra [email protected] chester.edu

and

yy Dipartimento di Scienze Fisiche, Universit a di Nap oli

and I.N.F.N. Sez. di Nap oli,

Mostra d'Oltremare Pad. 19, 80125 Nap oli ITALY.

e-mail: [email protected]

Abstract

We study a three dimensional conformal eld theory in terms of its partition function

on arbitrary curved spaces. The large N limit of the nonlinear sigma mo del at the non-

trivial xed p oint is shown to b e an example of a conformal eld theory, using zeta{

function regularization. We compute the critical prop erties of this mo del in various spaces

2 1 1 1 2 2 1 1 1 2 1

of constant curvature R S , S S R, S R, H R, S S S and S S

and we argue that what distinguishes the di erent cases is not the Riemann curvature

but the conformal class of the metric. In the case H R constant negative curvature,

the O N symmetry is sp ontaneously broken at the critical p oint and the correlation

length is in nite. In the case S R constant p ositive curvature we nd that the free

, although energy vanishes, consistent with conformal equivalence of this manifold to R

the correlation length is nite. In the zero curvature cases, the correlation length is nite

due to nite size e ects. These results describ e two dimensional quantum phase transitions

or three dimensional classical ones. 1

1. Intro duction

The correlation functions of statistical mechanical systems exhibit scale invariance

at a phase transition of second or higher order. It is p ossible to describ e such a phase

transition in terms of a scale invariant euclidean quantum eld theory,atypical classical

statistical system corresp onding to a regularized euclidean quantum eld theory. The

correlation length inverse mass of the particles of this eld theory will diverge as the

coupling constants approach a ` xed p oint'. The theory de ned by the xed p ointinthe

space of coupling constants will b e scale invariant and corresp onds to a phase transition

p oint of the statistical mechanical system. The immediate neighb orho o d of the xed p oint

describ es quantum eld theories with masses of the particles small compared to the cut{o .

At present, the only general way to construct quantum eld theories is as such limiting

cases of statistical mechanical systems. It is of great interest to construct scale invariant

quantum eld theories directly. This should b e p ossible since they are ` nite' i.e., cor-

resp ond to a xed p oint of the renormalization group; moreover it could eventually lead

to a de nition of a general quantum eld theory as a p erturbation to a scale invariant

quantum eld theory without the usual cumb ersome pro cedure of renormalization. Any

progress in that direction is of imp ortance to particle physics, where the standard mo del

should eventually b e constructed by suchanintrinsic pro cedure.

Scale invariant quantum eld theories are also interesting for phenomenological rea-

sons: they describ e exp erimentally accessible phase transition phenomena. The critical

exp onents of such transitions have b een calculated with great accuracy in many realistic

cases. Also, in the case of two dimensional systems, many systems have b een solved ex-

actly. That is, their partition and correlation functions have b een obtained in terms of the

sp ecial functions of classical mathematics.

The key to this success in two dimensions is that the theories often have a symmetry

much larger than scale invariance: conformal invariance. In two dimensions, in nitesimal

conformal transformations p osition dep endent scale transformations are the same as com-

plex analytic co{ordinate transformations. This large symmetry puts strong constraints

on the correlation functions; in the case of `minimal mo dels' these constraints are strong

enough to determine them completely. This is similar in spirit to the solution of classical

p otential problems in two dimensions by conformal Schwarz{Christo el transformations.

For example, the general b oundary value problem of the two dimensional Laplace equation

can b e solved by mapping the b oundary to one of a small class of standard b oundaries. 2

Most interesting classical phase transitions o ccur in three dimensional systems. One

should not exp ect a complete generalization of conformal techniques to three dimensions

as an analogy, it is not p ossible to solve the general b oundary value problem of the

Laplace equation in three dimension by conformal transformation to a standard b oundary.

One should exp ect that conformal invariance is a p owerful constrainteven in this case,

although not strong enough to completely determine the correlation functions. In this

pap er we will b egin a study of three dimensional conformal eld theory.We study the

O N non-linear sigma mo del. It is seen that in three dimensions also, the partition and

correlation functions at a second or higher order transition are invariant under conformal

transformations of the metric tensor. We will also compute the partition function in some

sp ecial cases to con rm this general picture.

In three dimensions, it is necessary to study eld theories in a curved space to fully

understand conformal invariance. In two dimensions all manifolds are conformally at,

so the only geometric information ab out the manifold that can a ect critical systems is a

nite number of Teichmuller mo duli parameters. This is replaced in the three dimensional

case by an in nite dimensional space of conformal structures see app endix B for a precise

de nition. The partition function of the system should b e viewed as a functional of the

metric density^g = g g , where g = detg ; it is the generating functional of the

ij ij ij

correlation functions of the stress tensor. The p oint is that conformal invariance alone

will not determine this functional, even for `minimal' mo dels. The correlation functions of

the stress tensor are physically measurable quantities; therefore the partition function on

curved spaces is of physical interest even when the true metric of space is at.

There is another reason to study critical phenomena in curved spaces. If we sub ject

a system to external stresses, we can deform the underlying microscopic structure such

as a lattice so that the e ective distance b etween p oints is no longer the usual one. At

a critical p oint, universality suggests that the details of the microscopic structures do not

matter; but the system is still sensitive to the deformation through the e ective metric

tensor density.

In three dimensions there is no conformal anomaly; hence a conformal eld theory

can b e de ned as one whose partition function is a function on the space of conformal

structures if there were an anomaly,itwould have b een a section of a real line bundle.

That is, its partition function should satisfy

Z [e g ]= Z [g ] 1 3

g b eing the metric tensor. It is not obvious that interacting theories of this kind exist; we

will show that the large N limit of the O N sigma mo del [1], [2], [3], [4], [5] is

such a conformal eld theory, at its non-trivial xed p oint. Moreover a primary eld is one

whose correlation functions transform homogeneously under a conformal transformation:

f x +f x

1 n

<x x > 2f = e <x x > 2

1 n 1 n g

e g

Again, the eld of the O N sigma mo del are an example of such a primary eld. These

de nitions are the appropriate generalizations of the p oint of view of Ref. [6].

Although there is no conformal anomaly in three dimensions, there could b e a parity

anomaly in general. The particular example we are studying, the O N sigma mo del do es

not have such an anomaly.We plan to return to this issue in the context of fermionic and

Grassmannian sigma mo dels [7].

2. ON non-linear sigma mo del in three dimensions

In this section we study the O N non-linear sigma mo del [1-5] on a Riemannian

manifold M; g. Of particular interest to us is the case M = R with b eing an

arbitrary 2{dimensional curved space: this describ es a quantum phase transition at zero

temp erature, the temp erature b eing the inverse radius of R.We will also brie y consider

some cases at nite temp erature, of the typ e, M = S .We study the theory in detail

at its non-trivial xed p oint for manifolds with zero, constant p ositive and constant

negative Riemannian curvature.

The euclidean partition function of the O N non-linear sigma mo del in 3{dimensions

in the presence of a background metric g xis given by,

3 i

d x g x[g x@ @ ]

Z [g ]= D []e 3

i i

where i =1; 2; ;N and the x satisfy the constraint x x=1. is a coupling

constant. As it stands, the action is not invariant under the conformal transformation of

the metric, g x ! xg x. The action b ecomes conformally invariant [8] when

the laplacian @ g xg x@ is mo di ed to the \conformal laplacian", =

@ g x g x@ + R, with transforming as x ! xx. R, in the

conformal laplacian, denotes the Ricci scalar and isanumerical constant. =

4d1

and is equal to 1=8 for dimensions d = 3. The generating functional then reads,

3 i

d x g [ ]

g i

Z [g ]= D []e 4

However, the constraint x x = 1 still violates conformal invariance. The constraint

on the elds can b e implemented by a Lagrange multiplier, in the form of an auxiliary

eld x with no dynamics, as follows:

3 i i

d x g [ 1] +

g i

Z [g ]= D []D [ ] e 4

Although, classically this is not conformally invariant, we will show that there is a non-

trivial xed p oint for the quantum theory at which it is conformally invariant. Following

Wilson's approach, we regularize the generating functional Z in the ultraviolet byintro-

ducing a cut-o , , in the momentum space. Before doing that, let us note the canonical

i i

dimensions of the elds and the couplings in our action. Rescaling to ,wehave,

i i 3

+ ] d x g [

g i

2 2 2

Z [g ]= D []D [ ] e

The canonical dimensions in mass units can b e read o from the action:

[]=1=2 ; [ ]= 2 ; [ ]=1

to so that the coupling constantisnow Let us rede ne our coupling constant

dimensionless. The regularized partition function can now b e formally written as,

i 3

] d x g [ +

g i

5 Z [g; ;]= D []D [ ]e

where D []= dk and similarly D [ ].

jk j<

The large N limit

Since it not p ossible to solve this theory exactly,wehave to use some approximation

metho ds. At this p oint there are a few di erent approximation metho ds one could use in

studying this problem. These are the 2 + or 4 expansion metho ds, or the large N

expansion. Wecho ose to study the problem in the large N limit. In this pap er wedoall

our analysis to the leading order in the large N expansion.

