2.1 Conventions

We work with schemes and stacks over the complex numbers. All schemes are Noetherian and separated. By linear algebraic group we mean a closed subgroup of $\operatorname {GL}(m\,;{\mathbb C})$ for some m.

2.2 Notation

We list here notation that occurs throughout the entire paper.

Remark 1 (zero loci)

We add the subscript ${\mathbb P}({\pmb w})$ when we refer to the zero locus in ${\mathbb P}({\pmb w})$ of a polynomial f which is ${\pmb w}$-weighted homogeneous. In this way, we have

$$\begin{align*}Z_{{\mathbb P}({\pmb w})}(f)=[U/{\mathbb G}_m],\qquad \text{ with } U=Z(f)\subset {\mathbb C}^n\setminus \pmb 0. \end{align*}$$

3.1 Nondegenerate polynomials

We consider quasi-homogeneous polynomials P of positive degree d and of positive weights $w_0,\dots ,w_n$ satisfying

$$ \begin{align*} P(\lambda^{w_0}x_0,\dots,\lambda^{w_n}x_n)=\lambda^dP(x_0,\dots,x_n), \end{align*} $$

for all $\lambda \in {\mathbb C}$. We assume that the polynomial P is nondegenerate; i.e., the choice of weights and degree is unique and the partial derivatives of W vanish simultaneously only at the origin. We consider the zero locus

$$ \begin{align*} \Sigma_P=Z_{{\mathbb P}({\pmb w})}(P)\subset {\mathbb P}({\pmb w}) \end{align*} $$

which is, by nondegeneracy, a smooth hypersurface within the weighted projective stack ${\mathbb P}({\pmb w})= [({\mathbb C}^{n+1}\setminus \pmb 0)/{\mathbb G}_m]$ with ${\mathbb G}_m$ acting with weights $w_0,\dots ,w_n$. The polynomial is of Calabi-Yau type if

(2)$$ \begin{align} \sum_{i=0}^n{w_i}=d. \end{align} $$

This implies that the canonical bundle of ${\Sigma _P}$ is trivial; we refer to $\Sigma _P$ as a Calabi–Yau orbifold.

Because P is nondegenerate, the group of its diagonal automorphisms

$$ \begin{align*} \operatorname{Aut}_P=\{\mathrm {diag}(\alpha_0,\dots,\alpha_n)\mid P(\alpha_0x_0,\dots, \alpha_nx_n)=P(x_0,\dots,x_n)\} \end{align*} $$

is finite. Indeed, the $h\times ({n+1})$ exponent matrix $E=(m_{i,j})$ defined by the condition $P=\sum _{i=1}^h c_i \prod _{j=0}^n x_j^{m_{i,j}}$ is left invertible as a consequence of the uniqueness of the vector $\left ({w_i}/{d}\right )_{i=0}^n=E^{-1}{\pmb 1}.$

Since we are working over ${\mathbb C}$, we adopt the notation

$$ \begin{align*} [a_0,\dots,a_n]=\mathrm {diag}(\exp(2\pi\mathtt{i} a_i))_{i=0}^n \end{align*} $$

for $a_i\in \mathbb Q\cap [0,1[$. The age of the diagonal matrix above is

(3)$$ \begin{align} \mathrm{age}[a_0,\dots,a_n]=\sum_{i=0}^na_i. \end{align} $$

The distinguished diagonal symmetry

$$ \begin{align*} j_P=\left[\frac{w_0}{d},\dots,\frac{w_n}{d}\right], \end{align*} $$

usually denoted by j, spans the intersection $\operatorname {Aut}_P\cap \,{\mathbb G}_m$, where ${\mathbb G}_m$ is the group of automorphisms of the form $\mathrm {diag}(\lambda ^{w_0},\dots ,\lambda ^{w_n})$. The group element $j_P$ has a natural interpretation as a monodromy operator of the fibration defined by P restricted to the complement in ${\mathbb C}^{n+1}$ of the zero locus $Z(P)$; see, for instance, [Reference Dimca17]. We will denote by $M_P$ the generic Milnor fibre

(4)

For any subgroup H of $\operatorname {Aut}_P$ containing $j_P$, we consider the Deligne–Mumford stacks

$$ \begin{align*} \Sigma_{P,H}=[\Sigma_P/H_0],\qquad M_{P,H}=[M_P/H], \end{align*} $$

where $H_0=H/(H\cap {\mathbb G}_m)=H/\langle j\rangle $ and acts faithfully on $\Sigma _P$. The orbifold $\Sigma _{P,H}$ is a smooth codimension-$1$ substack of $[{\mathbb P}({\pmb w})/H_0]$

$$ \begin{align*} \Sigma_{P,H}\subset [{\mathbb P}({\pmb w})/H_0], \end{align*} $$

and has trivial canonical bundle if P is Calabi–Yau in the sense of equation (2) and H lies in

$$ \begin{align*} \operatorname{SL}_P:=\operatorname{Aut}_P\cap \operatorname{SL}(n+1;{\mathbb C}). \end{align*} $$

3.2 Polynomials with automorphism

We focus on polynomials of Calabi–Yau type of the form

$$ \begin{align*} W(x_0,x_1,\dots,x_n)=(x_0)^k+f(x_1,\dots,x_n). \end{align*} $$

We have $\operatorname {Aut}_W={\pmb \mu }_k\times \operatorname {Aut}_f$, where, using again the choice $\exp (2\pi i/k)$, the first factor is regarded here as ${\mathbb Z}/k$, canonically generated by the order-k automorphism

(5)$$ \begin{align} s=\left[\frac1k,0,\dots,0\right]. \end{align} $$

The second factor is regarded as the subgroup of $\operatorname {Aut}_W$ formed by symmetries fixing the first coordinate. We have $j_W=s\cdot j_f$, with $s\in {\mathbb Z}/k$ and $j_f\in \operatorname {Aut}_f$. Notice that the group ${\mathbb Z}/k$ acts on the stack $\Sigma _{W,H}$

$$ \begin{align*} \mathsf{m}\colon ({\mathbb Z}/k)\times \Sigma_{W,H}\to \Sigma_{W,H}. \end{align*} $$

Instead of considering all groups $H\subseteq \operatorname {Aut} W\cap \operatorname {SL}(n+1;{\mathbb C})$ containing $j_W$, we can equivalently consider their intersections with $\operatorname {Aut}_f$. In this way, since the first coordinate of $(j_W)^h$ equals $1$ if and only if $h\in k{\mathbb Z}$, we obtain all the subgroups $K \subset \operatorname {Aut}_f$ satisfying

$$ \begin{align*} (j_f)^k\in K\subseteq \operatorname{SL}_f. \end{align*} $$

We recover the previous subgroups H of $\operatorname {Aut} W\cap \operatorname {SL}(n+1;{\mathbb C})$ as the subgroups of $\operatorname {Aut}_W$ spanned by $j_W$ and K.

Our mirror duality requires a slightly more general class of groups. Therefore, we consider the subgroup of $\operatorname {Aut}_W$

$$ \begin{align*} K[j_W,s]=\sum_{a,b=0}^{k-1} (j_W)^{a}(s)^{b}K, \end{align*} $$

with its natural $(\frac 1k{\mathbb Z}/{\mathbb Z})$-gradings

$$ \begin{align*} d_j=\frac{a}{k},\qquad d_s=\frac{b}{k}. \end{align*} $$

By the condition $\sum _{i=0}^n w_i=d$, the determinant of an element $g\in K[j_W,s]$ equals $\exp (2\pi \mathtt {i} d_s(g))$. The condition $d_s=0$ singles out the groups which we considered initially: $H\subseteq \operatorname {Aut} W\cap \operatorname {SL}(n+1;{\mathbb C})$ containing $j_W$.

4.1 A g-orbifolded cohomology

The g-orbifold cohomology is the cohomology of $I_g({\mathfrak X})$ defined in equation (6) shifted by the locally constant function ‘age’ given by ${\mathfrak a}_g={\mathfrak a}\circ {\mathsf p}$

(8)$$ \begin{align} {\mathfrak a}_g\colon I_g({\mathfrak X})\xrightarrow{{\mathsf p}} I[{\mathfrak X}/G]\xrightarrow{{\mathfrak a}} \mathbb Q, \end{align} $$

where ${\mathsf p}$ is the fibre diagram (7) and where ${\mathfrak a}$ assigns to each geometric point of the ordinary inertia stack $(x, g\in \operatorname {Aut}(x))$ the rational number $\operatorname {age}(g)$ from equation (3) applied to the diagonalization of the action of $g\in \operatorname {Aut}(x)$ on the tangent bundle $T({\mathfrak X})$ at x.

We assume that ${\mathfrak X}$ is smooth so that $I_g({\mathfrak X})$ is smooth and all coarse spaces are quasi-smooth; in particular cohomology groups admit a Hodge decomposition. Starting from a Hodge decomposition of weight n, for any $r\in {\mathbb Q}$, we can produce a new decomposition of weight $n-2r$ via $H(r)^{p,q}=H^{p+r,p+r}.$ We will systematically use this notation $(r)$ for bi-graded vector spaces.

Definition 7 (g-orbifold cohomology)

For any $g\in G$ the g-orbifold cohomology is defined as

$$ \begin{align*} H_g^*({\mathfrak X};{\mathbb C})=H^*(I_g({\mathfrak X});{\mathbb C})(-{\mathfrak a}_g). \end{align*} $$

We point out the slight abuse of notation: The $\operatorname {age}$ function is not constant in general, but, since it is locally constant, the shift operates independently on each cohomology group arising from each connected component. A precise notation should read

$$\begin{align*}H^{p,q}(\text{---};{\mathbb C})({\mathfrak a})=\bigoplus_{r\in \mathbb Q_{\ge 0}}H^{p,q}({\mathfrak a}^{-1}(r);{\mathbb C})(-r). \end{align*}$$

For $g=\operatorname {id}=1_G$, g-orbifold cohomology coincides with the cohomology of the inertia stack shifted by the age function. By definition, this is Chen–Ruan orbifold cohomology

$$ \begin{align*} H_{\operatorname{id}}^*({\mathfrak X};{\mathbb C})=H_{\mathrm{CR}}^*({\mathfrak X};{\mathbb C}). \end{align*} $$

In this paper, we often consider the relative version of orbifold Chen–Ruan cohomology; indeed when ${\mathfrak Z}$ is a substack of ${\mathfrak X}$ then $I({\mathfrak Z})$ is a substack of $I({\mathfrak X})$, and we set

$$ \begin{align*} H_{\operatorname{id}}^*({\mathfrak X},{\mathfrak Z};{\mathbb C})=H^*(I({\mathfrak X}),I({\mathfrak Z});{\mathbb C})(-{\mathfrak a}_{\operatorname{id}}), \end{align*} $$

where ${\mathfrak a}_{\operatorname {id}}$ is the age function on $I({\mathfrak X})$.

Yasuda [Reference Yasuda37] proves the identity of the dimensions of each term in the Hodge decomposition of orbifold Chen–Ruan cohomology under the following definition of K-equivalence: Two smooth Deligne–Mumford stacks ${\mathfrak X}$ and ${\mathfrak Y}$ are K-equivalent whenever there exists a smooth and proper Deligne–Mumford stack ${\mathfrak Z}$ with birational morphisms ${\mathfrak Z}\to {\mathfrak X}$ and ${\mathfrak Z}\to \mathfrak Y$ with $\omega _{{\mathfrak Z}/{\mathfrak X}}\cong \omega _{{\mathfrak Z}/\mathfrak Y}$. This equivalence holds for our Gorenstein orbifolds and for the crepant resolution of their coarse space by a simple argument which we detail in the proof of Proposition 9.

