Yang-Yang functions, monodromy and knot polynomials

We derive a structure of ℤ[t, t−1]-module bundle from a family of Yang-Yang functions. For the fundamental representation of the complex simple Lie algebra of classical type, we give explicit wall-crossing formula and prove that the monodromy representation of the ℤ[t, t−1]-module bundle is equivalent to the braid group representation induced by the universal R-matrices of Uh(g). We show that two transformations induced on the fiber by the symmetry breaking deformation and respectively the rotation of two complex parameters commute with each other.

The structure of a local system associated with a hypergeometric system of rank 9

We study a local system associated with a system [Formula: see text] of hypergeometric differential equations in two variables of rank [Formula: see text] with seven parameters [Formula: see text] and [Formula: see text]. We modify the fundamental system of solutions to [Formula: see text] given in [A system of hypergeometric differential equations in two variables of rank 9, Internat. J. Math. 28 (2017), 1750015, 34 pp] so that it is valid even in cases where [Formula: see text] satisfy some integral conditions. By using this fundamental system, we show the irreducibility of the monodromy representation of [Formula: see text] under some conditions on the parameters. We characterize the fundamental group of the base space of this local system as the group generated by three loops with four relations among them.

Analytic continuation of holonomy germs of Riccati foliations along Brownian paths

Consider a Riccati foliation whose monodromy representation is non-elementary and parabolic and consider a non-invariant section of the fibration whose associated developing map is onto. We prove that any holonomy germ from any non-invariant fibre to the section can be analytically continued along a generic Brownian path. To prove this theorem, we prove a dual result about complex projective structures. Let $\unicode[STIX]{x1D6F4}$ be a hyperbolic Riemann surface of finite type endowed with a branched complex projective structure: such a structure gives rise to a non-constant holomorphic map ${\mathcal{D}}:\tilde{\unicode[STIX]{x1D6F4}}\rightarrow \mathbb{C}\mathbb{P}^{1}$, from the universal cover of $\unicode[STIX]{x1D6F4}$ to the Riemann sphere $\mathbb{C}\mathbb{P}^{1}$, which is $\unicode[STIX]{x1D70C}$-equivariant for a morphism $\unicode[STIX]{x1D70C}:\unicode[STIX]{x1D70B}_{1}(\unicode[STIX]{x1D6F4})\rightarrow \mathit{PSL}(2,\mathbb{C})$. The dual result is the following. If the monodromy representation $\unicode[STIX]{x1D70C}$ is parabolic and non-elementary and if ${\mathcal{D}}$ is onto, then, for almost every Brownian path $\unicode[STIX]{x1D714}$ in $\tilde{\unicode[STIX]{x1D6F4}}$, ${\mathcal{D}}(\unicode[STIX]{x1D714}(t))$ does not have limit when $t$ goes to $\infty$. If, moreover, the projective structure is of parabolic type, we also prove that, although ${\mathcal{D}}(\unicode[STIX]{x1D714}(t))$ does not converge, it converges in the Cesàro sense.

The monodromy representation and twisted period relations for Appell’s hypergeometric function F 4

We consider the systemF4(a, b, c)of differential equations annihilating Appell's hypergeometric seriesF4(a,b,c;x). We find the integral representations for four linearly independent solutions expressed by the hypergeometric seriesF4. By using the intersection forms of twisted (co)homology groups associated with them, we provide the monodromy representation ofF4(a, b, c)and the twisted period relations for the fundamental systems of solutions ofF4.

The monodromy representation and twisted period relations for Appell’s hypergeometric function F4

We consider the system F4 (a, b, c) of differential equations annihilating Appell's hypergeometric series F4(a,b,c;x). We find the integral representations for four linearly independent solutions expressed by the hypergeometric series F4. By using the intersection forms of twisted (co)homology groups associated with them, we provide the monodromy representation of F4(a, b, c) and the twisted period relations for the fundamental systems of solutions of F4.

Invariants of 3-manifolds derived from covering presentations

By a covering presentation of a 3-manifold, we mean a labelled link (i.e., a link with a monodromy representation), which presents the 3-manifold as the simple 4-fold covering space of the 3-sphere branched along the link with the given monodromy. It is known that two labelled links present a homeomorphic 3-manifold if and only if they are related by a finite sequence of some local moves. This paper presents a method for constructing topological invariants of 3-manifolds based on their covering presentations. The proof of the topological invariance is shown by verifying the invariance under the local moves. As an example of such invariants, we present the Dijkgraaf–Witten invariant of 3-manifolds.