In the large N limit, by whichwe mean N !1 keeping N xed, the generating

functional can b e calculated using the saddle p oint approximation. We will therefore

rede ne N 1 in the action as which will remain xed as N !1.We also 5

rescale the eld to N 1 .We can nowintegrate out the rst N 1 comp onents

N N

of the eld and rewrite Z as,

N 1

TrLog d x g [ x] + + +

N g g N

2 2

Z [g; ;] = D [ ]D [ ]e

this amounts to rotating the N-comp onentvector xinto the direction of its N-th

comp onent using the O N symmetry; we are justi ed in doing this since we will b e

studying this mo del only on manifolds with constant curvature.

We rst discuss the general case of the O N sigma mo del on a manifold with an

arbitrary metric. In all the examples we consider in this pap er, the manifolds are of

constant curvature and we therefore lo ok for the uniform saddle p oint, x=m and

x=b. This is given by the following set of equations \gap equations" obtained by

extremizing the action with resp ect to xkeeping x xed and vice{versa.

+ m b =0 7

2 2

G x; x; m ;g 8 b =

2 2 1 0 2

where G x; x; m ;g=hxj + m jxi . Gx ;x; m ;g is the two p oint correlation

function of the elds.

The physical interpretation for m is that of the physical mass of the eld and for b

is that of the vacuum exp ectation value of the eld sp ontaneous magnetization. Once

we nd the saddle p oint solutions of the action, we can compute the free energy density,

expansion. W , of the system at the saddle p oint, to the leading order in the

2 3 2

W [g; ;] = [TrLog ] 9 + m d x gm

At the critical p oint, this will b e a conformally invariant quantity. This is easier to see in

the zeta function regularization in whichwe will see that the critical value of vanishes.

+ Therefore the free energy in this regularization scheme is just given by TrLog

. In app endix A we give a brief outline of the pro of of the conformal invariance of

TrLog + in o dd dimensions [9], [10].

g 6

3. Flat space R

We will see in the next section that the short distance divergences of the Green's

function, G x; x; m ;g, are indep endent of the curvature of the space. It is therefore

natural to rst study our theory on R . Though this is a very well studied case, [2], [4],

let us brie y recapitulate the study of the xed p oints in this theory.OnR , the Ricci

scalar is zero. Therefore the saddle p oint solutions are given by the gap equations,

m b =0 10

2 2

b = G x; x; m ;g 11

2 2 2

= @ + @ + @ . The sp ectrum of the conformal laplacian is In at space,

x y z

2 3

Sp =k where k isavector in R . Therefore,

d k 1

G x; x; m ;g=

3 2 2

2 k + m

2 2

The gap equation 10 admits 3 p ossible solutions with m =0;b 6=0 or m 6=0;b =0 or

b oth m and b vanish.

1 When m = 0 and b 6=0,

d k 1

b =

3 2

2 k

At the critical coupling given by

1 d k

= ;

3 2

2 k

b = vanishes.

2 When m 6= 0 and b =0,

2 3

m d k

3 2 2 2

2 k k + m

4 7

from which it is seen that m>0 only when > . This corresp onds to the unbroken

phase of the O N symmetry. When <, m< 0 which is unphysical. If wedoa

more careful analysis as in [2], we will b e able to see that m = 0 when < . This is

the broken phase where the O N symmetry is sp ontaneously broken and has given rise

to N 1 Goldstone b osons with m =0.At = , which separates the two phases, m

go es to zero. This is the non{trivial xed p oint of the theory, the trivial xed p oint b eing

=0. We therefore see that at m = 0 and b =0,wehave the critical theory at the

non{trivial xed p oint. This xed p oint is UV stable.

4. Short distance divergence structure of the Green's function

Wewould next like to study our theory on curved 3{dimensional manifolds M . Before

we pro ceed, wewould like to establish that the only divergences that arise in the Green's

function, in the gap equation in curved space, are those which arise on R , namely they

are indep endent on the long distance b ehaviour of the Green's function.

1 + t

G x; x; ;g=hxj + jxi = jxi 12 dthxje

+ . The Green's function Let and x b e the eigenvalues and eigenvectors of

g n n

can b e written as follows:

G x; x; ;g= dt ht; x; x

+ and has the formal expansion where ht; x; x is the heat kernel of the op erator

ht; x; x= e x x:

The divergence in the Green's function comes from short distance and hence from small

t region. Let us therefore isolate the divergent part, from the nite part in the Green's

function. This can b e done byinvoking the asymptotic expansion of the heat kernel, [11]

which, in three dimensions, is given by,

d x;y

ht; x; y = a x; y t 13

n=0 8

where dx; y is the Riemannian distance b etween p oints x and y on the manifold M . The

Green's function is then

G x; x; ;g= dt a xt

The leading term, with a x = 1, is the only divergent term in the ab ove expression.

Therefore the Green's function is given by,

G x; x; ;g= + nite part

We see that the divergent part of the Green's function is indep endent of the metric g x.

Thus the critical value of the coupling constant is indep endent of the metric. Having

established this, we will pro ceed to study the O N sigma mo del on some curved manifolds

at the critical p oint . At this critical p oint the theory is nite.

5. Zeta function regularization

Although regularizations such as the Pauli-Villars regularization are easier to under-

stand physically, the zeta function regularization is more tractable when we are in curved

space. We will thus b e using this regularization in our work. Since the critical value of

the coupling constant at which the theory b ecomes nite is indep endent of the background

3 2

metric, we compute this critical coupling on R . The Green's function, Gx; x; m ;g, is

regularized as,

2 s 2

+ m jxi = s G x; x; m ;g=hxj

g g s

where

s= j j

g n

6=0

with as the eigenvalues of + m and the sum includes degeneracies. If the

n g

eigenvalues are continuous the sum gets replaced byanintegral. Then,

Gx; x; m ;g = lim s: 14

s!1

The gap equation on R b ecomes then,

Z Z Z

3 s1

2 2

1 d k 1 t

2 k +m t 2 2

lim [ k dk e ] = b + = b + dt

3 2 2 s 2

s!1

s 2 k + m 2 s

0 9

where the regularized coupling in the Pauli-Villars regularization has b een replaced

by in the zeta function regularization. Here wehave used the Mellin transform to

analytically continue the zeta function. Note that,

d k

3 2 2 s

2 k + m

has no p ole at s =1.Itisnow easy to verify that,

32s

1 m

lim [ = b ]

s!1

s s

=2 Recall, by analytic continuation of the gamma function, we can see that

and thus,

1 m

lim = b + :

s!1

s 4

Wehave seen though from the analysis in the case of M = R that at the critical p oint,

m = 0 and b =0. m and b are physical quantities and are regularization indep endent.

=0.We will b e using this value of Therefore in the zeta function regularization, lim

s!1

the critical coupling in all our future calculations.

We are now ready to study the sp ecial cases where the manifolds have zero, constant

p ositive and constant negative curvature. The next two sp ecial cases weinvestigate are

2 1 1 1

the O N sigma mo del on the manifolds, R S and S S R, b oth of zero Riemann

curvature. They are however not conformally equivalenttoR .We will see that in these

cases m is non-zero at the critical p oint. In these cases, m is the inverse correlation length

and therefore the correlation length is nite at criticality. This is due to the nite size of

the manifold in some directions. More generally, whenever the manifold is not conformally

equivalenttoR ,we should exp ect the value of m at criticality to b e non-zero. Our

computations con rm this conjecture.

