arXiv subjects
Publications and source records attributed to Hessel Posthuma.
We explain a general procedure to compute characters of chiral algebras of 2d (S)CFTs geometrically from their coadjoint orbits, in analogy with the coadjoint orbit method for finite-dimensional Lie groups. We demonstrate that this method can be used to give concise rederivations of the characters of the Virasoro algebra, Kac-Moody algebras, as well as supersymmetric generalizations.
Let $\Gamma$ be a finitely generated discrete group acting properly and cocompactly on a smooth manifold M. By employing heat-kernel techniques we prove a geometric formula for the pairing of the index class associated to a $\Gamma$-equivariant Dirac operator $D$ with a delocalized cyclic cocycles $\tau$ in $HP^\bullet (\mathbb{C}\Gamma,\langle \gamma \rangle)$. Our formula takes place on the fixed point manifold $M^\gamma$ and should be regarded as a higher Lefschetz formula for $D$. The formula involves the Atiyah-Segal-Singer form and an explicit $Z_\gamma$-invariant form on $M^\gamma$ that is naturally associated to $\tau\in HP^\bullet (\mathbb{C}\Gamma,\langle \gamma \rangle)$
We compute the Hochschild cohomology of universal enveloping algebras of Lie-Rinehart algebras in terms of the Poisson cohomology of the associated graded quotient algebras. Central in our approach are two cochain complexes of "nonlinear Chevalley-Eilenberg" cochains whose origins lie in Lie-Rinehart modules "up to homotopy", one on the Hochschild cochains of the base algebra, another related to the adjoint representation. The Poincare-Birkhoff-Witt isomorphism is then extended to a certain intertwiner between such modules. Finally, exploiting the twisted Calabi-Yau structure, we obtain results for the dual Hochschild and cyclic homology.
Let $G$ be a connected, linear real reductive group and let $X$ be a cocompact $G$-proper manifold without boundary. We define delocalized eta invariants associated to a $L^2$-invertible perturbed Dirac operator $D_X+A$ with $A$ a suitable smoothing perturbation. We also investigate the case in which $D_X$ is not invertible but $0$ is isolated in the $L^2$-spectrum of $D_X$. We prove index formulas relating these delocalized eta invariants to Atiyah-Patodi-Singer delocalized indices on $G$-proper manifolds with boundary. In order to achieve this program we give a detailed account of both the large and small time behaviour of the heat-kernel of perturbed Dirac operators, as a map from the positive real line to the algebra of Lafforgue integral operators. We apply these results to the definition of rho-numbers associated to $G$-homotopy equivalences between closed $G$-proper manifolds and to the study of their bordism properties. We also define delocalized signatures of manifolds with boundary satisfying an invertibility assumption on the differential form Laplacian of the boundary in middle degree and prove an Atiyah-Patodi-Singer formula for these delocalized signatures.
Given a smooth action of a Lie group on a manifold, we give two constructions of the Chern character of an equivariant vector bundle in the cyclic cohomology of the crossed product algebra. The first construction associates a cycle to the vector bundle whose structure maps are closely related to Getzler's model for equivariant cohomology. The second construction uses a direct map between this model and the (periodic) cyclic cohomology localized at the unit element. Finally, it is shown that the two constructions are equivalent when the action is proper.
Let $G$ be a connected, linear real reductive group. We give sufficient conditions ensuring the well-definedness of the delocalized eta invariant $\eta_g (D_X)$ associated to a Dirac operator $D_X$ on a cocompact $G$-proper manifold $X$ and to the orbital integral $\tau_g$ defined by a semisimple element $g\in G$. Along the way, we give a detailed account of the large time behaviour of the heat kernel and of its short time bahaviour near the fixed point set of $g$. We prove that such a delocalized eta invariant enters as the boundary correction term in an index theorem computing the pairing between the index class and the 0-degree cyclic cocycle defined by $\tau_g$ on a $G$-proper manifold with boundary. More importantly, we also prove a higher version of such a theorem, for the pairing of the index class and the higher cyclic cocycles defined by the higher orbital integral $\Phi^P_g$ associated to a cuspidal parabolic subgroup $P<G$ with Langlands decomposition $P=MAN$ and a semisimple element $g\in M$. We employ these results in order to define (higher) rho numbers associated to $G$-invariant positive scalar curvature metrics.
We define a morphism from the deformation complex of a Lie groupoid to the Hochschild complex of its convolution algebra, and show that it maps the class of a geometric deformation to the algebraic class of the induced deformation in Hochschild cohomology. Applied to the adiabatic groupoid, we show that the van Est map to deformation cohomology of Lie algebroids is induced by taking the classical limit of a quantization map on the dual of the Lie algebroid.
Let G be a finitely connected Lie group and let K be a maximal compact subgroup. Let M be a cocompact G-proper manifold with boundary, endowed with a G-invariant metric which is of product type near the boundary. Under additional assumptions on G, for example that it satisfies the Rapid Decay condition and is such that G/K has nonpositive sectional curvature, we define higher Atiyah-Patodi-Singer C^*-indices associated to smooth group cocycles on G and to a generalized G-equivariant Dirac operator D on M with L^2-invertible boundary operator D_\partial. We then establish a higher index formula for these C^*-indices and use it in order to introduce higher genera for M, thus generalizing to manifolds with boundary the results that we have established in Part 1. Our results apply in particular to a semisimple Lie group G. We use crucially the pairing between suitable relative cyclic cohomology groups and relative K-theory groups.
