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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 the

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 [

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 trajectory

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 (

It is a consequence of (

Our aim in this paper is to present in full technical details, added with our original improvements, the work of A. M. Polyakov [

We give (also in details) the path integral meaning for the usual case of Nambu-Goto for (

This paper is organized as follows: in Section

A given surface

The Nambu-Goto area functional associated to a given surface vector position

The most important property of the functional equation (

Formal Euler-Lagrange equations associated to the surface action functional equation (

If one now choose the conformal gauge for the surface

The solution of the above-mentioned potential problem can be exactly given by conformal complex variable theory methods ([

Note that (

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 of

The classically equivalent area action function is now given by the massless fields

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 surface

In the important case of the metric field

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 (with

From the last classical motion equation (

Another important classical surface function, probably related to the still not completely understood functional

In the Polyakov’s version of two-dimensional

Boundary conditions to be imposed on the complete action

For instance, if one considers the case

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 [

We left to our readers to prove the Green’s formula for the fourth-order problem equations (

As a further comment and just for the reader’s curiosity, one has an analogue of representing (at least locally) two-dimensional harmonic functions on

If the surface

It is worth to recall that in the important case of a surface

Finally we remark that one could easily generalize all the above written results to the general case of the initial ambient space

After exposing some basic concepts of the classical surface theory in Section

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 (

Here

We aim now to evaluate explicitly the A. M. Polyakov’s integral equation (

Since at the classical level, the metric field decouples (one can choose it in the form

After these preliminaries considerations one is led to evaluate the following covariant path integral (after disregarding classical field contributions to the path integral):

It is worth recalling that there is a classical term (a functional depending on the loop boundary

The Feynman-Wiener measure

As a consequence the path integral equations ((

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:

By using the theory of invariant integration on Riemann Manifolds of ours [

Let us then choose the conformal gauge fixing to evaluate the above metric path integral successfully [

As a result the functional infinitesimal displacements

Let us rewrite the functional De-Witt metric in the form

Since there is only three independent components of the metric tensor

Here the explicitly expressions of the effective

So the De Witt metric on 2D is nonsingular only if

Let us substitute the general functional displacement equation (

Firstly, we get (one could choose from the beginning on (

Let us analyze those terms involving the conformal factor.

By noting that

As a consequence we have the full result:

As a consequence the functional volume element is written as

We have, for instance, the following sample calculation:

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)

In [

Let us point out the validity of the relationship below:

As our final result of this covariant Polyakov path integral, we get the 2D-induced quantum gravity Liouville model:

Here

Note that the quantum measure on the above-displayed Liouville model is not the usual flat Feynman measure

As a consequence one should rewrite it in terms of the canonical Goldstone boson field

Sometimes one can formally consider the variable change in the path measure of (

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 [

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 gauge

In order to give a path integral meaning for the symbolic Feynman continuum sum over surfaces’ histories (

After introducing again the complex-euclidean light-cone coordinates on the domain

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 [

Note the usual noncovariant function “flat” constraint relating the surface vector position to the auxiliary metric field

After realizing the immediate

Now we realize that one must choose

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:

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 (

Let us start by this appendix by considering the euclidean Dirac partition functional on a two-dimensional space time:

The two-dimensional euclidean matrixes on the Dirac operator (

Let us note the more suitable redefinition of the euclidean Dirac operator:

Let us note that (with a undetermined infinite phase!)

Since we have the result

In the general nongauge fixed case, one has the result

Let us show in detail the above calculation:

It is important to remark that if one had used the following self-adjoint Dirac euclidean operator,

At this point it is worth to recall the Seeley asymptotic expansion

One of the most celebrated (pure) mathematical theorems on modern geometry and topology of compact orientable manifolds on

As in the last reference [

An important remark to be done now originally due to ourselves [

On the basis of the above made remark (

After integrating the Grassmannian variables

Now the

At the classical limit

By grouping together (

The universal constant is adjusted by defining as unity the Chern induces or the Euler-Poincaré characteristic) of spheres

In two dimensions, (

We start this appendix by considering the Liouville quantum field model partition functional in terms of the correct

In order to define (

After that step has been taken, one has the result (with the new cosmological constant

Clearly at a perturbation level around the one-loop expansion Gaussian path integral, one can see that all the infinite set of perturbative vertices (higher terms

Note that we are indeed considering the Liouville-Model as the two-dimensional version of quantum gravity.

Just for completeness, we write the full expression of the one-loop Gaussian path integral associated to our proposed

Note again that one should choose the Liouville quantum fluctuations in such a way that the constraints between the field

As a general conclusion on this sigma model for two-dimensional quantum gravity, one can see that at least for applications on string theory for Q.C.D., one necessarily must set

However, it is worth to remark and keep on mind that the full Liouville path integral although being defined for