TY - JOUR

T1 - Two-dimensional topological gravity and equivariant cohomology

AU - Getzler, E.

N1 - Copyright:
Copyright 2007 Elsevier B.V., All rights reserved.

PY - 1994/7

Y1 - 1994/7

N2 - The analogy between topological string theory and equivariant cohomology for differentiable actions of the circle group on manifolds has been widely remarked on. One of our aims in this paper is to make this analogy precise. We show that topological string theory is the "derived functor" of semi-relative cohomology, just as equivariant cohomology is the derived functor of basic cohomology. That homological algebra finds a place in the study of topological string theory should not surprise the reader, granted that topological string theory is the conformal field theorist's algebraic topology. In [7], we have shown that the cohomology of a topological conformal field theory carries the structure of a batalin-Vilkovisky algebra (actually, two commuting such structures, corresponding to the two chiral sectors of the theory). In the second part of this paper, we describe the analogous algebraic structure on the equivariant cohomology of a topological conformal field theory: we call this structure a gravity algebra. This algebraic structure is a certain generalization of a Lie algebra, and is distinguished by the fact that it has an infinite sequence of independent operations {a1, ..., ak}, k≥2, satisfying quadratic relations generalizing the Jacobi rule. (The operad underlying the category of gravity algebras has been studied independently by Ginzburg-Kapranov [9].) The author is grateful to M. Bershadsky, E. Frenkel, M. Kapranov, G. Moore, R. Plesser and G. Zuckerman for the many ways in which they helped in the writing of this paper; also to the Department of Mathematics at Yale University for its hospitality while part of this paper was written.

AB - The analogy between topological string theory and equivariant cohomology for differentiable actions of the circle group on manifolds has been widely remarked on. One of our aims in this paper is to make this analogy precise. We show that topological string theory is the "derived functor" of semi-relative cohomology, just as equivariant cohomology is the derived functor of basic cohomology. That homological algebra finds a place in the study of topological string theory should not surprise the reader, granted that topological string theory is the conformal field theorist's algebraic topology. In [7], we have shown that the cohomology of a topological conformal field theory carries the structure of a batalin-Vilkovisky algebra (actually, two commuting such structures, corresponding to the two chiral sectors of the theory). In the second part of this paper, we describe the analogous algebraic structure on the equivariant cohomology of a topological conformal field theory: we call this structure a gravity algebra. This algebraic structure is a certain generalization of a Lie algebra, and is distinguished by the fact that it has an infinite sequence of independent operations {a1, ..., ak}, k≥2, satisfying quadratic relations generalizing the Jacobi rule. (The operad underlying the category of gravity algebras has been studied independently by Ginzburg-Kapranov [9].) The author is grateful to M. Bershadsky, E. Frenkel, M. Kapranov, G. Moore, R. Plesser and G. Zuckerman for the many ways in which they helped in the writing of this paper; also to the Department of Mathematics at Yale University for its hospitality while part of this paper was written.

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U2 - 10.1007/BF02101459

DO - 10.1007/BF02101459

M3 - Article

AN - SCOPUS:21344485518

VL - 163

SP - 473

EP - 489

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 3

ER -