Torus Fibrations and Localization of Index II

We give a framework of localization for the index of a Dirac-type operator on an open manifold. Suppose the open manifold has a compact subset whose complement is covered by a family of finitely many open subsets, each of which has a structure of the total space of a torus bundle. Under an acyclic condition we define the index of the Dirac-type operator by using the Witten-type deformation, and show that the index has several properties, such as excision property and a product formula. In particular, we show that the index is localized on the compact set.     

Higher-Degree Analogs of the Determinant Line Bundle

 In the first part of this paper, given a smooth family of Dirac-type operators on an odd-dimensional closed manifold, we construct an abelian gerbe-with-connection whose curvature is the three-form component of the Atiyah-Singer families index theorem. In the second part of the paper, given a smooth family of Dirac-type operators whose index lies in the subspace of the reduced K-theory of the parametrizing space, we construct a set of Deligne cohomology classes of degree i whose curvatures are the i-form component of the Atiyah-Singer families index theorem. Received: 27 September 2001 / Accepted: 5 April 2002 Published online: 22 August 2002     

High Energy Limits of Laplace-Type and Dirac-Type EigenFunctions and Frame Flows

We relate high-energy limits of Laplace-type and Dirac-type operators to frame flows on the corresponding manifolds, and show that the ergodicity of frame flows implies quantum ergodicity in an appropriate sense for those operators. Observables for the corresponding quantum systems are matrix-valued pseudodifferential operators and therefore the system remains non-commutative in the high-energy limit. We discuss to what extent the space of stationary high-energy states behaves classically. The first author was supported by NSERC, FQRNT and Dawson fellowship.     

Bundle gerbes and moduli spaces

In this paper, we construct the index bundle gerbe of a family of self-adjoint Dirac-type operators, refining a construction of Segal. In a special case, we construct a geometric bundle gerbe called the caloron bundle gerbe, which comes with a natural connection and curving, and show that it is isomorphic to the analytically constructed index bundle gerbe. We apply these constructions to certain moduli spaces associated to compact Riemann surfaces, constructing on these moduli spaces, natural bundle gerbes with connection and curving, whose 3-curvature represent Dixmier-Douady classes that are generators of the third de Rham cohomology groups of these moduli spaces.     

Dirac Operators on Taub-NUT Space: Relationship and Discrete Transformations

It is shown that the N = 4 superalgebra of the Dirac theory in Taub-NUT space has different unitary representations related among themselves through unitary U(2) transformations. In particular the SU(2) transformations are generated by the spin-like operators constructed with the help of the same covariantly constant Killing-Yano tensors which generate Dirac-type operators. A parity operator is defined and some explicit transformations which connect the Dirac-type operators among themselves are given. These transformations form a discrete group which is a realization of the quaternion discrete group. The fifth Dirac operator constructed using the non-covariant constant Killing-Yano tensor of the Taub-NUT space is quite special. This non-standard Dirac operator is connected with the hidden symmetry and is not equivalent to the Dirac-type operators of the standard N = 4 supersymmetry.     

Vertex Operators, Grassmannians, and Hilbert Schemes

We approximate the infinite Grassmannian by finite-dimensional cutoffs, and define a family of fermionic vertex operators as the limit of geometric correspondences on the equivariant cohomology groups, with respect to a one-dimensional torus action. We prove that in the localization basis, these are the well-known fermionic vertex operators on the infinite wedge representation. Furthermore, the boson-fermion correspondence, locality, and intertwining properties with the Virasoro algebra are the limits of relations on the finite-dimensional cutoff spaces, which are true for geometric reasons. We then show that these operators are also, almost by definition, the vertex operators defined by Okounkov and the author in Carlsson and Okounkov ( [math.AG], 2009), on the equivariant cohomology groups of the Hilbert scheme of points on \mathbb C²{\mathbb C^2} , with respect to a special torus action.     

