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 quantum ...

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 In theoretical physics, type I string theory is one of five consistent supersymmetric string theories in ten dimensions. It is the only one whose strings are unoriented (both orientations of a string are equivalent) and the only one which perturba ...

with the

gauge group A gauge group is a group of gauge symmetries of the Yang–Mills gauge theory of principal connections on a principal bundle. Given a principal bundle P\to X with a structure Lie group G, a gauge group is defined to be a group of its vertical ...

SO(32) In string theory, a heterotic string is a closed string (or loop) which is a hybrid ('heterotic') of a superstring and a bosonic string. There are two kinds of heterotic superstring theories, the heterotic SO(32) and the heterotic E8 ×&nbs ...

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 coo ...

of the

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 (physics), anomaly: it is a feature of quantum mechanics—usually a one-loop diagram—that invalidates the gauge symmetry of a quantum field theory; i.e. of a gauge theory. Al ...

, mixed anomalies, and gravitational anomalies 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 is de ...

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 introduced ...

. For the special choice of the

SO(32) or E8 x E8, however, the anomaly factorizes and may be cancelled by a tree diagram. In

string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...

, 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 Universe ''The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory'' is a book by Brian Greene published in 1999, which introduces string and superstring theory, and provides a comprehensive though non-technical asses ...

s 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 measure ...

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 In theoretical physics in general and string theory in particular, the Kalb–Ramond field (named after Michael Kalb and Pierre Ramond), also known as the Kalb–Ramond ''B''-field or Kalb–Ramond NS–NS ''B''-field, is a quantum field that tra ...

, 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