# Basics Polyakov’s Quantum Surface Theory on the Formalism of Functional Integrals and Applications

Authors: Luiz C. L.  Botelho
Journal: ISRN High Energy Physics
 (2012)

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## Abstract

We propose an alternative view to the covariant Polyakov’s string path integral.
In our new approach we clarified the role of the Liouville model on string
theory for Q.C.D. and Dual Models. In the appendices, we present additional material,
difficult to find in the specialized path integral literature on the detailed evaluation of the
(Q.C.D.) fermion determinant, the path integral proof of the Atiyah-Singer index theorem
on Riemannian Manifolds, and the role of the1/D expansion on Polyakov’s String path
integral, all containing important new results, insights on theses topics.


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## Body

## 1. Introduction

Since its inception seventy years ago, nonabelian gauge theories have been shown to be the most promising mathematical formalism for a realistic description of strong interactions and even formulated on its supersymmetric version; it became an attractive attempt for unify Physics.In strong interaction Physics the picture image of a mesonic quantum excitation is, for instance, a wave quantum mechanical functional assigned to a classical configuration of a space-time trajectory of a pair quark-antiquark bounding a space-time nonabelian gluon flux surface (of all topological genera) connecting both particle pairs: the famous t’Hooft-Feymman planar diagrams [1].It appears thus appealing for mathematical formulations to be considered directly as dynamical variables or wave functions in this Faraday line framework for nonabelian gauge theories, the famous quantum Wilson Loop, or (quantum) holonomy factor associated to a given space-time Feynman quark-antiquark closed trajectoryC in a SU(Nc) Yang-Mills quantum field theory. Namely,
(1.1)ψ[C]=〈W[C]〉=〈1Nc{TrSU(Nc)(c)[ℙ(exp ig∮C‍Aμdxμ)]}〉.
Here 〈⋯〉 is defined by the Yang-Mills quantum path integral and Aμ(x)=Aμc(x)λa denotes the quantized Yang-Mills (Gluon) connection.It thus searched loop space dynamical equations (at least on the formal level without considering those famous ultraviolet “renormalization problems”) for the new wave function equation (1.1). However, Lattice Yang-Mills theories have indicated that “string solutions” with a symbolic structure form
(1.2)ψ[C]=∑S,∂s=𝒞‍{[exp(-12πα'A(S))]ΦQCD[C]}
with ∑S;∂S=𝒞 denoting the Feynman-Wiener continuous sum over all surfaces S, with all topological genera and bounded by C; α' is the fundamental constant of strong force physics called the Regge slope parameter [1]; A(S) is the area functional “evaluated” at the sample surface S; Φ[C] is a functional related probably to the existence of a (neutral) nonabelian 2D intrinsic fermions structure on the surface S, called the Elfin fermionic functional [1].It is a consequence of (1.1)-(1.2) that nonabelian gauge theories (even on supersymmetric versions [2]) should be better reformulated as a dynamics of random surfaces (strings theories).Our aim in this paper is to present in full technical details, added with our original improvements, the work of A. M. Polyakov [1] to give a precise path integral meaning for (1.2) in a 2D gravitational context and only considering the case of trivial surface topology.We give (also in details) the path integral meaning for the usual case of Nambu-Goto for (1.2) in the pure surface context, our new results [1].This paper is organized as follows: in Section2 we survey some basic results of classical surface theory. In Section 3, we expose the physical toy model exactly soluble Polyakov’s framework. In Section 4, we expose the Nambu-Goto theory (our results).