2 1

6. Finite size e ects on R S

This case has b een well studied in the pap ers [3], [4] in the Pauli-Villars regularization.

We will study it in the zeta function regularization [12] to make later comparisons easier.

Also, the calculation is technically simpler in the zeta function regularization.

2 1 2

On R S wecho ose the metric tensor to b e g =1; 1; . The radius of the circle,

, can b e thoughtofasinverse temp erature for a system on the 2{dimensional at space 10

R . The conformal laplacian on this space is,

2 2 2

+ + @ = @ @

y x

2 1

where x; y are co-ordinates in R and is the lo cal co-ordinate on S . The sp ectrum of

2 2

4 n

Sp ; =k +

n =0; 1; 2; . The gap equations are

m b =0 15

1 1 d k

lim [b = ]: 16

2 2

4 n

2 2 2 s

s!1

2 k + + m

At the critical p oint the ab ove equation b ecomes,

1 d k

= b lim

2 2

4 n

2 s 2 2

s!1

+ m 2 k +

and, taking the Mellin transform, we get

1 1

Z Z

s1 2 2

2 2

4 n

t dk 1

k + +m t

dt lim ke = b :

s!1

s 2

0 0

Weintegrate over k and use the Poisson sum formula to separate out the divergent part

in the small t region of the sum so that we are now able to interchange the sum and the

integral. The Poisson sum formula in this case is,

2 2

X 2 2 X

4 n

= e 17 e

n n

On using this formula, the gap equation reduces to,

1 1

Z Z

5 5

s s

2 2 X

2 2

1 t t 2

m t+ m t 2

+ dt e dt e = b lim

3 3

s!1

s s

2 2

4 4

n=1

0 0

We use the standard integral,

1 + t

K 2 18 dt t e =2

0 11

for Re>0; Re >0, where K is the MacDonald's function. Note, s =1=1.The

gap equation is then:

s 3

32s 2 2

2 4

4 m 3 n

lim K m n s =b

3 3

s!1

4m 2

2 2

4 4

n=1

2 m n

1 1

e , in the limit s ! 1 the gap Noting that, K m n=K m n=

2m n

2 2

equation simpli es to,

m n

1 e m

+ = b

2 n 4

n=1

m n

and using, = Log1 e we get a very simple equation:

n=1

m 1

Log 2 sinh = b : 19

2 2

We know from equation 15 that either m =0orb = 0 or b oth. The equation ab ove

is not satis ed for m = 0 since b is always p ositive. Hence we require b = 0 to satisfy

1+ 5

equation 15. This gives us m =2T Log ,asin [3], where is the Golden mean

and T = and can b e interpreted as the temp erature of the system.

2 1

Free energy densityonR S

The regularized free energy density of the O N sigma mo del in the large N limit, in

the presence of a background metric g , on a generic manifold M , is given by

m N

2 3

[TrLog ]: + m lim d x g W g; s=

s!1

2 s

Observing that in the zeta function regularization,

2 0

TrLog + m = 0

and at the critical p oint, lim =0,we nd that the free energy density at the critical

s!1

p ointisgiven by

W 1

= 0:

N 2

2 1

,wehave then: In the case under consideration, namely M = R S

W d k d 1

= lim : 20

2 2

4 n

2 2 2 s

s!0

N ds 2

2 k + + m

n 12

We Mellin transform the r.h.s and simplify the ab ove expression to,

n W d 4 1

32s

= lim [m + K m n]

s!0

N ds s s 2m

n=1

Using the fact that in the limit s ! 0, s ! ,

0 0

2 X

1 s 3 s n W

32s

K m n] = lim [m s 4

2 2

s!0

N s 2 s 2m

n=1

= 1, Using, lim

s!0

2 X

W 2 2m 1

= m [ K m n]

3 3

N 3

2 2

4 n

n=1

3 1 1

Again, recalling K x=2 K x K x, we obtain,

2 2 2

1 1

3 m n m n

X X

1 m 2m 2 e e W

= [ + + ] 21

2 2 3 3

N 4 3 n n

n=1 n=1

We know though that at the critical p oint, m = log = log2 . We also recall

the p ower series representation of the p olylogarithm, =Li x. Putting all these

n=1

together, the free energy density, in agreement with [3], is,

W 1 1 2

= [ log 2 + log 2 Li 2 Li 2 ] = Li 1 22

2 3 3

3 3

N 2 6 5

The last equality can b e arrived at by using p olylogarithm identities and is derived in

Ref. [3]. Using the expression for free energy density obtained from hyp erscaling for a

d{dimensional slab geometry [13],

W = c;~

it is seen that,c ~ = a rational numb er. Note, here d is the Riemann zeta function.

7. Study of the O N sigma mo del on manifolds of the typ e R

In the next three subsections we will b e studying the O N sigma mo del on a manifold

M = R.We therefore discuss the general form of the gap equations on such manifolds

b efore we consider sp eci c cases of . 13

In general, on an arbitrary curved space of the typ e R, the eld b mayhave

several non{zero comp onents; the large N arguments are still valid if the numb er of non{

zero comp onents is small compared to N . The Green's function we need can b e written in

terms of the geometry of the 2-manifold :

2 2 s

G x; x; m ;g=hxj + m jxi

s g

2 2 2

s1 [r +R@ +m ]t

= t hxje jxi

2 2

1 dt 3

[r +R+m ]t s

= hxje jxi t

1 1

= s ;x;m

s 2

Here,

2 2 2 s

s; x; m =hxj[r + R + m ] jxi:

In general,m and b can dep end on x. The gap equations are,

+ m xb x=0 24

g i

2 2

lim [ G x; x; m ;g= b x] 25

s!1

At the critical p oint, the second equation can b e simpli ed to

1 1

2 2

;x;m : 26 b x =

2 2

The zeta function on the r.h.s. is analytic at s = so that this equation is nite the

singularities of the zeta function of an even dimensional manifold o ccur at negativeinteger

values of s.

Wenow go on to study sp eci c examples with having zero, constant p ositive and

constant negative curvature. Toevaluate s ;x;m , wehave to nd the sp ectrum

2 2

of the conformal laplacian, r + R + m , on the space . In this case we can lo ok for

solutions of the gap equations with m and b constant; then b can b e chosen to have only

i i

one non{zero comp onent. 14

1 1

7.1. S S R

For simplicity let us study the case where the radii of the two circles are the same

and are denoted by . This is a space of zero curvature. The Ricci scalar R = 0 and the

conformal laplacian is just the ordinary laplacian and is given by

2 2 2

= @ + @ + @

where and are lo cal co{ordinates on the two circles and z the co{ordinate on R. The

sp ectrum of the conformal laplacian is,

2 2 2

= Sp p + q +k

where, p; q =0; 1; 2; and k takes values on the real line. The gap equations on

1 1

S S R are,

m b =0 27

i h

1 1 1

p 1 1

=b 28 s lim

S S

s!1

s s 2

At the critical p oint, lim =0. Using the integral representation of the zeta

s!1

function,

2 s

2 2 2

1 1 t

p +q tm t

1 1

e s = dt

S S

p;q

at the critical p oint, 28 can b e written as,

i 2 2 h

2 4 p

3 dt 1

s 2 m t

t = b lim e e

s!1

p=1

Using the Poisson sum formula 17, this reduces to,

1 1

h i

2 2 2

X X

2 2

p +q

dt 1 5

m t s 2

2 4t 4t

lim e 1+4 e +4 e t = b

s!1

p=1 p;q =1

In the limit s ! 1,

1 1

Z Z

1 1

2 2 2

X 2 2 X

p +q

2 2

3 3 4 4

m t m t

2 4t 2 4t

dt t e dt t e +

3 3

2 2

4 4

p=1 p;q =1

0 0

1 m

b =

4 15

Recalling the standard integral 18, we can integrate the two terms on the l.h.s of the

ab ove expression and obtain,

2 2

1 1

p +q m

2 mp

X X

1 b e e

= + +

2 2

4 mp m

m p + q

p=1 p;q =1

On p erforming the rst sum in the ab ove expression, we get,

2 2

p +q m

1 e 1 b

Log1 e + 29 =

2 2

4 m m

m p + q

p;q =1

This equation is dicult to solve as the double sum we are left with is not an obvious

one. We can see that m 6= 0 without actually solving the equation. In essence, we put

in an ansatz m ! 0 and we will show that this is inconsistent with the gap equation and

hence m 6= 0 in this case at the critical p oint. If m were small, we could approximate the

double sum by a double integral and the equation 29 can b e written as,

1 1

Z 2 2 Z

p +q m

1 1 b e

Log1 e + = dp dq

2 2

4 m m

m p + q

0 0

The integral in the ab ove expression can b e easily p erformed by using p olar co-ordinates:

