This paper studies the possible Hodge groups of simple polarizable Q-Hodge structures with Hodge numbers (n, 0, . . . , 0, n). In particular, it generalizes earlier work of Ribet and MoonenZarhin to completely determine the possible Hodge groups of such Hodge structures when n is equal to 1, 4, or a prime p. In addition, the paper determines possible Hodge groups, under certain conditions on th...

This paper extends Kontsevich’s ideas on quantizing Poisson manifolds. A new differential is added to the Hodge decomposition of the Hochschild complex, so that it becomes a bicomplex, even more similar to the classical Hodge theory for complex manifolds. These notes grew out of the author’s attempt to understand Kontsevich’s ideas [Kon95a] on quantizing Poisson manifolds. We introduce a new di...

The spin can be described in the star product formalism by extending the bosonic Moyal product in the fermionic sector. The fermionic star product is then the Clifford product of geometric algebra and it is possible to formulate the fermionic star product formalism in analogy to the bosonic star product formalism. For the fermionic star product description of spin, one can then establish the re...

In this paper, we recast the fermionic ghost sector of Witten’s open bosonic string field theory in the language of noncommutative field theory. In particular, following the methods of hep-th/0202087, we find that in Siegel gauge Witten’s star product roughly corresponds to a continuous tensor product of Clifford Algebras, and we formulate important operators of the theory in this language, not...

For an operator generating a group on $$L^p$$ spaces transference results give bounds the Phillips functional calculus also known as spectral multiplier estimates. In this paper we consider specific generators which are abstraction of first order differential operators and prove similar estimates assuming only that is bounded $$L^2$$ rather than . We R-bounded Hörmander result by abstract Sobol...

The goal of this expository article is first of all to show that Hodge theory provides naturally defined subvarieties of any moduli space parameterizing smooth varieties, the “Hodge loci”, although only the Hodge conjecture would guarantee that these subvarieties are defined on a finite extension of the base field. We will show how these subsets can be studied locally in the Euclidean topology ...

A bstract We investigate a relationship between particular class of two-dimensional integrable non-linear σ -models and variations Hodge structures. Concretely, our aim is to study the classical dynamics λ -deformed G / model show that special solutions its equations motion precisely describes one-parameter variation find this obtained by identifying group-valued field -model with Weil operator...

Given a variation of Hodge structure over P with Hodge numbers (1, 1, . . . , 1), we show how to compute the degrees of the Deligne extension of its Hodge bundles, following Eskin–Kontsevich–Möller–Zorich, by using the local exponents of the corresponding Picard– Fuchs equation. This allows us to compute the Hodge numbers of Zucker’s Hodge structure on the corresponding parabolic cohomology gro...

Classical Hodge theory gives a decomposition of the complex cohomology of a compact Kähler manifold M , which carries the standard Hodge structure {H(M), p + q = k} of weight k. Deformations of M then lead to variations of the Hodge structure. This is best understood when reformulating the Hodge decomposition in an abstract manner. Let HC = HR ⊕ C be a complex vector space with a real structure...