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

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 the 1/𝐷 expansion on Polyakov’s String path integral, all containing important new results, insights on theses topics.

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 trajectory 𝐶 in a 𝑆𝑈(𝑁_𝑐) Yang-Mills quantum field theory. Namely, 𝜓[𝐶]=⟨𝑊[𝐶=1]⟩𝑁_𝑐Tr_𝑁^(𝑐)_𝑆𝑈𝑐ℙexpig_𝐶𝐴_𝜇𝑑𝑥_𝜇. (1.1) Here ⟨⋯⟩ is defined by the Yang-Mills quantum path integral and 𝐴_𝜇(𝑥)=𝐴^𝑐_𝜇(𝑥)𝜆_𝑎 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 𝜓[𝐶]=_{𝑆,𝜕𝑠=𝒞}−1exp2𝜋𝛼^′𝐴(𝑆)Φ^𝑄𝐶𝐷[𝐶] (1.2) 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 Section 2 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 surface 𝑆, with a boundary being a curve 𝐶 and embedded into euclidean space time 𝑅^𝐷, is usually described by a vector field 𝑋_𝜇(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(𝑋_𝜇(𝜉₁,𝜉₂))_𝜕Ω=𝐶).

The Nambu-Goto area functional associated to a given surface vector position (𝑋_𝜇(𝜉₁,𝜉₂)) is given by usual Riemann integral ((𝜉₁,𝜉₂)∶=𝜉): 𝐴𝑆_𝐶=_Ωdetℎ_𝑎𝑏(𝜉)^1/2𝑑²𝜉 (2.1) with ℎ_𝑎𝑏(𝜉)=(𝜕_𝑎𝑋^𝜇𝜕_𝑏𝑋_𝜇)(𝜉) denoting the metric tensor induced on the surface 𝑆_𝑐 (rigorously induced on the surface-manifold tangent bundle) by the parameterization {𝑋_𝜇(𝜉)}.

The most important property of the functional equation (2.1) is its invariance under the (formal) group of the reparameterizations of 𝑆. Namely, 𝜉₁,𝜉₂=𝜑₁𝜉^′₁,𝜉^′₂;𝜑₂𝜉^′₁,𝜉^′₂,𝑋^′_𝜇𝜉^′=𝑋_𝜇𝜑𝜉^′. (2.2)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): Δ_ℎ𝑋^𝜇(𝜉)𝑋=0,_𝜇(𝜉)_{𝜉∈𝜕Ω}=𝐶^𝜇, (2.3) where Δ_ℎ denotes the second-order elliptic operator called Laplace-Beltrami associated to the metric (ℎ_𝑎𝑏(𝜉))(ℎ(𝜉)=det{ℎ_𝑎𝑏(𝜉)}): Δ_ℎ=1√ℎ𝜕_𝑎√ℎℎ^𝑎𝑏𝑑_𝑏(𝜉). (2.4)

If one now choose the conformal gauge for the surface ℎ_𝑎𝑏(𝜉)=𝑒^𝜑(𝜉)𝛿_𝑎𝑏, formally one obtains that the surface vector position satisfies the Dirichlet problem in Ω: Δ_{ℎ=𝛿𝑎𝑏}𝑋^𝜇(𝜉)𝑋=0,_𝜇(𝜉)_{𝜉∈𝜕𝐷}=𝐶^𝜇. (2.5)

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 ℎ_𝑎𝑏(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 of 𝐷 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 {𝑋^𝜇(𝜉)} gravitation content: 𝐴𝑆_𝐶1∶=2_Ω√𝑔𝑔^𝑎𝑏𝜕_𝑎𝑋^𝜇𝜕_b𝑋_𝜇(𝜉)𝑑²𝜉, (2.6) where {𝑔_𝑎𝑏(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 surface 𝑆): 𝜉^′_𝑎−𝜉_𝑎=𝛿𝜉_𝑎=ℰ_𝑎(𝜉),𝛿𝑋_𝜇(𝜉)=ℰ_𝑎(𝜉)𝜕^𝑎𝑋_𝜇(𝜉),𝛿𝑔_𝑎𝑏(𝜉)=∇_𝑎ℰ_𝑏+∇_𝑏ℰ_𝑎(𝜉). (2.7) Here the surface’s reparameterization vector fields {ℰ_𝑎(𝜉)} are such that (∇^𝑎ℰ_𝑎)(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 ℰ_𝑎(𝜉) vanishing at the boundary of 𝑆 (i.e., ℰ_𝑎(𝜉)|_𝜕Ω≡0). The covariant derivative is usually defined by the Christoffel-Schwartz objects associated to the given metric tensor {𝑔_𝑎𝑏(𝜉)}:∇_𝑎ℰ_𝑏(𝜉)=𝑔_𝑎𝑎′∇^𝑎′ℰ_𝑏(𝜉)=𝜕_𝑎ℰ_𝑏−Γ^𝑐_𝑎𝑏ℰ_𝑐(𝜉)Γ_𝑐𝑎𝑏,(𝜉)=12𝑔^𝑐𝑑𝜕_𝑎𝑔_𝑏𝑑+𝜕_𝑏𝑔_𝑎𝑑−𝜕_𝑎𝑔_𝑎𝑏(𝜉). (2.8a)(2.8b)

In the important case of the metric field 𝑔_𝑎𝑏(𝜉) has the conformal form 𝑔_𝑎𝑏(𝜉)=𝑒^𝜑(𝜉)𝛿_𝑎𝑏; the above written objects take the simple forms written below:Γ^𝑐_𝑎𝑏(𝜉)=12𝛿^𝑐𝑑𝑒^−𝜑𝜕_𝑎𝜑𝑒^𝜑𝛿_𝑏𝑑+𝜕_𝑏𝜑𝑒^𝜑𝛿−𝜕_𝑑𝜑𝑒^𝜑𝛿_𝑎𝑏=12𝜕_𝑎𝜑𝛿^𝑐𝑏+12𝜕_𝑏𝜑𝛿^𝑐𝑎−12𝜕_𝑐𝜑𝛿_𝑎𝑏,∇_𝑎ℰ_𝑏=𝜕_𝑎ℰ_𝑏−12𝜕_𝑐𝜑ℰ^𝑐𝛿_𝑎𝑏−12𝜕_𝑎𝜑ℰ_𝑏−𝜕_𝑏𝜑ℰ_𝑎. (2.9a)(2.9b)

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 𝜆(𝜉)>0), crucial for the exactly model solubility: 𝑔_𝑎𝑏(𝜉)→𝜆(𝜉)𝑔_𝑎𝑏(𝜉). (2.10) The classical motion equations associated to the Howe-Brink-Polyakov functional are easily written down: Δ_𝑔𝑎𝑏𝑋^𝜇(𝜉)𝑋=0,_𝜇(𝜉)|_𝜕Ω=𝐶^𝜇,𝑇_𝑎𝑏(𝜉)=𝜕_𝑎𝑋_𝜇𝜕_𝑏𝑋^𝜇−12𝑔_𝑎𝑏𝑔^𝑐𝑑𝜕_𝑐𝑋^𝜇𝜕_𝑑𝑋_𝜇(𝜉)=0. (2.11)

From the last classical motion equation (2.11) for the intrinsic metric field, one obtains the result 𝑔_𝑎𝑏(𝜉)=𝜆(𝜉)𝜕_𝑎𝑋_𝜇𝜕_𝑏𝑋^𝜇(𝜉), (2.12) where the unknown scale 𝜆(𝜉) can be fixed to be the function 𝜆(𝜉)=1. Note that there is the additional boundary condition 𝑔_𝑎𝑏(𝜉)_𝜕Ω=ℎ_𝑎𝑏(𝜉)_𝜕Ω=𝜕_𝑎𝑋_𝜇𝜕_𝑏𝑋^𝜇_𝜕Ω. (2.13)

