[Paper Review] Polyakov's String: Twenty Five Years After

V. L. Pokrovsky, A. A. Belavin+2 more|ArXiv.org|Oct 25, 2005

Particle physics theoretical and experimental studiesPhysics and Astronomy33 references16 citations

TL;DR

This paper commemorates the 25th anniversary of Alexander Polyakov's seminal 1981 work on the quantum geometry of bosonic strings, presenting a collection of advanced theoretical physics contributions centered on conformal field theory, Liouville gravity, and two-dimensional quantum gravity. It explores moduli integrals, ground ring structures, and four-point functions in minimal Liouville gravity, while also addressing massive Majorana fermions coupled to 2D gravity and their relation to the Ising model on random lattices, highlighting Polyakov's enduring influence across statistical mechanics and quantum field theory.

ABSTRACT

An International Workshop dedicated to the anniversary of the Polyakov's String (of course today there is no need to remind the meaning and the role of this theory) was held in Chernogolovka in June 2005. Apart from the 25-th anniversary of the first appearance of the Polyakov's String theory, this conference, to our mind, might be also thought of as a 35 years from the discovery of the Conformal Invariance, 30 years of the Monopole and the Instantons and, finally, as the 20-th anniversary of the CFT. A case of mysterious coincidence, this year is also a jubilee of Sasha himself, whose contribution to the Theoretical Physics of 20-th century is far from being exhausted by the achievements listed above.

Motivation & Objective

• To honor and reflect on the foundational impact of Alexander Polyakov’s 1981 string theory and its 25-year legacy in theoretical physics.
• To investigate the mathematical and physical structure of minimal Liouville gravity, particularly moduli integrals and four-point functions.
• To explore the connection between 2D quantum gravity and the random lattice Ising model, focusing on massive Majorana fermions.
• To examine Polyakov’s contributions to statistical and condensed matter physics, including phase transitions, critical phenomena, and conformal invariance.
• To analyze the role of zeta regularization and contour integrals in handling divergent expressions in quantum field theory and matrix models.

Proposed method

• Utilizes zeta regularization to analytically continue divergent sums and integrals, particularly in the context of partition functions and spectral zeta functions.
• Applies contour integration techniques to evaluate complex integrals involving complete elliptic integrals K(x) and K(1−x), reducing them to products of one-dimensional integrals.
• Employs the canonical product representation of entire functions to relate the coefficients of a generating function to the zeta function of its zeros.
• Derives a parametric family of integrals I(ν) involving K(x)K(1−x) and power-law singularities, which are analytically continued for non-integer ν.
• Uses integral formulas involving gamma functions and trigonometric factors to express I(ν) in a compact closed form involving Γ(ν) and γ(ν−1/4).
• Analyzes the behavior of I(ν) near ν=1/6, showing a double zero that justifies the cancellation of logarithmic divergences in the grand partition function.

Experimental results

Research questions

• RQ1 How do moduli integrals and the ground ring structure constrain the four-point function in minimal Liouville gravity?
• RQ2 What is the role of the b²=3/4 point in the context of solvable matrix models and logarithmic divergences in the grand partition function?
• RQ3 How does the massive Majorana fermion coupled to 2D gravity relate to the random lattice Ising model?
• RQ4 In what way do Polyakov’s early works on phase transitions and conformal invariance anticipate developments in statistical mechanics and quantum field theory?
• RQ5 What is the significance of the double zero at ν=1/6 in the parametric integral I(ν), and how does it relate to the vanishing of logarithmic terms in the partition function?

Key findings

• The parametric integral I(ν) is expressed in closed form as I(ν) = 2^{4ν−9}π²(1−2ν)²(1−6ν)²γ²(ν−1/4)/[γ²(1+ν)γ²(3/4)], revealing a double zero at ν=1/6.
• The double zero at ν=1/6 confirms the expected cancellation of the logarithmic divergence m⁴μlogμ in the grand partition function, supporting the solvability of the model at b²=3/4.
• The coefficient z₄ in the grand partition function, estimated via sum rules, is numerically close to zero, consistent with the cancellation mechanism.
• The zeta function of the zeros of an entire function of order ρ∈(1,2) is defined via analytic continuation and related to the coefficient z₂ via z₂=ζ_z(1).
• The canonical product representation allows the reconstruction of the function from its zeros, with the zeta function of zeros providing the leading coefficient in the expansion.
• The contour integral representation via the Mellin transform and the sine function enables the inversion of the zeta function to recover the logarithm of the canonical product.