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 ...
in
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 th ...
.
Discovery
In 1984,
Michael Green and
John H. Schwarz realized that the
anomaly
Anomaly may refer to:
Science
Natural
*Anomaly (natural sciences)
** Atmospheric anomaly
** Geophysical anomaly
Medical
* Congenital anomaly (birth defect), a disorder present at birth
** Physical anomaly, a deformation of an anatomical struct ...
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 contains ...
with the
gauge group SO(32) cancels because of an extra "classical" contribution from a
2-form 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
gauge group of
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 contains ...
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; i.e. of a gauge theory
In physics, a gauge ...
,
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'' h ...
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 introdu ...
. For the special choice of the
gauge group SO(32) or
E8 x E8, 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
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 th ...
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
Susy is a feminine given name, sometimes a short form (hypocorism) of Susan, Susanne, Susannah, etc.
''Susy'' may refer to:
People
* Susy Andersen (born 1940), Italian actress
* Susanne Augustesen (b. 1956), Danish footballer
* Susy Avery (born ...
''D'' = 10 gauge theory is the hexagon which has a particular linear combination of the two-form gauge field strength and Ricci tensor,
, 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
, where the integral is over the 10 dimensions,
is the rank-two
Kalb–Ramond field, and
is
a gauge invariant combination of
(with space-time indices not contracted), which is precisely one of the factors appearing in the hexagon anomaly.
If the variation of
under the transformations of gauge field for
and under general coordinate transformations is appropriately specified, then
the Green–Schwarz term
, 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