Another important classical surface function, probably related to the still not completely understood functional Φ^𝐴𝐶𝐷[𝐶] on (1.2), is the so-called extrinsic functional which is defined by the square of the surface mean curvature (see (2.4)): ℋ²(𝑆)=∇_ℎ𝑋^𝜇2=−1√ℎ𝜕_𝑎√ℎℎ^𝑎𝑏𝜕_𝑏𝑋^𝜇2. (2.14) Namely, ℱ_extr(𝑆)=_Ω√ℏΔ_ℎ𝑋^𝜇2(𝜉)𝑑²𝜉. (2.15)

In the Polyakov’s version of two-dimensional Ω quantum gravity, the above functional can be replaced by the fourth-order scalar field action: ℱ_ext𝑋^𝜇(𝜉),𝑔_𝑎𝑏(𝜉)=_Ω√𝑔Δ_𝑔𝑋^𝜇2(𝜉)𝑑²𝜉+_Ω√𝑔𝜆^𝑎𝑏(𝜉)𝑔_𝑎𝑏(𝜉)−ℎ_𝑎𝑏𝜉,𝑋^𝜇(𝜉), (2.16) where 𝜆^𝑎𝑏(𝜉) denote lagrange multipliers insuring that at the classical level 𝑔_𝑎𝑏(𝜉)=(𝜕_𝑎𝑋^𝜇𝜕_𝑏𝑋_𝜇)(𝜉).

Boundary conditions to be imposed on the complete action 𝐴(𝑆)+ℱ_ext(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 case 𝑔_𝑎𝑏=𝛿_𝑎𝑏 and the associated biharmonic operatorΔ𝑋^𝜇2=𝜕⁴𝑋^𝜇𝜕𝜉⁴₁𝜕+2⁴𝑋^𝜇𝜕𝜉²₁𝜕𝜉²₂+𝜕⁴𝑋^𝜇𝜕𝜉⁴₂, (2.17a) one can associate the following system of boundary operators: 𝑋^𝜇(𝜉)_𝜕Ω=𝐶^𝜇(𝜉),−𝜕𝑋𝜕𝑛^𝜇(𝜉)_𝜕Ω=𝑀^𝜇(𝜉). (2.17b)Here the meaning of 𝑀^𝜇 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): _Ω𝑓Δ²𝑔𝑑²𝜉=_Ω𝑔Δ²𝑓𝑑²𝜉+_𝜕Ω−𝑓𝜕Δ𝑔+𝜕𝑛𝜕𝑓𝜕𝜕𝑛Δ𝑔+𝑔𝜕𝑛Δ𝑔−𝜕𝑔𝜕𝑛Δ𝑔𝑑𝑠. (2.18)

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 Ω by analytical complex variable functions; one still has the following representation (not unique!) for biharmonic functions [1, 5]: 𝒱𝜉₁,𝜉₂=Re𝑧𝜑(𝑧)+𝜌(𝑧), (2.19) where 𝑧=𝜉₁+𝑖𝜉₂ and (𝜛(𝑧),𝜌(𝑧)), a pair of complex variable analytic functions on Ω.

If the surface 𝑆 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 by𝑅ℎ_𝑎𝑏=1+𝑓²_𝑋1+𝑓²_𝑋2²⋅det𝑓_𝑎𝑏𝑋¹,𝑋². (2.20a) and for the mean curvature, we have the following result: 𝐻=1+𝑓²_𝑋1+𝑓²_𝑋2^−3/21+𝑓²_𝑋2𝑓_𝑋1𝑋1−2𝑓_𝑋1𝑓_𝑋2𝑓_𝑋1𝑋2+1+𝑓²_𝑋1𝑓_𝑋2𝑋2. (2.20b)

It is worth to recall that in the important case of a surface 𝑆 embedded in 𝑅³ (useful in Surface’s Statistical Physics), but with the general parameterization: 𝑋¹=𝑋¹𝜉₁,𝜉₂;𝑋²=𝑋²𝜉₁,𝜉₂;𝑋³=𝑋³𝜉₁,𝜉₂, (2.21) the expression for the area functional and the surface extrinsic functional can be given explicitly by (exercise for our diligent readers!)𝐴(𝑆)=_Ω𝐸𝐺−𝐹²(𝜉)𝑑²ℱ𝜉,_ext(𝑆)=_Ω𝐻²(𝜉)𝑑²𝜉, (2.22a)(2.22b) where the surface objects are given by 𝐸=𝜕𝑋^𝜇𝜕𝜉₁,𝜕𝑋_𝜇𝜕𝜉₁;𝐹=𝜕𝑋^𝜇𝜕𝜉₁,𝜕𝑋_𝜇𝜕𝜉₂;𝐺=𝜕𝑋^𝜇𝜕𝜉₂,𝜕𝑋_𝜇𝜕𝜉₂;1𝐻=𝐸𝐺−𝐹²𝐸𝐷^′′−2𝐷^′;1𝐹+𝐷𝐺𝐷=𝐸𝐺−𝐹²𝑋¹_𝜉1𝜉1𝑋²_𝜉1𝜉1𝑋³_𝜉1𝜉1𝑋¹_𝜉1𝑋²_𝜉1𝑋³_𝜉1𝑋¹_𝜉2𝑋²_𝜉2𝑋³_𝜉2,𝐷^′=1𝐸𝐺−𝐹²𝑋¹_𝜉1𝜉2𝑋²_𝜉1𝜉2𝑋³_𝜉1𝜉2𝑋¹_𝜉1𝑋²_𝜉1𝑋³_𝜉1𝑋¹_𝜉2𝑋²_𝜉2𝑋³_𝜉2,𝐷^′′=1𝐸𝐺−𝐹²𝑋¹_𝜉2𝜉2𝑋²_𝜉2𝜉2𝑋³_𝜉2𝜉2𝑋¹_𝜉1𝑋²_𝜉1𝑋³_𝜉1𝑋¹_𝜉2𝑋²_𝜉2𝑋³_𝜉2. (2.22c)

Finally we remark that one could easily generalize all the above written results to the general case of the initial ambient space 𝑅^𝐷 being a new general Riemann Manifold (𝑀,𝐺_𝜇𝜈(𝑋^𝛾)). In this case, the “induced” surface metric tensor ℎ_𝑎𝑏(𝑋(𝜉))=𝜕_𝑎𝑋^𝜇(𝜉)𝜕_𝑏𝑋_𝜈(𝜉) will be replaced by the new “induced” tensor: ℎ_𝑥𝛽𝑋(𝜉),𝐺_𝜇𝜈=𝜕_𝑎𝑋^𝜇(𝜉)𝐺_𝜇𝜈𝑋^𝛾(𝜉)𝜕_𝑏𝑋^𝜈(𝜉) (2.23) 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 Section 2, 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)):𝐺_NG(𝐶)=_{𝑋𝜇(𝜉)𝜉∈𝜕Ω=𝐶𝜇}𝑑𝜇𝑋^𝜇(𝜉−1)exp2𝜋𝛼^′𝐴𝑆_𝑐,𝐺_𝑃(𝐶)=_{𝑋𝜇(𝜉)𝜉∈𝜕Ω=𝐶𝜇𝑔𝑎𝑏(𝜉)𝜉∈𝜕Ω=0}𝑑𝜇𝑋^𝜇(𝜉),𝑔_𝑎𝑏(𝜉−1)exp2𝜋𝛼^′𝐴𝑆_𝑐×exp−𝜇²_0Ω√𝑔(𝜉)𝑑²𝜉. (3.1a)(3.1b)

