Mirror symmetry and automorphisms

TL;DR

This paper extends mirror duality for Calabi-Yau orbifolds by incorporating automorphisms, linking fixed loci and cohomology actions, and applies it to K3 surfaces to unify different mirror symmetry frameworks.

Contribution

It introduces an enhanced mirror duality involving automorphisms, connecting fixed loci and cohomology actions, and proves a relation between Berglund-Hübsch and K3 lattice mirror symmetry.

Findings

Automorphisms influence mirror symmetry via fixed loci and cohomology actions.

The new duality matches automorphism weights and fixed loci, not just cohomology classes.

Application to K3 surfaces confirms the equivalence of different mirror symmetry approaches.

Abstract

We show that there is an extra dimension to the mirror duality discovered in the early nineties by Greene-Plesser and Berglund-H\"ubsch. Their duality matches cohomology classes of two Calabi--Yau orbifolds. When both orbifolds are equipped with an automorphism $s$ of the same order, our mirror duality involves the weight of the action of $s^*$ on cohomology. In particular, it matches the respective $s$-fixed loci, which are not Calabi-Yau in general. When applied to K3 surfaces with non-symplectic automorphism $s$ of odd prime order, this provides a proof that Berglund-H\"ubsch mirror symmetry implies K3 lattice mirror symmetry replacing earlier case-by-case treatments.

Equations

$$h^{p,q}(X)=h^{d-p,q}(X^{\vee }).$$

$$H_{\mathrm{CR}}^{p,q}({\Sigma }_{W,H};\mathbb{C})=H_{\mathrm{CR}}^{d-p,q}({\Sigma }_{W^{\vee },H^{\vee }};\mathbb{C}),$$

$$H^{\ast }(,\mathbb{C}{)}_i=\{x\mid s^{\ast }x=i(s)x\}.$$

$$H_{orb}^{p,q}({\Sigma }_{W,H};\mathbb{C}{)}_0\cong \underset{i=1}{\overset{k-1}{⨁}}{H}_{orb}^{d-p,q}({\Sigma }_{W^{\vee },H^{\vee }};\mathbb{C}{)}_i,$$

$$\mathfrak{X}\underset{s,\mathfrak{X}\times \mathfrak{X},\mathrm{id}}{\times }\mathfrak{X},$$

$$H_{s^b}^{p,q}(\Sigma ){\left(\frac{b}{k}\right)}_t\cong H_{s^{-t}}^{d-p,q}({\Sigma }^{\vee }){\left(\frac{k-t}{k}\right)}_{-b},$$

$${\left[H_{s^j}^{p,q}(\Sigma )\left(\frac{j}{k}\right)\right]}^s\oplus {\overline{H}}_{\mathrm{id},j}^{p,q}(\Sigma )\cong {\left[H_{s^j}^{d-p,q}({\Sigma }^{\vee })\left(\frac{j}{k}\right)\right]}^s\oplus {\overline{H}}_{\mathrm{id},j}^{d-p,q}({\Sigma }^{\vee }).$$

$$Z_{\mathbb{P}(\mathbf{w})}(f)=[U/{\mathbb{G}}_m],\text{ with U=Z(f)⊂Cn∖0.}$$

$$W(x_0,x_1,\dots ,x_n)=x_0^k+f(x_1,\dots ,x_n)$$

$$P({\lambda }^{w_0}{x}_0,\dots ,{\lambda }^{w_n}{x}_n)={\lambda }^{d}P(x_0,\dots ,x_n),$$

$${\Sigma }_P=Z_{\mathbb{P}(\mathbf{w})}(P)\subset \mathbb{P}(\mathbf{w})$$

$$\sum _{i=0}^n{w}_i=d.$$

$${\mathrm{Aut}}_P=\{\mathrm{diag}({\alpha }_0,\dots ,{\alpha }_n)\mid P({\alpha }_0{x}_0,\dots ,{\alpha }_n{x}_n)=P(x_0,\dots ,x_n)\}$$

$$[a_0,\dots ,a_n]=\mathrm{diag}(\exp (2\pi \mathtt{i}{a}_i){)}_{i=0}^n$$

$$\mathrm{age}[a_0,\dots ,a_n]=\sum _{i=0}^n{a}_i.$$

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

$$M_P\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\Sigma }_{P,H}=[{\Sigma }_P/H_0],M_{P,H}=[M_P/H];$$

$${\Sigma }_{P,H}\subset [\mathbb{P}(\mathbf{w})/H_0],$$

$${\mathrm{SL}}_P:={\mathrm{Aut}}_P\cap \mathrm{SL}(n+1;\mathbb{C}).$$

$$W(x_0,x_1,\dots ,x_n)=(x_0{)}^k+f(x_1,\dots ,x_n)$$

$$s=\left[\frac{1}{k},0,\dots ,0\right].$$

$$\mathsf{m}:\mathbb{Z}/k\times {\Sigma }_{W,H}\to {\Sigma }_{W,H}.$$

$$(j_f{)}^k\in K\subseteq {\mathrm{SL}}_f,$$

$$K[j_W,s]=\sum _{a,b=0}^{k-1}(j_W{)}^a(s{)}^{b}K,$$

$$d_j=\frac{a}{k},d_s=\frac{b}{k}.$$

$$\mathsf{m}:G\times \mathfrak{X}\to \mathfrak{X}.$$

$$I_G(\mathfrak{X})\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\mathfrak{a}:I\mathfrak{X}\to \mathbb{Q}$$

$$I_G(\mathfrak{X})=(G\times \mathfrak{X}){\times }_{(\mathsf{m},\mathsf{p}{\mathsf{r}}_2),\mathfrak{X}\times \mathfrak{X},\Delta }\mathfrak{X}=\underset{g\in G}{⨆}{I}_g(\mathfrak{X})$$

Full text

1 Introduction

The earliest formulation of mirror symmetry relates pairs of $d$-dimensional Calabi–Yau manifolds $X,X^{\vee}$ with mirror Hodge diamonds:

[TABLE]

In the early 1990s, physicists Greene, Morrison, and Plesser found many such mirror pairs [19], starting with a Calabi–Yau (and Fermat) hypersurface in projective space and constructing a mirror, which is a resolution of the quotient of the same hypersurface by a finite group. In 1992, this construction was generalized by Berglund–Hübsch [6], starting with a Calabi–Yau given as a quotient of a more general hypersurface in weighted projective spaces by a finite group. The hypersurface is a Calabi–Yau orbifold defined as the zero locus of a quasi-homogenous polynomial $W=\sum_{i=0}^{n}\prod_{j=0}^{n}x_{j}^{m_{ij}}$ such that $W$ is non-degenerate and “invertible” (i.e. with as many variables as monomials). After quotienting out by a finite group $H$ of diagonal symmetries within $\operatorname{SL}(n+1;\mathbb{C})$ one obtains the orbifold $\Sigma_{W,H}$. The mirror $\Sigma_{W^{\vee},H^{\vee}}$ is another such quotient of a hypersurface modulo a finite group. The hypersurface is given by the polynomial $W^{\vee}$, defined by transposing the matrix of the exponents $E=[m_{ij}]$ of $W$. The group $H^{\vee}$ is a subgroup of $\operatorname{SL}(n+1;\mathbb{C})$ Cartier dual to $H$ and preserving $W^{\vee}$, see (13). Then, the mirror duality can be stated in terms of orbifold Chen–Ruan cohomology as

[TABLE]

which implies the same relation in ordinary cohomology whenever there exists crepant resolutions.

The striking mirror relation above becomes elementary when we look at it through the lenses of singularity theory or, in physics terminology, the Landau–Ginzburg (LG) model. This happens because mirror symmetry holds for LG models without any Calabi–Yau condition. In this paper we present this change of perspective through the LG model via the crepant resolution of a singularity, see Section 5. This not only allows us to simplify previous proofs of LG/CY correspondence by the first author with Ruan [9]; it also yields a new statement of mirror symmetry relating the fixed loci of powers of an isomorphism $s$ of $\Sigma$, the Hodge decomposition, and the weights the representation $s^{*}$ in cohomology.

Let $W=x_{0}^{k}+f(x_{1},\dots,x_{n})$ be a non-degenerate, quasi-homogenous, invertible polynomial. Let us consider again the automorphisms groups $H\subseteq\operatorname{Aut}W$ and its dual $H^{\vee}\in\operatorname{Aut}W$ within $\operatorname{SL}(n+1;\mathbb{C})$. The Calabi–Yau orbifolds $\Sigma_{W,H}$, $\Sigma_{W^{\vee},H^{\vee}}$ are equipped with the action by the group ${\boldsymbol{\mu}}_{k}$ of $k$th roots of unity spanned by $s\colon x_{0}\mapsto e^{2\pi i/k}x_{0}$. For $i$ in the group of characters $\mathbb{Z}/k=\operatorname{Hom}({\boldsymbol{\mu}}_{k};\mathbb{G}_{m})$ we consider the weight-$i$ term of cohomology

[TABLE]

The first statement is that the $s$-invariant cohomology mirrors the “moving” cohomology: the sum of all cycles of nonvanishing weight.

Theorem A (see Thm 35, part 1). Consider the mirror pair $s\colon\Sigma_{W,H}\to\Sigma_{W,H}$ and $s\colon\Sigma_{W^{\vee},H^{\vee}}\to\Sigma_{W^{\vee},H^{\vee}}$. We have

[TABLE]

where $d=n-1$ is the dimension of $\Sigma_{W,H}$.

