Riemann Surface Of Genusg Research Articles

Spin(8,C)-Higgs pairs over a compact Riemann surface

Abstract Let X X be a compact Riemann surface of genus g ≥ 2 g\ge 2 , G G be a semisimple complex Lie group and ρ : G → GL ( V ) \rho :G\to {\rm{GL}}\left(V) be a complex representation of G G . Given a principal G G -bundle E E over X X , a vector bundle E ( V ) E\left(V) whose typical fiber is a copy of V V is induced. A ( G , ρ ) \left(G,\rho ) -Higgs pair is a pair ( E , φ ) \left(E,\varphi ) , where E E is a principal G G -bundle over X X and φ \varphi is a holomorphic global section of E ( V ) ⊗ L E\left(V)\otimes L , L L being a fixed line bundle over X X . In this work, Higgs pairs of this type are considered for G = Spin ( 8 , C ) G={\rm{Spin}}\left(8,{\mathbb{C}}) and the three irreducible eight-dimensional complex representations which Spin ( 8 , C ) {\rm{Spin}}\left(8,{\mathbb{C}}) admits. In particular, the reduced notions of stability, semistability, and polystability for these specific Higgs pairs are given, and it is proved that the corresponding moduli spaces are isomorphic, and a precise expression for the stable and not simple Higgs pairs associated with one of the three announced representations of Spin ( 8 , C ) {\rm{Spin}}\left(8,{\mathbb{C}}) is described.

F 4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E 6-Higgs bundles over a compact Riemann surface

Abstract Let X X be a compact Riemann surface of genus g ≥ 2 g\ge 2 and ℳ ( E 6 ) {\mathcal{ {\mathcal M} }}\left({E}_{6}) be the moduli space of E 6 {E}_{6} -Higgs bundles over X X . We consider the automorphisms σ + {\sigma }_{+} of ℳ ( E 6 ) {\mathcal{ {\mathcal M} }}\left({E}_{6}) defined by σ + ( E , φ ) = ( E ∗ , − φ t ) {\sigma }_{+}\left(E,\varphi )=\left({E}^{\ast },-{\varphi }^{t}) , induced by the action of the outer involution of E 6 {E}_{6} in ℳ ( E 6 ) {\mathcal{ {\mathcal M} }}\left({E}_{6}) , and σ − {\sigma }_{-} defined by σ − ( E , φ ) = ( E ∗ , φ t ) {\sigma }_{-}\left(E,\varphi )=\left({E}^{\ast },{\varphi }^{t}) , which results from the combination of σ + {\sigma }_{+} with the involution of ℳ ( E 6 ) {\mathcal{ {\mathcal M} }}\left({E}_{6}) , which consists on a change of sign in the Higgs field. In this work, we describe the fixed points of σ + {\sigma }_{+} and σ − {\sigma }_{-} , as F 4 {F}_{4} -Higgs bundles, F 4 {F}_{4} -Higgs pairs associated with the fundamental irreducible representation of F 4 {F}_{4} , and PSp ( 8 , C ) {\rm{PSp}}\left(8,{\mathbb{C}}) -Higgs pairs associated with the second symmetric power or the second wedge power of the fundamental representation of Sp ( 8 , C ) {\rm{Sp}}\left(8,{\mathbb{C}}) . Finally, we describe the reduced notions of semistability and polystability for these objects.

Moduli Spaces of Generalized Hyperpolygons

Abstract We introduce the notion of generalized hyperpolygon, which arises as a representation, in the sense of Nakajima, of a comet-shaped quiver. We identify these representations with rigid geometric figures, namely pairs of polygons: one in the Lie algebra of a compact group and the other in its complexification. To such data, we associate an explicit meromorphic Higgs bundle on a genus-g Riemann surface, where g is the number of loops in the comet, thereby embedding the Nakajima quiver variety into a Hitchin system on a punctured genus-g Riemann surface (generally with positive codimension). We show that, under certain assumptions on flag types, the space of generalized hyperpolygons admits the structure of a completely integrable Hamiltonian system of Gelfand–Tsetlin type, inherited from the reduction of partial flag varieties. In the case where all flags are complete, we present the Hamiltonians explicitly. We also remark upon the discretization of the Hitchin equations given by hyperpolygons, the construction of triple branes (in the sense of Kapustin–Witten mirror symmetry), and dualities between tame and wild Hitchin systems (in the sense of Painlevé transcendents).

