A Map Between Moduli Spaces of Connections

Let M be a smooth complex projective toric variety equipped with an action of a torus T , such that the complement D of the open T-orbit in M is a simple normal crossing divisor.Let G be a complex reductive affine algebraic group.We prove that an algebraic principal G-bundle E G M admits a T-equivariant structure if and only if E G admits a logarithmic connection singular over D. If E H M is a T-equivariant algebraic principal H-bundle, where H is any complex affine algebraic group, then E H in fact has a canonical integrable logarithmic connection singular over D.

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 prove a conjecture about the moduli space S U C ( 3 ) \mathcal {SU}_C(3) of semi-stable rank 3 vector bundles with trivial determinant on a genus 2 curve C C , due to I. Dolgachev. Given C C a smooth projective curve of genus 2, and the embedding of the Jacobian J C JC into | 3 Θ | |3\Theta | , A. Coble proved, at the beginning of the 20th century, that there exists a unique cubic hypersurface C \mathcal {C} in | 3 Θ | ∗ ≃ P 8 |3\Theta |^* \simeq \mathbb {P}^8 , J C [ 3 ] JC[3] -invariant and singular along J C JC . On the other hand, we have a map of degree 2 from S U C ( 3 ) \mathcal {SU}_C(3) over | 3 Θ | ≃ P 8 ∗ |3\Theta | \simeq \mathbb {P}^{8*} , ramified along a sextic hypersurface B \mathcal {B} . Dolgachev’s conjecture affirms that the sextic B \mathcal {B} is the dual variety of Coble’s cubic C \mathcal {C} .

A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.

The logarithmic connections studied in the paper are direct images of regular connections on line bundles over genus-2 double covers of the elliptic curve. We give an explicit parametrization of all such connections, determine their monodromy, differential Galois group and the underlying rank-2 vector bundle. The latter is described in terms of elementary transforms. The question of its (semi)-stability is addressed.

We are interested in the stability of holomorphic rank 2 vector bundles of degree 0 over compact Riemann surfaces, which are provided with irreducible meromophic tracefree connections. In the case of a logarithmic connection on the Riemann sphere, such a vector bundle will be trivial up to the isomonodromic deformation associated to a small move of the poles, according to a result of A. Bolibruch. In the general case of meromorphic connections over Riemann surfaces of arbitrary genus, we prove that the vector bundle will be semi-stable, up to a small isomonodromic deformation. More precisely, the vector bundle underlying the universal isomonodromic deformation is generically semi-stable along the deformation, and even maximally stable. For curves of genus g ≥ 2, this result is non-trivial even in the case of non-singular connections.

Let [Formula: see text] be a smooth complex projective curve of genus [Formula: see text] and let [Formula: see text] be a line bundle on [Formula: see text] with [Formula: see text]. Let [Formula: see text] be the moduli space of semistable rank 2 [Formula: see text]-twisted Higgs bundles with trivial determinant on [Formula: see text]. Let [Formula: see text] be the moduli space of stable rank 2 [Formula: see text]-twisted Higgs bundles with determinant [Formula: see text] for some [Formula: see text] on [Formula: see text]. We construct a cycle in the product of a stack of rational maps from nonsingular curves to [Formula: see text] and [Formula: see text] by using Hecke modifications of a stable [Formula: see text]-twisted Higgs bundle in [Formula: see text].

We present normal forms for elliptic curves over a field of characteristic 2 analogous to Edwards normal form, and determine bases of addition laws, which provide strikingly simple expressions for the group law. We deduce efficient algorithms for point addition and scalar multiplication on these forms. The resulting algorithms apply to any elliptic curve over a field of characteristic 2 with a 4-torsion point, via an isomorphism with one of the normal forms. We deduce algorithms for duplication in time 2M + 5S + 2m c and for addition of points in time 7M + 2S, where M is the cost of multiplication, S the cost of squaring, and m c the cost of multiplication by a constant. By a study of the Kummer curves \(\mathcal{K}\) = E/{[±1]}, we develop an algorithm for scalar multiplication with point recovery which computes the multiple of a point P with 4M + 4S + 2m c + m t per bit where m t is multiplication by a constant that depends on P.

Two-dimensional Riemann problem for rigid representations on an elliptic curve

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

A generalized bc-system associated to a Hermitian vector bundle over a Riemann surface is introduced in close analogy to the usual rank one case. Some of the geometric analogies to the well-known case are studied. In particular, if there are no zero-modes, the “non-Abelian” theta divisor appears. In the general case where zero-modes exist, it seems to be more difficult to find a natural description. It is discussed in detail why no such system exists in low genus that is on the Riemann sphere and on elliptic curves.

We give the first examples of smooth Fano and Calabi–Yau varieties violating the (narrow) canonical strip hypothesis, which concerns the location of the roots of Hilbert polynomials of polarized varieties. They are given by moduli spaces of rank 2 bundles with fixed odd-degree determinant on curves of sufficiently high genus, hence our Fano examples have Picard rank 1, index 2, are rational, and have moduli. The hypotheses also fail for several other closely related varieties.

This chapter considers the moduli space of rank 2 Higgs bundles with fixed determinant over a smooth projective curve X of genus 2 over ℂ, and studies involutions defined by tensoring the vector bundle with an element α of order 2 in the Jacobian of the curve, combined with multiplication of the Higgs field by ±1. It describes the fixed points of these involutions in terms of the Prym variety of the covering of X defined by α, and gives an interpretation in terms of the moduli space of representations of the fundamental group.

We consider varieties over an algebraically closed field k of characteristicp>0. Given a linear representation of a reductive group, we prove that the ring of invariants is F-regular provided the associated projective quotient is Frobenius-split, the twisting sheaves are Cohen-Macaulay (C-M), and a mild technical condition is met. As an example of how this can be used, we show that the ring of invariants (under the adjoint action of SL (3)) ofg copies ofM3 is C-M. (HereM3 denotes the vector space of 3×3 matrices over k andp>3.) The method of proof involves an induction, and is potentially of wide applicability. As a corollary we obtain that the moduli space of rank 3 and degree 0 bundles on a smooth projective curve of genusg is C-M.

We consider g=N−1 coinciding M-5-branes on top of each other, in the multiple KK monopole background Q. The worldvolume of an M-5-brane is the local product of the four-dimensional spacetime R 1,3 and an elliptic curve. Taken together, these genus-one Riemann surfaces are supposed to give a single (Seiberg-Witten) hyperelliptic curve Σ g in the coincidence limit, where the gauge symmetry is known to be enhanced to SU(N). We make this gauge symmetry enhancement manifest by analyzing the corresponding hypermultiplet LEEA which is given by the N=2 non-linear sigma-model having Q as the target space. The hyper-Kahler manifold Q is known to be given by the multicentre Taub-NUT space, which in the coincidence limit amounts to the multiple Eguchi-Hanson (ALE) space Q mEH. The latter is most naturally described by using the hyper-Kahler coset construction in harmonic superspace, in terms of the auxiliary N=2 vector superfields as Lagrange multipliers in the presence of FI terms. The Maldacena limit, in which the LEEA is given by the N=4 SYM with the gauge group SU(N), corresponds to sending all the FI terms to zero at large N, when all the auxiliary N=2 vector superfields become dynamical.