Here 𝛼^′ and 𝜇²₀ are constants to be identified with certain physical parameters of the theory. Note that 𝛼^′ has dimension of inverse if mass square and [𝜇²₀]=[𝛼^′]⁻¹. It is an important remark that the Feynman-Wiener path measures 𝑑𝜇[𝑋^𝜇(𝜉)] and 𝑑𝜇[𝑋^𝜇(𝜉),𝑔_𝑎𝑏(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 𝑔_𝑎𝑏(𝜉)(𝑔_𝑎𝑏(𝜉)|_𝜕Ω=0). Those results lead us to the effective action for the surface dynamics weight function: 𝐴𝑆_𝑐=12_Ω√𝑔𝑋^𝜇−Δ_𝑔𝑋_𝜇(𝜉)𝑑²𝜉. (3.1c)

Since at the classical level, the metric field decouples (one can choose it in the form 𝑔_𝑎𝑏=𝑒^𝜑𝛿_𝑎𝑏) and considers the usual classical plus quantum decomposition 𝑋_𝜇(𝜉)=𝑋^CL_𝜇(𝜉)+(𝜋𝛼^′)𝑋^𝑞_𝜇(𝜉), where the quantum correction 𝑋^𝑞_𝜇(𝜉) is such that 𝑋^𝑞_𝜇(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): 𝐼𝑔_𝑎𝑏=𝑑𝜇𝑋^𝜇(𝜉−1)exp2_Ω√𝑔𝑋^𝜇−Δ_𝑔𝑋_𝜇(𝜉)𝑑²𝜉. (3.1d)

It is worth recalling that there is a classical term (a functional depending on the loop boundary 𝐶) coming from the partial integration to arrive at ((3.1a), (3.1b), (3.1c), and (3.1d)) for the case 𝑔_𝑎𝑏=𝛿_𝑎𝑏, which clearly does not satisfy the quantum-fluctuating nature of these nonconstant 𝑔_𝑎𝑏(𝜉): _Ω𝜕_𝑎𝑋^𝜇_el𝜕_𝑎𝑋_𝜇,el(𝜉)𝑑²𝜉=𝐶_𝜕Ω^{classicalcontribution𝜇}𝜕𝐶𝜕𝑛^𝜇(𝜎)𝑑𝜎. (3.2)

The Feynman-Wiener measure 𝑑𝜇[𝑋^𝜇(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]: 𝑑𝜇𝑋^𝜇(𝜉)∶=_𝜉∈Ω⁴𝑔(𝜉)𝑑𝑋^𝜇(𝜉). (3.3)

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: 𝐼𝑔_𝑎𝑏=det^−𝐷/2−Δ_𝑔. (3.4)

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: 𝐼=𝑑𝜇𝑔_𝑎𝑏det^−𝐷/2−Δ_𝑔𝑒^{−𝜇20∫Ω√𝑔𝑑2𝜉}. (3.5)

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[𝑔_𝑎𝑏] as the (functional) element of volume of the following functional metric (so-called De Witt metric) with (𝑐≠−1/2): 𝛿𝑔_𝑎𝑏²=_Ω√𝑔𝛿𝑔_𝑎𝑏𝑔^𝑎𝑎′𝑔^𝑏𝑏′+𝑐𝑔^𝑎𝑏𝑔^{𝑎′𝑏′}𝛿𝑔_{𝑎′𝑏′}(𝜉)𝑑²𝜉. (3.6)

Let us then choose the conformal gauge fixing to evaluate the above metric path integral successfully [1], but already considering the metric positivity: 𝑔_𝑎𝑏(𝜉)=𝑒^𝜑(𝜉)𝛿_𝑎𝑏. (3.7)

As a result the functional infinitesimal displacements 𝛿𝑔_𝑎𝑏(𝜉) can be written in the following general form on the functional manifold of the metric [𝑔_𝑎𝑏(𝜉)] (on the formal infinite-dimensional functional tangent bundle): 𝛿𝑔_𝑎𝑏(𝜉)=𝛿𝜑(𝜉)𝑔_𝑎𝑏(𝜉)+∇_𝑎ℰ_𝑏+∇_𝑏ℰ_𝑎(𝜉). (3.8)

Let us rewrite the functional De-Witt metric in the form𝛿𝑔_𝑎𝑏²=_Ω𝛿𝑔_𝑎𝑏(𝜉)𝛾^{(𝑎𝑏,𝑎′𝑏′)}𝑔_𝑐𝑑(𝜉)𝛿𝑔_{𝑎′𝑏′}(𝜉),𝛾^{(𝑎𝑏,𝑎′𝑏′)}𝑔_𝑐𝑑(𝜉)=√𝑔𝑔^𝑎𝑎′𝑔^𝑏𝑏′+𝑐𝑔^𝑎𝑏𝑔^{𝑎′𝑏′}(𝜉). (3.9a)(3.9b)

Since there is only three independent components of the metric tensor 𝑔_𝑎𝑏(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: 𝛿𝑔_𝑎𝑏²=_Ω𝛿𝑔₁₁,𝛿𝑔₁₂,𝛿𝑔₂₂𝛾𝛿𝑔₁₁𝛿𝑔₁₂𝛿𝑔₂₂(𝜉)𝑑²𝜉=𝛿𝑔₁₁𝛾₁₁𝛿𝑔₁₁+𝛿𝑔₁₂𝛾₂₂𝛿𝑔₂₂+𝛿𝑔₂₂𝛾₃₃𝛿𝑔₂₂+𝛿𝑔₁₁𝛾₁₂+𝛾₂₁𝛿𝑔₁₂+𝛿𝑔₁₁𝛾₁₃+𝛾₃₁𝛿𝑔₂₂+𝛿𝑔₁₂𝛾₂₃+𝛾₃₂𝛿𝑔₂₂(𝜉)×𝑑²𝜉. (3.10)

Here the explicitly expressions of the effective [𝛾]-matrix are related to those of (3.9b) by the results 𝛾₁₁=𝛾^(11,11)=𝑒^𝜑𝑒^−𝜑𝛿¹¹𝑒^−𝜑𝛿¹¹+𝑐𝑒^−𝜑𝛿¹¹𝑒^−𝜑𝛿¹¹=(1+𝑐)𝑒^−𝜑;𝛾₁₂=𝛾^(11,12)=0;𝛾₂₁=𝛾^(12,11)=0;𝛾₁₃=𝛾^(11,22)=𝑐𝑒^−𝜑;𝛾₃₁=𝛾^(22,11)=𝑐𝑒^−𝜑;𝛾₂₃=𝛾^(12,22);𝛾₃₃=𝛾^(22,12)=(1+𝑐)𝑒^−𝜑. (3.11) As a consequence det𝛾^{(𝑎𝑏,𝑎′𝑏′)}=det𝛾_𝑖𝛿=(1+𝑐)²𝑒^3𝜑−𝑐²𝑒^3𝜑=(1+𝑐)𝑒^−3𝜑. (3.12)

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) 𝛿𝑔_𝑎𝑏²=_Ω𝑑²𝜉𝑒^𝜑(𝜉)𝛿𝜑𝑔^𝑎_𝑏+∇^𝑎ℰ_𝑏+∇_𝑏ℰ^𝑎⋅𝛿𝜑𝑔^𝑏_𝑎+∇^𝑏ℰ_𝑎+∇_𝑎ℰ^𝑏+4𝑐∇_𝑐ℰ^𝑐+𝛿𝜑². (3.13)

Let us analyze those terms involving the conformal factor.