The locus of geometric points of $\Sigma_{W,H}$ which are fixed by $s$ also exhibits a mirror phenomenon. Since $\Sigma_{W,H}$ is a stack, let us provide a definition for this $s$-fixed locus. For $s$ a finite order automorphism acting on a smooth Deligne–Mumford orbifold, we consider the graph of $\Gamma_{s}\colon\mathfrak{X}\to\mathfrak{X}\times\mathfrak{X}$ and its intersection with the graph of the identity (the diagonal morphism)

[TABLE]

(we write $s$ and $\operatorname{id}$ instead of the respective graphs). We recall that orbifold cohomology is simply the (age-shifted) cohomology of this product for $s=\operatorname{id}$. The $s$-orbifold cohomology is defined as the age-shifted cohomology of the above fibred product in general (see Defn. 5). This is a bi-graded vector space and, if the coarse space $X$ of $\mathfrak{X}$ admits a crepant resolution $\widetilde{X}$ where $s$ lifts, there is a bidegree-preserving isomorphism $H_{s}^{*}(\mathfrak{X};\mathbb{C})\cong H_{s}^{*}(\widetilde{X};\mathbb{C}),$ where the right hand side is the age-shifted cohomology of the $s$-fixed locus in $\widetilde{X}$, see Prop. 6.

Under the same conditions on $W$ and $H$ as above, set $\Sigma=\Sigma_{W,H}$ and $\Sigma^{\vee}=\Sigma_{W^{\vee},H^{\vee}}$. If the order $k$ of $s$ is not prime, then $s$ acts non-trivially on the fixed locus of powers of $s$. The $s$-moving cohomology of the fixed locus of powers of $s$ mirrors the same on $\Sigma^{\vee}$, interweaving the weight and the exponent of the power of $s$.

Theorem B (see Thm 35, part 3). *Let $b,t\neq 0$. Then, we have *

[TABLE]

where $d=n-2$, the largest dimension of the components of the $s$-fixed locus.

Finally, also the fixed cohomology of each power $s^{j}$ exhibits a mirror phenomenon, but only after adding certain moving cycles in $\Sigma$. Namely, the cycles we add are all those whose weight differs from [math] (i.e. moving cycles) and from $j$ (the exponent of $s$). We denote this group by $\overline{H}^{p,q}_{\operatorname{id},j}(\Sigma)$, see (26).

Theorem C (see Thm 35, part 2). * For $0<j<k$, we have*

[TABLE]

The correcting terms $\overline{H}^{*}$ disappear when $k=2$ (for $k=2$, we have $s^{j}=s$ and there is no positive weight except $1$). This shows how the statement above specialises to the construction of Borcea–Voisin mirror pairs (see [12]).

In dimension 2, and after resolving, these results are about mirror symmetry for K3 surfaces with non-symplectic automorphisms. Suppose $X$ and $X^{\vee}$ are crepant resolutions of $\Sigma_{W,H}$ and $\Sigma_{W^{\vee},H^{\vee}}$ where $W$ is a polynomial in $4$ variables. The above mirror theorems imply that the topological invariants of the fixed locus of the K3 surface $X$ controls that of $X^{\vee}$; we refer to Corollary 42 for simple formulae on the number of fixed points and the genera of the fixed curves. The automorphism $s$ also gives the K3 surface a lattice polarization: $H^{2}(X,\mathbb{Z})^{s}$. There is another version of mirror symmetry for lattice polarized K3 surfaces, arising from the work of Nikulin [24], Dolgachev [15], Voisin [31], and Borcea [7]. When the order of $s$ is odd and prime, this lattice is characterised by the invariants $(r,a)$: the rank and the discriminant. Families of lattice polarized K3 surfaces come in mirror pairs, and in the odd prime case this mirror symmetry takes a lattice with invariants $(r,a)$ to $(20-r,a)$. The following corollary is a theorem of Comparin, Lyons, Priddis, and Suggs [14] proven by case-by-case analysis. Here, it is shown directly from the above statements (see Thm. 44).

Corollary ([14]). *Let $p$ be prime and different from $2$. Let $\Sigma_{W,H}$ and $\Sigma_{W^{\vee},H^{\vee}}$ be mirror K3 orbifolds with order-$p$ automorphisms $s,s^{\vee}$, and let $\Sigma$ and $\Sigma^{\vee}$ be crepant resolutions with automorphisms also denoted $s,s^{\vee}$. Then $\Sigma$ and $\Sigma^{\vee}$ are mirror as lattice polarized K3 surfaces. *

1.1 Relation to previous work

This paper generalises the results of [12]. There, only involutions were considered; here the mirror theorems apply to automorphism of any order. There, Theorems A and C are simpler (invariant classes mirror anti-invariant classes in Theorem A and no extra terms appear in Theorem C). Theorem B does not apply in the involution case. In the above Corollary, we do not consider the order-2 case treated in [12]; in the present paper this allows us to deduce the lattice mirror symmetry statement of [14] in full.

Section 5 restates and recasts the proof of mirror symmetry through LG models and the correspondence between cohomology and LG models in terms of resolutions of singularities (see Theorem 23). This may be regarded as the outcome of the work of many authors, we refer to [22], [8], [21] [9], [18], [17] and [16] and [13] validating over the years the approach of the physicists Intriligator–Vafa [20] and Witten [32]. It is also worth mentioning that the main object of our study, a polynomial $W=x_{0}^{k}+f(x_{1},...,x_{n})$ with the cyclic symmetry group of $k$th roots of unity acting on $x_{0}$, was used in Varchenko’s proof of semicontinuity of Steenbrink’s spectra of singularities ([30] and [27]). We hope that this may lead to further explanations of mirror symmetry in the framework of singularity theory. In particular, our setup only concerns hypersurfaces in weighted projective space, it would be interesting to see if it extends to other contexts where mirror constructions are known.

Finally, it is worth mentioning that the work of Comparin, Lyons, Priddis, and Suggs [14], earlier work of Artebani, Boissière and Sarti [3] and more generally Nikulin’s classification [24] yield several tables summarising explicit treatments of K3 surfaces via resolution of singularities. Much of these data are now embodied into the $s$-weighted Hodge numbers of Theorems A, B, and C. We provide some examples for this in the tables at the end of §7.

1.2 Structure of the paper

Section 2 states notation and terminology. Section 3 presents the Berglund–Hübsch mirror symmetry construction. Section 4 sets up our generalisation of orbifold cohomology sensing the $s$-fixed locus: $s$-orbifold cohomology. Section 5 illustrates and reproves the transition to Landau-Ginzburg models which is crucial in the proof. In particular it provides a straightforward description of the LG/CY correspondence from the crepant resolution conjecture without using the combinatorial model of [9]. Section 6 is the technical heart of the paper; it proves the main theorem (Theorem 34) on the LG side. Section 7 translates the result from the LG side to the CY side. It contains Theorem 35 proving the statements A, B, and C and Theorem 44 specialising to K3 surfaces.

Aknowledgements

We are grateful to Behrang Noohi, whose explanations clarified inertia stacks to us. We are grateful to Baohua Fu, Lie Fu, Dan Israel, and Takehiko Yasuda for many helpful conversations. We thank Davide Cesare Veniani, with whom we started studying involutions of Calabi–Yau orbifolds, for continuing to share his insights and expertise.

2 Terminology

Deligne–Mumford orbifolds are smooth separated Deligne–Mumford stacks with a dense open subset isomorphic to an algebraic variety.

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 $m$. We often need to identify a stack locally. In order to avoid repeated mention of étale localization or strict Henselizations, we often use the expression “the local picture of the stack $\mathfrak{X}$ at the geometric point $x\in\mathfrak{X}$ is the same as $\mathfrak{U}$ at $u\in\mathfrak{U}$”. By this we mean that the strict Henselization of $\mathfrak{X}$ at $x$ is the same of that of $\mathfrak{U}$ at $u$. Often it is enough to say that there is an étale neighbourhood $X^{\prime}$ of $x$ and an isomorphism $X^{\prime}\to U^{\prime}$ with an étale neighbourhood $U^{\prime}$ of $u$. We refer to [25, 54.33.2] for a definition of the strict Henselization and to [1, §1.2,5] for further discussion (see in particular the “algebra-to-analysis translation”, where strict Henselizations are described analytically as the germ of $X$ at $x$).

2.2 Notation

We list here notation that occurs throughout the entire paper.

[TABLE]

Remark 1* (zero loci).*

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

[TABLE]

Remark 2* (degree shift).*

We often write $H(a)$ for $H(a,a)$.

Remark 3* (cohomology coefficients).*

We only consider cohomology with $\mathbb{C}$ coefficients; therefore, we sometimes write $H^{*}(X;\mathbb{C})$ as $H^{*}(X).$

Remark 4* (graphs and maps).*

Given an automorphism $\alpha$ of $\mathfrak{X}$, we write $\Gamma_{\alpha}$ for the graph $\mathfrak{X}\to\mathfrak{X}\times\mathfrak{X}$. However, to simplify formulæ, we often abuse notation and use $\alpha$ for the graph $\Gamma_{\alpha}$ as well as the automorphism. In this way, in subscripts, the diagonal $\Delta\colon\mathfrak{X}\to\mathfrak{X}\times\mathfrak{X}$ will be often written as $\operatorname{id}_{\mathfrak{X}}$ or simply $\operatorname{id}$.

