Compactification Of Moduli Space Research Articles

Multi-fractional instantons in SU(N) Yang-Mills theory on the twisted T4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathbbm{T}}^4 $$\\end{document}

We construct analytical self-dual Yang-Mills fractional instanton solutions on a four-torus T4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathbbm{T}}^4 $$\\end{document} with ’t Hooft twisted boundary conditions. These instantons possess topological charge Q=rN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ Q=\\frac{r}{N} $$\\end{document}, where 1 ≤ r < N. To implement the twist, we employ SU(N) transition functions that satisfy periodicity conditions up to center elements and are embedded into SU(k) × SU(ℓ) × U(1) ⊂ SU(N), where ℓ + k = N. The self-duality requirement imposes a condition, kL1L2 = rℓL3L4, on the lengths of the periods of T4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathbbm{T}}^4 $$\\end{document} and yields solutions with abelian field strengths. However, by introducing a detuning parameter ∆ ≡ (rℓL3L4 – kL1L2)/L1L2L3L4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\sqrt{L_1{L}_2{L}_3{L}_4} $$\\end{document}, we generate self-dual nonabelian solutions on a general T4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathbbm{T}}^4 $$\\end{document} as an expansion in powers of ∆. We explore the moduli spaces associated with these solutions and find that they exhibit intricate structures. Solutions with topological charges greater than 1N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{1}{N} $$\\end{document} and k ≠ r possess non-compact moduli spaces, along which the O∆\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O}\\left(\\Delta \\right) $$\\end{document} gauge-invariant densities exhibit runaway behavior. On the other hand, solutions with Q=rN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ Q=\\frac{r}{N} $$\\end{document} and k = r have compact moduli spaces, whose coordinates correspond to the allowed holonomies in the SU(r) color space. These solutions can be represented as a sum over r lumps centered around the r distinct holonomies, thus resembling a liquid of instantons. In addition, we show that each lump supports 2 adjoint fermion zero modes.

Analysis, Geometry and Topology of Singular PDE

This is a report on the Oberwolfach conference “Analysis, geometry and topology of singular PDE”, June 2-12, 2021. This workshop, held in an hybrid format, focused on the topology, geometry and geometric analysis of certain spaces with singularities: stratified spaces, compactifications of moduli spaces, spaces carrying a (singular) foliation, etc., and on the microlocal techniques, either in their classical forms or in more recent versions developed to handle singular PDE, or via the groupoid approach.

The moduli space of the tropicalizations of Riemann surfaces

In this paper we study the moduli space of the tropicalizations of Riemann surfaces. We first tropicalize a smooth pointed Riemann surface by its pair of pants decomposition. Then we can construct the moduli space of tropicalizations based on a fixed regular tropicalization, and compactify it by adding strata parametrizing weighted contractions. This, in the covering level, is analogous to adding the frontier set, subordinate to a pants decomposition, to the Teichmüller space. We show that this compact moduli space is also Hausdorff. In the end, we compare this moduli space with the moduli space of Riemann surfaces, establishing a partial order-preserving correspondence between the stratifications of these two moduli spaces.

Stable log surfaces, admissible covers, and canonical curves of genus 4

We explicitly describe the KSBA/Hacking compactification of a moduli space of log surfaces of Picard rank 2. The space parametrizes log pairs ( S , D ) (S, D) where S S is a degeneration of P 1 × P 1 \mathbb {P}^1 \times \mathbb {P}^1 and D ⊂ S D \subset S is a degeneration of a curve of class ( 3 , 3 ) (3,3) . We prove that the compactified moduli space is a smooth Deligne–Mumford stack with 4 boundary components. We relate it to the moduli space of genus 4 curves; we show that it compactifies the blow-up of the hyperelliptic locus. We also relate it to a compactification of the Hurwitz space of triple coverings of P 1 \mathbb {P}^1 by genus 4 curves.

Harmonic tropical morphisms and approximation

Harmonic amoebas are generalisations of amoebas of algebraic curves immersed in complex tori. Introduced by Krichever in 2014, the consideration of such objects suggests to enlarge the scope of tropical geometry. In the present paper, we introduce the notion of harmonic morphisms from tropical curves to affine spaces and show how these morphisms can be systematically described as limits of families of harmonic amoeba maps on Riemann surfaces. It extends previous results about approximation of tropical curves in affine spaces and provides a different point of view on Mikhalkin’s approximation Theorem for regular phase-tropical morphisms, as stated e.g. by Mikhalkin in 2006. The results presented here follow from the study of imaginary normalised differentials on families of punctured Riemann surfaces and suggest interesting connections with compactifications of moduli spaces.

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 $$\mathcal {P}_n$$.

On the associativity of gluing

This paper studies the associativity of gluing of trajectories in Morse theory. We show that the associativity of gluing follows from the existence of compatible manifold with face structures on the compactified moduli spaces. Using our previous work, we obtain the associativity of gluing in certain cases. In particular, associativity holds when the ambient manifold is compact and the vector field is Morse–Smale.