By noting that ∇_𝑐𝑔_𝑎𝑏=∇_𝑐𝑔^𝑎𝑏=0 and (∇_𝑎ℰ_𝑏)𝑔^𝑎𝑎′𝑔^𝑏𝑏′(∇_𝑏′ℰ_𝑎′)=∇^𝑎′ℰ_𝑏∇^𝑏ℰ_𝑎′, added with ∇_𝑎ℰ^𝑎=0; ∇_𝑎′(𝑔^𝑏𝑏′ℰ_𝑏′)=𝑔^𝑏𝑏′∇_𝑎′ℰ_𝑏′=∇_𝑎′ℰ^𝑏, one obtains the chain of results written below 𝛿𝑔_𝑎𝑏²=_Ω𝑑²𝜉𝑒^𝜑(𝜉)𝛿𝜑𝑔^𝑎_𝑏+∇^𝑎ℰ_𝑏+∇_𝑏ℰ^𝑎⋅𝛿𝜑𝑔^𝑏_𝑎+∇^𝑏ℰ_𝑎+∇_𝑎ℰ^𝑏+4𝑐∇_𝑐ℰ^𝑐+𝛿𝜑²𝜉)=_Ω(𝑑²𝜉𝑒^𝜑(𝜉)∇^𝑎ℰ_𝑏+∇_𝑏ℰ^𝑎∇^𝑏ℰ_𝑎+∇_𝑎ℰ^𝑏−2∇_𝑑ℰ^𝑑2+2(1+2𝑐)∇_𝑑ℰ^𝑑+𝛿𝜑²𝜉)()(=21+2𝑐_Ω𝑑²𝜉𝑒^𝜑(𝜉)∇_𝑑ℰ^𝑑+𝛿𝜑²(𝜉)−2_Ω𝑑²𝜉𝑒^𝜑(𝜉)ℰ^𝑎∇_𝑑∇^𝑑+∇^𝑏,∇_𝑎ℰ_𝑏, (3.14) where we have used the covariant by parts, integration rule (𝑔|_𝜕Ω=0): _Ω𝑑²𝜉√𝑔𝑓∇_𝑑ℎ=−_Ω𝑑²𝜉√𝑔∇_𝑑𝑓ℎ. (3.15)

As a consequence we have the full result: 𝛿𝑔_𝑎𝑏²=(1+2𝑐)_Ω𝑑²𝜉𝑔(𝜉)(𝛿𝒱(𝜉))²+_Ω𝑑²𝜉𝑔(𝜉)ℰ₁(𝜉),ℰ₂(𝜉)ℒℰ₁(𝜉)ℰ₂(𝜉), (3.16) where 𝛿𝒱(𝜉)=𝛿𝜑(𝜉)+∇^𝑐ℰ⁼⁰_𝑐(𝜉)=𝛿𝜑(𝜉) (3.17) and the Faddeev-Popov operator acting on the two-dimensional vector field generators of the local group of coordinate transformations is explicitly given by ℒℰ_𝑎=−∇^𝑑∇_𝑑𝛿^𝑏_𝑎+∇^𝑏,∇_𝑎ℰ_𝑏(𝜉). (3.18)

As a consequence the functional volume element is written as 𝑑𝜇𝑔_𝑎𝑏(𝜉)=𝑑^cov𝜑(𝜉)𝑑^covℰ₁(𝜉)𝑑^covℰ₂(𝜉)det^1/2[ℒ]. (3.19) Here 𝑑^cov𝜑(𝜉)=_(𝜉∈Ω)𝑒^{(𝜑(𝜉)/2)}𝛿𝜑(𝜉)=_𝜉∈Ω𝛿2𝑒^{(𝜑(𝜉)/2)}=𝐷^𝐹𝑒^{(𝜑(𝜉)/2)}𝑑^covℰ_𝑎(𝜉)=_𝜉∈Ω𝑑ℰ_𝑎(𝜉)=𝐷^𝐹ℰ_𝑎(𝜉). (3.20) 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: _Ω𝑑²𝜉√𝑔∇_𝑎ℰ_𝑏+∇_𝑏ℰ_𝑎𝑔^𝑎𝑎′𝑔^𝑏𝑏′∇_𝑎′ℰ_𝑏′+∇_𝑏′ℰ_𝑎=𝑑²𝜉√𝑔∇_𝑎ℰ_𝑏𝑔^𝑎𝑎′𝑔^𝑏𝑏′∇_𝑎′ℰ_𝑏′𝜉)=𝑑(+similarterms₂𝜉√𝑔∇^𝑎ℰ_𝑏,∇_𝑎ℰ^{𝑔′𝑏𝑏∇𝑎′ℰ𝑏′=∇𝑎′𝑔′𝑏𝑏ℰ𝑏′=∇𝑎′ℰ𝑏𝑏}+∇_𝑎ℰ^𝑏∇_𝑏ℰ^𝑎+∇_𝑏ℰ^𝑎∇_𝑎ℰ^𝑏+∇_𝑏ℰ^𝑎∇^𝑏ℰ_𝑎(𝜉)=𝑑²𝜉√𝑔ℰ_𝑏−∇^𝑎∇_𝑎ℰ^𝑏−ℰ^𝑏∇_𝑎∇_𝑏ℰ^𝑎−ℰ^𝑎∇_𝑏∇_𝑎ℰ^𝑏−ℰ^𝑎∇_𝑏∇^𝑏ℰ_𝑎=−𝑑²𝜉√𝑔2ℰ_𝑏−∇_𝑎∇_𝑎ℰ^𝑏−ℰ∇_𝑎∇_𝑏+∇_𝑎∇_𝑏ℰ^{𝑏2∇𝑎∇𝑏=2∇𝑏∇𝑎+2∇,∇𝑏−𝑎}=−2𝑑²𝜉√𝑔ℰ_𝑎∇^𝑐∇_𝑐𝛿_𝑎𝑏+∇_𝑏,∇^{∇𝑎ℰ𝑎=0}_𝑎−ℰ_𝑏. (3.21)

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) Δ_{(𝑔=𝑒𝜑𝛿𝑎𝑏)}=𝑒^{−𝜑(𝜉)}𝜕_𝑧𝜕_𝑧,ℒ_{(𝑔=𝑒𝜑𝛿𝑎𝑏)}=𝑒^{−2𝜑(𝜉)}𝜕_𝑧𝑒^𝜑(𝜉)𝜕_𝑧, (3.22) 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]: 𝜕=𝜕𝜕𝑧𝜕𝜉₁𝜕−𝑖𝜕𝜉₂𝜕𝜕𝑧=𝜕𝜕𝜉₁𝜕+𝑖𝜕𝜉₂.𝑖=√−1, (3.23)

In [3], we present the explicit evaluation of elliptic operators of the form (for 𝑗 being a positive integer) considered below: ℒ_𝑗=−𝑒^{−(𝑗+1)𝜑(𝜉)}𝜕_𝑧𝑒^{𝑗𝜑(𝜉)}𝜕_𝑧. (3.24) The result is det_𝐹ℒ_𝑗=exp−1+6𝑗𝑗+112𝜋_Ω𝑑²𝜉12𝜕_𝑎𝜑²(𝜉)−lim_𝜀→0−112𝜋𝜀_Ω𝑑²𝜉𝑒^𝜑(𝜉). (3.25)

Let us point out the validity of the relationship below: det_𝐹^−𝐷/2Δ_{𝑔=𝑒𝜑𝛿𝑎𝑏}=det_𝐹^−𝐷/4ℒ_𝑗=0,det^+1/2𝐿_{𝑔=𝑒𝜑𝛿𝑎𝑏}=det_𝐹^+1/2ℒ_𝑗=1. (3.26)

As our final result of this covariant Polyakov path integral, we get the 2D-induced quantum gravity Liouville model: 𝐼=𝐷^𝐹𝑒^{(𝜑/2)(𝜉)}−()exp26−𝐷48𝜋_Ω12𝜕_𝑎𝜑²(𝜉)𝑑²𝜉×𝑒^−𝜇2ren_Ω𝑒^𝜑(𝜉)𝑑²𝜉. (3.27)

Here 𝜇²_𝑟=𝜇²₀+lim_𝜀→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)=𝐷^𝐹[𝜑(𝜉)] as initially supposed by A. M. Polyakov.

