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Topological string theory
In theoretical physics, topological string theory is a version of string theory. Topological string theory appeared in papers by theoretical physicists, such as Edward Witten and Cumrun Vafa, by analogy with Witten's earlier idea of topological quantum field theory.
There are two main versions of topological string theory: the topological A-model and the topological B-model. The results of the calculations in topological string theory generically encode all holomorphic quantities within the full string theory whose values are protected by spacetime supersymmetry. Various calculations in topological string theory are closely related to Chern–Simons theory, Gromov–Witten invariants, mirror symmetry, geometric Langlands Program, and many other topics.
The operators in topological string theory represent the algebra of operators in the full string theory that preserve a certain amount[clarification needed] of supersymmetry. Topological string theory is obtained by a topological twist of the worldsheet description of ordinary string theory: the operators are given different spins. The operation is fully analogous to the construction of topological field theory which is a related concept. Consequently, there are no local degrees of freedom in topological string theory.
The fundamental strings of string theory are two-dimensional surfaces. A quantum field theory known as the N = (1,1) sigma model is defined on each surface. This theory consist of maps from the surface to a supermanifold. Physically the supermanifold is interpreted as spacetime and each map is interpreted as the embedding of the string in spacetime.
Only special spacetimes admit topological strings. Classically, one must choose a spacetime such that the theory respects an additional pair of supersymmetries[why?], making the spacetime an N = (2,2) sigma model[further explanation needed]. A particular case of this is if the spacetime is a Kähler manifold and the H-flux is identically equal to zero. Generalized Kähler manifolds can have a nontrivial H-flux.
Ordinary strings on special backgrounds are never topological[why?]. To make these strings topological, one needs to modify the sigma model via a procedure called a topological twist which was invented by Edward Witten in 1988. The central observation[clarification needed] is that these[which?] theories have two U(1) symmetries known as R-symmetries, and the Lorentz symmetry may be modified by mixing rotations and R-symmetries. One may use either of the two R-symmetries, leading to two different theories, called the A model and the B model. After this twist, the action of the theory is BRST exact[further explanation needed], and as a result the theory has no dynamics. Instead, all observables depend on the topology of a configuration. Such theories are known as topological theories.
Classically this procedure is always possible.[further explanation needed]
Quantum mechanically, the U(1) symmetries may be anomalous, making the twist impossible. For example, in the Kähler case with H = 0[clarification needed] the twist leading to the A-model is always possible but that leading to the B-model is only possible when the first Chern class of the spacetime vanishes, implying that the spacetime is Calabi–Yau[clarification needed]. More generally (2,2) theories have two complex structures and the B model exists when the first Chern classes of associated bundles sum to zero whereas the A model exists when the difference of the Chern classes is zero. In the Kähler case the two complex structures are the same and so the difference is always zero, which is why the A model always exists.
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Topological string theory
In theoretical physics, topological string theory is a version of string theory. Topological string theory appeared in papers by theoretical physicists, such as Edward Witten and Cumrun Vafa, by analogy with Witten's earlier idea of topological quantum field theory.
There are two main versions of topological string theory: the topological A-model and the topological B-model. The results of the calculations in topological string theory generically encode all holomorphic quantities within the full string theory whose values are protected by spacetime supersymmetry. Various calculations in topological string theory are closely related to Chern–Simons theory, Gromov–Witten invariants, mirror symmetry, geometric Langlands Program, and many other topics.
The operators in topological string theory represent the algebra of operators in the full string theory that preserve a certain amount[clarification needed] of supersymmetry. Topological string theory is obtained by a topological twist of the worldsheet description of ordinary string theory: the operators are given different spins. The operation is fully analogous to the construction of topological field theory which is a related concept. Consequently, there are no local degrees of freedom in topological string theory.
The fundamental strings of string theory are two-dimensional surfaces. A quantum field theory known as the N = (1,1) sigma model is defined on each surface. This theory consist of maps from the surface to a supermanifold. Physically the supermanifold is interpreted as spacetime and each map is interpreted as the embedding of the string in spacetime.
Only special spacetimes admit topological strings. Classically, one must choose a spacetime such that the theory respects an additional pair of supersymmetries[why?], making the spacetime an N = (2,2) sigma model[further explanation needed]. A particular case of this is if the spacetime is a Kähler manifold and the H-flux is identically equal to zero. Generalized Kähler manifolds can have a nontrivial H-flux.
Ordinary strings on special backgrounds are never topological[why?]. To make these strings topological, one needs to modify the sigma model via a procedure called a topological twist which was invented by Edward Witten in 1988. The central observation[clarification needed] is that these[which?] theories have two U(1) symmetries known as R-symmetries, and the Lorentz symmetry may be modified by mixing rotations and R-symmetries. One may use either of the two R-symmetries, leading to two different theories, called the A model and the B model. After this twist, the action of the theory is BRST exact[further explanation needed], and as a result the theory has no dynamics. Instead, all observables depend on the topology of a configuration. Such theories are known as topological theories.
Classically this procedure is always possible.[further explanation needed]
Quantum mechanically, the U(1) symmetries may be anomalous, making the twist impossible. For example, in the Kähler case with H = 0[clarification needed] the twist leading to the A-model is always possible but that leading to the B-model is only possible when the first Chern class of the spacetime vanishes, implying that the spacetime is Calabi–Yau[clarification needed]. More generally (2,2) theories have two complex structures and the B model exists when the first Chern classes of associated bundles sum to zero whereas the A model exists when the difference of the Chern classes is zero. In the Kähler case the two complex structures are the same and so the difference is always zero, which is why the A model always exists.