THE ISING MODEL

T h e Ising model, formulated as a two-matrix model, has a double-scaling limit which, in the formalism of ref. [9] is described in terms of a third-order Lax operator, the continuum limit of the multiplication by A on orthogonal polynomi- als, Q ( + ) . We may construct macroscopic loops in terms of this o p e r a t o r as follows*:

( T r Q( + ),/a) ~ ( w ( + l ) ) =

f ; d x ( x l e x p [ +I(K3D 3 + c2KD +

L',)]lx)

h=O= c o n s t × K

' f : d x ( - c 2 ) ' / e e ±'''K,/3(III(. - c 2 ) 3 / 2 ) ,

(6.2)

* Some related formulae have been obtained in ref. [39].

G. Moore et al. / Two-dimensional quantum gracity 697 where

u 2 ( x , t / , ~ ) = - ( 2 x ) 1 / 3 / 4 + ...

is the specific heat and

vl(x,t~, ~)

is the magnetization. To each Ising operator ~r#(~,,) associate a fraction

v = n + a / 3

and an integer s = l for n + a even and s = 2 for n + a odd. Using ref. [32] we differentiate (6.2) to obtain

(o-,(~'~)w(/)) = K -

(sgnl)Slll d y y ~ K ~ / 3 ( y ) .

(6.3)

such that

a . ( G o ) + c . . . . + - . .

( ~,(G,~) w (l)) = (sgn

l)SK - ' I x " / 2 K v ( f - ~ ) .

(6.5) As in the one-matrix model we can express

w(l)

in terms of ~ and

I,,+,~/3.

We can find the coefficients by first working out the two-loop amplitude. The genus-zero two-loop amplitudes are (l i > 0)

(w(ll)w(12) ) = ( / i / 2 ) - 1 / 3

f~dn

n Ai(12/31zl/3 -

1~

I / 3 r / ) A i ( 12/31zl/3 +

12

'/3n), JO

( w(-ll)w(12)) = - (lit2)-1/3

fo dn n Ai(12/3l.LI/3 +

l l'/3n)mi(l~/31zl/3 + 12

I/3n).

(6.6) These integrals are rather difficult, but using eq. (B.11) the integral may be evaluated exactly for vanishing cosmological constant:

( w(ll)w(12))u=o = sgn(ll)sgn(12)(lll2 ) ,/3 II/3 + ll/3

(6.7) Ii +12

Following the reasoning of sect. 4 we can use this to determine the coefficients in the expansion

w ( l ) = • [ ( - s g n l ) " ( n + ~)~n(d~'l)lj,-n/2-1/6ln+l/3(¢Si~ ) n>~O

__ ( sgn l)"+ l(n + 3) .(~'2)/x __ 2 ~ - n / 2 - 1 ~ 3 i n

+2/3( ~1 "~ ) (6.8)

,

from which one may obtain the

IK

expansion of the propagator, as predicted from (6.4) Using eq. (B.12) and the properties (B.6) of Bessel functions, it easily follows that there is an analytic lower-triangular transformation

698 (;. Moore ct al. ,/ Two-dimensiomd quantttm grarity

sect. 5. We recall here the discussion at the cnd of subsecI. 5.2 where comparison of the matrix model and Liouville revealed some mysterious signs in the sum over matter states. The same p h e n o m e n o n occurs here, which is, perhaps, a hint that these signs are not connected to the nonunitary nature of the (2,

2m -

1) confor- mal field theories.

6.2. U N I T A R Y D I S C R E T E S E R I E S

In this section we will calculate the loop and the wave function of the cosmologi- cal constant in the general (n, n + 1) model. Using the solution of ref. [32] we can express the Lax o p e r a t o r of the nth model as

n = ["/2](n-k-1)!(k.~,, y'~ ~'(n~21¢) ' -Tu) 2k

tl ) n/2

=i" - 2 T"(P/(-Zu)'/2)'

(6.9)

where T,z is a Tchebyshev polynomial, which satisfies

and

T , ( i s i n h 0 ) = ( - 1 ) °'

l~/2isinh(nO)

T,(i

sinh 0) = ( -

1)"/2cosh(nrO)

for n odd,

for n even.

Thus we may compute the macroscopic loop amplitude

z c

w(I) = f~ dx(xle±tC),,Ix)

: ' - ° c o n s t

x f / dxxI/2"KI/,,(2/x12 ),

(6.11))

from which we obtain that the loop is given by

l ~K~+~/,(2~/I.tl 2 )

and the wave function of the cosmological constant is

tx~/2"K~/,,(2y/~l 2).