2 1 1 1

Z 2 2 Z Z Z

p +q m

e e 1

d RdR dq dp =

2 2

4 mR

m p + q

0 0 0 0

Using this result we nally nd:

1 1 b

Log 1 e + : =

4 m 2m m

When m ! 0, we can approximate Log1 e toLogm and the ab ove equation

reduces to a transcendental equation for m,

2 2

+mLog m = b m 30

4 2

It is immediately apparent from the ab ove equation that m = 0 cannot b e one of its

solutions. This implies that b has to b e zero at the critical p oint in order to satisfy the gap 16

1 1

equation 27. Hence the solutions to the gap equations in the case of M = S S R

are m 6= 0 and b = 0 as we exp ected and m is the solution to

2 2

p +q m

1 1 e

=0 Log1 e +

2 2

4 m

m p + q

p;q =1

Once we solve for m, the free energy density can b e computed at the critical p oint with

this value of m.

7.2. S R - example of a space of constant p ositive curvature

Nowwe will study the case of a sphere of radius . Since the space S has nite

volume, we might tend to exp ect m to b e non-zero at criticality. But we will see that this

2 3

is not true; the manifold S R is conformal to R f0g and it turns out that in fact

m =0.Thus we see that what matters, for m to b e zero or otherwise, is the conformal

class of the metric and not the `size' of the system. m = 0 do es not however mean an

in nite correlation length on S R; the correlation length at criticality is in fact nite.

In order to nd the conformal laplacian on S R,we need to calculate the Ricci scalar on

2 1

S . This is a standard calculation and will in the end yield R = and therefore R = .

2 2

Since conformal curvature plays an imp ortant role in our study of the critical theory,itis

2 3

worthwhile to demonstrate this equivalence of S R to R f0g.

2 3

Let the metric on S R b e denoted by g and that on R f0g by g . The line element

R is, on S

2 2 2 2

ds = du + d

S R

where, u is the co-ordinate on R and d is the solid angle. The line elementon R f0g

in spherical p olar co-ordinates is,

2 2 2 2

ds = dr + r d

R f0g

u 3

On writing r as r = e , the line elementonR f0g b ecomes,

2 2u 2 2 2

= e du + d ds

R f0g

We immediately see that the metrics g and g are related by a conformal transformation,

2f 2 3

g = e g with f x=u. Thus the manifolds S R and R f0g are conformally

1 17

equivalent. We can now use this fact to x R on S R.For a conformal transformation,

g ! g = e g , the scalar curvature term R transforms as follows [10]:

f f

d+2 d2

2 2

R = e e

1 g

2 3

= r since the Ricci scalar is zero for R f0g. Also, in our case, d = 3 and Note,

f x is a constant and is equal to u. Hence,

5u u

2 2

R = e r e

1 g

where,

1 1

2 2 2

r = @ + @ @ +

r sin

2 2 2u 2 2

@ = e @ + @ +

sin

This gives R = . The conformal laplacian on S R is therefore given by,

1 2

= r +

S R

The sp ectrum of the conformal laplacian on S R is:

l +

+ k ] 31 =[ Sp

S R

where l =0; 1; 2; and k 2 R with degeneracy 2l + 1. Notice that the conformal

laplacian on S R has no zero mo des.

The gap equations on S R are,

+ m b =0 32

1 1 1

lim [ s =b ] 33

s!1

s s 2

From equation 32 we see that b = 0 since m + cannot b e equal to zero b ecause

b oth m and are p ositive. At the critical p oint, the gap equation 33 reduces to,

1 1

s s

1 2l 1 1

2 2

p p

lim [ =0] s =

s!1

s 2 s

4 4

+ m

2 18

On Mellin transforming the ab ove, we obtain,

m t

=0 34 lim e dt 2le

s!1

As b efore we see that the sum is divergent in the small t region and we therefore haveto

separate out the divergent piece in the sum b efore we can interchange the sum and the

integral over t.To do this we need an analog of the Poisson sum formula for the case of

the sum over half-integers l which turns out to b e the following:

2 2 2

x 1 x x

dx 35 + P cosec 1e 2le =

2 4t 2 2

x x

where, byP dx cosec 1e we mean the principal value of the integral;

2 2

x x

cosec 1 has simple p oles at all non-zero integral multiples of 2.We give a brief

2 2

derivation of this formula in app endix C. On using the Poisson sum formula 35, the gap

equation reduces to,

1 1 1

Z Z Z

2 2

1 x 5 x x

m t m t s s3

4t 2

lim [ dx e =0] P + dt t cosec 1 dt t e

s!1

2 2

0 0

Let us rescale to x so that in the limit, s ! 1, wehave,

1 1

Z Z

2 2

2 m t

dx xcosecx 1 P =0 36 dt t e + m

The integral over t can b e easily p erformed using 18 and will give a Macdonald's

function. But this waywe nd it hard to extract the solution for m from the expression

we get. We exp ect m = 0 to b e the solution, though, since we showed that S R is

conformally equivalentto R f0g. Let us therefore try putting the ansatz m = 0 as the

solution in the l.h.s of the ab ove gap equation and see if this is a consistent solution. On

putting m = 0 in equation 36, we get,

1 1

Z Z

2 2

L:H:S= P dxxcosec x 1 dt t e

1 1 1

= dx P cosec x

x x

1 19

We can easily check that,

1 1

P dx cosecx =0 ;

x x

thus giving us the l.h.s of the gap equation to b e zero, consistent with the r.h.s. Hence

m = 0 is indeed the correct solution to the gap equation in this case. We therefore nd

that at the critical p oint, m = 0 and b =0 on S R.

Although m = 0 at criticality, the correlation length of this system is not in nite. If

we consider the correlation function <x; u y; 0 > as a function of x; y 2 S and

i j

u 2 R, it will decay like an exp onential in u as u !1. In the x; y directions the space

has nite radius. This is b ecause the op erator r + R has no zero mo des. This is to

b e contrasted with the situation on H R where, we will see that m 6= 0 at criticality,

yet the correlation length is in nite.

Free energy densityonS R

The regularized free energy densityonS R is given by,

Z Z

2 2

N m 1 l

2 3 2

W = + m lim d x g dk [ 2l Log k + ] 37

2 2

s!1

2 2 s

In the zeta function regularization,

2 2 0

Logk + + m = 0

and therefore,

Z Z

W m d 1 2l

dk lim d x g = lim ]

2 l

2 s 2

s!1 s!0

N ds 4 s

+ m k +

Using the analytic continuation of the zeta function, this can b e written as,

1 1

Z Z Z

2 s1

p 2 2

m W t d 1

3 k t m t

d x g 2l e lim dk e dt = lim e ]

s!1 s!0

N ds 4 s s

1 0

2 20

On integrating over k and using the Poisson sum formula 35, we get,

1 1 1

Z Z Z

s s3

2 2

W t t x x dt dt d x

m t m t

= lim e + e P dx cosec 1e

s!0

N ds 8 s 16 s 2 2

0 0

lim d x g

s!1

which simpli es to,

1 1

Z Z

32s s3

W x m 1 x d t x

m t

= lim + e P cosec 1e dt dx

s!0

N ds 8 s 16 s 2 2

0 1

lim d x g

s!1

At the critical p oint, lim = 0 and m = 0 and the free energy densityisgiven by,

s!1

1 1

Z Z

d 1 x x x W t

= lim P dx cosec 1e 38 dt

s!0

N ds 16 s 2 2

, in the limit s ! 0, we get, Changing variables, t !

