Singer Theorem Research Articles

A cohomological formula for the Atiyah–Patodi–Singer index on manifolds with boundary

The main result of this paper is a new Atiyah–Singer type cohomological formula for the index of Fredholm pseudodifferential operators on a manifold with boundary. The nonlocality of the chosen boundary condition prevents us to apply directly the methods used by Atiyah and Singer in [4, 5]. However, by using the K-theory of C*-algebras associated to some groupoids, which generalizes the classical K-theory of spaces, we are able to understand the computation of the APS index using classic algebraic topology methods (K-theory and cohomology). As in the classic case of Atiyah–Singer ([4, 5]), we use an embedding into a Euclidean space to express the index as the integral of a true form on a true space, the integral being over a C∞-manifold called the singular normal bundle associated to the embedding. Our formula is based on a K-theoretical Atiyah–Patodi–Singer theorem for manifolds with boundary that is inspired by Connes' tangent groupoid approach, it is not a groupoid interpretation of the famous Atiyah–Patodi–Singer index theorem.

AMBROSE–SINGER THEOREM ON DIFFEOLOGICAL BUNDLES AND COMPLETE INTEGRABILITY OF THE KP EQUATION

In this paper, we start from an extension of the notion of holonomy on diffeological bundles, reformulate the notion of regular Lie group or Frölicher Lie groups, state an Ambrose–Singer theorem that enlarges the one stated in [J.-P. Magnot, Structure groups and holonomy in infinite dimensions, Bull. Sci. Math.128 (2004) 513–529], and conclude with a differential geometric treatment of KP hierarchy. The examples of Lie groups that are studied are principally those obtained by enlarging some graded Frölicher (Lie) algebras such as formal q-series of the quantum algebra of pseudo-differential operators. These deformations can be defined for classical pseudo-differential operators but they are used here on formal pseudo-differential operators in order to get a differential geometric framework to deal with the KP hierarchy that is known to be completely integrable with formal power series. Here, we get an integration of the Zakharov–Shabat connection form by means of smooth sections of a (differential geometric) bundle with structure group, some groups of q-deformed operators. The integration obtained by Mulase [Complete integrability of the Kadomtsev–Petviashvili equation Adv. Math.54 (1984) 57–66], and the key tools he developed, are totally recovered on the germs of the smooth maps of our construction. The tool coming from (classical) differential geometry used in this construction is the holonomy group, on which we have an Ambrose–Singer-like theorem: the Lie algebra is spanned by the curvature elements. This result is proved for any connection a diffeological principal bundle with structure group a regular Frölicher Lie group. The case of a (classical) Lie group modeled on a complete locally convex topological vector space is also recovered and the work developed in [J.-P. Magnot, Difféologie du fibré d'Holonomie en dimension infinie, Math. Rep. Canadian Roy. Math. Soc.28(4) (2006); J.-P. Magnot, Structure groups and holonomy in infinite dimensions, Bull. Sci. Math. 128 (2004) 513–529] is completed.

ALGEBRAIC ASPECTS OF INTEGRABILITY FOR POLYNOMIAL DIFFERENTIAL SYSTEMS

In this article we summarize the results on algebraic aspects of integrability for polynomial differential systems and its application, which include the Darboux, elementary and Liouvelle integrability. Darboux theory of integrability was found by Darboux in 1878, and it becomes extremely useful in study of the center focus problem, of bifurcation, of limit cycle problem and of global dynamics. The importance of Darboux theory of integrability is also presented by the Singer's theorem for planar polynomial differential system. That is, if a polynomial system is Liouville integrable, then it is Darboux integrable, i.e. the system has a Darboux first integral or a Darboux integrating factor.

DIRAC OPERATOR ON COMPLEX MANIFOLDS AND SUPERSYMMETRIC QUANTUM MECHANICS

We explore a simple [Formula: see text] supersymmetric quantum mechanics (SQM) model describing the motion over complex manifolds in external gauge fields. The nilpotent supercharge Q of the model can be interpreted as a (twisted) exterior holomorphic derivative, such that the model realizes the twisted Dolbeault complex. The sum [Formula: see text] can be interpreted as the Dirac operator: the standard Dirac operator if the manifold is Kähler and the Dirac operator involving certain particular extra torsions for a generic complex manifold. Focusing on the Kähler case, we give new simple physical proofs of the two mathematical facts: (i) the equivalence of the twisted Dirac and twisted Dolbeault complexes and (ii) the Atiyah–Singer theorem.

