Topological string theory is a branch of string theory that focuses on studying certain aspects of string theory using topological methods. It provides a framework to investigate the mathematical properties and relations of strings without considering their dynamics in spacetime. In other words, topological string theory is concerned with the topological aspects of string theory rather than the detailed physics of particles and forces.
One of the key features of topological string theory is its connection to certain geometric and topological invariants known as Donaldson-Thomas invariants and Gromov-Witten invariants. These invariants encode information about the counting of curves and their intersections on Calabi-Yau manifolds, which are important mathematical objects in string theory. By studying these invariants, researchers gain insights into the mathematical structure underlying string theory and its relation to other branches of mathematics, such as algebraic geometry and differential topology.
Topological string theory is particularly useful for understanding certain limits of string theory, such as the large volume limit, where the effects of gravity are weak. In this limit, the theory simplifies, and calculations become more tractable. By analyzing the topological aspects, researchers can gain a better understanding of the underlying mathematical structure and explore connections between string theory and other areas of mathematics.
Moreover, topological string theory has provided valuable insights into the duality relationships in string theory. Dualities are important symmetries that relate seemingly different physical theories, and they have been instrumental in unraveling the non-perturbative aspects of string theory. Topological string theories have played a role in uncovering new dualities, such as the geometric Langlands duality and the AGT correspondence, which relate string theory to other areas of physics and mathematics.
In summary, topological string theory focuses on the mathematical and topological aspects of string theory, providing valuable insights into the underlying structure and connections to other branches of mathematics. It offers a complementary perspective to the more traditional approaches in string theory and has contributed to our understanding of the theory's mathematical foundations, duality relationships, and non-perturbative aspects.