Topological phases for free fermions in systems with crystal symmetry are classified by the topology of the valence band viewed as a vector bundle over the Brillouin zone. Additional symmetries, such as crystal symmetries which act non-trivially on the Brillouin zone, or time-reversal symmetry, endow the vector bundle with extra structure. These vector bundles are classified by a suitable version of K-theory. While relatively easy to define, these K-theory groups are notoriously hard to compute in explicit examples. In this paper we describe in detail how one can compute these K-theory groups starting with a decomposition of the Brillouin zone in terms of simple submanifolds on which the symmetries act nicely. The main mathematical tool is the Atiyah-Hirzebruch spectral sequence associated to such a decomposition, which will not only yield the explicit result for several crystal symmetries, but also sheds light on the origin of the topological invariants. This extends results that have appeared in the literature so far. We also describe examples in which this approach fails to directly yield a conclusive answer, and discuss various open problems and directions for future research.
Let G be a Lie group with finitely many connected components and let K be a maximal compact subgroup. We assume that G satisfies the rapid decay (RD) property and that G/K has non-positive sectional curvature. As an example, we can take G to be a connected semisimple Lie group. Let M be a G-proper manifold with compact quotient M/G. In this paper we establish index formulae for the C^*-higher indices of a G-equivariant Dirac-type operator on M. We use these formulae to investigate geometric properties of suitably defined higher genera on M. In particular, we establish the G-homotopy invariance of the higher signatures of a G-proper manifold and the vanishing of the A-hat genera of a G-spin, G-proper manifold admitting a G-invariant metric of positive scalar curvature.
We study Lie algebroids from the point of view noncommutative geometry. More specifically, using ideas from deformation quantization, we use the PBW-theorem for Lie algebroids to construct a Fedosov-type resolution for the associated sheaves of Weyl algebras. This resolution enables us to construct a "character map" --in the derived category-- from the sheafified cyclic chain complexes to the Chevalley--Eilenberg complex of the Lie algebroid. The index theorem computes the evaluation of this map on the trivial cycle in terms of the Todd--Chern characteristic class. Finally, we show compatibility of the character map with the Hochschild--Kostant--Rosenberg morphism.
We compute the bi-Hamiltonian cohomology of an arbitrary dispersionless Poisson pencil in a single dependent variable using a spectral sequence method. As in the KdV case, we obtain that $BH^p_d(\hat{F}, d_1,d_2)$ is isomorphic to $\mathbb{R}$ for $(p,d)=(0,0)$, to $C^\infty (\mathbb{R})$ for $(p,d)=(1,1)$, $(2,1)$, $(2,3)$, $(3,3)$, and vanishes otherwise.
We prove that the bihamiltonian cohomology of a semisimple pencil of Poisson brackets of hydrodynamic type vanishes for almost all degrees. This implies the existence of a full dispersive deformation of a semisimple bihamiltonian structure of hydrodynamic type starting from any infinitesimal deformation.
Using spectral sequences techniques we compute the bihamiltonian cohomology groups of the pencil of Poisson brackets of dispersionless KdV hierarchy. In particular this proves a conjecture of Liu and Zhang about the vanishing of such cohomology groups.
For a possibly singular subset of a regular Poisson manifold we construct a deformation quantization of its algebra of Whitney functions. We then extend the construction of a deformation quantization to the case where the underlying set is a subset of a not necessarily regular Poisson manifold which can be written as the quotient of a regular Poisson manifold on which a compact Lie group acts freely by Poisson maps. Finally, if the quotient Poisson manifold is regular as well, we show a "quantization commutes with reduction" type result. For the proofs, we use methods stemming from both singularity theory and Poisson geometry.
To each partition function of cohomological field theory one can associate an Hamiltonian integrable hierarchy of topological type. The Givental group acts on such partition functions and consequently on the associated integrable hierarchies. We consider the Hirota and Lax formulations of the deformation of the hierarchy of N copies of KdV obtained by an infinitesimal action of the Givental group. By first deforming the Hirota quadratic equations and then applying a fundamental lemma to express it in terms of pseudo-differential operators, we show that such deformed hierarchy admits an explicit Lax formulation. We then compare the deformed Hamiltonians obtained from the Lax equations with the analogous formulas obtained in [1, 2], to find that they agree. We finally comment on the possibility of extending the Hirota and Lax formulation on the whole orbit of the Givental group action.
We give a systematic description of the cyclic cohomology theory of Hopf algebroids in terms of its associated category of modules. Then we introduce a dual cyclic homology theory by applying cyclic duality to the underlying cocyclic object. We derive general structure theorems for these theories in the special cases of commutative and cocommutative Hopf algebroids. Finally, we compute the cyclic theory in examples associated to Lie-Rinehart algebras and étale groupoids.
We consider the construction of Frobenius manifolds associated to projective special geometry and analyse the dependence on choices involved. In particular, we prove that the underlying F-manifold is canonical. We then apply this construction to integrable systems of Hitchin type.