Faber and Grunsky operators corresponding to bordered Riemann surfaces

LetR\mathfrak {R}be a compact Riemann surface of finite genusg>0\mathfrak {g}>0and letΣ\Sigmabe the subsurface obtained by removingn≥1n\geq 1simply connected regionsΩ1+,…,Ωn+\Omega _1^+, \dots , \Omega _n^+fromR\mathfrak {R}with non-overlapping closures. Fix a biholomorphismfkf_kfrom the unit disc ontoΩk+\Omega _k^+for eachkkand letf=(f1,…,fn)\mathbf {f}=(f_1, \dots , f_n). We assign a Faber and a Grunsky operator toR\mathfrak {R}andf\mathbf {f}when all the boundary curves ofΣ\Sigmaare quasicircles inR\mathfrak {R}. We show that the Faber operator is a bounded isomorphism and the norm of the Grunsky operator is strictly less than one for this choice of boundary curves. A characterization of the pull-back of the holomorphic Dirichlet space ofΣ\Sigmain terms of the graph of the Grunsky operator is provided.

Anomaly inflow, accidental symmetry, and spontaneous symmetry breaking

We consider the 6d (1,0) SCFT on a stack of N M5-branes probing a ℂ2/ℤ2 singularity. In particular, we study its compactifications to four dimensions on a smooth genus-g Riemann surface with non-trivial flavor flux, yielding a family of 4d CFTs. By tracking the M-theory origin of the global symmetries of the 4d CFTs, we detect the emergence of an accidental symmetry and the spontaneous symmetry breaking of a U(1) generator. These effects are visible from geometric considerations and not apparent from the point of view of the compactification of the 6d field theory. These phenomena leave an imprint on the ’t Hooft anomaly polynomial of the 4d CFTs, which is obtained from recently developed anomaly inflow methods in M-theory [1]. In the large-N limit, we identify the gravity dual of the 4d setups to be a class of smooth AdS5 solutions first discussed by Gauntlett-Martelli-Sparks-Waldram. Using our anomaly polynomial, we compute the conformal central charge and a non-Abelian flavor central charge at large N , finding agreement with the holographic predictions.

Mathcal{N} = 1 supersymmetric indices and the four-dimensional A-model

We compute the supersymmetric partition function of mathcal{N} = 1 supersymmetric gauge theories with an R-symmetry on {mathrm{mathcal{M}}}_4cong {mathrm{mathcal{M}}}_{g,p}times {S}^1 , a principal elliptic fiber bundle of degree p over a genus-g Riemann surface, Σg . Equivalently, we compute the generalized supersymmetric index {I_{mathrm{mathcal{M}}}}_{{}_{g,p}} , with the supersymmetric three-manifold {mathrm{mathcal{M}}}_{g,p} as the spatial slice. The ordinary mathcal{N} = 1 supersymmetric index on the round three-sphere is recovered as a special case. We approach this computation from the point of view of a topological A-model for the abelianized gauge fields on the base Σg. This A-model — or A-twisted two-dimensional mathcal{N} = (2, 2) gauge theory — encodes all the information about the generalized indices, which are viewed as expectations values of some canonically-defined surface defects wrapped on T2 inside Σg × T2. Being defined by compactification on the torus, the A-model also enjoys natural modular properties, governed by the four-dimensional ’t Hooft anomalies. As an application of our results, we provide new tests of Seiberg duality. We also present a new evaluation formula for the three-sphere index as a sum over two-dimensional vacua.

THE COMPLEXITY OF RIEMANN SURFACES AND THE HURWITZ EXISTENCE PROBLEM

AbstractThecomplexityof a branched cover of a Riemann surfaceMto the Riemann sphereS2is defined as its degree times the hyperbolic area of the complement of its branching set inS2. ThecomplexityofMis defined as the infimum of the complexities of all branched covers ofMtoS2. We prove that ifMis a connected, closed, orientable Riemann surface of genusg≥1, then its complexity equals 2π(mmin+2g−2) , wheremminis the minimum total length of a branch datum realisable by a branched coverp:M→S2.

