Searcharxiv⌕ Search

arXiv subjects

Eckhard Meinrenken

Publications and source records attributed to Eckhard Meinrenken.

51 records · Page 3Linked to original sources
arXiv2000

Linearization of Poisson actions and singular values of matrix products

We prove that the linearization functor from the category of Hamiltonian K-actions with group-valued moment maps in the sense of Lu, to the category of ordinary Hamiltonian K-actions, preserves products up to symplectic isomorphism. As an application, we give a new proof of the Thompson conjecture on singular values of matrix products and extend this result to the case of real matrices. We give a formula for the Liouville volume of these spaces and obtain from it a hyperbolic version of the Duflo isomorphism.

math.SG↗
arXiv2000

Quasi-Poisson Manifolds

A quasi-Poisson manifold is a G-manifold equipped with an invariant bivector field whose Schouten bracket is the trivector field generated by the invariant element in $\wedge^3 \g$ associated to an invariant inner product. We introduce the concept of the fusion for such manifolds, and we relate quasi-Poisson manifolds to the previously introduced quasi-Hamiltonian manifolds with group-valued moment maps.

math.DG↗
arXiv2000

Formulas of Verlinde type for non-simply connected groups

We derive Verlinde's formula from the fixed point formula for loop groups proved in the companion paper "A fixed point formula for loop group actions", and extend it to compact, connected groups that are not necessarily simply-connected.

math.SG↗
arXiv1999

Group-valued equivariant localization

We prove a localization formula for group-valued equivariant de Rham cohomology of a compact G-manifold. This formula is a non-trivial generalization of the localization formula of Berline-Vergne and Atiyah-Bott for the usual equivariant de Rham cohomology. As an application, we obtain a version of the Duistermaat-Heckman formula for Hamiltonian spaces with group-valued moment maps.

math.DG↗
arXiv1999

The non-commutative Weil algebra

Let G be a connected Lie group with Lie algebra g. The Duflo map is a vector space isomorphism between the symmetric algebra S(g) and the universal enveloping algebra U(g) which, as proved by Duflo, restricts to a ring isomorphism from invariant polynomials onto the center of the universal enveloping algebra. The Duflo map extends to a linear map from compactly supported distributions on the Lie algebra g to compactly supported distributions on the Lie group G, which is a ring homomorphism for G-invariant distributions. In this paper we obtain analogues of the Duflo map and of Duflo's theorem in the context of equivariant cohomology of G-manifolds. Our result involves a non-commutative version of the Weil algebra and of the de Rham model of equivariant cohomology.

math.DG↗
arXiv1997

Singular Reduction and Quantization

Consider a compact prequantizable symplectic manifold M on which a compact Lie group G acts in a Hamiltonian fashion. The ``quantization commutes with reduction'' theorem asserts that the G-invariant part of the equivariant index of M is equal to the Riemann-Roch number of the symplectic quotient of M, provided the quotient is nonsingular. We extend this result to singular symplectic quotients, using partial desingularizations of the symplectic quotient to define its Riemann-Roch number. By similar methods we also compute multiplicities for the equivariant index of the dual of a prequantum bundle, and furthermore show that the arithmetic genus of a Hamiltonian G-manifold is invariant under symplectic reduction.

dg-ga↗
arXiv1997

Lie Group Valued Moment Maps

We develop a theory of "quasi"-Hamiltonian G-spaces for which the moment map takes values in the group G itself rather than in the dual of the Lie algebra. The theory includes counterparts of Hamiltonian reductions, the Guillemin-Sternberg symplectic cross-section theorem and of convexity properties of the moment map. As an application we obtain moduli spaces of flat connections on an oriented compact 2-manifold with boundary as quasi-Hamiltonian quotients of the space G^2 x ... x G^2.

dg-ga↗
arXiv1997

Fusion of Hamiltonian loop group manifolds and cobordism

We construct an oriented cobordism between moduli spaces of flat connections on the three holed sphere and disjoint unions of toric varieties, together with a closed two-form which restricts to the symplectic forms on the ends. As applications, we obtain formulas for mixed Pontrjagin numbers and Witten's formulas for symplectic volumes.

dg-ga↗
arXiv1996

Symplectic Surgery and the Spin-C Dirac operator

Let $G$ be a compact connected Lie group, and $M$ a compact Hamiltonian $G$-space, with moment map $J$. For each $G$-equivariant Hermitian vector bundle $E$ over $M$, one has an associated twisted Spin-C Dirac operator, whose equivariant index is a symplectic invariant of $E$. In the present paper, we study gluing properties of the equivariant index under "symplectic cutting" operations. Our main application is a proof of the Guillemin-Sternberg conjecture, which says that if $E=L$ is a quantizing line bundle and $0$ a regular value of $J$, the multiplicity of the trivial representation in the equivariant index is equal to the Riemann-Roch number of the symplectic quotient. This generalizes previous results for the case that $G=T$ is abelian.

dg-ga↗
arXiv1996

A symplectic proof of Verlinde factorization

We prove a multiplicity formula for Riemann-Roch numbers of reductions of Hamiltonian actions of loop groups. This includes as a special case the factorization formula for the quantum dimension of the moduli space of flat connections over a Riemann surface.

dg-ga↗
arXiv1996

Non-abelian convexity by symplectic cuts

In this paper we extend the results of Kirwan et alii on convexity properties of the moment map for Hamiltonian group actions, and on the connectedness of the fibers of the moment map, to the case of non-compact orbifolds. Our motivation is twofold. First, the category of orbifolds is important in symplectic geometry because, generically, the symplectic quotient of a symplectic manifold is an orbifold. Second, our proof is conceptually very simple since it reduces the non-abelian case to the abelian case.

dg-ga↗
arXiv1994

On Riemann-Roch Formulas for Multiplicities

A Theorem due to Guillemin and Sternberg about geometric quantization of Hamiltonian actions of compact Lie groups $G$ on compact Kaehler manifolds says that the dimension of the $G$-invariant subspace is equal to the Riemann-Roch number of the symplectically reduced space. Combined with the shifting-trick, this gives explicit formulas for the multiplicities of the various irreducible components. One of the assumptions of the Theorem is that the reduction is regular, so that the reduced space is a smooth symplectic manifold. In this paper, we prove a generalization of this result to the case where the reduced space may have orbifold singularities. Our proof uses localization techniques from equivariant cohomology, and relies in particular on recent work of Jeffrey-Kirwan and Guillemin. Since there are no complex geometry arguments involved, the result also extends to non Kaehlerian settings.

alg-geom↗
arXiv1994

Toeplitz Quantization of Kähler Manifolds and $gl(N)$ $N\to\infty$

For general compact Kähler manifolds it is shown that both Toeplitz quantization and geometric quantization lead to a well-defined (by operator norm estimates) classical limit. This generalizes earlier results of the authors and Klimek and Lesniewski obtained for the torus and higher genus Riemann surfaces, respectively. We thereby arrive at an approximation of the Poisson algebra by a sequence of finite-dimensional matrix algebras $gl(N)$, $N\to\infty$.

hep-th↗