Volume 194, number 1
PHYSICS LETTERSB
30 July 1987
C O N D I T I O N S OF WEYL INVARIANCE OF T H E T W O - D I M E N S I O N A L S I G M A M O D E L
F R O M EQUATIONS OF STATIONARITY OF T H E "CENTRAL C H A R G E " ACTION
A.A. TSEYTLIN
Department of TheoreticalPhysics. P.N. LebedevPhysicalInstitute, Leninsky prospect 53 Moscow 117924, USSR
Received 3 April 1987
Generalizing the "c-theorem" of Zamolodchikovto the case of the bosonica-model with the dilaton couplingwe prove that the
conditions of its Weyl invariance (presumably equivalent to the tree-level Bose string equations of motion) can be derived as
equations of motion from some action. A simple expression for the action is found.
1. One of the basic elements of the presently wellappreciated connection between string theories and
two-dimensional a-models is the conjecture of
equivalence of the tree-level effective equations of
motion for the massless string modes and the conditions of Weyl invariance of the corresponding amodel [1-3]. This conjecture was checked by a
number ofperturbative calculations [2-9] and arguments were put forward suggesting that it should be
true to all orders in a ' [ 10-13,6]. The validity of the
equivalence conjecture implies that the a-model Weyl
invariance conditions (the vanishing of "//-functions" [ 14,15 ]) can be derived as stationarity conditions of some action. This statement should be true
for the a-model irrespective of the string theory. The
crucial step on a way to its proof was made in ref.
[ 12 ] (see also ref. [ 13 ]). Here we generalize the "ctheorem" of ref. [ 12] to the case of the bosonic amodel with the dilaton coupling [ 16 ]. We find that
in agreement with the proposal of refs. [2,5,6] the
corresponding action can be chosen as
S=
dDy x / ~ e 2¢,]~¢,
cussions of the a-model on a curved 2-space
[6,14,15] (see also refs. [ 17-20]). The bare a-model
action is
Io= l f d2a x/~{O~xUO~x~Gou~(x)
+ (i/x/~) eabOax"ObX~Bou,(X) + a'R(~)Oo(X)}
= f d2a Aoi.~Oio,
(1)
where aa={al,a2}, ~o'={Gu,, Bu~, ¢}, 2 = 2 h a ' . The
couplings ~0~are functions on RD= {y} and the summation over i includes integration over yU(A~ =
(1/22)x/-gO~xuO~x"6(°)(x(a)-y), etc.). 0/0~0~ will
always be the functional derivative 8/8~0~(y). The
dilaton coupling is essential for the renormalizability
of the model [ 14,15 ] and hence for the finiteness of
the energy-momentum tensor operator Tab=
(2/x/g) 8Io/8g~b. The expression for its trace in terms
of finite composite operators ("normal products"
[...]) is given by [14,15]
0 = T,~ = [At] f f + E
where /~o is the basic Weyl anomaly ("central
charge") coefficient ( ( T~, ) ~ff°R(2) +...). Modulo
the equivalence conjecture we thus represent the
string theory effective action as an object in the amodel theory.
= ( 1 / 2 ) 0 { [ 0a x u O,x , -p~~ ( x ) ]
+ (i/x/g) e~b[O~X~'ObX"~Bu~(X)]
+ a'R (2)[]~(x) l } + E ,
(2)
2. The present work is based on the previous dis0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V.
(North-Holland Physics Publishing Division)
63
V o l u m e 194, n u m b e r 1
PHYSICS LETTERS B
30 J u l y 1987
As follows from (2) [ 6 ]
-]/~B + ½H..~M :a + OI~,L~,
- B __
(3)
( 0 ) = (1/4n) ffOR (2) + non-local structures,
(9)
ao,
I~=~ D-¼ a ' R +...,
M, =2a'a,o+ W~,
H~,~)~= 30o, B~zl,
(4)
where ]/'=d~Tdt ( t = l n St, rf are the renormalized
couplings) are the ordinary RG ]/-functions, W~,and
L~ are determined by the renormalization matrix for
the operator AG and AB and E=EU(x(a))
× 8Io/SX~'(a) (E is not renormalized and ( E ) = 0 in
dimensional regularization).