## 2. Elementary Results on the Classical (Bosonic) Surfaces Theory

A given surfaceS, with a boundary being a curve C and embedded into euclidean space time RD, is usually described by a vector field Xμ(ξ1,ξ2)(μ=1,…,D) and a given two-dimensional domain Ω, compatible with the topology of S in the case of the surface’s hypothesis smoothness. It is imposed also that such vector field, called from now on surface vector position field, when restricts to the boundary of Ω should be a reparameterization of C(Xμ(ξ1,ξ2))∂Ω=C).The Nambu-Goto area functional associated to a given surface vector position(Xμ(ξ1,ξ2)) is given by usual Riemann integral ((ξ1,ξ2):=ξ):
(2.1)A(SC)=∫Ω‍(det{hab(ξ)})1/2d2ξ
with hab(ξ)=(∂aXμ∂bXμ)(ξ) denoting the metric tensor induced on the surface Sc (rigorously induced on the surface-manifold tangent bundle) by the parameterization {Xμ(ξ)}.The most important property of the functional equation (2.1) is its invariance under the (formal) group of the reparameterizations of S. Namely,
(2.2)(ξ1,ξ2)=φ1(ξ1′,ξ2′);φ2(ξ1′,ξ2′),Xμ′(ξ′)=Xμ(φ(ξ′)).φ(ξ′)=:(φ1(ξ′),φ2(ξ′)) Here φ(ξ') denotes a two-dimensional intrinsic C1 vector field on Ω with everywhere nonzero Jacobian.Formal Euler-Lagrange equations associated to the surface action functional equation (2.1) can be easily written and produce the boundary value problem on Ω for each μ-component (μ=1,…,D):
(2.3)|ΔhXμ(ξ)=0,Xμ(ξ)|ξ∈∂Ω=Cμ,
where Δh denotes the second-order elliptic operator called Laplace-Beltrami associated to the metric (hab(ξ))(h(ξ)=det{hab(ξ)}):
(2.4)Δh=(1h∂a(hhabdb))(ξ).If one now choose the conformal gauge for the surfacehab(ξ)=eφ(ξ)δab, formally one obtains that the surface vector position satisfies the Dirichlet problem in Ω:
(2.5)|Δh=δabXμ(ξ)=0,Xμ(ξ)|ξ∈∂D=Cμ.The solution of the above-mentioned potential problem can be exactly given by conformal complex variable theory methods ([2], Chapter 1).Note that (2.5) produces minimal surfaces (i.e., classical surfaces which minimize the area functional locally) as the associated classical surfaces actions on the Nambu-Goto theory (2.1). Note that all the above-displayed discussion remains correct to the case of general topological genera (i.e., χ(S)=(1/4π)∫Ω‍(hR(h))dξ2≠1; here R(h) denoting the scalar of curvature associated to the induced metric hab(ξ)).However, at the classical level, there is a quadratic area functional due to Howe-Brink-Polyakov equivalent to the above result related to the classical aspects Nambu-Goto action. It is the functional associated to a theory ofD scalar classical massless fields on Ω but interacting with intrinsic (news) dynamical degrees of freedom given by the infinite-dimensional manifold of two-dimensional metrical structures on Ω.The classically equivalent area action function is now given by the massless fields{Xμ(ξ)} gravitation content:
(2.6)A¯(SC):=12∫Ω‍(ggab∂aXμ∂bXμ)(ξ)d2ξ,
where {gab(ξ)}a=1,2b=1,2 belongs to the infinite-dimensional space manifold of all possible metrics admissible by the surface S the famous exactly soluble string Polyakov’ model.That more rich (from a dynamically point of view) surface action functional is now invariant (formally) under the extended group of surface reparameterizations (the group of local diffeomorphism of the surfaceS):
(2.7)ξa′-ξa=δξa=ℰa(ξ),δXμ(ξ)=ℰa(ξ)∂aXμ(ξ),δgab(ξ)=(∇aℰb+∇bℰa)(ξ).
Here the surface’s reparameterization vector fields {ℰa(ξ)} are such that (∇aℰa)(ξ)=0, a necessary restriction in order to take out from the above written reparameterizations, all those which act on as a simply metrical rescalings. Note that we have imposed further the vector field ℰa(ξ) vanishing at the boundary of S (i.e., ℰa(ξ)|∂Ω≡0). The covariant derivative is usually defined by the Christoffel-Schwartz objects associated to the given metric tensor {gab(ξ)}:(2.8a)(∇aℰb)(ξ)=(gaa′∇a′)(ℰb)(ξ)=(∂aℰb-Γabcℰc)(ξ),(2.8b)Γabc(ξ)=12{gcd(∂agbd+∂bgad-∂agab)}(ξ).In the important case of the metric fieldgab(ξ) has the conformal form gab(ξ)=eφ(ξ)δab; the above written objects take the simple forms written below:(2.9a)Γabc(ξ)=12δcde-φ[∂aφeφδbd+∂bφeφδ-∂dφeφδab]=12(∂aφ)δcb+12(∂bφ)δca-12∂cφδab,(2.9b)∇aℰb=∂aℰb-12(∂cφ)ℰcδab-12[(∂aφ)ℰb-(∂bφ)ℰa].Note that one has in addition to the usual reparameterization invariance one further pivotal (point dependent) new symetry called the Weyl conformal symetry which acts solely on the new degree of freedom (withλ(ξ)>0), crucial for the exactly model solubility:
(2.10)gab(ξ)→λ(ξ)gab(ξ).
