Punctured Riemann Surface Research Articles

Gunning–Narasimhan’s Theorem with a Growth Condition

Given a compact Riemann surface X and a point x_0 in X, we construct a holomorphic function without critical points on the punctured Riemann surface R = X - x_0 which is of finite order at the point x_0. This complements the result of Gunning and Narasimhan from 1967 who constructed a noncritical holomorphic function on every open Riemann surface, but without imposing any growth condition. On the other hand, if the genus of X is at least one, then we show that every algebraic function on R admits a critical point. Our proof also shows that every cohomology class in H^1(X;C) is represented as a de Rham class by a nowhere vanishing holomorphic one-form of finite order on the punctured surface X-x_0.

$ {\text{SL}}\left( {2,\mathbb{R}} \right) $ Chern-Simons, Liouville, and gauge theory on duality walls

We propose an equivalence of the partition functions of two different 3d gauge theories. On one side of the correspondence we consider the partition function of 3d SL(2,R) Chern-Simons theory on a 3-manifold, obtained as a punctured Riemann surface times an interval. On the other side we have a partition function of a 3d N=2 superconformal field theory on S^3, which is realized as a duality domain wall in a 4d gauge theory on S^4. We sketch the proof of this conjecture using connections with quantum Liouville theory and quantum Teichmuller theory, and study in detail the example of the once-punctured torus. Motivated by these results we advocate a direct Chern-Simons interpretation of the ingredients of (a generalization of) the Alday-Gaiotto-Tachikawa relation. We also comment on M5-brane realizations as well as on possible generalizations of our proposals.

Darboux coordinates, Yang-Yang functional, and gauge theory

The moduli space of SL(2) flat connections on a punctured Riemann surface with the fixed conjugacy classes of the monodromies around the punctures is endowed with a system of holomorphic Darboux coordinates, in which the generating function of the variety of SL(2)-opers is identified with the universal part of the effective twisted superpotential of the corresponding four dimensional N=2 supersymmetric theory subject to the two-dimensional Omega-deformation. This allows to give a definition of the Yang-Yang functionals for the quantum Hitchin system in terms of the classical geometry of the moduli space of local systems for the dual gauge group, and connect it to the instanton counting of the four dimensional gauge theories, in the rank one case.

Mirrors of 3d Sicilian theories

We consider the compactification of the 6d $$ \mathcal{N} = \left( {2,0} \right) $$ theories, or equivalently of M-theory 5-branes, on a punctured Riemann surface times a circle. This gives rise to what we call 3d $$ \mathcal{N} = 4 $$ Sicilian theories, and we find that their mirror theories are star-shaped quiver gauge theories. We also discuss an alternative construction of these 3d theories through 4d $$ \mathcal{N} = 4 $$ SYM on a graph, which allows us to obtain the 3d mirror via 4d S-duality.

Hitchin equation, singularity, and N = 2 superconformal field theories

We argue that Hitchin's equation determines not only the low energy effective theory but also describes the UV theory of four dimensional N=2 superconformal field theories when we compactify six dimensional $A_N$ $(0,2)$ theory on a punctured Riemann surface. We study the singular solution to Hitchin's equation and the Higgs field of solutions has a simple pole at the punctures; We show that the massless theory is associated with Higgs field whose residual is a nilpotent element; We identify the flavor symmetry associated with the puncture by studying the singularity of closure of the moduli space of solutions with the appropriate boundary conditions. For the mass-deformed theory the residual of the Higgs field is a semi-simple element, we identify the semi-simple element by arguing that the moduli space of solutions of mass-deformed theory must be a deformation of the closure of the moduli space of the massless theory. We also study the Seiberg-Witten curve by identifying it as the spectral curve of the Hitchin's system. The results are all in agreement with Gaiotto's results derived from studying the Seiberg-Witten curve of four dimensional quiver gauge theory.

Topology of character varieties and representations of quivers

In Hausel et al. (2008) [10] we presented a conjecture generalizing the Cauchy formula for Macdonald polynomial. This conjecture encodes the mixed Hodge polynomials of the character varieties of representations of the fundamental group of a punctured Riemann surface of genus g. We proved several results which support this conjecture. Here we announce new results which are consequences of those in Hausel et al. (2008) [10].