A topological index theorem for manifolds with corners

AbstractWe define an analytic index and prove a topological index theorem for a non-compact manifoldM0with poly-cylindrical ends. Our topological index theorem depends only on the principal symbol, and establishes the equality of the topological and analytical index in the groupK0(C*(M)), whereC*(M) is a canonicalC*-algebra associated to the canonical compactificationMofM0. Our topological index is thus, in general, not an integer, unlike the usual Fredholm index appearing in the Atiyah–Singer theorem, which is an integer. This will lead, as an application in a subsequent paper, to the determination of theK-theory groupsK0(C*(M)) of the groupoidC*-algebra of the manifolds with cornersM. We also prove that an elliptic operatorPonM0has an invertible perturbationP+Rby a lower-order operator if and only if its analytic index vanishes.

DIMENSION OF THE MODULI SPACE AND HAMILTONIAN ANALYSIS OF BF FIELD THEORIES

By using the Atiyah–Singer theorem through some similarities with the instanton and the antiinstanton moduli spaces, the dimension of the moduli space for two- and four-dimensional BF theories valued in different background manifolds and gauge groups scenarios is determined. Additionally, we develop Dirac's canonical analysis for a four-dimensional modified BF theory, which reproduces the topological YM theory. This framework will allow us to understand the local symmetries, the constraints, the extended Hamiltonian and the extended action of the theory.

Stable functions and common stabilizations of Heegaard splittings

We present a new proof of Reidemeister and Singer's Theorem that any two Heegaard splittings of the same 3-manifold have a common stabilization. The proof leads to an upper bound on the minimal genus of a common stabilization in terms of the number of negative slope inflection points and type-two cusps in a Rubinstein-Scharlemann graphic for the two splittings.

Brownian Chen series and Atiyah–Singer theorem

The purpose of this work is to give a new and short proof of the Atiyah–Singer local index theorem for the Dirac operator on the spin bundle. This proof is obtained by using heat semigroups approximations based on the truncation of Brownian Chen series.

Structure groups and holonomy in infinite dimensions

We give a theorem of reduction of the structure group of a principal bundle P with regular structure group G. Then, when G is in the classes of regular Lie groups defined by T. Robart in [Can. J. Math. 49 (4) (1997) 820–839], we define the closed holonomy group of a connection as the minimal closed Lie subgroup of G for which the previous theorem of reduction can be applied. We also prove an infinite dimensional version of the Ambrose–Singer theorem: the Lie algebra of the holonomy group is spanned by the curvature elements.

A generalization of Weyl's theorem on projectively related affine connections

From a physical point of view, the geodesics in a four-dimensional Lorentzian spacetime are really significant only as point sets. In 1921 Weyl proved that two torsion-free covariant derivative operators D M and D M on a manifold M have the same geodesics with possibly different parametrizations if and only if there is a 1-form α on M such that D = D + α ⊗ 1 + 1 ⊗ α , where 1 is the identity (1,1) tensor on M. By a theorem of Ambrose, Palais and Singer [1], torsion-free covariant derivative operators are generated by affine sprays, and vice versa. More generally, any (not necessarily affine) spray induces a number of covariant derivatives in the tangent bundle τ of M, or in the pull-back bundle τ ∗τ . We show that in the context of sprays, similarly to Weyl's relation, a correspondence between the Yano derivatives can be detected.

CHIRAL FERMIONS ON THE LATTICE AND INDEX RELATIONS

Comparing recent lattice results on chiral fermions and old continuum results for the index puzzling questions arise. To clarify this issue we start with a critical reconsideration of the results on finite lattices. We then work out various aspects of the continuum limit. After determining bounds and norm convergences we obtain the limit of the anomaly term. Collecting our results the index relation of the quantized theory gets established. We then compare in detail with the Atiyah–Singer theorem. Finally we analyze conventional continuum approaches.

