Chern–Simons classes for a superconnection

This chapter describes a general group-theoretical property enjoyed by fundamental groups of compact Kahler manifolds. In general, the condition that an infinitesimal deformation be tangent to an analytic path is equivalent to the existence of a formal power series deformation with the infinitesimal deformation as its leading term. If a k-jet of a deformation exists, the condition that this k-jet of a deformation extend to a -jet of a deformation is given by the vanishing of a certain cohomology class, which depends on the previous k terms of the series. The chapter discusses the Lemma C is proved, the canonical isomorphisms between de Rham cohomology and Eilenberg-MacLane cohomology, Theorem A is reduced to Theorem D. The proofs of Theorems A and C involve the harmonic theory of M with coefficients in a flat vector bundle. The flat vector bundle we shall need is the complexification of the flat vector bundle ad P, denote this flat complex vector bundle by E.

This article is devoted to a study of flat orbifold vector bundles. We construct a bijection between the isomorphic classes of proper flat orbifold vector bundles and the equivalence classes of representations of the orbifold fundamental groups of base orbifolds. We establish a Bismut-Zhang like anomaly formula for the Ray-Singer metric on the determine line of the cohomology of a compact orbifold with coefficients in an orbifold flat vector bundle. We show that the analytic torsion of an acyclic unitary flat orbifold vector bundle is equal to the value at zero of a dynamical zeta function when the underlying orbifold is a compact locally symmetric space of the reductive type, which extends one of the results obtained by the first author for compact locally symmetric manifolds.

A flat complex vector bundle (E,D) on a compact Riemannian manifold (X,g) is stable (resp. polystable) in the sense of Corlette [C] if it has no D -invariant subbundle (resp. if it is the D -invariant direct sum of stable subbundles). It has been shown in [C] that the polystability of (E,D) in this sense is equivalent to the existence of a so-called harmonic metric in E . In this paper we consider flat complex vector bundles on compact Hermitian manifolds (X,g) . We propose new notions of g -(poly-)stability of such bundles, and of g -Einstein metrics in them; these notions coincide with (poly-)stability and harmonicity in the sense of Corlette if g is a Kähler metric, but are different in general. Our main result is that the g -polystability in our sense is equivalent to the existence of a g -Hermitian-Einstein metric. Our notion of a g -Einstein metric in a flat bundle is motivated by a correspondence between flat bundles and Higgs bundles over compact surfaces, analogous to the correspondence in the case of Kähler manifolds [S1], [S2], [S3].

We summarize the global geometric formulation of Einstein-Scalar-Maxwell theories twisted by flat symplectic vector bundle which encodes the duality structure of the theory. We describe the scalar-electromagnetic symmetry group of such models, which consists of flat unbased symplectic automorphisms of the flat symplectic vector bundle lifting those isometries of the scalar manifold which preserve the scalar potential. The Dirac quantization condition for such models involves a local system of integral symplectic spaces, giving rise to a bundle of polarized Abelian varieties equipped with a symplectic flat connection, which is defined over the scalar manifold of the theory. Generalized Einstein-Scalar-Maxwell models arise as the bosonic sector of the effective theory of string/M-theory compactifications to four-dimensions, and they are characterized by having non-trivial solutions of "U-fold" type.

The direct image of a flat fibration with complex fibers

The Partition Function for Topological Field Theories

We present a new proof, as well as a C/Q extension (and also certain C/Z extension), of the Riemann–Roch–Grothendieck theorem of Bismut–Lott for flat vector bundles. The main techniques used are the computations of the adiabatic limits of η-invariants associated to the so-called sub-signature operators. We further show that the Bismut–Lott analytic torsion form can be derived naturally from transgressions of η-forms appearing in the adiabatic limit computations.

We show an equality between the analytic torsion and the absolute value at zero of the Ruelle dynamical zeta function on a closed odd dimensional locally symmetric space twisted by an acyclic flat vector bundle obtained by the restriction of a representation of the underlying Lie group. This generalises author’s previous result for unitarily flat vector bundles, and the results of Bröcker, Müller, and Wotzke on closed hyperbolic manifolds.

