Kalb–Ramond field

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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.[4][5]

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.[6]


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

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

See also[edit]


  1. ^ M. Kalb and Pierre Ramond, "Classical direct interstring action." Phys. Rev. D 9 (1974), 2273–2284.
  2. ^ Andrei S. Losev, Andrei Marshakov, Anton M. Zeitlin, "On First Order Formalism in String Theory", p. 1.
  3. ^ Alejandro Gaona, J. Antonio Garcia, "First Order Actions and Duality", p. 13.
  4. ^ Michael Kalb and Pierre Ramond (1974). "Classical direct interstring action". Phys. Rev. D. 9 (8): 2273–2284. Bibcode:1974PhRvD...9.2273K. doi:10.1103/PhysRevD.9.2273.
  5. ^ See also: Ogievetsky V. I., Polubarinov I. V. (1967). Sov. J. Nucl. Phys. 4. 156 (Yad. Fiz 4, 216).
  6. ^ Neveu, A., Schwarz, J. (1971). "Tachyon-free dual model with a positive-intercept trajectory." Physics Letters, 34B, 517–518.