arXiv subjects

Ivan Horozov

Publications and source records attributed to Ivan Horozov.

24 records · Page 2Linked to original sources

arXiv2012

Parallel Transport on Higher Loop Spaces

We construct a parallel transport on higher loop spaces of a manifold in term of a higher dimensional generalization of iterated path integrals. Under mild assumptions, we define a de Rham complex on higher loop spaces and we recover a known result of Hain of a de Rham structure on higher homotopy groups of a manifold. The key ingredient is a new definition of iterated integrals on membranes, which also have applications in number theory, algebraic geometry and mathematical physics.

math.AT↗

arXiv2012

Iterated integrals over higher dimensional loops

We give a definition of higher dimensional iterated integrals based on integration over membranes. We prove basic properties of this definition and formulate a conjecture which extends Chen's de Rham Theorem for iterated integrals to the membrane case.

math.DG↗

arXiv2010

A refinement of the Parshin symbol for surfaces

On an algebraic curve there are Tate symbols, which satisfy Weil reciprocity law. The analogues in higher dimensions are the Parshin symbols, which satisfy Kato-Parshin reciprocity laws. We give a refinement of the Parshin symbol for surfaces, using iterated integrals in the sense of Chen. The product of the refined symbol over the cyclic permutations of the functions recovers the Parshin symbol. Also, we construct a logarithmic version of the Parshin symbol. We prove reciprocity laws for both the refined symbol and a logarithm of the Parshin symbol.

math.AG↗

arXiv2010

Non-abelian reciprocity laws on a Riemann surface

On a Riemann surface there are relations among the periods of holomorphic differential forms, called Riemann's relations. If one looks carefully in Riemann's proof, one notices that he uses iterated integrals. What I have done in this paper is to generalize these relations to relations among generating series of iterated integrals. Since the main result is formulated in terms of generating series, it gives infinitely many relations - one for each coefficient of the generating series. The lower order terms give the well known classical relations. The new result is reciprocity for the higher degree terms, which give non-trivial relations among iterated integrals on a Riemann surface. As an application we refine the definition of Manin's noncommutative modular symbol in order to include Eisenstein series. Finally, we have to point out that this paper contains some constructions needed for multidimensional reciprocity laws like a refinement of one of the Kato-Parshin reciprocity laws.

math.AG↗

arXiv2006

Non-commutative Two Dimensional Modular Symbol

We define iteration over a two dimensional manifold as analog of iteration over a path defined by Chen. We give several applications. Some of them include constructions of non-abelian modular symbol for $SL(3,\Z)$ and for $SL_{2/K}$, where $K$ is a real quadratic field. We construct the latter as generating series of iterated integrals over a two-dimensional manifold of Hilbert modular forms. We give a motivic interpretation of this non-abelian modular symbol. Other applications give motivic interpretations of certain iterated completed Dedekind zeta functions.

math.NT↗

arXiv2006

Cohomology of GL(4,Z) with Non-trivial Coefficients

In this paper we compute the cohomology groups of GL(4,Z) with coefficients in symmetric powers of the standard representation twisted by the determinant. This problem arises in Goncharov's approach to the study of motivic multiple zeta values of depth 4. The techniques that we use include Kostant's formula for cohomology groups of nilpotent Lie subalgebras of a reductive Lie algebra, Borel-Serre compactification, a result of Harder on Eisenstein cohomology. Finally, we need to show that the ghost class, which is present in the cohomology of the boundary of the Borel-Serre compactification, disappears in the Eisenstein cohomology of GL(4,Z). For this we use a computationally effective version for the homological Euler characteristic of GL(4,Z) with non-trivial coefficients.

math.NT↗