Analytic torsion and holomorphic determinant bundles

In this paper, we derive the main properties of Kähler fibrations. We introduce the associated Levi-Civita superconnection to construct analytic torsion forms for holomorphic direct images. These forms generalize in any degree the analytic torsion of Ray and Singer. In the case of acyclic complexes of holomorphic Hermitian vector bundles, such forms are calculated by means of Bott-Chern classes.Supported by NSF Grant DMS 850248 and by the Sloan Foundation     

Analytic torsion and holomorphic determinant bundles I. Bott-Chern forms and analytic torsion

We attach secondary invariants to any acyclic complex of holomorphic Hermitian vector bundles on a complex manifold. These were first introduced by Bott and Chern [Bot C]. Our new definition uses Quillen's superconnections. We also give an axiomatic characterization of these classes. These results will be used in [BGS2] and [BGS3] to study the determinant of the cohomology of a holomorphic vector bundle.     

Generalized holomorphic bundles and the B-field action

On a generalized complex manifold, there is an associated definition of a generalized holomorphic bundle, introduced by Gualtieri. In the case of an ordinary complex structure, this notion yields an object which we call a co-Higgs bundle, and we consider the B-field action of a closed form of type (1,1)$(1,1)$, both local and global. The effect makes contact with both Nahm’s equations and holomorphic gerbes.     

Equivariant determinant bundles and supersymmetric closed strings

This work is to point out the relationship between the geometric approach to the string theories of Bowick and Rajeev, and Pilch and Warner and the recent research on the index of the Dirac operator in loop space. We consider character-valued Dirac index bundles instead of the vacuum bundle over some parameter space such as Diff S¹/S¹. Thus, some necessary conditions for the absence of the Virasoro anomaly can be derived from some formulas presented here.     

Uniqueness of balanced metrics on holomorphic vector bundles

Let E→M$E\to M$ be a holomorphic vector bundle over a compact Kähler manifold (M,ω)$(M,\omega )$. We prove that if E$E$ admits a ω$\omega$-balanced metric (in X. Wang’s terminology (Wang, 2005 [3])) then it is unique. This result together with Biliotti and Ghigi (2008) [14] implies the existence and uniqueness of ω$\omega$-balanced metrics of certain direct sums of irreducible homogeneous vector bundles over rational homogeneous varieties. We finally apply our result to show the rigidity of ω$\omega$-balanced Kähler maps into Grassmannians.     

Curvature on determinant bundles and first Chern forms

The Quillen–Bismut–Freed construction associates a determinant line bundle with connection to an infinite dimensional super vector bundle with a family of Dirac-type operators. We define the regularized first Chern form of the infinite dimensional bundle, and relate it to the curvature of the Bismut–Freed connection on the determinant bundle. In finite dimensions, these forms agree (up to sign), but in infinite dimensions there is a correction term, which we express in terms of Wodzicki residues.

We illustrate these results with a string theory computation. There is a natural super vector bundle over the manifold of smooth almost complex structures on a Riemannian surface. The Bismut–Freed superconnection is identified with classical Teichmüller theory connections, and its curvature and regularized first Chern form are computed.     

Curvature of determinant bundles for degenerate families

We calculate the (1, 1) curvature of the Beilinson Schechtman connection for the determinant bundle associated to a family of Riemann surfaces with ordinary singularities. As consequences we obtain generalizations of theorems of Bismut and Bost.Supported in part by NSF Grant No DMS-9201022.Supported in part by NSC Grant No 83-0208-M-002-039, Republic of China.     

Hermitian–Einstein metrics on holomorphic vector bundles over Hermitian manifolds

In this paper, we prove the long-time existence of the Hermitian–Einstein flow on a holomorphic vector bundle over a compact Hermitian (non-Kähler) manifold, and solve the Dirichlet problem for the Hermitian–Einstein equations. We also prove the existence of Hermitian–Einstein metrics for holomorphic vector bundles on a class of complete non-compact Hermitian manifolds.     

Two-dimensional topological gravity and intersection theory on the moduli space of holomorphic bundles

We define a two-dimensional topological Yang-Mills theory for an arbitrary compact simple Lie group. This theory is defined in terms of intersection theory on the moduli space of flat connections on a two-dimensional surface and corresponds physically to a two-dimensional reduction and truncation of four-dimensional topological Yang-Mills theory. Two-dimensional topological Yang-Mills theory defines a topological matter system and may be naturally coupled to two-dimensional topological gravity. This topological Yang-Mills theory is also closely related to Chern-Simons gauge theory in 2 + 1 dimensions. We also discuss a relation between SL (2, ) Chern-Simons theory and two-dimensional topological gravity.     

Classical solutions of the chiral model,unitons, and holomorphic vector bundles

This paper deals with classical solutions of theSU(2) chiral model on ², and of a generalized chiral model on ²⁺¹. Such solutions are shown to correspond to certain holomorphic vector bundles over minitwistor space. With an appropriate boundary condition, the solutions (called 1-unitons in [9]) correspond to bundles over a compact 2-dimensional complex manifold, and the problem becomes one of algebraic geometry.     

On the geometry of Dirac determinant bundles in two dimensions

The gauge and diffeomorphism anomalies are used to define the determinant bundles for the left-handed Dirac operator on a two-dimensional Riemann surface. Three different moduli spaces are studied: (1) the space of vector potentials modulo gauge transformations; (2) the space of vector potentials modulo bundle automorphisms; and, (3) the space of Riemannian metrics modulo diffeomorphisms. Using the methods earlier developed for the studies of affine Kac-Moody groups, natural geometries are constructed for each of the three bundles.This work was supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under contract #DE-AC02-76ER03069     

Current algebras ind+1-dimensions and determinant bundles over infinite-dimensional Grassmannians

We extend the methods of Pressley and Segal for constructing cocycle representations of the restricted general linear group in infinite-dimensions to the case of a larger linear group modeled by Schatten classes of rank 1p<. An essential ingredient is the generalization of the determinant line bundle over an infinite-dimensional Grassmannian to the case of an arbitrary Schatten rank,p1. The results are used to obtain highest weight representations of current algebras (with the operator Schwinger terms) ind+1-dimensions when the space dimensiond is any odd number.This work is supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under contract #DE-AC02-76ER03069     

The analysis of elliptic families. I. Metrics and connections on determinant bundles

In this paper, we construct the Quillen metric on the determinant bundle associated with a family of elliptic first order differential operators. We also introduce a unitary connection on and calculate its curvature. Our results will be applied to the case of Dirac operators in a forthcoming paper.     

Real and holomorphic SuSy algebras

We propose a new constructive scheme for real SuSy algebras within the general formalism of real Clifford algebra theory, and use it to characterize a natural holomorphic extension admitted by a subclass of real SuSy algebras in signaturess–t=3 mod 4. The resulting holomorphic SuSy algebras are the abstract models of a new type of supersymmetry algebras, already known to be relevant in the development of canonical or Hamiltonian methods in superspace.Research supported in part by CNPq-Brazil.     

Affine holomorphic quantization

We present a rigorous and functorial quantization scheme for affine field theories, i.e., field theories where local spaces of solutions are affine spaces. The target framework for the quantization is the general boundary formulation, allowing to implement manifest locality without the necessity for metric or causal background structures. The quantization combines the holomorphic version of geometric quantization for state spaces with the Feynman path integral quantization for amplitudes. We also develop an adapted notion of coherent states, discuss vacuum states, and consider observables and their Berezin–Toeplitz quantization. Moreover, we derive a factorization identity for the amplitude in the special case of a linear field theory modified by a source-like term and comment on its use as a generating functional for a generalized S$S$-matrix.