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Jaesuk Park - One of the best experts on this subject based on the ideXlab platform.
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Enhanced homotopy theory for period integrals of smooth projective hypersurfaces
'International Press of Boston', 2018Co-Authors: Jaesuk Park, Park J.Abstract:The goal of this paper is to reveal hidden structures on the singular cohomology and the Griffiths period integral of a smooth projective hypersurface in terms of BV(Batalin-Vilkovisky) algebras and homotopy Lie theory (so called, L∞-homotopy theory). Let XG be a smooth projective hypersurface in the complex projective space Pn defined by a homogeneous polynomial G(x) of degree d ≥ 1. Let H = Hn-1 prim(XG, C) be the middle dimensional primitive cohomology of XG. We explicitly construct a BV algebra BVX = (AX,QX,KX) such that its 0-th cohomology H0 K X (AX) is Canonically isomorphic to H. We also equip BVX with a decreasing filtration and a bilinear pairing which realize the Hodge filtration and the cup product polarization on H under the Canonical Isomorphism. Moreover, we lift C[γ]: H → C to a cochain map Cγ: (AX, KX) → (C, 0), where C[γ] is the Griffiths period integral given by ω → ∫γ ω for [γ] ε Hn-1(XG, Z). We use this enhanced homotopy structure on H to study an extended formal deformation of XG and the correlation of its period integrals. If XG is in a formal family of Calabi-Yau hypersurfaces XGT, we provide an explicit formula and algorithm (based on a Gröbner basis) to compute the period matrix of XGT in terms of the period matrix of XG and an L∞-morphism K which enhances C[γ] and governs deformations of period matrices.1321sciescopu
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deformations of coisotropic submanifolds and strong homotopy lie algebroids
Inventiones Mathematicae, 2005Co-Authors: Jaesuk ParkAbstract:In this paper, we study deformations of coisotropic submanifolds in a symplectic manifold. First we derive the equation that governs C∞ deformations of coisotropic submanifolds and define the corresponding C∞-moduli space of coisotropic submanifolds modulo the Hamiltonian isotopies. This is a non-commutative and non-linear generalization of the well-known description of the local deformation space of Lagrangian submanifolds as the set of graphs of closed one forms in the Darboux-Weinstein chart of a given Lagrangian submanifold. We then introduce the notion of strong homotopy Lie algebroid (or L∞-algebroid) and associate a Canonical Isomorphism class of strong homotopy Lie algebroids to each pre-symplectic manifold (Y,ω) and identify the formal deformation space of coisotropic embeddings into a symplectic manifold in terms of this strong homotopy Lie algebroid. The formal moduli space then is provided by the gauge equivalence classes of solutions of a version of the Maurer-Cartan equation (or the master equation) of the strong homotopy Lie algebroid, and plays the role of the classical part of the moduli space of quantum deformation space of coisotropic A-branes. We provide a criterion for the unobstructedness of the deformation problem and analyze a family of examples that illustrates that this deformation problem is obstructed in general and heavily depends on the geometry and dynamics of the null foliation.
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deformations of coisotropic submanifolds and strong homotopy lie algebroids
arXiv: Symplectic Geometry, 2003Co-Authors: Jaesuk ParkAbstract:In this paper, we study deformations of coisotropic submanifolds in a symplectic manifold. First we derive the equation that governs $C^\infty$ deformations of coisotropic submanifolds and define the corresponding $C^\infty$-moduli space of coisotropic submanifolds modulo the Hamiltonian isotopies. This is a non-commutative and non-linear generalization of the well-known description of the local deformation space of Lagrangian submanifolds as the set of graphs of {\it closed} one forms in the Darboux-Weinstein chart of a given Lagrangian submanifold. We then introduce the notion of {\it strong homotopy Lie algebroid} (or {\it $L_\infty$-algebroid}) and associate a Canonical Isomorphism class of strong homotopy Lie algebroids to each pre-symplectic manifold $(Y,\omega)$ and identify the formal deformation space of coisotropic embeddings into a symplectic manifold in terms of this strong homotopy Lie algebroid. The formal moduli space then is provided by the gauge equivalence classes of solutions of a version of the {\it Maurer-Cartan equation} (or the {\it master equation}) of the strong homotopy Lie algebroid, and plays the role of the classical part of the moduli space of quantum deformation space of coisotropic $A$-branes. We provide a criterion for the unobstructedness of the deformation problem and analyze a family of examples that illustrates that this deformation problem is obstructed in general and heavily depends on the geometry and dynamics of the null foliation.
Valentin Ovsienko - One of the best experts on this subject based on the ideXlab platform.