The classical motion equations associated to the Howe-Brink-Polyakov functional are easily written down:
(2.11)ΔgabXμ(ξ)=0,Xμ(ξ)|∂Ω=Cμ,Tab(ξ)=(∂aXμ∂bXμ-12gabgcd∂cXμ∂dXμ)(ξ)=0.From the last classical motion equation (2.11) for the intrinsic metric field, one obtains the result
(2.12)gab(ξ)=λ¯(ξ)(∂aXμ∂bXμ)(ξ),
where the unknown scale λ¯(ξ) can be fixed to be the function λ¯(ξ)=1. Note that there is the additional boundary condition
(2.13)(gab(ξ))∂Ω=hab(ξ)|∂Ω=(∂aXμ∂bXμ)|∂Ω.Another important classical surface function, probably related to the still not completely understood functionalΦACD[C] on (1.2), is the so-called extrinsic functional which is defined by the square of the surface mean curvature (see (2.4)):
(2.14)(ℋ2(S))=(∇hXμ)2={-1h∂a(hhab∂b)(Xμ)}2.
Namely,
(2.15)ℱextr(S)=∫Ω‍ℏ((ΔhXμ)2(ξ))d2ξ.In the Polyakov’s version of two-dimensionalΩ quantum gravity, the above functional can be replaced by the fourth-order scalar field action:
(2.16)ℱext[Xμ(ξ),gab(ξ)]=∫Ω‍(g(ΔgXμ)2(ξ))d2ξ+∫Ω‍(gλab(ξ)(gab(ξ)-hab(ξ,Xμ)))(ξ),
where λab(ξ) denote lagrange multipliers insuring that at the classical level gab(ξ)=(∂aXμ∂bXμ)(ξ).Boundary conditions to be imposed on the complete actionA(S)+ℱext(S) are mathematical subtle in order to guarantee the mathematical good property of strong elliptic of the associated boundary value problem ([3], Chapter 5).For instance, if one considers the casegab=δab and the associated biharmonic operator(2.17a)(ΔXμ)2=∂4Xμ∂ξ14+2∂4Xμ∂ξ12∂ξ22+∂4Xμ∂ξ24,
one can associate the following system of boundary operators:
(2.17b)Xμ(ξ)|∂Ω=Cμ(ξ),-∂∂nXμ(ξ)|∂Ω=Mμ(ξ).Here the meaning of Mμ is not so clear as an independent dynamical parameter of the loop C (string).In the following discussions we will always regard the fourth-order actions as effective actions coming from the integrating out fermionic intrinsic degrees of freedom [1].We left to our readers to prove the Green’s formula for the fourth-order problem equations (2.17a) and (2.17b):
(2.18)∫Ω‍(fΔ2g)d2ξ=∫Ω‍(gΔ2f)d2ξ+∫∂Ω‍(-f∂Δg∂n+∂f∂nΔg+g∂∂nΔg-∂g∂nΔg)ds.As a further comment and just for the reader’s curiosity, one has an analogue of representing (at least locally) two-dimensional harmonic functions onΩ by analytical complex variable functions; one still has the following representation (not unique!) for biharmonic functions [1, 5]:
(2.19)𝒱(ξ1,ξ2)=Re{z¯φ(z)+ρ(z)},
where z=ξ1+iξ2 and (ϖ(z),ρ(z)), a pair of complex variable analytic functions on Ω.If the surfaceS is on R3 and given in “nonparametric form” X3=f(X2,X2), we can introduce parametric coordinates ξ1=X1, ξ2=X2 and obtain a result convenient for computations (exercises for the diligent reader); the Gauss curvature is given explicitly by(2.20a)R(hab)=(1+fX12+fX22)2·det{fab(X1,X2)}.