Real embeddings and the Atiyah–Patodi–Singer index theorem for Dirac operators

We present the details of our embedding proof, which was announced in [DZ1], of the Atiyah-Patodi-Singer index theorem for Dirac operators on manifolds with boundary [APS1]. Introduction. The index theorem of Atiyah, Patodi and Singer [APS1, (4.3)] for Dirac operators on manifolds with boundary has played important roles in various problems in geometry, topology as well as mathematical physics. Not surprisingly then, there are by now quite a number of proofs of this index theorem other than Atiyah, Patodi and Singer’s original proof [APS1]. Among these proofs we mention those of Cheeger [C1, 2] (see also Chou [Ch]), Bismut-Cheeger [BC1] and Melrose [M]. One common point underlying all these proofs (including the original one) is that they can all be viewed, in one way or another, as certain extensions to manifolds with boundary of the heat kernel proof of the local index theorem for Dirac operators on closed manifolds (cf. [BeGV]). That is, one starts with a Mckean-Singer type formula and then studies the small time asymptotics of the corresponding heat kernels. In particular, one makes use of the explicit formulas for the heat kernel of the Laplace operators on the cylinder ([APS1], [M]) and/or cone ([BC1], [C1, 2], [Ch]) (being attached the boundary) for the analysis near the boundary. The η-invariant on the boundary, which was first defined in [APS1], appears naturally during the process. Now recall that Atiyah and Singer [AS] also have a K-theoretic proof of their index theorem for elliptic operators on closed manifolds. In such a proof, one transforms the problem, through direct image constructions in K-theory, to a sphere and then applies the Bott periodicity theorem on the sphere to establish the result. It is thus natural to ask whether the strategy of Atiyah-Singer’s K-theoretic ideas can be used to prove the Atiyah-Patodi-Singer index theorem for manifolds with boundary. The purpose of this paper is to present such a proof, of which an announcement of basic ideas has already appeared in [DZ1]. Briefly speaking, we embed the manifold with boundary under consideration into a ball, instead of a sphere, so that it maps the boundary of the original manifold to the boundary sphere of the ball, and reduce the problem to the ball. Now since any vector bundle on the ball is topologically trivial, one obtains the result immediately. This works even when the original manifold has no boundary, giving a proof of the Atiyah-Singer index theorem for Dirac operators. The Bott periodicity theorem is thus not needed. Observe that in [AS], Atiyah and Singer made heavy use of the techniques of pseudodifferential operators, which is not suitable for treating directly the global elliptic boundary problems. This is the first serious difficulty in extending directly the arguments in [AS] to deal with the Atiyah-Patodi-Singer boundary problems. ∗Received February 25, 2000; accepted for publication March 5, 2000. Partially supported by NSF grant DMS-9022140 when both authors were visiting MSRI in 1994. †Department of Mathematics, University of California, Santa Barbara, California 93106, USA (dai@math.ucsb.edu). Partially supported by NSF and Alfred P. Sloan Foundation. ‡Nankai Institute of Mathematics, Nankai University, Tianjin 300071, People’s Republic of China (weiping@sun.nankai.edu.cn). Partially supported by the CNNSF, EMC and the Qiu Shi Foundation. 1See also the book of Lawson-Michelsohn [LM] for a comprehensive treatment of this approach.

On a Theorem of Deutsch and Singer

We introduce the notion of a starshaped set-valued function, and relate it to convex set-valued functions. We prove a theorem which, in the convex case, implies that single-valuedness is exceptional if the function is not everywhere single-valued. A second result shows a remarkable costantness of images for starshaped set-valued functions. Both theorems were inspired from a result of Deutsch and Singer, which they strengthen.

The Extended Monodromy Group and Liouvillian First Integrals

We present a connection between the differential Galois theory and topological Galois theory. The first one is represented by Singer's theorem about integration of polynomial planar vector fields and the second is Khovanskii's monodromy group of a multivalued analytic first integral of such vector field. We show that the monodromy maps together with certain additional maps, which are analogues of the Stokes operators, generate a solvable group.

