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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Chiral equivariant cohomology II
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by Bong H. Lian, Andrew R. Linshaw and Bailin Song PDF
Trans. Amer. Math. Soc. 360 (2008), 4739-4776

Abstract:

This is the second in a series of papers on a new equivariant cohomology that takes values in a vertex algebra. In an earlier paper, the first two authors gave a construction of the cohomology functor on the category of $O({\mathfrak {s}}{\mathfrak {g}})$ algebras. The new cohomology theory can be viewed as a kind of “chiralization” of the classical equivariant cohomology, the latter being defined on the category of $G^*$ algebras a là H. Cartan. In this paper, we further develop the chiral theory by first extending it to allow a much larger class of algebras which we call ${\mathfrak {s}}{\mathfrak {g}}[t]$ algebras. In the geometrical setting, our principal example of an $O({\mathfrak {s}}{\mathfrak {g}})$ algebra is the chiral de Rham complex ${\mathcal {Q}}(M)$ of a $G$ manifold $M$. There is an interesting subalgebra of ${\mathcal {Q}}(M)$ which does not admit a full $O({\mathfrak {s}}{\mathfrak {g}})$ algebra structure but retains the structure of an ${\mathfrak {s}}{\mathfrak {g}}[t]$ algebra, enough for us to define its chiral equivariant cohomology. The latter then turns out to have many surprising features that allow us to delineate a number of interesting geometric aspects of the $G$ manifold $M$, sometimes in ways that are quite different from the classical theory.
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Additional Information
  • Bong H. Lian
  • Affiliation: Department of Mathematics, Brandeis University, Waltham, Massachusetts 02254-9110
  • Andrew R. Linshaw
  • Affiliation: Department of Mathematics, MS 050, Brandeis University, Waltham, Massachusetts 02454-9110
  • Address at time of publication: Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, Dept. 0112, La Jolla, California 92093-0112
  • MR Author ID: 791304
  • Bailin Song
  • Affiliation: Department of Mathematics, MS 050, Brandeis University, Waltham, Massachusetts 02454-9110
  • Address at time of publication: Department of Mathematics, University of California, Los Angeles, Box 951555, Los Angeles, California 90095-1555
  • Received by editor(s): July 10, 2006
  • Published electronically: April 7, 2008
  • © Copyright 2008 by the authors
  • Journal: Trans. Amer. Math. Soc. 360 (2008), 4739-4776
  • MSC (2000): Primary 57R91; Secondary 17B69
  • DOI: https://doi.org/10.1090/S0002-9947-08-04504-2
  • MathSciNet review: 2403703