The Determinant Line Bundle for Fredholm Operators: Construction, Properties, and Classification

We investigate functorial properties of two hermitian line bundles over the moduli space of flat SU(n)-connections on a closed oriented surface; that is, of the Chern–Simons line bundle and the determinant line bundle. We investigate actions of cyclic subgroups of the mapping class group on them. As a consequence, we show that if we modify the determinant line bundle by the Hodge bundle over the moduli space of Riemann surfaces, then these line bundles are functorially isomorphic. This implies two quantum Hilbert spaces defined by the Chern–Simons line bundle and the modified determinant line bundle are functorially isomorphic.

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.

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.

This article deals with two topics: the first, which has a general character, is a variation formula for the determinant line bundle in non-Kahlerian geometry. This formula, which is a consequence of the non-Kahlerian version of the Grothendieck–Riemann–Roch theorem proved recently by Bismut [Bi], gives the variation of the determinant line bundle corresponding to a perturbation of a Fourier–Mukai kernel E on a product B×X by a unitary flat line bundle on the fiber X. When this fiber is a complex surface and E is a holomorphic 2-bundle, the result can be interpreted as a Donaldson invariant.

Let M(r) be the moduli space of rank r vector bundles with trivial determinant on a Riemann surface X . This space carries a natural line bundle, the determinant line bundle L . We describe a canonical isomorphism of the space of global sections of L^k with a space known in conformal field theory as the ``space of conformal blocks, which is defined in terms of representations of the Lie algebra sl(r, C((z))).

We investigate the analytic properties of torsion isomorphisms (determinants) of mapping cone triangles of Fredholm complexes. Our main tool is a generalization to Fredholm complexes of the perturbation isomorphisms constructed by R. Carey and J. Pincus for Fredholm operators. A perturbation isomorphism is a canonical isomorphism of determinants of homology groups associated to a finite rank perturbation of Fredholm complexes. The perturbation isomorphisms allow us to establish the invariance properties of the torsion isomorphisms under finite rank perturbations. We then show that the perturbation isomorphisms provide a holomorphic structure on the determinant lines over the space of Fredholm complexes. Finally, we establish that the torsion isomorphisms and the perturbation isomorphisms provide holomorphic sections of certain determinant line bundles.

Let M be a closed oriented smooth manifold, G a compact Lie group consisting of diffeomorphisms of M , P → Z M,P \to Z a principal G-bundle with a connection and D a G-equivariant elliptic operator. Then a locally constant family of elliptic operators and its determinant line bundle over Z are naturally defined by D. Moreover the holonomy of the determinant line bundle is defined by the connection in P. In this note, we give an explicit formula to calculate the holonomy (Theorem 1.4) and give a proof of the Witten holonomy formula (Theorem 1.7) in the special case above.

Let C C be an elliptic curve, w ∈ C w\in C , and let S ⊂ C S\subset C be a finite subset of cardinality at least 3 3 . We prove a Torelli type theorem for the moduli space of rank two parabolic vector bundles with determinant line bundle O C ( w ) \mathcal O_C(w) over ( C , S ) (C,S) which are semistable with respect to a weight vector ( 1 2 , … , 1 2 ) \big (\frac {1}{2}, \dots , \frac {1}{2}\big ) .

We describe a physics derivation of theorems due to Dai and Freed about the Atiyah-Patodi-Singer eta-invariant which is important for anomalies and topological phases of matter. This is done by studying a massive fermion. The key role is played by the wave function of the ground state in the Hilbert space of the fermion in the large mass limit. The ground state takes values in the determinant line bundle and has nontrivial Berry phases which characterize the low energy topological phases.

The construction of families of Sato Grassmannians, their determinant line bundles and the extensions induced by them are given. The base scheme is an arbitrary scheme.

