Equation for string theory
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Equations in String Theory: A Comprehensive Overview
String Equations in Matrix Models and 2D Gravity
The string equations in the context of hermitian and unitary matrix models of 2D gravity are interpreted as flatness conditions. These conditions can be seen as consistency requirements for isomonodromic deformation of an equation with an irregular singularity. The partition function of the matrix model is identified as the tau function for isomonodromic deformation, with the physical parameters defining the string equation interpreted as moduli of meromorphic gauge fields. This interpretation allows the equations to be viewed as compatibility conditions for transport on the "quantum moduli space" of correlation functions in a theory of free fermions .
Kinetic Theory and Transport Equations for Strings
In the study of string dynamics, a distribution function defined on a 9+1D phase space is used to derive a transport equation analogous to the Boltzmann transport equation for particles. This equation governs the evolution of long strings with Nambu-Goto dynamics, including reconnections. Additionally, a system of coupled transport equations, similar to the BBGKY hierarchy for particles, is derived to describe both long strings and simple loops. This formalism is applicable to the non-linear dynamics of fundamental strings, D-brane strings, or field theory strings .
String Equation for Compactified 2D String Theory
A specific string equation is derived for compactified 2D string theory, which, together with the Toda Lattice hierarchy, determines the integrable structure of the theory. The form of this constraint depends on the compactification radius ( R ). This string equation is used to calculate the free energy and correlation functions in the dispersionless limit, revealing a phase diagram with two UV critical points, one of which describes infinitely strong vortex or tachyon perturbation .
Evolution Equation in Bosonic String Field Theory
Within the field theory of bosonic strings formulated in the light-cone gauge, a stringy version of the Altarelli-Parisi equation is provided. This equation is used to study the behavior of the decay function of strings under changes in the reference scale, particularly under the assumption of large transverse momentum. The behavior of the n-th moment of the decay function in this context can differ significantly from that in Quantum Chromodynamics (QCD) .
Scattering Equations and String Theory Amplitudes
Scattering equations for tree-level amplitudes are explored within string theory, leading to the definition of a new dual model whose amplitudes align with string theory in both the small and large (\alpha') limit. This model, which supports only the scattering equations, can be solved exactly, providing a resummed model for (\alpha') corrections to all orders. The approach is extended to amplitudes involving fermions and mixtures of scalars, gluons, and fermions, showing exact agreement with known results .
Fundamental String Solutions in Open String Field Theories
In Witten's open cubic bosonic string field theory and Berkovits' superstring field theory, solutions to the equations of motion with appropriate source terms are investigated. These solutions correspond to the Callan-Maldacena solution in Born-Infeld theory, representing fundamental strings ending on D-branes. The solutions are provided in an order-by-order manner, with some full-order properties demonstrated in the (\alpha') expansion. In the superstring case, the solution is shown to be 1/2 BPS in full order .
Conclusion
The equations governing string theory span a wide range of applications and interpretations, from matrix models and 2D gravity to kinetic theory and field theory formulations. These equations provide critical insights into the dynamics, integrable structures, and scattering amplitudes of strings, contributing to our understanding of fundamental physics and the behavior of strings in various theoretical frameworks.
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