Determinant bundles,manifolds with boundary and surgery

We define determinant bundles associated to the following data: (i) a family of generalized Dirac operators on even dimensional manifolds with boundary, (ii) the choice of a spectral section for the family of Dirac operators induced on the boundary. Under the assumption that the operators of the boundary family have null spaces of constant dimension we define, through the notion ofb-zeta function, a Quillen metric. We also introduce the analogue of the Bismut-Freed connection. We prove that the curvature of a natural perturbation of the Bismut-Freed connection equals the 2-form piece in the right-hand side of the family index formula, thus extending to manifolds with boundary results of Quillen, Bismut and Freed. Given a closed fibration, we investigate the behaviour of the Quillen metric and of the Bismut-Freed connection under the operation of surgery along a fibering hypersurface. We prove, in particular, additivity formulae for the curvature and for the logarithm of the holonomy.     

Stability in Yang-Mills theories

At a solution of the Yang-Mills equations onS ⁴, or the Yang-Mills-Higgs equation on ?³, the hessian of the action functional defines a natural second order, elliptic operator. The number of negative eigenvalues of this operator is bounded below by a multiple of the relevant topological charge. The proof of this assertion requires a relative index theorem for Dirac-type operators on ? ⁿ ,n≧3.     

Partition functions and physical states in two-dimensional quantum gravity and supergravity

We review the Liouville theory calculation of the genus-one path integral for c 1 conformal models coupled to two-dimensional gravity. From the modular integrand we derive the existence of an infinite number of physical operators which are in one-to-one correspondence with the conformal primary fields and null states of the matter theory. We also calculate the torus path integral and find the spectrum of physical operators for superconformal models coupled to supergravity. The amplitude in the odd spin structure requires a special treatment and is found to be proportional to the Witten index of the matter theory.     

The Feynman–Kac formula for Schrödinger operators on vector bundles over complete manifolds

Methods from stochastic analysis are combined with functional analytic methods in order to prove a Feynman–Kac formula för Schrödinger type operators with nonnegative locally square integrable potentials on vector bundles over complete Riemannian manifolds. In particular, we obtain a Feynman–Kac–Itô formula on manifolds for Schrödinger operators with magnetic fields.     

Homoclinic tangency and chaotic attractor disappearance in a dripping faucet experiment

A sequence of attractors, reconstructed from interdrops time series data of a leaky faucet experiment, is analyzed as a function of the mean dripping rate. We established the presence of a saddle point and its manifolds in the attractors and we explained the dynamical changes in the system using the evolution of the manifolds of the saddle point, as suggested by the orbits traced in first return maps. The sequence starts at a fixed point and evolves to an invariant torus of increasing diameter (a Hopf bifurcation) that pushes the unstable manifold towards the stable one. The torus breaks up and the system shows a chaotic attractor bounded by the unstable manifold of the saddle. With the attractor expansion the unstable manifold becomes tangential to the stable one, giving rise to the sudden disappearance of the chaotic attractor, which is an experimental observation of a so called chaotic blue sky catastrophe.     

Hidden symmetries and killing tensors on curved spaces

Higher-order symmetries corresponding to Killing tensors are investigated. The intimate relation between Killing-Yano tensors and nonstandard supersymmetries is pointed out. In the Dirac theory on curved spaces, Killing-Yano tensors generate Dirac-type operators involved in interesting algebraic structures as dynamical algebras or even infinite dimensional algebras or superalgebras. The general results are applied to space-times which appear in modern studies. One presents the infinite dimensional superalgebra of Dirac type operators on the 4-dimensional Euclidean Taub-NUT space that can be seen as a twisted loop algebra. The existence of the conformal Killing-Yano tensors is investigated for some spaces with mixed 3-Sasakian structures.     