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differential operators on supercircle conformally equivariant quantization and symbol calculus
Letters in Mathematical Physics, 2007Co-Authors: Hichem Gargoubi, Najla Mellouli, Valentin OvsienkoAbstract:We consider the supercircle S 1|1 equipped with the standard contact structure. The Lie superalgebra K (1) of contact vector fields contains the M¨ obius superalgebra osp(1|2). We study the space of linear differential operators on weighted densities as a module over osp(1|2). We introduce the Canonical Isomorphism between this space and the corresponding space of symbols and find all cases where such an Isomorphism does not exist.
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differential operators on supercircle conformally equivariant quantization and symbol calculus
arXiv: Mathematical Physics, 2006Co-Authors: Hichem Gargoubi, Najla Mellouli, Valentin OvsienkoAbstract:We consider the supercircle $S^{1|1}$ equipped with the standard contact structure. The conformal Lie superalgebra K(1) acts on $S^{1|1}$ as the Lie superalgebra of contact vector fields; it contains the M\"obius superalgebra $osp(1|2)$. We study the space of linear differential operators on weighted densities as a module over $osp(1|2)$. We introduce the Canonical Isomorphism between this space and the corresponding space of symbols and find interesting resonant cases where such an Isomorphism does not exist.
Hichem Gargoubi - One of the best experts on this subject based on the ideXlab platform.
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differential operators on supercircle conformally equivariant quantization and symbol calculus
Letters in Mathematical Physics, 2007Co-Authors: Hichem Gargoubi, Najla Mellouli, Valentin OvsienkoAbstract:We consider the supercircle S 1|1 equipped with the standard contact structure. The Lie superalgebra K (1) of contact vector fields contains the M¨ obius superalgebra osp(1|2). We study the space of linear differential operators on weighted densities as a module over osp(1|2). We introduce the Canonical Isomorphism between this space and the corresponding space of symbols and find all cases where such an Isomorphism does not exist.
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differential operators on supercircle conformally equivariant quantization and symbol calculus
arXiv: Mathematical Physics, 2006Co-Authors: Hichem Gargoubi, Najla Mellouli, Valentin OvsienkoAbstract:We consider the supercircle $S^{1|1}$ equipped with the standard contact structure. The conformal Lie superalgebra K(1) acts on $S^{1|1}$ as the Lie superalgebra of contact vector fields; it contains the M\"obius superalgebra $osp(1|2)$. We study the space of linear differential operators on weighted densities as a module over $osp(1|2)$. We introduce the Canonical Isomorphism between this space and the corresponding space of symbols and find interesting resonant cases where such an Isomorphism does not exist.
Najla Mellouli - One of the best experts on this subject based on the ideXlab platform.
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differential operators on supercircle conformally equivariant quantization and symbol calculus
Letters in Mathematical Physics, 2007Co-Authors: Hichem Gargoubi, Najla Mellouli, Valentin OvsienkoAbstract:We consider the supercircle S 1|1 equipped with the standard contact structure. The Lie superalgebra K (1) of contact vector fields contains the M¨ obius superalgebra osp(1|2). We study the space of linear differential operators on weighted densities as a module over osp(1|2). We introduce the Canonical Isomorphism between this space and the corresponding space of symbols and find all cases where such an Isomorphism does not exist.
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differential operators on supercircle conformally equivariant quantization and symbol calculus
arXiv: Mathematical Physics, 2006Co-Authors: Hichem Gargoubi, Najla Mellouli, Valentin OvsienkoAbstract:We consider the supercircle $S^{1|1}$ equipped with the standard contact structure. The conformal Lie superalgebra K(1) acts on $S^{1|1}$ as the Lie superalgebra of contact vector fields; it contains the M\"obius superalgebra $osp(1|2)$. We study the space of linear differential operators on weighted densities as a module over $osp(1|2)$. We introduce the Canonical Isomorphism between this space and the corresponding space of symbols and find interesting resonant cases where such an Isomorphism does not exist.
Julian Rosen - One of the best experts on this subject based on the ideXlab platform.
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a choice free absolute galois group and artin motives
arXiv: Number Theory, 2017Co-Authors: Julian RosenAbstract:Proofs that an arbitrary field has a separable closure are necessarily non-constructive, and separable closures are unique only up to non-Canonical Isomorphism. This means that the absolute Galois group of a field is defined only up to inner automorphism. Here we construct a profinite algebraic group which is an inner form of the absolute Galois group. Our construction uses no form of the axiom of choice, and the group is defined up to Canonical Isomorphism. We also show that the Frobenius associated with a prime of a number field unramified in an extension, which is classically defined only up to conjugation, has a uniquely-defined analogue in terms of our group. We give a construction of the category of Artin motives with coefficients in an arbitrary field, and we give an interpretation of our absolute Galois group in terms of this category.