The hypotheses in our setup are slightly stronger and yield the statement below. Indeed, we consider a smooth Gorenstein orbifold ${\mathfrak X}$: The local picture at each point $x\in {\mathfrak X}$ is $[{\mathbb C}^{\dim {\mathfrak X}}/H]$ with $H\in \operatorname {SL}(\dim {\mathfrak X};{\mathbb C})$. In particular, the orbifold cohomology groups $H^{p,q}$ of ${\mathfrak X}$ vanish if $(p,q)\not \in {\mathbb Z}^2$.

Let $G=\langle s\rangle $ be a cyclic group of order k whose generator s acts on each tangent space $T_x$ of the Gorenstein orbifold ${\mathfrak X}$ with age $a_x\in 1/k+{\mathbb Z}$ (the age is defined up to automorphisms of x whose age is integer). Let us assume that the coarse space X of ${\mathfrak X}$ admits a crepant resolution $\widetilde X$ where we can lift the G-action induced by ${\mathfrak X}$ on X. By Yasuda’s theorem we have a bidegree-preserving isomorphism

$$ \begin{align*} H^*({\mathfrak X};{\mathbb C})\cong H^*(\widetilde X;{\mathbb C}). \end{align*} $$

The aim of the following proposition, is to point out how this isomorphism relating ${\mathfrak X}$ and $\widetilde X$ extends to g-orbifold cohomology.

Proposition 9. We assume the above conditions on ${\mathfrak X}$ (smooth Gorenstein Deligne–Mumford stack), on X (the coarse space) and on $\widetilde X$ (the crepant resolution). In particular, we consider the order-k cyclic group $G=\langle s\rangle $ acting compatibly on ${\mathfrak X}, X,$ and $\widetilde X$. The age $a_x$ of the action of s on each tangent space $T_x$ of ${\mathfrak X}$ satisfies $a_x\in 1/k+{\mathbb Z}$. Then, for any $g\in G$, we have an isomorphism preserving the bidegree

$$ \begin{align*} H_g^*({\mathfrak X};{\mathbb C})\cong H_g^*(\widetilde X;{\mathbb C}). \end{align*} $$

In particular, the isomorphism identifies $H_g^*({\mathfrak X};{\mathbb C})$ with $H^*(\widetilde X_g;{\mathbb C})(-\widetilde {{\mathfrak a}}_g)$, where $\widetilde {{\mathfrak a}}_g$ is the composite of $\widetilde X_g\to [\widetilde X/G]$ and of the age function $[\widetilde X/G]\to \mathbb Q$.

In special cases where $\widetilde {{\mathfrak a}}_g$ is constant, the above theorem allows us to relate the g-orbifold cohomology to the cohomology of the g-fixed locus of the resolution via a constant shift by $\widetilde {{\mathfrak a}}_g$. The following example generalises the case of antisymplectic involutions of orbifold K3 surfaces considered in [Reference Chiodo, Kalashnikov and Veniani12] (this case occurs in Section 7 for $k=2$ and $4$).

Example 11. Consider a proper, smooth, Gorenstein, Deligne–Mumford orbifold ${\mathfrak X}$ of dimension 2 satisfying the Calabi–Yau condition $\omega \cong \mathcal O$. We refer to this as a K3 orbifold because there exists a minimal resolution $\widetilde X$ which is a K3 surface. Consider the volume form $\Omega $ of $\widetilde X$, which descends on ${\mathfrak X}$. We assume that g is an order-k automorphism of ${\mathfrak X}$ whose induced action on $\Omega $ is multiplication by $e^{2 \pi i (k-1)/k}$. Then, g naturally lifts to the minimal resolution $\widetilde X$; furthermore, locally at each fixed point of $\widetilde X$, the action of g can be diagonalized and expressed as $(x,y)\mapsto (\xi _k^a x, \xi _k^b y)$ where $a+b\equiv k-1\mod k$. In fact, $a+b$ equals $k-1$ without reduction mod k (this happens because the case $a+b=2k-1$ is impossible for $a,b\in \{0,\dots , k-1\}$). In this way, the age shift ${\mathfrak a}_g$ at the fixed loci always equals $1-\frac 1k$

$$ \begin{align*} H_g^*({\mathfrak X};{\mathbb C})=H^*(\widetilde X_g ;{\mathbb C})(\textstyle{\frac1k-1}).\end{align*} $$

5.1 K-equivalence

Consider the rank-$(n+1)$ vector bundle

$$ \begin{align*} {\mathbb V}=\mathcal O_{{\mathbb P}(d)}(-w_0)\oplus \dots\oplus \mathcal O_{{\mathbb P}(d)}(-w_n)=[{\mathbb C}^{n+1}/\langle j\rangle], \end{align*} $$

where we recall that ${\mathbb P}(d)$ is simply the special case of a zero-dimensional projective stack isomorphic to $B{\pmb \mu }_d$ (see Remark 2). Its coarse space is $X={\mathbb C}^{n+1}/\langle j\rangle $ with a singularity given by the ${\pmb \mu }_d$-action $[\frac {w_0}d,\frac {w_1}d,\dots ,\frac {w_m}d]$.

We also consider the smooth Deligne–Mumford stack

$$ \begin{align*} {\mathbb L}=\mathcal O_{{\mathbb P}({\pmb w})}(-d) \end{align*} $$

defined as the (stack-theoretic) total space ${\mathbb L}$ of the line bundle of degree $-d$ on ${\mathbb P}({\pmb w})$. Because ${\mathbb L}$ is a weighted blow up of X in the stack-theoretic sense, there is a natural map from ${\mathbb L}$ to X.

The stacks ${\mathbb V}$ and ${\mathbb L}$ are the two stack-theoretic GIT quotients of ${\mathbb C}\times {\mathbb C}^{n+1}$ modulo ${\mathbb G}_m$ operating with weights $(-d,w_0,\dots ,w_{n+1})$ on the two open subsets obtained by removing the origin of the left- and right-hand side factors of the product ${\mathbb C}\times {\mathbb C}^{n+1}$. Notice that ${\mathbb V}$ without the origin coincides with the line bundle ${\mathbb L}$ without the zero section: ${\mathbb V}^{\times } ={\mathbb L}^{\times }$.

As in equation (4), we consider the generic fibre $M_P$ of $P\colon {\mathbb C}^{n+1}\to {\mathbb C}$. The morphism P descends to a morphism from ${\mathbb V}$ to ${\mathbb C}$, whose generic fibre is isomorphic to ${\mathbb F}=[M_P/\langle j \rangle ]$, a substack of ${\mathbb V}^{\times }$ which we may regard, via the above identification, as a substack of ${\mathbb L}^{\times }$.

We assume the Calabi–Yau condition $\sum _{i=0}^n w_i=d$. Then, the canonical bundle of ${\mathbb V}$ is j-invariant and descends to X and its pullback to ${\mathbb L}$ coincides with $\omega _{{\mathbb L}}$. Following the above arguments, i.e., by Proposition 9 and ultimately by Yasuda [Reference Yasuda37], we have

(9)$$ \begin{align} \Phi\colon H^{p,q}_{g}({\mathbb V};{\mathbb C})\xrightarrow{\cong} H^{p,q}_{g}({\mathbb L};{\mathbb C}) \end{align} $$

for any $p,q\in \mathbb Q$ and for any $g=s^b$ for $b\in \{0,\dots , k-1\}$ and $g=[0,\frac 1k,0\dots ,0]$ (notice that we extended g trivially on the fibre of ${\mathbb L}$).

The isomorphism $H^*_{g}({\mathbb V};{\mathbb C})\to H^*_{g}({\mathbb L};{\mathbb C})$ is not canonical, the claim of its existence is simply an identity between dimensions of vector spaces. The argument here consists in choosing an isomorphism $\Phi $ in such a way that it commutes with the restrictions $H^r_{g}({\mathbb V};{\mathbb C})\to H^r_{g}({\mathbb F};{\mathbb C})$ and $H^r_{g}({\mathbb L};{\mathbb C}) \to H^r_{g}({\mathbb F};{\mathbb C})$ and with the identity on $H^r_{g}({\mathbb F};{\mathbb C})$ in the diagram below in all degrees $r\in \mathbb Q$

(10)

We detail this isomorphism $\Phi $ in §5.4 after studying more carefully the two horizontal exact sequences involving ${\mathbb L}$ and ${\mathbb V}$ in §5.2 and §5.3, respectively. This allows us to conclude that there is a bidegree-preserving isomorphism

$$ \begin{align*} H^*_{g}({\mathbb V},{\mathbb F};{\mathbb C})\cong H^*_{g}({\mathbb L},{\mathbb F};{\mathbb C}) \end{align*} $$

making the above diagram commute. Based on §5.2 (using [Reference Chiodo and Nagel13, Prop. 3.5]), the right-hand side is naturally identified via the Thom isomorphism to the Chen–Ruan cohomology of $\Sigma _{P,H}$ up to a $(-1)$-shift. Based on §5.3, the left-hand side is naturally identified to an orbifold version of the Jacobi ring known as the FJRW or Landau–Ginzburg state space.

5.2 Thom isomorphism in orbifold cohomology

Consider $P\colon {\mathbb L}\to {\mathbb C}$ and its generic fibre ${\mathbb F}=P^{-1}(t)$ for $t\neq 0$. We have an isomorphism of Hodge structures

(11)$$ \begin{align} H^*({\mathbb L}, {\mathbb F};{\mathbb C})\cong H^*(\Sigma_{P};{\mathbb C})(-1). \end{align} $$

Indeed, the left-hand side can be regarded after retraction as

$$\begin{align*}H^*({\mathbb P}({\pmb w}),{\mathbb P}({\pmb w})\setminus \Sigma_P;{\mathbb C})\end{align*}$$

which is isomorphic to the $(-1)$-shifted cohomology of $\Sigma _P$ by the Thom isomorphism

(12)$$ \begin{align} H^*({\mathbb P}({\pmb w}),{\mathbb P}({\pmb w})\setminus \Sigma_P;{\mathbb C})\cong H^*(\Sigma_{P};{\mathbb C})(-1). \end{align} $$

Equation (11) suggests that the orbifold cohomology $H^{p,q}_{\operatorname {id}}({\mathbb L}_H,{\mathbb F}_{P,H};{\mathbb C})$ is related to the orbifold cohomology of $\Sigma _{P,H}$. However, due to the age shift, the argument above does not yield an isomorphism in orbifold cohomology. In fact, the two isomorphisms

(13)$$ \begin{align} H^{*}({\mathbb L}_H,{\mathbb F}_{P,H};{\mathbb C})\cong H^*({\mathbb P}({\pmb w}),{\mathbb P}({\pmb w})\setminus \Sigma_{P,H};{\mathbb C})\cong H^*(\Sigma_{P,H};{\mathbb C})(-1) \end{align} $$

may not respect the orbifold cohomology bidegree. However, Jan Nagel and the first author proved that the first and the third term of equation (13) match in orbifold cohomology with their respective bigrading even when the second isomorphism (the ordinary Thom isomorphism) does not hold in orbifold cohomology. For these reasons, we regard the special case where $g=\operatorname {id}$

$$ \begin{align*} H_{\operatorname{id}}^*({\mathbb L}, {\mathbb F};{\mathbb C})\cong H_{\operatorname{id}}^*(\Sigma_{P};{\mathbb C})(-1) \end{align*} $$

as the correct formulation of Thom isomorphism in orbifold cohomology.

Theorem 12 (the orbifold Thom isomorphism, [Reference Chiodo and Nagel13])

For any $H\subseteq \operatorname {Aut}_P$ containing $j_P$, for any g in $\operatorname {Aut}_P$ and for any $p,q\in \mathbb Q$, we have

(14)$$ \begin{align} H^{p,q}_{g}({\mathbb L}_H, {\mathbb F}_{P,H};{\mathbb C})\cong H^{p,q}_{g}(\Sigma_{P,H};{\mathbb C})(-1). \end{align} $$

Before proving the statement, we illustrate it with a simple example allowing us to describe the two cases of the proof.