3 Setup

We recall the general setup of non-degenerate polynomials $P$ where the theory of Jacobi rings applies. Then we introduce polynomials of the special form

[TABLE]

for $n>0$.

3.1 Non-degenerate polynomials

We consider quasi-homogeneous polynomials $P$ of degree $d$ and of weights $w_{0},\dots,w_{n}$

[TABLE]

for all $\lambda\in\mathbb{C}$. We assume that that the polynomial $P$ is non-degenerate; 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

[TABLE]

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

[TABLE]

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

Because $P$ is non-degenerate, the group of its diagonal automorphisms

[TABLE]

is finite. Indeed, the $n\times h$ matrix $E=(m_{i,j})$ of the exponents of $P=\sum_{i=1}^{h}c_{i}\prod_{j=0}^{n}x_{j}^{m_{i,j}}$ is left invertible as a consequence of the uniqueness of the vector $\left({w_{i}}/{d}\right)_{i=0}^{n}=E^{-1}\boldsymbol{1}.$ Since we are working over $\mathbb{C}$, we adopt the notation

[TABLE]

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

[TABLE]

The distinguished diagonal symmetry

[TABLE]

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 automorphism $j_{P}$ is the monodromy operator of the fibration defined by $W$ restricted to the complement in $\mathbb{C}^{n+1}$ of the zero locus $Z(P)$; we will denote by $M_{P}$ the generic Milnor fibre

[TABLE]

For any subgroup $H$ of $\operatorname{Aut}_{W}$ containing $j_{P}$ we consider the Deligne–Mumford stacks

[TABLE]

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 $[\mathbb{P}(\boldsymbol{w})/H_{0}]$

[TABLE]

and has trivial canonical bundle as soon as $P$ is Calabi–Yau and $H$ lies in

[TABLE]

3.2 Polynomials with automorphism

More specifically, we focus on polynomials of Calabi–Yau type of the form

[TABLE]

We have $\operatorname{Aut}_{W}={\boldsymbol{\mu}}_{k}\times\operatorname{Aut}_{f}$, 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

[TABLE]

We have a $\mathbb{Z}/k$-action on the stack $\Sigma_{W,H}$

[TABLE]

We have $j_{W}=s\cdot j_{f}$; where $j_{f}$ is regarded as an element of $\operatorname{Aut}_{f}$.

Instead of $H\subseteq\operatorname{Aut}W\cap\operatorname{SL}(n+1;\mathbb{C})$ containing $j_{W}$, we can equivalently work with subgroups $K\subset\operatorname{Aut}_{f}$ satisfying

[TABLE]

(we recover $H$ by considering the subgroup of $\operatorname{Aut}W$ spanned by $j_{W}$ and $K$). More generally we consider the subgroup of $\operatorname{Aut}_{W}$

[TABLE]

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

[TABLE]

By (2), $\exp(2\pi\mathtt{i}d_{s}(g))$ is the determinant of an element $g\in K[j_{W},s]$.

4 Inertia

We consider a finite group $G$ acting on a Deligne–Mumford orbifold $\mathfrak{X}$

[TABLE]

We consider the $G$-inertia stack $I_{G}(\mathfrak{X})$ fitting in the following fibre diagram

[TABLE]

When $G$ is a trivial group, $I\mathfrak{X}$ is the ordinary inertia of $\mathfrak{X}$. There is a locally constant function

[TABLE]

which assigns to each geometric point $(x\in\mathfrak{X},g\in\operatorname{Aut}(x))\in I\mathfrak{X}$ the rational number $\operatorname{age}(g)$ (see [2] and [12, §4.2]).

In this way we have

[TABLE]

where $I_{g}(\mathfrak{X})$ is the $g$-inertia orbifold

[TABLE]

The $G$-inertia stack of $\mathfrak{X}$ fits in the fibre diagram

[TABLE]

and we may regard $\mathsf{p}$ as a $G$-torsor by pullback of $\mathfrak{X}\to[\mathfrak{X}/G]$.

The $G$-action on $\bigsqcup_{g\in G}I_{g}\mathfrak{X}$ is given by conjugation on the indices and by $F_{h}\colon I_{g}\mathfrak{X}\to I_{hgh^{-1}}\mathfrak{X}$ on the components, where $F_{h}$ acts by the effect of $h$ on the first factor of (5) and by the identity on the second.

4.1 A $g$-orbifolded cohomology

The cohomology of the $g$-inertia stack coincides with the cohomology of the $g$-fixed locus when $\mathfrak{X}$ is representable. In the spirit of orbifold cohomology we define $g$-orbifold cohomology groups which are invariant under $K$-equivalence.

The $g$-orbifold cohomology is the cohomology of (5) shifted by the locally constant function “age” given by

[TABLE]

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 denote by $(r)$ the isomorphism induced by the identity at the level of the vector spaces; it identifies the Hodge decomposition of weight $n$ with the Hodge decomposition of weight $n-2r$.

We can now provide the definition of $g$-orbifolded cohomology.

Definition 5** ($g$-orbifold cohomology).**

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

[TABLE]

We point out the slight abuse of notation: $\operatorname{age}$ 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

[TABLE]

For $g=\operatorname{id}=1_{G}$, the above definition coincides with Chen–Ruan orbifold cohomology

[TABLE]

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

[TABLE]

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

Yasuda [33] proves the invariance of the Hodge decomposition of Chen–Ruan cohomology of smooth Deligne–Mumford stacks $\mathfrak{X}$ and $\mathfrak{Y}$ whenever there exists a smooth and proper Deligne–Mumuford stack $\mathfrak{Z}$ with birational morphisms $\mathfrak{Z}\to\mathfrak{X}$ and $\mathfrak{Z}\to\mathfrak{Y}$ with $\omega_{\mathfrak{Z}/\mathfrak{X}}\cong\omega_{\mathfrak{Z}/\widetilde{X}}$. In particular, for Gorenstein orbifolds Chen–Ruan cohomology coincides with the cohomology of any crepant resolution of the coarse space. Furthermore, we have the following proposition.

Proposition 6**.**

Let $G$ by a finite group acting on a Gorenstein orbifold $\mathfrak{X}$. 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 $X$. Then, for any $g\in G$ we have a bidegree-preserving isomorphism

[TABLE]

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 $\widetilde{X}_{g}\to[\widetilde{X}/G]$ and of the age function $[\widetilde{X}/G]\to\mathbb{Q}$.

Proof.

The stack $\mathfrak{X}$ and its resolution $\widetilde{X}$ are $K$-equivalent. In order to see this, we consider the $\mathfrak{Z}=\mathfrak{X}\times_{X}\widetilde{X}$ and the associated reduced stack. Then, there exists a proper birational morphism $\mathfrak{Z}^{\prime}\to\mathfrak{Z}$ such that $\mathfrak{Z}^{\prime}$ is smooth. This is explained in Sect. 4.5, §2, of Yasuda’s paper [33] (this is essentially due to Villamayor papers [28] and [29] showing the existence of resolutions compatible with smooth, in particular étale, morphisms). Actually, in his recent generalization [34], Yasuda proves that it suffices to consider the reduction and the normalization of $\mathfrak{Z}^{\prime}$, without any resolution. This happens because his new statements allows us to extend the definition of orbifold cohomology to singular or wild (in positive characteristic) Deligne–Mumford stacks.

Now we consider the abelian group $H=\langle g\rangle$. Then $\mathfrak{A}^{\prime}=[\mathfrak{X}/H]$ and $\mathfrak{A}^{\prime\prime}=[\widetilde{X}/H]$ are $K$-equivalent by the same argument. Indeed the action of $H$ descends compatibly to the coarse space $X$ and we can consider the stack $\mathfrak{A}=[X/H]$ and the morphisms $\mathfrak{A}^{\prime}\to\mathfrak{A}$ and $\mathfrak{A}^{\prime\prime}\to\mathfrak{A}$. Then, the reduced stack associated to the fibred product $\mathfrak{A}^{\prime}\times_{\mathfrak{A}}\mathfrak{A}^{\prime\prime}$ can be resolved and yields a smooth Deligne–Mumford stack $\mathfrak{Z}$ mapping to $\mathfrak{A}^{\prime}$ and $\mathfrak{A}^{\prime\prime}$. As above, the fact that the canonical bundles of $\mathfrak{X}$ and $\widetilde{X}$ are the pullback of $\omega_{X}$ is enough to show that $\mathfrak{Z}\to\mathfrak{A}^{\prime}=[\mathfrak{X}/H]$ and $\mathfrak{Z}\to\mathfrak{A}^{\prime}=[\mathfrak{X}/H]$ is a $K$-equivalence.

The desired claim follows because the cohomology of $I_{g}\mathfrak{X}$ and that of $I_{g}\widetilde{X}$ appear as summands of the Chen–Ruan cohomology groups of $[\mathfrak{X}/H]$ and of $[\widetilde{X}/H]$. Indeed they arise as the cohomology groups of the sectors attached to $g$ whose cohomology are the $g$-invariant classes of $I_{g}(\mathfrak{X})$ and $I_{g}(\widetilde{X})$. Since $g$ operates trivially on these sectors, we can regard these contributions as $H^{*}(I_{g}(\mathfrak{X});\mathbb{C})$ and $H^{*}(I_{g}(\widetilde{X});\mathbb{C})$. We should further mention that we obtain an identification at the level of the age-shifted $g$-orbifolded cohomology $H_{g}^{*}(\ \ ;\mathbb{C})$ due to the fact that the age is a rational function factoring through the age function of $[\mathfrak{X}/H]$ and of $[\widetilde{X}/H]$. ∎

Remark 7*.*

The proposition above only claims the existence of an isomorphism. In special cases in dimension $2$ we have proven the existence of an explicit isomorphism, see [12].

