Ezra Getzler

Batalin-Vilkovisky algebras and two-dimensional topological field theories

By a Batalin-Vilkovisky algebra, we mean a graded commutative algebraA, together with an operator Δ:A⊙→A⊙+1 such that Δ2 = 0, and [Δ,a]−Δa is a graded derivation ofA for alla∈A. In this article, we show that there is a natural structure of a Batalin-Vilkovisky algebra on the cohomology of a topological conformal field theory in two dimensions. We make use of a technique from algebraic topology: the theory of operads.

Lie theory for nilpotent L∞-algebras

The Deligne groupoid is a functor from nilpotent differential graded Lie algebras concentrated in positive degrees to groupoids; in the special case of Lie algebras over a field of characteristic zero, it gives the associated simply connected Lie group. We generalize the Deligne groupoid to a functor γ from L ∞ -algebras concentrated in degree > -n to n-groupoids. (We actually construct the nerve of the n-groupoid, which is an enriched Kan complex.) The construction of gamma is quite explicit (it is based on Duponts proof of the de Rham theorem) and yields higher dimensional analogues of holonomy and of the Campbell-Hausdorff formula. In the case of abelian L ∞ algebras (i.e., chain complexes), the functor γ is the Dold-Kan simplicial set.

A SHORT PROOF OF THE LOCAL ATIYAH-SINGER INDEX THEOREM*

(Received 24 April 1985) IN THIS paper, we will give a simple proof of the local Atiyah-Singer index theorem first proved by Patodi [9]; in fact, his earlier proof of the Gauss-Bonnet-Chern theorem (Patodi [8]) is quite close to ours. (Perhaps he did not find the proof for Dirac operators given in this paper because he was unaware of the symbol calculus for Clifford algebras.) A paper of Kotake [S] contains a proof of the Riemann-Roth theorem for Riemann surfaces along similar lines, and a recent paper of Bismut [3] is also very closely related. It might be helpful to give a short history of this theorem. As explained in Atiyah et al. [l], all of the common geometric complexes, namely, the twisted Dirac operators, &operators, signature operators and the De Rham complex, are, locally, Dirac operators. We shall refer to all of these operators as Dirac operators, although this is not globally correct on non-spin manifolds. The index theorem for Dirac operators was first proven, at least for Kahler manifolds, by Hirzebruch using cobordism theory. A few years later, McKean and Singer gave their famous formula for the index of the Dirac operator: Index (D) = Str efo2( = Tr erp-e’ - Tr erpcm-)

Pseudodifferential operators on supermanifolds and the Atiyah-Singer index theorem

Fermionic quantization, or Clifford algebra, is combined with pseudodifferential operators to simplify the proof of the Atiyah-Singer index theorem for the Dirac operator on a spin manifold.

A Darboux theorem for Hamiltonian operators in the formal calculus of variations

We prove a Darboux theorem for formal deformations of Hamiltonian operators of hydrodynamic type (Dubrovin-Novikov). Not all deformations are equivalent to the original operator: there is a moduli 2-stack of normal forms. The paper utilizes three main concepts: 1) dg Lie algebras concentrated in degrees [-1,\infty) such as the Schouten algebra - these give a convenient language for describing deformation problems; 2) the Deligne 2-groupoid associated to such a dg Lie algebra, which represents the moduli of formal deformations; 3) a refined version of the Schouten bracket in the formal calculus of variations, due to V. O. Soloviev (hep-th/9305133).

Virasoro constraints and the Chern classes of the Hodge bundle

Abstract We analyse the consequences of the Virasoro conjecture of Eguchi, Hori and Xiong for Gromov-Witten invariants, in the case of zero degree maps to the manifolds P 1 and P 2 (or more generally, smooth projective curves and smooth simply connected projective surfaces). We obtain predictions involving intersections of psi and lambda classes on M g,n . In particular, we show that the Virasoro conjecture for P 2 implies the numerical part of Fabers conjecture on the tautological Chow ring of Mg.

The odd chern character in cyclic homology and spectral flow

If A is a Banach ∗-algebra, an odd theta-summable Fredholm module over A consists of the following data: a Hilbert space H, a continuous ∗-representation ρ of A on H, and a self-adjoint operator D on H such that (1) there is a constant C such that ‖[D, ρ(a)]‖ ≤ C‖a‖ for all a ∈ A, and (2) if t > 0, then the operator e−tD 2 is trace class. Such a Fredholm module determines a class [D] in the group K−1(A), which might be thought of as an odd-dimensional chain on the non-commutative space with function algebra A. Dually, unitary matrices g ∈ UN (A) with entries in A represent elements of the group K1(A). If A0 and A1 are self-adjoint operators on H with the same spectrum (including multiplicities), the spectral flow sf(A0, A1), introduced by Atiyah-Patodi-Singer [1], is the integer which counts the number of eigenvalues crossing zero from below minus the number crossing zero from above, as we travel along the family Au = (1− u)A0 + uA1 in the direction of increasing u. If D is an odd theta-summable Fredholm module over D and g ∈ UN (A) is a unitary matrix with entries in A, the spectral flow defines a pairing

Differential forms on loop spaces and the cyclic bar complex

In this article, we present a model for the differential graded algebra (dga) of differential forms on the free loop space LX of a smooth manifold X and show how to construct certain important differential forms in terms of this model. Our motivation is an observation of Witten (described by Atiyah in [2]) that the index theorem for the Dirac operator can be thought of as an application of the localization (or fixed point) theorem in T-equivariant homology, suitably generalised to the infinite dimensional case of the free loop space; here, T is the circle group. We can summarise our main results as follows. We are really concerned with equivariant differential forms and equivariant currents and we show how to reformulate these geometric objects as cyclic chains and cochains over the differential graded algebra Ω(X) of differential forms on X. In fact, the cyclic chain complex of Ω(X), if it is normalized correctly, is a sub-complex of the complex of equivariant differential forms on LX, and is a good enough approximation that it allows us to compute the ordinary and equivariant cohomology of LX, and to write down explicitly certain important differential forms and currents. We begin by explaining the motivation more carefully. Let S be a Clifford module on X with Dirac operator D. Witten observed that it should be possible to associate to D an equivariantly closed (inhomogeneous) current μD on the free loop space of X. The basic property of this current is that the index of D is given by pairing μD with the differential form 1 ∈ Ω(LX):

Operads and Moduli Spaces of Genus 0 Riemann Surfaces

In this paper, we study two dg (differential graded) operads related to the homology of moduli spaces of pointed algebraic curves of genus 0. These two operads are dual to each other, in the sense of Kontsevich [21] and Ginzburg and Kapranov [14].

The cyclic homology of crossed product algebras

In this article, we give a new derivation of this spectral sequence, and generalize it to negative and periodic cyclic homology HC. (yl) and HP· (A). The method of proof is itself of interest, since it involves a natural generalization of the notion of a cyclic module, in which the condition that the morphism τ 6 Λ (n, n) is cyclic of order n + l is relaxed to the condition that it be invertible. We call this category the paracyclic category.