Let $X$ be a non-compact geometrically finite hyperbolic 3-manifold without cusps of rank 1. The deformation space $\mc{H}$ of $X$ can be identified with the Teichmuller space $\mc{T}$ of the conformal boundary of $X$ as the graph of a section in $T^*\mc{T}$. We construct a Hermitian holomorphic line bundle $\mc{L}$ on $\mc{T}$, with curvature equal to a multiple of the Weil-Petersson symplectic form. This bundle has a canonical holomorphic section defined by $e^{\frac{1}{\pi}{\rm Vol}_R(X)+2\pi i\CS(X)}$ where ${\rm Vol}_R(X)$ is the renormalized volume of $X$ and $\CS(X)$ is the Chern-Simons invariant of $X$. This section is parallel on $\mc{H}$ for the Hermitian connection modified by the $(1,0)$ component of the Liouville form on $T^*\mc{T}$. As applications, we deduce that $\mc{H}$ is Lagrangian in $T^*\mc{T}$, and that ${\rm Vol}_R(X)$ is a Kahler potential for the Weil-Petersson metric on $\mc{T}$ and on its quotient by a certain subgroup of the mapping class group. For the Schottky uniformisation, we use a formula of Zograf to construct an explicit isomorphism of holomorphic Hermitian line bundles between $\mc{L}^{-1}$ and the sixth power of the determinant line bundle.

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 .

Super Riemann surfaces are the arena for two-dimensional superconformal field theory. They can be regarded as smooth 212-dimensional supermanifolds equipped with a reduction of their structure group to the group of upper triangular 2 × 2 complex matrices. The integrability conditions for such a reduction turn out to be (most of) the famous torsion constraints of 2d supergravity. The other torsion constraints are merely conditions to fix some of the gauge freedom in this description, or to specify a particular connection on such a manifold, analogous to the Levi-Ci vita connection in Riemannian geometry. Unlike ordinary Riemann surfaces, a super Riemann surface \(\hat{X}\) cannot be regarded as having only one complex dimension; instead it has one commuting and one anticommuting complex coordinate. Nevertheless, in certain important aspects SRS behave as nicely as if they had only one dimension. In particular they possess an analog of the Cauchy-Riemann operator \(\bar{\partial }\) on an ordinary Riemann surface X. The latter is a first-order differential operator with values in a canonical line bundle ω º Ω1X over the surface; this bundle in turn is in a sense a square root of the bundle of volume forms. Similarly super Riemann surfaces have a canonically-defined first-order operator \(\hat{\bar{\partial }}\) taking values in the bundle \(\hat{\omega } \equiv BER{{\Omega }^{1}}\hat{X}\) of half-volume forms. This observation allows us to write a superstring world-sheet action depending only on a “superconformal structure” analogous to a conformal structure on an ordinary 2-surface. Furthermore, the operator \(\hat{\bar{\partial }}\) furnishes a short resolution of the structure sheaf of the super Riemann surface, making possible a Quillen theory of determinant line bundles. The material in this talk, and further discussion, can be found in the following papers: 1. S. Giddings and P. Nelson, “Torsion constraints and super Riemann surfaces,” Phys. Rev. Lett. 59, 2619 (1987). 2. P. Nelson, “Holomorphic coordinates for supermoduli space,” Comm. Math. Phys. 115, 167 (1988). 3. S. Giddings and P. Nelson, “The geometry of super Riemann surfaces,” Comm. Math. Phys. (1988)

We introduce and study the GIT cone of $\overline{M}{0,n}$, which is generated by the pullbacks of the natural ample line bundles on the GIT quotients $(\mathbb P^1)^n/!/SL(2)$. We give an explicit formula for these line bundles and prove a number of basic results about the GIT cone. As one application, we prove unconditionally that the log canonical models of $\overline{M}{0,n}$ with a symmetric boundary divisor coincide with the moduli spaces of weighted curves or with the symmetric GIT quotient, extending the result of Matt Simpson.

We give an explicit formula of the normalized Mumford form which expresses the second tautological line bundle by the Hodge line bundle defined on the moduli space of algebraic curves of any genus. This formula is represented as an infinite product which is a higher genus version of the Ramanujan delta function under the trivialization by normalized abelian differentials and Eichler integrals of their products. Furthermore, this formula gives a universal expression of the normalized Mumford form as a computable power series with integral coefficients by the moduli parameters of algebraic curves. Therefore, one can describe the behavior of this form and hence of the Polyakov string measure around the Deligne-Mumford boundary.