A Class of Topological Actions

We review definitions of generalized parallel transports in terms of Cheeger-Simons differential characters. Integration formulae are given in terms of Deligne-Beilinson cohomology classes. These representations of parallel transport can be extended to situations involving distributions as is appropriate in the context of quantized fields.Comment: 41 pages, no figure

Reformulating Supersymmetry with a Generalized Dolbeault Operator

The conditions for N=1 supersymmetry in type II supergravity have been previously reformulated in terms of generalized complex geometry. We improve that reformulation so as to completely eliminate the remaining explicit dependence on the metric. Doing so involves a natural generalization of the Dolbeault operator. As an application, we present some general arguments about supersymmetric moduli. In particular, a subset of them are then classified by a certain cohomology. We also argue that the Dolbeault reformulation should make it easier to find existence theorems for the N=1 equations.Comment: 30 pages, no figures. v2: minor correction

Cyclic homology and the Macdonald conjectures

Let A+(k) denote the ring ℂ[ t ]/ t k+1 and let G be a reductive complex Lie algebra with exponents m 1 , ..., m n . This paper concerns the Lie algebra cohomology of G ⊗ A + ( k ) considered as a bigraded algebra (here one of the gradings is homological degree and the other, which we call weight , is inherited from the obvious grading of G ⊗ A + ( k )). We conjecture that this Lie algebra cohomology is an exterior algebra with k +1 generators of homological degree 2 m s +1 for s =1,2, ..., n . Of these k +1 generators of degree 2 m s +1, one has weight 0 and the others have weights ( k +1) m s +t for t =1,2, ..., k .Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46617/1/222_2005_Article_BF01391498.pd

New constructions of twistor lifts for harmonic maps

We show that given a harmonic map $\varphi$ from a Riemann surface to a classical compact simply connected inner symmetric space, there is a $J_2$-holomorphic twistor lift of $\varphi$ (or its negative) if and only if it is nilconformal. In the case of harmonic maps of finite uniton number, we give algebraic formulae in terms of holomorphic data which describes their extended solutions. In particular, this gives explicit formulae for the twistor lifts of all harmonic maps of finite uniton number from a surface to the above symmetric spaces.Comment: Some minor changes and a correction of Example 8.