Determinant Line Bundle Research Articles

Local analog of the Deligne-Riemann-Roch isomorphism for line bundles in relative dimension 1

We prove a local analog of the Deligne-Riemann-Roch isomorphism in the case of line bundles and relative dimension $1$. This local analog consists in computation of the class of $12$th power of the determinant central extension of a group ind-scheme $\mathcal G$ by the multiplicative group scheme over $\mathbb Q$ via the product of $2$-cocyles in the second cohomology group. These $2$-cocycles are the compositions of the Contou-Carrère symbol with the $\cup$-product of $1$-cocycles. The group ind-scheme $\mathcal{G}$ represents the functor which assigns to every commutative ring $A$ the group that is the semidirect product of the group $A((t))^*$ of invertible elements of $A((t))$ and the group of continuous $A$-automorphisms of $A$-algebra $A((t))$. The determinant central extension naturally acts on the determinant line bundle on the moduli stack of geometric data (proper quintets). A proper quintet is a collection of a proper family of curves over $\operatorname{Spec} A$, a line bundle on this family, a section of this family, a relative formal parameter at the section, a formal trivialization of the bundle at the section that satisfy further conditions.

Convergence in Conformal Field Theory

Convergence and analytic extension are of fundamental importance in the mathematical construction and study of conformal field theory. The author reviews some main convergence results, conjectures and problems in the construction and study of conformal field theories using the representation theory of vertex operator algebras. He also reviews the related analytic extension results, conjectures and problems. He discusses the convergence and analytic extensions of products of intertwining operators (chiral conformal fields) and of q-traces and pseudo-q-traces of products of intertwining operators. He also discusses the convergence results related to the sewing operation and the determinant line bundle and a higher-genus convergence result. He then explains conjectures and problems on the convergence and analytic extensions in orbifold conformal field theory and in the cohomology theory of vertex operator algebras.

Renormalization of determinant lines in quantum field theory

On a compact manifold $M$, we consider the affine space $A$ of non self-adjoint perturbations of some invertible elliptic operator acting on sections of some Hermitian bundle, by some differential operator of lower order. We construct and classify all complex analytic functions on the Fr\'echet space $A$ vanishing exactly over non invertible elements, having minimal order and which are obtained by local renormalizations, a concept coming from quantum field theory, called renormalized determinants. The additive group of local polynomial functionals of finite degrees acts freely and transitively on the space of renormalized determinants. We provide different representations of the renormalized determinants in terms of spectral zeta determinants, Gaussian Free Fields, infinite product and renormalized Feynman amplitudes in perturbation theory in position space \`a la Epstein-Glaser. Specializing to the case of Dirac operators coupled to vector potentials and reformulating our results in terms of determinant line bundles, we prove our renormalized determinants define some complex analytic trivializations of some holomorphic line bundle over $A$ relating our results to a conjectural picture from some unpublished notes by Quillen [52] from April 1989.

Extension of the curvature form of the relative canonical line bundle on families of Calabi–Yau manifolds and applications

Given a proper, open, holomorphic map of Kahler manifolds, whose general fibers are Calabi-Yau manifolds, the volume forms for the Ricci-flat metrics induce a hermitian metric on the relative canonical bundle over the regular locus of the family. We show that the curvature form extends as a closed positive current. Consequently the Weil-Petersson metric extends as a positive current. In the projective case, the Weil-Petersson form is known to be the curvature of a certain determinant line bundle, equipped with a Quillen metric. As an application we get that after blowing up the singular locus, the determinant line bundle extends, and the Quillen metric extends as singular hermitian metric, whose curvature is a positive current.

An analytic application of Geometric Invariant Theory

Given a compact Kähler manifold, Geometric Invariant Theory is applied to construct analytic GIT-quotients that are local models for a classifying space of (poly)stable holomorphic vector bundles containing the coarse moduli space of stable bundles as an open subspace. For local models invariant generalized Weil-Petersson forms exist on the parameter spaces, which are restrictions of symplectic forms on smooth ambient spaces. If the underlying Kähler manifold is of Hodge type, then the Weil-Petersson form on the moduli space of stable vector bundles is known to be the Chern form of a certain determinant line bundle equipped with a Quillen metric. It gives rise to a holomorphic line bundle on the classifying GIT space together with a continuous hermitian metric.

