Locus Of Moduli Space Research Articles

Group actions on local moduli space of holomorphic vector bundles

AbstractWe prove that actions of complex reductive Lie groups on a holomorphic vector bundle over a complex compact manifold are locally extendable to its local moduli space.

Conical Metrics on Riemann Surfaces, II: Spherical Metrics

Abstract We continue our study, initiated in [34], of Riemann surfaces with constant curvature and isolated conic singularities. Using the machinery developed in that earlier paper of extended configuration families of simple divisors, we study the existence and deformation theory for spherical conic metrics with some or all of the cone angles greater than $2\pi $. Deformations are obstructed precisely when the number $2$ lies in the spectrum of the Friedrichs extension of the Laplacian. Our main result is that, in this case, it is possible to find a smooth local moduli space of solutions by allowing the cone points to split. This analytic fact reflects geometric constructions in [37, 38].

Boundary of the pyramidal equisymmetric locus of $${\mathcal M}_n$$

The augmented moduli space $$\widehat{\mathcal M}_n$$ is a compactification of moduli space $$\mathcal M_n$$ obtained by adding stable hyperbolic surfaces. The different topological types of the added stable surfaces produce a stratification of $$\partial \widehat{\mathcal M}_n$$. Let $$\mathcal {P}_n\subset {\mathcal M}_n$$ be the pyramidal locus in moduli space, i.e., the set of hyperbolic surfaces of genus n such that the topological action of its preserving-orientation isometry group is the pyramidal action of the dihedral group $$D_n$$. The purpose of this paper is to state the complete list of strata in the boundary of $$\mathcal {P}_n$$.

Dn Dynkin quiver moduli spaces

We study 3d quiver gauge theories with gauge nodes forming a Dn Dynkin diagram and their relation to nilpotent varieties in . The class of good Dn Dynkin quivers is completely characterised and the moduli space singularity structure fully determined for all such theories. The class of good Dn Dynkin quivers is denoted where is an integer, and are integer partitions and denotes membership of one of two broad subclasses. Small subclasses of these quivers are known to realise some nilpotent varieties with their moduli space branches. We fully determine which nilpotent varieties are realisable as Dn Dynkin quiver moduli spaces and which are not. Quiver addition is introduced and is used to give large subclasses of Dn Dynkin quivers poset structure. The partial ordering is determined by inclusion relations for the moduli space branches. The resulting Hasse diagrams are used to both classify Dn Dynkin quivers and determine the moduli space singularity structure for an arbitrary good theory. The poset constructions and local moduli space analyses are complemented throughout by explicit checks utilising moduli space dimension matching.

A heterotic standard model with B − L symmetry and a stable proton

We consider heterotic Calabi-Yau compactifications with S(U(4)xU(1)) background gauge fields. These models lead to gauge groups with an additional U(1) factor which, under certain conditions, can combine with hypercharge to a B-L symmetry. The associated gauge boson is automatically super-massive and, hence, does not constitute a phenomenological problem. We illustrate this class of compactifications with a model based on the monad construction, which leads to a supersymmetric standard model with three families of quarks and leptons, one pair of Higgs doublets, three right-handed neutrinos and no exotics charged under the standard model group. The presence of the B-L symmetry means that the model is safe from proton decay induced by dimension four operators. Due to the presence of a special locus in moduli space where the bundle structure group is Abelian and the low-energy symmetry enhances we can also show the absence of dimension five proton-decay inducing operators.

The morphism induced by Frobenius push-forward

Let X be a smooth projective curve of genus g ⩾ 2 over an algebraically closed field k of characteristic p > 0, and F: X → X (1) the relative Frobenius morphism. Let \(\mathfrak{M}_X^s (r,d)\) (resp. \(\mathfrak{M}_X^{ss} (r,d)\)) be the moduli space of (resp. semi-)stable vector bundles of rank r and degree d on X. We show that the set-theoretic map \(S_{Frob}^{ss} :\mathfrak{M}_X^{ss} (r,d) \to \mathfrak{M}_{X^{(1)} }^{ss} (rp,d + r(p - 1)(g - 1))\) induced by Open image in new window is a proper morphism. Moreover, the induced morphism \(S_{Frob}^s :\mathfrak{M}_X^s (r,d) \to \mathfrak{M}_{X^{(1)} }^s (rp,d + r(p - 1)(g - 1))\) is a closed immersion. As an application, we obtain that the locus of moduli space \(\mathfrak{M}_{X^{(1)} }^s (p,d)\) consisting of stable vector bundles whose Frobenius pull backs have maximal Harder-Narasimhan polygons is isomorphic to the Jacobian variety JacX of X.

