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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 Amodel and the topological Bmodel. 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 twodimensional 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 Hflux is identically equal to zero. Generalized Kähler manifolds can have a nontrivial Hflux.
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 Rsymmetries, and the Lorentz symmetry may be modified^{[ clarification needed ]} by mixing rotations and Rsymmetries. One may use either of the two Rsymmetries, 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 Amodel is always possible but that leading to the Bmodel is only possible when the first Chern class of the spacetime vanishes, implying that the spacetime is CalabiYau ^{[ 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.
There is no restriction on the number of dimensions of spacetime, other than that it must be even because spacetime is generalized Kähler. 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 complexdimensional space. The topological string theory is not equivalent to the physical string theory, even on the same space, but certain^{[ which? ]} supersymmetric quantities agree in the two theories.
The topological Amodel comes with a target space which is a 6 realdimensional generalized Kähler spacetime. In the case in which the spacetime is Kähler, the theory describes two objects. There are fundamental strings, which wrap two realdimensional holomorphic curves. Amplitudes for the scattering of these strings depend only on the Kähler 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 Amodel closed strings is known as Kähler gravity, and was introduced by Michael Bershadsky and Vladimir Sadov in Theory of Kähler Gravity.
In addition, there are D2branes 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 Kähler form to the submanifold vanishes. The worldvolume theory on a stack of N D2branes is the string field theory of the open strings of the Amodel, which is a U(N) Chern–Simons theory.
The fundamental topological strings may end on the D2branes. While the embedding of a string depends only on the Kähler 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 Kähler form and the holomorphic 3form is zero. 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. These were first introduced by Anton Kapustin and Dmitri Orlov in Remarks on ABranes, Mirror Symmetry, and the Fukaya Category
The Bmodel also contains fundamental strings, but their scattering amplitudes depend entirely upon the complex structure and are independent of the Kähler 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 Bmodel is known as the Kodaira–Spencer theory of gravity and was developed by Michael Bershadsky, Sergio Cecotti, Hirosi Ooguri and Cumrun Vafa in Kodaira–Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes.
The Bmodel also comes with D(1), D1, D3 and D5branes, which wrap holomorphic 0, 2, 4 and 6submanifolds respectively. The 6submanifold is a connected component of the spacetime. The theory on a D5brane 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 CalabiYau case. The Lagrangian densities of the theories on the lowerdimensional branes may be obtained from holomorphic Chern–Simons theory by dimensional reductions.
Topological Mtheory, which enjoys a sevendimensional spacetime, is not a topological string theory, as it contains no topological strings. However topological Mtheory on a circle bundle over a 6manifold has been conjectured to be equivalent to the topological Amodel on that 6manifold.
In particular, the D2branes of the Amodel lift to points at which the circle bundle degenerates, or more precisely Kaluza–Klein monopoles. The fundamental strings of the Amodel lift to membranes named M2branes in topological Mtheory.
One special case that has attracted much interest is topological Mtheory on a space with G_{2} holonomy and the Amodel on a CalabiYau. In this case, the M2branes wrap associative 3cycles. Strictly speaking, the topological Mtheory conjecture has only been made in this context, as in this case functions introduced by Nigel Hitchin in The Geometry of ThreeForms 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..
The 2dimensional worldsheet theory is an N = (2,2) supersymmetric sigma model, the (2,2) supersymmetry means that the fermionic generators of the supersymmetry algebra, called supercharges, may be assembled into a single Dirac spinor, which consists of two Majorana–Weyl spinors of each chirality. 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 Rsymmetries. The Rsymmetry group of a 2dimensional N = (2,2) field theory is U(1) × U(1), twists by the two different factors lead to the A and B models respectively. The topological twisted construction of topological string theories was introduced by Edward Witten in his 1988 paper.^{ [1] }
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 Qinvariant operators are independent of the metric. In this sense, the theory is topological.
