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Laboratoire de Physique Th´eorique ENSLAPP †

arXiv:hep-th/9212032v1 4 Dec 19923D-Ising Model as a String Theoryin Three Dimensional Euclidean SpaceA. Sedrakyan∗Laboratoire de Physique Th´eorique ENSLAPP †Chemin de Bellevue, BP 110, F - 74941 Annecy-le-Vieux Cedex, FranceAbstractA three dimensional string model is analyzed in the strong coupling regime.The contribution of surfaces with different topology to the partition function isessential.

A set of corresponding models is discovered. Their critical indices,whichdepend on two integers (m, n), are calculated analytically.

The critical indices ofthe three dimensional Ising model should belong to this set. A possible connectionwith the chain of three dimensional lattice Pott’s models is pointed out.ENSLAPP-A-410/92November 1992∗permanent address : Yerevan Physics Institute, Br.

Alikhanian, st.2, Yerevan 36, Armenia†URA 14-36 du CNRS, associ´ee `a l’E.N.S. de Lyon, et au L.A.P.P.

(IN2P3-CNRS) d’Annecy-le-Vieux0

1IntroductionThe understanding and construction of strings in noncritical dimensions ispresently attracting interest of string theorists. For example, the string theoryin four dimensional space-time is probably a good framework for a theory ofstrong interactions of the elementary particles.

This framework first appearedat the end of seventies,motivated by the dual resonance approach [1]. Moreover,the three dimensional Ising model (3DIM), being a gauge theory with the groupZ2, is, in fact, a theory of random surfaces.

Polyakov has put forward the idea[2] of its equivalence with a fermionic string near the point of the second orderphase transition. An important element in understanding of the 3DIM as a stringtheory is the fact that closed surfaces with different topologies contribute to thepartition function, and that the module of the string coupling constant is one.Thus, we have a string in the strong coupling regime.There have been many attempts to construct a fermionic string on the lattice[2]-[9] which corresponds to the 3DIM, but the construction suggested in [8]differs essentially from the others.

There, a naive continuum limit (lattice spacinga →0) of the lattice action exists and three dimentional Dirac fermions appearin the action quadratically. This fact allowed the author to develop in [10] theidea of equivalence of 3DIM with the theory of some kind of matter fields whichinteract with 2D quantum gravity.The essential ingredient of the approach considered in [10] is the evaluation ofthe contribution of the two dimensional manifolds, immersed into 3D-euclideanspace and singular at the end points of selfintersection lines, to the functionalintegral over all surfaces (partition function of the 3DIM).

The singular surfacesare essential for the 3DIM, because some of them appear in the partition functionwith the weight -1, ensuring cancellation of the contribution of a part of surfaces.On the other hand, some of singular surfaces are nothing else but unorientedsurfaces immersed into 3D-euclidean space (sphere with M¨obius cups).It happens that the spins of Dirac fermions in the presence of singularities aremodified and, also, that the original vacuum must be changed by filling it withthem. The change of the spin of the fermions diminishes the central charge ofthe matter fields and that is why the KPZ equation [11] for the SL(2.R) couplingconstant of the 2D quantum gravity, which is the condition of restoration of thereparametrization invariance has a solution for D = 3.In this article I shall further develop the approach presented in ref [10] cor-recting at the same time the errors made in that article.

It will be shown thatthe matter sector of the model, consists of three scalars (surface coordinates ⃗X)and a spin (0,1) fermionic (b, c) system (corresponding to singularities) with theirtotal central charge equal to one. In conformal gauge the 2D-gravity representedby Liouville acton and the ghost fields has to be introduced.

The sum over allpossible singular surfaces (and at the same time over all topologies) in the par-tition function induces a new term in effective action of the model while keeping1

the vacuum unchanged, namely, inteaction of Liouville field with the spin (0,1)bosonized fermionic (b, c) system. The true vacuum now differs from the originalone which corresponded to a simple topology.

The set of models is discovered,for which the contribution of the singular surfaces in partition function is essen-tial. Then the scaling behaviour will be analyzed and the critical indices, whichdepend on two integers (m, n), analitically calculated.