1 1 1

Z Z Z

W x x 1 x x x tlog t 4

= P cosec 1 e cosec 1 dx dt P dx

N 16 2 2 s x 2 2

1 0 1

. We can again p erform the The rst term vanishes since in the limit s ! 0,s !

integral over x in the second term and verify that it is zero. We see therefore that the

regularized free energy density,

=0 39

2 2

on S R. This just means that the free energy densityon S R is the same as that

on R at the critical p oint which is what we should exp ect from general considerations of

2 3

conformal equivalence of the spaces S R and R f0g.

7.3. H R - example of a space of constant negative curvature

Let us consider as an example of constant negative curvature M = H R, where

H is a two{dimensional hyp erb oloid. Let the co-ordinate on R b e denoted by u and the

hyp erb oloid b e parameterized as

H = fz =x; y :x 2 R; 0

with line element,

2 2 2

ds = dx + dy

and laplacian

2 2 2

= r @ + @

x y

2 1

The Ricci scalar on this space is R = . Therefore R = . The gap equations are,

2 2

+ m b =0 40

1 1 1

p 2

lim [ ] = b 41 s

s!1

s s 2

1 1

. At the critical p oint, lim = 0 and the gap Wehavetonow calculate s

2 s

s!1

equation 41 reduces to,

1 1

p 2

lim =b 42 s

s!1

s 2

, we need to nd the sp ectrum of the conformal laplacian, To determine s

r + R on the hyp erb olic space. This sp ectrum is continuous and therefore wehave

to nd the density of states . The ob ject that is of interest to us is,

1 1 1 1 1

2 2 s 2 s

2 2

= d [ s =[Trr + m ] + m ]

2 2

2 4 4

for simplicity,we assume that = 1. If we de ne by = 1 , the resolvent,

1 2

1 ] 43 R=[r

is a single valued function of . This means that the resolvent has a branch cut on the

complex plane with = as the branch p oint. The explicit form of the resolventis

given in [14]:

0 0

dx dy

0 0

uz; z f z Rf z =

y>0

0 2

jz z j

where, z =x; y , uz; z = and,

4yy

u= dt [t1 t] t + u 44

0 22

If we consider the matrix elements,

0 0

hz jRjz i = uz; z

we can get the density of states which is given by the discontinuity of the resolvent

of the Laplace op erator across the branch cut,

0 0

45 = [hz jR + ijz ihz jR ijz i]

z =z

to b e,

1 1 1

= + tanh + 46

8 4 4

We are now ready to compute s .

1 1

Z Z

1 t 1 1

[ +m ]t

dt = d tanh + e s

2 4

Wechange variables, k = + .

1 1 1

Z Z Z

h i

2 2 2

5 1 1 3

s s k t m t m t

2 2

dt t s = s dt t kdke [tanh k 1] lim e +2 e

s!1

2 2

0 0 0

In the limit s ! 1, this gives,

k 1

dk =2m +2 [tanh k 1] s lim

2 2

s!1

k + m

We can now put this backinto the gap equation 42 and obtain,

dk [m + 1 tanh k] = b 47

2 2

k + m

We see that each term in the l.h.s is manifestly p ositive . Therefore, b = 0 cannot b e

a solution when m 6=0.If m = 0, again it is easy to check that the l.h.s is non-zero it is

ln 2

equal to and hence b = 0 cannot satisfy the ab ove equation. Therefore, in order to

satisfy the gap equation 40, we require,

m =

2 23

since we set =1, m = at the critical p oint. We see that negative curvature has

induced the symmetry to b e sp ontaneously broken at the critical p oint and there is a

non-zero sp ontaneous magnetization given by the order parameter b.

2 2

+ R + m has a zero{mo de kernel for that The value m = is sp ecial in that r

value. This means that the correlation functions fall o with distance likeapower law,

not an exp onential this can b e seen by using the integral representation for the resolvent.

Thus even though m is non{zero, the correlation length is in fact in nite. This is consistent

with Goldstone's theorem, since we nd that the O N symmetry is sp ontaneously broken

at the critical p oint.

We can nowevaluate the order parameter b at criticality from 47. We expand

2 2

k +m

binomially and simplify 47 to the following:

2r +1

] b =[m +2 dk

r e +1

r =0

The integral over k can b e p erformed and and after restoring the appropriate factors of ,

we can express b at criticality as,

2r 1 2

[1 + 2r + 1!1 2 2r + 2] 48 b =

2r +2

2 r

r =0

this sum can also b e written as a similar sum over Bernoulli numb ers.

Free energy densityonH R

The free energy densityonH R is given by,

Z Z Z

N m 1 1

3 2 2

W = x d g + + m lim + dp d Log p [ ] 49

2 2

s!1

2 2 4 s

At the critical p oint, the free energy densityisgiven bywe again set = 1,

r 1 1

Z Z Z

2 2

d 1 t 1 1 W

p + +m t

d tanh = lim dp dt + e

s!0

N ds 4 s 4

. where m =

2 24

We rst p erform the integral over p and then change variables, k = + to obtain,

1 1 1

Z Z Z

5 3

s s

2 2

2 2 2

d W 1 t t

k t m t m t

kdke tanh k 1 = lim dt e +2 dt e

s!0

N ds 4 s s

0 0 0

Integrating over t gives us the following expression for the free energy density:

3 1

s s

W 1 d 1

2 2 s 32s

2 2

dk m + k tanh k 1 = lim m +2

s!0

N ds 4 s s

s 2 2

and p erform the integral over We once again use the binomial expansion of m + k

k and obtain,

W 1 d

32s

= lim m

s!0

N ds s

2r 2s+1

s m

2r 1

2r + 1!1 2 2r +2 4

2r +2

r s 2

r =0

After restoring the appropriate factors of , this simpli es to the following expression:

W 1 1 1

2r 1

= 2r + 1!1 2 2r +2 51

3 3 2r +2

N 24 4 r

r =0

8. O N sigma mo del at nite temp erature

We next formulate the nite temp erature versions of the O N sigma mo del on some

three dimensional manifolds M with constant curvature. The sp eci c examples we will

1 1 1 2 1

consider are S S S a torus, which has zero curvature and S S constant p ositive

curvature.

1 1 1

8.1. S S S

The torus is again at and therefore the conformal laplacian is the same as the ordinary

laplacian. We consider the simple case where the radii of the circles are the same. The

sp ectrum of the conformal laplacian is,

2 2 2

p + q + r Sp =

where, p; q ; r =0; 1; 2; . The gap equations in this case are,

m b =0 52

i h

1 1 1

= b 53 lim

2 2 2 2 s

s!1

p + q + r +m ] [

p;q ;r

Again, at the critical p oint, lim =0. As b efore, we Mellin transform the zeta

s!1

function to obtain,

2 s1

2 2 2 2

1 t

p +q +r tm t

lim = b 54 e dt

s!1

p;q ;r

1 1

We can argue as in the case of S S R that b = 0 and m can b e determined from

the equation 54 which, after using the Poisson sum formula, 17, can b e rewritten in

terms of functions as follows:

1 t

m t 3

lim dt e 0;e =0

s!1

where 0;q has the p ower series representation, 0;q= q ; jq j < 1. This expres-

3 3

sion, as it is written, is divergent in the small t region. We can separate the divergent

piece which can then b e given a meaning by analytic continuation and rewrite the ab ove

as a nite expression in the limit s ! 1 as follows:

3 1

m t 3

2 4t

dt t e [ 0;e 1] + m =0 55

where =2 . The critical value of m is a solution to the ab ove equation. We

can compute the free energy using this critical value of m.