The partition function corresponding to (1) is
(lO)
where we have used that in dimensional regularization ( O"xUO~x~) = - ¼a' R(:)G~(y) +.., [23] and
fixed the renormalization scheme (the definitions of
~*) so that no additional terms appear in (lO). If
W - - l n ~, then (O)=(2/v/-g) gob 8W/Sgob and
hence the dependence of W on the conformal factor
of the metric is determined by (9)
W=
[- JYR(2)z~-- 1R (2) + other structures,
d
Zo(q~o(A), A, g) = j ~ x e -~0 = Z(~(St), St, g),
P-'~'Yl "+Y2 ln(A/st 2) +y3[ln(a/st2)] 2+ ....
dZ/dt=O (A is an ultra-violet cut-off). We can formally split f ~ x = J l - I , 4 x ( a ) ~ ( x ( a ) )
into an
A= - (1/x/g)
integral over constant and non-constant functions:
x=y+~, y=const.,
z = f dDy
G~
e-Z°(Y)2(y),
Z = f [~{1 e -'-°[¢'-'1,
(5)
Oa(,v~gabOb)
( 11 )
---- -- 172 ,
where yn=y,(q~(y,#)), 7t = - (1/16n)/~° and dv/dt=O
implies that 72 = ½d7 t/dt, etc. The key point [ 12,13]
is to introduce the "running coupling" ~i(z),
d~'/dln z =ff(~(z)).
Then v=v(0) = y , ( ~ ( a ) ) ,
0=_x/~. To justify (11) we may also formally consider the expansion near a flat 2-space: gob= 5oh+ hob.
Then in the momentum representation
W = [d (d2k) gab(k) Pobcd(k) h-~d(-k),
+ a'R(2) O,,Oo(y) ~*' +...],
where
(6)
[ ~¢]
contains the "gauge condition"
3~°~(f d2a x/g() [21] or 6(m(~(ao)) [22] (ao is an
arbitrary point of the 2-space M2). We assumed M 2
to have the spherical topology (so that f d 2 o " x/g
xR(2)=8n). Note that x ( a ) = y is not a stationary
point of Io when R t 2) ~ 0 and 0o # const, but this will
not be important for us here (the value of Z is independent of an expansion point). We shall use the following averages
((F)) = f ~xe-~°F(x)
=f d°y.f6e-:~2 <V>.
(d2k) _=d2 k/(2n) 2,
where ~ a b = f d 2 a e - i k a hab(ff). P is symmetric and
transverse (kaPaa.d= 0). On dimensional and Lorentz
invariance grounds
5
P=~&n.,
;'/=1
n l abcd = k.bcd/k2 ,
ka...b = ka...kb,
n 2abcd= riabkckd + ~cdk~ kb,
n 3abed= ~,ckb kd + 6bckokc+ 8be ko ka + rSaakbkc,
(7)
;g 4 a b c d = k
2
(~ab(~ca,
;g 5abcd ---- k 2 ( (~ac~bd + (~ bc(Jad) •
( F ) = Z - ' f [ ~ ] e-r°F(y+~).
,/
64
(12)
(8)
( 1 3)
,Lnare dimensionless functions of x / ~ . The trans-
Volume 194, number l
PHYSICS LETTERS B
30 July 1987
8 T c d ( a ' ) / ~ g a b ( ( l ) = ( 1 / 2 ) {O(2)(a-a')
versality implies that 21-'t-22+223=0, 22+24=0,
,~3"~25=0. Also, in two dimensions ku~b8
~ d = 0 and
X [ -- ½(gcdOaX'UObxV--ga(cgd)bOeX'UOe x v ) Go,uv(X )
hence rr3-~rs=2(ztz-n~). As a result,
- a ' (ga(~ga)t, V2 - g c d Va Vb) ¢0 ]
eabcd =,~l ( R l -- Tt2 di- n 4 )abcd
- a ' V ' 8 ~ 2 ) ( a - a ') Vk¢o
= [2,(k)lk 2] (kakb(14) -C~abk2) (k~ka-8~ak2),
× (g~(~gd)(agb)k --g~(agb)kg~d
(14)
and hence (12) is in agreement with (11) if
2)(k)=v(k), v ( k ) = - ( 1 / 1 6 n ) ~ o ( ~ o ( k ) ) ~ . Note
that eq. (9) can also be rewritten as
( 0 ) = (l/4rt) ffo(tp(O)) R ( 2 ) + . . . .
3. Let us now consider the correlators of the two
energy-momentum tensors which are used in the
proof of the "c-theorem" of ref. [ 12 ]. As we already
noted, ((Tab)) is finite if we correctly account for the
renormalization of the dilaton coupling. However,
additional infinities may appear in
((Tab(a) T~a(a') ))
Therefore, the infinite part of (15) is necessarily local,
i.e. is proportional to O(2~(a-a ') or OaO(2)(a-a').