New F-theory lifts II: permutation orientifolds and enhanced singularities

In this paper, a procedure is developed to construct compact F-theory fourfolds corresponding to perturbative IIB O7/O3 models on CICY threefolds with permutation involutions. The method is explained in generality, and then applied to specific examples where the involution permutes two Del Pezzo surfaces. The fourfold construction is successfully tested by comparing the D3 charges predicted by F-theory and IIB string theory. The constructed smooth fourfolds are then taken to the locus in moduli space where they have enhanced SU(5) singularities. A general, intuitive method is developed for engineering the desired singularities in Weierstrass models for complicated D7-brane setups.

Symmetry and Variation of Hodge Structures

The Torelli theorem (cf.[Tor13], [Andr58], [Weil-57]) states that two algebraic curves are isomorphic if and only if their Jacobian varieties are isomorphic as polarized abelian varieties. Andre Weil ([Weil-CPII]) set up a program for doing arithmetics on K3 surfaces, based on a Torelli type theorem, which was later proven through the effort of several authors (for this long story and related references we refer to Chapter VIII of the book [BPV]). This result is crucial for answering questions about the existence of K3 surfaces or families thereof possessing certain curve configurations. The general Torelli question, as set up by Griffiths ([Grif68], [Grif70], cf. also [Grif-Schmid], [Grif84]), is to associate to each projective variety X of general type its Hodge structure of weight n = dim(X), and ask whether the corresponding ψ n is injective on the local moduli space (or Kuranishi space of X). It was known since long time that, as soon as the dimension is at least two, there are families of varieties without Hodge structures, which are not rigid. Surfaces of general type with q = pg = 0 were constructed in the 30's by Campedelli and Godeaux ([Cam32], [God35]), and for instance, in the case of the Godeaux surfaces, the Kuranishi family has dimension 8. A natural question which arises is: under which hypothesis on X is a local Torelli theorem valid for the Hodge structure of weight n = dim X? In other words, when is the local period map ψ n a local embedding?

Generalized Calabi-Yau Manifolds

A geometrical structure on even-dimensional manifolds is defined which generalizes the notion of a Calabi-Yau manifold and also a symplectic manifold. Such structures are of either odd or even type and can be transformed by the action of both diffeomorphisms and closed 2-forms. In the special case of six dimensions we characterize them as critical points of a natural variational problem on closed forms, and prove that a local moduli space is provided by an open set in either the odd or even cohomology.

The local moduli ofSasakian 3‐manifolds

The Newman‐Penrose‐Perjes formalism is applied to Sasakian 3‐manifolds and the local form of the metric and contact structure is presented. The local moduli space can be parameterised by a single function of two variables and it is shown that, given any smooth function of two variables, there exists locally a Sasakian structure with scalar curvature equal to this function. The case where the scalar curvature is constant (η‐Einstein Sasakian metrics) is completely solved locally. The resulting Sasakian manifolds include S 3, Nil, and , as well as the Berger spheres. It is also shown that a conformally flat Sasakian 3‐manifold is Einstein of positive scalar curvature.

Mumford curves in a specialized pencil of sextics

We discuss Mumford curves in the pencil on a Del Pezzo quintic surface constructed by Edge [Ed1]. The abstract group structures of the normalizer of the corresponding Schottky groups are described, which give us some knowledges on Mumford loci in moduli space of curves.

A note on Campedelli surfaces

In this paper we construct a 6-dimensional family of surfaces of general type with pg=pa=0 and K2=2, classically known as Campedelli surfaces. We start from a well known 4-dimensional family and obtain a family which is of maximal dimension. The construction presented here allows an explicit description of the algebraic structure of the corresponding component of the coarse moduli space. As a by-product, we also show that some surfaces with pg=pa=4 and K2=10 have a smooth local moduli space of dimension 30.

Small volumes in compactified string theory

We discuss some of the classical and quantum geometry associated to the degeneration of cycles within a Calabi-Yau compactification. In particular, we focus on the definition and properties of quantum volume, especially as it applies to identifying the physics associated to loci in moduli space where non-perturbative effects become manifest. We discuss some unusual features of quantum volume relative to its classical counterpart.

On the period maps of projective hypersurfaces

This paper gives an explicit infinitesimal (to all orders) description of the period map associated to a smooth projective hypersurface, as well as related objects such as the full Hodge filtration on the middle cohomology, the local moduli space and the Gauss–Manin connection and its iterates.

Topological strings on elliptic fibrations

We study topological string theory on elliptically fibered Calabi-Yau manifolds using mirror symmetry. We compute higher genus topological string amplitudes and express these in terms of polynomials of functions constructed from the special geometry of the deformation spaces. The polynomials are fixed by the holomorphic anomaly equations supplemented by the expected behavior at special loci in moduli space. We further expand the amplitudes in the base moduli of the elliptic fibration and find that the fiber moduli dependence is captured by a finer polynomial structure in terms of the modular forms of the modular group of the elliptic curve. We further find a recursive equation which captures this finer structure and which can be related to the anomaly equations for correlation functions. ∗alim@physics.harvard.edu †emanuel.scheidegger@math.uni-freiburg.de 1 ar X iv :1 20 5. 17 84 v3 [ he pth ] 2 1 Ju n 20 13