Topological string theory

In theoretical physics, topological string theory is a simplified version of string theory. The operators in topological string theory represent the algebra of operators in the full string theory that preserve a certain amount of supersymmetry. Topological string theory may be 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.

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, and many other topics. As you can see, these are mathematical topics. Indeed, topological string theory cannot be understood as a realistic theory to describe the physical world.

Topological string theory was established and is studied by physicists like Edward Witten and Cumrun Vafa.

Admissible spacetimes

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, and so is in fact an N=(2,2) sigma model. This will be the case for example if the spacetime is Kahler and the H-flux is identically equal to zero, although there are more general cases in which the target is a generalized Kahler manifold and the H-flux is nontrivial.

So far we have described ordinary strings on special backgrounds. These strings are never topological. 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 is that these theories have two U(1) symmetries known as R-symmetries, and one may modify the Lorentz symmetry 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, and as a result the theory has no dynamics, instead all observables depend on the topology of a configuration. Such theories are known are topological theories.

While classically this procedure is always possible, quantum mechanically the U(1) symmetries may be anomalous. In this case the twisting is not possible. For example, in the Kahler case with H=0 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. 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 Kahler case the two complex structures are the same and so the difference is always zero, which is why the A model always exists.

There is no restriction on the number of dimensions of spacetime, other than that it must be even because spacetime is generalized Kahler. 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 supersymmetric quantities agree in the two theories.

The objects

The A-model

The topological A-model comes with a target space which is a 6 real-dimensional generalized Kahler spacetime. In the case in which the spacetime is Kahler, the theory describes two objects. There are fundamental strings, which wrap 2 real-dimensional holomorphic curves. Amplitudes for the scattering of these strings depend only on the Kahler form of the spacetime, and not on the complex structure. 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. The string field theory of the A model closed strings is known as Kahler gravity, and was introduced by Michael Bershadsky and Vladimir Sadov in [http://xxx.lanl.gov/abs/hep-th/9410011 Theory of Kahler Gravity] .

In addition there are D2-branes which wrap Lagrangian submanifolds of spacetime. These are submanifolds whose dimensions are one half that of space time, and such that the pullback of the Kahler form to the submanifold vanishes. 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. Notice that while the embedding of a string depends only on the Kahler form, the embeddings of the branes depends entirely on the complex structure. In particular, when a string ends on a brane the intersection will always be orthogonal, as the wedge product of the Kahler form and the homolomorphic 3-form is zero. In the physical string this is necessary for the stability of the configuration, but here it is a property of Lagrangian and homolomorphic cycles on a Kahler manifold.

Away from the Calabi-Yau case there may also be coisotropic branes in various dimensions. These were first introducted by Anton Kapustin and Dmitri Orlov in [http://xxx.lanl.gov/abs/hep-th/0109098| Remarks on A-Branes, Mirror Symmetry, and the Fukaya Category]

The B-model

The B-model also contains fundamental strings, but their scattering amplitudes depend entirely upon the complex structure and are independent of the Kahler structure. 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 string field theory of the closed strings of the B model is known as The Kodaira-Spencer theory of gravity and was developed by Michael Bershadsky, Sergio Cecotti, Hirosi Ooguri and Cumrun Vafa in [http://xxx.lanl.gov/abs/hep-th/9309140 Kodaira-Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes] .

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

Topological M-theory, which enjoys a 7-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. The fundamental strings of the A-model lift to membranes named M2-branes in topological M-theory.

One special case that has attracted a lot of interest is topological M-theory on a space with G₂ 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 [http://xxx.lanl.gov/abs/math.dG/0010054 The Geometry of Three-Forms in Six and Seven Dimensions] and [http://xxx.lanl.gov/abs/math.dG/0107101 Stable Forms and Special Metrics] provide a candidate low energy effective action.

Dualities

Dualities between TSTs

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. Also a combination of the A-model and a sum of the B-model and its conjugate are related to topological M-theory by a kind of dimensional reduction. Here the degrees of freedom of the A-model and the B-models appear to not be simultaneously observable, but rather to have a relation similar to that between position and momentum in quantum mechanics.