Using an argument analogous to the case of the Ising model one can show that the operators ~,,(~'L) exist. In particular, from the equation for the loop we see that the only operators which have one-point functions which are nonanalytic in ~ are o~0(~' ~) and ~2(~'~).

As in subsect. 4.3, one linear combination is the Liouville equation of motion.

T h e analytic terms in /x in (6.10) lead to one-point functions of ~r2k+~(:/,, t) (e.g., the energy o p e r a t o r in the Ising model) as found in ref. [32]. We expect that the ~,,(cY r) operators exist for all r in all p, q models but have not carried out the

G. Moore et al. / Two-dimensional quantum gracity 699 detailed proof. It would follow immediately from an

IK

expansion of the two-loop amplitude that the nondiagonal two-point functions (~rk(~qr)~rs(@ . _ , 3 ) for k + s an odd integer are nonzero and analytic in ~ as found in ref. [32].

6.3. c = l

Recently macroscopic loop amplitudes have been c o m p u t e d for an uncompacti- fled c = 1 system [40]. T h e two-loop formula at genus zero is

k

F(-Iqn-k) ( l,12 )

×KFal+2k(2V//z(l 2 + l 2) ), (6.11)

where

wq(l)

is a macroscopic loop carrying m o m e n t u m q with loop length l and we have only kept terms with fractional powers of l (we limit ourselves to the case q ~ Z). This complicated formula can be substantially simplified by use of the G e g e n b a u e r formula, along the lines of subsect. 4.4. Applying the G e g e n b a u e r formula to each term in (6.11) we obtain an

IK

expansion. T h e coefficient of each term Ilql+2kKiql+2k is a sum of factorials, and one can show that, except for the first term, this sum vanishes. Thus a

single

term in the

IK

expansion survives and we obtain the simple result

sin ~-]q] 11 < l 2 . (6.12)

As an immediate consequence of this we discover that the wave functions of ~'~

are [40]

(6.13) Interestingly, the expansion (6.12) has many good properties lacking in the c < 1 expressions we have seen thus far. For one thing, the embarrassing sum over an infinite set of states has disappeared. In ref. [40] it was proposed that the spectrum at c = 1 had the characteristic of a topological field theory, with operators ~rzn(~q).

Even if that is correct, eq. (6.12) is a strong hint that the higher r e d u n d a n t operators are automatically decoupled from macroscopic loops. To settle this issue we must check the higher-point functions. O n e lesson that we may draw from eqs.

(6.12) and (6.13) is that one should study macroscopic loop amplitudes in terms of

physical

wave functions rather than merely expanding them in powers of I. Only

then does the physics b e c o m e clear.

7()() (;. Moore et al. / Two-dimensioNal quantum gratio'

6.4. S T R I N G S IN --2 D I M E N S I O N S

As a final example we check some of the p r e c e d i n g ideas by investigating a model with a s o m e w h a t different flavor from what we have thus far studied, namely, we consider r a n d o m surface e m b e d d e d in D = - 2 dimensions [41]. N o t e that this theory is different from the m = 1 one-matrix model, although in both cases we have c = - 2 . T h e m = 1 t h e o r y is actually a t h e o r y o f loops spanning a w o r l d - s h e e t of zero area (see appendix A). O n the o t h e r hand, D = - 2 has a clear i n t e r p r e t a t i o n as tree-like polymers living on the world-sheet [42]. T h e partition function for closed surfaces of spherical topology is

Z(/.~,,) = Y'~ e -u'''' T ( G ) , (6.14)

{G}

where we sum over a class of p l a n a r g r a p h s having n vertices. In o u r case we will use planar ~0 3 g r a p h s . / x 0 is the bare cosmological constant a n d T ( G ) is the n u m b e r o f spanning trees of the g r a p h G. T h e partition sum is e v a l u a t e d by rewriting (6.14) a s

zcp.~) = ~ e-U"" x {contractions}, (6.15) {T}

i.e. we first sum over t h r e e - c o o r d i n a t e d trees and then wire up the ends of the trees to form a ~ p l a n a r g r a p h as in fig. 9.

I n t r o d u c i n g a loop o f length L into the r a n d o m g r a p h is simple: As usual we i n t r o d u c e one supervertex with L legs. T h e loop partition sum simply b e c o m e s

Z ( # o, L ) = 1 / L ~ e "'"' × {contractions} , (6.16) {7+!+' >}

' " ' , , , , , , . , " : ... ,,,"

Fig. 9. A tree with planar contractions forms a

&&random graph.

