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3D Ising model and 2D induced gravity
Volume 260, number 1,2
PHYSICS LETTERS B
9 May 1991
3D Ising model and 2D induced gravity A.G. Sedrakyan Laboratoire d'Annecy-le-Vieux de Physique des Particules, IN2P3-CNRS, Chemin de Bellevue, BP 110, F-74941 Annecy-le-Vieux Cedex, France
Received 16 November 1990
The 3D Ising model ( 3DIM ) is investigated near the critical point. It will be shown that in this limit in 3DIM there are induced gravitational modes, which are described by the SL(2, ~) current algebra with coupling constant ~. The singularities of the twodimensional surfaces, immersed in 3D euclidean space are taken into account. Their contribution to the partition function is equivalent to the contribution of the anyons with charges + ½(K+ 4 ) and conformal spin ½[ 1+ ½(K+ 4 ) ] included in the problem under consideration. The Knizhnik-Polyakov-Zamolodchikov equation for the renormalized matter-field coupling constant K is changed and the solution D= 3, K= 0 is found.
1. Introduction
The interest in the 3D Ising model was raised by Polyakov [ 1 ] in the context o f constructing string theory in noncritical dimensions. It is known [ 2 ] that the 3D Ising model ( I M ) has a phase transition point o f second order and, so, there must be some continuum field theory corresponding to it at this point. On the other hand the 3DIM is dual to the 3D gauge IM with the 7~2 symmetry and it has been shown [ 3 ] that the partition function reduces to the summation o f the statistical weights of surfaces over all closed surfaces on the lattice. In this representation of the partition function of the 3DIM statistical weights o f the surfaces appear with the sign + 1 (sign factor q~ ( X ) ) which ensures cancellation o f self-intersecting surfaces (an analogous situation exists in the 2 D I M which is equivalent to the theory of free Majorana fermions). This fact lies at the basis o f the Polyakov hypothesis [ 1,4,5 ] that near the critical point (continuum limit) the 3 D I M is equivalent to string theory with some fermionic structure. There are many attempts to construct fermionic strings on the lattice [ 3-8 ] which corresponds to the 3DIM, but the construction suggested in ref. [ 7 ] essentially differs from the others in two aspects. First, there exists a naive continuum limit of the lattice action and second, the fermions are external and appear in the action quadratically. In ref. [7 ] it was shown that the naive continuum limit (lattice spacing a o 0) o f the lattice action is the Dirac action on a two-dimensional surface induced from the spinor structure of the 3D euclidean space (see ref. ( 1 4 ) ) . Developing this picture for the 3DIM, the authors o f refs. [9,10] have investigated quantum fluctuations o f fermions in the induced Dirac action and have found that because of anomalies o f the quantum theory there are additional modes, namely induced gravitational modes, which we need to add to the induced Dirac action to restore the original classical symmetries o f the 3DIM. The action obtained in ref. [ 10 ] for the induced 2D gravity modes in the chiral gauge is the W Z N W action for the SL(2, R) algebra with some coupling constant ~ , which coincides with the Polyakov results obtained in ref. [ 11 ] because o f the universality o f the anomaly calculus in two dimensions. It was shown [ 10,1 1 ] that the renormalized coupling constant ~ o f the SL (2, R ) W Z N W model is related to the dimensionality o f the external space (or to the number of degrees o f freedom o f the fields under consideraPermanent address: Yerevan Physics Institute, Br. Alikhanyan street. 2, SU-375 036 Yerevan 36, Armenia, USSR. Elsevier Science Publishers B.V. (North-Holland)
45
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9 May 1991
tion) and a consistent theory exists for d < 1 or d > 25. The results of refs. [ 10,11 ] has been obtained assuming that we have immersed (the determinant of the metric tensor is not zero) surfaces. But in the 3DIM problem the singular surfaces (immersed besides the same discrete points, the end-points of the self-intersection lines of the surfaces) play an essential role, because just for these surfaces we can have a factor - 1 in the partition function. So we need to take into account the singularities of the surfaces and find their contribution in the 2D gravity action. In this sense the approach with the external fermions developed in refs. [ 7,9,10 ] has an advantage over the approach using internal fermions. In this article we investigate the contribution of the singularities of surfaces to the induced 2D gravity action. It will be shown, that like the approach developed in ref. [ 12 ] for high-Tc superconductivity, the existence of singularities is equivalent to the inclusion of the interaction with gravity anyons (particles with fractal statistics) in the problem under consideration. The statistics of anyons depends on the coupling constant ~ and the condition of restoration of the original symmetries of the 3DIM provides a relation between the coupling constant and dimension D of space-time, different from one, obtained in ref. [ 11 ]: ½+D-~-~.3K(K+8)-
13+ ~
6
+6(K+2)=0,
(1.1)
where K = - (~ff+ 4) is the SL (2, ~) coupling constant of the matter fields. This equation has a solution D = 3 and K = 0. Recently, very intriguing conjectures have been proposed [ 13 ], implying that there is an interesting connection between the topological and conventional formulation of 2D gravity, and it seems that in the case K = 0 [ 14 ] indeed we have one. But as we will see below, in the case of 3DIM we must consider the K = 0 theory in the modified vacuum (filled with anyons) which probably corresponds to another phase of the theory. In section 2 the basic definitions of the 3DIM will be given. Section 3 presents the gravity action induced by external fermions (in the problem). In section 4 we calculate the contribution of anyons (singularities of the surfaces in 3D space) to the 2D gravity action.
