arXiv research

We explain the bundle structures of the {\it Determinant line bundle} and the {\it Quillen determinant line bundle} considered on the connected component of the space of Fredholm operators including the identity operator in an intrinsic way. Then we show that these two are isomorphic and that they are non-trivial line …

Quantization of fermions yields determinant line bundle.

In \cite{BR1}, \cite{BR2}, a parabolic determinant line bundle on a moduli space of stable parabolic bundles was constructed, along with a Hermitian structure on it. The construction of the Hermitian structure was indirect: The parabolic determinant line bundle was identified with the pullback of the determinant line b…

We provide a thorough construction of a system of compatible determinant line bundles over spaces of Fredholm operators, fully verify that this system satisfies a number of important properties, and include explicit formulas for all relevant isomorphisms between these line bundles. We also completely describe all possi…

We compute the curvature of the determinant line bundle on a family of Dirac operators for a noncommutative two torus. Following Quillen's original construction for Riemann surfaces and using zeta regularized determinant of Laplacians, one can endow the determinant line bundle with a natural Hermitian metric. By using …

Let $M$ be an irreducible smooth complex projective variety equipped with an action of a compact Lie group $G$, and let $({\mathcal L},h)$ be a $G$-equivariant holomorphic Hermitian line bundle on $M$. Given a compact connected Riemann surface $X$, we construct a $G$-equivariant holomorphic Hermitian line bundle $(L\,,…

In their study of the representation theory of loop groups, Pressley and Segal introduced a determinant line bundle over an infinite dimensional Grassmann manifold. Mickelsson and Rajeev subsequently generalized the work of Pressley and Segal and in the process introduced for any p >=1 another infinite dimensional Gras…

It is shown that the determinant line bundle associated to a family of Dirac operators over a closed partitioned manifold has a canonical Hermitian metric with compatible connection whose curvature satisfies an additivity formula with contributions from the families of Dirac operators over the two halves. This curvatur…

We study the geometry of determinant line bundles associated to Dirac operators on compact odd dimensional manifolds. Physically, these arise as (local) vacuum line bundles in quantum gauge theory. We give a simplified derivation of the commutator anomaly formula using a construction based on noncyclic trace extensions…

The paper establishes conditions for optimal sampling configurations on complex manifolds.

We show that any compact Kahler manifold with integral Kahler form, parametrizes a natural holomorphic family of Cauchy-Riemann operators on the Riemann sphere such that the Quillen determinant line bundle of this family is isomorphic to a sufficiently high tensor power of the holomorphic line bundle determined by the …

Let $X \rightarrow S$ be a smooth projective surjective morphism, where $X$ and $S$ are integral schemes over complex numbers. Let L_0, L_1, .... L_{n-1}, L_{n} be line bundles over $X$. There is a natural isomorphism of the Deligne pairing $<L_{0},...,L_{n}>$ with the determinant line bundle ${\rm Det}(\otimes_{i=0}^{…

The Seiberg-Witten family of elliptic curves defines a Jacobian rational elliptic surface $\Z$ over $\mathbb{C}\mathrm{P}^1$. We show that for the $\bar{\partial}$-operator along the fiber the logarithm of the regularized determinant $-1/2 \log \det' (\bar\partial^* \bar\partial)$ satisfies the anomaly equation of the …

In this paper we prequantize the moduli space of non-abelian vortices. We explicitly calculate the symplectic form arising from the $L^2$ metric and we construct a prequantum line bundle whose curvature is proportional to this symplectic form. The prequantum line bundle turns out to be Quillen's determinant line bundle…

Let $E$ be a holomorphic vector bundle on a compact Kähler manifold $X$. If we fix a metric $h$ on $E$, we get a Laplace operator $Δ$ acting upon smooth sections of $E$ over $X$. Using the zeta function of $Δ$, one defines its regularized determinant $det'(Δ)$. We conjectured elsewhere that, when $h$ varies, this deter…

We show that there is a canonical construction of a zeta (Bismut-Quillen) connection on the determinant line bundle of a family of APS elliptic boundary problems and that it has curvature equal to the 2-form part of a relative eta form.

Abstract: Generalizes modular forms to family case and finds new anomaly cancellation formulas.

This is a note for the conference proceedings Topological and Geometrical Problems related to Quantum Field Theory. We summarize our joint work with Dai about eta invariants on manifolds with boundary. Then we apply these results to prove the curvature and holonomy formulas for the natural connection on the determinant…

The paper classifies and constructs differential symmetry breaking operators from a line bundle to a vector bundle over real projective spaces.

The purpose of this paper is to compute determinant index bundles of certain families of Real Dirac type operators on Klein surfaces as elements in the corresponding Grothendieck group of Real line bundles in the sense of Atiyah. On a Klein surface these determinant index bundles have a natural holomorphic description …

We introduce logarithmic Picard algebroids, a natural class of Lie algebroids adapted to a simple normal crossings divisor on a smooth projective variety. We show that such algebroids are classified by a subspace of the de Rham cohomology of the divisor complement determined by its mixed Hodge structure. We then solve …

The paper studies the zeta-regularized determinant of a pseudo-Laplacian on a cuspidal end with flat unitary line bundle.

