## 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.## References

- Henri Cartan,
*La transgression dans un groupe de Lie et dans un espace fibré principal*, Colloque de topologie (espaces fibrés), Bruxelles, 1950, Georges Thone, Liège; Masson & Cie, Paris, 1951, pp. 57–71 (French). MR**0042427** - M. Duflo, S. Kumar, and M. Vergne, Sur la Cohomologie Équivariante des Variétés Différentiables, Astérisque 215 (1993).
- E. Frenkel and M. Szczesny, Chiral de Rham Complex and Orbifolds, math.AG/0307181.
- Daniel Friedan, Emil Martinec, and Stephen Shenker,
*Conformal invariance, supersymmetry and string theory*, Nuclear Phys. B**271**(1986), no. 1, 93–165. MR**845945**, DOI 10.1016/0550-3213(86)90356-1 - Vassily Gorbounov, Fyodor Malikov, and Vadim Schechtman,
*Gerbes of chiral differential operators*, Math. Res. Lett.**7**(2000), no. 1, 55–66. MR**1748287**, DOI 10.4310/MRL.2000.v7.n1.a5 - Victor W. Guillemin and Shlomo Sternberg,
*Supersymmetry and equivariant de Rham theory*, Mathematics Past and Present, Springer-Verlag, Berlin, 1999. With an appendix containing two reprints by Henri Cartan [ MR0042426 (13,107e); MR0042427 (13,107f)]. MR**1689252**, DOI 10.1007/978-3-662-03992-2 - P. J. Hilton and U. Stammbach,
*A course in homological algebra*, 2nd ed., Graduate Texts in Mathematics, vol. 4, Springer-Verlag, New York, 1997. MR**1438546**, DOI 10.1007/978-1-4419-8566-8 - Bong H. Lian and Andrew R. Linshaw,
*Chiral equivariant cohomology. I*, Adv. Math.**209**(2007), no. 1, 99–161. MR**2294219**, DOI 10.1016/j.aim.2006.04.008 - F. Malikov, and V. Schectman, Chiral de Rham Complex, II, math.AG/9901065.
- Fyodor Malikov, Vadim Schechtman, and Arkady Vaintrob,
*Chiral de Rham complex*, Comm. Math. Phys.**204**(1999), no. 2, 439–473. MR**1704283**, DOI 10.1007/s002200050653 - Edward Witten,
*Two-dimensional models with $(0,2)$ supersymmetry: perturbative aspects*, Adv. Theor. Math. Phys.**11**(2007), no. 1, 1–63. MR**2320663**

## 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