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 List of natural phenomena, natural phenomena. This is in contrast to experimental p ...
in general and
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 ...
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 transforms as a two-
form
Form is the shape, visual appearance, or configuration of an object. In a wider sense, the form is the way something happens.
Form may also refer to:
*Form (document), a document (printed or electronic) with spaces in which to write or enter dat ...
, i.e., an antisymmetric
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
field with two indices.
The adjective "NS" reflects the fact that in the
RNS formalism, these fields appear in the NS–NS sector in which all vector fermions are anti-periodic. Both uses of the word "NS" refer to
André Neveu and
John Henry Schwarz, who studied such boundary conditions (the so-called
Neveu–Schwarz boundary conditions) and the fields that satisfy them in 1971.
Details
The Kalb–Ramond field generalizes the
electromagnetic potential but it has two indices instead of one. This difference is related to the fact that the electromagnetic potential is integrated over one-dimensional
worldlines of particles to obtain one of its contributions to the action while the Kalb–Ramond field must be integrated over the two-dimensional
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 as a direct generalization of the world line concept for a point particle in special an ...
of the string. In particular, while the
action for a charged particle moving in an electromagnetic
potential is given by
:
that for a string coupled to the Kalb–Ramond field has the form
:
This term in the action implies that the fundamental string of string theory is a source of the NS–NS ''B''-field, much like charged particles are sources of the electromagnetic field.
The Kalb–Ramond field appears, together with the
metric tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
and
dilaton
In particle physics, the hypothetical dilaton is a particle of a scalar field \varphi that appears in theories with extra dimensions when the volume of the compactified dimensions varies. It appears as a radion in Kaluza–Klein theory's compa ...
, as a set of massless excitations of a
closed string.
See also
*
Curtright field
*
p-form electrodynamics
In theoretical physics, -form electrodynamics is a generalization of Maxwell's theory of electromagnetism.
Ordinary (via. one-form) Abelian electrodynamics
We have a 1-form \mathbf, a gauge symmetry
:\mathbf \rightarrow \mathbf + d\alpha ,
where ...
*
Ramond–Ramond field
In theoretical physics, Ramond–Ramond fields are differential form fields in the 10-dimensional spacetime of type II supergravity theories, which are the classical limits of type II string theory. The ranks of the fields depend on which type II t ...
References
String theory
Gauge bosons
{{string-theory-stub