2 1

8.2. M = S S

2 1

In the case where the manifold M = S S ,

1 1

2 2

@ = r @ + R =

2 2

2 1 2 1

R on S S is the same as that on S R since S is at. The sp ectrum of the conformal

laplacian is,

2 2

l +

4 n

2 1

Sp =[ + ]

S S

2 2

where l =0; 1; 2; and n =0; 1; 2; and the degeneracy is 2l + 1. As in the case

2 2 1

of S R, the conformal laplacian on S S has no zero mo des. The gap equations in

this case are,

+ m b =0 56

2 2

lim [ G x; x; m =b ] 57

s!1

cannot b e equal to zero. Since The equation 56 tells us that b = 0 since m +

lim = 0, the gap equation at the critical p ointis

s!1

X X

1 2l

lim [G x; x; m = =0]

2 2 2

2 2

l 4 n

2 s

s!1

+ + m

2 2

Mellin transform the ab ove, to obtain,

1 1

2 s1 2 2

X X

l 4 n

t t

m t

2 2

=0 58 2le e lim e dt

s!1

n=1

On using the Poisson sum formulae 17 and 35, to factor out the divergent pieces in the

sums in the small t region, the gap equation reduces to,

1 1

Z Z

X 2 2 2

x +n

x x 1

s3 m t

dx cosec 1 dt t e lim [

s!1

2 2

n=1

X 2 2

m t s

2 4t

=0] + e dt t

n=1

The second integral over t in the ab ove expression is divergent in the small t region for

n =0.We can again separate the term with the zero mo de and that term can b e given a

well de ned meaning by analytic continuation. After this is done, in the limit s ! 1, the

equation 59 can b e written as,

1 1

Z Z

2 2 2

x +n

x 1 x

2 m t

dx P [ dt t e cosec 1

2 2

n=1

2 2 X

m t

2 4t

dt t e +2 + m =0]

n=1

0 27

Using the standard integral 18, we can simplify this further to,

1 p

2 2 2

K mx + n

x 4 x m

P dx cosec 1 Log 2 sinh =0 61

2 2 2

2 2 m 2

x + n

n=1

where K x is a MacDonald function. The critical value of m is a solution to the ab ove

nite expression and the free energy can b e computed with this value of m.

App endix A{Outline of pro of for the conformal invariance of TrLog + x

in o dd dimensions

f d 2f 1

, the action is invariant. If we gd x If g ! e g and ! e

g ij ij

2f d

assign the transformation ! e , the action [ + ] gd x is still conformally

invariant. We can de ne a determinant for this op erator by

+ ] 6=0 0; if dimker[

det [ + ]=

v 0

+ ]=0 1 e ; if dimker[

where v is the numb er of negative eigenvalues assumed to b e nite in numb er of + .

j j , b eing the eigenvalues of Also, s= + .

n n g

6=0

Theorem. Given a d dimensional compact manifold M , det + is conformally

2f x 2f

invariant under the transformation of the metric g ! e g and ! e ,ifM is

o dd dimensional. If M is even dimensional,

+ ]=4 + ] 2f xtr a d x: det[ det[

g g f

a d x is one of the co ecients in the asymptotic expansion of the heat kernel ht; x; y

which is a solution of the di usion equation,

@ + ht; x; y =0; t>0;

t g

ht; x; y = x; y :

The asymptotic expansion of ht; x; y is given by,

d x;y

ht; x; y = a xt

k =0 28

where dx; y is the distance b etween the p oints x and y on M.

A pro of of this theorem in the case = 0, in the zeta function formulation is given in

[10]. This pro of can b e extended without much diculty to our case where the op erator

is + x. The explicit expression for a x will change, but this is not of interest in

g k

the o dd dimensional case.

Though the theorem is proved for the case of a compact manifold M ,we consider non-

compact spaces as the limiting case of the compact manifold.

The large N limit of the O N sigma mo del at the xed p oint can b e viewed as a classical

eld theory with action

1 1

log det[ [b ;]= b [ gd x + + ]b + ] 62

i i g i g

2 2

The gap equations are the Euler{Lagrange equations of this non{lo cal action. This action

is conformally invariant under the ab ove transformations. In this sense the large N limit

of the O N sigma mo del is a conformal eld theory. The eld b is a primary eld in the

sense that its correlation functions are conformally covariant.

App endix B: Conformal Geometry in Three Dimensions

Let M; g b e an oriented Riemannian manifold of dimension d.We are mostly inter-

ested in metrics of p ositive Euclidean signature. Two Riemann metrics g; g~ are said to

b e conformally equivalent if there is a smo oth function f : M ! R such thatg ~ = e g .

Under such a transformation of the metric, g ! e g , the angles b etween twovectors will

b e unchanged, but the lengths of vectors will change. A metric is conformally at if it

is conformally equivalent to a at metric.The set of conformal transformations form an

ab elian group, the group of smo oth functions C M under addition.

We can de ne a `conformal structure' on an oriented manifold M to b e a non{

degenerate symmetric tensor density^g of weight 2 and with detg ^ =1. We nd it

convenient to use the convention that the volume is a densityofweight d. The determi-

nantof^g is a scalar, so the condition that it b e equal to one is di eomorphism invariant.

Given a Riemannian metric, a conformal structure is determined by^g = g g , where g

ij ij

is the determinantof g . Clearlyg ^ is invariantifwechange g ! e g .^g determines

ij ij ij ij ij 29

the equivalence class of the metric tensor under conformal transformations. It is reason-

able to callg ^ the metric tensor density, since it determines the angle u; v between two

vectors:

i j

g^ u v

cos u; v = : 63

kl k l mn m n

[^g u u g^ v v ]

k l

This is a scalar although the `length' of a vector [^g u u ] is a scalar densityofweight

If is the space of conformal structures on M , the group Di M acts on it by

pull{back:

^ ^

2 Di M; : ! ; g^ 7! g~ = g:^ 64

Explicitly in terms of co{ordinates,

k l

@ @

g~ x = [det @] g^ x : 65

ij kl

i j

@x @x

Two metric densities that di er only by such a di eomorphism are to b e regarded as

equivalent, as they only di er by`change of co{ordinates'. We will thus b e interested

in ob jects of conformal geometry that are covariant under this action of the group of

di eomorphisms Di M of M . These ob jects will b e de ned on the space = Di M of

equivalence classes of conformal structures under the di eomorphism group. This space

however, is not in general a manifold, since the action of the group can have xed p oints.

It is useful to study rst the change ofg ^ under the action of an in nitesimal di eo-

morphism. This is the Lie derivative with resp ect to a vector eld,

k k k k

@ v g^ : 66 [L g^] = v @ g^ + @ v g^ + @ v g^

k ij v ij k ij i kj j ik

The last term arises from the fact thatg ^ is a tensor density,ofweight 2.

By the way, this might also b e thoughtofasacovariant derivative that maps the

covariantvector density^v =^g v to a symmetric traceless tensor density. After some

i ij

calculation,

kl k

[L g^] =D v^ = @ v^ + @ v^ @ v^ g^ g^ 2 v^ 67

v ij ij i j j i k l ij k

Here,

2 1

kl mn k

g^ [@ g^ + @ g^ g^ g^ @ g^ ]: 68 =

i lj j il ij m nl

2 d

Our construction shows that D is a covariant derivative that maps covariantvector densities

are of weight 2 to symmetric traceless covariant tensor densities of weight 2.

ij 30

the conformal geometric analogues of the Christo el symb ols of Riemannian geometry.

It would b e interesting to understand conformal curvature as a `commutator' of such

conformally covariant derivatives.

If [L g^] =0, v is a conformal Killing vector; then this in nitesimal di eomorphism

v ij

hasg ^ as a xed p oint. The compact manifold with the maximum numb er of such con-

formal Killing vectors is S ; they form the Lie algebra O n +1; 1. This space can also b e

thoughtofasR ; the conformal killing vectors corresp ond then to translations, rotations,

dilatations and some `sp ecial conformal transformations':

j 2

v = a + x + x + x b 2x b:x: 69

i i ij i i i

The quotient space = Di M will not then b e a Hausdor top ological space and hence

not a manifold. This diculty can b e avoided by restricting to the op en dense subset of

which do es not haveany conformal Killing vectors. Alternatively,we can restrict to the

subgroup Di M which agrees with the identity map up two derivatives, at one p oint

p.Even in the case with largest numb er of conformal Killing vectors, this condition will

remove all of them. This will remove all conformal Killing vectors when M is compact.

There could still b e xed p oints due to nite conformal isometries; but they only lead to

`orbifold' typ e singularities in the quotient which can b e removed by passing to a covering

space. All ob jects of interest in conformal geometry will b e sections of vector bundles over

the space Q = = Di M . A similar p oint of view was found to b e very useful in studying

Yang{Mills theories [15], [16].

