Green-Schwarz mechanism
Encyclopedia
The Green–Schwarz mechanism (sometimes called the Green-Schwarz anomaly cancellation mechanism) is the main discovery that started the first superstring revolution in superstring theory
.
and John H. Schwarz realized that the anomaly
in type I string theory
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
of the gauge group of type I string theory
must be 496
and then demonstrated this to be so.
In the original calculation, gauge anomalies, mixed anomalies, and gravitational anomalies were expected to arise from a hexagon Feynman diagram
. 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
in which the anomaly cancellation is more transparent.
As recounted in The Elegant Universe
's TV version, in the second episode, "The String's the Thing", section "Wrestling with String Theory", Green describes the night of the finding 496 on each side of the equals sign to 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
".
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,
, 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.
Superstring theory
Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modelling them as vibrations of tiny supersymmetric strings...
.
Discovery
In 1984, Michael GreenMichael Green (physicist)
Michael Boris Green FRS is a British physicist and one of the pioneers of string theory. Currently a professor in the Department of Applied Mathematics and Theoretical Physics and a Fellow in Clare Hall at the University of Cambridge in England, he succeeded Stephen Hawking on 1 November 2009...
and John H. Schwarz realized that the anomaly
Anomaly (physics)
In quantum physics an anomaly or quantum anomaly is the failure of a symmetry of a theory's classical action to be a symmetry of any regularization of the full quantum theory. In classical physics an anomaly is the failure of a symmetry to be restored in the limit in which the symmetry-breaking...
in type I string theory
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 and which contains not only closed strings, but also open strings.The classic 1976 work of Ferdinando Gliozzi, Joel Scherk and...
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
Dimension
In physics and mathematics, the dimension of a 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 because only one coordinate is needed to specify a point on it...
of the gauge group of type I string theory
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 and which contains not only closed strings, but also open strings.The classic 1976 work of Ferdinando Gliozzi, Joel Scherk and...
must be 496
496 (number)
Four hundred [and] ninety-six is the natural number following four hundred [and] ninety-five and preceding four hundred [and] ninety-seven.-In mathematics:...
and then demonstrated this to be so.
In the original calculation, gauge anomalies, mixed anomalies, and gravitational anomalies were expected to arise from a hexagon Feynman diagram
Feynman diagram
Feynman diagrams are a pictorial representation scheme for the mathematical expressions governing the behavior of subatomic particles, first developed by the Nobel Prize-winning American physicist Richard Feynman, and first introduced in 1948...
. For the special choice of the gauge group SO(32) or E8 x E8
E8 (mathematics)
In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8...
, however, the anomaly factorizes and may be cancelled by a tree diagram. In string theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
, 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
Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modelling them as vibrations of tiny supersymmetric strings...
in which the anomaly cancellation is more transparent.
As recounted in The Elegant Universe
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 assessment of the theory and some of its shortcomings...
's TV version, in the second episode, "The String's the Thing", section "Wrestling with String Theory", Green describes the night of the finding 496 on each side of the equals sign to 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
Theory of everything
A theory of everything is a putative theory of theoretical physics that fully explains and links together all known physical phenomena, and predicts the outcome of any experiment that could be carried out in principle....
".
Details
Anomalies in quantum theory arise from one-loop diagrams, with a chiral fermion in the loop and gauge fields, Ricci tensors, 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,
, 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 isa 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 thetransformations of gauge field for
and under general coordinate transformations is appropriately specified, thenthe 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.