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 anti-symplectic involutions of orbifold K3 surfaces considered in [12] (this case occurs below for $k=2$).

Example 8*.*

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 written as $\frac{1}{k}[a,b]$ with $a+b=k-1$ (this happens because the case $a+b=2k-1$ is impossible). In this way the age shift $\mathfrak{a}_{g}$ at the fixed loci always equals $1-1/k$

[TABLE]

5 Landau–Ginzburg state space

The expression “Landau–Ginzburg” comes from physics and is often used for $\mathbb{C}$-valued functions defined on vector spaces possibly equipped with the action of a group. More generally the definition is extended to vector bundles on a stack. In this paper we only use it for the above setup $P\colon[\mathbb{C}^{n+1}/H]\to\mathbb{C},$ where $P$ is a non-degenerate polynomial and $j\in H\subseteq\operatorname{Aut}_{P}$. Indeed this may be regarded as a $\mathbb{C}$-valued function defined on a rank-$(n+1)$ vector bundle on the stack $BH=[\operatorname{Spec}\mathbb{C}/H]$. We show how this geometric setup is naturally connected to $\Sigma_{W,H}$ via $K$-equivalence.

5.1 $K$-equivalence

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

[TABLE]

its coarse space $X=\mathbb{C}^{n+1}/\langle j\rangle$, and the smooth Deligne–Mumford stack

[TABLE]

total space $\mathbb{L}$ of the line bundle of degree $-d$ on $\mathbb{P}(\boldsymbol{w})$. The stacks $\mathbb{V}$ and $\mathbb{L}$ are the two GIT quotients of $\mathbb{C}\times\mathbb{C}^{n+1}$ modulo $\mathbb{G}_{m}$ operating with weights $(-d,w_{0},\dots,w_{n+1})$. Notice that $\mathbb{V}$ without the origin coincides with the line bundle $\mathbb{L}$ without the zero section: $\mathbb{V}^{\times}=\mathbb{L}^{\times}$.

We assume that $P$ is of Calabi–Yau type in the sense of (2). Then, the canonical bundle of $\mathbb{V}$ descends to $X$ and its pullback to $\mathbb{L}$ coincides with $\omega_{\mathbb{L}}$. Following the same argument as above, by Yasuda [33], we have

[TABLE]

for any $p,q\in\mathbb{Q}$ and for any $g\in\operatorname{Aut}_{W}$.

The isomorphism $H^{d}_{g}(\mathbb{V};\mathbb{C})\to H^{d}_{g}(\mathbb{L};\mathbb{C})$ is not canonical; notice, however, that we can at least impose a compatibility with respect to the restrictions $H^{d}_{g}(\mathbb{V};\mathbb{C})\to H^{d}_{g}(\mathbb{F};\mathbb{C})$ and $H^{d}_{g}(\mathbb{L};\mathbb{C})\to H^{d}_{g}(\mathbb{F};\mathbb{C})$ for $\mathbb{F}=[M_{P}/\langle j\rangle]$ included in $\mathbb{V}^{\times}=\mathbb{L}^{\times}\subseteq\mathbb{V},\mathbb{L}$. This happens because the fundamental classes of the inertia stacks $I_{g}({\mathbb{V}})$, $I_{g}(\mathbb{L})$ and $I_{g}(\mathbb{F})$ attached to the same automorphism $\beta=g\cdot(\lambda^{-d},\lambda^{w_{1}},\dots,\lambda^{w_{n}})$ with $\lambda\in\cup_{j=1}^{m}{\boldsymbol{\mu}}_{w_{j}}$ can be identified since their bidegree equal $(\operatorname{age}(\beta),\operatorname{age}(\beta))$ by construction. In this way, we can require that (7) respects the canonical identification between the fundamental classes of $I_{g}({\mathbb{V}})$ and $I_{g}(\mathbb{L})$ and this is enough to insure that the following diagram commutes

[TABLE]

and yields a bidegree-preserving isomorphism

[TABLE]

Note that $\mathbb{F}$ can be regarded as the generic fibre of $P\colon\mathbb{V}\to\mathbb{C}$ as well as the generic fibre of $P\colon\mathbb{L}\to\mathbb{C}$. 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

[TABLE]

for any $p,q$ and for any $g\in\operatorname{Aut}_{W}$.

The left hand side is naturally identified via the Thom isomorphism to the Chen–Ruan cohomology of $\Sigma_{P,H}$ up to a (-1)-shift whereas the left hand side is naturally identified to an orbifold version of the Jacobi ring known as the FJRW or Landau–Ginzburg state space. We detail these two aspects in the next two sections.

5.2 Thom isomorphism

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

[TABLE]

This happens because the left hand side can be regarded after retraction as

[TABLE]

which is isomorphic to the $(-1)$-shifted cohomology of $\Sigma_{P}$ by the Thom isomorphism.

Equation (9) 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 $\Sigma_{P}$. However, the argument above does not yield an isomorphism respecting the orbifold cohomology bidegree. This happens because

[TABLE]

may fail in orbifold cohomology. However the first author and Nagel proved that Equation (9) holds in orbifold cohomology without changes even when the Thom isomorphism $H_{\operatorname{id}}^{*}(\mathbb{P}(\boldsymbol{w}),\mathbb{P}(\boldsymbol{w})\setminus\Sigma_{P};\mathbb{C})\cong H_{\operatorname{id}}^{*}(\Sigma_{P};\mathbb{C})(-1)$ does not. We regard

[TABLE]

as the correct formulation of Thom isomorphism in orbifold cohomology. We refer to the theorem below.

For the benefit of the reader we illustrate the study of $H_{\operatorname{id}}^{*}(\mathbb{P}(\boldsymbol{w}),\mathbb{P}(\boldsymbol{w})\setminus\Sigma_{P};\mathbb{C})$ and of $H_{\operatorname{id}}^{*}(\Sigma_{P};\mathbb{C})(-1)$ sector-by-sector. We distinguish two cases. For ordinary sectors (such as the untwisted sector) $\Sigma_{P,g}$ is a codimension-$1$ hypersurface in $\mathbb{P}(\boldsymbol{w}_{g})$. Then, there is an identification $H^{*}(\mathbb{P}(\boldsymbol{w}_{g}),\mathbb{P}(\boldsymbol{w}_{g})\setminus\Sigma_{P,g};\mathbb{C})\cong H^{*}(\Sigma_{P,g};\mathbb{C})(-1)$ and the age shift of $\mathbb{P}(\boldsymbol{w}_{g})$ coincides with that of $\Sigma_{P,g}$ since $g$ acts trivially on the normal bundle $N_{\Sigma_{P,g}/\mathbb{P}(\boldsymbol{w}_{g}))}$. On the other hand, it may happen that $\mathbb{P}(\boldsymbol{w}_{g})$ and $\Sigma_{P,g}$ coincide as we illustrate in Example 10. In these cases we have $H^{*}(\mathbb{P}(\boldsymbol{w}_{g}),\mathbb{P}(\boldsymbol{w}_{g})\setminus\Sigma_{P,g};\mathbb{C})=H^{*}(\mathbb{P}(\boldsymbol{w}_{g});\mathbb{C})\cong H^{*}(\Sigma_{P,g};\mathbb{C})$ without Tate shift $(-1)$. Furthermore the difference between the age shift of the sector $\mathbb{P}(\boldsymbol{w}_{g})$ and that of $\Sigma_{P,g}$ is strictly positive: it equals the age $q\in]0,1[$ of the character operating via $g$ on $N_{\Sigma_{P,g}/\mathbb{P}(\boldsymbol{w}_{g}))}$. In these cases we have $H^{*}(\mathbb{P}(\boldsymbol{w}_{g}),\mathbb{P}(\boldsymbol{w}_{g})\setminus\Sigma_{P,g};\mathbb{C})\cong H^{*}(\Sigma_{P,g};\mathbb{C})(-q)$. It is now possible to observe that we have

[TABLE]

as desired. The result is proven in [13] for all complete intersections. We get

Theorem 9** (Thom isomorphism, [13]).**

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

[TABLE]

The following example is added here in the sake of clarity, but plays no essential role in the rest of the text; it illustrates in a simple way the issue arising for a Calabi–Yau embedded in a nonGorenstein $\mathbb{P}(\boldsymbol{w})$.