Example 13. We provide an elementary and yet very rich example of a Calabi–Yau orbifold embedded in a non-Gorenstein ${\mathbb P}({\pmb w})$.

The isomorphism above matches the orbifold cohomology of a hypersurface $\Sigma _{P}$ and the relative cohomology of $({\mathbb L}, {\mathbb F})$. Therefore, on the one side, we consider the hypersurface $\Sigma _P$ defined by $P=x^3+xy=0$ in ${\mathbb P}(1,2)$. There are two components in the ambient orbifold ${\mathbb P}(1,2)$: the untwisted sector (labelled with u) attached to the identity and the twisted sector (labelled with t) attached to the nontrivial symmetry $(-1,1)$.

Within the untwisted sector ${\mathbb P}(1,2)_u={\mathbb P}(1,2)$, the equation $x^3+xy=0$ cuts out a codimension-one hypersurface which can be described as the disjoint union of an ordinary point, the ${\mathbb G}_m$-orbit $p=(-\lambda ,\lambda ^2)$ and a point with nontrivial ${\pmb \mu }_2$-automorphism: the locus where the first coordinate vanishes $(x=0)\cong {\mathbb P}(2)$.

Within the twisted sector ${\mathbb P}(1,2)_t={\mathbb P}(2)$, the equation $x^3+xy=0$ identifies a hypersurface which is not transversal to ${\mathbb P}(2)$. In other words, the twisted sector of $\Sigma _P$ is the entire twisted sector ${\mathbb P}(1,2)_t={\mathbb P}(2)$.

These two conditions, $\Sigma _{P,\beta }\subsetneq {\mathbb P}({{\pmb w}}_{\beta })$ and $\Sigma _{P,\beta }={\mathbb P}({\pmb w}_{\beta })$, play two different roles in the orbifold Thom isomorphism as we illustrate in this example and detail further in general in the proof. The right-hand side of the orbifold Thom isomorphism reads $H^{*}_{\operatorname {orb}}(\Sigma _{P};{\mathbb C})(-1)$ (for $g=\operatorname {id}$) and can be easily computed in this example: ${\mathbb C}^{\oplus 3}(-1)$. This happens because the inertia stack consists of three points

(15)$$ \begin{align} I(\Sigma_P)=\Sigma_{P,u}\sqcup \Sigma_{P,t}=(2{\mathrm{\,pts}})\sqcup {\mathbb P}(2), \end{align} $$

(the labels u and t stand again for untwisted and twisted). The age-shift plays no role for a zero dimensional stack (the tangent bundle vanishes). We get

(16)$$ \begin{align} H^*_{\operatorname{orb}}(\Sigma_{P};{\mathbb C})(-1)\cong {\mathbb C}^{\oplus 3}(-1). \end{align} $$

The left-hand side of the orbifold Thom isomorphism above reads $H^*_{\operatorname {orb}}({\mathbb L}, {\mathbb F}_{P};{\mathbb C})$, which, unlike the orbifold cohomology group $H^*_{\operatorname {orb}}({\mathbb P}(1,2),{\mathbb P}(1,2)\setminus \Sigma )$, matches equation (16). We compute both $H^*_{\operatorname {orb}}({\mathbb P}(1,2),{\mathbb P}(1,2)\setminus \Sigma )$ and $H^*_{\operatorname {orb}}({\mathbb L}, {\mathbb F}_{P};{\mathbb C})$.

For $H^*_{\operatorname {orb}}({\mathbb P}(1,2),{\mathbb P}(1,2)\setminus \Sigma )$, we have $I({\mathbb P}(1,2))={\mathbb P}(1,2)_u\sqcup {\mathbb P}(1,2)_t={\mathbb P}(1,2)\sqcup {\mathbb P}(2)$. Using equation (15) and writing ${\mathbb P}({\pmb w})={\mathbb P}(1,2)$ for short, we get

$$ \begin{align*} H^*_{\operatorname{orb}}({\mathbb P}({\pmb w}),{\mathbb P}({\pmb w})\setminus \Sigma) &=H^*({\mathbb P}({\pmb w})_u,{\mathbb P}({\pmb w})_u\setminus \Sigma_{P,u}) \oplus H^*({\mathbb P}({\pmb w})_t,{\mathbb P}({\pmb w})_t\setminus \Sigma_{P,t})(-{\textstyle\frac12})\\ &= H^*({\mathbb P}({\pmb w}),{\mathbb P}({\pmb w})\setminus \{2\mathrm{\ pts}\}) \oplus H^*({\mathbb P}(2),\varnothing)(-{\textstyle\frac12})\\ &= H^*(2{\mathrm{\ pts}})(-1) \oplus H^0({\mathbb P}(2))(-{\textstyle\frac12})\\&= {\mathbb C}^{\oplus 2}(-1)\oplus {\mathbb C}(-{\textstyle\frac12}).\end{align*} $$

On the other hand, for $H^*_{\operatorname {orb}}({\mathbb L},{\mathbb F})$, the ambient orbifold ${\mathbb L}$ induces an extra $\frac 12$ age-shift on the twisted sector because the stabilizer of ${\mathbb P}(2)$ acts nontrivially in the direction of the fibre of the line bundle ${\mathbb L}\to {\mathbb P}(1,2)$. Therefore, we get

$$ \begin{align*} H^*_{\operatorname{id}}({\mathbb L},{\mathbb F})= {\mathbb C}^2(-1)\oplus H^*({{\mathbb P}(2)})(-\textstyle 1)\cong {\mathbb C}^{\oplus 3}(-1) \end{align*} $$

as in equation (16).

Proof. We study $H_{g}^*({\mathbb L},{\mathbb F};{\mathbb C})$, $H_{g}^*({\mathbb P}({\pmb w}),{\mathbb P}({\pmb w})\setminus \Sigma _P;{\mathbb C})$, and $H_{g}^*(\Sigma _{P};{\mathbb C})(-1)$ sector-by-sector, i.e., by restricting to each symmetry $\beta =g(\lambda ^{w_0},\lambda ^{w_1}\dots ,\lambda ^{w_n})$ operating on ${\mathbb C}^{n+1}$ diagonally. We distinguish two cases: (1) the ordinary sectors where $\Sigma _{P,\beta }$ is a codimension-$1$ hypersurface in ${\mathbb P}({\pmb w}_{\beta })$ (this happens, for instance, for the untwisted sector in Example 13); (2) the case where ${\mathbb P}({\pmb w}_{\beta })$ and $\Sigma _{P,\beta }$ coincide (as in the twisted sector of Example 13). The two cases above may also be distinguished as follows: In case (1) we consider diagonal automorphisms $\beta $ fixing points of $\Sigma _P$ and the fibre of the normal bundle $N_{\Sigma _P/{\mathbb P}({\pmb w})}$ ($\beta $ fixes the points of ${\mathbb P}({\pmb w}_{\beta })$); in case (2) we consider automorphisms $\beta $ fixing points of $\Sigma _P$ and operating nontrivially on the normal bundle $N_{\Sigma _P/{\mathbb P}({\pmb w})}$. Notice also the following simple reformulation; since the restriction of ${\mathbb L}=\mathcal O_{{\mathbb P}({\pmb w})}(d)$ to $\Sigma _P$ is dual to the normal bundle of $\Sigma _P$ within ${\mathbb P}({\pmb w})$, in case (1) we have $\Sigma _{P,\beta }\subsetneq {\mathbb P}({\pmb w}_{\beta }) \subsetneq {\mathbb L}_{\beta }$ and each inclusion has codimension 1, whereas in case (2) we have $\Sigma _{P,\beta }= {\mathbb P}({\pmb w}_{\beta }) = {\mathbb L}_{\beta }$.

For ordinary sectors where $\Sigma _{P,\beta }\subsetneq {\mathbb P}({\pmb w}_{\beta }) \subsetneq {\mathbb L}_{\beta }$, we have $H^*({\mathbb P}({\pmb w}_{\beta }),{\mathbb P}({\pmb w}_{\beta })\setminus \Sigma _{P,\beta };{\mathbb C})\cong H^*(\Sigma _{P,\beta };{\mathbb C})(-1)$ and the age shift of ${\mathbb P}({\pmb w}_{\beta })$ coincides with that of $\Sigma _{P,\beta }$ since $\beta $ acts trivially on the normal bundle $N_{\Sigma _{P,\beta }/{\mathbb P}({\pmb w}_{\beta })}$. No age-shift difference occurs also when we pass to ${\mathbb L}_{\beta }$, which is a vector bundle on ${\mathbb P}({\pmb w}_{\beta })$: The age-shifted classes of $H^*({\mathbb L}_{\beta },{\mathbb F}_{\beta })$ match those of $H^*({\mathbb P}({\pmb w}_{\beta }),{\mathbb P}({\pmb w}_{\beta })\setminus \Sigma _{P,\beta })$ and, by the above argument, those of $H^*(\Sigma _{P,\beta };{\mathbb C})(-1)$.

For the remaining sectors, when $\Sigma _{P,\beta }= {\mathbb P}({\pmb w}_{\beta }) = {\mathbb L}_{\beta }$, the cohomology classes of the sectors labelled by $\beta $ within ${\mathbb L}, {\mathbb P}({\pmb w})$ and $\Sigma _P$ match via the trivial identities

$$ \begin{align*} H^*({\mathbb L}_{\beta},{\mathbb F}_{\beta})&= H^*({\mathbb P}({\pmb w}_{\beta}),{\mathbb P}({\pmb w}_{\beta})\setminus \Sigma_{P,\beta};{\mathbb C})\\ &= H^*({\mathbb P}({\pmb w}_{\beta}),\varnothing;{\mathbb C})\\&= H^*({\mathbb P}({\pmb w}_{\beta});{\mathbb C})\\ &=H^*(\Sigma_{P,\beta};{\mathbb C}). \end{align*} $$

Hence, in ordinary cohomology, the desired equation (14) holds without any $(-1)$-shift on the right-hand side. The $(-1)$-shift arises when we compare the age-shifts of the sector ${\mathbb L}_{\beta }$ and that of $\Sigma _{P,\beta }$; i.e., when we compare the age of the action of $\beta $ on the normal bundles $N_{\Sigma _{P,\beta }/\Sigma _P}$ and $N_{{\mathbb L}_{\beta }/{\mathbb L}}$ restricted to ${\mathbb L}_{\beta }=\Sigma _{P,\beta }$. Let $q\in ]0,1[$ be the nontrivial character of the action of $\beta $ on the normal bundle $N_{\Sigma _P/{\mathbb P}({\pmb w})}$. In these cases, we have $H^*_{\operatorname {orb}}({\mathbb P}({\pmb w}_{\beta }),{\mathbb P}({\pmb w}_{\beta })\setminus \Sigma _{P,\beta };{\mathbb C}) \cong H_{\operatorname {orb}}^*(\Sigma _{P,\beta };{\mathbb C})(-q)$. Since the restriction of ${\mathbb L}=\mathcal O_{{\mathbb P}({\pmb w})}(d)$ to $\Sigma _P$ is dual to the normal bundle $N_{\Sigma _P/{\mathbb P}({\pmb w})}$, the age of $\beta $ acting on ${\mathbb L}$ at a point p of $\Sigma _{P,\beta }={\mathbb L}_{\beta }$ differs from the age of the representation on the tangent bundle $T_{\Sigma _P, p}$ by $q+(1-q)$. We have

$$ \begin{align*} H^*({\mathbb L}_{\beta},{\mathbb F}_{\beta};{\mathbb C})({\mathfrak a}^{{\mathbb L}})\cong H^*(\Sigma_{P,\beta};{\mathbb C}) ({\mathfrak a}^{\Sigma}-q-(1-q))= H^*(\Sigma_{P,\beta};{\mathbb C})(a^{\Sigma}-1) \end{align*} $$

as desired (we wrote ${\mathfrak a}^{{\mathbb L}}$ and ${\mathfrak a}^{\Sigma }$ for the locally constant age functions on the orbifolds ${\mathbb L}$ and $\Sigma _P$).

