Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. H. Cartan, La Transgression dans un groupe de Lie et dans un espace fibré principal, Colloque de Topologie, C.B.R.M., Bruxelles 57-71 (1950). MR 0042427 (13:107f)
  • 2. M. Duflo, S. Kumar, and M. Vergne, Sur la Cohomologie Équivariante des Variétés Différentiables, Astérisque 215 (1993).
  • 3. E. Frenkel and M. Szczesny, Chiral de Rham Complex and Orbifolds, math.AG/0307181.
  • 4. D. Friedan, E. Martinec, and S. Shenker, Conformal Invariance, Supersymmetry and String Theory, Nucl. Phys. B271 (1986) 93-165. MR 845945 (87i:81202)
  • 5. V. Gorbounov, F. Malikov, and V. Schectman, Gerbes of Chiral Differential Operators, Math. Res. Lett. 7 (2000) no. 1, 55-66. MR 1748287(2002c:17040)
  • 6. V. Guillemin and S. Sternberg, Supersymmetry and Equivariant de Rham Theory, Springer, 1999. MR 1689252 (2001i:53140)
  • 7. P. Hilton, and U. Stammbach, A Course in Homological Algebra, 2nd ed., Springer-Verlag, New York 1997. MR 1438546 (97k:18001)
  • 8. B. Lian, and A. Linshaw, Chiral Equivariant Cohomology I, Adv. Math. 209, 99-161 (2007). MR 2294219 (2008c:17021)
  • 9. F. Malikov, and V. Schectman, Chiral de Rham Complex, II, math.AG/9901065.
  • 10. F. Malikov, V. Schectman, and A. Vaintrob, Chiral de Rham Complex, Comm. Math. Phys, 204, 439-473 (1999). MR 1704283 (2000j:17035a)
  • 11. E. Witten, Two-Dimensional Models With (0,2) Supersymmetry: Perturbative Aspects, Adv. Theor. Math. Phys. 11 (2007) no. 1, 1-63. MR 2320663

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 57R91, 17B69

Retrieve articles in all journals with MSC (2000): 57R91, 17B69


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

American Mathematical Society