Example 10*.*

Consider the hypersurface $\Sigma$ defined by $x^{3}+xy=0$ in $\mathbb{P}(1,2)$. It consists of two points, the orbit $p=(-x^{2},x)$ and the point $(x=0)=\mathbb{P}(2)$. We have $I(\Sigma)=\Sigma_{0}\sqcup\mathbb{P}(2)_{1}$ and $I(\mathbb{P}(1,2))=\mathbb{P}(1,2)_{0}\sqcup\mathbb{P}(2)_{1}$, where the label $j=0,1$ denotes the root of unity $\exp(2\pi\mathtt{i}j/2)$ acting as $(\lambda,\lambda^{2})$ with nonempty fixed locus. The orbifold cohomology of $\Sigma$ is $\mathbb{C}^{3}$ concentrated in degree $(0,0)$ (the age-shift does not intervene for a [math]-dimensional stack). Since the Thom isomorphism holds after a $(-1)$-shift, we compare $\mathbb{C}^{3}(-1)$ to

[TABLE]

On the other hand, we set $\mathbb{F}$ via $P=x^{3}+xy$ as above; then $\mathbb{C}^{3}(-1)$ matches

[TABLE]

We refer to [13, Prop. 3.4].

Remark 11*.*

The ambient cohomology of $\Sigma_{P,H}$ is Poincaré dual to the image of the the homology of $I_{g}(\Sigma_{P,H})$ within the homology of $I_{g}(\mathbb{P}(\boldsymbol{w}))$. By the identification above we may regard it also as the image of the morphism

[TABLE]

We can also consider the primitive cohomology of $I_{g}(\Sigma_{P,H})$ whose direct image vanish in $I_{g}(\mathbb{P}(\boldsymbol{w}))$. Than the kernel of the morphism (11) above matches the primitive cohomology of $\Sigma_{P,H}$ in $H^{*}(\Sigma_{P,H};\mathbb{C})(1)$

[TABLE]

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

[TABLE]

regarded as a $\mathbb{C}$-vector space, has dimension $\prod_{j}\frac{d-w_{j}}{w_{j}}$ (due to the non-degeneracy of the polynomial $P$) and is isomorphic to $H^{*}(\mathbb{C}^{n},M_{P};\mathbb{C})$. The natural monodromy action of ${\boldsymbol{\mu}}_{d}=\langle j\rangle$ from (3), and more generally the action of any $\mathrm{diag}(\alpha_{0},\dots,\alpha_{n})\in\operatorname{Aut}_{P}$,

[TABLE]

allows us to write

[TABLE]

where the subscript ${p,q}$ denotes the elements

[TABLE]

The above claim is due to Steenbrink [26] in the present weighted homogenous setup, see also [11, Appendix A].

Remark 12*.*

The action of $\operatorname{Aut}_{P}$ on $\operatorname{Jac}(P)$ is well defined because any automorphism $\mathrm{diag}(\alpha_{0},\dots,\alpha_{n})$ operates on each monomial in $\partial_{i}P$ by multiplication by $\alpha_{i}^{-1}$. This happens because $\mathrm{diag}(\alpha_{0},\dots,\alpha_{n})$ fixes $x_{i}\partial_{i}P$ since it fixes $P$.

Furthermore, the grading $(p,q)$ is well defined simply because $\deg(x_{i})=\frac{w_{i}}{d}$ yields a $\mathbb{Q}$-grading on $\mathbb{C}[x_{0},\dots,x_{n}]$, which descends to a $\mathbb{Q}$-grading of $\operatorname{Jac}(P)$ because the Jacobi ideal $(\partial_{0}P,\dots,\partial_{n}P)$ is homogeneous (each monomial in $\partial_{i}P$ has degree $d-\frac{w_{i}}{d}$).

This calls for the following definition.

Definition 13**.**

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

[TABLE]

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.

Remark 14*.*

Notice that, as a consequence of the non-degeneracy of $P$, the restriction $P_{g}$ is still a non-degenerate polynomial.

Remark 15*.*

When $g$ is the identity we recover the FJRW state space $\mathcal{H}_{P,H}$.

Remark 16*.*

We immediately have

[TABLE]

Remark 17*.*

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. In complete analogy with ambient cohomology, we can identify the narrow and broad classes to the image and the kernel of the morphism

[TABLE]

We have

[TABLE]

5.4 Landau–Ginzburg/Calabi–Yau correspondence

The above equations (8), (10), and (12) add up to a simple proof of the so called Landau–Ginzburg/Calabi–Yau correspondence based on Yasuda’s theorem and $K$-equivalence (insured by the Calabi–Yau condition).

Theorem 18** ([10, 13]).**

For any non-degenerate 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

[TABLE]

Since the above isomorphism follows from $K$-equivalence, it is not explicitly given. In [10] we provide an explicit automorphism. In [13] we generalize it to complete intersections. Notice that, by a slight abuse of notation, we adopted the same notation for the above isomorphism as for $\Phi\colon H_{g}^{*}(\mathbb{V};\mathbb{C})\to H_{g}^{*}(\mathbb{L};\mathbb{C})$ from (7).

6 Unprojected mirror symmetry

6.1 Mirror duality

The mirror construction due to Berglund and Hübsch [6] is elementary. It applies to non-degenerate 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

[TABLE]

by transposing the matrix of exponents $E$.

Remark 19*.*

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

It exchanges $J_{P}$ with $\operatorname{SL}_{P^{\vee}}$. It reverses the inclusions.

We define the unprojected state space

[TABLE]

by summing over all diagonal symmetries and without taking any invariant. For each summand $\operatorname{Jac}(P_{h})$ there exists an $\operatorname{Aut}_{P^{\vee}}$-grading defined by

[TABLE]

where all the products run over the set $F_{h}$ of labels of variables fixed by $h$. The map is well defined because each monomial in $\partial_{i}P_{h}$ maps to $J_{P^{\vee}}(\overline{\rho}_{i})^{-1}$. This happens because each monomial appearing in $\partial_{i}P_{h}dx_{i}$ maps to the same automorphism as each monomial appearing in $P$. Furthermore, the automorphism obtained in this way is the identity by the relation $\sum_{j}m_{i,j}\overline{\rho}_{j}$ discussed above. We can finally conclude that $\partial_{i}P_{h}\bigwedge_{j}dx_{j}$ maps to the same automorphism as $\bigwedge_{j\neq i}dx_{j}$: namely $\prod_{j\neq i}\overline{\rho}_{j}=J_{P^{\vee}}(\overline{\rho}_{i})^{-1}$. In this way the unprojected state space admits a double decomposition

[TABLE]

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

[TABLE]

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 20**.**

The vector space $U_{H}^{K}$ is the $K^{\vee}$-invariant subspace of $U_{H}$

[TABLE]

In particular we have

[TABLE]

Proof.

This happens because, for any form $f$ in $\operatorname{Jac}(P_{h})(-\operatorname{age}(h))\subseteq U_{P}$ the following equivalence holds. We have $\ell_{h}(f)\in K$ if and only if $f$ is invariant with respect to $K^{\vee}$. This is just another way to phrase the definition of $K^{\vee}$. ∎

Theorem 21** (Krawitz [22], Borisov [8]).**

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

[TABLE]

yielding an explicit isomorphism

[TABLE]

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 22*.*

Let $P=x^{k}$. Then $\operatorname{Aut}_{P}$ equals $\mathbb{Z}_{k}$ (because we fix a primitive $k$th root $\xi$). For $1\in{\boldsymbol{\mu}}_{k}=\mathbb{Z}_{k}$, we have $P_{1}=P$, so

[TABLE]

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

[TABLE]

By mapping $x^{i-1}dx\in\operatorname{Jac}(P_{1})$ to the generator $1_{\xi^{i}}$ of $\operatorname{Jac}(P_{x^{i}})$ 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 non-degenerate polynomials $P^{\prime}$ and $P^{\prime\prime}$ involving disjoint sets of variables we clearly have $\operatorname{Aut}_{P}=\operatorname{Aut}_{P^{\prime\prime}}\times\operatorname{Aut}_{P^{\prime\prime}}$. This and the theorem above imply the following crucial properties of the mirror map $M_{P}$.

Thom–Sebastiani.

If $P$ is a polynomial which can be expressed as the sum of two invertible and non-degenerate polynomials

[TABLE]

involving two disjoint sets of variables, then we have

[TABLE]

Group actions.

For any $H,K\subseteq\operatorname{Aut}_{P}$, the restriction of $M_{P}$ yields an isomorphism

[TABLE]

This mirror construction is due to Berglund and Hübsch [6]. The result presented here appeared first in this form in Krawitz [22]; we should also refer to Berglund and Henningson [5] for the group duality, to Kreuzer and Skarke [23] for a systematic study, and to Borisov [8] for a reinterpretation of the setup and further generalizations in terms of vertex algebræ.

There are many consequences of the existence of $M_{P}$ 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 [10] and it should be regarded as a combination of the mirror map $M_{P}$ of Krawitz and Borisov [22, 8] and of the LG/CY isomorphism $\Phi$ of the first named author with Ruan [10]. In the present setup, it is extremely elementary to prove its main statement.

Theorem 23** (Mirror Symmetry for CY models).**

For any invertible, non-degenerate $P$ 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

[TABLE]

Proof.

Since $M_{P}$ satisfies the above property with respect to group actions we have $M_{P}(\mathcal{H}_{P,H,\operatorname{id}})=\mathcal{H}_{P^{\vee},H^{\vee},\operatorname{id}}$. In order to apply Theorem 18 we need the Calabi–Yau condition on $P$ and the conditions $H\ni j_{P}$ and $H^{\vee}\ni j_{P^{\vee}}$. The last equation is equivalent to $H^{\vee}\subseteq\operatorname{SL}_{P}$. The claim follows. ∎

The Thom–Sebastiani property applies to $P^{\prime}=x_{0}^{k}$ and $P^{\prime\prime}=f$ adding up to

[TABLE]

The aim of this paper is to study the relation between the above cohomological mirror symmetry and the the symmetry $s=[\frac{1}{k},0,\dots,0]$.