Given a holomorphic Hilbertian bundle on a compact complex manifold, we introduce the notion of holomorphic $L^2$ torsion, which lies in the determinant line of the twisted $L^2$ Dolbeault cohomology and represents a volume element there. Here we utilise the theory of determinant lines of Hilbertian modules over finite…

For a symplectic manifold with quantizing line bundle, a choice of almost complex structure determines a Laplacian acting on tensor powers of the bundle. For high tensor powers Guillemin-Uribe showed that there is a well-defined cluster of low-lying eigenvalues, whose distribution is described by a spectral density fun…

Let $X\rightarrow S$ be a smooth projective surjective morphism of relative dimension $n$, where $X$ and $S$ are integral schemes over $\mathbb C$. Let $L\rightarrow X$ be a relatively very ample line bundle. For every sufficiently large positive integer $m$, there is a canonical isomorphism of the Deligne pairing $\la…

In this note we specialize and illustrate the ideas developed in the paper math.DG/0201112 of the first author ("Index theory, eta forms, and Deligne cohomology ") in the case of the determinant line bundle. We discuss the surgery formula in the adiabatic limit using the adiabatic decomposition formula of the zeta regu…

We prove that the $L^2$ metric on the direct image of an adjoint positive line bundle by a locally trivial submersion between projective manifolds is Nakano positive, under the assumption that the typical fiber has zero first Betti number. As a consequence, we get that the symmetric powers of an ample vector bundle ten…

Geometric Invariant Theory applied to Kähler manifolds yields analytic models for vector bundles.

Generalizes Tulczyjew triples for contact manifolds in Hamiltonian and Lagrangian formalisms.

Recently, for a family of ungraded Dirac operators over some space $B$ J. Lott constructed an index gerbe. In the present paper we show (in analogy to the holonomy formula for the determinant bundle in the graded case) that the holonomy of the index gerbe (a priori a hermitean line bundle with connection over the free …

New system solves curvature for ample vector bundles, proving Griffiths conjecture.

We find the entropy's infinite-size behavior in complex manifold sections.

We compute the log canonical thresholds of non-negatively curved singular hermitian metrics on ample linearized line bundles on bi-equivariant group compactifications of complex reductive groups. To this end, we associate to any such metric a convex function whose asymptotic behavior determines the log canonical thresh…

We compute the relative zeta-function metric on the determinant line bundle for a family of elliptic boundary value problems of Dirac-type. To do this we prove a general formula relating the zeta-determinant to a Fredholm determinant over the boundary for a class of higher-order elliptic boundary value problems.

Study on conical Laplacian operators on Riemann surfaces, focusing on determinants and moduli spaces.

This is a slightly expanded version of the talk given by Ch.O. at the conference "Instantons in complex geometry", at the Steklov Institute in Moscow. The purpose of this talk was to explain the algebraic results of our paper "Abelian Yang-Mills theory on Real tori and Theta divisors of Klein surfaces". In this paper w…

We study eta-invariants on odd dimensional manifolds with boundary. The dependence on boundary conditions is best summarized by viewing the (exponentiated) eta-invariant as an element of the (inverse) determinant line of the boundary. We prove a gluing law and a variation formula for this invariant. This yields a new, …

This article deals with two topics: the first, which has a general character, is a variation formula for the the determinant line bundle in non-Kählerian geometry. This formula, which is a consequence of the non-Kählerian version of the Grothendieck-Riemann Roch theorem proved recently by Bismut, gives the variation of…

In this paper we first show that on projective manifolds (M, ω), there are holomorphic determinant bundles (in the sense of Knusden-Mumford used by Bismut, Gillet, Soule) which play the role of the geometric quantum bundle, namely one for each input data of a Hermitian holomorphic line bundle L of non-trivial Chern cla…

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…

Using properties of the determinant line bundle for a family of elliptic boundary value problems, we explain how the Fock space functor defines an axiomatic quantum field theory which formally models the Fermionic path integral. The 'sewing axiom' of the theory arises as an algebraic pasting law for the determinant of …

Study automorphism equivariant Hitchin index for Riemann surfaces.

Let L be a line bundle over a compact complex manifold X and endow L and TX with Hermitian metrics. Our main result provides a formula for the average distribution of the exponentially small eigenvalues of the corresponding Dolbeault Laplacians associated to high tensor powers of L; which in physics terminology is a me…

In this paper, we extend Deligne's functorial Riemann-Roch isomorphism for hermitian holomorphic line bundles on Riemann surfaces to the case of flat, not necessarily unitary connections. The Quillen metric and star-product of Gillet-Soule are replaced with complex valued logarithms. On the determinant of cohomology si…

In this note we quantize the usual symplectic (Kähler) form on the vortex moduli space by modifying the Quillen metric of the Quillen determinant line bundle.

We construct a canonical element, called the refined analytic torsion, of the determinant line of the cohomology of a closed oriented odd-dimensional manifold M with coefficients in a flat complex vector bundle E. We compute the Ray-Singer norm of the refined analytic torsion. In particular, if there exists a flat Herm…

In this paper, we suggest a construction of determinant lines of finitely generated Hilbertian modules over finite von Neumann algebras. Nonzero elements of the determinant lines can be viewed as volume forms on the Hilbertian modules. Using this, we study both $L^2$ combinatorial and $L^2$ analytic torsion invariants …

Torsors over moduli spaces of vector bundles with fixed determinant.