A Closed Bosonic String Field Theory Based on the Fokker–Planck Formalism
© The Physical Society of Japan
This article is on
The Fokker–Planck formalism for closed bosonic strings
(PTEP Editors' Choice)
Prog. Theor. Exp. Phys. 2023, 023B05 (2023).
A string field theory for closed bosonic strings is formulated using pants decomposition and the Fokker–Planck formalism.
String theory is a framework where elementary particles and their interactions are replaced with strings as the basic building blocks of matter. Similar to how a particle sweeps out a worldline as it travels in spacetime, strings sweep out worldsheets as they move through spacetime. String field theory (SFT) is a formulation of string theory that reframes the dynamics of relativistic strings in terms of a quantum field theory. To construct an SFT, one must specify rules to cut the worldsheets into their fundamental units, namely propagators and vertices (Figure 1).
Figure 1
Unlike the particle theories, however, there are many choices for such rules and we can construct an SFT starting from one, in principle. The problem is that the SFT thus constructed becomes very complicated In general. The rule should, therefore, be carefully chosen to make for a simple SFT.
In this paper, I formulated an SFT for closed bosonic strings based on the “pants decomposition,” in which a bordered or punctured Riemann surface (describing a worldsheet for a closed bosonic string) can be decomposed into a pair of pants (Figure 2), which is analogous to threestring vertices. Every Riemann surface with a genus g and n punctures or boundaries admits a hyperbolic metric with a constant negative curvature if it satisfies the condition 2g  2 + n > 0. This hyperbolic metric allows for a pants decomposition that, in turn, may be exploited to construct an SFT for closed bosonic strings.
Figure 2
However, it was pointed out in the 1980’s that such an idea does not work. If one constructs the action for an SFT based on the pants decomposition, the amplitudes are expressed by integrals in which the fundamental domain of the modular group is covered infinite times, resulting in incorrect amplitudes.
In my paper, I circumvent this problem using Mirzakhani’s scheme for computing integrals over the moduli space of bordered Riemann surfaces. By applying this scheme, I derive a recursion relation satisfied by the offshell amplitudes of closed bosonic strings. This recursion relation is related to the loop equations of minimal string theory, which can be described by an SFT through the Fokker–Planck formalism.
Accordingly, I convert the recursion relation into an SFT using a Fokker–Planck Hamiltonian consisting of kinetic terms and threestring vertices. However, the resulting SFT lacks the worldsheet BRST (BecchiRouetStoraTyutin) symmetry. Fortunately, the theory can be made BRST invariant by introducing auxiliary fields.
This study thus offers an approach to constructing a simple action for closed bosonic string theory that uses the Fokker–Planck formalism instead of the traditional path integral formulation. Hopefully, this idea would open doors to constructing an SFT for superstrings.
The Fokker–Planck formalism for closed bosonic strings
(PTEP Editors' Choice)
Prog. Theor. Exp. Phys. 2023, 023B05 (2023).
Share this topic
Fields
Related Articles

Novel Insights Into Bulk Reconstruction in the Antide Sitter/ Conformal Field Theory Correspondence
Theoretical Particle Physics
202361
Bulk reconstruction in antide Sitter/conformal field theory is fundamental to our understanding of quantum gravity. We show that contrary to popular belief, bulk reconstruction is rather simple and intuitive.

The Stolz–Teichner Conjecture and Supermoonshine
Theoretical Particle Physics
2023425
We check the validity of the "divisibility property," predicted by the Stolz–Teichner conjecture, for several infinite families of theories built from Duncan’s Supermoonshine module. Along the way, we develop the tools necessary to construct the socalled "periodicity class."

Fractional Topological Charge Construction in a U(1) Lattice Gauge Theory Framework
Theoretical Particle Physics
2023327
We have constructed a fractional topological charge in the lattice Abelian gauge theory formalism, which avoids the issue of infinite degrees of freedom in continuum spacetime quantum field theory.

Theoretical Assessment of Fwave Bottom Mesons and Their Properties
Theoretical Particle Physics
20221124
We use heavy quark effective theory to study Fwave bottom mesons. Using theory and experimental data, we calculate their masses and analyze their decay widths to estimate the upper bound to the associated couplings.

Representations of KBc Algebra for Generating String Field Theory Solutions
Theoretical Particle Physics
20221031
We propose a general method for generating solutions in string field theory from a solution constructed using KBc algebra, and reproduce known solutions constructed earlier using different methods.