Proposition 24**.**

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

[TABLE]

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 $M_{W}(\phi,h)$ is of the form $(\phi^{\prime},h^{\prime})$ with $h^{\prime}=[\frac{i}{k},a_{1},\dots,a_{n}]$. In particular $M_{W}$ maps invariant elements to non-invariant elements.

Proof.

This happens because $s$ spans $\operatorname{Aut}_{x_{0}^{k}}$, whose dual group is trivial. The claim follows by $M_{W}=M_{x_{0}^{k}}\otimes M_{f}$ (see Example 22). ∎

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

[TABLE]

Set

[TABLE]

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 (of immediate proof) describes how $K$ and $K[j_{W},s]$ behave with respect to the the group duality.

Proposition 25**.**

Consider $K\in\operatorname{Aut}_{f}$ satisfying $(j_{f})^{k}\in K\subseteq\operatorname{SL}_{f}.$ Then we have

[TABLE]

Furthermore, we have a mirror isomorphism

[TABLE]

∎

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.

Proposition 26**.**

We have

[TABLE]

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

[TABLE]

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 $p,q\in\mathbb{Z}$. ∎

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

[TABLE]

We also write $M$ for the mirror map $M_{W}$.

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

[TABLE]

is an eigenvector with respect to the diagonal symmetry $\alpha=[p_{0},\dots,p_{n}]$: the eigenvalue is $\exp(2\pi\mathtt{i}\sum_{j}b_{j}p_{j}).$ 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:

[TABLE]

We decompose $\mathbb{H}$ as

[TABLE]

and we can consider the following $\mathbb{Q}/\mathbb{Z}$-valued gradings on the set of generators

[TABLE]

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

[TABLE]

The following proposition further simplifies the decomposition.

Proposition 27** (the moving subspace, the fixed subspace)).**

For any element $(\phi,g)$ we have either (i) $d_{s}=-d_{j}$, or (ii) $Q_{s}\neq 0$.

Proof.

This happens because $Q_{s}=0$ if and only if $g\cdot x_{0}=x_{0}$, i.e. $0\in I$. By definition of $d_{s}$ and $d_{j}$ we have $g\in j^{kd_{j}}s^{kd_{s}}K$ and $g\cdot x_{0}=\exp(2\pi\mathtt{i}(d_{j}+d_{s}))x_{0}$. We conclude that $Q_{s}=0$ if and only if $d_{s}+d_{j}\in\mathbb{Z}$. ∎

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

[TABLE]

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

[TABLE]

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

[TABLE]

is motivated by the following fact.

Proposition 28** (twist).**

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

[TABLE]

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

Proof.

Indeed the above homomorphism exchanges $d_{j}+d_{s}$ with $Q_{s}$ and preserves $d_{j}$ and $Q_{s}-Q_{j}$. ∎

Remark 29*.*

The index $Q_{s}-Q_{j}$ may be regarded as the (opposite of) the $j$-charge of the form $\phi$ restricted to $(x_{0}=0)$.

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 30** (elevators).**

For any $0\leq X,Y<k$ and $0<Z^{\prime}<Z^{\prime\prime}<k$ we have the isomorphisms

[TABLE]

with $e^{m}_{Z^{\prime},Z^{\prime\prime}}$ and $e^{f}_{Z^{\prime},Z^{\prime\prime}}$ transforming $(p,q)$-classes into classes whose bidegrees equal $(p-(Z^{\prime\prime}-Z^{\prime})/k,q+(Z^{\prime\prime}-Z^{\prime})/k)$ and $(p+(Z^{\prime\prime}-Z^{\prime})/k,q+(Z^{\prime\prime}-Z^{\prime})/k)$, respectively.∎

For $0<Z^{\prime}<Z^{\prime\prime}<k$, we set $e^{m}_{Z^{\prime\prime},Z^{\prime}}=(e^{m}_{Z^{\prime},Z^{\prime\prime}})^{-1}$ and $e^{f}_{Z^{\prime\prime},Z^{\prime}}=(e^{f}_{Z^{\prime},Z^{\prime\prime}})^{-1}$.

Proposition 25 specializes to the following statement.

Proposition 31**.**

The mirror isomorphism $M$ yields isomorphisms

[TABLE]

Proof.

Let us consider $M$ as a morphism mapping $U_{W}$ to $U_{W^{\vee}}$. For $W=(x_{0})^{k}+f$ we have $W^{\vee}=(\overline{x}_{0})^{k}+f^{\vee}$. Using (15), every state of the form $(\phi,g)\in U_{W}$ can be regarded as an element of

[TABLE]

with $a_{1}\in\operatorname{Aut}_{(x_{0})^{k}}=\mathbb{Z}/k$, $b_{1}\in\operatorname{Aut}_{(\overline{x}_{0})^{k}}=\mathbb{Z}/k$, $a_{2}\in\operatorname{Aut}_{f}$ and $b_{2}\in\operatorname{Aut}_{f^{\vee}}.$ Example 22 shows that there are only two possibilities: (1) $b_{1}=1$ or (2) $a_{1}=1$. 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 $s$, $a_{1}$ 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 ${\boldsymbol{\mu}}_{k}$-characters. The claim $(X,Y,Z)\mapsto(Y,X,Z)$ follows from

[TABLE]

where ${\boldsymbol{\mu}}_{k}$-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

[TABLE]

from (14): 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 ${\boldsymbol{\mu}}_{k}$-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.

Remark 32*.*

For Fermat potentials all the above discussion can be carried out more explicitly because the group elements $\boldsymbol{a}=\prod_{i=1}^{n}\rho_{i}^{a_{i}}$ coincide with $\frac{1}{d}[a_{1}w_{1},\dots,a_{n}w_{n}]$. By adopting this notation, the space $U_{a}^{b}(W)$ may be regarded as the one-dimensional space spanned by

[TABLE]

where

[TABLE]

Mirror symmetry is simply an exchange of the $w\mathbb{Z}/d\mathbb{Z}$-valued vectors $\boldsymbol{a}$ and $\boldsymbol{b}$. The bidegree $(p,q)$ coincides with

[TABLE]

where $\#(\boldsymbol{b})$ 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 27 reads $b_{0}=0$ is a special case of (19). Furthermore we have

[TABLE]

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 26, we obtain the $j$-invariant contribution by setting $Q_{j}=0$. By (18), this amounts to imposing $Y=Z$ within $\mathbb{H}^{m}$ and $Y=0$ within $\mathbb{H}^{f}$. We get

[TABLE]

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 $\mathbb{H}^{f}$

[TABLE]

(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 26). By (18), we obtain it by setting $X=-b$ within $\mathbb{H}^{m}$ and $Z=X+b$ within $\mathbb{H}^{f}$

[TABLE]

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

[TABLE]

with the convention $(e^{f}_{i,j})=(e^{f}_{j,i})^{-1}$ if $j<i$. By Proposition 26 the above data correspond to $H_{s^{b}}^{*}(\Sigma_{W,K[j_{W}]};\mathbb{C})$ under the Calabi–Yau condition.

Proposition 6 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 33**.**

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

[TABLE]

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

Proof.

We prove that

[TABLE]

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

[TABLE]

with $I^{\prime}=\{k\geq 1\mid j^{b}g\cdot x_{k}=x_{k}\}\subseteq\{1,\dots,N\}$. Using $w_{0}/d=1/k$, we get

[TABLE]

Let us write $g$ as $[0,p_{1}\dots,p_{n}]\in\operatorname{Aut}_{f}$; then we have $k\in I^{\prime}$ if and only if

[TABLE]

Then $k$ divides $bt$, because

[TABLE]

where the last relation holds since $\phi$ is $g$-invariant. ∎

6.4 Mirror symmetry on the Landau–Ginzburg side

In this section, we derive an interpretation of Proposition 31 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

[TABLE]

Their mirrors 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 31. Let

[TABLE]

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

[TABLE]

where $H=K[j]$.

We now study $\mathbb{H}^{f}_{0,0,i}$. The subspace $\mathbb{H}^{f}_{0,0,i}$ mirrors $\mathbb{H}^{m}_{0,0,i}$. By applying the twist $\tau$ from Proposition 28, we land again on $\mathbb{H}^{f}_{0,0,i}$ which is a part of the $j$-invariant state space $\mathcal{H}_{W^{\vee},H^{\vee},s^{i}}$.

Using the elevators of Proposition 30, we obtain a homomorphism

[TABLE]

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

[TABLE]

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

[TABLE]

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 (21) (recalling that $d_{j}$ and $d_{s}$ switch under mirror symmetry) we can write

[TABLE]

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

[TABLE]

where $0<j<k.$ Finally we focus on the moving part of $\mathcal{H}_{W,K[j_{W}],s^{b}}$ which, by (20) 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

[TABLE]

Therefore we have

[TABLE]

Notice that the map on the bidegrees is the composite of

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

In the following statement we apply to (21), (24), and (25) to the geometric interpretation (12) 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 non-degenerate 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 $s=[\frac{1}{k},0,\dots,0]\colon\mathbb{V}_{H}\to\mathbb{V}_{H}$, 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

[TABLE]

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

Theorem 34** (mirror theorem for Landau–Ginzburg models).**

Let $W=x_{0}^{k}+f(x_{1},\dots,x_{n})$ be a quasi-homogeneous, non-degenerate, invertible polynomial and $H$ a group of symmetries satisfying $j_{W}\in H\subseteq\operatorname{SL}_{W}$. As above, the polynomial $W$ descends to $\mathbb{V}_{H}=[\mathbb{V}/H_{0}]\to\mathbb{C}$ and its generic fibre is $\mathbb{F}_{W,H}$.

