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Transactions of the American Mathematical Society

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

   
 

 

Chiral equivariant cohomology II


Authors: Bong H. Lian, Andrew R. Linshaw and Bailin Song
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
Published electronically: April 7, 2008
MathSciNet review: 2403703
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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

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

DOI: https://doi.org/10.1090/S0002-9947-08-04504-2
Received by editor(s): July 10, 2006
Published electronically: April 7, 2008
Article copyright: © Copyright 2008 by the authors