Constantin Teleman

Loop groups and twisted K-theory I

This is the first in a series of papers investigating the relationship between the twisted equivariant K-theory of a compact Lie group G and the “Verlinde ring” of its loop group. In this paper we set up the foundations of twisted equivariant K-groups, and more generally twisted K-theory of groupoids. We establish enough basic properties to make effective computations. We determine the twisted equivariant K-groups of a compact connected Lie group G with torsion free fundamental group. We relate this computation to the representation theory of the loop group at a level related to the twisting.

The quantization conjecture revisited

A strong version of the quantization conjecture of Guillemin and Sternberg is proved. For a reductive group action on a smooth, compact, polarized variety (X, L), the cohomologies of L over the GIT quotient X//G equal the invariant part of the cohomologies over X. This generalizes the theorem of [GS] on global sections, and strengthens its subsequent extensions ([JK], [li]) to RiemannRoch numbers. Remarkable by-products are the invariance of cohomology of vector bundles over X//G under a small change in the defining polarization or under shift desingularization, as well as a new proof of Boutots theorem. Also studied are equivariant holomorphic forms and the equivariant Hodgeto-de Rham spectral sequences for X and its strata, whose collapse is shown. One application is a new proof of the Borel-Weil-Bott theorem of [Ti] for the moduli stack of C-bundles over a curve, and of analogous statements for the moduli stacks and spaces of bundles with parabolic structures. Collapse of the Hodge-to-de Rham sequences for these stacks is also shown.

Twisted equivariant K-theory with complex coefficients

Using a global version of the equivariant Chern character, we describe the complexified twisted equivariant K-theory of a space with a compact Lie group action in terms of fixed-point data. We apply this to the case of a compact group acting on itself by conjugation and relate the result to the Verlinde algebra and to the Kac numerator at q=1. Verlindes formula is also discussed in this context.

Lie algebra cohomology and the fusion rules

We prove a vanishing theorem for Lie algebra cohomology which constitutes a loop group analogue of Kostants Lie algebra version of the Borel-Weil-Bott theorem. Consider a complex semi-simple Lie algebra and an integrable, irreducible, negative energy representation ℋ of. Givenn distinct pointszk in ℂ, with a finite-dimensional irreducible representationVk of assigned to each, the Lie algebra of-valued polynomials acts on eachVk, via evaluation atzk. Then, the relative Lie algebra cohomologyH* is concentrated in one degree. As an application, based on an idea of G. Segals, we prove that a certain “homolorphic induction” map from representations ofG to representations ofLG at a given level takes the ordinary tensor product into the fusion product. This result had been conjectured by R. Bott.

Loop groups and twisted

Elliptic operators appear in different guises in the representation theory of compact Lie groups. The Borel-Weil construction [BW], phrased in terms of holomorphic functions, has at its heart the ∂̄ operator on Kähler manifolds. The ∂̄ operator and differential geometric vanishing theorems figure prominently in the subsequent generalization by Bott [B]. An alternative approach using an algebraic Laplace operator was given by Kostant [K3]. The Atiyah-Bott proof [AB] of Weyl’s character formula uses a fixed point theorem for the ∂̄ complex. On a spin Kähler manifold ∂̄ can be expressed in terms of the Dirac operator. This involves a shift, in this context the ρ-shift whose analog for loop groups appears in our main theorem. Dirac operators may be used instead of ∂̄ in these applications to representation theory, and indeed they often appear explicitly. In this paper we introduce a new construction: the Dirac family attached to a representation of a Lie group G which is either compact or the loop group of a compact Lie group; in the latter case the representation is restricted to having positive energy. The Dirac family is a collection of Fredholm operators parametrized by an affine space, equivariant for an affine action of a central extension of G by the circle group T. For an irreducible representation the support of the family is the coadjoint orbit given by the Kirillov correspondence, and the entire construction is reminiscent of the Fourier transform of the character [K2, Rule 6]. The Dirac family represents a class in twisted equivariant K-theory, so we obtain a map from representations to K-theory. For compact Lie groups it is a nonstandard realization of the twisted equivariant Thom homomorphism, which was proved long ago to be an isomorphism. Our main result, Theorem 3.44, is that this map from representations to K-theory is an isomorphism when G is a loop group. The existence of a construction along these lines was first suggested by Graeme Segal; cf. [AS, §5].

K

AbstractIn the first section of this note, we show that Theorem 1.8.1 of Bayer-Manin can be strengthened in the following way: If the even quantum cohomology of a projective algebraic manifold V is generically semisimple, then V has no odd cohomology and is of Hodge-Tate type. In particular, this answers a question discussed by G. Ciolli. In the second section, we prove that an analytic (or formal ) supermanifold M with a given supercommutative associative

An update on semisimple quantum cohomology and F-manifolds

\mathcal{O}_M

Self-extensions of Verma modules and differential forms on opers

-bilinear multiplication on its tangent sheaf