Then, for $b$ and $t\neq 0$, we have

$H^{p,q}_{\operatorname{id}}(\mathbb{V}_{H},\mathbb{F}_{W,H})_{\chi_{s}=0}\cong\bigoplus_{i=1}^{k-1}H^{n+1-p,q}_{\operatorname{id}}(\mathbb{V}_{H^{\vee}},\mathbb{F}_{W^{\vee},H^{\vee}})_{\chi_{s}=i};$** 2. 2.

Let $\mathbb{F}:=\mathbb{F}_{W,H}$ and $\mathbb{F}^{\vee}:=\mathbb{F}_{W^{\vee},H^{\vee}}$.

For $0<i<k$,

[TABLE] 3. 3.

$H^{p,q}_{s^{b}}(\mathbb{V}_{H},\mathbb{F}_{W,H})\left(\frac{b}{k}\right)_{\chi_{s}=t}\cong H^{n-p,q}_{s^{-t}}(\mathbb{V}_{H^{\vee}},\mathbb{F}_{W^{\vee},H^{\vee}})\left(\frac{k-t}{k}\right)_{\chi_{s}=-b}.$**

Proof.

Since $H$ equals $K[j_{W}]$ for a suitable $K\subseteq\operatorname{SL}_{f}$ containing $j_{W}^{k}$, we can conclude via $\mathcal{H}_{P,H,g}^{p,q}=H_{g}^{p,q}(\mathbb{V}_{H};\mathbb{F}_{P,H})$. ∎

7 Geometric mirror symmetry

If $W$ is of Calabi–Yau type, via the Landau–Ginzburg/Calabi–Yau correspondence of Theorem 18 based on $\Phi\colon H^{*}(\mathbb{V};\mathbb{C})\to H^{*}(\mathbb{L};\mathbb{C})$, we provide an equivalent statement on the Calabi–Yau side.

The existence of the isomorphism $\Phi$ is guaranteed by the Calabi–Yau condition (ensuring $K$-equivalence). As before, for $0<i<k$, define $\overline{H}^{*}_{\operatorname{id},i}(\Sigma_{W,H})$ to be the bigraded vector space

[TABLE]

where again, $j$ runs between $1$ and $k-1$. Then we have the following statement.

Theorem 35** (mirror theorem for CY orbifolds with automorphism $s$).**

Let $W=x_{0}^{k}+f(x_{1},\dots,x_{n})$ be a be a quasi-homogeneous, non-degenerate, invertible, Calabi–Yau polynomial and $H$ a group of symmetries satisfying $j_{W}\in H\subseteq\operatorname{SL}_{W}$. Let $\Sigma=\Sigma_{W,H}$ and $\Sigma^{\vee}=\Sigma_{W^{\vee},H^{\vee}}.$ Then the following holds for $b,t\neq 0$.

Let $d=n-1$. Then $H^{p,q}_{\operatorname{id}}(\Sigma)_{\chi_{s}=0}\cong\bigoplus_{i=1}^{k-1}H^{d-p,q}_{\operatorname{id}}(\Sigma^{\vee})_{\chi_{s}=i};$ 2. 2.

Let $d=n-2$. For $0<i<k$,

[TABLE] 3. 3.

Let $d=n-2$. Then $H^{p,q}_{s^{b}}(\Sigma)_{\chi_{s}=t}\left(\frac{b}{k}\right)\cong H^{d-p,q}_{s^{-t}}(\Sigma^{\vee})\left(\frac{k-t}{k}\right)_{\chi_{s}=-b}.$

Proof.

This follows immediately from Theorem 34 and the LG/CY correspondence. ∎

Remark 36*.*

In the theorem, $d$ denotes the maximum of the dimensions of the components of the inertia stack considered in each case.

For $k=2$, the second equation of the statement of Theorem 35 can be stated as a mirror symmetry statement involving the cohomology groups $H^{*}_{s}$. Notice that the first statement says that Berglund–Hübsch mirror symmetry exchanges invariant $(p,q)$-classes for $\Sigma_{W,H}$ and anti-invariant $(n-1-p,q)$-classes of $\Sigma_{W^{\vee},H^{\vee}}$ (and vice versa). Finally the third statement is trivial because both sides vanish by Proposition 33. In this way we recover the main theorem of [12].

Corollary 37**.**

Let $W=x_{0}^{2}+f(x_{1},\dots,x_{n})$ be a be a quasi-homogeneous, non-degenerate, invertible, Calabi–Yau polynomial and $H$ a group of symmetries satisfying $j_{W}\in H\subseteq\operatorname{SL}_{W}$. Then, we have

[TABLE]

Example 38*.*

Let us consider $E=(x^{6}+y^{3}+z^{2}=0)$ within $\mathbb{P}(1,2,3)$ with its order-$6$ symmetry $s=[\frac{1}{6},0,0]$. In this case the “Calabi–Yau orbifold” is represented by an elliptic curve. The cohomology groups $H_{s^{b}}^{*}$ describe the cohomology of the $s^{b}$-fixed loci $E_{b}$, shifted by $(\frac{b}{6},\frac{b}{6})$. Furthermore, the mirror of $E$ coincides with $E$, because the defining equation is of Fermat type and $J$ equals $\operatorname{SL}$ (the order of $\operatorname{SL}$ is $w_{x}w_{y}w_{z}/\deg$ and equals the order $\deg$ of $J$). This example allows us to test $\mathbb{H}$ as a state space computing the cohomology of $E$, and the cohomology of its fixed spaces satisfying $E_{1}=E_{2}\cap E_{3}$ and $E_{2}=E_{4}$. Since $E$ is the elliptic curve with order-$6$ complex multiplication, $E_{1}$ is the origin and the fixed spaces $E_{3}(=E[2])$ and $E_{2}$ are respectively a set of $4$ points and 3 points interecting at the origin. Clearly $E_{3}\setminus E_{1}$ is the unique order-$3$ orbit and $E_{2}\setminus E_{1}$ is the unique order-$2$ orbit.

The $b$th row in the table below represents the ranks of contributions of $\mathbb{H}[d_{s}=\frac{b}{6}]$ whereas the $a$th column represents the contributions to the state space of $\mathbb{H}[d_{j}=\frac{a}{6}]$. Notice that, by means of the elevators, all rows are identical except for the anti-diagonal entries of the form $\mathbb{H}[d_{j}+d_{s}=0]$, which we underlined.

[TABLE]

The [math]th row is the $4$-dimensional cohomology of the elliptic curve $E$ organised in its $2$-dimensional primitive part (spanned by the forms $dx\wedge dy\wedge dz$ and $x^{4}ydx\wedge dy\wedge dz$) and its $2$-dimensional ambient part arising in the state space $\mathbb{H}[d_{s}=0,d_{j}=a/6]$ for $a=1$ and $a=5$ ($j$ and $j^{5}$ correspond to the only narrow sectors of the state space, i.e. the only powers of $[\frac{1}{6},\frac{1}{3},\frac{1}{2}]$ fixing only the origin). On the row corresponding to $H^{*}_{s}$ there is a single contribution for $d_{j}=1/5$. This happens because $E_{1}$ is a point. Furthermore $\mathbb{H}[d_{j}=5/6,d_{s}=1/6]=\mathbb{H}[d_{j}=1/6,d_{s}=5/6]$ vanish by Proposition 33. The remaining anti-diagonal terms are $\mathbb{H}[d_{j}=4/6,d_{s}=2/6]=\mathbb{H}[d_{j}=2/6,d_{s}=4/6]=\langle x^{2}dx\wedge dz\rangle$ and $\mathbb{H}[d_{j}=3/3,d_{s}=3/3]=\langle xydx\wedge dy,x^{3}dx\wedge dy\rangle$.

The above mirror symmetry statement (1) involves the first row and claims that all fixed cohomology classes appearing for $d_{j}=\frac{1}{6},\dots,\frac{5}{6}$ match the classes of $\mathbb{H}[d_{j}=0,d_{s}=0]$; we already noticed that this identifies two $2$-dimensional spaces of ambient and primitive cohomology. Statement (2), for $i=1$, says that the 1 dimensional space $\mathbb{H}[d_{s}=0,q_{j}=1,2,3,4]$ (spanned by the class $dx\wedge dy\wedge dz$) matches the cohomology class spanned by $x^{4}dx\wedge dy\wedge dz$.

Statement (3) is a map $M\colon xydx\wedge dy\mapsto x^{2}dx\wedge dz\in\mathbb{H}[d_{j}=\frac{2}{6},d_{s}=\frac{4}{6}]$ and a map $M\colon x^{3}dx\wedge dy\mapsto x^{2}dx\wedge dz\in\mathbb{H}[d_{j}=\frac{4}{6},d_{s}=\frac{2}{6}]$. In this way

[TABLE]

In geometric terms mirror symmetry matches the order-$2$ orbit to the order-$3$ orbit. More precisely, the mirror statement (3) claims that there are as many eigenvectors of eigenvalue $(\chi_{6})^{2}$ and $(\chi_{6})^{4}$ in the cohomology of $E_{3}$ as eigenvectors of eigenvalue $(\chi_{6})^{3}$ in the cohomology of $E_{2}$ and of $E_{4}$.