More generally, any Dterm 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. Yet more generally, in the B model any term which may be written as an integral over the fermionic coordinates does not contribute, whereas in the Amodel any term which is an integral over or over does not contribute. This implies that A model observables are independent of the superpotential (as it may be written as an integral over just ) but depend holomorphically on the twisted superpotential, and vice versa for the B model.
A number of dualities relate the above theories. The Amodel and Bmodel on two mirror manifolds are related by mirror symmetry, which has been described as a Tduality on a threetorus. The Amodel and Bmodel on the same manifold are conjectured to be related by Sduality, which implies the existence of several new branes, called NS branes by analogy with the NS5brane, which wrap the same cycles as the original branes but in the opposite theory. Also a combination of the Amodel and a sum of the Bmodel and its conjugate are related to topological Mtheory by a kind of dimensional reduction. Here the degrees of freedom of the Amodel and the Bmodels appear to not be simultaneously observable, but rather to have a relation similar to that between position and momentum in quantum mechanics.
The sum of the Bmodel 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 Bmodel suffers from a holomorphic anomaly, which states that the dependence on complex quantities, while classically holomorphic, receives nonholomorphic quantum corrections. In 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 Bmodel and its conjugate one reobtains all of the missing degrees of freedom and also eliminates the dependence on the arbitrary choice of polarization.
There are also a number of dualities that relate configurations with Dbranes, which are described by open strings, to those with branes the branes replaced by flux and with the geometry described by the nearhorizon geometry of the lost branes. The latter are described by closed strings.
Perhaps the first such duality is the GopakumarVafa duality, which was introduced by Rajesh Gopakumar and Cumrun Vafa in On the Gauge Theory/Geometry Correspondence. This relates a stack of N D6branes on a 3sphere in the Amodel on the deformed conifold to the closed string theory of the Amodel 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 Amodel is described by the Kähler gravity.
Although the conifold is said to be resolved, the area of the blown up twosphere is zero, it is only the Bfield, 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 threesphere 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 2cycle in the resolved conifold to closed strings in the B model on the deformed conifold. Open strings in the Bmodel 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.
In the paper Quantum CalabiYau and Classical Crystals, Andrei Okounkov, Nicolai Reshetikhin and Cumrun Vafa conjectured that the quantum Amodel is dual to a classical melting crystal at a temperature equal to the inverse of the string coupling constant. This conjecture was interpreted in 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 Amodel. In particular this suggests that the Amodel Kähler form should be quantized.
Amodel 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 Bmodel, 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.
Mtheory is a theory in physics that unifies all consistent versions of superstring theory. Edward Witten first conjectured the existence of such a theory at a stringtheory conference at the University of Southern California in the spring of 1995. Witten's announcement initiated a flurry of research activity known as the second superstring revolution.
In physics, string theory is a theoretical framework in which the pointlike particles of particle physics are replaced by onedimensional objects called strings. String theory describes how these strings propagate through space and interact with each other. On distance scales larger than the string scale, a string looks just like an ordinary particle, with its mass, charge, and other properties determined by the vibrational state of the string. In string theory, one of the many vibrational states of the string corresponds to the graviton, a quantum mechanical particle that carries gravitational force. Thus string theory is a theory of quantum gravity.
In algebraic geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has properties, such as Ricci flatness, yielding applications in theoretical physics. Particularly in superstring theory, the extra dimensions of spacetime are sometimes conjectured to take the form of a 6dimensional Calabi–Yau manifold, which led to the idea of mirror symmetry. Their name was coined by Candelas et al. (1985), after Eugenio Calabi who first conjectured that such surfaces might exist, and ShingTung Yau (1978) who proved the Calabi conjecture.
The Chern–Simons theory is a 3dimensional topological quantum field theory of Schwarz type developed by Edward Witten. It was discovered firstly by a mathematical physicist Albert Schwarz. It is named after mathematicians ShiingShen Chern and James Harris Simons who introduced the Chern–Simons 3form. In the Chern–Simons theory, the action is proportional to the integral of the Chern–Simons 3form.