Let (...)re, denote the average (...) in which such
local terms are dropped. Then
(Tab(a) Tcd(a') )reg
= - (2/v/g))618gab(a) (Tca(a') )
=
-((2/v/g) 8/Sg#b)~ ((2/x/g) 8/Sgcd)o, W.
(18)
( ( O ( a ) O ( a ' ) ))reg = ~ i ~ i j ( a ,
(15)
(we consider the connected part of the correlator).
Since the second term is finite (being derivative of
a finite quantity) the infinities may come only from
the first term (see e.g. ref. [24]). By definition (here
for simplicity we set Bu, = O)
Tab = (2/x/~) alo/ag ab
= (112) [(OaxUOhx~ - ½gal~OCXUOcXv) Gou,
--Ot'(VaVh--gae Vz) ~0],
(17)
In the case of the two traces 0 (2)
= ((((2/x/g) 8/Sgab)~, T~a(a') ))
- ((Z/x/g) 8/~gab)o ((Tcd(a') ))
- ~1 g,kga(~ga)b + ½g, kgabgca) }.
i
iT'),
(19)
g o - (( [Ai(a)] [A;(a')] ))r~,,
(20)
where we have used that
(( O( a) EU(x( a') ) 6Io/SxU( a ') ))r~g
= - (((8/SxU(a')) O(a) EU(x(a')) ))~g =0.
Substituting (I l ) into (18) and taking the flatspace limit after the functional differentiation we get
(Tab(a) Tcd(a') )reg,fl
(16)
=-17(0) []-'(OaO~--~abU]) (OcOa--~aE])
×O~2~(a-a'),
(21)
o(O) = (l/Sn)/7~(~(y, O)), 3= -2v,
or
in
the
momentum
(22)
representation
(Tab(a) = f (d2k) ei*~T~b(k))) ,2
"~ In two dimensions f In AR ~-'~= f R(2~A- ~R (2~ as can be checked
in the conformal gauge or using the expansion near a flat space.
Thus the terms f (In A)"R t 21can be represented as f R t 2~j - ~R t'
×(lnA)~...d-lR('-k
,2 The same representation is true also for the full correlator with
singular terms included. This follows from the analysis of the
possible structure of a transverse symmetric fourth-rank symmetric tensor Papa in two dimensions (see the discussion
above).
65
Volume 194, number 1
PHYSICS LETTERS B
(T~,,(k) T~a(q) )~g.a = (2rt)2a(2)(k+q) &b~d(k),
30 July 1987
04
U= 16z 4 0-~/2,
P,t,,d=~(k) k-2(k~kh-~a~k 2)
3 - 03
× (k,.ka- acak2).
(23)
The same relation is true for ((T~b(k) Tea(q)))~g,n
with 17(k) replaced by (see (7), (8))
iT(k) = f dDy x/G e 2°0(~0(y, k)).
0
X= - 1 6 z Z~z3~--~I2,
02 02
y = 16zZZ 2 0Z 2 082 if2'
(28)
(24)
~2=K2(Izl)=f
(dZk) e ik~ tY(k)k
2
Note that while we have taken the flat-space limit in
the correlator of the energy-momentum tensors here
we included the e 2O-factor corresponding to the
spherical topology ( W of course vanishes in the flatspace limit). This seems to be necessary in order to
correctly account for the infrared effects. Introducing the complex coordinates z, 2=a~+itr 2, ~==0,
a_-e= ½, so that k~tr4=k:z+k~g, k2=4k:ke and
where Jo is the zeroth Bessel function. Using the
variable r / = - l n l z l it is straightforward to rewrite
(28) as
0=4T____, T=4T_-:=TIt-T22-2iTI2,
U= (d/dr/+tr) (d/dr/+4) (d/dr/+2)dO/dr/,
(25)
we define, following ref. [12] (tr' =0)
_ 1
2g
I
dkJo(klzl)
o
v(k)
,
k
(29)
t2=t2(e-D,
U=z4 (( T(17) T(0))) n,
X= - (d/dr/+ 4) (d/dr/+ 2) d 212/dr/2,
X=z3e ((T(tr) 0(0)))n,
Y= (d/dr/+ 2) 2 d2/2/dr/2,
Y = z 2 2 2 ((0((7) 0 ( 0 ) ))fl.