The Weil–Petersson current on Douady spaces

AbstractThe Douady space of compact subvarieties of a Kähler manifold is equipped with the Weil–Petersson current, which is everywhere positive with local continuous potentials, and of class when restricted to the locus of smooth fibers. There a Quillen metric is known to exist, whose Chern form is equal to the Weil–Petersson form. In the algebraic case, we show that the Quillen metric can be extended to the determinant line bundle as a singular hermitian metric. On the other hand the determinant line bundle can be extended in such a way that the Quillen metric yields a singular hermitian metric whose Chern form is equal to the Weil–Petersson current. We show a general theorem comparing holomorphic line bundles equipped with singular hermitian metrics which are isomorphic over the complement of a snc divisor B. They differ by a line bundle arising from the divisor and a flat line bundle. The Chern forms differ by a current of integration with support in B and a further current related to its normal bundle. The latter current is equal to zero in the case of Douady spaces due to a theorem of Yoshikawa on Quillen metrics for singular families over curves.

On spin structures and orientations for gauge-theoretic moduli spaces

Let X be a compact manifold, G a Lie group, P→X a principal G-bundle, and BP the infinite-dimensional moduli space of connections on P modulo gauge, as a topological stack. For a real elliptic operator E• we previously studied orientations on the real determinant line bundle over BP, twisting E• by connections ∇Ad(P). These are used to construct orientations in the usual sense on smooth gauge theory moduli spaces, and have been extensively studied since the work of Donaldson [15–17].Here we consider complex elliptic operators F• and introduce the idea of spin structures, square roots of the complex determinant line bundle of F• twisted over BP. These may be used to construct spin structures in the usual sense on smooth complex gauge theory moduli spaces. We study the existence and classification of such spin structures.Our main result identifies spin structures on X with orientations on X×S1. Thus, if P→X and Q→X×S1 are principal G-bundles with Q|X×{1}≅P, we relate spin structures on (BP,F•) to orientations on (BQ,E•) for a certain class of operators F• on X and E• on X×S1.Combined with [25], we obtain canonical spin structures for positive Diracians on spin 6-manifolds and gauge groups G=U(m),SU(m). In a sequel [26] we will apply this to define canonical orientation data for all Calabi–Yau 3-folds X over C, as in Kontsevich and Soibelman [28, §5.2], solving a long-standing problem in Donaldson–Thomas theory.

Meromorphic connections, determinant line bundles and the Tyurin parametrization

We develop a holomorphic equivalence between on one hand the space of pairs (stable bundle, flat connection on the bundle) and the of holomorphic (the sheaf of splittings of the one-jet sequence) for the determinant (Quillen) line bundle over the moduli space of vector bundles on a compact connected Riemann surface. This equivalence is shown to be holomorphically symplectic. The equivalences, both holomorphic and symplectic, seem to be quite general, in that they extend to other general of holomorphic bundles and holomorphic connections, in particular those arising from families of stable bundles over the surface. These generalize the Tyurin parametrization of stable vector bundles $E$ over a compact connected Riemann surface, and one can build above them spaces of (equivalence classes of) connections, which are again symplectic. These spaces are also symplectically biholomorphically equivalent to the sheaf of connections for the determinant bundle over the Tyurin family. The last portion of the paper shows how this extends to moduli of framed bundles.

Beilinson–Drinfeld Schubert varieties and global Demazure modules

AbstractWe compute the spaces of sections of powers of the determinant line bundle on the spherical Schubert subvarieties of the Beilinson–Drinfeld affine Grassmannians. The answer is given in terms of global Demazure modules over the current Lie algebra.

Refined Verlinde formulas for Hilbert schemes of points and moduli spaces of sheaves on K3 surfaces

We compute generating functions for elliptic genera with values in line bundles on Hilbert schemes of points on surfaces. As an application we also compute generating functions for elliptic genera with values in determinant line bundles on moduli spaces of sheaves on K3 surfaces.