and for the mean curvature, we have the following result:
(2.20b)H=(1+fX12+fX22)-3/2{(1+fX22)fX1X1-2fX1fX2fX1X2+(1+fX12)fX2X2}.It is worth to recall that in the important case of a surfaceS embedded in R3 (useful in Surface’s Statistical Physics), but with the general parameterization:
(2.21)X1=X1(ξ1,ξ2);X2=X2(ξ1,ξ2);X3=X3(ξ1,ξ2),
the expression for the area functional and the surface extrinsic functional can be given explicitly by (exercise for our diligent readers!)(2.22a)A(S)=∫Ω‍(EG-F2)(ξ)d2ξ,(2.22b)ℱext(S)=∫Ω‍H2(ξ)d2ξ,
where the surface objects are given by
(2.22c)E=(∂Xμ∂ξ1),(∂Xμ∂ξ1);F=(∂Xμ∂ξ1),(∂Xμ∂ξ2);G=(∂Xμ∂ξ2),(∂Xμ∂ξ2);H=1(EG-F2)(ED′′-2D′F+DG);D=1EG-F2|Xξ1ξ11Xξ1ξ12Xξ1ξ13Xξ11Xξ12Xξ13Xξ21Xξ22Xξ23|,D′=1EG-F2|Xξ1ξ21Xξ1ξ22Xξ1ξ23Xξ11Xξ12Xξ13Xξ21Xξ22Xξ23|,D′′=1EG-F2|Xξ2ξ21Xξ2ξ22Xξ2ξ23Xξ11Xξ12Xξ13Xξ21Xξ22Xξ23|.Finally we remark that one could easily generalize all the above written results to the general case of the initial ambient spaceRD being a new general Riemann Manifold (M,Gμν(Xγ)). In this case, the “induced” surface metric tensor hab(X(ξ))=∂aXμ(ξ)∂bXν(ξ) will be replaced by the new “induced” tensor:
(2.23)hxβ(X(ξ),[Gμν])=∂aXμ(ξ)Gμν(Xγ(ξ))∂bXν(ξ)
on the exposed formulae. The full classical and quantum theory of the surfaces σ-models can be found in [1].


## 3. Path Integral Quantization on Polyakov’s Theory of Surface (or How to Quantize 2D Massless Scalar Fields in the Presence of 2D Quantum Gravity)

After exposing some basic concepts of the classical surface theory in Section2, we now pass to the vital problem of quantization on the formalism of Feynman-Wiener path integrals.Following R. P. Feynman in his theory of path integration sum over histories, a mathematical meaning for the continuous sum over (now), euclidean quantum (random) surfaces for (1.2), should be given by following path integrals for the Nambu-Goto case and of A. M. Polyakov, respectively, (see (2.1)–(2.6)):(3.1a)GNG(C)=∫Xμ(ξ)|ξ∈∂Ω=Cμdμ[Xμ(ξ)]exp{-12πα′A(Sc)},(3.1b)GP(C)=∫Xμ(ξ)|ξ∈∂Ω=Cμgab(ξ)|ξ∈∂Ω=0dμ¯[Xμ(ξ),gab(ξ)]exp{-12πα′A¯(Sc)}×exp{-μ02∫Ω‍g(ξ)d2ξ}.Hereα' and μ02 are constants to be identified with certain physical parameters of the theory. Note that α' has dimension of inverse if mass square and [μ02]=[α′]-1. It is an important remark that the Feynman-Wiener path measures dμ[Xμ(ξ)] and dμ¯[Xμ(ξ),gab(ξ)] must be defined in such a way in order to be fully invariant under the action of the local diffeomorphism group of both descriptions (see (2.2) and (2.7)).We aim now to evaluate explicitly the A. M. Polyakov’s integral equation (3.1b). As a first step one recall that we have imposed the “fluctuating” zero Dirichlet boundary condition for the intrinsic metric field gab(ξ)(gab(ξ)|∂Ω=0). Those results lead us to the effective action for the surface dynamics weight function:
(3.1c)A¯(Sc)=12∫Ω‍(gXμ(-Δg)Xμ)(ξ)d2ξ.Since at the classical level, the metric field decouples (one can choose it in the formgab=eφδab) and considers the usual classical plus quantum decomposition Xμ(ξ)=XμCL(ξ)+(πα')Xμq(ξ), where the quantum correction Xμq(ξ) is such that Xμq(ξ)|∂Ω=0.After these preliminaries considerations one is led to evaluate the following covariant path integral (after disregarding classical field contributions to the path integral):(3.1d)I[gab]=∫‍dμ[Xμ(ξ)]exp{-12∫Ω‍(gXμ(-Δg)Xμ)(ξ)d2ξ}.It is worth recalling that there is a classical term (a functional depending on the loop boundaryC) coming from the partial integration to arrive at ((3.1a), (3.1b), (3.1c), and (3.1d)) for the case gab=δab, which clearly does not satisfy the quantum-fluctuating nature of these nonconstant gab(ξ):
(3.2)∫Ω‍(∂aXelμ∂aXμ,el)(ξ)d2ξ=∫∂Ω‍(Cμ∂∂nCμ)︷classical  contribution(σ)dσ.The Feynman-Wiener measuredμ[Xμ(ξ)) on (3.1d) must be chosen in order to be invariant under (2.7). The only local candidate is given by the “weighted” product Feynman measure [1]:
(3.3)dμ[Xμ(ξ)]:=∏ξ∈Ω‍(g(ξ)4dXμ(ξ)).As a consequence the path integral equations ((3.1a), (3.1b), (3.1c), and (3.1d)) turns out to be Gaussian and metric dependent. It yields the immediate result:
(3.4)I[gab]=det-D/2(-Δg).We thus have reduced the “explicitly” evaluation of the Polyakov’s path integral to the functional integration of the above written function over all fluctuating metric fields:(3.5)I=∫‍dμ[gab]det-D/2(-Δg)e-μ02∫Ω‍gd2ξ.By using the theory of invariant integration on Riemann Manifolds of ours [1], one can insure automatically the preservation of the diffeomorphism group by defining the above mentioned metric dμ[gab] as the (functional) element of volume of the following functional metric (so-called De Witt metric) with (c≠-1/2):
(3.6)∥δgab∥2=∫Ω‍[g{(δgab)(gaa′gbb′+cgabga′b′)(δga′b′)}](ξ)d2ξ.Let us then choose the conformal gauge fixing to evaluate the above metric path integral successfully [1], but already considering the metric positivity:
(3.7)g¯ab(ξ)=eφ(ξ)δab.As a result the functional infinitesimal displacementsδgab(ξ) can be written in the following general form on the functional manifold of the metric [gab(ξ)] (on the formal infinite-dimensional functional tangent bundle):
(3.8)δgab(ξ)=δφ(ξ)g¯ab(ξ)+(∇aℰb+∇bℰa)(ξ).Let us rewrite the functional De-Witt metric in the form(3.9a)∥δgab∥2=∫Ω‍(δgab(ξ))γ(ab,a'b')[gcd(ξ)](δga′b′(ξ)),(3.9b)γ(ab,a'b')[gcd(ξ)]=(g(gaa′gbb′+cgabga′b′))(ξ).Since there is only three independent components of the metric tensorgab(ξ) in two dimensions (g12=g21), the metric coefficients {γ¯ij}1≤i≤31≤j≤3 are effectively a 3×3 matrix on the functional manifold of the intrinsic 2D metric fields:
(3.10)∥δgab∥2=∫Ω‍{(δg11,δg12,δg22)[γ¯](δg11δg12δg22)}(ξ)d2ξ=∫‍{(δg11)(γ¯11)(δg11)+(δg12)(γ¯22)(δg22)+(δg22)(γ¯33)(δg22)+(δg11)(γ¯12+γ¯21)(δg12)+(δg11)(γ¯13+γ¯31)(δg22)+(δg12)(γ¯23+γ¯32)(δg22)}(ξ)×d2ξ.Here the explicitly expressions of the effective[γ¯]-matrix are related to those of (3.9b) by the results