As a consequence one should rewrite it in terms of the canonical Goldstone boson field 𝜌(𝜉)=𝑒^{(𝜑/2)(𝜉)} which leads us to a kind of “noncompact” 𝜎-model with a “renormalized” mass term: 𝐼=𝐷^𝐹𝜌(𝜉)exp−26−𝐷12𝜋_Ω𝜕_𝑎𝜌²𝜌²𝑑²𝜉−𝜇²_𝑟Ω𝜌²(𝜉)𝑑²𝜉 (3.28) 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): 𝐷^𝐹𝑒^{(𝜑/2)(𝜉)}=det_𝐹𝑒^𝜑/2𝛿_𝑎𝑏𝛿𝜉−𝜉^{FunctionalJacobian′}𝐷^𝐹𝜑(𝜉=)lim_𝜀→0+expTrℓ𝑔𝑒^𝜑/2𝛿_𝑎𝑏𝑒^−ℰΔ𝑔=𝑒^𝜑×𝐷^𝐹𝜑(𝜉)=lim_𝜀→0+exp_Ω𝑑²𝜉𝑒^𝜑(𝜉)𝜑(𝜉)21+14𝜋𝜀𝑅𝜑12𝜋×𝐷^𝐹𝜑(𝜉)=lim_𝜀→0+1exp8𝜋𝜀_Ω𝑒^𝜑(𝜉)𝜑(𝜉)𝑑²𝜉−1×exp24𝜋_Ω𝜑(−Δ)𝜑(𝜉)𝑑²𝜉×𝐷^𝐹𝜑(𝜉) (3.29) 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): 𝜑→48𝜋𝐷26−𝐷𝜑,_𝐹𝑒^𝜑/2→𝐷^𝐹𝑒−−−−−−−^{(48𝜋/(26−𝐷))𝜑𝐷→−∞}→𝐷^𝐹[0]=classical. (3.30)

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 gauge 𝑔_𝑎𝑏(𝜉)=𝑒^𝜑(𝜉)𝛿_𝑎𝑏. 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: Δ⁻¹_{𝑔=𝑒𝜑𝛿𝑎𝑏}𝜉,𝜉^{′𝜀=(𝜉−𝜉′)→0}∼1−12𝜋𝜀𝑒2𝜋ℓ𝑔^𝜑(𝜉)𝜉−𝜉^′∼112𝜋𝜀−1𝜑(𝜉)−12𝜋2𝜋ℓ𝑔𝜉−𝜉^′. (3.31) This lead us to consider the regularized from of the Green function used on dual models based on the Polyakov’s path integral: 𝜉Δ⁻¹_{𝑔=𝑒𝜑𝛿𝑎𝑏}𝑒^{−ℰΔ𝜑𝛿𝑔=𝑒𝑎𝑏}𝜉^′=−12𝜋ℓ𝑔𝜉−𝜉^′−1𝜉≠𝜉'ℓ4𝜋^𝑔1𝜀+1𝜑(𝜉)4𝜋𝜉=𝜉^′. (3.32) Note that we have used the following formulae to write explicitly the functional trace of the general two-dimensional (strongly) elliptic operators [3]:𝐴=−𝑔₁₁𝜕²𝜕𝜉²₁+𝑔₂₂𝜕²𝜕𝜉²₂𝟏_2×2−𝐴₁𝜕𝜕𝑥₁−𝐴₂𝜕𝜕𝑥₂−𝐴₀lim_𝑡→0+tr𝑒^−𝑡𝐴∼1𝑑2𝜋𝑡²𝜉√𝑔₁₁𝑔₂₂+12𝜋𝑑²𝜉√𝑔₁₁𝑔₂₂−16𝑅𝑔×𝑑²𝜉−121√𝑔₁₁𝑔₂₂Tr_matrix𝜕𝜕𝑥₁√𝑔₁₁𝑔₂₂𝐴₁+𝜕𝜕𝑥₂√𝑔₁₁𝑔₂₂𝐴₂+𝑑²𝜉−14𝐴Tr¹²𝑔₁₁+𝐴₂²𝑔₂₂+𝐴₀+𝑂(𝑡). (3.33a)(3.33b)

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]: 𝐺_NG(𝐶)=𝑑𝜇𝑥^𝜇(𝜉),𝑔_𝑎𝑏(𝜉−1)exp2𝜋𝛼^′_Ω𝑑²𝜉√𝑔𝑔^𝑎𝑏𝜕_𝑎𝑋^𝜇𝜕_𝑏𝑋_𝜇(𝜉)×𝛿^(𝐹)_cov𝑔_𝑎𝑏−𝜕_𝑎𝑋^𝜇𝜕_𝑏𝑋_𝜇. (4.1) Note that our constraint is rewritten as a covariant (diffeomorphism) invariant delta functional: 𝛿^(𝐹)_cov𝑔_𝑎𝑏−𝜕_𝑎𝑋^𝜇𝜕_𝑏𝑋_𝜇=𝐷^cov𝜆_𝑎𝑏𝑒^{𝑖{∫Ω𝑑2√𝜉(𝑔𝜆𝑎𝑏(𝜕𝑎𝑋𝜇𝜕𝑏𝑋𝜇−𝑔𝑎𝑏))(𝜉)}}, (4.2) where the covariant Feynman-Wiener measure on (4.2) is the element of volume of the functional metric: 𝑑𝑠²=𝛿𝜆_𝑎𝑏²=√𝑔𝛿𝜆_𝑎𝑏𝑔^𝑎𝑎′𝑔^𝑏𝑏′𝛿𝜆_𝑎′𝑏(𝜉)𝑑²𝜉. (4.3)

After introducing again the complex-euclidean light-cone coordinates on the domain Ω: 𝜕₊=𝜕_𝜉1−𝑖𝜕_𝜉22=𝜕_𝑧,𝜕₋=𝜕_𝜉1+𝑖𝜕_𝜉22=𝜕_𝑧 (4.4) and imposing the surface conformal gauge 𝑔_𝑎𝑏(𝜉)=𝜌(𝜉)𝛿_𝑎𝑏, which means that 𝑔₁₁=𝑔₂₂, 𝑔₁₂=𝑔₂₁=0, or equivalently using light-cone coordinates again, 𝜕₊𝑋^𝜇𝜕₊𝑋_𝜇=𝜕₋𝑋^𝜇𝜕₋𝑋_𝜇⟺𝜕_𝜉1𝑋^𝜇𝜕_𝜉2𝑋_𝜇=𝑔₁₂=𝑔₂₁,√=0𝑔=𝜕₊𝑋^𝜇𝜕₋𝑋^𝜇𝑔₁₁=𝜕₊𝑋^𝜇𝜕₋𝑋_𝜇=𝑔₂₂≠0. (4.5)

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): 𝐺_NG(𝐶)=𝐷^𝐹𝜌(𝜉)𝐷^𝐹𝑋^𝜇(𝜉−())×exp26−𝐷12𝜋_Ω𝑑²𝜉12𝜕_𝑎𝜌𝜌²+lim_𝜀→0(2−𝐷)2𝜋𝜀_Ω𝑑²𝜉𝜌²−1×exp2𝜋𝛼^′_Ω𝑑²𝜉𝜕_𝑎𝑋^𝜇2(𝜉)×𝛿^(𝐹)𝜌²(𝜉)𝛿_𝑎𝑏−ℎ_𝑎𝑏𝑋^𝜇(𝜉^{=𝜕+𝑋𝜇𝜕−𝑋𝜇}). (4.6)

Note the usual noncovariant function “flat” constraint relating the surface vector position to the auxiliary metric field 𝑔_𝑎𝑏(𝜉) on the conformal gauge.