An uplifting discussion of T-duality

It is well known that string theory has a T-duality symmetry relating circle compactifications of large and small radius. This symmetry plays a foundational role in string theory. We note here that while T-duality is order two acting on the moduli space of compactifications, it is order four in its action on the conformal field theory state space. More generally, involutions in the Weyl group W (G) which act at points of enhanced G symmetry have canonical lifts to order four elements of G, a phenomenon first investigated by J. Tits in the mathematical literature on Lie groups and generalized here to conformal field theory. This simple fact has a number of interesting consequences. One consequence is a reevaluation of a mod two condition appearing in asymmetric orbifold constructions. We also briefly discuss the implications for the idea that T-duality and its generalizations should be thought of as discrete gauge symmetries in spacetime.

Limits of equisymmetric $1$-complex dimensional families of Riemann surfaces

We describe the limit surfaces of some equisymmetric $1$-complex dimensional families of Riemann surfaces in the boundary of the Deligne-Mumford compactification of moduli space. We provide a description of such nodal Riemann surfaces in terms of the group of automorphisms defining the family. We apply our method to some known examples.

Geometric compactification of moduli spaces of half-translation structures on surfaces

In this paper, we give an equivariant compactification of the space $${\mathbb {P}}{\text {Flat}}(\Sigma )$$ of homothety classes of half-translation structures on a compact, connected, orientable surface $$\Sigma $$ . We introduce the space $${\mathbb {P}}{\text {Mix}}(\Sigma )$$ of homothety classes of mixed structures on $$\Sigma $$ , that are $${\text {CAT}}(0)$$ tree-graded spaces in the sense of Drutu and Sapir, with pieces which are $${\mathbb {R}}$$ -trees and completions of surfaces endowed with half-translation structures. Endowing $${\text {Mix}}(\Sigma )$$ with the equivariant Gromov topology, and using asymptotic cone techniques, we prove that $${\mathbb {P}}{\text {Mix}}(\Sigma )$$ is an equivariant compactification of $${\mathbb {P}}{\text {Flat}}(\Sigma )$$ , thus allowing us to understand in a geometric way the degenerations of half-translation structures on $$\Sigma $$ . We finally compare our compactification to the one of Duchin–Leininger–Rafi, based on geodesic currents on $$\Sigma $$ , by the mean of the translation distances of the elements of the covering group of $$\Sigma $$ .

Surjectivity of a gluing construction in special Lagrangian geometry

This paper is motivated by a relatively recent work by Joyce in special Lagrangian geometry, but the basic idea of the present paper goes back to an earlier pioneering work of Donaldson in Yang--Mills gauge theory; Donaldson discovered a global structure of a (compactified) moduli space of Yang--Mills instantons, and a key step to that result was the proof of surjectivity of Taubes' gluing construction. In special Lagrangian geometry we have currently no such a global understanding of (compactified) moduli spaces, but in the present paper we determine a neighbourhood of a `boundary' point. It is locally similar to Donaldson's result, and in particular as Donaldson's result implies the surjectivity of Taubes' gluing construction so our result implies the surjectivity of Joyce's gluing construction in a certain simple case.

An ℓ-th root of a test configuration of exponent ℓ

Abstract Let (X, L) be a polarized algebraic manifold. Then for every test configuration μ = (X, L,Ψ) for (X, L) of exponent ℓ, we obtain an ℓ-th root (κ, D) of μ and Gm-equivariant desingularizations ι : ^X → X and η : ^X → Y, both isomorphic on^X \^X 0, such that whereκ= (Y, Q, η) is a test configuration for (X, L) of exponent 1, and D is an effective Q-divisor on^X such that ℓD is an integral divisor with support in the fiber X0. Then (κ, D) can be chosen in such a way that where C1 and C2 are positive real constants independent of the choice of μ and ℓ. This plays an important role in our forthcoming papers on the existence of constant scalar curvature Kähler metrics (cf. [6]) and also on the compactified moduli space of test configurations (cf. [5],[7]).

Isomorphism of Compactifications of Vector Bundles Moduli: Nonreduced Moduli

We continue the study of the compactification of the moduli scheme for Gieseker-semistable vector bundles on a nonsingular irreducible projective algebraic surface S with polarization L, by locally free sheaves. The relation of main components of the moduli functor or admissible semistable pairs and main components of the Gieseker – Maruyama moduli functor (for semistable torsion-free coherent sheaves) with the same Hilbert polynomial on the surface S is investigated. The compactification of interest arises when families of Gieseker-semistable vector bundles E on the nonsingular polarized projective surface (S, L) are completed by vector bundles E on projective polarized schemes (S, L) of special form. The form of the scheme S, of its polarization L and of the vector bundle E is described in the text. The collection ((S, L), E) is called a semistable admissible pair. Vector bundles E on the surface (S, L) and E on schemes (S, L) are supposed to have equal ranks and Hilbert polynomials which are compute with respect to polarizations L and L, respectively. Pairs of the form ((S, L), E) named as S-pairs are also included into the class under the scope. Since the purpose is to study the compactification of moduli space for vector bundles, only families which contain S-pairs are considered. We build up the natural transformation of the moduli functor for admissible semistable pairs to the Gieseker – Maruyama moduli functor for semistable torsion-free coherent sheaves on the surface (S, L), with same rank and Hilbert polynomial. It is demonstrated that this natural transformation is inverse to the natural transformation built in the preceding paper and defined by the standard resolution of a family of torsion-free coherent sheaves with a possibly nonreduced base scheme. The functorial isomorphism constructed determines the scheme isomorphism of compactifications of moduli space for semistable vector bundles on the surface (S, L).