(3.11)γ¯11=γ(11,11)=eφ(e-φδ11e-φδ11+ce-φδ11e-φδ11)=(1+c)e-φ;γ¯12=γ(11,12)=0;γ¯21=γ(12,11)=0;γ¯13=γ(11,22)=ce-φ;γ¯31=γ(22,11)=ce-φ;γ¯23=γ(12,22);γ¯33=γ(22,12)=(1+c)e-φ.
As a consequence
(3.12)det[γ(ab,a′b′)]=det[γ¯iδ]=(1+c)2e3φ-c2e3φ=(1+c)e-3φ.So the De Witt metric on 2D is nonsingular only if(C≠-1/2).Let us substitute the general functional displacement equation (3.8) on (3.9a)-(3.9b).Firstly, we get (one could choose from the beginning on (3.9b)
(3.13)∥δgab∥2=∫Ωd2ξeφ(ξ){(δφgba+∇aℰb+∇bℰa)·(δφgab)+∇bℰa+∇aℰbd2ξeφ(ξ)+4c(∇cℰc+δφ)2}.Let us analyze those terms involving the conformal factor.By noting that∇cgab=∇cgab=0 and (∇aℰb)gaa'gbb'(∇b'ℰa')=∇a'ℰb∇bℰa', added with ∇aℰa=0; ∇a'(gbb'ℰb')=gbb'∇a'ℰb'=∇a'ℰb, one obtains the chain of results written below
(3.14)∥δgab∥2=∫Ω‍d2ξeφ(ξ){∫Ω‍(δφgba+∇aℰb+∇bℰa)·(δφgab)∫Ω‍d2ξeφ(ξ)+∇bℰa+∇aℰb+4c(∇cℰc+δφ)2∫Ω‍}(ξ)=∫Ω‍d2ξeφ(ξ){(∇dℰd)2(∇aℰb+∇bℰa)(∇bℰa+∇aℰb)∫Ω‍d2ξeφ(ξ)-2(∇dℰd)2+2(1+2c)(∇dℰd+δφ)2}(ξ)=2(1+2c)∫Ω‍d2ξeφ(ξ)(∇dℰd+δφ)2(ξ)-2∫Ω‍d2ξeφ(ξ){ℰa(∇d∇d+[∇b,∇a])ℰb},
where we have used the covariant by parts, integration rule (g|∂Ω=0):
(3.15)∫Ω‍d2ξgf(∇dh)=-∫Ω‍d2ξg(∇df)h.As a consequence we have the full result:(3.16)∥δgab∥2=(1+2c)∫Ω‍d2ξg(ξ)(δ𝒱(ξ))2+∫Ω‍d2ξg(ξ)|(ℰ1(ξ),ℰ2(ξ))[ℒ^](ℰ1(ξ)ℰ2(ξ))|,
where
(3.17)δ𝒱(ξ)=δφ(ξ)+(∇cℰc)(ξ)︷=0=δφ(ξ)
and the Faddeev-Popov operator acting on the two-dimensional vector field generators of the local group of coordinate transformations is explicitly given by
(3.18)(ℒ^ℰ)a=-((∇d∇dδab+[∇b,∇a])ℰb)(ξ).As a consequence the functional volume element is written as(3.19)dμ[gab(ξ)]=dcov[φ(ξ)]dcov[ℰ1(ξ)]dcov[ℰ2(ξ)]det1/2[ℒ].
Here
(3.20)dcov(φ(ξ))=∏(ξ∈Ω)‍e(φ(ξ)/2)δφ(ξ)=∏ξ∈Ω‍δ(2e(φ(ξ)/2))=DF[e(φ(ξ)/2)]dcov[ℰa(ξ)]=∏ξ∈Ω‍dℰa(ξ)=DF[ℰa(ξ)].
Just for pedagogical purpose let us evaluate in details the relevant term for the Fadeev-Popov operator in the above formulae.We have, for instance, the following sample calculation:(3.21)∫Ωd2ξg(∇aℰb+∇bℰa)gaa′gbb′(∇a′ℰb′+∇b′ℰa)=(∫d2ξ{g(∇aℰb)gaa′gbb′(∇a′ℰb′)}(ξ))+similar  terms=∫d2ξg[(∇aℰb,∇aℰb)︷gbb′∇a′ℰb′=∇a′(gbb′ℰb′)=∇a′ℰb+∇aℰb∇bℰa+∇bℰa∇aℰb+∇bℰa∇bℰa](ξ)=∫d2ξg{ℰb(-∇a∇a)ℰb-ℰb(∇a∇b)ℰa-ℰa(∇b∇a)ℰb-ℰa(∇b∇b)ℰa}=-∫d2ξg{2ℰb(-∇a∇a)ℰb-ℰb(∇a∇b+∇a∇b)ℰa︷2∇a∇b=2∇b∇a+2[∇,∇b]-}=-2∫d2ξgℰa[(∇c∇c)δab+[∇b,∇a]-︷(∇aℰa=0)]ℰb.As an exercise to our diligent readers, the form of the above elliptic operator in the conformal gauge is given by (action on two-component objects)(3.22)Δ(g=eφδab)=e-φ(ξ)∂z¯∂z,ℒ^(g=eφδab)=e-2φ(ξ)∂z¯(eφ(ξ)∂z),
where we have introduced the complex plane notation and the above displayed operators now acting on complex functions (isomorphic to two-component objects = vector fields) [8]:
(3.23)∂∂z=∂∂ξ1-i∂∂ξ2∂∂z¯=∂∂ξ1+i∂∂ξ2.(i=-1),In [3], we present the explicit evaluation of elliptic operators of the form (for j being a positive integer) considered below:
(3.24)ℒj=-e-(j+1)φ(ξ)∂z(ejφ(ξ)∂z¯).