Example 39*.*

We consider the genus-$3$ curve $C$ defined by the degree-$4$ Fermat quartic $x_{1}^{4}+x_{2}^{4}+x_{3}^{4}=0$ in $\mathbb{P}^{2}$. The $4$-fold cover of $\mathbb{P}^{2}$ ramified on $C$ is a K3 surface defined as the vanishing locus of the polynomial $W=x_{0}^{4}+x_{1}^{4}+x_{2}^{4}+x_{3}^{4}$. In this example, the Calabi–Yau orbifold $\Sigma_{W}$ is again representable and we can treat the cohomologies $H_{\operatorname{id}}^{*}$ and $H_{s}^{*}$ as ordinary cohomologies of the K3 surface and of the ramification locus. As in the previous example, we display the cohomological data in a table. The $b$th row in the table below represents the ranks of contributions of $\mathbb{H}[d_{s}=\frac{b}{4}]$ whereas the $a$th column represents the contributions to the state space of $\mathbb{H}[d_{j}=\frac{a}{4}]$.

[TABLE]

The colors in the table refer to the weight of $s$: cohomology in red has character $1$, blue has character $2$, and green has character $3$. Statement (2) involves, on one side, the cohomology of the curve $C$ ($H_{s}$) and the moving cohomology of the K3 surface with weights $1$ and $2$. The total cohomology on one side of Statement (2) is thus

[TABLE]

We notice that the only $\operatorname{SL}$-invariant broad cohomology classes in the entire unprojected state space $U(W)$ are contained in $U(W)_{\operatorname{id}}$; this implies $U(W)_{sg}^{\operatorname{SL}}=0$. Hence $H^{*}_{\mathrm{prim},s}$ vanishes. One can compute the mirror table as

[TABLE]

From this, we see that the mirror $s$-fixed locus is four projective curves and 12 isolated fixed points. The mirror Hodge diamond for Statement (2) is

[TABLE]

Note that despite being an order 2 automorphism, the Hodge diamonds of the fixed loci of $s^{2}$ do not mirror each other.

Clearly mirror symmetry should also yield a relation between the quantum invariants of the primitive classes of the curve and the orbifold quantum invariants of these sectors.

The structure of the of the above example is shared by all K3 orbifolds of this type with order 4 automorphism. Combining the mirror theorem with the fact that $s$ and $s^{3}$ have the same fixed locus (and hence cohomology of the same dimension), we can see that for any $W=x_{0}^{4}+f(x_{1},x_{2},x_{3})$ and group $G$, the table for $\Sigma_{W,G}$ is given by

[TABLE]

.

The table for the $\Sigma_{W^{\vee},G^{\vee}}$ is obtained from this table by replacing $x\mapsto x^{\vee}.$

Using this table, we can find relationships between the topological invariants of the fixed loci of crepant resolutions of $\mathfrak{X}:=\Sigma_{W,G}$ and its mirror $\mathfrak{X}^{\vee}$. Example 8 shows that there is an isomorphism between the $s^{3}$-orbifold cohomology of $\mathfrak{X}$ and the cohomology of the $s$ fixed locus in the resolution $\widetilde{X}$. Recall that this is because for K3 surfaces, the age function is constant (of $3/4$) on the $s^{3}$-orbifold cohomology of the resolution. By similar reasoning, the $s^{2}$-orbifold cohomology of $\widetilde{X}$ also has a constant age function (of $1/2$).

Now consider the following invariants for $i=1,2$:

• •

$f_{i},$ the number of isolated fixed points of $s^{i}$;

• •

$g_{i},$ the sum of the genera of the fixed curves of $s^{i}$;

• •

$N_{i},$ the number of curves in the fixed locus of $s^{i}$.

A superscript $\vee$ indicates the invariants of the mirror K3.

Corollary 40**.**

We have

$N_{1}=g_{1}^{\vee}+1$; 2. 2.

$N_{2}+g_{2}+f_{1}=20-N_{2}^{\vee}-g_{2}^{\vee}-f_{1}^{\vee}.$**

Proof.

The table above implies that $2a+b+2a^{\vee}+b^{\vee}=24$, and that $g_{1}=g,$ $N_{1}=g^{\vee}+1$, and $f_{1}=a^{\vee}+b^{\vee}-2$. The statements for the mirror invariants are obtained by $x\leftrightarrow x^{\vee}$: for example, $N_{1}^{\vee}=g+1.$

Similarly, $N_{2}-g_{2}^{\vee}=(g^{\vee}+c+a^{\vee})-(g^{\vee}+c)=a^{\vee}$, which implies the statement. ∎

The same analysis also works for K3 surfaces with prime order automorphisms. Let $W$ be a Calabi–Yau polynomial of the form $W=x_{0}^{p}+f(x_{1},x_{2},x_{3})$ for $p$ prime. Then the Landau–Ginzburg state space breaks down as

[TABLE]

The following lemma follows immediately from considering this table.

Lemma 41**.**

Suppose $\Sigma_{W,H}$ is a K3 orbifold with $W=x_{0}^{p}+f(x_{1},x_{2},x_{3})$. Then $p-1|24.$

Let $\tilde{X}$ be a crepant resolution of $\mathfrak{X}=\Sigma(W,G)$, and $\tilde{X}^{\vee}$ a crepant resolution of the mirror. The fixed locus of $s$ is a disjoint union of curves an isolated fixed points. As before, let $f_{1}$ be the number of isolated fixed points, $N_{1}$ the number of curves, and $g_{1}$ the sum of the genera of the curves.

Corollary 42**.**

Suppose $p>2$. Then $N_{1}=g_{1}^{\vee}+1$ and

[TABLE]

Proof.

Using the table, it is easy to see

[TABLE]

Additionally,

[TABLE]

Combining this with $(p-1)a+(p-1)a^{\vee}=24$, we obtain the statement in the theorem. ∎

This corollary implies that Berglund–Hübsch mirror symmetry agrees with mirror symmetry for lattice polarised K3 surfaces. We briefly recall the latter.

Given a smooth K3 surface $\Sigma$, $\Lambda=H^{2}(\Sigma,\mathbb{Z})$ is equipped with a lattice structure via the cup product taking values in $H^{4}(\Sigma;\mathbb{Z})=\mathbb{Z}.$ Let $S_{\Sigma}:=\Lambda\cap H^{1,1}(\Sigma;\mathbb{C})$ be the Picard lattice of $\Sigma$.

Let $M$ be a hyperbolic lattice with signature $(1,t-1)$. A K3 surface $\Sigma$ is called $M$-polarized if there exists a primitive embedding $M\hookrightarrow S_{\Sigma}$. Given a non-symplectic automorphism $s$ of $\Sigma$, the invariant sublattice $S(s):=\Lambda^{s}$ is in fact a primitive sublattice of the Picard lattice.

Definition 43**.**

Given $M$ a primitive hyperbolic sublattice of $\Lambda=H^{2}(\Sigma,\mathbb{Z})$ of rank at most 19 such that

[TABLE]

$M^{\vee}$* is defined to be the mirror lattice to $M$.*

Recall that we have restricted to the case where $s$ has prime order $p>2$ (we have discussed $p=2$ in [12]). We now show that if two K3 surfaces with prime order automorphisms arise as crepant resolutions of a mirror pair of Berglund-Hübsch orbifolds, they have mirror lattices. In this case, $M:=S(s)$ is $p$-elementary. That is, $M^{*}/M=(\mathbb{Z}/p\mathbb{Z})^{\oplus a}$, and it is completely classified by its rank $r$ and $a$. Then, by [4], the fixed locus of $s$ is either just isolated points or a disjoint union of $N$ curves, of which $N-1$ are rational and the remaining one has genus $g$, and $f$ isolated points. Set $m=\frac{22-r}{p-1}$. Moreover, [4] states (in a slightly different form) that for $p=3,5,7,13$, if the fixed locus contains a curve,

• •

$m=2g+a,-g+N=\frac{r-11+p}{p-1}.$

Notice that these are the only prime orders we need to consider, as we have shown that $p-1|24$.

Lattice mirror symmetry exchanges $(r,a)$ with $(20-r,a)$.

Theorem 44**.**

Let $\Sigma_{W,H}$ and $\Sigma_{W^{\vee},H^{\vee}}$ be mirror K3 orbifolds with prime order $p>2$ automorphisms $s,s^{\vee}$, and let $\Sigma$ and $\Sigma^{\vee}$ be crepant resolutions with automorphisms also denoted $s,s^{\vee}$. Then $\Sigma$ and $\Sigma^{\vee}$ are mirror as lattice polarized K3 surfaces.

Proof.

Corollary 42 relates the invariants $(g,N,f)$ and $(g^{\vee},N^{\vee},f^{\vee})$. It is enough to show that these relations give the mirror relations on $(r,a)$, namely that

[TABLE]

Notice that there is always a fixed curve when the K3 is a hypersurface in weighted projective space of this form. Therefore, we see that

[TABLE]

Finally, this implies

[TABLE]

Using that $N=\frac{r-11+p}{p-1}+g,$ we obtain that

[TABLE]

∎