The Green–Schwarz mechanism (sometimes called the Green–Schwarz anomaly cancellation mechanism) is the main discovery that started the

first superstring revolution The history of string theory spans several decades of intense research including two superstring revolutions. Through the combined efforts of many researchers, string theory has developed into a broad and varied subject with connections to quantu ...

superstring theory 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. 'Superstring theory' is a shorthand for supersymmetric string t ...

Discovery

In 1984, Michael Green and John H. Schwarz realized that the anomaly in type I string theory with the

gauge group In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie group ...

SO(32) cancels because of an extra "classical" contribution from a

2-form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...

field. They realized that one of the necessary conditions for a superstring theory to make sense is that the

dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...

of the

of type I string theory must be 496 and then demonstrated this to be so. In the original calculation,

gauge anomalies In theoretical physics, a gauge anomaly is an example of an anomaly: it is a feature of quantum mechanics—usually a one-loop diagram—that invalidates the gauge symmetry of a quantum field theory In theoretical physics, quantum field theo ...

, mixed anomalies, and

gravitational anomalies In theoretical physics, a gravitational anomaly is an example of a gauge anomaly: it is an effect of quantum mechanics — usually a one-loop diagram—that invalidates the general covariance of a theory of general relativity combined with som ...

were expected to arise from a

hexagon In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A '' regular hexagon'' has ...

Feynman diagram In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduc ...

. For the special choice of the

SO(32) or

E8 x E8 In non-technical terms, M-theory presents an idea about the basic substance of the universe. As of 2022, science has produced no experimental evidence to support the conclusion that M-theory is a description of the real world. Although a complet ...

, however, the anomaly factorizes and may be cancelled by a tree diagram. In string theory, this indeed occurs. The tree diagram describes the exchange of a virtual quantum of the B-field. It is somewhat counterintuitive to see that a tree diagram cancels a one-loop diagram, but in reality, both of these diagrams arise as one-loop diagrams in

in which the anomaly cancellation is more transparent. As recounted in '' The Elegant Universes TV version, in the second episode, "The String's the Thing", section "Wrestling with String Theory", Green describes finding 496 on each side of the equals sign during a stormy night filled with lightning, and fondly recalls joking that "the gods are trying to prevent us from completing this calculation". Green soon entitled some of his subsequent lectures " The Theory of Everything".

Details

Anomalies in quantum theory arise from one-loop diagrams, with a chiral fermion in the loop and gauge fields,

Ricci tensor In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measur ...

s, or global symmetry currents as the external legs. These diagrams have the form of a triangle in 4 spacetime dimensions, which generalizes to a hexagon in ''D'' = 10, thus involving 6 external lines. The interesting anomaly in SUSY ''D'' = 10 gauge theory is the hexagon which has a particular linear combination of the two-form gauge field strength and Ricci tensor,

F^6,\ F^4 R^2,\ F^2 R^4,\ R^6

, for the external lines. Green and Schwarz realized that one can add a so-called Chern–Simons term to the classical action, having the form

S_ = \int B_\wedge X_8

, where the integral is over the 10 dimensions,

B_

is the rank-two Kalb–Ramond field, and

X_8

is a gauge invariant combination of

F^4,\ F^2 R^2,\ R^4

(with space-time indices not contracted), which is precisely one of the factors appearing in the hexagon anomaly. If the variation of

B_

under the transformations of gauge field for

F_

and under general coordinate transformations is appropriately specified, then the Green–Schwarz term

S_

, when combined with a trilinear vertex through exchange of a gauge boson, has precisely the right variation to cancel the hexagon anomaly.

References

{{DEFAULTSORT:Green-Schwarz Mechanism Anomalies (physics) Quantum gravity String theory