2. Basic definitions The gauge 3DIM is the system of spins an~,= + 1 placed on the links (n,/t) of the 3D regular lattice (n is the enumeration of the sites of the lattice and # = + 1, + 2, + 3 is the direction of the link attached to the site n) with the nearest neighbour interaction
H=J
~_,
an#trn+~.~tr#+u+ ~ _~an~.
(2.1)
plaquettes
It has been shown in refs. [ 3,6,7] that the partition function of the 3DIM can be represented as a sum of statistical weights of the free parametrized surfaces E:
Z=
~
all parametrized surfaces
exp(Z4) @ ( £ ) = ~ e x p ( - H / K T ) , {a}
(2.2)
where 2 = In th J / K T and A is the area of the surface E. The factor @(~) is taking values in 712={+ 1} and it is called the sign factor of the 3DIM. It ensures the cancellation of self-intersecting surfaces in the statistical sum and it is the origin of the fermionic structure (without this factor we will have a bosonic string). In ref. [ 3 ] it has been proved that when we take @(E) = ( -- l ) t where l is the length of the self-intersection line, then there is a correct cancellation of self-interacting surfaces. Near the critical point one just has a functional integral over the parametrized surfaces X ( ~ ~2) and we need to 46
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represent qb(X) through the differential geometric characteristics of the E. This was done in ref. [7 ] #1 In eq. (2.2) only the surfaces with an open line of self-intersections can have the minus sign (because the closed ones have even length on the regular lattice). The open self-intersection lines have end-points and it has been shown by Whitney [ 16] that they are the only stable singularities (at this point the induced metric g~a=O,~XOaXis degenerate) at the maps Y,2-->E3 (fig. 1 ). They are stable by the fact that the linking number of the strip C surrounding the point 0 is the topological invariant and takes the value - 1. One can represent the linking number by the formula (2.3)
~ ( C ) = ½Tr P e x p ( ! S - ' d S ) ,
where the matrix S is defined as follows. Let us consider at this point the orthonormalized vectors Ya (~) tangent to the surface, as well as the normal n ( O ,
ya(~)=Oo, X'eT,
c~=1,2,
a=l,2,
n(~)=i~hxy2,
(2.4)
where the e J are the zweibeins corresponding to the induced metric g~p:
eao,epa=g,~a=O,~XOaX.
(2.5)
Then S is the operator which rotates the matrices y~= Ya"tr and n = n. tr (tr are the Pauli matrices) into oz.
Sy,,S-l=aa,
SnS-l=a3
(2.6)
and
S-I dS=~(yadYa +n dn)=lrab[ya, Yb] +~ha[yan],
a, b= 1, 2,
(2.7)
where 1-'ab=l-'~b d~ '~ is a usual spinor connection in 2D gravity and ha=h~ d~ ~, the second quadratic form of the embedded surface. F "b and h satisfy the Peterson-Codazzi (PC) equations a
dFab+hahb=O, dha-l-'abhb=O.
(2.8)
The field S-1 dS looks like a fiat gauge connection and one can think that ~o(C) in formula (2.3) is equal to one. But when C surrounds the singularity 0, where d ( S - 1 dS) - S - ~ d S ^ S - t dS= 2in82 (~)az d2~,
(2.9)
#1 In ref. [ 15 ] a slightly different definition of the sign factor ~ ( X ) for surfaces on the lattice is provided and a detailed analysis of its properties is carried out.
x~
Fig. 1.
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PHYSICSLETTERSB
then we have ~ ( C ) = point 0 (fig. 1 )
X~=~,~z,
9 May 1991
1. This can be shown using Whitney's [15 ] parametrization of the surface near the
X3=¼~2.
X2=~2,
(2.10)
It is easy to see (by use of eqs. (2.5), (2.7), (2.9) ) that near the point 0
U
h. . .1.. . g = t c d - (- ¢d 'a~)c2 .
(2.11)
.h 2 i.n g = 0
.
Now let us cover the closed surface in the lattice by systems of closed curves which cross the links of the surface once. This can be achieved by drawing two parallel lines, connecting the midpoints of neighbouring links on each plaquette of the surface (for details see ref. [7] ). One thus obtains a class of coverings (A is the area of the surface). It was shown in refs. [ 7,15 ] that the product
2A/a~
q~(Y) =M,q~(C,) = ( - 1 )'
(2.12)
2A/a2
does not depend on types of coverings and is equal to ( - 1 )l. In ref. [7 ] the lattice action for the fermionic string, which reproduces the sign factor (2.12) and has the following classical continuum limit (lattice spacing a ~ 0 ) :
S=½i f
d2~x/g ~({) ( ~ 0 ~ - 0"Y")¢({) '
(2.13)
has been found. In eq. (2.13), q>, ~t are 3D fermionic fields living on the two-dimensional surface X(~), g = det 13,,X 0~XI and y'~, a = 1, 2 are defined as follows: ~ = 0,~X'tr.
(2.14)
This is the induced Dirac action. So, the expected partition function for the 3DIM near the critical point is
Z= f ~X~/exp(-S-i~Z y dZ¢x/g) .
