N=2 LANDAU-GINZBURG VS. CALABI-YAU σ-MODELS: NON-PERTURBATIVE ASPECTS

1991 ◽  
Vol 06 (10) ◽  
pp. 1749-1813 ◽  
Author(s):  
S. CECOTTI
Keyword(s):  
Algebraic Geometry ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Catastrophe Theory ◽  
Intersection Theory ◽  
Spectral Flow ◽  
Hodge Structures ◽  
Absolute Normalization ◽  
Chiral Ring ◽  
Mixed Hodge Structures

We discuss some nonperturbative aspects of the correspondence between N=2 Landau-Ginzburg orbifolds and Calabi-Yau σ-models. We suggest that the correct framework is Deligne’s theory of mixed Hodge structures (closely related to catastrophe theory). We derive a general topological formula for the chiral ring OPE coefficients of any Landau-Ginzburg model, including the absolute normalization. This follows from the identification of spectral flow with Grothendieck’s local duality. Wherever the LG model has a CY interpretation, its OPE coefficients are equal to those of the σ-model as given by intersection theory, including normalization. We discuss at length the tricky case of a number of LG fields greater than c/3+2, presenting explicit examples. In passing, we get many results about the geometry of moduli spaces for such conformal theories. We explain the beautiful algebraic geometry connected with a remarkable model pointed out by Vafa, and its relations with moduli space geometry.

MIXED HODGE STRUCTURES OF THE MODULI SPACES OF PARABOLIC CONNECTIONS

2016 ◽  
Vol 225 ◽  
pp. 185-206
Author(s):  
ARATA KOMYO
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Higgs Bundles ◽  
Hodge Structures ◽  
Character Varieties ◽  
Generic Eigenvalues ◽  
Hodge Polynomials ◽  
Mixed Hodge Structures

In this paper, we investigate the mixed Hodge structures of the moduli space of $\boldsymbol{\unicode[STIX]{x1D6FC}}$-stable parabolic Higgs bundles and the moduli space of $\boldsymbol{\unicode[STIX]{x1D6FC}}$-stable regular singular parabolic connections. We show that the mixed Hodge polynomials are independent of the choice of generic eigenvalues and the mixed Hodge structures of these moduli spaces are pure. Moreover, by the Riemann–Hilbert correspondence, the Poincaré polynomials of character varieties are independent of the choice of generic eigenvalues.

scholarly journals On the boundary of the moduli spaces of log Hodge structures: Triviality of the torsor

2010 ◽  
Vol 198 ◽  
pp. 173-190 ◽  
Author(s):  
Tatsuki Hayama
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Hodge Structures

AbstractThis paper examines the moduli spaces of log Hodge structures introduced by Kato and Usui. This moduli space is a partial compactification of a discrete quotient of a period domain. This paper treats the following two cases: (A) where the period domain is Hermitian symmetric, and (B) where the Hodge structures are of the mirror quintic type. Especially it addresses a property of the torsor.

A boundary divisor in the moduli spaces of stable quintic surfaces

2017 ◽  
Vol 28 (04) ◽  
pp. 1750021 ◽  
Author(s):  
Julie Rana
Keyword(s):  
Algebraic Geometry ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Deformation Theory ◽  
General Type ◽  
Topological Invariants ◽  
Surfaces Of General Type ◽  
Wide Range ◽  
Stable Surfaces ◽  
Standing Question

We give a bound on which singularities may appear on Kollár–Shepherd-Barron–Alexeev stable surfaces for a wide range of topological invariants and use this result to describe all stable numerical quintic surfaces (KSBA-stable surfaces with [Formula: see text]) whose unique non-Du Val singularity is a Wahl singularity. We then extend the deformation theory of Horikawa to the log setting in order to describe the boundary divisor of the moduli space [Formula: see text] corresponding to these surfaces. Quintic surfaces are the simplest examples of surfaces of general type and the question of describing their moduli is a long-standing question in algebraic geometry.

scholarly journals HODGE POLYNOMIALS OF THE MODULI SPACES OF PAIRS

2007 ◽  
Vol 18 (06) ◽  
pp. 695-721 ◽  
Author(s):  
VICENTE MUÑOZ ◽  
DANIEL ORTEGA ◽  
MARIA-JESÚS VÁZQUEZ-GALLO
Keyword(s):  
Moduli Spaces ◽  
Holomorphic Section ◽  
Complex Numbers ◽  
Projective Curve ◽  
Hodge Structures ◽  
Smooth Projective Curve ◽  
Holomorphic Bundle ◽  
Rank 2 ◽  
Hodge Polynomials ◽  
Mixed Hodge Structures

Let X be a smooth projective curve of genus g ≥ 2 over the complex numbers. A holomorphic pair on X is a couple (E, ϕ), where E is a holomorphic bundle over X of rank n and degree d, and ϕ ∈ H0(E) is a holomorphic section. In this paper, we determine the Hodge polynomials of the moduli spaces of rank 2 pairs, using the theory of mixed Hodge structures. We also deal with the case in which E has fixed determinant.

scholarly journals A gravitino distance conjecture

2021 ◽  
Vol 2021 (8) ◽  
Author(s):  
Alberto Castellano ◽  
Anamaría Font ◽  
Alvaro Herráez ◽  
Luis E. Ibáñez
Keyword(s):  
Moduli Space ◽  
Complex Structure ◽  
Hubble Constant ◽  
Low Energy ◽  
Hodge Structures ◽  
Supergravity Theory ◽  
Gravitino Mass ◽  
Mixed Hodge Structures ◽  
Range Of Values ◽  
Weak Gravity