After realizing the immediate 𝜌(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(𝑋^𝜇(𝑧,𝑧)))×𝑑𝑧𝑑𝑧}, can be assigned to exp{0}=1 in the dimensional regularization scheme): 𝐺_𝑁(𝐶)=_{(𝜇,𝜉,𝜉′)}𝑑𝑋^𝜇(𝑧,𝑧)⁴ℎ(𝑋^𝜇(𝑧,𝑧))×𝛿^(𝐹)𝜕₊𝑋^𝜇𝜕₊𝑋_𝜇𝛿^(𝐹)𝜕₋𝑋^𝜇𝜕₋𝑋^𝜇−1exp2𝜋𝛼^′_renΩ𝑑𝑧𝑑𝑧𝜕₊𝑋^𝜇𝜕₋𝑋_𝜇(𝑧,𝑧−())exp26−𝐷48𝜋_Ω𝑑𝑧𝑑𝑧(1/2)𝜕₊𝜕₊𝑋^𝜇𝜕₋𝑋_𝜇𝜕₊𝑋^𝜇𝜕₋𝑋_𝜇×𝜕₋𝜕₊𝑋^𝜇𝜕₋𝑋_𝜇𝜕₊𝑋^𝜇𝜕₋𝑋_𝜇(𝑧,𝑧). (4.7)

Now we realize that one must choose 𝐷=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: 𝑉=√ℎℎ^{[𝑎1,𝑎2]}⋯ℎ^{𝑎2𝑚−1,𝑎2𝑛}𝑋^𝜇(𝑧.𝑧)×𝜕_𝑎1𝑋^𝜇⋯𝜕_𝑎2𝑛𝑋^{𝜇2𝜇⋅spinpolarizationtensors}ℰ_{𝜇1…𝜇2𝑚}⋅𝑒^{𝑖𝐾𝜇𝑋𝜇}(𝑧,𝑧), (4.8) 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).

Appendices

A. On the Formal Evaluation of the Euclidean Dirac Functional Determinant on Two Dimensions

Let us start by this appendix by considering the euclidean Dirac partition functional on a two-dimensional space time: _𝑍𝐴𝜇¹_{=∶𝑍𝐴𝜇×=0𝐷𝜓𝐷𝜓𝑒}^{−∫(𝜓𝐷(𝐴)𝜓)}/^{(𝜉)𝑑2𝜉}_=∶det^𝐷/^{(𝐴)=det1/2}/^𝐷(𝐴)2, (A.1) where the (nonself adjoint) euclidean Dirac operator is explicitly given by (including the massive case!) ^𝐷/^(𝐴)=𝛾_𝜇^𝜕_𝜇^+𝑔𝐴_𝜇^+im. (A.2)

The two-dimensional euclidean matrixes on the Dirac operator (A.2) satisfy the relationship below: 𝛾_𝜇,𝛾_𝜈=𝛾_𝜇𝛾_𝜈+𝛾_𝜈𝛾_𝜇=2𝛿_𝜇𝜈,𝛾_𝜇𝛾₅=𝑖ℰ_𝜇𝜈𝛾_𝜈,𝛾⁵=𝑖𝛾₀𝛾₁,ℰ₀₁=−ℰ₁₀=1. (A.3)

Let us note the more suitable redefinition of the euclidean Dirac operator: ^𝐷/^(𝐴)=𝑖/^{𝐷(𝐴)==𝑖𝛾𝜇𝜕𝜇+𝑖𝑔𝜕𝜇𝐴𝜇𝑒𝑖𝑔𝛾5}/^𝐷_{(𝑖𝛾𝜇𝜕𝜇)𝑒}^𝑖𝑔𝛾₅/^𝐷_, (A.4) where the background abelian gauge field configurations are chosen on the landau gauge 𝜕_𝜇𝐴_𝜇=0: 𝐴_𝜇=ℰ_𝜇𝜈𝜕_𝜈𝜑. (A.5)

Let us note that (with a undetermined infinite phase!) ^det_𝐹^𝐷/^(𝐴)=det/^𝐷(𝐴),_det𝐹^𝐷/^(𝐴)2=det/^𝐷(𝐴)2. (A.6)

Since we have the result ^𝐷/^{(𝐴)2=−𝜕2−2𝑔𝐴}_𝜇^(𝜉)𝜕_𝜇^{−𝑔2𝐴2}_𝜇^(𝜉)_{−𝑔4[𝛾𝜇,𝛾𝜈]𝜎𝜇𝜈⋅𝐹𝜇𝜈(𝐴(𝜉𝜕))−𝑔𝜇𝐴𝜇(𝜉)} (A.7) thus ^lim_𝑡→0+^{exp−𝑡𝐷(𝐴)2𝜉}_{∼1𝟏4𝜋}/_{𝑡2×2+1𝑔4𝜋2𝐴2𝜇+𝑔4𝜎𝜇𝜈𝐹𝜇𝜈+𝑔𝜕𝜇𝐴𝜇−142𝑔𝐴𝜇2−12𝜕𝜇2𝑔𝐴𝜇,} (A.8) one can straightforwardly apply the Romanov-Schwartz theorem [1] to write the following (formal) differential equation for the functional determinant under analysis: ^2𝑑ℓ_𝑑𝜎𝑔^{det𝐷𝐴𝜎}_=lim𝜀→0/_{+4𝑑2𝜉TrDirac𝑖𝑔𝛾5𝜙1+𝑔4𝜋𝜀𝜎16𝜋=𝛾𝜇,𝛾𝜈𝜎𝜇𝜈𝐹𝜇𝜈(𝐴).} (A.9) Here 𝐴^(𝜎)=𝜎ℰ_𝜇𝜈𝜕_𝛾𝜑. (A.10) Since Tr_Dirac(𝛾₅)=0, one has the immediate result ^det𝐷(𝐴/⁾_det/^{𝜕=𝑒−(𝑔2/2𝜋)∫(𝜕𝜙)2𝑑2𝜉=𝑒−(𝑔2/2𝜋)∫𝐴2𝜇(𝜉)𝑑2𝜉×𝛿(𝐹)𝜕𝜇𝐴𝜇.} (A.11)

In the general nongauge fixed case, one has the result ^{det𝐷(𝐴)}/_det/^{𝜕=𝐷𝐹[𝜔(𝑥𝑒)]−(𝑔2/2𝜋)∫(𝐴𝜇+𝜕𝜇𝜔)2𝑑2𝜉}_{=𝑒−(𝑔2/2𝜋)∫𝑑2𝜉[𝐴𝜇(𝛿𝜇𝜈−((𝜕𝜇𝜕𝜈)/(−𝜕2)))𝐴𝜈](𝜉).} (A.12)