(2.15,
The action (2.9) is obtained in a classical continuum limit. In a quantum continuum limit, in general we can get some additional terms (anomalies) in the action, which we have to add to the matter field action (2.13 ) in partition function (2.15). The general principle to follow in order to find these terms is to find the action induced by anomalies of the quantum fluctuations of matter fields and add it to (2.13) with renormalized coupling constants ensuring the original reparametrization (diffeomorphism) and Weyl invariances.
3 . 2 D induced gravity
The 2D gravity induced by quantum fluctuations of external 3D fermions which have the action (2.13) has been investigated in refs. [ 9,10]. There it is assumed that the 2D surfaces X(~) do not have the singularities mentioned above. It has been shown that the effective action for fermion field fluctuations looks as follows [ 10 ]:
"(f (Fa-½iTrtY3,~-l~a)L)(l'#-½iTra3)~-l~#,~)ga#v/-gd2~
W ( X ) = ~--~ ½
Z
+i ~ FdF-½ f F~Tr(a32-'O~2)~Pd2~),
(3.1)
T-t
where F~ = 48
F~bEabis the spinor connection for 2D gravity, the variable 2 = exp (ia3~o)~ SO (2)
is the SO (2) ro-
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tation matrix in the plane tangent to the surface Z, F=Fo,d(~'+Ftdt, a one-form in the manifold Zt extended in one dimension, the boundary of which is the surface X(~) _=Z under consideration, and o,ugis the coupling constant. The second terms in (3.1) is a multivalued functional, whose integrand is a density of the Hopf topological invariant, the map $3--,S z. Thus the partition function, improved by contributions from anomalies, becomes
z= f ~X~exp(-S-zt2 ~d2,x/~-W(X)). Now one needs to change the integration parameter X(~) in Z and introduce zweibeins from the relation
(3.2)
e. a. It is easy to see
e,~aeap= O,X OaX
(3.3)
that the integration measures of e,~a and X are related as follows: ~ X = ~E,~ 4. (det A~<')) --DI2,
(3.4)
where Ao(e) is the 0-form laplacian, which is equivalent to the contribution of free bosonic fields 43 into the partition function
z= f ~e,,'~'~'~exp(-S(e) - & .... -W(e)-lt2 f dZ~v/g).
(3.5)
In refs. [ 9,10 ] it has been shown that when in the induced gravity action (3.1) we fix the gauge det e,~ 4= 1
(3.6)
and 0~P
×
D
0,, (v/gJ~) = 0 ,
(3.7)
where J~ = e'~rgrp/x/g is the complex structure and f a are isothermal coordinates, we obtain the usual WessZumino-Novikov-Witten (WZNW) action with the SL(2, ~) group elements e,~a, together with some term which ensures the gauge invariance under the left SO (2) transformations of the e, a. Thus, we have an SL (2, N ) right current algebra action with coupling constant o~ together with a free spin ½fermion, three scalars and ghost actions.
4. Inclusion of singularities In order to find the contribution of singularities of the surfaces to the partition function of 3DIM, one needs to rewrite the Hopf invariant term in (3.1) in terms of the second quadratic form ha((). By use of the PC equations (2.8) we have
FdF=~-~n hadh a.
(4.1)
Zt
The connection h a contains regular (h a) and singular (h a ) parts. When we have 2N singularities in the surface, located at points ~k, then the singular parts h a are defined by the equation 49
Volume 260, number 1,2 2,,
h~ = ~ C d k=,
PHYSICS LETTERSB
9 May 1991
(¢_¢~)~ I¢-¢kl 2
d~ c
(4.2)
and 2N
d^h~ =2n ~,, 62(~--~k)d2~.
(4.3)
k=l
Then in the Hopf term in the effective action we have
f hadh ~= S h 7 d h a + 2 f hadhf+ f hsdhs= f hfdhf+ k~= l 4n~ h~ d t +
~t
Et
Zt
~t
f hsdhs.
(4.4)
Ck
In the foregoing we used eq. (4.3) and the second term in (4.4) transformed into the integrals along the contours Ck, which connect 2N singular points. So in the partition the function Z we have terms coming from singularities: Z~exp
_i.
4in~m f hi d t + W
f h~ dh~
).
(4.5)
C,n
The first term in (4.5) is obtained using Whitney parametrization (2.10) and the particular disposition of singularities in E 3. The covariant form of the corresponding terms is
Z-½Pexp(-i -2~=1f ~r u h~'b'ab)exp(-~n~ f hsdh~) "
(4.6)
C,n
It is possible to reproduce the first term in (4.6) introducing the new fermionic field 0 in three-dimensional space, which interacts with the field h a~b~.abwith the action ,~= iYd f d2~ x//g ~ya( Oa+ ½ihaTb'ab)O
(4.7)
and considering the correlation functions of 2N fermionic fields placed at the points ~ (n = 1..... 2N)
I
dEal ... d2~N (x/~)E~V(/7~ (~1)...~7~u(~N)O~(~+L)...O,~(~EN
) e-~) .
(4.8)
As for the second term in (4.6) with singular h~, the situation is similar to the one considered in ref. [ 12] in connection with high-temperature superconductivity. If one considers the fermionic fields 0 interacting with the U( 1 ) gauge field h~, the action of which is the Chern-Simons (CS) action ( X / 8 n ) f h~dh~,then the solutions of the classical equations of motion tell us that fermions create magnetic vortices (by eq. (4.2)) and they can be considered as free particles but with different statistics. They become anyons with _+ charges, which corresponds to the two end points of the line of self-intersection of the surface (in this case anyons interact with gravity). The change of statistics is defined by the charge of fermions (in this case q = + ½o,~) and by the U ( 1 ) CS action coupling constant ½# = ~ff/8n. So, the spin of the anyons is [ 12 ] 1 q2 As= ~ 2n---~, s = l ( l + ½ X )
.
