Abstract
In this paper, we formulate a generalization of the classical BRST construction which applies to the case of the reduction of a Poisson manifold by a submanifold. In the case of symplectic reduction, our procedure generalizes the usual classical BRST construction which only applies to symplectic reduction of a symplectic manifold by a coisotropic submanifold, i.e. the case of reducible “first class” constraints. In particular, our procedure yields a method to deal with “second-class” constraints. We construct the BRST complex and compute its cohomology. BRST cohomology vanishes for negative dimension and is isomorphic as a Poisson algebra to the algebra of smooth functions on the reduced Poisson manifold in zero dimension. We then show that in the general case of reduction of Poisson manifolds, BRST cohomology cannot be identified with the cohomology of vertical differential forms.
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Batalin, I.A., Vilkovisky, G.A.: Phys. Lett.69B, 309 (1977); Batalin, I.A., Fradkin, E.S.: Phys. Lett.122B, 2 (1983); Henneaux, M.: Phys. Rep.126, 1 (1985); McMullan, D.: J. Math. Phys.28, 428 (1987); Browning, A.D., McMullan, D.: J. Math. Phys.28, 438 (1987)
Fisch, J.M.L., Henneaux, M., Stasheff, J., Teitelboim, C.: Commun. Math. Phys.120, 379 (1989)
Stasheff, J.: Bull. Am. Math. Soc.19, 287 (1988)
Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer School of Science Monographs Series, Yeshiva University, 1964
Dubois-Violette, M.: Ann. Inst. Fourier37, 4, 45 (1987)
Wilbour, D., Arms, J.M.: Reduction Procedures for Poisson Manifolds. Washington preprint (1991)
Wilbour, D.: U. Washington thesis
Tate, J.: Ill. J. Math.1, 14 (1957)
Lang, S.: Algebra. Reading MA: Addison-Wesley, 1984
Figueroa-O'Farrill, J.M., Kimura, T.: Homological Approach to Symplectic Reduction. Leuven/Austin preprint (1991)
Warmer, F.W.: Foundations of Differentiable Manifolds and Lie Groups: Scott, Foresman, and Co. 1971
Rinehart, R.G.: Trans. Am. Math. Soc.108, 195 (1963)
Henneaux, M., Teitelboim, C.: Commun. Math. Phys.115, 213 (1988)
Stasheff, J.: Homological Reduction of Constrained Poisson Algebras. J. Diff. Geom. (to appear)
Kostant, B., Sternberg, S.: Ann. Phys.176, 49 (1987)
Frenkel, I.B., Garland, H., Zuckerman, G.J.: Proc. Natl. Acad. Sci. USA83, 8442 (1986)
Feigin, B., Frenkel, E.: Commun. Math. Phys.137, 617 (1991)
Kimura, T.: Prequantum BRST Cohomology. Contemporary Mathematics in the Proceedings of the 1991 Joint Summer Research Conference on Mathematical Aspects of Classical Field Theory. Gotay, M.J., Marsden, J.E., Moncrief, V.E. (eds.)
Figueroa-O'Farrill, J.M., Kimura, T.: Commun. Math. Phys.136, 209 (1991)
Huebschmann, J.: Graded Lie-Rinehart algebras, graded Poisson algebras, and BRST-quantization I. The Finitely Generated Case. (preprint) Heidelberg 1991
Duval, C., Elhadad, J., Gotay, M.J., Sniatycki, J., Tuynman, G.M.: Ann. Phys.206, 1 (1991)
Tuynman, G.M.: Geometric Quantization of the BRST Charge. Commun. Math. Phys. (to appear)
Duval, C., Elhadad, J., Tuynman, G.M.: Commun. Math. Phys.126, 535 (1990)
Kimura, T.: (in preparation)
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Communicated by K. Gawedzki
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Kimura, T. Generalized classical BRST cohomology and reduction of Poisson manifolds. Commun.Math. Phys. 151, 155–182 (1993). https://doi.org/10.1007/BF02096751
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DOI: https://doi.org/10.1007/BF02096751