There is another p oint of view on conformal structures in low dimensions that is

i i

1 m

useful. Recall that on any oriented manifold the Levi{Civita symb ols and

i i

1 m

are natural anti{symmetric tensor densities of weight d and d resp ectively. If d =2,

b cb

this allows us to describ e a conformal structure also by a tensor J =^g .We can see

b c ec c

now that J J = detg ^ = . This shows that a conformal structure is the same

a a

as a complex structure in dimension two the integrability condition on J is trivial in

lj k

two dimensions. The analogue when m = 3 is a tensor density c =^g of weight1

satisfying

jl jk jk

]=1: 70 c + cyclic [jkl]= 0; det [c c c

i i i

This will de ne a `cross pro duct' among covariantvector densities of weight 1. The

ij ki

conformal metric can b e recovered from the contractiong ^ = c c . Thus in three

dimensions a conformal structure is the same as a Lie algebra structure, isomorphic to 31

SU 2, or SU 1; 1 onvector densities of weight 1. This p oint of view needs to b e

investigated further.

Returning to the study of Q, let us rst get an idea of how big Q is. Wehavea

^ ^

principal bundle Di M ! !Q. Let us ask what conditions a one{form in must

satisfy in order that it b e the pullback of a one{form in Q by the natural pro jection

: !Q. This will give an idea of the size of the cotangent space of Q. A tangentvector

to is a traceless symmetric covariant tensor densityofweight 2inM . A cotangent

vector one-form in is a traceless symmetric contravariant tensor densityofweight d +2.

The contraction of a vector h in with a one{form t is then,

ij d

i t = t h d x: 71

h ij

Given a vector eld in M , there is a vertical vector eld in , given by the in nitesimal

action:

h =[D v^] 72

ij ij

where D is the covariant derivativeof^v =^g v de ned earlier. If a one{form t in is

i ij

the pull{back of a one{form in Q it must annihilate all vertical vector elds:

ij d

t [Dv ] d x =0 73

This implies that the covariant divergence of t should b e zero:

[D t] =0 74

Strictly sp eaking, [D t] must b e zero when contracted with vector elds that vanish

up two derivatives at p. This means that [D t] is a combination of derivatives of the delta

function concentrated at p. If the manifold do es not admit conformal Killing vectors, this

subtlety can b e ignored.

This covariant divergence [D t] can b e de ned in terms ofg ^ alone:

i ij ik

[D t] = @ t t 75

b eing the conformal analogues of the Christo el symb ols de ned earlier. The covariant

divergence of such a tensor density has a meaning within conformal geometry, without any

reference to Riemann metric. This divergence is in fact a vector densityofweight d +2. 32

So far wehave considered t as a one{form at a p oint^g of . If it is the pull{backof

ij ij

a one{form in Q, t must change along the vertical direction in a way that is determined

by the action of Di M :

ij d

[L t] + y [D v^ ] y d y =0: 76

v kl

Conversely,any one{form in satisfying the ab ove conditions is the pullbackofa

one{form in Q.

The size of Q is given by the numb er of indep endent solutions to the equation, [D t] =

d+2d1

0, among traceless symmetric tensor densities. A traceless tensor density has

indep endent comp onents at each p ointof M . The condition of having zero divergence is

given by d equations at each p ointof M . Hence the numb er of indep endent comp onents

d2d+1

in a one{form of Q is .Away to formalize this statement is that Q is a manifold

d2d+1

mo delled over a vector space [C M ] .

The conditions of b eing traceless and divergence free are familar prop erties of stress

tensor of a conformal eld theory. Indeed the stress tensor of a conformal eld theory is

a one-form on Q, when there is no `conformal anomaly'for example in three dimensions.

Moreover, in general, the stress tensor will b e a closed 1{form; it is not always exact. If the

parity anomaly vanishes it is exact. Similar considerations arise in the canonical formalism

of general relativity [17].

In particular, if d = 2,there are no lo cal degrees of freedom in sucha t : the equations

[D t] = 0 form an elliptic system. This do es not mean that Q is trivial, just that it is

nite dimensional. The dimension of Q the numb er of indep endent solutions of [D t] =0

is6h 6 for a compact manifold of genus h 2. The fact that Q has no lo cal degrees

of freedom do es imply that all 2{manifolds are lo cally conformally equivalent and hence

conformally at.

In the case of most interest to us, d = 3, there are two indep endent degrees of freedom

p er p ointof M . Thus there must b e a conformal curvature tensor, which measures the

lo cal deviation from conformal atness. We will discuss this tensor so on.

Conformal Geometry from Riemannian Geometry

We can view the space as the quotient of the space of Riemannian metrics by the 33

1 1

group C M . The principal bundle C M ! ! is de ned by the group action

g ! e g : 77

ij ij

Thus it is also p ossible to study conformal geometry by lo oking at structures in Riemannian

geometry invariant under the semi{direct pro duct Di M C M .

The curvature or Riemann tensor transforms as follows under a conformal transfor-

mation [18]:

R ! R = e [R + g f + g f g f g f

ij k l ij k l ij k l il jk jk il ik jl jl ik

+g g g g jdf j :

il jk ik jl

Here,

2 mn

f = r @ f @ f@ f; jdf j = g f f : 78

ij i j i j ;m ;n

For explicit computations there is still nothing b etter than the classical co{ordinate no-

tation of tensor calculus. From this, it follows that the Ricci tensor and Ricci scalar

transform as follows:

R ! R = R +d 2f + g [f +d 2jdf j ]: 79

ij ij ij ij ij

2f 2

R ! R = e [R +2d 1f +d 1d 2jdf j ] 80

Here, f = g f is the Laplacian. The traceless part of the Riemann tensor,

;ij

1 1

h h h h h h h h

C = R + [ [ R g R + g R g ]R 81 g R ]+

ik ij ij ik ik ij

ij k ij k j k k j j k

d 2 d 1d 2

is called the Weyl tensor. It is identically zero unless d 4. The Weyl tensor is invariant

under conformal transformations:

h h

C ! C : 82

ij k ij k

If d 4, a manifold is conformally at if and only if the Weyl tensor vanishes.

If d =1every manifold is at, so in particular conformally at. If d =2,every

manifold is lo cally conformally equivalent to at space. This was already seen in the

previous discussion on Q.

When d = 3, the situation is more subtle: the Weyl tensor vanishes, but not every 3{

manifold is even lo cally conformally at. This is already clear from the previous discussion 34

of the numb er of degrees of freedom of Q. There is a tensor density, sp ecial to three

dimensions that measures conformal curvature [18]:

1 1

j j

ij ik l

= g r [R R] 83

l l

ij k

Here is the Levi{Civita tensor which dep ends on the choice of orientation on M

and g = det g . This is a tensor densityofweight 5 that is invariant under conformal

transformations.

ij 1

Thus is a geometric ob ject on the space ==C M : it is in fact a one{form.

Moreover, a 3{manifold is lo cally conformally at i = 0. In fact it is now p ossible to

verify that it satis es the conditions

ij ji ij i

= C ; g^ =0; [D ] =0: 84

Moreover, it satis es the condition that it is a geometrical tensor, dep ending only on g ,

3 ij

L g y d y =0: 85 L +

v kl v

g y

These conditions have a simple interpretation: de nes a 1{form on Q. In fact it is

a closed form. This can b e veri ed by explicit computation of its exterior derivative. A

b etter strategy is to use the fact [19] that the conformal curvature is the derivativeof

the Chern{Simons term.

Thus there is something sp ecial ab out conformal geometry in three dimensions: the

conformal curvature is given by a tensor density p eculiar to three dimensions. Also, the

space of conformal structures carries a 1{form that measures the deviation of each p oint

from lo cal conformal atness. There are two indep endent comp onents in p er p ointof

M .

b eing zero only implies lo cal conformal atness. Typically, for compact M , there

is a nite dimensional manifold of inequivalent conformal structures with =0on

some manifold M . This nite dimensional space is analogous to Teichmuller space. The

equations = 0 are the conformal analogues of the eld equations of Chern{ Simons

theory; the solutions are parameterized by conjugacy classes of homomorphisms of the

fundamental group of M to the conformal isometry group of R , whichisO 4; 1 recall

that the Teichmuller space of a Riemann surface surface is the set of conjugacy classes of

homomorphisms ! O 2; 1 However, the study of the theory on such conformally

1 35

at manifolds do es not seem to give suciently detailed information on phase transition.

