Target space duality of Calabi-Yau spaces with two moduli (Q1902445)

The mathematical content of this paper is a computation of the fundamental group of \(\mathbb{P}^2 \backslash\) (critical locus of the family \(W)\), \(W\) the family of quintic threefolds over \(\mathbb{P}^2\): \[ c(y^5_1 + \cdots + y^5_5) - 5y^2_4 y^2_5 (ay_4 + by_5) = 0 \] \((a,b, c\) homogeneous coordinates on \(\mathbb{P}^2\), \(y_i\) homogeneous coordinates on \(\mathbb{P}^5)\) and its monodromy representation on a certain three-dimensional space of periods (of a holomorphic 3-form). The result is to identify the fundamental group \(\Gamma\) with the braid group \(B_5 \), this is obtained by an explicit computation of the critical locus and its singularities, thus the generators correspond to the set of critical points of a Lefschetz pencil, and the relations are found by van Kampen's theorem [\textit{E. R. van Kampen}, Am. J. Math. 55, 255-260 (1933; Zbl 0006.41502)]. The monodromy representation is computed by an suitable explicit description of the Gauß-Manin connection. It leads to a projective \(U(1,2)\)-representation of \(\Gamma\). The introduction contains furthermore comments about the relation of this result to supersymmetric conformal field theory.