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Local BRST cohomology in the antifield formalism: I. General theorems

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Abstract

We establish general theorems on the cohomologyH * (s/d) of the BRST differential modulo the spacetime exterior derivative, acting in the algebra of localp-forms depending on the fields and the antifields (=sources for the BRST variations). It is shown thatH −k (s/d) is isomorphic toH k (δ/d) in negative ghost degree−k (k>0), where δ is the Koszul-Tate differential associated with the stationary surface. The cohomology groupH 1 (δ/d) in form degreen is proved to be isomorphic to the space of constants of the motion, thereby providing a cohomological reformulation of Noether's theorem. More generally, the groupH k (δ/d) in form degreen is isomorphic to the space ofn−k forms that are closed when the equations of motion hold. The groupsH k (δ/d)(k>2) are shown to vanish for standard irreducible gauge theories. The groupH 2 (δ/d) is then calculated explicitly for electromagnetism, Yang-Mills models and Einstein gravity. The invariance of the groupsH k (s/d) under the introduction of non-minimal variables and of auxiliary fields is also demonstrated. In a companion paper, the general formalism is applied to the calculation ofH k (s/d) in Yang-Mills theory, which is carried out in detail for an arbitrary compact gauge group.

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Author notes

  1. Glenn Barnich (Aspirant)

    Present address: Fonds National de la Recherche Scientifique, (Belgium)

  2. Marc Henneaux

    Present address: Centro de Estudios Científicos de Santiago, Chile

Authors and Affiliations

  1. Faculté des Sciences, Université Libre de Bruxelles, Campus Plaine C.P. 231, B-1050, Bruxelles, Belgium

    Glenn Barnich (Aspirant) & Marc Henneaux

  2. NIKHEF-H, Postbus 41882, NL-1009 DB, Amsterdam, The Netherlands

    Friedemann Brandt

Authors
  1. Glenn Barnich
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  2. Friedemann Brandt
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  3. Marc Henneaux
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Communicated by G. Felder

Supported by Deutsche Forschungsgemeinschaft

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Barnich, G., Brandt, F. & Henneaux, M. Local BRST cohomology in the antifield formalism: I. General theorems. Commun.Math. Phys. 174, 57–91 (1995). https://doi.org/10.1007/BF02099464

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