Let us show in detail the above calculation: ¹₀^ℓ_𝑔^{det𝐷𝐴(𝜎)2𝑑𝜎}_=l/_{im𝜀→0+4𝑑2𝜉tr𝑖𝑔𝛾5𝜑1+𝑔4𝜋𝜀𝜎16𝜇,𝜈𝐹𝜇𝜈(𝐴×)10.𝑑𝜎𝜎} (A.13) So _ℓ𝑔^{det𝐷(𝐴)}/_det/^{𝜕=𝑔2𝑑16𝜋2𝜉TrDirac𝑖𝛾5−2𝑖ℰ𝜇𝜈𝛾5𝜙×𝐹𝜇𝜈(𝐴)}_{=𝑔2𝑑4𝜋2𝜉𝜙ℰ𝜇𝜈𝐹𝜇𝜈(𝐴)=𝑔2𝑑2𝜋2𝜉𝜙𝜕2𝑔𝜙=−2𝑑2𝜋2𝜉𝜕𝜙2𝑔=−2𝑑2𝜋2𝜉𝐴𝜇2.} (A.14)

It is important to remark that if one had used the following self-adjoint Dirac euclidean operator, ^𝐷/^{(𝐴)=𝑒𝑒𝛾5𝜙𝑖𝛾}_𝜇^𝜕_𝜇^{𝑒𝑔𝛾5𝜙} (A.15) one would obtain a “Tachyon” (imaginary mass) term for the dynamically generated mass term for the abelian gauge field: ^det_𝐹^𝐷(𝐴/⁾_det/^{𝜕+𝑔=exp22𝜋𝑑2𝜉𝐴𝜇2.} (A.16)

At this point it is worth to recall the Seeley asymptotic expansion lim_𝑡→0+𝜉(𝜉=1exp−𝑡𝐴)𝟏4𝜋𝑡^2×2+1𝑉4𝜋⁰−𝑉²_𝜇4−𝜕_𝜇𝑉_𝜇2 (A.17) for a formal second order operator of the form on 𝑅²: 𝐴=−𝜕²−𝑉_𝜇𝜕_𝜇−𝑉₀. (A.18)

B. On Atiyah-Singer Index Theorem in the Framework of Feynman

B.1. Pseudoclassical Path Integrals: Author’s Original Remarks

One of the most celebrated (pure) mathematical theorems on modern geometry and topology of compact orientable manifolds on 𝑅^𝑁 is the so-called index theorem [4]. By the other side, it is widely known that the techniques to obtain such results are extremely intricated and based on the most difficult aspects of the theory of elliptic operators on closed compact orientable 𝑁-dimensional manifolds (see [5], Chapter 11). Our aim in this section is to somewhat simplify a very important index theorem on the Laplacian operator acting on forms [6].

As in the last reference [6], one can argue that the index of the Laplacian operator Δ_𝑓 acting on differential forms on a 𝑁-dimensional Riemannian compact orientable manifold ℳ is given by the object called supertrace which by its turn can be written as a Grassmannian (supersymmetric) euclidean path integral [4]: indexΔ_𝑓=∶lim_{ℏ→0𝑋𝜇(0)=𝑋𝜇(1)=𝑥𝜇}^{≡𝐷𝐹cov[𝑋𝜇(𝑠)]}_{0≤𝑠≤11≤𝑛≤𝑁}det^1/2[𝐺_𝜇𝜈(𝑋^𝜇(𝑠))]𝑑𝑋^𝜇×(𝑠)_{≡𝐷𝐹cov(𝜉𝑖(𝛼)(𝜉)]0≤𝑠≤1𝛼=1,21≤𝑖≤2𝑁}det^1/2√𝐺𝑋^𝜇(𝑠)𝑑𝜉^𝑖_(𝛼)(𝑠)𝑑𝜉^∗𝑖_(𝛼)(𝛼)−1×expℏ¹₀1𝑑𝑠2𝐺_𝜇𝜈𝑋^𝛼(𝑠)𝑑𝑋^𝜇(𝑠)𝑑𝑠𝑑𝑋^𝜇(𝑠)+𝑖𝑑𝑠2𝐺_𝜇𝜈𝑋^𝛼(𝑠)𝜉^𝑖_(𝛼(𝑠)𝑑𝜉𝑑𝑠^𝑖(𝛼)+18𝑅_{𝑖𝑗𝑘ℓ}𝑋^𝛼(𝑠)𝜉^𝑖₍₁₎(𝑠)𝜉₍₂₎(𝑠)^𝑘𝜉^𝑗₍₁₎(𝑠)𝜉^ℓ₍₂₎(𝑠). (B.1)

An important remark to be done now originally due to ourselves [7] is about the “bosonic” behavior of the Grassmannian composite products 𝜉^𝑖₍₁₎(𝑠)𝜉^𝑗₍₂₎=𝜎^𝑖𝑗(𝑠) as can be easily verified from the computation below ([𝑎,𝑏]₋=𝑎𝑏−𝑏𝑎; [𝑎,𝑏]₊=𝑎𝑏+𝑏𝑎): 𝜉^𝑖₍₁₎𝜉^𝑗₍₂₎,𝜉^𝑝₍₁₎𝜉^𝑞₍₂₎₋=𝜉^𝑖₍₁₎𝜉^𝑝₍₁₎𝜉^𝑞₍₂₎,𝜉^𝑗₍₂₎₊−𝜉^𝑝₍₁₎𝜉^𝑞₍₂₎,𝜉^𝑖₍₁₎₊𝜉^𝑗₍₂₎=𝜉^𝑖_(𝑗)𝜉^𝑝₍₁₎𝜉^𝑞₍₂₎𝜉^𝑗₍₂₎−𝜉^𝑝₍₁₎𝜉^𝑞₍₂₎𝜉^𝑖₍₁₎𝜉^𝑗₍₂₎+𝜉^𝑖₍₁₎𝜉^𝑗₍₂₎𝜉^𝑝₍₁₎𝜉^𝑞₍₂₎−𝜉^𝑖₍₁₎𝜉^𝑝₍₁₎𝜉^𝑞₍₂₎𝜉^𝑗₍₂₎=0+0=0. (B.2)

On the basis of the above made remark (B.2) one can rewrite the Grassmannian path integral equation (B.1) in the terms of “composite bosonic operators” [8]: indexΔ_𝑓=lim_{𝑡→0𝑋𝜇(0)=𝑋𝜇(1)=𝑋𝜇}𝐷^𝐹_cov𝑋^𝜇(𝑠)_{𝜉𝑖𝛼(0)=−𝜉𝑖𝛼(1)}𝐷^𝐹𝜉^𝑖_𝛼(𝑠×_{𝜎𝑖𝑘(0)=𝜎𝑖𝑘(1)})𝐷[𝜎_𝑖𝑘(𝑠)]^𝐹_cov^{BosonicVariable}_{𝜆𝑖𝑘(0)=𝜆𝑖𝑘(1)}𝐷[𝜆^{𝐹BosonicVariable}_𝑖𝑘𝑖(𝑠)]×expℎ₀𝑑𝑠𝜆^𝑖𝑘𝜉^𝑖₍₁₎(𝑠)𝜉^𝑘₍₂₎(𝑠)−𝜎_𝑖𝑘(𝑠)−1×exp4ℎ¹₀12𝑅_{𝑖𝑗𝑘ℓ}𝑋^𝛼(𝑠)𝜎^𝑖𝑘_(𝑠)𝜎^𝑗ℓ_(𝑠). (B.3)