Fig. 10. A tree with a vertex of coordination number L makes a random surface with a loop of size L. (The contractions are omitted in this

figure.)

G. Moore et aL / Two-dimensional quantum gracity 701 where

{T (L)}

is the class of all three-coordinated trees possessing one supervertex of size L as illustrated in fig. 10.

Since the number of spanning trees on a planar graph equals the number of spanning trees on the dual graph; we are thus indeed evaluating the partition sum of planar triangulations (with a hole of size L ) embedded in D -- - 2 dimensions.

Z(/z0, L) is easily calculated using the methods of refs. [41,42]. The generating function for ¢3-trees with one marked boundary point is

T(z)

= Jz(1 - x/1 - 4 z ) . (6.17)

Furthermore, a generating function for the number of planar contractions of a k-legged vertex is

f_2 dA p(A)Ak ' (6.18)

2

with p ( A ) = 1/~--A 2 . T h e reader will recognize p(A) as the Wigner eigenvalue distribution of the m = 1 one-matrix model. It is now easily seen that eq. (6.16) can be written as (g = e -~''')

1 f[ dA ~ ( 1 ~ ) L , (6.19)

L 2

since

(T(z)) L

is the generating function for the class {T~L)}. It is straightforward to scale (6.19). The critical point is gc = ½; so introduce a cutoff a and put g = gc - aZ/.t as well as l =

La.

We also note that the singularity in (6.19) comes from the integration region close to the branchpoint at Ac = 2. Changing variables A = 2 -

aZx

and neglecting nonuniversal terms we calculate

Z(/z0, L ) ~ l / L f l 2 d A

2 V ~ Z A - A ( 1 - ~ ) L

~ l / a -

o e

- a 4 7 f °

d x x / x e t ~ . (6.20)

As explained before we define the loop function

(w(l))= lZ(l~, l)

and the wave function of the puncture operator

(O/Ott)(w(l))

and we find

0 = l f ~ d x v'x ~ e - ' f i - ~ (6.21)

7112 (;. Moore ct al. / Two-dimen~'i(mal qttattlum grat i O'

L L2

Fig. 11. T h e t r e e s k e l e t o n ( c o n t r a c t i o n s arc o m i t t e d ) of a t r c c with two v e r t i c e s of c o o r d i n a t i o n n u m b e r s L I a n d L , .

This integral is a Bessel function representation and we obtain

~( w( l))

0 = I / 2 t x ' / 2 K , ( lt.tl/2),

(6.22)

as expected. By a standard identity between Bessel functions the wave function of the boundary o p e r a t o r is

/ ( w ( z ) ) = (6.23)

It is straightforward to extend the above m e t h o d of calculation to more loops and to added handles (even nonperturbative expressions may be obtained). H e r e we will be content to present the two-loop function at genus zero. We have two supervertices of size L 1 and L 2 which are connected by a backbone [42]. T h e tree skeleton is shown in fig. 11. T h e blobs indicate a tree. These diagrams are g e n e r a t e d by

Z(tx,,,L,,L2)= f2 dap(a)T(ga)L'l_2T(ga) T(ga)":. 1 (6.24)

Note that T ( g c a c) = ~, so the backbone becomes macroscopic in the scaling limit.

Using the above method one readily derives

~ vfx- e-(/, +/-')( ~ - . (6.25)

Z ( / 3 ~ o , L I , L 2 ) ~ a 2 d x x¢~-+~-

So, defining ( w ( l t )w(12)} = Itl2Z(Ix, Ii,/2), we obtain 1 Ill 2

( w ( l l ) w ( 1 2 ) ) - 2 l, + 12 t~'/2K'[(l'

+/2)/'£1/2]

^{" (6.26)}

G. Moore et a L / Two-dimensional quantum gracity Using Gegenbauer's theorem this may be expanded as

703

( w ( l , ) w ( 1 2 ) ) : E ( - 1)J(J + 1)2I,+i(1,~'/2) K, +j(/2/z'/2) • j o

(6.27)

We would like to thank T. Banks, C. Crnkovid, M. Douglas, D. Friedan, D.

Kutasov, B. Lian, E. Martinec, A. Morozov, S. Shenker, A.B. Zamolodchikov, and G. Zuckerman for very useful discussions.

