Global anomaly
Encyclopedia
In theoretical physics
, a global anomaly is a type of anomaly
: in this particular case, it is a quantum effect that invalidates a large gauge transformation
s that would otherwise be preserved in the classical theory. This leads to an inconsistency in the theory because the space of configurations which is being integrated over in the functional integral involves both a configuration and the same configuration after a large gauge transformation has acted upon it and the sum of all such contributions is zero and the space of configurations cannot be split into connected components for which the integral is nonzero.
Alternatively, the existence of a global anomaly implies that the measure of Feynman
's functional integral cannot be defined globally.
The adjective "global" refers to the properties of a group
that are not visible locally. For example, all features of a discrete group (as opposed to a Lie group
) are global in character.
A famous example is an SU(2) Yang-Mills theory in 4D with an odd number of chiral fermions transforming as doublets under SU(2).
Many types of global anomalies must cancel for a theory to be consistent. An example is modular invariance
, the requirement of anomaly cancellation for a part of a gravitational anomaly
that deals with the large diffeomorphism
s over two dimensional worldsheet
s of genus
1 or more.
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
, a global anomaly is a type of 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 this particular case, it is a quantum effect that invalidates a large gauge transformation
Large gauge transformation
Let's say we have a topological space M and a topological group G and a principal G-bundle over M. A global section of this principal bundle is a gauge fixing and the process of replacing one section by another is a gauge transformation...
s that would otherwise be preserved in the classical theory. This leads to an inconsistency in the theory because the space of configurations which is being integrated over in the functional integral involves both a configuration and the same configuration after a large gauge transformation has acted upon it and the sum of all such contributions is zero and the space of configurations cannot be split into connected components for which the integral is nonzero.
Alternatively, the existence of a global anomaly implies that the measure of Feynman
Feynman
Feynman may refer to:* Richard Feynman , physicist** Feynman diagram** Feynman graph** Feynman–Kac formula** The Feynman Lectures on Physics** Feynman integral, see Path integral formulation** Feynman parametrization...
's functional integral cannot be defined globally.
The adjective "global" refers to the properties of a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
that are not visible locally. For example, all features of a discrete group (as opposed to a Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
) are global in character.
A famous example is an SU(2) Yang-Mills theory in 4D with an odd number of chiral fermions transforming as doublets under SU(2).
Many types of global anomalies must cancel for a theory to be consistent. An example is modular invariance
Modular invariance
In theoretical physics, modular invariance is the invariance under the group such as SL of large diffeomorphisms of the torus. The name comes from the classical name modular group of this group, as in modular form theory....
, the requirement of anomaly cancellation for a part of a gravitational anomaly
Gravitational anomaly
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 some other fields. The adjective "gravitational" is derived from...
that deals with the large diffeomorphism
Large diffeomorphism
In mathematics and theoretical physics, a large diffeomorphism is a diffeomorphism that cannot be continuously connected to the identity diffeomorphism ....
s over two dimensional worldsheet
Worldsheet
In string theory, a worldsheet is a two-dimensional manifold which describes the embedding of a string in spacetime. The term was coined by Leonard Susskind around 1967 as a direct generalization of the world line concept for a point particle in special and general relativity.The type of string,...
s of genus
Genus
In biology, a genus is a low-level taxonomic rank used in the biological classification of living and fossil organisms, which is an example of definition by genus and differentia...
1 or more.