The Deligne–Mumford compactification of the real multiplication locus and Teichmüller curves in genus 3

In the moduli space $$ \mathcal{M} $$ g of genus-g Riemann surfaces, consider the locus $$ \mathcal{R}{\mathcal{M}_{\mathcal{O}}} $$ of Riemann surfaces whose Jacobians have real multiplication by the order $$ \mathcal{O} $$ in a totally real number field F of degree g. If g = 3, we compute the closure of $$ \mathcal{R}{\mathcal{M}_{\mathcal{O}}} $$ in the Deligne–Mumford compactification of $$ \mathcal{M} $$ g and the closure of the locus of eigenforms over $$ \mathcal{R}{\mathcal{M}_{\mathcal{O}}} $$ in the Deligne–Mumford compactification of the moduli space of holomorphic 1-forms. For higher genera, we give strong necessary conditions for a stable curve to be in the boundary of $$ \mathcal{R}{\mathcal{M}_{\mathcal{O}}} $$ . Boundary strata of $$ \mathcal{R}{\mathcal{M}_{\mathcal{O}}} $$ are parameterized by configurations of elements of the field F satisfying a strong geometry of numbers type restriction. We apply this computation to give evidence for the conjecture that there are only finitely many algebraically primitive Teichmuller curves in $$ \mathcal{M} $$ 3. In particular, we prove that there are only finitely many algebraically primitive Teichmuller curves generated by a 1-form having two zeros of order 3 and 1. We also present the results of a computer search for algebraically primitive Teichmuller curves generated by a 1-form having a single zero.

On the topological types of symmetries of elliptic-hyperelliptic Riemann surfaces

LetX be a Riemann surface of genusg. The surfaceX is called elliptic-hyperelliptic if it admits a conformal involutionh such that the orbit spaceX/〈h〉 has genus one. The involutionh is then called an elliptic-hyperelliptic involution. Ifg>5 then the involutionh is unique, see [A]. We call symmetry to any anticonformal involution ofX. LetAut±(X) be the group of conformal and anticonformal automorphisms ofX and letσ, τ be two symmetries ofX with fixed points and such that {σ, hσ} and {τ, hτ} are not conjugate inAut±(X). We describe all the possible topological conjugacy classes of {σ, σh, τ, τh}. As consequence of our study we obtain that, in the moduli space of complex algebraic curves of genusg (g even >5), the subspace whose elements are the elliptic-hyperelliptic real algebraic curves is not connected. This fact contrasts with the result in [Se]: the subspace whose elements are the hyperelliptic real algebraic curves is connected.

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.

Counting higher genus curves in a Calabi-Yau manifold

We explicitly evaluate the low energy coupling F g in a d = 4, N = 2 compactification of the heterotic string. The holomorphic piece of this expression provides the information not encoded in the holomorphic anomaly equations, and we find that it is given by an elementary polylogarithm with index 3 - 2g, thus generalizing in a natural way the known results for g = 0, 1. The heterotic model has a dual Calabi-Yau compactification of the type 11 string. We compare the answer with the general form expected from curve-counting formulae and find good agreement. As a corollary of this comparison we predict some numbers of higher genus curves in a specific Calabi-Yau, and extract some intersection numbers on the moduli space of genus-g Riemann surfaces.

Rational curves on moduli spaces of vector bundles

We identify the spaces Homi(ℙ1,M) fori = 1, 2, whereM is the moduli space of vector bundles of rank 2 and determinant isomorphic to $$\mathcal{O}(x_0 )$$ ,x 0 ∈X, on a compact Riemann surface of genusg ≥ 2.

Triangulations and moduli spaces of Riemann surfaces with group actions

We study that subset of the moduli space $$\bar M^g $$ of stable genusg,g>1, Riemann surfaces which consists of such stable Riemann surfaces on which a given finite groupF acts. We show first that this subset is compact. It turns out that, for general finite groupsF, the above subset is not connected. We show, however, that for ℤ2 actions this subsetis connected. Finally, we show that even in the moduli space ofsmooth genusg Riemann surfaces, the subset of those Riemann surfaces on which ℤ2 actsis connected. In view of deliberations of Klein ([8]), this was somewhat surprising. These results are based on new coordinates for moduli spaces. These coordinates are obtained by certainregular triangulations of Riemann surfaces. These triangulations play an important role also elsewhere, for instance in approximating eigenfunctions of the Laplace operator numerically.