Finally, the partition function for 3DIM is found to be
50
(4.9)
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Z= N=~ 1 f ~eaa~Ct~¢~o f x/F~ld2~l...N/g2Nd2~2N~°tl(~l)...OOtl(~N+l)... Xexp(-Smatt-Sgrav(e,-lt2f~d2~),
(4.10,
W(e) +
where Smart includes a free chiral fermion q/, scalars ¢ and anyon 0, Sgrav= Wghost, and the summation accounts for the all possible amounts of singularities corresponding to the pairs of anyons. One can find the equation for the renormalized coupling constant ~ff following refs. [ 10,11 ]: Ctot
3J¢" =C~,+C~+Co-28+ X+~--2 -6a'f'=0,
(4.11)
where
C¢=½, C~=D,
Co=+~(4-3Jd2).
(4.12)
If instead of off one introduces the SL (2, R) coupling constant K = - (o,~ff+4) for the matter fields, we will obtain eq. ( 1.1 ): ~+ D - ~ - ~ . 3 K ( K + 8 ) -
13+ ~
6
+6(K+2)=0,
(4.13)
which has the solution D = 3, K = 0.
5. Conclusion We have seen that the singularities of the two dimensional surfaces, immersed in 3D euclidean space which are essential for 3DIM become chiral anyons (in the case of D = 3 they are fermions with conformal spins 3). So, we can consider 3 D I M as 2D gravity model with one fermion ~u (spin ½), three scalars ~ (spin 0) and the 3 conformal spin anyon matter fields. Their c o m m o n central charge is equal to - 2. C~,~,~ =
- 2.
(SA)
It was argued in ref. [ 14 ] that in this case we have topological gravity [ 13 ] with N = 2 supersymmetry. But in the case of 3 D I M we need to modify the vacuum and fill it by spin 3 anyons. Let us mention here that as shown in ref. [ 17 ] the renormalization equation for charge K in the case of the SO (2) 2D supergravity model also admits the solution D = 3, K = 0.
Acknowledgement I would like to acknowledge the Theory Division of LAPP for hospitality, R. Stora for discussions and E. Buturovi6 for reading the manuscript.
References [ 1 ] A. Polyakov, Phys. Lett. B 82 (1979) 247.
[2] C. Domb, D.S. Gaunt and A.J. Guttmann, in: Phase transitions and critical phenomena, eds. C. Domb and M.S. Green, Vol. 3 (Academic Press, New York ).
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[ 3 ] A. Polyakov, Gauge fields and strings, Contemporary Concepts in Physics, Vol. 3; W. Dotsenko, thesis, Landau Institute ( 1981 ); E. Fradkin, M. Srednicki and L. Susskind, Phys. Rev. D 21 (1980) 2885; S. Samuel, J. Math. Phys. 21 (1980) 2806, 2815, 2820. [4] A. Polyakov, Phys. Lett. B 103 ( 1981 ) 211. [5] W. Dotsenko, Nucl. Phys. B 285 [FSI9] (1987) 45. [6] A. Casher, D. Foerster and P. Widney, Nucl. Phys. B 251 [FSI3] (1985) 29; C. Itzykson, Nucl. Phys. B 210 [FS6] (1982) 477. [7] A.R. Kavalov and A.G. Sedrakyan, Phys. Lett. B 173 ( 1986 ) 449; Nucl. Phys. B 285 [FS19 ] (1987) 264. [ 8 ] P. Orland, Strings in the three-dimensional Ising model, Boston preprint BU-HEP-88-2. [9] A.R. Kavalov, I. Kostov and A. Sedrakyan, Phys. Lett. B 175 (1986) 331; A. Sedrakyan and R. Stora, Phys. Lett. B 188 (1987) 442. [ 10] D. Karakhanyan and A. Sedrakyan, Phys. Lett. B 236 (1989) 140; B 260 ( 1991 ) 53. [ 11 ] A. Polyakov, Mod. Phys. Lett. A 2 (1987) 893; V.G. Knizhnik, A. Polyakov and A. Zamolodchikov, Mod. Phys. Lett. A 3 ( 1988 ) 819. [ 12] Y.H. Chen, F. Wilczek, E. Witten and B. Halperin, Intern. J. Mod. Phys. B 3 (1989) 1001, and references therein. [ 13 ] E. Witten, On the structure of the topological phase of two dimensional gravity, IAS preprint IASSNS-HEP-89/66; R. Dijkgraaf and E. Witten, Princeton/IAS preprint PUPT-1166, IASSNS-HEP-90/18; E. Verlinde and H. Verlinde, IAS/Princeton preprint IASSNS-HEP-90/40, PUPT- 1176 (1990). [ 14] J. Distler, Nucl. Phys. B 342 (1990) 523. [ 15 ] N. Dolbilin et al., Dokl. Akad. Nauk SSSR 295 ( 1987 ) 19. [ 16 ] V. Arnold, A. Varchenko and S. Goussein-Zade, Singularit6s des applications diff6rentiables (MIR, Moscow, 1986 ). [ 17] A. Polyakov and A.B. Zamolodchikov, Mod. Phys. Lett. A 3 (1988) 1213.
