Kalb–Ramond field

In theoretical physics in general and string theory in particular, the Kalb–Ramond field (named after Michael Kalb and Pierre Ramond),^[1] also known as the Kalb–Ramond B-field^[2] or Kalb–Ramond NS–NS B-field,^[3] is a quantum field that transforms as a two-form, i.e., an antisymmetric tensor field with two indices.^[1][4]

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.^[5]

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 of the string. In particular, while the action for a charged particle moving in an electromagnetic potential is given by

${\displaystyle -q\int dx^{\mu }A_{\mu }}$

that for a string coupled to the Kalb–Ramond field has the form

${\displaystyle -\int dx^{\mu }dx^{\nu }B_{\mu \nu }}$

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 and dilaton, as a set of massless excitations of a closed string.

• Curtright field
• p-form electrodynamics
• Ramond–Ramond field

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