Remark 15 (definition of ambient and primitive cohomology)

For a hypersurface $\Sigma $ within a projective space ${\mathbb P}$, we usually refer to the Poincaré dual of the image of the homology of $\Sigma $ within the homology of ${\mathbb P}$ as the ambient cohomology of $\Sigma $. In orbifold cohomology, we say that the ambient cohomology of $\Sigma _{P,H}$ is Poincaré dual to the image of the homology of $I_g(\Sigma _{P,H})$ within the homology of $I_g({\mathbb P}({\pmb w}))$. The identification above allows us to provide another description. We may regard the ambient cohomology of $\Sigma _{P,H}$ as the image of the morphism

(17)$$ \begin{align} H^r_{g}({\mathbb L}_H, {\mathbb F}_{P,H};{\mathbb C})\to H^r_{g}({\mathbb L}_H;{\mathbb C}). \end{align} $$

Similarly, we can consider the primitive cohomology, which is the kernel after Poincaré duality of the direct image mapping the homology of $\Sigma $ within the cohomology of ${\mathbb P}$. In orbifold cohomology, the primitive cohomology of $\Sigma _{P,H}$ is the kernel of the morphism (17) in $H^*(\Sigma _{P,H};{\mathbb C})(1)$. We summarize these notations as follows.

Definition 16.

$$ \begin{align*} H_{g,\mathrm{amb}}^r(\Sigma_{W,H};{\mathbb C})(-1)&=\operatorname{im}\big(H^r_{g}({\mathbb L}_H, {\mathbb F}_{P,H};{\mathbb C})\to H^r_{g}({\mathbb L}_H;{\mathbb C})\big),\\ H_{g,\mathrm{prim}}^r(\Sigma_{W,H};{\mathbb C})(-1)&=\ker\big(H^r_{g}({\mathbb L}_H, {\mathbb F}_{P,H};{\mathbb C})\to H^r_{g}({\mathbb L}_H;{\mathbb C})\big). \end{align*} $$

We now turn to the LG side, where the image of the analogue morphism allows us to describe the so called ‘narrow’ and ‘broad’ sectors.

5.3 Jacobi ring

The Jacobi ring

$$ \begin{align*} \operatorname{Jac} P = dx_0\wedge\dots\wedge dx_n{\mathbb C}[x_0,\dots, x_n]/ (\partial_0P, \dots, \partial_n P), \end{align*} $$

regarded as a ${\mathbb C}$-vector space, has dimension $\prod _j \frac {d-w_j}{w_j}$ (due to the nondegeneracy of the polynomial P) and is isomorphic to $H^*({\mathbb C}^{n+1},M_P;{\mathbb C})$. The natural monodromy action of ${\pmb \mu }_d=\langle j\rangle $ from equation (4), and more generally the action of any $\mathrm {diag}(\alpha _0,\dots , \alpha _n)\in \operatorname {Aut}_P$,

$$ \begin{align*} \mathrm {diag}(\alpha_0,\dots, \alpha_n)\cdot \left(\prod_{j=0}^n x_j^{b_j-1} \bigwedge_{j=0}^n dx_j\right) = \prod_{j=0}^n \alpha _j^{b_j} \left(\prod_{j=0}^n x_j^{b_j-1} \bigwedge_{j=0}^ndx_j\right), \end{align*} $$

allows us to write

$$ \begin{align*} [\operatorname{Jac}(P)^{j_P}]_{p,q}=H^{p,q}({\mathbb V};{\mathbb F}), \end{align*} $$

where the subscript ${p,q}$ denote the subgroup spanned by elements of the form

$$ \begin{align*} \left(\prod_{j=0}^n x_j^{b_j-1} \bigwedge_{j=0}^n dx_j\right)\quad \text{with } \quad (p,q)=\left(n-\sum_j b_j\frac{w_j}d ,\sum_j b_j\frac{w_j}d \right). \end{align*} $$

The above claim is due to Steenbrink [Reference Steenbrink30] in the present weighted homogenous setup; see also [Reference Chiodo, Iritani and Ruan11, Appendix A].

We can introduce the following slight generalization of the FJRW state space.

Definition 18. For a quasi-homogenous polynomial P of degree d and weight $w_0,\dots ,w_n$ and for any $H\subseteq \operatorname {Aut}_P$ containing $j_P$, the g-orbifolded Landau–Ginzburg state space is

$$ \begin{align*} \mathcal H_{P,H,g}= \bigoplus_{h\in g H} \left(\operatorname{Jac} P_{h}\right)^H(-\operatorname{age}(h)), \end{align*} $$

where, for any diagonal symmetry $h\in H$, we consider the Jacobi ring $\operatorname {Jac} P_{h},$ where $P_{h}$ is the restriction of P to the ring of polynomials in the h-fixed variables (and the age is given by equation (3)).

Remark 21. We have

(18)$$ \begin{align} \mathcal H _{P,H,g}^{p,q} =H_{g}^{p,q}({\mathbb V}_H;{\mathbb F}_{P,H}).\end{align} $$

Indeed for $H=\langle J_P\rangle $ this is the well-known identification between the middle cohomology of the fibre ${\mathbb F}$ and the monodromy invariant part of the cohomology of the Milnor fibre (see, for instance, [Reference Dimca16]). The general statement follows from taking the invariant classes on the two sides with respect to the action of the finite group $H_0=H/\langle j_P \rangle $.

Remark 22 (definition of narrow and broad sectors)

The elements h for which no variables are fixed yield a summand $\operatorname {Jac}(P_{h})={\mathbb C}$; these are special elements in the FJRW state space; they span the subspace of the so-called narrow classes. In FJRW theory, the remaining summands are referred to as broad classes (see [Reference Fan, Jarvis and Ruan22]). In complete analogy with the notation for ambient and primitive cohomology, we can identify the narrow and broad classes to the image and the kernel of the morphism

$$ \begin{align*}H^r_{g}({\mathbb V}_H, {\mathbb F}_{P,H};{\mathbb C})\to H^r_{g}({\mathbb V}_H;{\mathbb C}).\end{align*} $$

Definition 23. We have

$$ \begin{align*} \mathcal H_{P,H,g}^{r,\mathrm{nar}}=\operatorname{im}\big(H^r_{g}({\mathbb V}_H, {\mathbb F}_{P,H};{\mathbb C}\big)\to H^r_{g}({\mathbb V}_H;{\mathbb C})),\\ \mathcal H_{P,H,g}^{r,\mathrm{broad}}=\ker\big(H^r_{g}({\mathbb V}_H, {\mathbb F}_{P,H};{\mathbb C}\big)\to H^r_{g}({\mathbb V}_H;{\mathbb C})). \end{align*} $$

5.4 Landau–Ginzburg/Calabi–Yau correspondence

The above equations (14) and (18) add up to a simple proof of the so-called Landau–Ginzburg/Calabi–Yau correspondence based on Yasuda’s theorem and K-equivalence (ensured by the Calabi–Yau condition).

Theorem 24 ([Reference Chiodo and Ruan10, Reference Chiodo and Nagel13])

For any nondegenerate quasi-homogeneous polynomial P of Calabi–Yau type, for any group $H\subseteq \operatorname {Aut}_P$ containing $j_P$ and for any $g\in \operatorname {Aut}_P$, we have

$$ \begin{align*}\Phi\colon H_g^{p,q}(\Sigma_{P,H};{\mathbb C})(-1) \xrightarrow{ \ \sim\ } \mathcal H_{P,H,g}^{p,q}.\end{align*} $$

Proof. Let us consider the case $H=\langle j\rangle $ for simplicity first. In the statement, the right-hand side is isomorphic to $H^{*}_g({\mathbb L},{\mathbb F})$ and the left-hand side is isomorphic to $H^{*}_g({\mathbb V},{\mathbb F})$; hence, we only need to define an isomorphism $H_g^*({\mathbb V})\to H_g^*({\mathbb L})$ making the diagram (10) commute. The relative cohomology sequence allows us to split $H_g^*({\mathbb V})$ and $H_g^*({\mathbb L})$ into two direct summands:

$$ \begin{align*}H_g^*(A)=\operatorname{coker}( H_g^*(A, {\mathbb F})\to H_g^*(A))\oplus \ker( H_g^*(A)\to H_g^*({\mathbb F})),\end{align*} $$

for $A={\mathbb V}$ and ${\mathbb L}$. We show that there is a bidegree-preserving isomorphism matching the above cokernels within $H^*({\mathbb V})$ and $H^*({\mathbb L})$. Since any extension on the kernel of the restriction morphism makes equation (10) commute, the claim follows.

We need to treat separately two cases depending on whether the isomorphism $\gamma =g(\lambda ^{-d},\lambda ^{w_0},\lambda ^{w_1},\dots , \lambda ^{w_n})$ yields an empty sector ${\mathbb F}_{\gamma }$ or not. If ${\mathbb F}_{\gamma }$ is empty, then the restrictions morphisms $H^r({\mathbb V}_{\gamma })\to H^r({\mathbb F}_{\gamma })$ and $H^r({\mathbb L}_{\gamma } )\to H^r( {\mathbb F}_{\gamma })$ vanish. Therefore, these sectors only contribute to the above mentioned kernel of the restrictions morphisms mapping to $H_g^*({\mathbb F})$.

In the remaining cases, as argued in the proof of Theorem 12, the sector $\Sigma _{P, \gamma } $ of $\Sigma _P$ is a codimension-$1$ subspace of the $\gamma $-fixed coordinate subspace ${\mathbb P}({\pmb w}_{\gamma })$ and ${\mathbb L}_{\gamma }$ is the bundle $\mathcal O(-d)$ over ${\mathbb P}({\pmb w}_{\gamma })$. In these cases, the restriction morphism is nonzero only on the fundamental classes of ${\mathbb V}_{\gamma }$ and ${\mathbb L}_{\gamma }$ (these vanishing property are obvious for ${\mathbb V}_{\gamma }$, whereas for ${\mathbb L}_{\gamma }$ we may refer to a more detailed study of the Lefschetz hyperplane theorem in this setup in [Reference Chiodo and Nagel13, Lem. 3.6]). We notice that, in these cases, the fundamental classes of ${\mathbb V}_{\gamma }$ and ${\mathbb L}_{\gamma }$ share the same bidegree $(\operatorname {age}(\gamma ), \operatorname {age}(\gamma ))$.

The existence of a bidegree-preserving isomorphism $H_g^*({\mathbb V})\cong H_g^*({\mathbb L})$, alongside with the bidegree-preserving identification ${\pmb 1}_{{\mathbb V}_{\gamma }}\mapsto {\pmb 1}_{{\mathbb L}_{\gamma }}$ (for ${\mathbb F}_{\gamma }\neq \varnothing $), which matches the cokernels within $H_g^*({\mathbb V})$ and $H_g^*({\mathbb L})$, ensures the existence of an isomorphism $\Phi \colon H_g^*({\mathbb V})\cong H_g^*({\mathbb L})$ extending ${\pmb 1}_{{\mathbb V}_{\gamma }}\mapsto {\pmb 1}_{{\mathbb L}_{\gamma }}$ on the kernels of the restriction morphisms. By construction $\Phi $ is compatible with equation (10), and, as a consequence, there exists a bidegree-preserving isomorphism $H^d_{g}({\mathbb V},{\mathbb F};{\mathbb C})\cong H^d_{g}({\mathbb L},{\mathbb F};{\mathbb C}).$

The complete statement involves a more general group action with H replacing $\langle j\rangle $. This is a minor generalization. Indeed, we note that ${\mathbb F}$ can be regarded as the generic fibre of the two morphisms induced by the polynomial P: ${\mathbb V}\to {\mathbb C}$ and ${\mathbb L}\to {\mathbb C}.$ If we consider any group $H\subseteq \operatorname {Aut}_P$ containing j, we can apply the above claim to ${\mathbb V}_H=[{\mathbb V}/H_0]$, ${\mathbb L}_H=[{\mathbb L}/H_0]$ and ${\mathbb F}_{P,H} =[{\mathbb F}/H_0]$. We get $H^{p,q}_{g}({\mathbb V}_{H},{\mathbb F}_{P,H};{\mathbb C})\cong H^{p,q}_{g}({\mathbb L}_H,{\mathbb F}_{P,H};{\mathbb C}),$ for any $p,q$ and for any $g\in \operatorname {Aut}_W$. The above dichotomy distinguishing two types of sectors applies without changes to isomorphisms of the form $g(\lambda ^{-d},\alpha _0\lambda ^{w_0}, \dots ,\alpha _n\lambda ^{w_n})$, for $\alpha _i, \lambda \in {\mathbb G}_m$.