The critical indices of the3DIM should belong to this set.It is known [11],[22], that c = 1 string theory lies on the boundary betweenthe ”weak” and ”strong” interacting phases of 2D quantum gravity. Perhaps theapproach presented here may open a way to cross the c = 1 barrier.22D-gravity structure of the 3DIMIn order to reveal the structure of 3DIM near the critical point we shall workwith the lattice formulation of the fermionic string, where the three dimensionalDirac fermions ψL and ψR are placed in the middle points of links of the latticeas explained in ref [8].

The general ideology is that of ref. [10].As it was shown in ref.

[8], this action reproduces exactly the 3DIM parti-tion function at arbitrary temperature, along with the correct sign-factor for thesurfaces. The classical continuum limit (lattice spacing a →0) isS0 = λ · arrea + i2Zd2ξ√g ¯ψ(ξ)(γα⃗∂α −⃗∂αγα)ψ(ξ).

(2.1)In eq. (2.1) ¯ψ, ψ are 3D Dirac fields living on the two-dimensional surface ⃗X(ξ) ;gαβ = ∂α ⃗X · ∂β ⃗X is an induced metric , γα = ∂α ⃗X · ⃗σ (⃗σ are the Pauli matrices)and λ =1a2 ln th J/KT, where J is the 3DIM coupling constant and T is thetemperature.Here, the following remark is in order.

In ref. [8], taking classical continuumlimit of the lattice action, we did not distinguish left (or right) fermions, placedat the opposite links of the plaquettes (see fig.3 in [8]) and therefore translationalinvariance of the theory was broken.

This distinction should be made, as onecan see by making diagonalization of the action in the momentum space of a flatmanifold. However, we will then have a complicated expression for the actionin the continuum limit even in a flat case, which is free of anomalies, but likethe Nambu action has some other problems.

By identifying some of the fieldswe obtain a simple massless action (2.1) which now has gravitational (or SO(3)rotational) anomalies, absent in 3DIM. Following the general procedure we cancorrect the situation by adding the 2D-gravity action induced (due to anomalies)by quantum fluctuations of ψ (and also of ⃗X) and restore the original symmetriesof the theory 1.

In principle, one is doing the same, when replacing the Nambuaction for bosonic string by the Polyakov’s action.1I thank A. Polyakov for a criticism at this point.2

Therefore, the candidate for the action of 3DIM near the critical point isS1 = λ · area + i2Zd2ξ√g ¯ψΩ−1σaeαa(∂α + Γα)Ωψ + Wgrav( ⃗X) + S(ghost) (2.2)In (2.2) we have made an algebraic transformation of the fermionic action(2.1) (see [12] for details).Here, eaα and Γα are the zweibeins and SO(2) spinorconnection, corresponding to induced metric, Ωis the element of Spin(3) anddefines the matrix which rotates ortonormalized Frene basic vectors of the surfaceinto the flat ones.The 2D-gravity action Wgrav( ⃗X) with the SL(2, R) couplingconstant k (or with the charge QL in conformal gauge) should be added, ensuringrestoration of all anomalous symmetries of the theory. This corresponds to theKPZ [11] equation Ctot = 0.Now we should investigate the contribution of Whitney singularities of thetwo dimensional surfaces, immersed in 3D euclidean space in the string functionalintegral.

Let us represent ΩasΩ= Ωs · Ωr,(2.3)where Ωr is a regular rotation matrix, making a surface flat, but not affectingsingularities and Ωs is the singular part of Ω.Since the rotation and reparametrization invariances are restored by additionof Wgrav, we can represent the fermionic action in the form (Ωr dissapears andmetric becomes flat)i2Zd2ξ ¯ψΩ−1s σa∂aΩsψ. (2.4)It is easy to calculate Ωs.