A family of non-restricted D=11 geometric supersymmetries

We construct a two parameter family of eleven-dimensional indecomposable Cahen-Wallach spaces with irreducible, non-flat, non-restricted geometric supersymmetry of fraction $\nu=\tfrac{3}{4}$. Its compactified moduli space can be parametrized by a compact interval with two points corresponding to two non-isometric, decomposable spaces. These singular spaces are associated to a restricted $N=4$ geometric supersymmetry with $\nu=\tfrac{1}{2}$ in dimension six and a non-restricted $N=2$ geometric supersymmetry with $\nu=\tfrac{3}{4}$ in dimension nine.

Connecting $$p$$ p -gonal loci in the compactification of moduli space

Consider the moduli space \(\mathcal {M}_{g}\) of Riemann surfaces of genus \(g\ge 2\) and its Deligne–Mumford compactification \(\overline{\mathcal {M}_{g}}\). We are interested in the branch locus \({\mathcal {B}_{g}}\) for \(g>2\), i.e., the subset of \(\mathcal {M}_{g}\) consisting of surfaces with automorphisms. It is well-known that the set of hyperelliptic surfaces (the hyperelliptic locus) is connected in \(\mathcal {M}_{g}\) but the set of (cyclic) trigonal surfaces is not. By contrast, the set of (cyclic) trigonal surfaces is connected in \(\overline{\mathcal {M}_{g}}\). We exhibit an explicit nodal surface that lies in the completion of every equisymmetric set of \(3\)-gonal Riemann surfaces providing an alternative proof of a result of Achter and Pries (Math Ann 338:187–206, 2007). For \(p>3\) the connectivity of the \(p\)-gonal loci becomes more involved. We show that for \(p\ge 11\) prime and genus \(g=p-1\) there are one-dimensional strata of cyclic \(p\)-gonal surfaces that are completely isolated in the completion \(\overline{\mathcal {B}_{g}}\) of the branch locus in \(\overline{\mathcal {M}_{g}}\).

Moduli spaces in genus zero and inversion of power series

This note shows, using elementary properties of ribbon trees, that the universal formula for the inversion of power series can be obtained by counting strata in the compactified moduli space \overline{\mathcal M}_{0,n} .

Counting lattice points in compactified moduli spaces of curves

We define and count lattice points in the moduli space of stable genus g curves with n labeled points. This extends a construction of the second author for the uncompactified moduli space. The enumeration produces polynomials with top degree coefficients tautological intersection numbers on the compactified moduli space and constant term the orbifold Euler characteristic of the compactified moduli space. We also prove a recursive formula which can be used to effectively calculate these polynomials.

The Calabi Conjecture and K-stability

We algebraically prove K-stability of polarized Calabi–Yau varieties and canonically polarized varieties with mild singularities. In particular, the “stable varieties” introduced by Kollar–Shepherd-Barron [“Threefolds and deformation of surface singularities.” Inventiones Mathematicae 91 (1988): 299–338] and Alexeev [“Moduli spaces M g,n (W) for surfaces.” Proceeding of “Higher-dimensional complex varieties (Trento, 1994)”, de Gruyter, Berlin, 1996, 1–22], which form compact moduli space, are proven to be K-stable although it is well known that they are not necessarily asymptotically (semi) stable. As a consequence, we have orbifold counterexamples to the folklore conjecture “K-stability implies asymptotic stability”. They have Kahler–Einstein (orbifold) metrics, so the result of Donaldson [“Scalar curvature and projective embeddings. I.” Journal of Differential Geometry 59, no. 3 (2001): 479–522] does not hold for orbifolds.

ON MODULI SPACES AND CW STRUCTURES ARISING FROM MORSE THEORY ON HILBERT MANIFOLDS

This paper proves some results on negative gradient dynamical systems of Morse functions on Hilbert manifolds. It contains the compactness of flow lines, manifold structures of certain compactified moduli spaces, orientation formulas, and CW structures of the underlying manifolds.

Counting lattice points in the moduli space of curves

We show how to define and count lattice points in the moduli space $\modm_{g,n}$ of genus $g$ curves with $n$ labeled points. This produces a polynomial with coefficients that include the Euler characteristic of the moduli space, and tautological intersection numbers on the compactified moduli space.