The result is
(3.25)detF(ℒj)=exp{-1+6j(j+1)12π∫Ω‍d2ξ12(∂aφ)2(ξ)-limε→0-112πε∫Ω‍d2ξeφ(ξ)}.Let us point out the validity of the relationship below:(3.26)detF-D/2(Δg=eφδab)=detF-D/4(ℒj=0),det+1/2(L^g=eφδab)=detF+1/2(ℒj=1).As our final result of this covariant Polyakov path integral, we get the 2D-induced quantum gravity Liouville model:(3.27)I=∫‍DF[e(φ/2)(ξ)]exp{-(26-D)48π∫Ω‍12(∂aφ)2(ξ)d2ξ}×{e-μren2∫Ω‍eφ(ξ)d2ξ}.Hereμr2=μ02+limε→0+((2-D)/8πε).Note that the quantum measure on the above-displayed Liouville model is not the usual flat Feynman measure∏ξ∈Ωdφ(ξ)=DF[φ(ξ)] as initially supposed by A. M. Polyakov.As a consequence one should rewrite it in terms of the canonical Goldstone boson fieldρ¯(ξ)=e(φ/2)(ξ) which leads us to a kind of “noncompact” σ-model with a “renormalized” mass term:
(3.28)I=∫‍DF[ρ¯(ξ)]exp{-26-D12π∫Ω‍(∂aρ¯)2ρ¯2d2ξ-μr2∫Ω‍ρ¯2(ξ)d2ξ}
and signaling the dynamical breaking of the conformal group [1].Sometimes one can formally consider the variable change in the path measure of (3.27):
(3.29)DF[e(φ/2)(ξ)]=detF(eφ/2δabδ(ξ-ξ′))DF[φ(ξ)]︷Functional  Jacobian=(limε→0+exp Tr[ℓg(eφ/2δab)e-ℰΔg=eφ])×DF[φ(ξ)]=limε→0+exp{∫Ω‍d2ξeφ(ξ)(φ(ξ)2)[14πε+112πR(φ)]}×DF[φ(ξ)]=limε→0+{exp[18πε∫Ω‍eφ(ξ)φ(ξ)d2ξ]×exp[-124π∫Ω‍(φ(-Δ)φ)(ξ)d2ξ]}×DF[φ(ξ)]
which would lead to the same expression (3.28) but with the “reduced” anomaly conformal coefficient 26D to 25D, if one could disregard the infinite piece on (3.29) and if the “Fujikawa-like” evaluation of the functional jacobian could be done more invariant. At this point it appears that the above 2D-Liouville-Polyakov path integral only makes sense at the classical level which is formally equivalent to evaluate all the observables in the φ(ξ)-theory at the limit of D→-∞ [1]. One can see this after considering the rescale (see Appendix C for a complete analysis of this study):
(3.30)φ→48π26-Dφ,DF[eφ/2]→DF[e(48π/(26-D))φ]→D→-∞DF[0]=classical.However, we should point out that all there questions on two-dimensional quantum gravity as a well-defined problem still are not well-understood since its inception in 1981 [1].Finally it is very important on applications to the Dual model theory for strong interactions (off-shell Scattering Amplitudes) to have a covariant regularized form for the formal Green function of the Beltrami-Laplace operator in the conformal gaugegab(ξ)=eφ(ξ)δab. It is well known from Hadamard theory of parametrix singular solutions that one has the following covariant behavior for the Beltrami-Laplace operator for small distances:
(3.31)(Δg=eφδab-1)(ξ,ξ′)ε=(ξ-ξ′)→0~12πε-12πℓg(eφ(ξ)|ξ-ξ′|)~12π(1ε)-12πφ(ξ)-12πℓg|ξ-ξ′|.
This lead us to consider the regularized from of the Green function used on dual models based on the Polyakov’s path integral:
(3.32)(ξ|Δg=eφδab-1e-ℰΔg=eφδab|ξ')={-12πℓg|ξ-ξ′|ξ≠ξ'-14πℓg(1ε)+14πφ(ξ)ξ=ξ′.
Note that we have used the following formulae to write explicitly the functional trace of the general two-dimensional (strongly) elliptic operators [3]:(3.33a)A=-(g11∂2∂ξ12+g22∂2∂ξ22)12×2-(A1)∂∂x1-(A2)∂∂x2-A0(3.33b)limt→0+tr(e-tA)~12πt(∫‍d2ξg11g22)+{12π(∫‍d2ξg11g22(-16R(g)))×∫‍d2ξ(-121g11g22)Trmatrix[(∂∂x1(g11g22A1)+∂∂x2(g11g22A2))]+∫‍d2ξ(-14Tr[(A1)2g11+(A2)2g22]+A0)}+O(t).


## 4. Path-Integral Quantization of the Nambu-Goto Theory of Random Surfaces

In order to give a path integral meaning for the symbolic Feynman continuum sum over surfaces’ histories (3.1a), we start rewriting it in the following Polyakov’v form, but in the presence of a constraint [1]:
(4.1)GNG(C)=∫‍dμ¯[xμ(ξ),gab(ξ)]exp{-12πα′∫Ω‍d2ξ(ggab∂aXμ∂bXμ)(ξ)}×δcov(F)(gab-∂aXμ∂bXμ).