In theoretical physics, Tduality is an equivalence of two physical theories, which may be either quantum field theories or string theories. In the simplest example of this relationship, one of the theories describes strings propagating in an imaginary spacetime shaped like a circle of some radius , while the other theory describes strings propagating on a spacetime shaped like a circle of radius proportional to . The idea of Tduality was first noted by Bala Sathiapalan in an obscure paper in 1987. The two Tdual theories are equivalent in the sense that all observable quantities in one description are identified with quantities in the dual description. For example, momentum in one description takes discrete values and is equal to the number of times the string winds around the circle in the dual description.
In gauge theory and mathematical physics, a topological quantum field theory is a quantum field theory which computes topological invariants.
In algebraic geometry and theoretical physics, mirror symmetry is a relationship between geometric objects called Calabi–Yau manifolds. The term refers to a situation where two Calabi–Yau manifolds look very different geometrically but are nevertheless equivalent when employed as extra dimensions of string theory.
Ftheory is a branch of string theory developed by Cumrun Vafa. The new vacua described by Ftheory were discovered by Vafa and allowed string theorists to construct new realistic vacua — in the form of Ftheory compactified on elliptically fibered Calabi–Yau fourfolds. The letter "F" supposedly stands for "Father".
Cumrun Vafa is an IranianAmerican theoretical physicist and the Hollis Professor of Mathematics and Natural Philosophy at Harvard University.
Andrew Eben Strominger is an American theoretical physicist who is the director of Harvard's Center for the Fundamental Laws of Nature. He has made significant contributions to quantum gravity and string theory. These include his work on Calabi–Yau compactification and topology change in string theory, and on the stringy origin of black hole entropy. He is a senior fellow at the Society of Fellows, and is the Gwill E. York Professor of Physics.
Homological mirror symmetry is a mathematical conjecture made by Maxim Kontsevich. It seeks a systematic mathematical explanation for a phenomenon called mirror symmetry first observed by physicists studying string theory.
In mathematics, specifically in symplectic topology and algebraic geometry, Gromov–Witten (GW) invariants are rational numbers that, in certain situations, count pseudoholomorphic curves meeting prescribed conditions in a given symplectic manifold. The GW invariants may be packaged as a homology or cohomology class in an appropriate space, or as the deformed cup product of quantum cohomology. These invariants have been used to distinguish symplectic manifolds that were previously indistinguishable. They also play a crucial role in closed type IIA string theory. They are named after Mikhail Gromov and Edward Witten.
In string theory, Ktheory classification refers to a conjectured application of Ktheory to superstrings, to classify the allowed Ramond–Ramond field strengths as well as the charges of stable Dbranes.
In theoretical physics a nonrenormalization theorem is a limitation on how a certain quantity in the classical description of a quantum field theory may be modified by renormalization in the full quantum theory. Renormalization theorems are common in theories with a sufficient amount of supersymmetry, usually at least 4 supercharges.
The Geometry Festival is an annual mathematics conference held in the United States.
Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modeling them as vibrations of tiny supersymmetric strings.
In string theory and related theories such as supergravity theories, a brane is a physical object that generalizes the notion of a point particle to higher dimensions. Branes are dynamical objects which can propagate through spacetime according to the rules of quantum mechanics. They have mass and can have other attributes such as charge.
This page is a glossary of terms in string theory, including related areas such as supergravity, supersymmetry, and high energy physics.
In mathematics and theoretical physics, and especially gauge theory, the deformed Hermitian Yang–Mills (dHYM) equation is a differential equation describing the equations of motion for a Dbrane in the Bmodel of string theory. The equation was derived by MariñoMinasianMooreStrominger in the case of Abelian gauge group, and by LeungYauZaslow using mirror symmetry from the corresponding equations of motion for Dbranes in the Amodel of string theory.