Abstract
The cohomology theory of supermanifolds is developed. Its basic properties are established and simple examples given. The Wess-Zumino term in the Green-Schwarz covariant superstring action is interpreted as a nontrivial class in the “supersymmetric cohomology” of flat superspace. A quotient supermanifold with nontrivial topology reflecting this class is constructed. It is shown that there is no topological quantization condition for the coefficient of the Wess-Zumino term. The superstring differs from conventional sigma models in this respect because its action is Grassmann-valued and its group manifold (superspace) is noncompact.
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References
Green, M.B., Schwarz, J.H.: Covariant description of superstrings. Phys. Lett.136B, 367 (1984)
Henneaux, M., Mezincescu, L.: A σ-model interpretation of Green-Schwarz covariant superstring action. Phys. Lett.152B, 340 (1985)
Martinec, E.: Unpublished
Witten, E.: Global aspects of current algebra. Nucl. Phys.B 223, 422 (1983)
Braaten, E., Curtright, T.L., Zachos, C.K.: Torsion and geometrostasis in nonlinear sigma models. Nucl. Phys. B260, 630 (1985)
Kostant, B.: In: Differential geometric methods in mathematical physics. Lecture Notes in Mathematics, Vol. 570. Bleuler, K., Reetz, A. (eds.). Berlin, Heidelberg, New York: Springer 1977
Picken, R.F.: Supergroups have no extra topology. J. Phys. A16, 3457 (1983)
Berezin, F.A.: Differential forms on supermanifolds. Sov. J. Nucl. Phys.30, 605 (1979)
Rogers, A.: A global theory of supermanifolds. J. Math. Phys.21, 1352 (1980)
Rabin, J.M.: Berezin integration on general fermionic supermanifolds. Commun. Math. Phys.103, 431–439 (1986)
Rabin, J.M., Crane, L.: Global properties of supermanifolds. Commun. Math. Phys.100, 141 (1985)
Rabin, J.M., Crane, L.: How different are the supermanifolds of Rogers and DeWitt? Commun. Math. Phys.102, 123 (1985)
DeWitt, B.S.: Supermanifolds. Cambridge: Cambridge University Press 1984
Rogers, A.: Graded manifolds, supermanifolds, and infinite-dimensional Grassmann algebras. Commun. Math. Phys.105, 375 (1986)
Bott, R., Tu, L.W.: Differential forms in algebraic topology. Berlin, Heidelberg, New York: Springer 1983, p. 33
Rogers, A.: On the existence of global integral forms on supermanifolds. J. Math. Phys.26, 2749 (1985)
Curtright, T.L., Mezincescu, L., Zachos, C.K.: Geometrostasis and torsion in covariant superstrings. Phys. Lett.161B, 79 (1985)
Greub, W., Halperin, S., Vanstone, R.: Connections, curvature, and cohomology, Vol. 2. New York: Academic Press 1973, Chap. 4
Kostelecký, V.A., Rabin, J.M.: A superspace approach to lattice supersymmetry. J. Math. Phys.25, 2744 (1984)
Gross, D., Harvey, J., Martinec, E., Rohm, R.: Heterotic string theory. I. The free heterotic string. Nucl. Phys. B256, 253 (1985)
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Communicated by S.-T. Yau
Enrico Fermi Fellow. Research supported by the NSF (PHY 83-01221) and DOE (DE-AC02-82-ER-40073)
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Rabin, J.M. Supermanifold cohomology and the Wess-Zumino term of the covariant superstring action. Commun.Math. Phys. 108, 375–389 (1987). https://doi.org/10.1007/BF01212316
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DOI: https://doi.org/10.1007/BF01212316