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