In the case of flat surface the Whitney singularity lookslike in Fig.1.Fig. 13

One can parametrize the surface as followsX3 = 0,w = z2,w = X1 + iX2,z = ξ1 + iξ2.The straightforward calculations, by use of formula (2.7) in [10], givesΩ−1s ∂Ωs=14zσ3Ω−1s ¯∂Ωs=−14¯zσ3(2.5)andΩs = z|z|!−1/2,(2.6)which means that there is a magnetic flux in the point.Then, redefining the fieldsb = ¯ψLΩ−1s , c = ΩsψL(2.7)one obtains a free action1πZd2zb∂c + c.c. (2.8)The spins of b, c fields are equal to 12(1 ± Q), where Q = 1.2Thus, in the presence of vortices (singularities) the conformal spin of fermionschanges and the correspodning central charge of the conformal theory becomes1 −3Q2.Besides modifying the spin of the fermions, as was shown in [10], one shouldfill the vacuum state of the system with the (b, c) pairs and the correspondingcorrelator will appear in the expression for free energy.This means that when acting by (b, c) operators on the vacuum state we createthe singularities on the sphere transforming it into the surface with a differenttopology.Next, following ref.

[10], instead of induced metric gαβ = ∂α ⃗X∂β ⃗X, we in-troduce the independent zweibeins eaα and fix the gauge. The ghost fields willappear.

For our purpose, to calculate the critical indices of the system, the con-formal gauge gαβ = eϕδαβ and the technic, developed by F. David [14] and J.Distler, H. Kawai [15] (DDK), seem to be most appropriate.2Here, I would like to mention that in ref. [10] a mistake of taking Q = k/2 has been made.Along with a renormalization of the coupling constant k the field h has also to be renormalized.This should be done consistently with the fact, that the topological origin of the sign-factorof 3DIM (which takes values ±1) prohibits its renormalization.

This imposes the conditionqk2dhren ≃δ(2)(ξ), which implies that Q = 1, instead of Q = k/2. Besides, there is no need tointroduce an additional fermion.4

Finally, the free energy F of the 3DIM near the critical point isF=∞XN=0ZD ⃗XD(b, c)DghostDϕ ·1(N! )2 ·Zd2zb(z)Zd2z′c(z′)Ne−S, where(2.9)S = 12πZd2z∂⃗X ¯∂⃗X + 1πZd2zb¯∂c + c.c.

+ S(ghost)+ 12πZd2z(∂ϕ¯∂ϕ −14QL√g ˆRϕ) + µZd2zeAϕ(2.10)µ = λ0 −λ,A = −QL2 +qQ2L −8,In eq. (2.9), N is the number of pairs of singularities of the surface, the presenceof the factor (N!

)2 is a consequence of having N identical b and N identical cfields, λ0 is bare cosmological constant and QL is the Liouville background charge.The expression for QL, in terms of SL(2, R) couling cosntant k, isQL2 =kk + 2 −2k −1. (2.11)The KPZ equation [11]ctot = 3 + (1 −3Q2) −26 + (1 + 3Q2L) = 0,(2.12)which is the condition for restoration of all anomalous symmetries, can be solved.For Q = 1 we obtain k = −3 and QL = 2√2.

The corresponding central chargeof the matter fields of the theory is equal to one.3Scaling properties. Specific heat index αIn order to calculate the critical indices consider the bosonized anticommuting(b, c) system [16].

The action is14Zd2zb¯∂c + c.c. = 12πZd2z(∂φ¯∂φ −i4Q ˆRφ)(3.1)where b = e−iφ(z) and c = eiφ(z).As they appear in vacuum (see (2.9)), the (b, c) fields are polarized by thegravitational filed ϕ and also by φ (selfpolarization),so that the ground state isreparametrization invariant.

After this dressing,b = eBϕeiDφ,c = e¯Bϕei ¯Dφ,(3.2)5

the conformal dimensions of both b and c field should be (1,1):−B(B + QL)2+D(D + Q)2= 1−¯B( ¯B + QL)2+¯D( ¯D + Q)2= 1(3.3)Let us introduce the string coupling constant Λ, by writing Λ2N in front of theexponent in (2.9) and take the sum in F over (b, c) pairs. In order to simplify theexpression for the effective action Seff we change the sum over N of the series in(2.9) into the product of two independent exponential series for b and c. One cando that, because the condition of mutual cancellation of the (b, c), cosmologicaland background charges (see (3.5)) will select the equal number of b and c fieldsin the expression for the path integral (2.9).