Birth of bilayered torus and torus breakdown in a piecewise-smooth dynamical system

Border-collision bifurcations arise when the periodic trajectory of a piecewise-smooth system under variation of a parameter crosses into a region with different dynamics. Considering a three-dimensional map describing the behavior of a DC/DC power converter, the Letter discusses a new type of border-collision bifurcation that leads to the birth of a “bilayered torus”. This torus consists of the union of two saddle cycles, their unstable manifolds, and a stable focus cycle. When changing the parameters, the bilayered torus transforms through a border-collision bifurcation into a resonance torus containing the stable cycle and a saddle. The Letter also presents scenarios for torus destruction through homoclinic and heteroclinic tangencies.     

Determinant Bundles, Manifolds with Boundary and Surgery¶II. Spectral Sections and Surgery Rules for Anomalies

Let be a closed fibration of Riemannian manifolds and let , be a family of generalized Dirac operators. Let be an embedded hypersurface fibering over B; . Let be the Dirac family induced on . Each fiber in is the union along of two manifolds with boundary . In this paper, generalizing our previous work[16], we prove general surgery rules for the local and global anomalies of the Bismut–Freed connection on the determinant bundle associated to . Our results depend heavily on the b-calculus [12], on the surgery calculus [11] and on the APS family index theory developed in [13], in particular on the notion of spectral section for the family . Received: 23 October 1996 / Accepted: 28 July 1997     

Chiral Asymmetry and the Spectral Action

We consider orthogonal connections with arbitrary torsion on compact Riemannian manifolds. For the induced Dirac operators, twisted Dirac operators and Dirac operators of Chamseddine-Connes type we compute the spectral action. In addition to the Einstein-Hilbert action and the bosonic part of the Standard Model Lagrangian we find the Holst term from Loop Quantum Gravity, a coupling of the Holst term to the scalar curvature and a prediction for the value of the Barbero-Immirzi parameter.     

Šilnikov manifolds in coupled nonlinear Schrödinger equations

We consider two coupled nonlinear Schrödinger equations with even, periodic boundary conditions, that are damped and quasiperiodically forced. We prove the existence of invariant manifolds with Šilnikov-type dynamics that are homoclinic to a spatially independent invariant torus. Such manifolds appear to induce complex behavior in numerical experiments.     

A note on toric contact geometry

After observing that the well-known convexity theorems of symplectic geometry also hold for compact contact manifolds with an effective torus action whose Reeb vector field corresponds to an element of the Lie algebra of the torus, we use this fact together with a recent symplectic orbifold version of Delzant’s theorem due to Lerman and Tolman [E. Lerman, S. Tolman, Trans. Am. Math. Soc. 349 (10) (1997) 4201–4230] to show that every such compact toric contact manifold can be obtained by a contact reduction from an odd dimensional sphere.     

Axial anomalies and index theorems on open spaces

Using an approach inspired by the theory of the anomalous divergence of the axial vector current, we derive trace formulas for the resolvents of Dirac operators on open spaces of odd dimension. These formulas readily yield index theorems for these operators. As applications we determine the index of the Dirac operator for a particle of arbitrary isospin in the background field of a static system of SU(2) monopoles; and we find formulas in essentially closed form for certain determinants involving these operators.This work is supported in part through funds provided by the U.S. Department of Energy (DOE) under contract EY-76-C-02-3069     

On mod 2 and higher elliptic genera

In the first part of this paper, we construct mod 2 elliptic genera on manifolds of dimensions 8k+1, 8k+2 by mod 2 index formulas of Dirac operators. They are given by mod 2 modular forms or mod 2 automorphic functions. We also obtain an integral formula for the mod 2 index of the Dirac operator. As a by-product we find topological obstructions to group actions. In the second part, we construct higher elliptic genera and prove some of their rigidity properties under group actions. In the third part we write down characteristic series for all Witten genera by Jacobi theta-functions. The modular property and transformation formulas of elliptic genera then follow easily. We shall also prove that Krichever's genera, which come from integrable systems, can be written as indices of twisted Dirac operators forSU-manifolds. Some general discussions about elliptic genera are given.