# Real structure in the mixed Hodge structure associated to an isolated singularity

We know that a mixed Hodge structure on a complex vector space $H$ with an integral lattice $H_{\mathbb Z}$ consists of the weight filtration and the Hodge filtration. For an isolated hypersurface singularity represented by a function germ $f: ({\mathbb C}^n, 0) \to ({\mathbb C}, 0)$, we have the vanishing cohomology bundle

$${\mathscr H}^* = \bigcup_{t \in {\mathbb D}^*} H^{n-1}(V_t; {\mathbb C}) \to {\mathbb D}^*.$$ Here $V_t = f^{-1}(t)$. A mixed Hodge structure can be defined on the fibres of ${\mathscr H}^*$. Let $\{{\mathscr F}^p\}_{p=0}^n$ be the Hodge filtration. The integral lattice is spanned by the parallel integral cocycles with respect to the Gauss-Manin connection.

A simple case is that when $f$ is a quasi-homogeneous polynomial. Suppose $d_1,\ldots, d_n, D$ are integers such that $$f( t^{d_1} z_1, \ldots, t^{d_n} z_n) = t^D f(z_1, \ldots, z_n).$$ Denote $\nu_i = d_i / D$. Then for the holomorphic $n$-form $\omega = z_1^{k_1}\cdots z_n^{k_n} dz_1 \wedge \cdots \wedge dz_n$, we have the so-called geometric sections'' of the cohomology bundle, whose value at $t$ is the residue form

$$\big[\frac{\omega}{df}\big]|_{V_t}.$$

This section belongs to ${\mathscr F}^p$ if and only if the charge of $\omega$, given by $l = \sum_{i=1}^n (k_i+1) \nu_i$, is less than or equal to $n-p$. (One can refer to the book Singularity I'' of Arnold etc.)

Here is the question: for quasi-homogeneous singularity, what is the complex conjugation of a geometric section (which can be given explicitly)? In the paper "${\mathcal N}=2$ Landau-Ginzburg vs. Calabi-Yau $\sigma$-models: non-perturbative aspects" by S. Cecotti, there is a physics description of this complex conjugation on page 1766 which I don't quite understand (especially the term of spectral flow''). Is there any mathematical literature which contains such discussion?