52
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3D Ising model and 2D induced gravity A.G. Sedrakyan Laboratoire d'Annecy-le-Vieux de Physique des Particules, IN2P3-CNRS, Chemin de Bellevue, BP 110, F-74941 Annecy-le-Vieux Cedex, France
Received 16 November 1990
The 3D Ising model ( 3DIM ) is investigated near the critical point. It will be shown that in this limit in 3DIM there are induced gravitational modes, which are described by the SL(2, ~) current algebra with coupling constant ~. The singularities of the twodimensional surfaces, immersed in 3D euclidean space are taken into account. Their contribution to the partition function is equivalent to the contribution of the anyons with charges + ½(K+ 4 ) and conformal spin ½[ 1+ ½(K+ 4 ) ] included in the problem under consideration. The Knizhnik-Polyakov-Zamolodchikov equation for the renormalized matter-field coupling constant K is changed and the solution D= 3, K= 0 is found.
1. Introduction
The interest in the 3D Ising model was raised by Polyakov [ 1 ] in the context o f constructing string theory in noncritical dimensions. It is known [ 2 ] that the 3D Ising model ( I M ) has a phase transition point o f second order and, so, there must be some continuum field theory corresponding to it at this point. On the other hand the 3DIM is dual to the 3D gauge IM with the 7~2 symmetry and it has been shown [ 3 ] that the partition function reduces to the summation o f the statistical weights of surfaces over all closed surfaces on the lattice. In this representation of the partition function of the 3DIM statistical weights o f the surfaces appear with the sign + 1 (sign factor q~ ( X ) ) which ensures cancellation o f self-intersecting surfaces (an analogous situation exists in the 2 D I M which is equivalent to the theory of free Majorana fermions). This fact lies at the basis o f the Polyakov hypothesis [ 1,4,5 ] that near the critical point (continuum limit) the 3 D I M is equivalent to string theory with some fermionic structure. There are many attempts to construct fermionic strings on the lattice [ 3-8 ] which corresponds to the 3DIM, but the construction suggested in ref. [ 7 ] essentially differs from the others in two aspects. First, there exists a naive continuum limit of the lattice action and second, the fermions are external and appear in the action quadratically. In ref. [7 ] it was shown that the naive continuum limit (lattice spacing a o 0) o f the lattice action is the Dirac action on a two-dimensional surface induced from the spinor structure of the 3D euclidean space (see ref. ( 1 4 ) ) . Developing this picture for the 3DIM, the authors o f refs. [9,10] have investigated quantum fluctuations o f fermions in the induced Dirac action and have found that because of anomalies o f the quantum theory there are additional modes, namely induced gravitational modes, which we need to add to the induced Dirac action to restore the original classical symmetries o f the 3DIM. The action obtained in ref. [ 10 ] for the induced 2D gravity modes in the chiral gauge is the W Z N W action for the SL(2, R) algebra with some coupling constant ~ , which coincides with the Polyakov results obtained in ref. [ 11 ] because o f the universality o f the anomaly calculus in two dimensions. It was shown [ 10,1 1 ] that the renormalized coupling constant ~ o f the SL (2, R ) W Z N W model is related to the dimensionality o f the external space (or to the number of degrees o f freedom o f the fields under consideraPermanent address: Yerevan Physics Institute, Br. Alikhanyan street. 2, SU-375 036 Yerevan 36, Armenia, USSR. Elsevier Science Publishers B.V. (North-Holland)
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tion) and a consistent theory exists for d < 1 or d > 25. The results of refs. [ 10,11 ] has been obtained assuming that we have immersed (the determinant of the metric tensor is not zero) surfaces. But in the 3DIM problem the singular surfaces (immersed besides the same discrete points, the end-points of the self-intersection lines of the surfaces) play an essential role, because just for these surfaces we can have a factor - 1 in the partition function. So we need to take into account the singularities of the surfaces and find their contribution in the 2D gravity action. In this sense the approach with the external fermions developed in refs. [ 7,9,10 ] has an advantage over the approach using internal fermions. In this article we investigate the contribution of the singularities of surfaces to the induced 2D gravity action. It will be shown, that like the approach developed in ref. [ 12 ] for high-Tc superconductivity, the existence of singularities is equivalent to the inclusion of the interaction with gravity anyons (particles with fractal statistics) in the problem under consideration. The statistics of anyons depends on the coupling constant ~ and the condition of restoration of the original symmetries of the 3DIM provides a relation between the coupling constant and dimension D of space-time, different from one, obtained in ref. [ 11 ]: ½+D-~-~.3K(K+8)-
13+ ~
6
+6(K+2)=0,
(1.1)
where K = - (~ff+ 4) is the SL (2, ~) coupling constant of the matter fields. This equation has a solution D = 3 and K = 0. Recently, very intriguing conjectures have been proposed [ 13 ], implying that there is an interesting connection between the topological and conventional formulation of 2D gravity, and it seems that in the case K = 0 [ 14 ] indeed we have one. But as we will see below, in the case of 3DIM we must consider the K = 0 theory in the modified vacuum (filled with anyons) which probably corresponds to another phase of the theory. In section 2 the basic definitions of the 3DIM will be given. Section 3 presents the gravity action induced by external fermions (in the problem). In section 4 we calculate the contribution of anyons (singularities of the surfaces in 3D space) to the 2D gravity action.