Analytic torsion for surfaces with cusps II. Regularity, asymptotics and curvature theorem

We study the Quillen metric on the determinant line bundle associated with a family of complex singular curves with hyperbolic cusp singularities.More precisely, we fix a family of complex curves, which admit at most double-point singularities. We endow the fibers of this family with Kähler metrics, which are defined away from a finite set of points, a divisor on the total space of the family, in the vicinity of which the metric has Poincaré-type singularities. We fix a holomorphic vector bundle over the total space of the family and endow it with a Hermitian metric with at most logarithmic singularities on the divisor, coming from the power of the relative canonical line bundle twisted by the divisor. An important example is the family of stable surfaces with marked points endowed with constant scalar curvature metric.The associated determinant line bundle is naturally endowed with the Quillen metric defined using the analytic torsion from the first paper of this series, generalizing the classical definition of Ray-Singer. We study the regularity of this Quillen metric near the locus of singular curves. The singularities turn out to be reasonable enough, so that the curvature of the Chern connection of the determinant line bundle endowed with the Quillen norm is well-defined as a current. We derive the explicit formula for this current, which gives a refinement of Riemann-Roch-Grothendieck theorem at the level of currents. This generalizes the curvature theorems of Takhtajan-Zograf and Bismut-Bost.

Hermitian-Yang-Mills approach to the conjecture of Griffiths on the positivity of ample vector bundles

Given a vector bundle of arbitrary rank with ample determinant line bundle on a projective manifold, we propose a new elliptic system of differential equations of Hermitian-Yang-Mills type for the curvature tensor. The system is designed so that solutions provide Hermitian metrics with positive curvature in the sense of Griffiths — and even in the dual Nakano sense. As a consequence, if an existence result could be obtained for every ample vector bundle, the Griffiths conjecture on the equivalence between ampleness and positivity of vector bundles would be settled. Bibliography: 15 titles.

The Batalin–Vilkovisky formalism and the determinant line bundle

Given a smooth family of massless free fermions parametrized by a base manifold B, we show that the (mathematically rigorous) Batalin–Vilkovisky quantization of the observables of this family gives rise to the determinant line bundle for the corresponding family of Dirac operators.

Notes on diagonals of the product and symmetric variety of a surface

Let X be a smooth quasi-projective algebraic surface and let Δn be the big diagonal in the product variety Xn. We study cohomological properties of the ideal sheaves IΔnk and their invariants (IΔnk)Sn by the symmetric group, seen as ideal sheaves over the symmetric variety SnX. In particular we obtain resolutions of the sheaves of invariants (IΔn)Sn for n=3,4 in terms of invariants of sheaves over Xn whose cohomology is easy to calculate. Moreover, we relate, via the Bridgeland-King-Reid equivalence, powers of determinant line bundles over the Hilbert scheme to powers of ideals of the big diagonal Δn. We deduce applications to the cohomology of double powers of determinant line bundles over the Hilbert scheme with 3 and 4 points and we give universal formulas for their Euler-Poincaré characteristic. Finally, we obtain upper bounds for the regularity of the sheaves IΔnk over Xn with respect to very ample line bundles of the form L⊠⋯⊠L and of their sheaves of invariants (IΔnk)Sn on the symmetric variety SnX with respect to very ample line bundles of the form DL.

Verlinde-type formulas for rational surfaces

K -theoretic Donaldson invariants are holomorphic Euler characteristics of determinant line bundles on moduli spaces of rank 2 sheaves on surfaces. We develop an algorithm which determines the generating functions of the K -theoretic Donaldson invariants on the projective plane and more general rational surfaces, and apply it in many cases to get explicit formulas. We relate the results to Le Potier’s strange duality conjecture.

The infinitesimal multiplicities and orientations of the blow-up set of the Seiberg–Witten equation with multiple spinors

I construct multiplicities and orientations of tangent cones to any blow-up set Z for the Seiberg–Witten equation with multiple spinors. This is used to prove that Z determines a homology class, which is shown to be equal to the Poincaré dual of the first Chern class of the determinant line bundle. I also obtain a lower bound for the 1-dimensional Hausdorff measure of Z.