(26)
As is easy to see from (15), (17) these correlators are
finite since they do not contain singular terms due
to the explicit powers ofz and ~ (hence ((...)) in (26)
can be freely replaced by ((...)) reg). It is easy to prove
that U, X, Y are real and depend only on Iz l. This
implies that they are functions of the "running" coupling ~0(1/Izl ) (additional dependence on Izl is ruled
out because U, X, Y are dimensionless). From
(21)-(24) we find
k~ke
X= - 1 6 z 3 f f (d2k)e i k a ~-5--~(k),
Y= 16zZz2
(d2k) eika k2- #(k).
As a consequence,
66
k~k~
thus proving the reality of U, X and Y. We are now
free to set their argument to be Izl = 1//~, r/=t. The
U, X, Y are functions of ~0(/~) (more precisely the
functionals of ~o(y, #) depending on/~). Introducing
their combination [ 12 ]
C = ~ ( U - 2 X - 3 Y ) = ( d / d t + 2 ) 2 dg2/dt,
(31)
we find from (30)
dC/dt= Y or ffOC/O~o'= Y.
(32)
The combination (31 ) is chosen just in order to have
the derivative proportional to Y. Let us now discuss
some consequences of (28)-(32). If O=const., then
/2 in (29) is simply proportional to the massless
propagator, I2= - (1/2n) Pin Izl+const., and hence
k4
U = 167.4 ~ (d2k) e i*~ ~ O(k),
f
(30)
(27)
X= Y=0,
U= (24/n)~,
C= (2/n)O,
a=const.
(33)
If Y=0 (e.g. if 0 = 0 ) then solving eq. (30) we get
Volume 194, number 1
PHYSICS LETTERS B
~ = ( a l rlq-a2) e -2, + a3 r/q_a4 '
U= - 16a~ e-2"+48a3,
X = - 8 a ~ e -2',
C=4a3.
(34)
The general restrictions on the behaviour of g probably imply that a~ =a2=0, i.e. that X = 0 is a consequence of Y= 0. one of such restrictions is the
positivity, according to the standard argument
(( O(k ) O(q) ))n = ( 27t)2d(2) ( k + q) k2ty(k )
X I <<010(0) Jn>>n [2
The positivity must also be the property of Y. In view
of (19), (20) we have
(35)
where we used a, fl (instead of i, j) to numerate only
the "flat-space" (G~,~and Bu~) couplings (the dilaton
terms drop out of (19), (20) in the flat-space limit
since A o ~ R ( 2 ) ) . The positivity of the metric in the
space of states seems to imply that x~p is a positive
definite matrix (for all the values of Iz[ ) [ 12 ]. Since
g, U, X, Y and hence C are certainly invariant under
the general coordinate transformations in the y-space
(SG~= ~(,r/~), etc.)
fl' OC/O~oi = ff OC/O~oi,
( 36 )
where we used that f~ in (3), (4) differ from i f just
on a coordinate transformation 8~0~ (and a gauge
transformation of BuD. Note that the sums over i in
(32) and (36) include the dilaton terms.
4. If OC/O~o'=O(i.e. if OC/O(~=OC/OG=OC/OB=O)
then, according to (32), Y=0 and hence in view of
the positivity ofx~p (35), f'~ =0 (i.e. fig =fiB =0).
At first sight, we are lacking the condition rio =0.
Remarkably, as follows from the non-renormalization of 0, f ~ = 0 imply that Oufi°=O, i.e. that
fO=const. [19,15] ,3. Moreover, since C can be
represented as f d O y x / ~ e -2~ ~ ( ~ 0 ( y , / 2 ) ) where
#3 It
is
easy
to
see
that
~
is
also /z-independent:
dfl~/dt=ff( O/O(o') 1~ - ½M'O,~ ~ =l~°( O/OO)/ ~ = 0 where we
used that fl" =0, 0 ~ ° = 0 and t h a t / ~ may depend on 0 only
through its derivatives. Note also that if/7* = const, g= 0 implies
l~=Osince f d°yx/Ge-~o>O.
depends on 0 only through its derivatives, OC/O0= 0
implies in fact that C= 0 and hence flo = 0. Thus we
have derived the a-model Weyl invariance conditions f f = 0 from the stationarity of the "action" C.