6.1 Mirror duality

In this section, we review the mirror construction due to Berglund and Hübsch [Reference Berglund and Hübsch5]. The mirror construction appeared first in the form given below in Krawitz [Reference Krawitz26]. Other key references are to Berglund and Henningson [Reference Berglund and Henningson4] for the group duality, Kreuzer and Skarke [Reference Kreuzer and Skarke27] for a systematic study and Borisov [Reference Borisov7] for a reinterpretation of the setup and further generalizations in terms of vertex algebræ.

The Berglund–Hübsch mirror construction [Reference Berglund and Hübsch5] applies to nondegenerate polynomials P of invertible type, i.e., having as many monomials as variables. Up to rescaling the variables, these polynomials are entirely encoded by the exponent matrix $E=(m_{i,j})$ and are paired to a second polynomial of the same type

$$\begin{align*}P(x_0,\dots, x_n)=\sum_{i=0}^n \prod_{j=0}^n x_j^{m_{i,j}},\qquad P^{\vee}(x_0,\dots, x_n)=\sum_{i=0}^n\prod_{j=0}^n x_j^{m_{j,i}},\end{align*}$$

by transposing the matrix of exponents E.

Remark 26. The square matrix E is invertible because the vector $(w_i/d)_{i=0}^n$ is uniquely determined as a consequence of the nondegeneracy condition. The inverse matrix $E^{-1}=(m^{i,j})$ allows a simple description of $\operatorname {Aut}_P$: The columns express the symmetries

$$ \begin{align*} \rho_j=[m^{0,j},m^{1,j},\dots,m^{n,j}] \end{align*} $$

spanning $\operatorname {Aut}_P$. It is also easy to see that the columns of E express all the relations $\sum _i m_{i,j} \rho _i$ among these generators. Naturally, the rows of $E^{-1}$ provide an expression for the symmetries $\overline \rho _i$ generating $\operatorname {Aut}_{P^{\vee }}$ under the relations provided by the rows $\sum _j m_{i,j} \overline \rho _j$ of E. In particular, we have a canonical isomorphism

$$ \begin{align*} \operatorname{Aut}_{P^{\vee}} = (\operatorname{Aut}_P)^*, \end{align*} $$

where $(G)^*$ denotes the Cartier dual $\operatorname {Hom}(G,{\mathbb G}_m)$. The identification matches the symmetry $[q_0,\dots , q_n]$ to the homomorphism mapping $\rho _i$ to $\exp (2\pi \mathtt {i} q_i) \in \mathbb Q/{\mathbb Z}$. Based on this identification, for any subset $S\subseteq \operatorname {Aut}_P$, we set $S^{\vee } \subseteq \operatorname {Aut}_{P^{\vee }}$ as follows

$$ \begin{align*} S^{\vee}= \{\varphi \in (\operatorname{Aut}_P)^*\ \mid \ \varphi\!\mid_S=0\}. \end{align*} $$

This is a duality exchanging subgroups of $\operatorname {Aut}_P$ and subgroups of $\operatorname {Aut}_{P^{\vee }}$; for any group $H\subseteq \operatorname {Aut}_P$, we can write

(19)$$ \begin{align} H^{\vee}=\ker \left(\operatorname{Aut}_{P}\twoheadrightarrow \operatorname{Hom}(H;{\mathbb G}_m)\right).\end{align} $$

The dual of the group generated by $J_P$ is $\operatorname {SL}_{P^{\vee }}$. The duality reverses inclusions: If $H_1 \subset H_2$, then $H_2^{\vee } \subset H_1^{\vee }.$

The unprojected state space $U_P$ is defined [Reference Krawitz26] as

$$ \begin{align*} U_P=\bigoplus_{h\in \operatorname{Aut}_P} \operatorname{Jac}(P_h)(-\operatorname{age}(h)), \end{align*} $$

where the sum is over all diagonal symmetries and without taking any invariant [Reference Krawitz26]. For each summand $\operatorname {Jac}(P_h)$, there exists an $\operatorname {Aut}_{P^{\vee }}$-grading defined by

(20)$$ \begin{align} \ell_h\colon \operatorname{Jac}(P_h)&\to \operatorname{Aut}_{P^{\vee}}\nonumber\\ \prod_j x_j^{b_j-1}\bigwedge_{j} dx_j&\mapsto \prod_j \overline \rho_j^{b_j}, \end{align} $$

where all the products run over the set $F_h$ of labels of variables fixed by h.

In this way, the unprojected state space admits a double decomposition

(21)$$ \begin{align} U_P=\bigoplus_{h\in \operatorname{Aut}_P} \operatorname{Jac}(P_h)(-\operatorname{age}(h))= \bigoplus_{h\in \operatorname{Aut}_P}\ \bigoplus_{k\in \operatorname{Aut}_{P^{\vee}}} U_h^k(P), \end{align} $$

where $U_h^k(P)$ is the k-graded component of $\operatorname {Jac}(P_h)(-\operatorname {age}(h))$. We write

$$ \begin{align*} U_H^K(P)=\bigoplus_{h\in H}\ \bigoplus_{k\in K} U_h^k(P), \end{align*} $$

for any set $H\subseteq \operatorname {Aut}_P$ and $K\subseteq \operatorname {Aut}_{P^{\vee }}$. When a subscript H or a supscript K is omitted, we assume that H or K equal $\operatorname {Aut}_P$ or $\operatorname {Aut}_{P^{\vee }}$. When no ambiguity may occur, we omit the polynomial P in the notation.

Proposition 28. The vector space $U_H^K$ is the $K^{\vee }$-invariant subspace of $U_H$

$$ \begin{align*} U_H^K(P)=\left[U_H(P)\right]^{K^{\vee}}. \end{align*} $$

In particular, we have

$$ \begin{align*} {\mathcal H}_{P,H,g}= U_{gH}^{H^{\vee}}(P). \end{align*} $$

Theorem 29 (Krawitz [Reference Krawitz26], Borisov [Reference Borisov7])

For any $h\in \operatorname {Aut}_P$ and $k\in \operatorname {Aut}_{P^{\vee }}$, we have an explicit isomorphism

$$ \begin{align*} U_h^k(P)\cong U_k^h(P^{\vee}), \end{align*} $$

yielding an explicit isomorphism

$$ \begin{align*} {\mathrm{Mirror}}_P\colon U_P &\longrightarrow U_{P^{\vee}} \end{align*} $$

mapping $(p,q)$-classes to $(n+1-p,q)$ classes.

We illustrate the isomorphism explicitly in the special case where $P=x^k$. It is elementary, and it plays a crucial role in this paper.

Example 30. Let $P=x^k$. Then $\operatorname {Aut}_P$ equals ${\mathbb Z}_k$ (because we fix a primitive kth root $\xi $). For $1\in {\pmb \mu }_k={\mathbb Z}_k$, we have $P_1=P$, so

$$ \begin{align*} \operatorname{Jac}(P_1)=dx{\mathbb C}[x]/(x^{k-1})= \sum_{h=1}^{k-1} U_{1}^{\xi^h} \end{align*} $$

with $U_{1}^{\xi ^h}$ spanned by $x^{h-1}dx$. Furthermore, for $i\neq 0$, we have $P_{\xi ^i}=0$, so

$$ \begin{align*} \operatorname{Jac}(P_{\xi^i})={\mathbb C}=U_{\xi^i}^1\ \ (\text{for } i=1,\dots, k-1). \end{align*} $$

By mapping $x^{i-1}dx\in \operatorname {Jac}(P_1)$ to the generator $1_{\xi ^i}$ of $\operatorname {Jac}(P_{\xi ^i})$, we have defined a map matching $(1-\frac {i}k,\frac {i}k)$-classes to $(\frac {i}k,\frac {i}k)$-classes.

If P is a polynomial which can be expressed as the sum of two invertible and nondegenerate polynomials $P'$ and $P''$ involving disjoint sets of variables, we clearly have $\operatorname {Aut}_P=\operatorname {Aut}_{P''}\times \operatorname {Aut}_{P''}$. This and the theorem above imply the following crucial properties of the mirror map ${\mathrm {Mirror}}_P$.

• Thom–Sebastiani. If P is a polynomial which can be expressed as the sum of two invertible and nondegenerate polynomials

  $$ \begin{align*} P=P'(x^{\prime}_0,\dots,x^{\prime}_{n_1})+P''(x^{\prime\prime}_0,\dots,x^{\prime\prime}_{n_2}) \end{align*} $$
  involving two disjoint sets of variables, then we have
  $$ \begin{align*} {\mathrm{Mirror}}_P={\mathrm{Mirror}}_{P'}\otimes {\mathrm{Mirror}}_{P''}. \end{align*} $$

• Group actions. For any $H, K \subseteq \operatorname {Aut}_P$, the restriction of ${\mathrm {Mirror}}_P$ yields an isomorphism

  (22)$$ \begin{align} {\mathrm{Mirror}}_P\colon [U_H(P)]^K \cong [U_{H^{\vee}}(P^{\vee})]^{K^{\vee}}. \end{align} $$

There are many consequences of the existence of ${\mathrm {Mirror}}_P$ and of its properties with respect to group actions. We list a few of them, starting from the first, most transparent, application. It appeared in [Reference Chiodo and Ruan10], and it should be regarded as a combination of the mirror map ${\mathrm {Mirror}}_P$ of Krawitz and Borisov [Reference Krawitz26, Reference Borisov7] and of the LG/CY isomorphism $\Phi $ of the first named author with Ruan [Reference Chiodo and Ruan10]. There the isomorphism is deduced by a combinatorial model. Here, we presented both sides of the correspondence in terms of relative orbifold cohomology, and we deduced the main theorem from K-equivalence between the ambient orbifolds. In this sense, the present setup clarifies [Reference Chiodo and Ruan10].

Theorem 31 (mirror symmetry for CY models)

For any invertible, nondegenerate P of Calabi–Yau type and for any $H\subseteq \operatorname {Aut}_P$ satisfying $j_P\in H\subseteq \operatorname {SL}_P$, we have an isomorphism

$$ \begin{align*} H_{\operatorname{id}}^{p,q}(\Sigma_{P,H};{\mathbb C})\cong H_{\operatorname{id}}^{n-1-p,q}(\Sigma_{P^{\vee},H^{\vee}};{\mathbb C}) \qquad (p,q\in \mathbb Q). \end{align*} $$

Proof. Recall Definition 18:

$$\begin{align*}\mathcal H_{P,H,\operatorname{id}}=\bigoplus_{h\in H} \left(\operatorname{Jac} P_{h}\right)^H(-\operatorname{age}(h)).\end{align*}$$

We get

$$\begin{align*}\mathcal H_{P,H,\operatorname{id}}=[U_{H}(P)]^{H} \text{ and } \mathcal H_{P^{\vee},H^{\vee},\operatorname{id}}=[U_{H^{\vee}}(P)]^{H^{\vee}}\end{align*}$$

Therefore, by equation (22), ${\mathrm {Mirror}}_P$ gives an isomorphism ${\mathrm {Mirror}}_P(\mathcal H_{P,H,\operatorname {id}}) \cong \mathcal H_{P^{\vee },H^{\vee },\operatorname {id}}$. By assumption, P satisfies the Calabi–Yau condition. We also have that $H\ni j_P$ and $H^{\vee } \ni j_{P^{\vee }}$—the latter follows from the assumption that $H \subseteq \operatorname {SL}_P$. We can therefore apply Theorem 24 to both sides to obtain the claim, at least without the grading. The fact that a $(p,q)$-graded element is mapped to an $(n-1-p,q)$-graded element follows from Theorem 29 and by the fact that the LG/CY correspondence in Theorem 24 preserves the bidegree.

