The Partition Function for Topological Field Theories

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.

This thesis is devoted to the study of a new construction of two-dimensional topological conformal field theories by gauge fixing two-dimensional topological gauge theories. We study in detail the Lorenz-gauged abelian and non-abelian BF theory, which are topological conformal classically and on the quantum level. We find that the abelian model corresponds to Witten’s B-model with a parity shifted flat target space. It is therefore obtained by twisting a N = (2,2) supersymmetric model, while no such twist exists in the non-abelian case. Furthermore, we study an analogue of Gromov-Witten periods in the abelian model. Finally, we show that the non-abelian model allows non-trivial Jordan blocks of the Hamiltonian and thus defines a logarithmic conformal field theory. The existence of infinite-dimensional Jordan blocks allows an explicit construction of primary fields, whose conformal weights are subject to quantum corrections.

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

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.

Chern–Simons classes for a superconnection

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.

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.

The leading terms in the tree-level effective action for the massless fields of the bosonic open string are calculated by integrating out all massive fields in Witten's cubic string field theory. In both the abelian and nonabelian theories, field redefinitions make it possible to express the effective action in terms of the conventional field strength. The resulting actions reproduce the leading terms in the abelian and nonabelian Born-Infeld theories, and include (covariant) derivative corrections.

We present a method to quantize SU([ital N]) gauge theories which does not use gauge fixing, and therefore circumvents the problematics around gauge fixing and Gribov copying. The method is illustrated in the Abelian theory where no Gribov ambiguity is present, and it is shown to be equivalent to Landau gauge fixing in this case. It is shown that there are fundamental differences between Abelian and non-Abelian theories. These differences imply that in Abelian theories the gauge particle must be massless, but this is not necessarily true in non-Abelian theories.

The structure of leading nonperturbative corrections to the static Coulomb potential at small distances in Abelian and non-Abelian theories is analyzed. Related problems of validity of the Dirac quantization condition for running charges in Abelian theory and significance of the quantity 〈A2〉 are briefly discussed.