Abstract
The chiral equivariant cohomology contains and generalizes the classical equivariant cohomology of a manifold M with an action of a compact Lie group G. For any simple G, there exist compact manifolds with the same classical equivariant cohomology, which can be distinguished by this invariant. When M is a point, this cohomology is an interesting conformal vertex algebra whose structure is still mysterious. In this paper, we scratch the surface of this object in the case G = SU(2).
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Akman, F.: The semi-infinite Weil complex of a graded Lie algebra. Ph. D. Thesis, Yale University, 1993
Borcherds R.: Vertex operator algebras, Kac-Moody algebras and the monster. Proc. Nat. Acad. Sci. USA 83, 3068–3071 (1986)
Duflo, M., Kumar, S., Vergne, M.: Sur la Cohomologie Équivariante des Variétés Différentiables. Astérisque 215, Paris: Soc. Math. France, 1993
Eck D.: Invariants of k-jet actions. Houston J. Math. 10(2), 159–168 (1984)
Ein, L., Mustata, M.: Jet schemes and singularities. Algebraic geometry—Seattle 2005. Part 2, Proc. Sympos. Pure Math., 80, Part 2, Providence, RI: Amer. Math. Soc., 2009, pp. 505–546
Frenkel, E., Ben-Zvi, D.: Vertex Algebras and Algebraic Curves. Math. Surveys and Monographs, Vol. 88, Providence, RI: Amer. Math. Soc., 2001
Feigin B., Frenkel E.: Semi-Infinite Weil Complex and the Virasoro Algebra. Commun. Math. Phys. 137, 617–639 (1991)
Frenkel I.B., Lepowsky J., Meurman A.: Vertex Operator Algebras and the Monster. Academic Press, New York (1988)
Friedan D., Martinec E., Shenker S.: Conformal invariance, supersymmetry and string theory. Nucl. Phys. B271, 93–165 (1986)
Gorelik M., Kac V.: On simplicity of vacuum modules. Adv. Math. 211, 621–677 (2007)
Guillemin V., Sternberg S.: Supersymmetry and Equivariant de Rham Theory. Springer, Berlin-Heidelberg-NewYork (1999)
Kac, V.: Vertex Algebras for Beginners. University Lecture Series, Vol. 10. Providence, RI: Amer. Math. Soc., 1998
Li H.: Local systems of vertex operators, vertex superalgebras and modules. J. Pure Appl. Algebra 109(2), 143–195 (1996)
Li H.: Vertex algebras and vertex Poisson algebras. Commun. Contemp. Math. 6, 61–110 (2004)
Lian B., Linshaw A.: Howe pairs in the theory of vertex algebras. J. Algebra 317, 111–152 (2007)
Lian B., Linshaw A.: Chiral equivariant cohomology I. Adv. Math. 209, 99–161 (2007)
Lian B., Linshaw A., Song B.: Chiral equivariant cohomology II. Trans. Am. Math. Soc. 360, 4739–4776 (2008)
Lian B., Linshaw A., Song B.: Chiral equivariant cohomology III. Amer. J. Math. 132(6), 1549–1590 (2010)
Lian B., Zuckerman G.J.: Commutative quantum operator algebras. J. Pure Appl. Algebra 100(1–3), 117–139 (1995)
Lian B., Zuckerman G.J.: New perspectives on the BRST-algebraic structure of string theory. Comm. Math. Phys. 154, 613–646 (1993)
Perez Alvarez J.: Jet invariants of compact Lie groups. J. Geom. Phys. 57, 293–295 (2006)
Malikov F., Schechtman V., Vaintrob A.: Chiral de Rham complex. Commun. Math. Phys. 204, 439–473 (1999)
Mustata M.: Jet schemes of locally complete intersection canonical singularities. Invent. Math. 145(3), 397–424 (2001)
Thielemanns K.: A Mathematica package for computing operator product expansions. Int. Mod. Phys. C2, 787 (1991)
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Communicated by Y. Kawahigashi
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Linshaw, A.R. Chiral Equivariant Cohomology of a Point: A First Look. Commun. Math. Phys. 306, 381–417 (2011). https://doi.org/10.1007/s00220-011-1284-z
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DOI: https://doi.org/10.1007/s00220-011-1284-z