The Thom–Sebastiani property applies to $P'=x_0^k$ and $P''=f$ adding up to

$$ \begin{align*} W=x_0^k+f(x_1,\dots, x_n). \end{align*} $$

The aim of this paper is to study the relation between the above cohomological mirror symmetry and the symmetry $s=[\frac 1k,0,\dots ,0]$. Note that $s=\rho _0$ in the notation of Remark 26.

Proposition 32. Let $(\phi ,h)$ be a monomial element

$$ \begin{align*} (\phi,h)=\left(\prod_{j=0}^n z_j^{b_j-1}\bigwedge_{j=0}^n dz_j\right) \end{align*} $$

in $\operatorname {Jac}(P^{\vee }_h)(-\operatorname {age}(h))$. Let $\xi =\exp (2\pi \mathtt {i} /k)$. Then, $s^*(\phi ,h)=\xi ^i(\phi ,h)$ if and only if ${\mathrm {Mirror}}_W(\phi ,h)$ is of the form $(\phi ',h')$ with $h'=[\frac {i}k,a_1,\dots ,a_n]$. In particular, ${\mathrm {Mirror}}_W$ maps invariant elements to noninvariant elements.

6.2 Unprojected states and automorphisms

We study the behaviour of $U_W$ with respect to s. We begin by restricting to a conveniently large state space ${\mathbb H}_W$ within $U_W$.

Let us consider $W=x_0^k+f(x_1,\dots ,x_n)$ and, as in §3.2, a subgroup $K \subset \operatorname {Aut}_f$ satisfying

$$ \begin{align*} (j_f)^k\in K\subseteq \operatorname{SL}_f. \end{align*} $$

We define

(23)$$ \begin{align} {\mathbb H}_{K[j_W,s]}^K(W)=\left[U_{K[j_W,s]}(W)\right]^K=\bigoplus_{h\in K[j_W,s]} \operatorname{Jac}(P_h)^K(-\operatorname{age}(h)). \end{align} $$

If no ambiguity arises, when the polynomial W and the group K are fixed, we write simply ${\mathbb H}$.

In the above setup, we have three groups: K, $K[j_W]$ and $K[j_W,s]$. Only $K[j_W]$ satisfies the two conditions of mirror symmetry theorems: Namely, it contains $j_W$ and is contained in $\operatorname {SL}_W$. Its mirror group $K[j_W]^{\vee }$ has the same properties. The following proposition describes how K and $K[j_W,s]$ behave with respect to the group duality. We omit the proof as it is identical to the proof in Proposition 6.2.2 in [Reference Chiodo, Kalashnikov and Veniani12].

Proposition 34. Consider $K\in \operatorname {Aut}_f$ satisfying $(j_f)^k\in K\subseteq \operatorname {SL}_f.$ Then we have

$$ \begin{align*} (j_{f^{\vee}})^k\in (K[j_W,s])^{\vee}\subseteq \operatorname{SL}_{f^{\vee}}\quad\text{and}\quad K^{\vee}=(K[j_W,s])^{\vee}[j_{W^{\vee}},s]. \end{align*} $$

Furthermore, we have a mirror isomorphism

(24)$$ \begin{align} {\mathrm{Mirror}}_W\colon \left({\mathbb H}_{K[j_W,s]}^K(W)\right)^{p,q}= \left({\mathbb H}_{K^{\vee}}^{K[j_W,s]^{\vee}}(W^{\vee})\right)^{n+1-p,q}. \end{align} $$

The unprojected state space projects to the sum of state spaces of the form $\mathcal H_{W,H,g}$ after taking $j_W$-invariant elements.

Corollary 35. We have

$$ \begin{align*} {\mathbb H}^{j_W}=\bigoplus_{b=0}^{k-1} \mathcal H_{W,K[j_W],s^b}. \end{align*} $$

In particular, if W is Calabi–Yau, we have

$$ \begin{align*} {\mathbb H}^{j_W}=\bigoplus_{b=0}^{k-1} H_{s^b}^*(\Sigma_{W,K[j_W]};{\mathbb C}), \end{align*} $$

where $(\frac {b}k+p,\frac {b}k+q)$-classes in $H_{s^b}^*(\Sigma _{W,K[j_W]};{\mathbb C})$ match $(\frac {b}k+p,\frac {b}k+q)$-classes in $\mathcal H_{W,K[j_W],s^b}$ for any $p,q\in {\mathbb Z}$.

Proof. Tracking through the definition,

$$ \begin{align*} {{\mathbb H}}_{K[j_W,s]}^K(W)=\bigoplus_{h\in K[j_W,s]} \operatorname{Jac}(P_h)^{K[j_W]}(-\operatorname{age}(h)), \end{align*} $$

which is precisely

$$ \begin{align*} \bigoplus_{b=0}^{k-1} \mathcal H_{W,K[j_W],s^b}. \end{align*} $$

Now we can apply equation (22) to obtain the claim. The statement about grading follows from the age shift.

6.3 The twist and the elevators

Throughout this section, the polynomial W and the group K will be fixed; we simplify the notation and write

$$ \begin{align*} {\mathbb H}:={\mathbb H}_{K[j_W,s]}^K(W),\qquad j:=j_W,\qquad {\mathbb H}^{\vee}:={\mathbb H}_{K^{\vee}}^{K[j_W,s]^{\vee}}(W^{\vee}),\qquad j^{\vee}=j_{W^{\vee}}. \end{align*} $$

We also write M for the mirror map $\mathrm {Mirror}_W$.

Note that the monomial element $(\phi ,h)\in {\mathbb H}$ with

$$ \begin{align*} \phi=\prod_{i \in I} x_i^{b_i-1}\bigwedge_{i\in I} dx_i\qquad \text{and}\qquad I=\{i\mid h\cdot x_i=x_i\} \end{align*} $$

is an eigenvector with respect to the diagonal symmetry $\alpha {\,=\,}[p_0,\dots ,p_n]$: The eigenvalue is $\exp (2\pi \mathtt {i} \sum _j b_jp_j).$ It is natural to attach to each $(\phi ,g)$ and $\alpha $ the so-called $\alpha $-charge of the form $\phi $ defined on the g-fixed space:

$$ \begin{align*} Q_{\alpha}\colon (\phi,g)\mapsto Q_{\alpha}(\phi,g)= \sum_{j\in J} b_jp_j\quad\mod {\mathbb Z}. \end{align*} $$

Definition 36. We define several important $\mathbb Q/{\mathbb Z}$-valued gradings on ${\mathbb H}$. To do this, we decompose ${\mathbb H}$ as

$$ \begin{align*} {\mathbb H}= \bigoplus_{a=0}^{k-1} \bigoplus_{b=0}^{k-1} \bigoplus_{g\in j^{a}s^{b} K} \left(\operatorname{Jac} W_g\right)^K. \end{align*} $$

The four $\mathbb Q/{\mathbb Z}$-valued gradings on the set of generators

$$ \begin{align*} \left(\phi=\prod_{i\in I} x_i^{b_i-1}\bigwedge_{i\in I} dx_i,\quad g=[p_0,p_1,\dots,p_n]\in j^as^bK\right) \end{align*} $$

are defined as

1. 1. the j-charge $Q_j=Q_j\colon (\phi ,g)\mapsto Q_j(\phi ,g);$

2. 2. the j-degree $d_j=\frac {a}k;$

3. 3. the s-charge $Q_s=Q_s\colon (\phi ,g)\mapsto Q_s(\phi ,g);$

4. 4. the s-degree $d_s=\frac {b}k.$

We can now decompose

$$ \begin{align*} {\mathbb H}=\bigoplus_{{0\le a,b,c,d\le k-1}} \left[{\mathbb H}\mid (d_j,d_s,Q_j,Q_s)=\frac1k(a,b,c,d)\right]. \end{align*} $$

The following proposition further simplifies the decomposition. By definition, the condition $Q_s=0$ identifies the fixed space with respect to the action of s. The condition $Q_s\neq 0$ identifies the space spanned by the moving states. We make the following observation.

In other words, ${\mathbb H}$ decomposes into an s-moving part ${\mathbb H}^m$ ($Q_s\neq 0$) and an s-fixed part ${\mathbb H}^f$ ($Q_s=0$)

$$ \begin{align*} {\mathbb H}={\mathbb H}^m\oplus {\mathbb H}^f=[{\mathbb H}\mid Q_s\neq 0]\oplus [{\mathbb H}\mid Q_s=0], \end{align*} $$

and the first summand is $[{\mathbb H}\mid d_j+d_s=0]$; hence, the three parameters $d_j,Q_j,Q_s$ suffice for decomposing ${\mathbb H}^m$, and the three parameters $d_j,d_s,Q_j$ suffice for decomposing ${\mathbb H}^f$. We write

$$ \begin{align*}{\mathbb H}^m&= \bigoplus_{\substack{{0\le X,Y<k}\\{0< Z< k}}} \left[{\mathbb H}\mid (d_j,Q_s-Q_j,Q_s)=\frac1k(X,Y,Z)\right]=\bigoplus_{\substack{{0\le X,Y< k}\\{0< Z<k}}}{\mathbb H}^m_{X,Y,Z} ,\\ {\mathbb H}^f&= \bigoplus_{\substack{{0\le X,Y< k}\\{0< Z< k}}} \left[{\mathbb H}\mid (d_j,-Q_j,d_j+d_s)=\frac1k(X,Y,Z)\right] =\bigoplus_{\substack{{0\le X,Y< k}\\{0< Z< k}}}{\mathbb H}^f_{X,Y,Z},\end{align*} $$

where the choice of the three parameters in $\{0,\dots , k-1\}$ modulo $k{\mathbb Z}$ are given by

(25)$$ \begin{align} X=kd_j,\ \ Y=\begin{cases}k(Q_s-Q_j)& {\text{in }{\mathbb H}^m}\\ k(Q_s-Q_j)=-kQ_j & {\text{in }{\mathbb H}^f}\end{cases}, \ \ Z =\begin{cases}kQ_s& {\text{in }{\mathbb H}^m}\\ k(d_j+d_s) & {\text{in }{\mathbb H}^f}. \end{cases} \end{align} $$

Note that the definition of Z depends on the sector. These choices are motivated by the following proposition.

Proposition 38 (twist)

For $Z=1,\dots , k-1$, we have an isomorphism

$$ \begin{align*} \tau\colon {\mathbb H}^m_{X,Y,Z}&\to {\mathbb H}^f_{X,Y,Z}\\ (x_0^{Z-1}dx_0\wedge\phi,g)&\mapsto (\phi,s^{Z}g) \end{align*} $$

transforming $(p,q)$-classes into $(p-1+2Z/k,q)$-classes.

There are natural isomorphisms matching ${\mathbb H}^f_{X,Y,1}\cong {\mathbb H}^f_{X,Y,2}\cong \dots \cong {\mathbb H}^f_{X,Y,k-1}$ and ${\mathbb H}^m_{X,Y,1}\cong {\mathbb H}^m_{X,Y,2}\cong \dots \cong {\mathbb H}^m_{X,Y,k-1}$. We refer to them as ‘elevators’.