Decomposable representations and Lagrangian submanifolds of moduli spaces associated to surface groups

The importance of explicit examples of Lagrangian submanifolds of moduli spaces is revealed by papers such as Dostoglou and Salamon (Ann. of Math (2), 139(3), 581–640, 1994) and Salamon (Proceedings of the international congress of mathematicians, vol.1, 2 (Zurich, 1994), pp. 526–536. Birkhauser, Basel, 1995): given a 3-manifold M with boundary ∂M = Σ, Dostoglou and Salamon use such examples to obtain a proof of the Atiyah-Floer conjecture relating the symplectic Floer homology of the representation space Hom(π1(Σ = ∂M), U)/U (associated to an explicit pair of Lagrangian submanifolds of this representation space) and the instanton homology of the 3-manifold M. In the present paper, we construct a Lagrangian submanifold of the space of representations \({\mathcal{M}_{g,l}:=Hom_\mathcal{C}(\pi_{g,l}, U)/U}\)of the fundamental group πg,lof a punctured Riemann surface Σg,l into an arbitrary compact connected Lie group U. This Lagrangian submanifold is obtained as the fixed-point set of an anti-symplectic involution\({\hat{\beta}}\) defined on \({\mathcal{M}_{g,l}}\) . We show that the involution \({\hat{\beta}}\) is induced by a form-reversing involution β defined on the quasi-Hamiltonian space \({(U\times U)^g \times\mathcal{C}_1\times\cdots\times \mathcal{C}_l}\) . The fact that \({\hat{\beta}}\) has a non-empty fixed-point set is a consequence of the real convexity theorem for group-valued momentum maps proved in Schaffhauser (A real convexity theorem for quasi-Hamiltonian actions, submitted, 25 p, 2007. http://arxiv.org/abs/math/0705.0858). The notion of decomposable representation provides a geometric interpretation of the Lagrangian submanifold thus obtained.

Quantum hall conductivity in a Landau type model with a realistic geometry II

We use a mathematical framework that we introduced in a previous paper to study geometrical and quantum mechanical aspects of a Hall system with finite size and general boundary conditions. Geometrical structures control possibly the integral or fractional quantization of the Hall conductivity depending on the value of NB/2π ( N is the number of charge carriers and B is the magnetic field). When NB/2π is irrational, we show that monovaluated wave functions can be constructed only on the graph of a free group with two generators. When NB/2π is rational, the relevant space becomes a punctured Riemann surface. We finally discuss our results from a phenomenological viewpoint.

A complete conformal metric of preassigned negative Gaussian curvature for a punctured hyperbolic Riemann surface

Leth be a complete metric of Gaussian curvature K0 on a punctured Riemann surface of genusg ≥ 1 (or the sphere with at least three punctures). Given a smooth negative functionK withK =K 0 in neighbourhoods of the punctures we prove that there exists a metric conformal toh which attains this function as its Gaussian curvature for the punctured Riemann surface. We do so by minimizing an appropriate functional using elementary analysis.

Fredholm theory and transversality for noncompact pseudoholomorphic mapsin symplectizations

AbstractWe study pseudoholomorphic maps from a punctured Riemann surface into the symplectization of a contact manifold. A Fredholm theory yields the virtual dimension of the moduli spaces of such maps in terms of the Euler characteristic of the Riemann surface and the asymptotics data given by the periodic solutions of the Reeb vector field associated to the contact form. The transversality results establish the existence of additional structure for these spaces. To be more precise, we prove that these spaces are generically smooth manifolds, and therefore their virtual dimension coincides with their actual dimension. © 2004 Wiley Periodicals, Inc.

Complex Hyperbolic Quasi-Fuchsian Groups and Toledo's Invariant

We consider discrete, faithful, type-preserving representations of the fundamental group of a punctured Riemann surface into PU(21), the holomorphic isometry group of complex hyperbolic space. Our main result is that there is a continuous family of such representations which interpolates between ℂ-Fuchsian representations and ℝ-Fuchsian representations. Moreover, these representations take every possible (real) value of the Toledo invariant. This contrasts with the case of closed surfaces where ℂ-Fuchsian and ℝ-Fuchsian representations lie in different components of the representation variety. In that case the Toledo invariant lies in a discrete set and indexes the components of the representation variety.

Existence of algebraic minimal surfaces for an arbitrary puncture set

We will show that any punctured Riemann surface can be conformally immersed into a Euclidean 3-space as a branched complete minimal surface of finite total curvature called an algebraic minimal surface.

Representations with Weighted Frames and Framed Parabolic Bundles

AbstractThere is a well-known correspondence (due to Mehta and Seshadri in the unitary case, and extended by Bhosle and Ramanathan to other groups), between the symplectic variety Mh of representations of the fundamental group of a punctured Riemann surface into a compact connected Lie group G, with fixed conjugacy classes h at the punctures, and a complex variety of holomorphic bundles on the unpunctured surface with a parabolic structure at the puncture points. For G = SU(2), we build a symplectic variety P of pairs (representations of the fundamental group into G, “weighted frame” at the puncture points), and a corresponding complex variety of moduli of “framed parabolic bundles”, which encompass respectively all of the spaces Mh, , in the sense that one can obtain Mh from P by symplectic reduction, andMh from by a complex quotient. This allows us to explain certain features of the toric geometry of the SU(2) moduli spaces discussed by Jeffrey and Weitsman, by giving the actual toric variety associated with their integrable system.

