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 The central observation [ clarification needed ] is that these [ which?
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.
Quantum mechanically, the U 1 symmetries may be anomalous , making the twist impossible. 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. However, all correlation functions with worldsheets that are not spheres vanish unless the complex dimension of the spacetime is three, and so spacetimes with complex dimension three are the most interesting. This is fortunate for phenomenology , as phenomenological models often use a physical string theory compactified on a 3 complex-dimensional space.
The topological string theory is not equivalent to the physical string theory, even on the same space, but certain [ which? There are fundamental strings, which wrap two real-dimensional holomorphic curves. Classically these correlation functions are determined by the cohomology ring. There are quantum mechanical instanton effects which correct these and yield Gromov—Witten invariants , which measure the cup product in a deformed cohomology ring called the quantum cohomology.
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In addition, there are D2-branes which wrap Lagrangian submanifolds of spacetime. The worldvolume theory on a stack of N D2-branes is the string field theory of the open strings of the A-model, which is a U N Chern—Simons theory. The fundamental topological strings may end on the D2-branes. In the physical string this is necessary for the stability of the configuration, but here it is a property of Lagrangian and holomorphic cycles on a Kahler manifold. There may also be coisotropic branes in various dimensions other than half dimensions of Lagrangian submanifolds. In particular, they are insensitive to worldsheet instanton effects and so can often be calculated exactly.
Mirror symmetry then relates them to A model amplitudes, allowing one to compute Gromov—Witten invariants. The B-model also comes with D -1 , D1, D3 and D5-branes, which wrap holomorphic 0, 2, 4 and 6-submanifolds respectively. The 6-submanifold is a connected component of the spacetime.
The theory on a D5-brane is known as holomorphic Chern—Simons theory. The Lagrangian density is the wedge product of that of ordinary Chern—Simons theory with the holomorphic 3,0 -form, which exists in the Calabi-Yau case.
The Lagrangian densities of the theories on the lower-dimensional branes may be obtained from holomorphic Chern—Simons theory by dimensional reductions. Topological M-theory, which enjoys a seven-dimensional spacetime, is not a topological string theory, as it contains no topological strings.
However topological M-theory on a circle bundle over a 6-manifold has been conjectured to be equivalent to the topological A-model on that 6-manifold. In particular, the D2-branes of the A-model lift to points at which the circle bundle degenerates, or more precisely Kaluza—Klein monopoles.
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The fundamental strings of the A-model lift to membranes named M2-branes in topological M-theory. One special case that has attracted much interest is topological M-theory on a space with G 2 holonomy and the A-model on a Calabi-Yau.
In this case, the M2-branes wrap associative 3-cycles. Strictly speaking, the topological M-theory conjecture has only been made in this context, as in this case functions introduced by Nigel Hitchin in The Geometry of Three-Forms in Six and Seven Dimensions and Stable Forms and Special Metrics provide a candidate low energy effective action.
These functions are called " Hitchin functional " and Topological string is closely related to Hitchin's ideas on generalized complex structure , Hitchin system , and ADHM construction etc.. This sigma model is topologically twisted, which means that the Lorentz symmetry generators that appear in the supersymmetry algebra simultaneously rotate the physical spacetime and also rotate the fermionic directions via the action of one of the R-symmetries.
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The topological twisted construction of topological string theories was introduced by Edward Witten in his paper. The topological twist leads to a topological theory because the stress—energy tensor may be written as an anticommutator of a supercharge and another field. As the stress—energy tensor measures the dependence of the action on the metric tensor , this implies that all correlation functions of Q-invariant operators are independent of the metric. In this sense, the theory is topological.
More generally, any D-term in the action, which is any term which may be expressed as an integral over all of superspace , is an anticommutator of a supercharge and so does not affect the topological observables. A number of dualities relate the above theories. The A-model and B-model on two mirror manifolds are related by mirror symmetry , which has been described as a T-duality on a three-torus. The A-model and B-model on the same manifold are conjectured to be related by S-duality , which implies the existence of several new branes, called NS branes by analogy with the NS5-brane , which wrap the same cycles as the original branes but in the opposite theory.
Towards a Classification of Conformal Field Theories. Knot Theory and Quantum Groups. Beyond the Planck Length. String Field Theory. Nonpolynomial String Field Theory. Geometric String Field Theory. Topological Field Theory. Back Matter Pages About this book Introduction Following on the foundations laid in his earlier book "Introduction to Superstrings", Professor Kaku discusses such topics as the classification of conformal string theories, the non-polynomial closed string field theory, matrix models, and topological field theory.
The presentation of the material is self-contained, and several chapters review material expounded in the earlier book.