Proposition 39 (elevators)

For any $0\le X,Y< k$ and $0<Z'<Z''<k$, we have the isomorphisms

$$ \begin{align*} e^m_{Z',Z''}\colon{\mathbb H}^m_{X,Y,Z'}&\to {\mathbb H}^m_{X,Y,Z''}&&&&&e^f_{Z',Z''}\colon{\mathbb H}^f_{X,Y,Z'}&\to {\mathbb H}^f_{X,Y,Z''}\\ (\phi, g)&\mapsto (x_0^{Z''-Z'}\phi, g)&&&&&(\phi, g)&\mapsto (\phi, s^{Z''-Z'}g), \end{align*} $$

with $e^m_{Z',Z''}$ and $e^f_{Z',Z''}$ transforming $(p,q)$-classes into classes whose bidegrees equal $(p-(Z''-Z')/k,q+(Z''-Z')/k)$ and $(p+(Z''-Z')/k,q+(Z''-Z')/k)$, respectively. □

For $0<Z'<Z''<k$, we set $e^m_{Z'',Z'}=(e^m_{Z',Z''})^{-1}$ and $e^f_{Z'',Z'}=(e^f_{Z',Z''})^{-1}$.

Proposition 34 specializes to the following statement.

Proposition 40. The mirror isomorphism M yields isomorphisms

$$ \begin{align*} M\colon {\mathbb H}^f_{X,Y,Z}\xrightarrow{\sim} ({\mathbb H}^{\vee})^m_{Y,X,Z},\qquad \qquad M\colon {\mathbb H}^m_{X,Y,Z}&\xrightarrow{\sim} ({\mathbb H}^{\vee})^f_{Y,X,Z}. \end{align*} $$

Proof. Let us consider M as a morphism mapping $U_W$ to $U_{W^{\vee }}$. For $W=(x_0)^k+f$, we have $W^{\vee }=((x_0)^k)^{\vee }+f^{\vee }=(x_0)^k+f^{\vee }$. Using equation (21) and Thom–Sebastiani, every state of the form $(\phi ,g)\in U_W$ can be regarded as an element of

$$ \begin{align*} (\phi,g)\in U^{b_1}_{a_1}\otimes U^{b_2}_{a_2} \end{align*} $$

with $a_1\in \operatorname {Aut}_{(x_0)^k}={\mathbb Z}/k$, $b_1\in \operatorname {Aut}_{((x_0)^k)^{\vee }}={\mathbb Z}/k$, $a_2\in \operatorname {Aut}_{f}$ and $b_2\in \operatorname {Aut}_{f^{\vee }}.$ Example 30 shows that there are only two possibilities: (1) $b_1$ is the identity element or (2) $a_1$ is the identity element. More precisely, in case (1), $(\phi ,g)$ is in ${\mathbb H}^f$, it is fixed by s, $b_1$ is the trivial symmetry $1\in \operatorname {Aut}_{(x_0)^k}$ and $a_1$ is the nontrivial character corresponding to $kd_s\in {\mathbb Z}/k\setminus \{0\}$. In case (2), $(\phi ,g)$ is in ${\mathbb H}^m$, it is not fixed by s and $a_1$ is trivial whereas $b_1$ is the nontrivial character $kQ_s \in {\mathbb Z}/k\setminus \{0\}$. Since M exchanges $a_1$ and $b_1$, this proves that M exchanges ${\mathbb H}^m$ and ${\mathbb H}^f$ and preserves the coordinate Z which coincides with $kd_s$ and $kQ_s$ within ${\mathbb H}^m$ and ${\mathbb H}^f$.

Furthermore, M maps $U_{a_2}^{b_2}(f)$ to $U_{b_2}^{a_2}(f^{\vee })$ with $a_2\in \operatorname {SL}_f[j_f]$ and $b_2\in \operatorname {SL}_{f^{\vee }}[j_{f^{\vee }}]$. We recall that $j_f^k\in \operatorname {SL}$ on both sides; therefore, $\det a_2$ and $\det b_2$ are ${\pmb \mu }_k$-characters. The claim $(X,Y,Z)\mapsto (Y,X,Z)$ follows from

$$ \begin{align*} \det a_2=-kd_j, \qquad \det b_2=-kQ_s+kQ_j, \end{align*} $$

where ${\pmb \mu }_k$-characters are identified with elements of ${\mathbb Z}/k$. The first identity is immediate: $a_2$ is related to $(\phi ,g)\in U_W$ by $a_2=g\!\mid _{x_0=0}$. The identity follows from $\det (j\!\!\mid _{x_0=0})=\xi _k^{-1}$ by the Calabi–Yau condition. The second identity follows from the definition of

$$ \begin{align*} \ell_{a_2}\colon \operatorname{Jac}(f_{a_2})\to \operatorname{Aut}(f^{\vee}), \qquad \prod_j x_j^{b_j-1}\bigwedge_{j} dx_j\mapsto \prod_j \overline \rho_j^{b_j} \end{align*} $$

from equation (20): The determinant of $\overline \rho _j$ is $\xi _d^{w_j}$; hence, $\det (\prod _j \overline \rho _j^{b_j})$ is identified with the $j_f$-charge $Q_{j_f}$ of the form $\phi $ restricted to $(x_0=0)$. This yields an identification between $\det b_2$ and the ${\pmb \mu }_k$-character $kQ_j-kQ_s$.

In view of the above proposition, mirror symmetry operates as a plane symmetry exchanging the two blocks; see Figure 1.

Figure 1 Two blocks (with elevators) representing the coordinates of the moving subspace ${\mathbb H}^m$ and of the fixed subspace ${\mathbb H}^f$. The condition $Q_j=0$ defines a plane cutting the diagonal D of the left-hand side face of the moving block; D represents the moving part of $H^*_{\operatorname {id}}(\Sigma _{W,H};{\mathbb C})$. On the fixed block, the same condition $Q_j=0$ defines the face on the right-hand side; within it, the diagonal $D'$ is symmetrical to D and represents the fixed part of $H^*_{\operatorname {id}}(\Sigma _{W,H};{\mathbb C})$.

Remark 41. For Fermat potentials, all the above discussion can be carried out more explicitly because the group elements $\pmb a=\prod _{i=1}^n \rho _i^{a_i}$ coincide with $\frac 1d[a_1w_1,\dots ,a_nw_n]$. By adopting this notation, the space $U_{a}^{b}(W)$ may be regarded as the one-dimensional space spanned by

$$ \begin{align*} \left (\phi=\prod_{b_i>0} x_i^{b_i-1} \bigwedge_{b_i>0} dx_i, \ \ \pmb a=\prod_{i=0}^n\rho_i^{a_i}\right), \end{align*} $$

where

(26)$$ \begin{align} a_i=0 \Leftrightarrow b_i\neq 0. \end{align} $$

Mirror symmetry is simply an exchange of the $w{\mathbb Z}/d{\mathbb Z}$-valued vectors $\pmb a$ and $\pmb b$. By unravelling Definition 23, the bidegree $(p,q)$ is given by

$$ \begin{align*} \left(\#(\pmb b)-\sum_{i=0}^n b_i \frac{w_i}d + \sum_{i=0}^n a_i \frac{w_i}d, \sum_{i=0}^n b_i \frac{w_i}d+ \sum_{i=0}^n a_i \frac{w_i}d,\right), \end{align*} $$

where $\#(\pmb b)$ is the number of elements i such that $b_i\neq 0$. Notice that $Q_s$ is $b_0/k$ and $d_s+d_j$ is $a_0/k$; therefore, the equivalence in Proposition 37 reads $b_0=0$ is a special case of equation (26). Furthermore, we have

$$ \begin{align*}(X,Y,Z)=\begin{cases} (a_0-|\pmb a|,b_0-|\pmb b|,b_0)&\text {on the moving side,}\\ (a_0-|\pmb a|,b_0-|\pmb b|,a_0)&\text {on the fixed side.} \end{cases} \end{align*} $$

It is now clear that M exchanges the moving side with the fixed side, $(X,Y,Z)$ with $(Y,X,Z)$ and $(p,q)$ with $(n+1-p,q)$.

In view of Proposition 35, we obtain the j-invariant contribution by setting $Q_j=0$. By equation (25), this amounts to imposing $Y=Z$ within ${\mathbb H}^m$ and $Y=0$ within ${\mathbb H}^f$. We get

$$ \begin{align*} \left(\bigoplus_{\substack{{0\le X< k}\\{0< Z< k}}} {\mathbb H}^m_{X,Z,Z} \right)\oplus \left(\bigoplus_{\substack{{0\le X< k}\\{0< Z< k}}} {\mathbb H}^f_{X,0,Z} \right) \end{align*} $$

We get a picture of the j-invariant state space $\mathcal H_{W,K[j_W],\operatorname {id}}$ by setting $X=0$ within ${\mathbb H}^m$ and $X=Z$ within ${\mathbb H}^f$

$$ \begin{align*} \mathcal H_{W,K[j_W],\operatorname{id}}=\left(\bigoplus_{\substack{{0< t< k}}} {\mathbb H}^m_{0,t,t} \right)\oplus\left(\bigoplus_{\substack{{0<t< k}}} {\mathbb H}^f_{t,0,t}\right) \end{align*} $$

(we refer to Figure 1). More generally, the $(d_s=b)$-part of ${\mathbb H}^{j_W}$ is the state space $\mathcal H_{W,K[j_W],s^b}$ (see Proposition 35). By equation (25), we obtain it by setting $X=-b$ within ${\mathbb H}^m$ and $Z=X+b$ within ${\mathbb H}^f$

(27)$$ \begin{align} \mathcal H_{W,K[j_W],s^b}=\left(\bigoplus_{\substack{{0< t< k}}} {\mathbb H}^m_{-b,t,t} \right)\oplus\left(\bigoplus_{\substack{{0\le t< k}\\t+b\neq 0}} {\mathbb H}^f_{t,0,t+b}\right). \end{align} $$

Notice that the second summand only depends on ${\mathbb H}^f_{0,0,1}$ and ${\mathbb H}^f_{1,0,1},\dots {\mathbb H}^f_{k-1,0,k-1}$ since, for $b\neq 0$, it equals

(28)$$ \begin{align} \mathcal H_{W,K[j_W],s^b}= \left(\bigoplus_{\substack{{0< t< k}}} {\mathbb H}^m_{-b,t,t} \right)\oplus\left(e^f_{1,b}({\mathbb H}^f_{0,0,1})\oplus \bigoplus_{\substack{{0< t< k}\\t+b\neq 0}} e^f_{t,t+b}({\mathbb H}^f_{t,0,t})\right), \end{align} $$

with the convention $(e^f_{i,j})=(e^f_{j,i})^{-1}$ if $j<i$. By Proposition 35, the above data correspond to $H_{s^b}^*(\Sigma _{W,K[j_W]};{\mathbb C})$ under the Calabi–Yau condition.

Proposition 9 relates it to the cohomology of an $s^b$-fixed locus within a crepant resolution. Using this geometric picture, we can predict some vanishing conditions, which we prove in general, without relying on any Calabi–Yau condition in the next proposition. The first guess is immediate: Since $\langle s\rangle $ operates trivially on an s-fixed locus, it is natural to expect that ${\mathbb H}^m_{-1,t,t}$ vanishes for all t. More generally, since $\langle s^b\rangle $ operates trivially on an $s^b$-fixed locus, we expect that ${\mathbb H}^m_{-b,t,t}$ vanishes if $tb\equiv 0$ mod ${\mathbb Z}$. We prove that this holds true regardless of any Calabi–Yau condition or existence of crepant resolution.

Proposition 42. Let $b\in \{0,\dots , k-1\}$. We have

$$ \begin{align*} {\mathbb H}^m_{b,t,t}= 0, \end{align*} $$

unless t is a multiple of $k/\gcd (b,k)$ in $k{\mathbb Z}$.

