The development of the mathematical framework behind string theory involved the contributions of many physicists over several decades. Here is a general overview of the mathematical aspects and the key steps in the evolution of string theory:
String Basics and Vibrations: The idea of string theory emerged in the late 1960s when physicists discovered that replacing point-like particles with tiny, one-dimensional strings could provide a consistent quantum theory of gravity. The behavior of these strings was initially described using classical mechanics, with the strings allowed to vibrate in different modes. The mathematics involved in this stage was relatively straightforward and built upon concepts from classical mechanics.
Conformal Field Theory (CFT): In the 1980s, physicists realized that the behavior of strings required the use of two-dimensional quantum field theories known as conformal field theories. Conformal symmetry, which describes the preservation of angles under scale transformations, is a crucial property of these theories. The study of conformal field theories, their correlation functions, and their properties became a central mathematical tool for understanding string theory.
Superstring Theory and Supersymmetry: Superstring theory, developed in the early 1970s, extended string theory by introducing supersymmetry—a symmetry that relates particles with integer and half-integer spin. This extension was crucial for addressing certain consistency issues in string theory. The mathematics of supersymmetry involves the study of superalgebras, superfields, and superspace, which are mathematical structures that incorporate fermionic and bosonic degrees of freedom.
Calabi-Yau Manifolds: In the mid-1980s, physicists realized that for string theory to describe our observed universe, the extra dimensions of space required in the theory must be compactified on special six-dimensional spaces called Calabi-Yau manifolds. Calabi-Yau manifolds have rich mathematical properties and are characterized by complex geometry. The mathematical study of Calabi-Yau manifolds, including their topological and geometric properties, became crucial for constructing realistic string vacua.
Duality and Gauge/Gravity Correspondence: The 1990s witnessed the discovery of various dualities within string theory. These dualities established surprising equivalences between seemingly different string theories and provided deeper insights into the nonperturbative aspects of the theory. The most famous example is the AdS/CFT correspondence, also known as the gauge/gravity duality, which relates certain string theories in a curved spacetime (Anti-de Sitter space) to certain conformal field theories without gravity. These dualities are highly nontrivial and have deep connections to areas such as quantum field theory, black holes, and quantum gravity.
String Perturbation Theory and Beyond: String theory is a highly challenging mathematical subject due to the intricate interactions of strings and the need to perform calculations at different orders of perturbation theory. Techniques such as worldsheet path integral formalism, conformal field theory techniques, and more advanced methods like scattering amplitudes have been developed to compute physical quantities within string theory. Nonperturbative aspects of the theory, such as D-branes, brane dynamics, and string dualities, also require sophisticated mathematical tools.
It's worth noting that the mathematical development of string theory is ongoing, and various mathematical areas, including algebraic geometry, differential geometry, topology, and group theory, continue to play essential roles in advancing our understanding of the theory. The mathematical challenges and complexities involved in string theory make it a rich and active field of research for both physicists and mathematicians.