Then, from eq. (2.9), we obtain:Seff = S + ΛZd2zeBϕeiDφ + ΛZd2ze¯Bϕei ¯Dφ(3.4)This action describes a Liouville theory interacting with a bosonized fermionic(b, c) system.

What is the geometrical meaning of the presence of b and c fields inthe vacuum ? Appearence of one pair of (b, c) in front of exp in (2.9) is equivalentto a change of vacuum charges Q(1−g) and QL(1−g) (g is the genus of Riemannsurface) of the fields φ and ϕ to Q(1−g+D+ ¯D) and QL(1−g+B+ ¯B) respectively,and can be interpreted as a change of g.There are two types of singularities of Riemann surfaces in 3D-space.

Onechanges the topology (Fig.2a) and the other (Fig. 2b) does not.Fig.2In 3DIM the surfaces with all topologies are present in the partition function.Therefore, the expression (3.4) should contain the dressed (b, c) system, whoseappeareance in the vacuum is equivalent to a change of genus by 12 (as in Fig.2a).6

Let us take n pairs of (b, c) fields, which, together with m cosmological terms,create the M¨obius cap on the manifold, as in Fig. 2a, which corresponds to thefollowing equationsn(D + ¯D) = −Q/2n(B + ¯B) + mA = −QL/2.

(3.5)Equations (3.3) and (3.5) determine B, ¯B, D, ¯D as functions of integers m andn.Following DDK [14, 15], we can evaluate specific heat index α by a simplescaling argument. Consider the scaling transformationsφ →φ + φ0,ϕ →ϕ + ϕ0A ,µ →µe−ϕ0andΛ →Λe−x(3.6)with constant φ0, ϕ0.

In order to find the scaling behaviour we should demandthe invariance of the interacting terms in eq. (3.4), which implies−x + BAϕ0 + iDφ0 = 0−x +¯BAϕ0 + i ¯Dφ0 = 0.

(3.7)Solving these equations, along with (3.3) and (3.5), one findsiφ0/ϕ0 =QA2(−QLA +mn−1/2) =12(2 +mn−1/2)(3.8)andxϕ0= p = 12n1 −m −14(2 +mn−1/2). (3.9)Then, using the scaling transformations, it is not hard to see thatSeff →Seff −QAϕ0 −iQφ0.

(3.10)EquivalentlyF(µe−ϕ0, Λe−x) = e( QA +Q iφ0ϕ0 )ϕ0F(µ, Λ),(3.11)i.e.F(µ, Λ) = µ−( QA +Q iφ0ϕ0 )F 1,Λµx/ϕ0!= µ2−α ˜F Λµp!. (3.12)Thus, the specific heat (or string susceptibility) index is (use also eq.

(3.8))3:α =12(2 +mn−1/2)(3.13)3For a review of the definition and old numerical calculations see, for example, [17].7

and Λ ∼µp. Since m ≥1 implies p < 0, we are in the strong coupling regime.If we introduce dressed (b, c) operators, which do not create topology (case ofFig.2b) and which depend on integers (m′, n′), their scaling equations (eq.

like(3.7)) should be consistent with eq. (3.7).

It is easy to find that the conditionof consistency is m′n′ =mn−1/2 and to see that the specific heat index α does notchange.4Correlation length index νIn order to calculate ν one should investigate long distance behaviour of thecorrelator of some operators Φ( ⃗X), placed at distance ⃗X. Then the correlationlength and its index ν are defined as followsR d ⃗X( ⃗X)2 < Φ( ⃗X)Φ(0) >R d ⃗X < Φ( ⃗X)Φ(0) >≡ξ2 ∼µ−2ν.

(4.1)This expression shows that we need to find the anomalous scaling dimensionof ⃗X. In the string picture the correlation length of operators in an externalspace is calculated by insertion of the operators of the typeΦ( ⃗X, ·) =Zd2ξΦ(·)δ(D)( ⃗X −⃗X(ξ))(4.2)into the partition function integral.To find the anomalous scaling dimension of ⃗X and then to calculate the cor-relation length index ν it is enough to consider the world-sheet operator 1, withthe conformal dimension 0.

