scholarly journals EXTENDING MIRROR CONJECTURE TO CALABI–YAU WITH BUNDLES

1999 ◽  
Vol 01 (01) ◽  
pp. 65-70 ◽  
Author(s):  
CUMRUN VAFA
Keyword(s):  
Riemann Surfaces ◽  
Hodge Structure ◽  
Stable Bundle ◽  
Type Ii ◽  
Holomorphic Maps ◽  
Variation Of Hodge Structure ◽  
Mirror Side ◽  
Calabi Yau Manifolds

We define the notion of mirror of a Calabi–Yau manifold with a stable bundle in the context of type II strings in terms of supersymmetric cycles on the mirror. This allows us to relate the variation of Hodge structure for cohomologies arising from the bundle to the counting of holomorphic maps of Riemann surfaces with boundary on the mirror side. Moreover it opens up the possibility of studying bundles on Calabi–Yau manifolds in terms of supersymmetric cycles on the mirror.

scholarly journals Kodaira–Saito vanishing via Higgs bundles in positive characteristic

2019 ◽  
Vol 2019 (755) ◽  
pp. 293-312
Author(s):  
Donu Arapura
Keyword(s):  
Positive Characteristic ◽  
Hodge Structure ◽  
Higgs Bundles ◽  
Vanishing Theorem ◽  
Variation Of Hodge Structure ◽  
Mixed Hodge Modules ◽  
Special Case

AbstractThe goal of this paper is to give a new proof of a special case of the Kodaira–Saito vanishing theorem for a variation of Hodge structure on the complement of a divisor with normal crossings. The proof does not use the theory of mixed Hodge modules, but instead reduces it to a more general vanishing theorem for semistable nilpotent Higgs bundles, which is then proved by using some facts about Higgs bundles in positive characteristic.

TOPOLOGICAL σ MODELS AND LARGE N MATRIX INTEGRAL

1995 ◽  
Vol 10 (29) ◽  
pp. 4203-4224 ◽  
Author(s):  
TOHRU EGUCHI ◽  
KENTARO HORI ◽  
SUNG-KIL YANG
Keyword(s):  
Matrix Model ◽  
Riemann Surfaces ◽  
Holomorphic Maps ◽  
Integrable Structure ◽  
Toda Hierarchy ◽  
Matrix Integral ◽  
Large N ◽  
The Matrix ◽  
Lax Operator ◽  
The One

In this paper we describe in some detail the representation of the topological CP1 model in terms of a matrix integral which we have introduced in a previous article. We first discuss the integrable structure of the CP1 model and show that it is governed by an extension of the one-dimensional Toda hierarchy. We then introduce a matrix model which reproduces the sum over holomorphic maps from arbitrary Riemann surfaces onto CP1. We compute intersection numbers on the moduli space of curves using a geometrical method and show that the results agree with those predicted by the matrix model. We also develop a Landau-Ginzburg (LG) description of the CP1 model using a superpotential eX + et0,Q e-X given by the Lax operator of the Toda hierarchy (X is the LG field and t0,Q is the coupling constant of the Kähler class). The form of the superpotential indicates the close connection between CP1 and N=2 supersymmetric sine-Gordon theory which was noted sometime ago by several authors. We also discuss possible generalizations of our construction to other manifolds and present an LG formulation of the topological CP2 model.

scholarly journals Lowest weights in cohomology of variations of Hodge structure

2012 ◽  
Vol 206 ◽  
pp. 1-24
Author(s):  
Chris Peters ◽  
Morihiko Saito
Keyword(s):  
Open Subset ◽  
Hodge Structure ◽  
Analytic Space ◽  
Mixed Hodge Structure ◽  
Zariski Open Subset ◽  
Weight Part ◽  
Variation Of Hodge Structure ◽  
Local Monodromy ◽  
Variations Of Hodge Structure ◽  
Complex Analytic Space

AbstractLetXbe an irreducible complex analytic space withj:U ↪ Xan immersion of a smooth Zariski-open subset, and let 𝕍 be a variation of Hodge structure of weightnoverU. Assume thatXis compact Kähler. Then, provided that the local monodromy operators at infinity are quasi-unipotent,IHk(X, 𝕍) is known to carry a pure Hodge structure of weightk+n, whileHk(U, 𝕍) carries a mixed Hodge structure of weight at leastk+n. In this note it is shown that the image of the natural mapIHk(X, 𝕍) →Hk(U, 𝕍) is the lowest-weight part of this mixed Hodge structure. In the algebraic case this easily follows from the formalism of mixed sheaves, but the analytic case is rather complicated, in particular when the complementX — Uis not a hypersurface.

scholarly journals THE TOPOLOGICAL CP1 MODEL AND THE LARGE-N MATRIX INTEGRAL

1994 ◽  
Vol 09 (31) ◽  
pp. 2893-2902 ◽  
Author(s):  
TOHRU EGUCHI ◽  
SUNG-KIL YANG
Keyword(s):  
Partition Function ◽  
Matrix Model ◽  
Riemann Surfaces ◽  
Hermitian Matrix ◽  
Coupling Constants ◽  
Holomorphic Maps ◽  
Matrix Integral ◽  
Large N

We discuss the topological CP 1 model which consists of the holomorphic maps from Riemann surfaces onto CP 1. We construct a large-N matrix model which reproduces precisely the partition function of the CP 1 model at all genera of Riemann surfaces. The action of our matrix model has the form [Formula: see text] where M is an N × N Hermitian matrix and tn,P(tn,Q), (n = 0, 1, 2, …) are the coupling constants of the nth descendant of the puncture (Kähler) operator.

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