In
theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
, modular invariance is the invariance under the
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
such as SL(2,Z) of
large diffeomorphism
In mathematics and theoretical physics, a large diffeomorphism is an equivalence class of diffeomorphisms under the equivalence relation where diffeomorphisms that can be continuously connected to each other are in the same equivalence class.
For ...
s of the
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
. The name comes from the classical name
modular group of this group, as in
modular form
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the Group action (mathematics), group action of the modular group, and also satisfying a grow ...
theory.
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 interac ...
, modular invariance is an additional requirement for
one-loop diagrams. This helps in getting rid of some
global anomalies such as the
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 ...
.
Equivalently, in
two-dimensional conformal field theory
A two-dimensional conformal field theory is a quantum field theory on a Euclidean two-dimensional space, that is invariant under local conformal transformations.
In contrast to other types of conformal field theories, two-dimensional conformal fie ...
the torus partition function must be invariant under the modular group
SL(2,Z).
String theory
Symmetry
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