2. Basic definitions The gauge 3DIM is the system of spins an~,= + 1 placed on the links (n,/t) of the 3D regular lattice (n is the enumeration of the sites of the lattice and # = + 1, + 2, + 3 is the direction of the link attached to the site n) with the nearest neighbour interaction
H=J
~_,
an#trn+~.~tr#+u+ ~ _~an~.
(2.1)
plaquettes
It has been shown in refs. [ 3,6,7] that the partition function of the 3DIM can be represented as a sum of statistical weights of the free parametrized surfaces E:
Z=
~
all parametrized surfaces
exp(Z4) @ ( £ ) = ~ e x p ( - H / K T ) , {a}
(2.2)
where 2 = In th J / K T and A is the area of the surface E. The factor @(~) is taking values in 712={+ 1} and it is called the sign factor of the 3DIM. It ensures the cancellation of self-intersecting surfaces in the statistical sum and it is the origin of the fermionic structure (without this factor we will have a bosonic string). In ref. [ 3 ] it has been proved that when we take @(E) = ( -- l ) t where l is the length of the self-intersection line, then there is a correct cancellation of self-interacting surfaces. Near the critical point one just has a functional integral over the parametrized surfaces X ( ~ ~2) and we need to 46
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represent qb(X) through the differential geometric characteristics of the E. This was done in ref. [7 ] #1 In eq. (2.2) only the surfaces with an open line of self-intersections can have the minus sign (because the closed ones have even length on the regular lattice). The open self-intersection lines have end-points and it has been shown by Whitney [ 16] that they are the only stable singularities (at this point the induced metric g~a=O,~XOaXis degenerate) at the maps Y,2-->E3 (fig. 1 ). They are stable by the fact that the linking number of the strip C surrounding the point 0 is the topological invariant and takes the value - 1. One can represent the linking number by the formula (2.3)
~ ( C ) = ½Tr P e x p ( ! S - ' d S ) ,
where the matrix S is defined as follows. Let us consider at this point the orthonormalized vectors Ya (~) tangent to the surface, as well as the normal n ( O ,
ya(~)=Oo, X'eT,
c~=1,2,
a=l,2,
n(~)=i~hxy2,
(2.4)
where the e J are the zweibeins corresponding to the induced metric g~p:
eao,epa=g,~a=O,~XOaX.
(2.5)
Then S is the operator which rotates the matrices y~= Ya"tr and n = n. tr (tr are the Pauli matrices) into oz.
Sy,,S-l=aa,
SnS-l=a3
(2.6)
and
S-I dS=~(yadYa +n dn)=lrab[ya, Yb] +~ha[yan],
a, b= 1, 2,
(2.7)
where 1-'ab=l-'~b d~ '~ is a usual spinor connection in 2D gravity and ha=h~ d~ ~, the second quadratic form of the embedded surface. F "b and h satisfy the Peterson-Codazzi (PC) equations a
dFab+hahb=O, dha-l-'abhb=O.
(2.8)
The field S-1 dS looks like a fiat gauge connection and one can think that ~o(C) in formula (2.3) is equal to one. But when C surrounds the singularity 0, where d ( S - 1 dS) - S - ~ d S ^ S - t dS= 2in82 (~)az d2~,
(2.9)
#1 In ref. [ 15 ] a slightly different definition of the sign factor ~ ( X ) for surfaces on the lattice is provided and a detailed analysis of its properties is carried out.
x~
Fig. 1.
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then we have ~ ( C ) = point 0 (fig. 1 )
X~=~,~z,
9 May 1991
1. This can be shown using Whitney's [15 ] parametrization of the surface near the
X3=¼~2.
X2=~2,
(2.10)
It is easy to see (by use of eqs. (2.5), (2.7), (2.9) ) that near the point 0
U
h. . .1.. . g = t c d - (- ¢d 'a~)c2 .
(2.11)
.h 2 i.n g = 0
.
Now let us cover the closed surface in the lattice by systems of closed curves which cross the links of the surface once. This can be achieved by drawing two parallel lines, connecting the midpoints of neighbouring links on each plaquette of the surface (for details see ref. [7] ). One thus obtains a class of coverings (A is the area of the surface). It was shown in refs. [ 7,15 ] that the product
2A/a~
q~(Y) =M,q~(C,) = ( - 1 )'
(2.12)
2A/a2
does not depend on types of coverings and is equal to ( - 1 )l. In ref. [7 ] the lattice action for the fermionic string, which reproduces the sign factor (2.12) and has the following classical continuum limit (lattice spacing a ~ 0 ) :
S=½i f
d2~x/g ~({) ( ~ 0 ~ - 0"Y")¢({) '
(2.13)
has been found. In eq. (2.13), q>, ~t are 3D fermionic fields living on the two-dimensional surface X(~), g = det 13,,X 0~XI and y'~, a = 1, 2 are defined as follows: ~ = 0,~X'tr.
(2.14)
This is the induced Dirac action. So, the expected partition function for the 3DIM near the critical point is
Z= f ~X~/exp(-S-i~Z y dZ¢x/g) .