After integrating the Grassmannian variables {𝜉^𝑖₍₁₎(𝑠),𝜉^𝑖₍₂₎}_{1≤𝑖≤𝑁}, one gets the below written functional determinant as outcome (prove it with the definition 𝜃(0)=0, here 𝜃(𝑠) denotes the heaviside step function-distribution): det_cov𝐺_𝑖𝑗𝑋^𝛼(𝑠)𝑑𝛿𝑑𝑠^𝑖𝑗+𝜆_𝑖𝑘(𝑠)=1. (B.4)

Now the 𝜎_𝑖𝑘 functional integral is Gaussian and yields the result −1exp2ℏ¹₀𝑑𝑠¹₀𝑑𝑠^′𝜆_𝑖𝑘(𝑠)12𝑅_{𝑝𝑞𝑟𝑠}𝑋^𝛼_(𝑠)()⋅⁻¹_{𝑖𝑘𝑗ℓ}𝛿𝑠−𝑠^′𝜆_𝑗ℓ(𝑠). (B.5)

At the classical limit ℏ→0, the closed periodic quantum trajectories 𝑋^𝜇(0)=𝑋^𝜇(𝑠)=𝑥^𝜇; all reduce to their initial common point 𝑥^𝜇. As a consequence one gets that at ℏ⟶0. 𝑋^𝜇(𝑠)→𝑥^𝜇,𝐷^𝐹_cov𝑋^𝜇(𝑠)⟶_ℳ𝑑^𝑁(𝑥)⋅det^1/2𝐺_𝜇𝜈𝑥^𝛼,𝐷^𝐹_cov𝜆_𝑝𝑞(𝑠−1)exp4tr¹₀𝑑𝑠¹₀𝑑𝑠^′𝜆𝑖𝑘_𝐴(𝑠)𝑅_(𝐴,𝐵)𝑋^𝑥(𝑠)⁻¹𝜆𝑗ℓ_𝐵𝑠^′⟶det^1/2𝑅_𝐴𝐵𝑥^𝜇=∶𝑃_𝑓𝑅_𝐴𝐵𝑥^𝜇. (B.6)

By grouping together (B.6), one gets the index of the Laplacian operator acting on differential forms defined on a compact orientable 𝑁-dimensional which is universally proportional to the famous Chern index of the endowed riemannian structure of M. Namely (see [8] for topological details), indexΔ_𝑓=𝜉_Chern(ℳ)=𝑒_ℳ𝑑^𝑁(𝑥)det^1/2𝐺_𝜇𝜈𝑥^𝜇×𝑃_𝑓𝑅_𝐴𝐵𝑥^{Pfaffinianformofthecurvaturetwo-form𝜇}. (B.7)

The universal constant is adjusted by defining as unity the Chern induces or the Euler-Poincaré characteristic) of spheres 𝑆^𝑁−1 with the usual 𝑅^𝑁 embedding metric (exercise to our differential geometric oriented readers!). Now it is well known that 𝜒_Chern(ℳ) are integer numbers.

In two dimensions, (B.7) reduces to the famous Gauss theorem for Riemann surfaces and its connections to the Euler topological surface genera: indexΔ_𝑓=𝜒_Chern(Σ)=_Σ√𝑔𝑅𝑔𝑥^𝜇𝑑²𝑥=2𝜋2−2𝑔, (B.8) where 𝑔 is the number of “holes” of Σ.

C. A Perturbative Analysis of the Liouville Quantum

C.1. Field Model on the Botelho Sigma Model Parametrization

We start this appendix by considering the Liouville quantum field model partition functional in terms of the correct 𝑝(𝑧)-field parametrization equation (3.28) of the text; called by is the 𝜎-model Liouville field theory: 𝑍=𝐷^𝐹𝑝(𝑧−())exp26−𝐷12𝜋_Ω12𝜕_𝑎𝑝𝑝²(𝜉)𝑑²𝜉−𝜇²_𝑅Ω𝑝²(𝑧)𝑑²𝜉. (C.1)

In order to define (C.1) perturbatively in the number of “inverse dimension” 1/𝐷 by a loop expansion, we consider the usual hypothesis of the field decomposition: 𝑝(𝜉)=𝑝₀(𝜉)+^{∶=𝜀theparameterloopexpansion}12𝜋26−𝐷^1/2ℎ(𝜉). (C.2) Here 𝑝₀(𝜉) is the classical background field configuration associated to the vanishing of the parameter loop expansion, that is, the well-known 𝐷→−∞ limit of the path-integral equation (C.1) [7].

After that step has been taken, one has the result (with the new cosmological constant 𝜇=(√2)𝜇_𝑅⋅𝜀) holding true for all orders of the loop expansion: 𝑍=𝐷^𝐹[4]−1exp𝜀²_Ω12𝜕𝑝₀𝑝₀^2∞_𝑛=1(−1)^𝑛−1⋅𝑛⋅𝜀^𝑛−1ℎ^𝑛−1𝑝₀^𝑛−1−1×exp𝜀_Ω12𝜕𝑝₀𝜕ℎ𝑝²₀^∞_𝑛=1(−1)^𝑛−1⋅𝑛⋅𝜀^𝑛−1ℎ^𝑛−1𝑝₀^𝑛−1−1×exp2_Ω𝜕ℎ𝑝²₀^2∞_𝑛=1(−1)^𝑛−1⋅𝑛⋅𝜀^𝑛−1ℎ^𝑛−1𝑝₀^𝑛−1−𝜇×exp²𝜀²_Ω12𝑝²₀+𝜀²ℎ²+2𝜀ℎ𝑝₀. (C.3)

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 𝜀^𝑝, with 𝑝≥2) renders the theory nonnormalizable, which by its turn is an already expected outcome from the result of nonperturbative renormalizability of quantum gravity theories in general.

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 (1/𝐷) expansion (or correctly the 𝜀 expansion with 𝜀=(12/(26−𝐷)); 𝐷<26): 𝑍^one-loop∼exp−^{classicalterm}1𝜀²_Ω12𝜕𝑝₀𝑝₀²+𝜇²𝑝₀²(𝜉)𝑑²𝜉×𝐷^𝐹[ℎ]−1exp2_Ω𝜕ℎ𝑝₀²+𝜇²ℎ²(𝜉)𝑑²𝜉×exp+_Ω𝜕𝑝₀𝑝³₀⋅ℎ²+1×exp𝜀_Ω𝜕𝑝₀²𝑝³₀−3⋅ℎ×exp2_Ω𝜕𝑝₀𝑝₀²⋅ℎ²𝑝²₀−𝜇×exp²𝜀ℎ𝑝₀. (C.4)

Note again that one should choose the Liouville quantum fluctuations in such a way that the constraints between the field ℎ and 𝑝₀ displayed below should vanish and leading thus to a massive scalar field ℎ (however, interacting with the background field configuration 𝑝₀ as one can see from (C.4)): 𝜕𝑝₀²𝑝³₀⋅ℎ=0=ℎ𝑝₀. (C.5)

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 𝜀=0 on the theory effective path integral averages like scattering amplitudes, or equivalently, one must take into account the theory already at the classical fix point 𝑝(𝜉)=𝑝₀(𝜉) theory path integral (the classical 𝐷→−∞ Liouville field configuration, see [1] for details in the fermionic string path integral).

However, it is worth to remark and keep on mind that the full Liouville path integral although being defined for 𝜀=0 must be defined for all topological surface genera (the old dual model unitarization) by taking into full account the quantization of the conformal surface metrical structures’ Teichmüller parameters degrees of freedom which makes the theory highly nontrivial [9].