Contact degree and the index of Fourier integral operators

An elliptic Fourier integral operator of order 0; associated to a homogeneous canonical dieomorphism, on a compact manifold is Fredholm on L2: The index may be expressed as the sum of a term, which we call the contact degree, associated to the canonical dieomorphism and a term, computable by the Atiyah-Singer theorem, associated to the symbol. The contact degree is shown to be dened for any oriented-contact dieomorphism of a contact manifold and is then reduced to the index of a Dirac operator on the mapping torus, also computable by the theorem of Atiyah and Singer. In this case, of an operator on a xed manifold, these results answer a question of Weinstein in a manner consistent with a more general conjecture of Atiyah.

About Systems of Differential Equations Related to Geodesic Double Differential Forms Mean Values and Harmonic Spaces Part I

In spaces of constant curvature P. Günther has introduced geodesic double differential forms and mean value operators for differential forms. As solutions of differential equations for forms (for instance, Weyl-De Rham equations) the form equations so as the mean values suffice certain ordinary systems of differential equations. We generalize such systems and study properties which are determined by differential geometric structures (parallel translation of double differential forms with respect to geodesic lines, the construction of closed, coclosed and harmonic components of differential forms and double forms and the telescopage theorem of McKean and Singer). As special cases we get the systems known for geodesic double differential forms or mean value operators in real and complex hyperbolic spaces by the application of structural information without any special information about these spaces.

Affine Versions of Singer's Theorem on Locally Homogeneous Spaces

A theorem of Singer says that an infinitesimaly homogeneous Riemannian manifold is locally homogeneous. We propose two result on affine connections similar to the theorem of Singer. As an application we prove a theorem giving a sufficient condition for local homogeneity in case of affine connections on 2-dimensional manifolds.

Algorithms and methods in differential algebra

Founded by J.F. Ritt, Differential Algebra is a true part of Algebra so that constructive and algorithmic problems and methods appear in this field. In this talk, I do not intend to give an exhaustive survey of algorithmic aspects of Differential Algebra but I only propose some examples to give an insight of the state of knowledge in this domain. Some problems are known to have an effective solution, others have an efficient effective solution which is implemented in recent computer algebra systems, and the decidability of some others is still an open question, which does not prevent computations from leading to interesting results. Liouville's theory of integration in finite terms and Risch's theorem are examples of problems that computer algebra systems now deal with very efficiently (implementation work by M Bronstein). In what concerns linear differential equations of arbitrary order, a basis for the vector space of all liouvillian solutions can “in principle” be computed effectively thanks to a theorem of Singer's [17, 22]; the complexity bound is actually awful and a lot of work is done or in progress, especially by M. Singer and F. Ulmer, to give realistic algorithms [20, 21] for third-order linear differential equations. Existence of liouvillian first integrals is a way to make precise the notion of integrability of vector fields. Even in the simplest case of three-dimensional polynomial vector fields, no decision procedure is known for this existence. Nevertheless, explicit computations with computer algebra yield interesting solutions for special examples. In this case, the process of looking for so-called Darboux curves can only be called a method but not an algorithm; for a given degree, this search is a classical algebraic elimination process but no bound is known on the degree of the candidate polynomials. This paper insists on the search of liouvillian first integrals of polynomial vector fields and a new result is given: the generic absence of such liouvillian first integrals for factorisable polynomial vector fields in three variables.

A signature theorem for disk bundles and the eta invariant

We apply the Signature Theorem of Atiyah, Patodi and Singer [2] to disk bundles over compact Riemannian manifolds. We show that the signature of a disk bundle can be expressed as the integral of a characteristic class over the manifold and a limiting eta invariant of the bounding sphere bundle. The characteristic class can be explicitly described and is modelled after the Hirzebruch L-class. The limiting process is essential and we show by a counter example that in the absence of the limit, the eta invariant is not a topological invariant. If the base manifold is Hodge, we obtain an expression for the limiting eta invariant in terms of our characteristic class and the bigraded Betti numbers of the base.

NON-PERTURBAITVE GREEN FUNCTIONS IN QUANTUM GAUGE THEORIES

Non-perturbative Green functions for gauge invariant variables are considered. The Green functions are found to be modified as compared with the usual ones in a definite gauge because of a physical configuration space (PCS) reduction. In the Yang-Mills theory with fermions, this phenomenon follows from the Singer theorem about the absence of a global gauge condition for the fields tending to zero at spatial infinity.