Note that our constraint is rewritten as a covariant (diffeomorphism) invariant delta functional:
(4.2)δcov(F)(gab-∂aXμ∂bXμ)=∫‍Dcov[λab]ei{∫Ω‍d2ξ(gλab(∂aXμ∂bXμ-gab))(ξ)},
where the covariant Feynman-Wiener measure on (4.2) is the element of volume of the functional metric:
(4.3)ds2=∥δλab∥2=∫‍(gδλab(gaa′gbb′)δλa′b)(ξ)d2ξ.After introducing again the complex-euclidean light-cone coordinates on the domainΩ:
(4.4)∂+=(∂ξ1-i∂ξ2)2=∂z,∂-=(∂ξ1+i∂ξ2)2=∂z¯
and imposing the surface conformal gauge gab(ξ)=ρ(ξ)δab, which means that g11=g22, g12=g21=0, or equivalently using light-cone coordinates again,
(4.5)∂+Xμ∂+Xμ=∂-Xμ∂-Xμ(⟺∂ξ1Xμ∂ξ2Xμ=g12=g21=0),g=∂+Xμ∂-Xμ(g11=∂+Xμ∂-Xμ=g22≠0).Now by taking into account the Jacobian relationship between the covariant functional measures and the associated pure Feynman-Wiener-Kac path measures, one obtains the following results.A careful discussion presented in [1]-chapter 12 leads us to the following result on the surface light-cone gauge (4.5):
(4.6)GNG(C)=∫‍DF[ρ¯(ξ)]DF[Xμ(ξ)]×exp{-(26-D)12π∫Ω‍d2ξ[12(∂aρ¯ρ¯)2]+limε→0(2-D)2πε∫Ω‍d2ξρ¯2}×exp{-12πα′∫Ω‍d2ξ(∂aXμ)2(ξ)}×δ(F)[(ρ¯)2(ξ)δab-hab(Xμ(ξ))︷=∂+Xμ∂-Xμ].Note the usual noncovariant function “flat” constraint relating the surface vector position to the auxiliary metric fieldgab(ξ) on the conformal gauge.After realizing the immediateρ¯(ξ) functional integral, one gets the final result, after a renormalization of the Regge slope constant (∂+=∂z, ∂-=∂z¯). (It is worth to remark that at a Feynman diagrammatic perturbative level, the measure “tad-pole” factor in (4.7), exp{((-δ(2)(0))/4)∫Ω‍ℓg(h(Xμ(z,z¯)))×dzdz¯}, can be assigned to exp{0}=1 in the dimensional regularization scheme):
(4.7)GN(C)=∫‍[∏(μ,ξ,ξ')‍dXμ(z,z¯)h(Xμ(z,z))4]×δ(F)(∂+Xμ∂+Xμ)δ(F)(∂-Xμ∂-Xμ)exp{-12πα′ren∫Ω‍dzdz¯(∂+Xμ)(∂-Xμ)(z,z¯)}exp{-(26-D)48π∫Ω‍dzdz¯[(1/2)∂+(∂+Xμ∂-Xμ)(∂+Xμ∂-Xμ)×∂-(∂+Xμ∂-Xμ)(∂+Xμ∂-Xμ)](z,z¯)}.Now we realize that one must chooseD=26 or introduce intrinsic Majorana fermions living on those Nambu-Goto random surfaces in order to vanish the nonrenormalizable Liouville surface term. The numbers of such Neutral Majorana 2D fields m must be such that 26=4+m on D=4. Note that complex pairing such neutral Majorana fermion one gets eleven complex fermion Dirac fields living on the random surface.It is worth (all the reader attention that in this framework of Nambu-Goto path integrals on surface light-cone gauge, the vertices are given by object:(4.8)V={hh[a1,a2]⋯ha2m-1,a2n}(Xμ(z.z¯))×{∂a1Xμ⋯∂a2nXμ2μ·ℰμ1…μ2m︷spin  polarization  tensors·eiKμXμ}(z,z¯),
which are free from tachyons, and so forth.A complete study of the spectrum of such Nambu-Goto strings with those tachyons-free “geometrical” vertices is still missing in literature and left to our diligent readers.Similar analysis can be straightforwardly implemented for the extrinsic surface path-integral weight functional equations (2.15)-(2.16).



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*Source: 674985-2012-11-19.xml*