The holomorphic anomaly

The sum of the B-model and its conjugate appears in the above duality because it is the theory whose low energy effective action is expected to be described by Hitchin's formalism. This is because the B-model suffers from a holomorphic anomaly, which states that the dependence on complex quantities, while classically holomorphic, receives nonholomorphic quantum corrections. In [http://xxx.lanl.gov/abs/hep-th/9306122 Quantum Background Independence in String Theory] , Edward Witten argued that this structure is analogous to a structure that one finds geometrically quantizing the space of complex structures. Once this space has been quantized, only half of the dimensions simultaneously commute and so the number of degrees of freedom has been halved. This halving depends on an arbitrary choice, called a polarization. The conjugate model contains the missing degrees of freedom, and so by tensoring the B-model and its conjugate one reobtains all of the missing degrees of freedom and also eliminates the dependence on the arbitrary choice of polarization.

Geometric transitions

There are also a number of dualities that relate configurations with D-branes, which are described by open strings, to those with branes the branes replaced by flux and with the geometry described by the near-horizon geometry of the lost branes. The latter are described by closed strings.

Perhaps the first such duality is the Gopakumar-Vafa duality, which was introduced by Rajesh Gopakumar and Cumrun Vafa in [http://xxx.lanl.gov/abs/hep-th/9811131 On the Gauge Theory/Geometry Correspondence] .This relates a stack of N D2-branes on a 3-sphere in the A-model on the deformed conifold to the closed string theory of the A-model on a resolved conifold with a B field equal to N times the string coupling constant.The open strings in the A model are described by a U(N) Chern-Simons theory, while the closed string theory on the A-model is described by the Kahler gravity.

Although the conifold is said to be resolved, the area of the blown up two-sphere is zero, it is only the B-field, which is often considered to be the complex part of the area, which is nonvanishing. In fact, as the Chern-Simons theory is topological, one may shrink the volume of the deformed three-sphere to zero and so arrive at the same geometry as in the dual theory.

The mirror dual of this duality is another duality, which relates open strings in the B model on a brane wrapping the 2-cycle in the resolved conifold to closed strings in the B model on the deformed conifold. Open strings in the B-model are described by dimensional reductions of homolomorphic Chern-Simons theory on the branes on which they end, while closed strings in the B model are described by Kodaira-Spencer gravity.

Dualities with other theories

Crystal melting, quantum foam and U(1) gauge theory

In the paper [http://xxx.lanl.gov/abs/hep-th/0309208 Quantum Calabi-Yau and Classical Crystals] , Andrei Okounkov, Nicolai Reshetikhin and Cumrun Vafa conjectured that the quantum A-model is dual to a classical melting crystal at a temperature equal to the inverse of the string coupling constant. This conjecture was interpreted in [http://xxx.lanl.gov/abs/hep-th/0312022 Quantum Foam and Topological Strings] , by Amer Iqbal, Nikita Nekrasov, Andrei Okounkov and Cumrun Vafa. They claim that the statistical sum over melting crystal configurations is equivalent to a path integral over changes in spacetime topology supported in small regions with area of order the product of the string coupling constant and α'.

Such configurations, with spacetime full of many small bubbles, dates back to John Archibald Wheeler in 1964, but has rarely appeared in string theory as it is notoriously difficult to make precise. However in this duality the authors are able to cast the dynamics of the quantum foam in the familiar language of a topologically twisted U(1) gauge theory, whose field strength is linearly related to the Kähler form of the A-model. In particular this suggests that the A-model Kahler form should be quantized.

Applications

A-model topological string theory amplitudes are used to compute prepotentials in N=2 supersymmetric gauge theories in four and five dimensions. The amplitudes of the topological B-model, with fluxes and or branes, are used to compute superpotentials in N=1 supersymmetric gauge theories in four dimensions. Perturbative A model calculations also count BPS states of spinning black holes in five dimensions.

References

[http://xxx.lanl.gov/abs/hep-th/0410178 Topological Strings and their Physical Applications] by Andrew Neitzke and Cumrun Vafa.

[http://xxx.lanl.gov/abs/hep-th/0411073 Topological M-theory as Unification of Form Theories of Gravity] by Robbert Dijkgraaf, Sergei Gukov, Andrew Neitzke and Cumrun Vafa.