On the structure of groups defined by Kim and Manturov
We study the structure of a series of groups $Γ_n^4$ defined by Kim and Manturov. We show that the groups are finite for all $n \ge 6$ and in fact they are 2-step nilpotent $2$-groups.
math.GR↗
arXiv subjects
Yuuki Tadokoro
Publications and source records attributed to Yuuki Tadokoro.
Minimal generating sets of groups of Kim-Manturov
We consider a series of groups defined by Kim and Manturov. These groups have their background in triangulations of a surface and configurations of points, lines or circles on the surface. They are expected to have relationships to many geometric objects. In this paper, we give a minimal generating set of the group and determine the abelianization. We also introduce some related groups which might be helpful to understand the structure of the original groups.
math.GT↗
Nonlinear $O(3)$ sigma model in discrete complex analysis
We present a discrete version of the two-dimensional nonlinear $O(3)$ sigma model examined by Belavin and Polyakov. We formulate it by means of Mercat's discrete complex analysis and its elaboration by Bobenko and Günther. We define a weighted discrete Dirichlet energy and area on a planar quad-graph and derive an inequality between them. We write $f$ for the complex function obtained from the unit vector field of the model. The inequality is saturated if and only if the $f$ is discrete (anti-)holomorphic. By using a weight $W$ obtained from a kind of tiling of the sphere $S^2$, the weighted discrete area ${\cal A}^{W}_{\diamondsuit}(f)$ admits a geometrical interpretation, namely, ${\cal A}^{W}_{\diamondsuit}(f)=4 πN $ for a topological quantum number $N \in π_2(S^2)$. This ensures the topological stability of the solution described by the $f$, and we have the quantized energy $E^{W}_{\diamondsuit}(f)=|{\cal A}^{W}_{\diamondsuit}(f)|=4 π|N| $. For quad-graphs with orthogonal diagonals, we show that the discrete (anti-)holomorphic function $f$ satisfies the Euler--Lagrange equation derived from the weighted discrete Dirichlet energy. On some rhombic lattices, the discrete power functions $z^{(N)}$ give the topological quantum number $N$. Moreover, the weighted discrete Dirichlet energy, area, and Euler--Lagrange equation tend to their continuous forms as the lattice spacings tend to zero.
hep-th↗
Pointed harmonic volume and its relation to the extended Johnson homomorphism
The period for a compact Riemann surface, defined by the integral of differential 1-forms, is a classical complex analytic invariant, strongly related to the complex structure of the surface. In this paper, we treat another complex analytic invariant called the pointed harmonic volume. As a natural extension of the period defined using Chen's iterated integrals, it captures more detailed information of the complex structure. It is also one of a few explicitly computable examples of complex analytic invariants. We obtain its new value for a certain pointed hyperelliptic curve. An application of the pointed harmonic volume is presented. We explain the relationship between the harmonic volume and first extended Johnson homomorphism on the mapping class group of a pointed oriented closed surface.
math.GT↗
Harmonic volume and its applications
The period is a classical complex analytic invariant for a compact Riemann surface defined by integration of differential 1-forms. It has a strong relationship with the complex structure of the surface. In this chapter, we review another complex analytic invariant called the harmonic volume. It is a natural extension of the period defined using Chen's iterated integrals and captures more detailed information of the complex structure. It is also one of a few explicitly computable examples of complex analytic invariants. As an application, we give an algorithm in proving nontriviality for a class of homologically trivial algebraic cycles obtained from special compact Riemann surfaces. The moduli space of compact Riemann surfaces is the space of all biholomorphism classes of compact Riemann surfaces. The harmonic volume can be regarded as an analytic section of a local system on the moduli space. It enables a quantitative study of the local structure of the moduli space. We explain basic concepts related to the harmonic volume and its applications of the moduli space.
math.GT↗
The period matrix of the hyperelliptic curve $w^2=z^{2g+1}-1$
A geometric algorithm is introduced for finding a symplectic basis of the first integral homology group of a compact Riemann surface, which is a $p$-cyclic covering of ${\mathbb C} P^1$ branched over 3 points. The algorithm yields a previously unknown symplectic basis of the hyperelliptic curve defined by the affine equation $w^2=z^{2g+1}-1$ for genus $g\geq 2$. We then explicitly obtain the period matrix of this curve, its entries being elements of the $(2g+1)$-st cyclotomic field. In the proof, the details of our algorithm play no significant role.
math.AG↗
Nontrivial algebraic cycles in the Jacobian varieties of some quotients of Fermat curves
We obtain the trace map image of the values of certain harmonic volumes for some quotients of Fermat curves. This provides the algorithm that the algebraic cycles called by the k-th Ceresa cycles are not algebraically equivalent to zero in the Jacobian varieties.
math.AG↗
A nontrivial algebraic cycle in the Jacobian variety of the Fermat sextic
We compute some value of the harmonic volume for the Fermat sextic. Using this computation, we prove that some special algebraic cycle in the Jacobian variety of the Fermat sextic is not algebraically equivalent to zero.
math.AG↗
The pointed harmonic volumes of hyperelliptic curves with Weierstrass base points
We give a explicit computation of the pointed harmonic volumes of hyperelliptic curves with Weierstrass base points, which are paraphrased into a combinatorial formula.
math.GT↗
The Harmonic Volumes of Hyperelliptic Curves
We determine the harmonic volumes for all the hyperelliptic curves. This gives a geometric interpretation of a theorem established by A. Tanaka.
math.GT↗
A nontrivial algebraic cycle in the Jacobian variety of the Klein quartic
We prove some value of the harmonic volume for the Klein quartic $C$ is nonzero modulo ${1/2}\{mathbb Z}$, using special values of the generalized hypergeometric function ${}_3F_2$. This result tells us the algebraic cycle $C-C^-$ is not algebraically equivalent to zero in the Jacobian variety $J(C)$.
math.AG↗