Note added in proof

After completion of this paper we were informed of ref. [49] which also discusses macroscopic loop amplitudes at genus zero. We also realized that the discussion of sect. 3 ignores contact terms when operators hit boundaries. Such contact terms can lead to negative powers of 1 which are nonanalytic in /~ [50].

Appendix A

MACROSCOPIC LOOPS IN TOPOLOGICAL FIELD THEORY

It is commonly said that analytic terms in IZ correspond to nonuniversal quantities. This leads to a paradox since in topological field theory all the quantities are analytic in t~ [43-45]. It would be ludicrous to assert that all topological field theory correlators are therefore meaningless. Nevertheless, it is worth understanding how one distinguishes "meaningful" analytic terms from

"meaningless" (i.e. nonuniversal) analytic terms. In subsect. 3.3 we showed how, from the point of view of Liouville theory, one can understand the existence of universal analytic terms in /x. In this appendix we explain the relation of that insight to the gaussian matrix model. Related remarks have been made in ref. [46].

In the continuum the m = 1 conformal field theory (the (1,2) minimal model) coupled to gravity does not have the identity operator in the spectrum. Therefore, the area of the surface is not well defined. It follows that we cannot fine-tune parameters of the cutoff theory to make large area surfaces. Instead, we can obtain continuum results by considering macroscopic loops, of length l / a in lattice units, and then letting a --* 0. More specifically, in the hermitian matrix model the m = 1 potential is simply the gaussian potential V(~b) = N tr 4, 2. It is technically easier to study loops by taking the Laplace transform of tr ~b l/a, thereby computing

1-1 ~ tr ~ (A.1)

704 G. Moore et aL / Two-dimensional quantum graciO'

The critical behavior is obtained as (i are tuned to their critical values, and not as p a r a m e t e r s in the potential are fine-tuned to critical values. For example the one-point function of the resolvent,

1

(A.2)

exhibits a square-root singularity in ( associated with the Wigner distribution.

Thus one can define a continuum limit amplitude unambiguously. T h e scaled version of ( is interpreted physically as the

boundary

cosmological constant p [30].

We can incorporate the double-scaling limit with a little more work. Since the discontinuity of the resolvent across the real axis is the eigenvalue density we see that, effectively, the m = 1 critical p h e n o m e n a comes from the edge of the eigenvalue distribution. Now, correlators of the resolvent o p e r a t o r may be com- puted readily in the fermionic formalism. On the lattice,

~1 tr[~l )--+fdA~*(A)~-~IA~(A)'N ~<,-d~

^(A.3)

where ~V(A) = ]Enan~,,(A) and I/G are the o r t h o n o r m a l wave functions made from the orthonormal polynomials. In the case of the gaussian model these are simply H e r m i t e functions. At the edge of the eigenvalue distribution the H e r m i t e func- tions behave like Airy functions [6,47,48] and the double-scaling limit of ~ is

fdza(z)Ai(z+h),

with a Fermi sea defined by a ( z ) l t z ) = 0 for z < p . and a(z)*lp~) = 0 for z > Iz. Since Airy functions are the Baker functions for the K d V theory with the potential

u(x)=x

we can obtain, from the gaussian model, the correlation functions predicted from the KdV theory at the m = 1 point.

In sum, by introducing loops, or equivalently, resolvents, we provide a cutoff in the matrix model which allows us to distinguish universal from nonuniversal quantities. This is the matrix-model version of the cutoff discussed in Liouville theory in sect. 3.

F r o m this discussion we see that in topological field theory t 0, the lowest coupling is in fact that boundary cosmological constant. Indeed at the m t h multicritical point cr m_ l is the boundary o p e r a t o r [30]. Moreover, drawing the F e y n m a n diagrams and their associated dual graphs in the gaussian model we see that the "surfaces" g e n e r a t e d when computing the resolvent correlators in (A.1) are completely degenerate, and more appropriately described as loops. We may connect these observations directly with the discussion of sect. 3 for the Liouville theory by considering (3.6) as A ~ 0. In the inside of the disk we obtain g = 0 (thus explicitly realizing Witten's idea that in topological field theory (g~t~) = 0.) Note, however, that the metric has nonzero support on the boundary.

G. Moore et al. / Two-dimensional quantum gracity

⁷⁰⁵

Appendix B

S O M E U S E F U L FACTS A B O U T BESSEL FUNCTIONS

Here we collect some formulae concerning the modified Bessel functions

l~(z)

and

K,,(z).