Proof. We prove that

$$ \begin{align*} {\mathbb H}^m_{b,t,t}\neq 0 \end{align*} $$

implies $bt\in k{\mathbb Z}$. Recall that $\frac {t}k$ equals $Q_s-Q_j$. For ${\mathbb H}^m_{b,t,t}\neq 0$, we can compute $Q_s-Q_j$ explicitly using $(\phi ,j^b g)\in {\mathbb H}^m_{b,t,t}$ with $g\in \operatorname {Aut}_f$ and $\phi $ a g-invariant form

$$ \begin{align*} \phi=x_0^{kQ_s-1}\prod_{l\in I'}x^{b_l-1} dx_0\wedge\bigwedge_{l\in I'}dx_l \end{align*} $$

with $I'=\{l\ge 1\mid j^b g\cdot x_l=x_l\}\subseteq \{1,\dots , N\}$. Using $w_0/d=1/k$, we get

$$ \begin{align*} Q_s-Q_j=Q_s-kQ_s\frac1k-\sum_{l\in I'} b_l\frac{w_l}d=-\sum_{l\in I'} b_l\frac{w_l}d. \end{align*} $$

Let us write g as $[0,p_1\dots , p_n]\in \operatorname {Aut}_f$; then we have $l\in I'$ if and only if

$$ \begin{align*} b\frac{w_l}d + p_l\in {\mathbb Z}. \end{align*} $$

Now note that

$$ \begin{align*} b\frac{t}k = -b\sum_{l\in I} b_l\frac{w_l}d= -\sum_{l\in I} b_l b\frac{w_l}d. \end{align*} $$

Up to an element of ${\mathbb Z}$, this is

$$ \begin{align*} \sum_{l\in I} b_l p_l\in {\mathbb Z}, \end{align*} $$

where the last relation holds since $\phi $ is g-invariant. So k divides $bt$.

6.4 Mirror symmetry on the Landau–Ginzburg side

In this section, we derive an interpretation of Proposition 40 in terms of the Landau–Ginzburg state space. This amounts to expressing both sides of the isomorphism ${\mathbb H}^m_{X,Y,Z}\cong {\mathbb H}^f_{Y,X,Z}$ in terms of $j_W$-invariant spaces.

Consider the $j_W$-invariant summands

$$ \begin{align*} {\mathbb H}^m_{X,Y,Y}\subset {\mathbb H}^m \ \ \text{and}\ \ {\mathbb H}^f_{X,0,Z}\subset {\mathbb H}^f. \end{align*} $$

Their mirrors (i.e., their image under M) are $({\mathbb H}^{\vee })^f_{Y,X,Y}$ and $({\mathbb H}^{\vee })^m_{0,X,Z}$ and lie in the $j_W$-invariant part if and only if $X=0$ and $X=Z$. This happens if and only if we consider the mirror of $[{\mathbb H}\mid Q_j=d_s=0]$ (imposing $X=0$ in ${\mathbb H}^m$ and $X=Z$ in ${\mathbb H}^f$ is the same as requiring $d_s=0$).

We obtain the first consequence of Proposition 40. Let

$$ \begin{align*} \left[\mathcal H_{W,H,\operatorname{id}}\right]^{p,q}_{\chi_s=i} \end{align*} $$

be the eigenspace on which s operates as the character $i\in {\mathbb Z}/k{\mathbb Z}$. For any $H\in \operatorname {Aut}_W$ containing $j_W$, we have

(29)$$ \begin{align} M\colon \left[\mathcal H_{W,H,\operatorname{id}}\right]^{p,q}_{\chi_s=0}\xrightarrow{ \ \ \cong\ \ } \bigoplus_{i=1}^{k-1} \left[\mathcal H_{W^{\vee},H^{\vee},\operatorname{id}}\right]^{n+1-p,q}_{\chi_s=i}, \end{align} $$

where $H=K[j]$.

We now study ${\mathbb H}^f_{0,0,i}$. By Proposition 40, the subspace $({\mathbb H}^{\vee })^f_{0,0,i}$ mirrors ${\mathbb H}^m_{0,0,i}$ under M. By applying the twist $\tau $ from Proposition 38, we land again on $({\mathbb H}^{\vee })^f_{0,0,i}$ which is a part of the j-invariant state space $\mathcal H_{W^{\vee },H^{\vee },s^i}$.

Recall that the elevator maps for $t \neq 0,k-i$

$$ \begin{align*} e^f_{t,t+i}:{\mathbb H}^f_{t,0,t} \xrightarrow{\ \ \ \ \ \ } {\mathbb H}^f_{t,0,t+i} \end{align*} $$

can be viewed as mapping into components of $\mathcal H_{ W,H,s^i}^s$ (see equation (27)). We can conclude from equation (28) that the homomorphism

(30)$$ \begin{align} el_i:=\bigoplus_{t\neq 0,k-i} e^f_{t,t+i} \colon \bigoplus_{\substack{{t \neq 0, k-i}}} {\mathbb H}^f_{t,0,t} \xrightarrow{\ \ \ \ \ \ } \left[\mathcal H_{ W,H,s^i}\right]^s \end{align} $$

has cokernel ${\mathbb H}_{0,0,i}^f$. Here $t+i$ is understood to be mod k. This means that the effect of the map on grading can be described as

$$\begin{align*}\operatorname{im}(el_i)(\tfrac{i}{k})\cong \bigoplus_{0<t<k-i} {\mathbb H}^f_{t,0,t} \oplus \bigoplus_{k-i<t<k} {\mathbb H}^f_{t,0,t}(1).\end{align*}$$

We write $el^{\vee }_i$ for the same construction on the mirror. We conclude

(31)$$ \begin{align} M\colon \left[\frac{\mathcal H_{W,H,s^i}(\tfrac{i}{k})^s}{\operatorname{im}(el_i)(\tfrac{i}{k})}\right]^{p,q} \xrightarrow{\ \ \cong\ \ } \left[\frac{\mathcal H_{W^{\vee},H^{\vee},s^i}(\tfrac{i}{k})^s}{\operatorname{im}(el^{\vee}_i)(\tfrac{i}{k})}\right]^{n-p,q}, \end{align} $$

where the bidegrees have been computed using $\mathcal H(\frac {i}{k})^{p,q}=\mathcal H^{p+i/k,q+i/k}$, the fact that $M_W$ transforms $(p,q)$-classes to $(n+1-p,q)$-classes and the twist $\tau $ maps $(p,q)$-classes to $(p-1+\frac {2i}{k},q)$-classes.

Note that using equation (29) (recalling that $d_j$ and $d_s$ switch under mirror symmetry) we can write

$$\begin{align*}[\operatorname{im}(el^{\vee}_i)(\tfrac{i}{k})]^{n-p,q}= \bigoplus_{0<j<k-i} \left[\mathcal H_{W,H,\operatorname{id}}(1,0)\right]^{p,q}_{\chi_s=j}\oplus \bigoplus_{k-i<j<k} \left[\mathcal H_{W,H,\operatorname{id}}(0,1)\right]^{p,q}_{\chi_s=j}.\end{align*}$$

Write $\mathcal H_{g}$ for $\mathcal H_{W,H,g}$ and $\mathcal H^{\vee }_g$ for $\mathcal H_{W^{\vee },H^{\vee },g}.$ We obtain

(32)$$ \begin{align} \begin{aligned} M\colon & [\mathcal H_{s^i}\left(\tfrac{i}{k}\right)]_{\chi_s=0}^{p,q} \oplus \bigoplus_{j<k-i} \left[\mathcal H_{\operatorname{id}}(1,0)\right]^{p,q}_{\chi_s=j}\oplus \bigoplus_{j>k-i} \left[\mathcal H_{\operatorname{id}}(0,1)\right]^{p,q}_{\chi_s=j} \xrightarrow{\ \ \cong\ \ } \\ &\quad [\mathcal H^{\vee}_{s^i}\left(\tfrac{i}{k}\right)]_{\chi_s=0}^{n-p,q} \oplus \bigoplus_{j<k-i} \left[\mathcal H^{\vee}_{\operatorname{id}}(1,0)\right]^{n-p,q}_{\chi_s=j}\oplus \bigoplus_{j>k-i} \left[\mathcal H^{\vee}_{\operatorname{id}}(0,1)\right]^{n-p,q}_{\chi_s=j}, \end{aligned} \end{align} $$

where $0<j<k.$ Finally, we focus on the moving part of $\mathcal H_{W,K[j_W],s^b}$ which, by equation (28) can be written as $\bigoplus _{0< t< k} {\mathbb H}^m_{k-b,t,t}$. This is the decomposition of $\mathcal H_{W,K[j],s^b}$ into eigenspaces corresponding to the s-action operating as the character $t\in {\mathbb Z}/k$. By applying the mirror map $M_W$, the twist $\tau ^{-1}$ and the elevator $e^f_{t,k-b}$ we get

$$ \begin{align*} {\mathbb H}^m_{k-b,t,t}\xrightarrow{\ M_W\ } ({\mathbb H}^{\vee})^f_{t,k-b,t}\xrightarrow{\ \tau^{-1}\ } ({\mathbb H}^{\vee})^m_{t,k-b,t}\xrightarrow{\ e^m_{t,k-b}\ } ({\mathbb H}^{\vee})^m_{t,k-b,k-b}. \end{align*} $$

Therefore, we have

(33)$$ \begin{align} \left[\mathcal H_{W,H,s^b}\left(\textstyle\frac{b}{k}\right)\right]^{p,q}_{\chi_s=t}\cong \left[\mathcal H_{W^{\vee},H^{\vee},s^t}\left(\textstyle\frac{k-t}{k}\right)\right]^{n-p,q}_{\chi_s=k-b}.\end{align} $$

Notice that the map on the bidegrees is the composite of

1. 1. a shift $(p,q)\mapsto (p+b/k,q+b/k)$,

2. 2. mirror symmetry $(p,q)\mapsto (n+1-p,q)$,

3. 3. $\tau ^{-1}$ yielding $(p,q)\mapsto (p+1-2t/k,q)$,

4. 4. the elevator yielding $(p,q)\mapsto (p+(b-(k-t))/k,q+(b+(k-t))/k)$,

5. 5. a shift backwards $(p,q)\mapsto (p-(k-t)/k,q-(k-t)/k)$,

inducing $(p,q)\mapsto (n-p,q)$.

We now are almost ready to state the theorem, which will follow from applying equations (29), (32) and (33) to the geometric interpretation (18) of the Landau–Ginzburg state space in terms of relative cohomology of $({\mathbb V}_H,{\mathbb F}_{W,H})$ (provided in §5.3).

Let $W=x_0^k+f(x_1,\dots ,x_n)$ be a quasi-homogeneous nondegenerate polynomial of degree d and weights $w_1,\dots ,w_n$. Assume $j_W\in H\subseteq \operatorname {SL}_W$ (in particular, $\sum _j w_j$ is a positive multiple of $d\mathbb N$). Then W descends to ${\mathbb V}_{H}=[{\mathbb V}/H_0]\to {\mathbb C}$ and its generic fibre is ${\mathbb F}_{W,H}$. Consider the automorphism $s=[\frac 1k,0,\dots ,0]\colon {\mathbb V}_H\to {\mathbb V}_H$, the orbifold cohomology groups $H_{\operatorname {id}}^*({\mathbb V}_H,{\mathbb F}_{W,H})$ and $H_{s}^*({\mathbb V}_H,{\mathbb F}_{W,H})$.

For $0<i<k$, define $\overline {H}^*_{\operatorname {id},i}({\mathbb V}_{H},{\mathbb F}_{W,H})$ to be the bigraded vector space

$$\begin{align*}\bigoplus_{j<k-i}\left[H^*_{\operatorname{id}}({\mathbb V}_{H},{\mathbb F}_{W,H})\left(1,0\right)\right]_{\chi_s=j} \oplus \bigoplus_{j>k-i}\left[H^*_{\operatorname{id}}({\mathbb V}_{H},{\mathbb F}_{W,H})\left(0,1\right)\right]_{\chi_s=j};\end{align*}$$

here $j \in \{1,\dots ,k-1\}.$ This is the padding needed to state the mirror theorem.