In this paper we discuss some geometric aspects concerning almost flat bundles, notion introduced by Connes, Gromov and Moscovici [< i> Conjecture de Novikov et fibrés presque plats< /i> , C. R. Acad. Sci. Paris Sér. I < b> 310< /b> (1990), 273–277]. Using a natural construction of [B. Hanke and T. Schick, < i> Enlargeability and index theory< /i> , preprint, 2004], we present here a simple description of such bundles. For this we modify the notion of almost flat structure on bundles over smooth manifolds and extend this notion to bundles over arbitrary CW-spaces using quasi-connections [N. Teleman, < i> Distance function, Linear quasi-connections and Chern character< /i> , IHES/M/04/27].< /p> < p> Connes, Gromov and Moscovici [< i> Conjecture de Novikov et fibrés presque plats< /i> , C. R. Acad. Sci. Paris Sér. I < b> 310< /b> (1990), 273–277] showed that for any almost flat bundle $\alpha$ over the manifold $M$, the index of the signature operator with values in $\alpha$ is a homotopy equivalence invariant of $M$. From here it follows that a certain integer multiple $n$ of the bundle $\alpha$ comes from the classifying space $B\pi_{1}(M)$. The geometric arguments discussed in this paper allow us to show that the bundle $\alpha$ itself, and not necessarily a certain multiple of it, comes from an arbitrarily large compact subspace $Y\subset B\pi_{1}(M)$ trough the classifying mapping.

We show that non-abelian quantum statistics can be studied using certain topological invariants which are the homology groups of configuration spaces. In particular, we formulate a general framework for describing quantum statistics of particles constrained to move in a topological space X. The framework involves a study of isomorphism classes of flat complex vector bundles over the configuration space of X which can be achieved by determining its homology groups. We apply this methodology for configuration spaces of graphs. As a conclusion, we provide families of graphs which are good candidates for studying simple effective models of anyon dynamics as well as models of non-abelian anyons on networks that are used in quantum computing. These conclusions are based on our solution of the so-called universal presentation problem for homology groups of graph configuration spaces for certain families of graphs.

In this paper we give a global version of the Bryant representation of surfaces of constant mean curvature one (cmc-1) in hyperbolic space. This allows to set the associated non-abelian period problem in the framework of flat unitary vector bundles on Riemann surfaces. We use this machinery to prove the existence of certain cmc-1 surfaces having prescribed global monodromy.

Gerbes and higher gerbes are geometric cocycles representing higher degree cohomology classes, and are attracting considerable interest in differential geometry and mathematical physics. We prove that a 2-gerbe has a torsion Dixmier–Douady class if and only if the gerbe has locally constant cocycle data. As an application, we give an alternative description of flat twisted vector bundles in terms of locally constant transition maps. These results generalize to n-gerbes for n=1 and n≥3, providing insights into the structure of higher gerbes and their applications to the geometry of twisted vector bundles.

We generalise Simpson’s nonabelian Hodge correspondence to the context of projective varieties with Kawamata log terminal (klt) singularities. The proof relies on a descent theorem for numerically flat vector bundles along birational morphisms. In its simplest form, this theorem asserts that given any klt variety $X$ and any resolution of singularities, any vector bundle on the resolution that appears to come from $X$ numerically, does indeed come from $X$ . Furthermore, and of independent interest, a new restriction theorem for semistable Higgs sheaves defined on the smooth locus of a normal, projective variety is established.

We investigate the nature of divergences in quantum field theory, showing that they are organized in the structure of a certain “motivic Galois group” U*, which is uniquely determined and universal with respect to the set of physical theories. The renormalization group can be identified canonically with a one-parameter subgroup of U*. The group U* arises through a Riemann-Hilbert correspondence. Its representations classify equisingular flat vector bundles, where the equisingularity condition is a geometric formulation of the fact that in quantum field theory the counterterms are independent of the choice of a unit of mass. As an algebraic group scheme, U* is a semidirect product by the multiplicative group G_m of a prounipotent group scheme whose Lie algebra is freely generated by one generator in each positive integer degree. There is a universal singular frame in which all divergences disappear. When computed as iterated integrals, its coefficients are certain rational numbers that appear in the local index formula of Connes-Moscovici. When working with formal Laurent series over ℚ, the data of equisingular flat vector bundles define a Tannakian category whose properties are reminiscent of a category of mixed Tate motives.

In a recent joint work with V. Turaev [6], we defined a new concept of combinatorial torsion which we called absolute torsion. Compared with the classical Reidemeister torsion, it has the advantage of having a well-determined sign. Also, the absolute torsion is defined for arbitrary orientable flat vector bundles, and not only for unimodular ones, as is classical Reidemeister torsion. In this paper I show that the sign behavior of the absolute torsion, under a continuous deformation of the flat bundle, is determined by the eta-invariant and the Pontrjagin classes. This result has a twofold significance. Firstly, it justifies the definition of the absolute torsion by establishing a relation to the well-known geometric invariants of manifolds. Viewed differently, the result of this paper allows to express (partially) the eta-invariant, which is defined using analytic tools, in terms of the absolute torsion, having a purely topological definition. The result may find applications in studying the spectral flow by methods of combinatorial topology.