They are linearly independent solutions of the Bessel equation

Its)

^Z --Z2--/.- '2

] Zv(z)=0.

(B.1) I~ may be expanded as

I~(z)

= . (B.2)

=~k!F(k +u+ l)

The expansion of K~ is then obtained from

K~(z)

2sin u~r [ l _ ~ ( z ) - I ~ ( z ) ] . (B.3)

Note that

K ~ ( z ) = K~(z)

for all u. but

I . ( z ) = I~(z)

only if v is an integer. For large Iz one has asymptotically

1 eZ (,argz, < 2 1

I~(z) ~

( 3 )

K v ~ e -z [arg zl < -~Tr . (B.4)

For half-integer values of the index Bessel functions are actually elementary:

1 [ k~= (--1)k(n+k)'

I+'"+'/z'(z) ~ eZ -k~n-Sk)!

( 2 z ) -k

)

___ ( _ 1 ) n + l e _ z ~

(n+k)!

( 2 z ) - *

]

'

k = 0

k = 0

706 (;. Moore el al. / I w o - d t m c t t s i o n a l qttantttm grt try

Some very useful identities betwecn Bessel functions of differing index arc

2vl.(z)=zl,, i ( z ) - z l , , + l ( z ), -2vK,,(z)--zK,, i ( z ) - z K , , ~ l ( z ) , 2 z ~ l . ( z) =zl,,_l( z ) + zl,,~l( z), d - 2 z ~ K , . ( z ) =zK,, ,(z)+zK,,~l(z ),

^d

d

Z ~ z l , , ( z )

= zt,,+ ~(z) _+ v l , , ( z ) ,

z - - K , , ( z) = -zK,.+~( z) + vK,( z).

d

dz

(B.6) T h e r e exists a large amount of integral representations for the modified Bessel functions. A representation particularly useful for transforming from fixed area to fixed cosmological constant is

K,,(z) = ~ ) dtt " le -'-:~-/4',

largzL < - 2 ' Re > 0 . (B.7)

For the calculation of the wave functions of the discrete unitary series one needs the representations

K,(~) -

cos

vrr/2 ) f~

dt c o s h ( u t ) c o s ( z sinh t ) ,

z = R e z > O .

- 1 < R e v < 1,

77"

K,,(z) =f() dtcosh(vt)e-:

... h(,), ]argz[ <--2_, (B.S)

where the two integrals apply to the odd and even m e m b e r s of the unitary series, respectively. In the special case of the lsing model the wave function is related to an Airy function defined by

A i ( z ) ^{= - -}

7 r 3 (B.9)

which in turn may be expressed as a Bessel function through

= _1 ,/~-_ K,/3(zz.,/:]

Ai(z) ~- V 3 3 ] ^(B.10)

F o r the calculation of the Ising p r o p a g a t o r at ~ = 0 we used the

G. Moore et aL / Two-dimensional quantum gravity Weber-Schafheitlin integral

707

fT dxx aKu(ax)K,,(bx)=

2 - 2-aa - v + a - lbv

r(1 - a )

× F 2 F 2

x F 2 F 2

1 - A + i x + v 1 - A - i . t + v

b 2 )

× F 2 ' 2 ' I - A ' I - ~ 2 '

(B.11)

where F is a hypergeometric series (in the Ising case it sums to a simple algebraic expression). To prove the existence of the operators d in the Ising case another integral is handy:

o c

1)

K.(a~/~) =r(iz)2~'a-~'K._u(a )

fl d x x - ~ ' / 2 ( x - " - | (B.12)

Bessel functions obey an addition law due to G e g e n b a u e r which, for K,, reads in its most general form

K~(z) ~ Ij+v(x ) Kj+~(y)

2vV( ,) (j+ Q(cos 4,),

2v j=0 X v yV

z 2 = x 2 + y 2 _ 2 x y c o s &. (B.13)

The G e g e n b a u e r polynomials

C~(t)

are defined as the coefficient of a j in the expansion of (1 -

2at

+ o~2) -v. Finally, we give an ascending series relevant to the consistency of our definition of the inner product:

l (z)L(z) =

5 o P(v +k +

1 ) F ( / z + k + 1 ) F ( v + ~ + k + 1)k!

(B.14)

7(18 (;. Moore el al. / -]ho-dimensional quantum gra~'ity

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在文檔中 Department of Physics and Astronomy, Rutgers Unit'ersity, Piscataway, NJ 08855-0849, USA (頁 32-45)