The mixed Hodge structure on the fundamental group of a punctured Riemann surface

Given a compact Riemann surface X of genus g and distinct points p and q on X, we consider the non-compact Riemann surface X := X {q} with basepoint p E X. The extension of mixed Hodge structures associated to the first two steps of iri(X, p) is studied. We show that it determines the element (2g q 2p K) in Pico (X), where K represents the canonical divisor of X as well as the corresponding extension associated to rn (X, p). Finally, we deduce a pointed Torelli theorem for punctured Riemann surfaces.

A Crossed Module Representing the Godbillon–Vey Cocycle

We exhibit crossed modules of Lie algebras (i.e. certain four-term exact sequences) corresponding to the Godbillon–Vey generator for the Lie algebras of formal vector fields in one variable, of vector fields on the circle, of differentiable vector fields of holomorphic type and of holomorphic vector fields on a punctured Riemann surface.

CUTTING AND PASTING OF RIEMANN SURFACES WITH ABELIAN DIFFERENTIALS I

We introduce a method of constructing once punctured Riemann surfaces by cutting the complex plane along "line segments" and pasting by "parallel transformations". The advantage of this construction is to give a good visualization of the deformation of complex structures of Riemann surfaces. In fact, given a positive integer g, there appears a family of once punctured Riemann surfaces of genus g which is complete and effectively parametrized at any point. Our construction naturally gives each of the resulting surfaces what we call a Lagrangian lattice Λ, a certain subgroup of the first homology. Furthermore Λ and the puncture determine an Abelian differential ωΛ of the second kind on the Riemann surface. Using Λ and ωΛ we consider the Kodaira–Spencer maps and some extension of the family to obtain any once punctured Riemann surface with a Lagrangian lattice. In particular we describe the moduli space of once punctured elliptic curves with Lagrangian lattices.

Harmonic metrics and connections with irregular singularities

We identify the holomorphic de Rham complex of the minimal extension of a meromorphic vector bundle with connexion on a compact Riemann surface X with the L 2 complex relative to a suitable metric on the bundle and a complete metric on the punctured Riemann surface. Applying results of C. Simpson, we show the existence of a harmonic metric on this vector bundle, giving the same L 2 complex.

The moduli space of complete embedded constant mean curvature surfaces

We examine the space of finite topology surfaces in ℝ3 which are complete, properly embedded and have nonzero constant mean curvature. These surfaces are noncompact provided we exclude the case of the round sphere. We prove that the spaceMk of all such surfaces withk ends (where surfaces are identified if they differ by an isometry of ℝ3) is locally a real analytic variety. When the linearization of the quasilinear elliptic equation specifying mean curvature equal to one has noL2-nullspace, we prove thatMk is locally the quotient of a real analytic manifold of dimension 3k−6 by a finite group (i.e. a real analytic orbifold), fork ≥ 3. This finite group is the isotropy subgroup of the surface in the group of Euclidean motions. It is of interest to note that the dimension ofMk is independent of the genus of the underlying punctured Riemann surface to which Σ is conformally equivalent. These results also apply to hypersurfaces of Hn+1 with nonzero constant mean curvature greater than that of a horosphere and whose ends are cylindrically bounded.

The plumbing of minimal area surfaces

We study the metric of minimal area on a punctured Riemann surface under the condition that all nontrivial homotopy closed curves be longer than or equal to 2π. By constructing deformations of admissible metrics we establish necessary conditions on minimal area metrics and a partial converse to Beurling's criterion for extremal metrics. We explicitly construct new minimal area metrics that do not arise from quadratic differentials. Under the physically motivated assumption of existence of the minimal area metrics, we show there exist neighborhoods of the punctures isometric to a flat semi-infinite cylinder of circumference 2π, allowing the definition of canonical complex coordinates around the punctures. The plumbing of surfaces with minimal area metrics is shown to induce a metric of minimal area on the resulting surface. This implies that minimal area string diagrams define a consistent quantum closed string field theory.

Spaces of RNA Secondary Structures

We prove two topological theorems in physical chemistry. Namely, we introduce a hybrid of transverse and tangential measures on train tracks to prove sphericity of various simplicial complexes which arise from certain idealized models of physical chemistry. These complexes are at once identified with Thurston′s space of projective geodesic laminations on an ideal polygon and with the analogue of a compactification (described elsewhere) of the moduli space of a punctured Riemann surface. The physical structures we study are various sub-collections of the set of all possible planar chemical bonds among the sites of a linear macromolecule. Each such collection we consider has a natural partial ordering, and the geometric realizations of appropriate posets are shown to be topological spheres. Such a topological statement encodes a wealth of combinatorial data, as we briefly discuss. In fact, our primary motivation here is to study secondary structures on RNA. This imposes the further restriction that there can be at most one base-pair supported at a given site of underlying linear macromolecule, and imposing this restriction leads to the class of "binary macromolecules." Our main results here assert the sphericity of certain topological spaces of both arbitrary and binary macromolecules, and it is the latter which we hope may have applications to RNA. Our techniques are largely elaborations of elementary topological techniques from Techmüller theory and the theory of train tracks.