Generating functions for 𝐾-theoretic Donaldson invariants and Le Potier’s strange duality

For a projective algebraic surface X X with an ample line bundle H H , let M H X ( c ) M_H^X(c) be the moduli space H H -semistable sheaves E \mathcal {E} of class c c in the Grothendieck group K ( X ) K(X) . We write c = ( r , c 1 , c 2 ) c=(r,c_1,c_2) or c = ( r , c 1 , χ ) c=(r,c_1,\chi ) with r r the rank, c 1 , c 2 c_1,c_2 the Chern classes, and χ \chi the holomorphic Euler characteristic. We also write M H X ( 2 , c 1 , c 2 ) = M X X ( c 1 , d ) M_H^X(2,c_1,c_2)=M_X^X(c_1,d) , with d = 4 c 2 − c 1 2 d=4c_2-c_1^2 . The K K -theoretic Donaldson invariants are the holomorphic Euler characteristics χ ( M H X ( c 1 , d ) , μ ( L ) ) \chi (M_H^X(c_1,d),\mu (L)) , where μ ( L ) \mu (L) is the determinant line bundle associated to a line bundle on X X . More generally for suitable classes c ∗ ∈ K ( X ) c^*\in K(X) there is a determinant line bundle D c , c ∗ \mathcal {D}_{c,c^*} on M H X ( c ) M^X_H(c) . We first compute some generating functions for K K -theoretic Donaldson invariants on P 2 \mathbb {P}^2 and rational ruled surfaces, using the wallcrossing formula of [Pure Appl. Math. Q. 5 (2009), pp. 1029–1111]. Then we show that Le Potier’s strange duality conjecture relating H 0 ( M H X ( c ) , D c , c ∗ ) H^0(M^X_H(c),\mathcal {D}_{c,c^*}) and H 0 ( M H X ( c ∗ ) , D c ∗ , c ) H^0(M^X_H(c^*),\mathcal {D}_{c^*,c}) holds for the cases c = ( 2 , c 1 = 0 , c 2 > 2 ) c=(2,c_1=0,c_2>2) and c ∗ = ( 0 , L , χ = 0 ) c^{*}=(0,L,\chi =0) with L = − K X L=-K_X on P 2 \mathbb {P}^2 , and L = − K X L=-K_X or − K X + F -K_X+F on P 1 × P 1 \mathbb {P}^1\times \mathbb {P}^1 and P 2 ^ \widehat {\mathbb {P}^2} with F F the fibre class of the ruling, and also the case c = ( 2 , H , c 2 ) c=(2,H,c_2) and c ∗ = ( 0 , 2 H , χ = − 1 ) c^*=(0,2H,\chi =-1) on P 2 \mathbb {P}^2 .

On the geometrical interpretation of locality in anomaly cancellation

A notion of local section of the determinant line bundle is defined giving necessary and sufficient conditions for anomaly cancellation compatible with locality. This definition gives an intrinsic geometrical interpretation of the local counterterms allowed in the renormalization program of quantum field theory. For global anomalies the conditions for anomaly cancellation are expressed in terms of the equivariant holonomy of the Bismut–Freed connection.

Curvature of the Determinant Line Bundle for the Noncommutative Two Torus

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 an analogue of Kontsevich-Vishik canonical trace, defined on Connes' algebra of classical pseudodifferential symbols for the noncommutative two torus, we compute the curvature form of the determinant line bundle by computing the second variation $\delta_{w}\delta_{\bar{w}}\log\det(\Delta)$.

Quantum Hall Effect and Quillen Metric

We study the generating functional, the adiabatic curvature and the adiabatic phase for the integer quantum Hall effect (QHE) on a compact Riemann surface. For the generating functional we derive its asymptotic expansion for the large flux of the magnetic field, i.e., for the large degree $k$ of the positive Hermitian line bundle $L^k$. The expansion consists of the anomalous and exact terms. The anomalous terms are the leading terms of the expansion. This part is responsible for the quantization of the adiabatic transport coefficients in QHE. We then identify the non-local (anomalous) part of the expansion with the Quillen metric on the determinant line bundle, and the subleading exact part with the asymptotics of the regularized spectral determinant of the Laplacian for the line bundle $L^k$, at large $k$. Finally, we show how the generating functional of the integer QHE is related to the gauge and gravitational (2+1)d Chern-Simons functionals. We observe the relation between the Bismut-Gillet-Soul\'e curvature formula for the Quillen metric and the adiabatic curvature for the electromagnetic and geometric adiabatic transport of the integer Quantum Hall state. Then we relate the adiabatic phase in QHE to the eta invariant and show that the geometric part of the adiabatic phase is given by the Chern-Simons functional.