The ma jor di erence b etween two and three dimensional conformal eld theory is that

in three dimension, the e ect of conformal curvature needs to b e taken into account.

This is whywe b elieve that generalizations of conformal eld theory to three or higher

dimensions based on invariance under O n +1; 1 describ e only part of the story.

The Yamab e problem and Classical Conformal Field Theory

No discussion of conformal geometry would b e complete without a mention of the

Yamab e problem [20]: Given a Riemannian manifold M; g, nd a function f : M ! R

such that the metric e g has constant Ricci scalar.

It has b een shown that this problem has a solution for all compact manifolds M . One

should exp ect that this has a close connection to conformal eld theory. Indeed there is

such a connection, but it is within the context of classical conformal eld theory.

Let M; g b e a Riemannian manifold, and g ! g a conformal transforma-

ij ij

tion. Here, is a p ositive smo oth function and p = . This parameterization is more

convenient for our present purp ose. Then, the scalar curvature changes as follows :

1 d 2

R ! [ + R]; = : 86

4d 1

Here = @ gg @ is the Laplace op erator. The op erator = + R is a

i j

conformal covariant analogue of the Laplace op erator. Supp ose we do the transformations,

g ! g~ = e g and ! e . This will leave^g and hence R invariant under the

ij ij ij ij

combined conformal transformation.Thus

21p

1p 1p

~ p2 p2

= e e : 87

Or,

d2 d+2

f f

2 2

e : 88 = e

The condition that the scalar curvature of g b e the constant b ecomes the

nonlinear eigenvalue problem Yamab e equation

= : 89

It is convenient to normalize by the requirement

p d

gd x =1 90 36

which removes the freedom under constant conformal transformations scale transforma-

tions. This requires the volume of M w.r.t. the rescaled metric g to b e one. The

solution to the Yamab e problem pro duces thus a conformal invariant `Yamab e invariant'

for each metric g on M .From our previous discussion, we see that this is a function ^g

on Q.

This is the Euler{Lagrange equation of the variational principle

p d

S []= 0; S = [ + jj ] gd x 91

2 p

In fact can b e viewed as the Lagrange multiplier for the normalization condition on .

This is the Lagrangian of the scale invariant classical scalar eld theory.If is viewed as an

order parameter in the sense of Landau and Ginzburg, S has the meaning of Free energy.

The p ower p = is the only one for which the theory is scale invariant. In particular,

p = 6 for three dimensions and p = 4 in four dimensions. It is imp ortant to note that

this is the same p ower for which the theory b ecomes p erturbatively renormalizable as a

quantum eld theory. What wehave seen that at least at the classical level this is also

conformally invariant. This conformal invariance is broken at the quantum level by e ects

of renormalization.

We also note another geometric interpretation of the ab ovevariational principle. Con-

R R

p p

d d

gd x as a function on the space of all gRd x + sider the Einstein{Hilb ert action

metric on M . Here is the cosmological constant. We can de ne a conformal invariant

by minimizing this over all metrics conformal to g . This is , the Yamab e invariantup

a sign. Thus itself can b e viewed as an action for conformal gravity induced by `inte-

grating out' the conformal uctuations from the Einstein{Hilb ert action. Such a study of

conformal uctuations is o ccasionally of interest in quantum gravity. The metric density

of least action w. r. t. is the standard metric densityonS , which is conformally at.

The Yamab e problem as de ned ab ove is, of course, trivial in one dimension. However,

an analogue of the Yamab e problem still exists in a generalization of Riemannian geometry

[21]. In this generalization, the metric tensor do es not transform as a tensor; instead it

has an inhomogeneous transformation law.

App endix C: Poisson sum formula on S

The Poisson sum formula 17 is quite standard. We give here a brief derivation of 37

the Poisson sum formula 35:

2 2

x 1 x

dx 2le + cosec 1e =

2 4t 2 2

where n =0; 1; 2; .We start with the general Poisson sum formula,

X X

4 t

x+n

1 1

2i n+

2 in

e 92 e e e =

n n

If wecho ose = and =2 ,we obtain from the ab ove general formula,

X X

x+2n

1 1

ixn+ tn+ in

2 2 4t

e e e = e

n n

Di erentiating b oth sides of the ab ove equation w.r.t x,we obtain,

X X

x+2n

i 1 1 1 1

tn+ ixn+ in

2 2 4t

n + e e x +2ne = e

2 2

2t 4t

n n

This equation can b e written as,

X X

x+2n

i 1

in tl ilx

e 93 x +2ne le e =

2t 4t

n=1

l= integers

Recall that,

X X

ilx ilx ilx

e e =icosec sgnl e =

1 1

l= integers l=

2 2

Therefore,

x i

ilx

dx cosec e 94 sgnl =

2 2

Use equation 94 to write 93 as,

X X

x+2n

1 1 x

l t n

e l sgnl e = 1 x +2ndxcosec

2 2

n=1

l= integers

On letting x +2n ! x and on further simpli cation, we get,

2n+2

1 1

X X

x 1 1 x x

l t

dx le = 95 cosec e

2 2 2

n=1

2 38

Rescaling t, t ! , and extracting completely the small t divergence of the integral,

2 2 2

1 x x x

le + = cosec 1e dx

2 2 2 4t

which is the required Poisson sum formula 35.

Acknowledgements We thank R. J. Henderson, F. Lizzi, S. Sachdev, G. Sparano and

O. T. Turgut for discussions. P.V. would like to thank the Dept. of Physics at the Univ.

of Ro chester for hospitality and S.G. thanks the organizers of the conference MRST-'94

Montreal for the opp ortunity to present part of this work. This work was supp orted in

part by the US Dept. of Energy, Grant No. DE-FG02-91ER40685.

REFERENCES

[1] I.Ya.Aref 'eva, Teor. Mat. Fiz. 311977 3; I.Ya. Aref 'eva and S.I.Azakov, Nucl. Phys.

B1621980 298

[2] A.M.Polyakov, Gauge elds and strings, Harwo o d Academic Publishers, 1987

[3] S.Sachdev, Phys. Lett. B3091993 285

[4] B.Rosenstein, B.J.Warr, S.H.Park, Nucl. Phys. B1161990 435

[5] J.Zinn-Justin, Quantum eld theory and critical phenomena, Oxford Univ. Press,

1989

[6] S.Jain, Int. J. Mo d. Phys. A3 1988 1759

[7] G.Ferretti and S.G.Ra jeev, Phys. Rev. Lett. 69 1992 2033

[8] N.D.Birrell and P.C.W.Davies, Quantum elds in curved space, Cambridge Univ.

Press, Cambridge, 1982

[9] K.Fujikawa, Phys. Rev. Lett. 441980 1733

[10] T.Parker and S.Rosenb erg, J.Di erential Geometry 251987 199

[11] E.B.Davies, Heat kernels and sp ectral theory, Cambridge Univ. Press, Cambridge,

1989 39

[12] A.H.Castro Neto and E.Fradkin, Nucl. Phys. B4001993 525

[13] J.L.Cardy, Nucl. Phys. B2901987 355; Nucl. Phys. B3661991 403

[14] S.Lang, SL R, Springer-Verlag New York co., 1985

[15] S.G.Ra jeev, Phys. Lett B212 1988 203

[16] K.S.Gupta, R.J.Henderson, S.G.Ra jeev and O.T.Turgut, Preprint UR-1327 1993,

to app ear in J. Di . Geometry

[17] C.W.Misner, K.S.Thorne, J.A.Wheeler, Gravitation, W.H.Freeman and Company,

1973

[18] L.P.Eisenhart, Riemannian Geometry, Princeton Univ. Press, 1949

[19] J.H.Horne and E.Witten, Phys. Rev. Lett. 62 1989 501

[20] T.Jerison, J. Math. Phys. 251987167

[21] R.J.Henderson and S.G.Ra jeev, Preprint-UR-1342 1994, to app ear in Class. Quant.

Grav. 40