Hodge theoretic aspects of mirror symmetry

By L.Katzarkov, M.Kontsevich, T.Pantev. 2008

310 citations recorded by the source corpus.

We discuss the Hodge theory of algebraic non-commutative spaces and analyze how this theory interacts with the Calabi-Yau condition and with mirror symmetry. We develop an abstract theory of non-commutative Hodge structures, investigate existence and variations, and propose explicit construction and classification techniques. We study the main examples of non-commutative Hodge structures coming from a symplectic or a complex geometry possibly twisted by a potential. We study the interactions of the new Hodge theoretic invariants with mirror symmetry and derive noncommutative analogues of some of the more interesting consequences of Hodge theory such as unobstructedness and the construction of canonical coordinates on moduli.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bec38ec2d65942285d9a430b7430c9f87a93d3f8

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Citations in the corpus

Family Floer cohomology and mirror symmetry (2014, background): “Much effort has gone in verifying this conjecture in certain examples [26, 27, 1], and extracting some of the classical statements of mirror symmetry from it [20, 8].”
Quantum periods for 3-dimensional Fano manifolds (2013, background): “The general outlines of such a mirror theorem are beginning to emerge [2,3,38,40,70], as are some promising approaches to proving it [24–27,42,43].”
Stability structures, motivic Donaldson-Thomas invariants and cluster transformations (2008, background): “sing the Calabi-Yau condition one expects that a triangulated compact homologically smooth A∞-category C(possibly Z/2Z-graded) admits (conjecturally) a non-commutative pure Hodge structure (see [39], [36], [42] about motivations, deﬁnitions as well as some conjectures and applications of this notion”
The structure of 2D semi-simple field theories (2007, background): “With different starting hypotheses, a vast extension of my classification has been reached by Kontsevich and collaborators in the framework of openclosed FTFTs (see [19] and sequels in preparation).”
Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants (2010, methodology): “Moreover, in [31] it was proven that any object of A0 is in fact a usual constructible sheaf shifted by [1].”
Mirror symmetry: from categories to curve counts (2015, methodology): “We will not consider the integral structure in this paper (however see [KKP08, §2.2.6] and [Iri09]).”
Quantum Cohomology and Periods (2011, methodology): “Generalizations of Hodge structure have been studied by many people and referred to in various ways: semi-infinite Hodge structure [2, 37], TERP structure [31] and noncommutative Hodge structure [40] etc.”
Matrix factorizations for nonaffine LG–models (2011, unknown): “This construction of matrix factorizations is also discussed in [5].”
The Crepant Transformation Conjecture for Toric Complete Intersections (2014, methodology): “This is a K0T (pt)-lattice in the space of flat sections for the equivariant quantum connection on X which is isomorphic to the integral equivariant K-group K0T (X): it generalizes the integral structure for non-equivariant quantum cohomology constructed by Iritani [51] and Katzarkov–Kontsevich–Pant”
Mirror Symmetry and Fano Manifolds (2012, unknown): “(1)It is natural to imagine that local systems with additional structures (realisations) subject to natural comparisons would be the object of a category of “quantum” motivic sheaves with a construction akin to [Del89, Jan90], see [KKP08].”
Derived Algebraic Geometry (2014, background): “This is the first step in the general construction of variations of noncommutative Hodge structures in the sense of [74] and is a far-reaching generalization of the noncommutative GaussManin connection constructed on flat families of algebras in [142].”
Global matrix factorizations (2011, methodology): “24 of [Orl1], we conclude that when the critical locus of W is proper, the category Inj(X, W ) is dg affine, proper, and homologically smooth as a differential Z/2Z-graded category [KatKonPan].”