It seems to us, that in the presence of backgroundLiouville and (b, c) charges QL and Q that is1 = e−QLϕe−iQφ(4.3)Imposing the scale invariance of thee−QLϕe−iQφ · δ(3)( ⃗X −⃗X(ξ))(4.4)under the transformations (3.6) along with ⃗X →ρ ⃗X and using the scaling argu-ments employed in the previous section we obtain:ν = 2 −α3. (4.5)5Discussion of resultsWe have obtained sets of critical indices which depend on two integers, m andn.

The common origin of all of them is the contribution of singular surfaces to8

the partition function of random surfaces. Among them one can find the criticalindices of 3DIM.An enormous and diverse work has been done on numerical calculations of3DIM critical indices.

A summary can be found in an article by J.-C. Le Guillouand J. Zinn-Justin [18].

The results quoted in this reference are:α = 0.11,γ = 1.239,ν = 0.631,β = 0.327,η = 0.0375(5.1)We can pick a choice of integers m and n, for example m = 4, n = 2, forwhich α ∼0.107, ν ∼0.631 and obtain the results within experimental errors.However, it seems to us that the case m = 1, n = 1, for whichα = 1/8,ν = 0.625(5.2)is more natural. These values coincide with some old estimates [17].

They alsoagree with a recent Monte Carlo renormalization group study of indices [19],where ν = 0.624(2) and η = 0.026(3) (we see, that the hardest index η becomesessentially smaller than one in (5.1)).The case m = 0, n = 1 with the indicesα = 1/4, ν = 7/12(5.3)is in good agreement with the three dimensional indices for a self-avoiding walk(SAW)problem [20]. In the limit, where n = 1, m →∞the indices get the valuesα = 0andν = 2/3,(5.4)which are in a good agreement with the 3D U(1)-model [13, 21].

One can con-jecture that the series n = 1 and arbitrary m corresponds to 3D Pott’s models.At the end of this analysis I would like to add a few words about magneticsusceptibility index γ. In order to calculate the magnetic susceptibility γ we needto analize the response of the system to the presence of the magnetic field h, i.e.to the presence of the additional term, h Pi σi, in the original lattice action ofthe 3DIM.

Then γ is defined by the anomalous scaling dimension of spin field σ.It is possible to develope a rough arguments (see also [17]) which gives the resultγ = 2ν (equivalently η = 0), but, to obtain a precise answer, one should representthe original spin variable σ in terms of stringy fields and calculate complicatedcorrelators, as suggested in ref. [2].

In principle, the direct and more precisenumerical calculation of η can be decisive for final determination of the criticalindices.Here we have calculated the indices starting from the spherical topology, butall nonorientable surfaces are taken into account by use of singularities. As fororientable ones, the carefull analysis of operator algebra is needed.The firstimpression is that the consistent introduction of operators which increase thegenus for one, does not change the scaling behaviour.9

Presumably, like two dimensional minimal models interacting with gravity,this set of models also corresponds to some topological field theory. This is aninteresting question and may be relevant to finding a correct topological definitionof the sign-factor of the 3DIM.In conclusion, I would like to make the last remark.

It seems to me thatthe picture presented here can be interpreted in the following way.We havea matter field ⃗X interacting with the ordinary Liouville field (representing thefluctuations of metric of 2D gravity) and also interacting with another ”Liouville”field originating from the bosonized fermionic (b, c) system and representing the”fluctuations” of topology. (There is also an interaction between two Liouvillefields).

It is tempting to conjecture that this interpretation of these two Liouvillefields may be valid for a general noncritical strings.I would like to thank for discussions J. Ambjorn, E. Buturovi´c, D. Gross,A. Kavalov, I. Klebanov, Y. Kogan, V. Kazakov, J.-C.

Le Guillou, M. Mkrtchian,H. Verlinde, E. Verlinde and the theory division of LAPP, where this work wasfinished.

I especially acknowledge A. Polyakov for discussions during many yearsand for criticism. I would especially like to thank J. Distler, conversations withwhom stimulated my interest to the DDK approach to 2D-gravity.10

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