(2.15,
The action (2.9) is obtained in a classical continuum limit. In a quantum continuum limit, in general we can get some additional terms (anomalies) in the action, which we have to add to the matter field action (2.13 ) in partition function (2.15). The general principle to follow in order to find these terms is to find the action induced by anomalies of the quantum fluctuations of matter fields and add it to (2.13) with renormalized coupling constants ensuring the original reparametrization (diffeomorphism) and Weyl invariances.
3 . 2 D induced gravity
The 2D gravity induced by quantum fluctuations of external 3D fermions which have the action (2.13) has been investigated in refs. [ 9,10]. There it is assumed that the 2D surfaces X(~) do not have the singularities mentioned above. It has been shown that the effective action for fermion field fluctuations looks as follows [ 10 ]:
"(f (Fa-½iTrtY3,~-l~a)L)(l'#-½iTra3)~-l~#,~)ga#v/-gd2~
W ( X ) = ~--~ ½
Z
+i ~ FdF-½ f F~Tr(a32-'O~2)~Pd2~),
(3.1)
T-t
where F~ = 48
F~bEabis the spinor connection for 2D gravity, the variable 2 = exp (ia3~o)~ SO (2)
is the SO (2) ro-
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9 May 1991
tation matrix in the plane tangent to the surface Z, F=Fo,d(~'+Ftdt, a one-form in the manifold Zt extended in one dimension, the boundary of which is the surface X(~) _=Z under consideration, and o,ugis the coupling constant. The second terms in (3.1) is a multivalued functional, whose integrand is a density of the Hopf topological invariant, the map $3--,S z. Thus the partition function, improved by contributions from anomalies, becomes
z= f ~X~exp(-S-zt2 ~d2,x/~-W(X)). Now one needs to change the integration parameter X(~) in Z and introduce zweibeins from the relation
(3.2)
e. a. It is easy to see
e,~aeap= O,X OaX
(3.3)
that the integration measures of e,~a and X are related as follows: ~ X = ~E,~ 4. (det A~<')) --DI2,
(3.4)
where Ao(e) is the 0-form laplacian, which is equivalent to the contribution of free bosonic fields 43 into the partition function
z= f ~e,,'~'~'~exp(-S(e) - & .... -W(e)-lt2 f dZ~v/g).
(3.5)
In refs. [ 9,10 ] it has been shown that when in the induced gravity action (3.1) we fix the gauge det e,~ 4= 1
(3.6)
and 0~P
×
D
0,, (v/gJ~) = 0 ,
(3.7)
where J~ = e'~rgrp/x/g is the complex structure and f a are isothermal coordinates, we obtain the usual WessZumino-Novikov-Witten (WZNW) action with the SL(2, ~) group elements e,~a, together with some term which ensures the gauge invariance under the left SO (2) transformations of the e, a. Thus, we have an SL (2, N ) right current algebra action with coupling constant o~ together with a free spin ½fermion, three scalars and ghost actions.
4. Inclusion of singularities In order to find the contribution of singularities of the surfaces to the partition function of 3DIM, one needs to rewrite the Hopf invariant term in (3.1) in terms of the second quadratic form ha((). By use of the PC equations (2.8) we have
FdF=~-~n hadh a.
(4.1)
Zt
The connection h a contains regular (h a) and singular (h a ) parts. When we have 2N singularities in the surface, located at points ~k, then the singular parts h a are defined by the equation 49
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h~ = ~ C d k=,
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(¢_¢~)~ I¢-¢kl 2
d~ c
(4.2)
and 2N
d^h~ =2n ~,, 62(~--~k)d2~.
(4.3)
k=l
Then in the Hopf term in the effective action we have
f hadh ~= S h 7 d h a + 2 f hadhf+ f hsdhs= f hfdhf+ k~= l 4n~ h~ d t +
~t
Et
Zt
~t
f hsdhs.
(4.4)
Ck
In the foregoing we used eq. (4.3) and the second term in (4.4) transformed into the integrals along the contours Ck, which connect 2N singular points. So in the partition the function Z we have terms coming from singularities: Z~exp
_i.
4in~m f hi d t + W
f h~ dh~
).
(4.5)
C,n
The first term in (4.5) is obtained using Whitney parametrization (2.10) and the particular disposition of singularities in E 3. The covariant form of the corresponding terms is
Z-½Pexp(-i -2~=1f ~r u h~'b'ab)exp(-~n~ f hsdh~) "
(4.6)
C,n
It is possible to reproduce the first term in (4.6) introducing the new fermionic field 0 in three-dimensional space, which interacts with the field h a~b~.abwith the action ,~= iYd f d2~ x//g ~ya( Oa+ ½ihaTb'ab)O
(4.7)
and considering the correlation functions of 2N fermionic fields placed at the points ~ (n = 1..... 2N)
I
dEal ... d2~N (x/~)E~V(/7~ (~1)...~7~u(~N)O~(~+L)...O,~(~EN
) e-~) .
(4.8)
As for the second term in (4.6) with singular h~, the situation is similar to the one considered in ref. [ 12] in connection with high-temperature superconductivity. If one considers the fermionic fields 0 interacting with the U( 1 ) gauge field h~, the action of which is the Chern-Simons (CS) action ( X / 8 n ) f h~dh~,then the solutions of the classical equations of motion tell us that fermions create magnetic vortices (by eq. (4.2)) and they can be considered as free particles but with different statistics. They become anyons with _+ charges, which corresponds to the two end points of the line of self-intersection of the surface (in this case anyons interact with gravity). The change of statistics is defined by the charge of fermions (in this case q = + ½o,~) and by the U ( 1 ) CS action coupling constant ½# = ~ff/8n. So, the spin of the anyons is [ 12 ] 1 q2 As= ~ 2n---~, s = l ( l + ½ X )
.