On the period matrix of a Riemann surface of large genus (with an Appendix by J.H. Conway and N.J.A. Sloane)

Riemann showed that a period matrix of a compact Riemann surface of genusg≧1 satisfies certain relations. We give a further simple combinatorial property, related to the length of the shortest non-zero lattice vector, satisfied by such a period matrix, see (1.13). In particular, it is shown that for large genus the entire locus of Jacobians lies in a very small neighborhood of the boundary of the space of principally polarized abelian varieties. We apply this to the problem of congruence subgroups of arithmetic lattices in SL2(ℝ). We show that, with the exception of a finite number of arithmetic lattices in SL2(ℝ), every such lattice has a subgroup of index at most 2 which is noncongruence. A notable exception is the modular groupSL 2(ℤ).

BRST quantization ofN = 2, 3, and 4 superconformal theories on higher-genus Riemann surfaces

Complex contour integral techniques, developed in a previous paper for theN=0 and 1 superconformal theories on higher-genus Riemann surfaces, are applied to a Becchi-Rouet-Stora-Tyutin (BRST) quantization procedure of superconformal theories withN=2, 3, and 4 super-Krichever-Novikov (KN) constraint algebras on a genus-g Riemann surface. The BRST charges of the superconformai theories are constructed and the nilpotency of the BRST charges is checked. The critical spacetime dimension and the “intercepts” are found for theN=2 and 4 cases. Also calculated are the central charge and the “intercept” for theN=3 case.

The KN algebras and the geometrical quantization of heterotic strings

By means of the KN algebra, the Virasoro and Kac-Moody superalgebras are extended on a genus-g Riemann surface. These are realized in the framework of heterotic superstring with the WZ term coupling the fiber bundle. Then we construct the corresponding BRST operators, which provide the critical dimension.

N=2 superconformal algebra on higher-genus Riemann surfaces

Using the Krichever-Novikov bases and the operator product expansions, the author constructs the N=2 superconformal algebra on a genus-g Riemann surface and the BRST charge corresponding to the superconformal algebra. He also checks the nilpotency of the BRST charge, and obtains the critical dimension of spacetime as D=2 for the N=2 superconformal theory on the higher-genus Riemann surface.

Quantum Yang-Mills on a Riemann surface

We obtain the quantum expectations of gauge-invariant functions of the connection on aG=SU(N) product bundle over a Riemann surface of genusg. We show that the spaceA/Gm of connections modulo those gauge transformations which are the identity at one point is itself a principal bundle with affine linear fiber. The base space Path2gG consists of 2g-tuples of paths inG subject to a relation on their endpoint values. Quantum expectations are iterated path integrals over first the fiber then over Path2gG, each with respect to the push-forward toA/Gm of the measuree−S(A)DA. Here,S(A) denotes the Yang-Mills action onA. We exhibit a global section ofA/Gm to define a choice of origin in each fiber, relative to which the measure on the fiber is Gaussian. The induced measure on Path2gG is the product of Wiener measures on the component paths, conditioned to preserve the endopoint relation. Conformal transformations of the metric onM act by reparametrizing these paths. We explicitly compute the partition function in the general case and the expectations of functions of certain products of Wilson loops in the caseg=1.

An algebraic geometry study of theb−c system with arbitrary twist fields and arbitrary statistics

We present an analysis of the generalb−c system (including the β−γ system) on a compact Riemann surface of arbitrary genusg≧0 by postulating that its correlation functions should only have the singularities imposed by the operator product expansion (OPE) of the system. Studying a very (in fact optimally) general form of theb−c system, we prove rigorously that the standard practice of eliminating zero modes, and even the standard lagrangian, follow from the analyticity structure dictated by the OPE alone. We extend the analysis to consider the most general case of the presence of twist (e.g. spin) fields. We then determine all the possible correlation functions of theb−c system, with statistics unspecified, compatible with the OPE. On imposing Fermi and Bose statistics, we obtain the correlation functions of the fermionicb−c and β−γ systems, respectively.

How covariant closed string theory solves a minimal area problem

For a given genusg Riemann surface withn≥0 punctures (n≥3 forg=0) we consider the problem of finding the metric of minimal area under the condition that the length of any nontrivial closed curve be greater or equal to 2π. The minimal area metrics are found for the case of all punctured genus zero surfaces and for many of the higher genus surfaces both with and without punctures. These metrics are induced by Jenkins-Strebel quadratic differentials. They arise from the string diagrams corresponding to restricted Feynman graphs of a closed string field theory action containing classical and quantum restricted polyhedra.