Abstract We conjecture that in a consistent supergravity theory with non-vanishing gravitino mass, the limit m3/2 → 0 is at infinite distance. In particular one can write Mtower ~ $$ {m}_{3/2}^{\delta } $$ m 3 / 2 δ so that as the gravitino mass goes to zero, a tower of KK states as well as emergent strings becomes tensionless. This conjecture may be motivated from the Weak Gravity Conjecture as applied to strings and membranes and implies in turn the AdS Distance Conjecture. We test this proposal in classical 4d type IIA orientifold vacua in which one obtains a range of values $$ \frac{1}{3} $$ 1 3 ≤ δ ≤ 1. The parameter δ is related to the scale decoupling exponent in AdS vacua and to the α exponent in the Swampland Distance Conjecture for the type IIA complex structure. We present a general analysis of the gravitino mass in the limits of moduli space in terms of limiting Mixed Hodge Structures and study in some detail the case of two-moduli F-theory settings. Moreover, we obtain general lower bounds δ ≥$$ \frac{1}{3},\frac{1}{4} $$ 1 3 , 1 4 for Calabi-Yau threefolds and fourfolds, respectively. The conjecture has important phenomenological implications. In particular we argue that low-energy supersymmetry of order 1 TeV is only obtained if there is a tower of KK states at an intermediate scale, of order 108 GeV. One also has an upper bound for the Hubble constant upon inflation H ≲ $$ {m}_{3/2}^{\delta }{M}_{\mathrm{P}}^{\left(1-\delta \right)} $$ m 3 / 2 δ M P 1 − δ .

scholarly journals On the boundary of the moduli spaces of log Hodge structures: Triviality of the torsor

2010 ◽  
Vol 198 ◽  
pp. 173-190
Author(s):  
Tatsuki Hayama
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Hodge Structures

AbstractThis paper examines the moduli spaces of log Hodge structures introduced by Kato and Usui. This moduli space is a partial compactification of a discrete quotient of a period domain. This paper treats the following two cases: (A) where the period domain is Hermitian symmetric, and (B) where the Hodge structures are of the mirror quintic type. Especially it addresses a property of the torsor.

scholarly journals The Geometric P=W Conjecture in the Painlevé Cases via Plumbing Calculus

2020 ◽  
Author(s):  
András Némethi ◽  
Szilárd Szabó
Keyword(s):  
Moduli Spaces ◽  
Hodge Structures ◽  
Mixed Hodge Structures

Abstract We use plumbing calculus to prove the homotopy commutativity assertion of the Geometric $P=W$ conjecture in all Painlevé cases. We discuss the resulting Mixed Hodge structures on Dolbeault and Betti moduli spaces.

scholarly journals HODGE STRUCTURES OF THE MODULI SPACES OF PAIRS

2010 ◽  
Vol 21 (11) ◽  
pp. 1505-1529 ◽  
Author(s):  
VICENTE MUÑOZ
Keyword(s):  
Vector Bundle ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Vector Bundles ◽  
Hodge Structure ◽  
Projective Curve ◽  
Hodge Structures ◽  
Smooth Projective Curve ◽  
Stable Vector Bundles ◽  
Algebraic Vector

Let X be a smooth projective curve of genus g ≥ 2 over ℂ. Fix n ≥ 2, d ∈ ℤ. A pair (E, ϕ) over X consists of an algebraic vector bundle E of rank n and degree d over X and a section ϕ ∈ H0(E). There is a concept of stability for pairs which depends on a real parameter τ. Let [Formula: see text] be the moduli space of τ-semistable pairs of rank n and degree d over X. Here we prove that the cohomology groups of [Formula: see text] are Hodge structures isomorphic to direct summands of tensor products of the Hodge structure H1(X). This implies a similar result for the moduli spaces of stable vector bundles over X.

scholarly journals Tevelev degrees and Hurwitz moduli spaces

2021 ◽  
pp. 1-32
Author(s):  
A. CELA ◽  
R. PANDHARIPANDE ◽  
J. SCHMITT
Keyword(s):  
Scattering Amplitudes ◽  
Exact Solutions ◽  
Projective Geometry ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Intersection Theory ◽  
Dyck Paths ◽  
Simple Recursion ◽  
Almost All ◽  
Two Parameter

Abstract We interpret the degrees which arise in Tevelev’s study of scattering amplitudes in terms of moduli spaces of Hurwitz covers. Via excess intersection theory, the boundary geometry of the Hurwitz moduli space yields a simple recursion for the Tevelev degrees (together with their natural two parameter generalisation). We find exact solutions which specialise to Tevelev’s formula in his cases and connect to the projective geometry of lines and Castelnuovo’s classical count of $g^1_d$ ’s in other cases. For almost all values, the calculation of the two parameter generalisation of the Tevelev degree is new. A related count of refined Dyck paths is solved along the way.

scholarly journals On the Voevodsky motive of the moduli space of Higgs bundles on a curve

2021 ◽  
Vol 27 (1) ◽  
Author(s):  
Victoria Hoskins ◽  
Simon Pepin Lehalleur
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Characteristic Zero ◽  
Rational Point ◽  
Triangulated Category ◽  
Higgs Bundles ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
De Rham

AbstractWe study the motive of the moduli space of semistable Higgs bundles of coprime rank and degree on a smooth projective curve C over a field k under the assumption that C has a rational point. We show this motive is contained in the thick tensor subcategory of Voevodsky’s triangulated category of motives with rational coefficients generated by the motive of C. Moreover, over a field of characteristic zero, we prove a motivic non-abelian Hodge correspondence: the integral motives of the Higgs and de Rham moduli spaces are isomorphic.

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