Finally, the partition function for 3DIM is found to be
50
(4.9)
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Z= N=~ 1 f ~eaa~Ct~¢~o f x/F~ld2~l...N/g2Nd2~2N~°tl(~l)...OOtl(~N+l)... Xexp(-Smatt-Sgrav(e,-lt2f~d2~),
(4.10,
W(e) +
where Smart includes a free chiral fermion q/, scalars ¢ and anyon 0, Sgrav= Wghost, and the summation accounts for the all possible amounts of singularities corresponding to the pairs of anyons. One can find the equation for the renormalized coupling constant ~ff following refs. [ 10,11 ]: Ctot
3J¢" =C~,+C~+Co-28+ X+~--2 -6a'f'=0,
(4.11)
where
C¢=½, C~=D,
Co=+~(4-3Jd2).
(4.12)
If instead of off one introduces the SL (2, R) coupling constant K = - (o,~ff+4) for the matter fields, we will obtain eq. ( 1.1 ): ~+ D - ~ - ~ . 3 K ( K + 8 ) -
13+ ~
6
+6(K+2)=0,
(4.13)
which has the solution D = 3, K = 0.
5. Conclusion We have seen that the singularities of the two dimensional surfaces, immersed in 3D euclidean space which are essential for 3DIM become chiral anyons (in the case of D = 3 they are fermions with conformal spins 3). So, we can consider 3 D I M as 2D gravity model with one fermion ~u (spin ½), three scalars ~ (spin 0) and the 3 conformal spin anyon matter fields. Their c o m m o n central charge is equal to - 2. C~,~,~ =
- 2.
(SA)
It was argued in ref. [ 14 ] that in this case we have topological gravity [ 13 ] with N = 2 supersymmetry. But in the case of 3 D I M we need to modify the vacuum and fill it by spin 3 anyons. Let us mention here that as shown in ref. [ 17 ] the renormalization equation for charge K in the case of the SO (2) 2D supergravity model also admits the solution D = 3, K = 0.
Acknowledgement I would like to acknowledge the Theory Division of LAPP for hospitality, R. Stora for discussions and E. Buturovi6 for reading the manuscript.
References [ 1 ] A. Polyakov, Phys. Lett. B 82 (1979) 247.
[2] C. Domb, D.S. Gaunt and A.J. Guttmann, in: Phase transitions and critical phenomena, eds. C. Domb and M.S. Green, Vol. 3 (Academic Press, New York ).
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[ 3 ] A. Polyakov, Gauge fields and strings, Contemporary Concepts in Physics, Vol. 3; W. Dotsenko, thesis, Landau Institute ( 1981 ); E. Fradkin, M. Srednicki and L. Susskind, Phys. Rev. D 21 (1980) 2885; S. Samuel, J. Math. Phys. 21 (1980) 2806, 2815, 2820. [4] A. Polyakov, Phys. Lett. B 103 ( 1981 ) 211. [5] W. Dotsenko, Nucl. Phys. B 285 [FSI9] (1987) 45. [6] A. Casher, D. Foerster and P. Widney, Nucl. Phys. B 251 [FSI3] (1985) 29; C. Itzykson, Nucl. Phys. B 210 [FS6] (1982) 477. [7] A.R. Kavalov and A.G. Sedrakyan, Phys. Lett. B 173 ( 1986 ) 449; Nucl. Phys. B 285 [FS19 ] (1987) 264. [ 8 ] P. Orland, Strings in the three-dimensional Ising model, Boston preprint BU-HEP-88-2. [9] A.R. Kavalov, I. Kostov and A. Sedrakyan, Phys. Lett. B 175 (1986) 331; A. Sedrakyan and R. Stora, Phys. Lett. B 188 (1987) 442. [ 10] D. Karakhanyan and A. Sedrakyan, Phys. Lett. B 236 (1989) 140; B 260 ( 1991 ) 53. [ 11 ] A. Polyakov, Mod. Phys. Lett. A 2 (1987) 893; V.G. Knizhnik, A. Polyakov and A. Zamolodchikov, Mod. Phys. Lett. A 3 ( 1988 ) 819. [ 12] Y.H. Chen, F. Wilczek, E. Witten and B. Halperin, Intern. J. Mod. Phys. B 3 (1989) 1001, and references therein. [ 13 ] E. Witten, On the structure of the topological phase of two dimensional gravity, IAS preprint IASSNS-HEP-89/66; R. Dijkgraaf and E. Witten, Princeton/IAS preprint PUPT-1166, IASSNS-HEP-90/18; E. Verlinde and H. Verlinde, IAS/Princeton preprint IASSNS-HEP-90/40, PUPT- 1176 (1990). [ 14] J. Distler, Nucl. Phys. B 342 (1990) 523. [ 15 ] N. Dolbilin et al., Dokl. Akad. Nauk SSSR 295 ( 1987 ) 19. [ 16 ] V. Arnold, A. Varchenko and S. Goussein-Zade, Singularit6s des applications diff6rentiables (MIR, Moscow, 1986 ). [ 17] A. Polyakov and A.B. Zamolodchikov, Mod. Phys. Lett. A 3 (1988) 1213.
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