String theory equation
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Understanding String Theory Equations
String theory, a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects known as strings, has led to the development of various equations and models. These equations are crucial for understanding the fundamental aspects of string theory and its applications in different physical contexts.
Geometry of String Equations
The string equations in hermitian and unitary matrix models of 2D gravity are interpreted as flatness conditions, which can be seen as consistency conditions 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. The physical parameters defining the string equation are viewed as moduli of meromorphic gauge fields, and the compatibility conditions define a "quantum" analog of a Riemann surface. These equations can also be interpreted as compatibility conditions for transport on the "quantum moduli space" of correlation functions in a theory of free fermions1.
Nonrelativistic String Theory and Newton-Cartan Geometry
Nonrelativistic string theory is described by a sigma model with a relativistic worldsheet and a nonrelativistic target spacetime geometry, known as string Newton-Cartan geometry. This geometry is obtained as a limit of the Riemannian geometry of general relativity with a fluxless two-form field. Applying this limit to relativistic string theory in curved background fields leads to nonrelativistic string theory in a string Newton-Cartan geometry coupled to a Kalb-Ramond and dilaton field background. This approach also helps in studying the spacetime equations of motion and T-duality transformations of nonrelativistic string theory2.
Counting String Theory Standard Models
An approximate analytic relation has been derived between the number of consistent heterotic Calabi-Yau compactifications of string theory with the exact charged matter content of the standard model of particle physics and the topological data of the internal manifold. This relation scales exponentially with the number of Kahler parameters and has been computationally verified for complete intersection Calabi-Yau threefolds (CICYs) with up to seven Kahler parameters. When extrapolated to the entire CICY list, the relation suggests approximately (10^{23}) string theory standard models; for Calabi-Yau hypersurfaces in toric varieties, it suggests around (10^{723}) standard models3.
Optimization Algorithms Inspired by String Theory
A new optimization algorithm based on String Theory, known as the String Theory Algorithm (STA), has been proposed. This meta-heuristic algorithm is inspired by the idea that all elemental particles in the universe are strings, and their vibrations create all existing particles. The STA uses equations based on the laws of physics stated in String Theory to generate potential solutions for optimization problems. The algorithm has been evaluated using traditional benchmark mathematical functions, benchmark functions of the CEC 2015 Competition, and the optimization of a fuzzy inference system (FIS)4.
Ambitwistor Strings and Scattering Equations
String theories admit chiral infinite tension analogues where only the massless parts of the spectrum survive. These are described by holomorphic maps to spaces of complex null geodesics, known as ambitwistor spaces. Quantization of these theories leads to formulas for tree-level scattering amplitudes of massless particles, which localize the vertex operators to solutions of the scattering equations. This localization emerges naturally from working on ambitwistor space, suggesting a way to extend these amplitudes to spinor fields and loop levels5.
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 correspond to the Callan-Maldacena solution in Born-Infeld theory, representing fundamental strings ending on D-branes. These solutions are given order by order, and in the superstring case, the solution is shown to be 1/2 BPS in full order6.
Conclusion
String theory equations play a pivotal role in understanding the fundamental aspects of the universe, from the geometry of string equations and nonrelativistic string theory to optimization algorithms and scattering equations. These equations not only provide insights into the theoretical underpinnings of string theory but also have practical applications in various fields of physics and mathematics.
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Geometry of the string equations
The string equations of 2D gravity can be viewed as compatibility conditions for transport on "quantum moduli space" of correlation functions in a theory of free fermions, potentially connecting to twistor theory.
Highly CitedString theory and string Newton–Cartan geometry
This paper demonstrates that string Newton-Cartan geometry can lead to nonrelativistic string theory in curved background fields, allowing for studies of beta-functions and T-duality transformations.
Highly CitedCounting string theory standard models
The number of string theory standard models scales exponentially with the number of Kahler parameters, giving 10 23 for complete intersection Calabi-Yau threefolds and 10 723 for Calabi-Yau hypersurfaces in toric varieties.
A new meta-heuristic optimization algorithm based on a paradigm from physics: string theory
The String Theory Algorithm (STA) outperforms other meta-heuristics in solving optimization problems, outperforming the Firefly Algorithm (FA) and Grey Wolf Optimizer (GWO) in various scenarios.
Ambitwistor strings and the scattering equations
Ambitwistor string theories are a natural extension of existing twistor string theories, allowing for chiral infinite tension analogues and a natural extension to tree-level scattering amplitudes of massless particles.
Highly CitedFundamental string solutions in open string field theories
The solutions in open string field theories show full order properties, with superstring solutions being 1/2 BPS in full order.
Thickening the string. I. The string perfect dust
The classical theory of the geometrical string is developed as a simple, surface-forming timelike bivector field in an arbitrary background space-time, with the stress-energy tensor for a perfect dust of such strings being written down.
Highly CitedDuality Symmetric String and M-Theory
Recent developments in duality symmetric string theory and M-theory provide a purely geometric higher dimensional origin to gauged supergravities, offering a new approach to supergravity.
Highly CitedSpacetime singularities in string theory.
Spacetime singularities in string theory can lead to classical solutions to Einstein's equation, but string propagation through these singular regions is not well-behaved.
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