Riemann Surface Of Positive Genus Research Articles

An integrable connection on the configuration space of a Riemann surface of positive genus

Let X be a Riemann surface of positive genus. Denote by X(n) the configuration space of n distinct points on X. We use the Betti–de Rham comparison isomorphism on H1(X(n)) to define an integrable connection on the trivial vector bundle on X(n) with fiber the universal algebra of the Lie algebra associated with the descending central series of π1 of X(n). The construction is inspired by the Knizhnik–Zamolodchikov system in genus zero and its integrability follows from Riemann period relations.

The prequantum line bundle on the moduli space of flat SU(N) connections on a Riemann surface and the homotopy of the large N limit

We show that the prequantum line bundle on the moduli space of flat $SU(2)$ connections on a closed Riemann surface of positive genus has degree 1. It then follows from work of Lawton and the second author that the classifying map for this line bundle induces a homotopy equivalence between the stable moduli space of flat $SU(N)$ connections, in the limit as $N$ tends to infinity, and $\mathbb{C}P^\infty$. Applications to the stable moduli space of flat unitary connections are also discussed.

The quiver at the bottom of the twisted nilpotent cone on $$\mathbb P^1$$ P 1

For the moduli space of Higgs bundles on a Riemann surface of positive genus, critical points of the natural Morse-Bott function lie along the nilpotent cone of the Hitchin fibration and are representations of $\mbox{A}$-type quivers in a twisted category of holomorphic bundles. The critical points that globally minimize the function are representations of $\mbox{A}_1$. For twisted Higgs bundles on the projective line, the quiver describing the bottom of the cone is more complicated. We determine it here. We show that the moduli space is topologically connected whenever the rank and degree are coprime, thereby verifying conjectural lowest Betti numbers coming from high-energy physics.

Holomorphic mappings and basic extremal lengths of once-holed tori

We are concerned with the set of once-holed tori from which there are handle-preserving holomorphic mappings into a given Riemann surface of positive genus. We investigate how the extremal lengths of homology classes of simple closed curves change under those holomorphic mappings of once-holed tori.

The supermembrane with central charges: [formula omitted] NCSYM, confinement and phase transition

The spectrum of the bosonic sector of the D = 11 supermembrane with central charges is shown to be discrete and with finite multiplicities, hence containing a mass gap. The result extends to the exact theory our previous proof of the similar property for the SU ( N ) regularised model and strongly suggest discreteness of the spectrum for the complete Hamiltonian of the supermembrane with central charges. This theory is a quantum equivalent to a symplectic non-commutative super-Yang–Mills in 2 + 1 dimensions, where the space-like sector is a Riemann surface of positive genus. In this context, it is argued how the theory in 4D exhibits confinement in the N = 1 supermembrane with central charges phase and how the theory enters in the quark–gluon plasma phase through the spontaneous breaking of the centre. This phase is interpreted in terms of the compactified supermembrane without central charges.

Exact discreteness and mass gap of N = 1 symplectic Yang‐Mills from M‐theory

AbstractIn this note we summarize some of the results found recently in [1]. We show the pure discreteness of the non‐perturbative quantum spectrum of a symplectic Yang‐Mills theory defined on a Riemann surface of positive genus, living in a target space that, in particular, can be 4D. This theory corresponds to the membrane with central charges. The presence of the central charge induces a confinement in the phase at zero temperature. When the energy rises, the center of the group breaks and the theory enters in a quark‐plasma phase after a topological transition, corresponding to the N = 4 wrapped supermembrane.

Normal CR structures on compact 3-manifolds

We study normal CR compact manifolds in dimension 3. For a choice of a CR Reeb vector field, we associate a Sasakian metric on them, and we classify those metrics. As a consequence, the underlying manifolds are topologically finite quotients of \(S^3\) or of a non-flat circle bundle over a Riemann surface of positive genus. In the latter case, we prove that their CR automorphisms group is a finite extension of \(S^1\), and we classify the normal CR structures on these manifolds.

Circle actions on symplectic four-manifolds

We complete the classification of Hamiltonian torus and circle actions on symplectic four-dimensional manifolds. Following work of Delzant and Karshon, Hamiltonian circle and 2-torus actions on any fixed simply connected symplectic four-manifold were characterized by Karshon, Kessler and Pinsonnault. What remains is to study the case of Hamiltonian actions on blowups of S^2-bundles over a Riemann surface of positive genus. These do not admit 2-torus actions. In this paper, we characterize Hamiltonian circle actions on them. We then derive combinatorial results on the existence and counting of these actions. As a by-product, we provide an algorithm that determines the g-reduced form of a blowup form. Our work is a combination of soft equivariant and combinatorial techniques, using the momentum map and related data, with hard holomorphic techniques, including Gromov-Witten invariants.