Arithmetic infinite Grassmannians and the induced central extensions

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 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 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 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 possible systems of compatible determinant line bundles and compare the conventions and approaches used elsewhere in the literature.

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 ) .

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.

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.

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 $X$ 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)

The purpose of this paper is to carry out the abelianization program pro posed by Atiyah [1] and Hitchin [9] for the geometric quantization of SU(2) Wess-Zumino-Witten model. Let C be a Riemann surface of genus g. Let Mg be the moduli space of semi-stable rank 2 holomorphic vector bundles on C with trivial determinant. For a positive integer k, let T(Mg, Ck) be the space of holomorphic sections of the fc-th tensor product of the determinant line bundle C on Mg. An element of T(Mg, Ck) is called a rank 2 theta function of level k. The main result of our abelianization is to give an explicit representation of a base of T{Mg, Ck) as well as its transformation formula in terms of classical Riemann theta functions with automorphic form coefficients defined on the Prym variety P associated with a two-fold branched covering surface C of C. r(Mg,Ck) can be identified with the conformai block of level k of the SU(2) WZW model ([5], [15]). The abelianization procedure enables us to de duce the various known results about the conformai block in a uniform way. Firstly, we construct a projectively flat connection on the vector bundle over the Teichm?ller space with fibre T{Mg,Ck). Secondly, making use of our ex plicit representation of rank 2 theta functions we construct a Hermitian product on the vector bundle preserved by the connection. Also our explicit represen tation enables us to prove that Y(Mg,Ck) has the predicted dimension from the Quantum Clebsh-Gordan conditions. A natural connection on the said vector bundle for the SU(iV) WZW model was first constructed by Hitchin [11]. It will turn out that the connection constructed in this paper coincides with the Hitchin connection. Laszlo [16] showed that the Hitchin connection coincides with the con nection constructed by Tsuchiya, Ueno and Yamada [21] through the above identification. On the other hand Kirillov [13], [14] constructed a Hermitian product on the conformai block compatible with the Tsuchiya-Ueno-Yamada connection using the representation theory of affine Lie algebras together with the theory of hermitian modular tensor categories; cf. [22]. Laszlo's result

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