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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.[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

\( -q\int dx^{\mu }A_{\mu } \)

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

\( -\int dx^{\mu }dx^{\nu }B_{{\mu \nu }} \)

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.
See also

Curtright field
p-form electrodynamics
Ramond–Ramond field


Kalb, Michael; Ramond, P. (1974-04-15). "Classical direct interstring action". Physical Review D. American Physical Society (APS). 9 (8): 2273–2284. doi:10.1103/physrevd.9.2273. ISSN 0556-2821.
Losev, Andrei S.; Marshakov, Andrei; Zeitlin, Anton M. (2006). "On first-order formalism in string theory". Physics Letters B. Elsevier BV. 633 (2–3): 375–381. arXiv:hep-th/0510065. doi:10.1016/j.physletb.2005.12.010. ISSN 0370-2693.
Gaona, Alejandro; García, J. Antonio (2007-02-10). "First-order Actions and Duality". International Journal of Modern Physics A. World Scientific Pub Co Pte Lt. 22 (04): 851–867. arXiv:hep-th/0610022. doi:10.1142/s0217751x07034386. ISSN 0217-751X.
See also: Ogievetsky V. I., Polubarinov I. V. (1967). Sov. J. Nucl. Phys. 4. 156 (Yad. Fiz 4, 216).

Neveu, A.; Schwarz, J.H. (1971). "Tachyon-free dual model with a positive-intercept trajectory". Physics Letters B. Elsevier BV. 34 (6): 517–518. doi:10.1016/0370-2693(71)90669-1. ISSN 0370-2693.


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