According to (29)-(31) the "action" C can be
represented in the following way:
C= - 2 ~ ~ [2k2 ,zl2jo(klz I ) _ k 3 ,zl3j~(klz[)
7~ d K
0
+4klzlJ~(klzl )]P(k),
k 2 / y ( k ) = (2tt) 2 ~ 6(2)(k-k.)
n
Y=fafPtCa#, /fa~ = ] z I 4 ( ~ o t f l ) n ,
30 July 1987
(37)
where J6=dJo(a)/da and we have used that a J 3 +
J6+aJo=0 (this integral is to be computed with a
damping factor e-'k). Observing that P(k)=
o(~o(~))+ff(Oo/O~o9 ln(k//z)+.., we may use (37)
to find the expression for C in terms of the derivatives of O. This, however, is unnecessary since we can
introduce a simpler action the stationarity of which
again leads to i f = 0 . In fact, as follows from (37),
O~/O~0i=0 implies OC/Oq~=0 and hence, by the above
arguments, f~ = 0. Thus the "central charge" ~ itself
can be taken as an action (cf ref. [13]). Recalling
(10), (11), (24) we finish with o = ( l / 8 n ) S>~0,
S= ; dOy x/-G e-2~ff~
= f dDy ~
e - 2 ~ ( f ~ _ kfG).
(38)
Substituting the expressions (2) for f i into (38) we
find that the Mu-dependent terms form precisely a
total derivative and hence the result is simply
r'.~ ~j,
S = f d~ x/G e-EO(fl o - ~t'u~laG
(39)
where fl~ and fig are the ordinary RG fl-functions.
The final observation is that (39) can be rewritten
in the following remarkably way
S= _1 dV/dt, V = f dOyx/-Ge -2~,
(40)
where we assumed that the couplings are taken at the
normalization point ~t and used that fl~=dq)/dt,
G =dGu#dt. This form of the action is certainly
flu~
suggestive.
The author is grateful to M. Tsypin and B. Voronov
for useful discussions.
67
Volume 194, number 1
PHYSICS LETTERS B
References
[ 1 ] C. Lovelace, Phys. Lett. B 135 (1984) 75.
[ 2 ] C.G. Callan, D. Friedan, E.J. Martinec and M.J. Perry, Nucl.
Phys. B 262 (1985) 593.
[3] A. Sen, Phys. Rev. D 32 (1985) 2102; Phys. Rev. Lett. 55
(1985) 1846.
[4] T. Banks, D. Nemeschansky and A. Sen, Nucl. Phys. B 277
(1986) 67.
[5] C.G. Callan, I.R. Klebanov and M.J. Perry, Nucl. Phys. B
278 (1986) 78.
[6] A.A. Tseytlin, Nucl. Phys. B 276 (1986) 391.
[7] D.J. Gross and E. Witten, Nucl. Phys. B 277 (1986) 1.
[ 8 ] M.T. Grisaru, A.E.M. van de Ven and D. Zanon, Phys. Lett.
B 173 (1986) 423;
M.D. Freeman and C.N. Pope, Phys. Lett. B 174 (1986) 48.
[ 9 ] R.R. Metsaev and A.A. Tseytlin, Lebedev Institute preprint
N48 (1987).
[ 10] C. Lovelace, Nucl. Phys. B 273 (1986) 413.
68
30 July 1987
[ 11 ] B.E. Fridling and A. Jevicki, Phys. Lett. B 174 (1986) 75.
[ 12] A.B. Zamolodchikov, JETP Lett. 43 (1986) 730.
[ 13 ] A.M. Polyakov, SLAC preprint Trans-0222 (1986).
[ 14] A.A. Tseytlin, Phys. Lett. B 178 (1986) 34.
[ 15] A.A. Tseytlin, Lebedev Institute preprint N 342 (1986).
[16] E.S. Fradkin and A.A. Tseytlin, Phys. Lett. B 158 (1986)
316.
[17] C.M. Hull and P.K. Townsend, Nucl. Phys. B 274 (1986)
349;
C.M. Hull, DAMTP preprint (1987).
[ 18] S,P. de Alwis, Phys. Rev. D 34 (1986) 3760;
G. Shore, Bern preprint BUTP-86/16 (1986).
[ 19] G. Curci and G. Paffuti, Nucl. Phys. B 286 (1987) 399.
[20] S. Randjbar-Daemi, A. Salam and J.A. Strathdee, Bern Preprint BUTP-87/3 (1987).
[21] D.H. Friedan, Phys. Rev. Lett. 51 (1980) 334; Ann. Phys.
(NY) 163 (1985) 318.
[22] F. David, Nucl. Phys. B 190 (1981) 205.
[23] S.P. de Alwis, Phys. Lett. B 168 (1986) 59.
[24] S.J. Hathrell, Ann. Phys. (NY) 139 (1982) 136.