Introduction

In mirror symmetry, one of the ways we approach the motivating conjectures (homological projective duality and general existence of mirrors for Calabi-Yau threefolds) is by collecting evidence, ie writing down invariants of Calabi-Yaus and their mirrors and finding different ways to calculate and compare them. For $X$ a simply connected Calabi-Yau threefold, there are a few types of invariants we are interested in:

• Diffeomorphism invariants
  • Hodge numbers, especially $h^{1,1}(X) = \text{rk} \ H^2(X, \mathbb{Z})$ and $h^{2,1}(X) = \text{rk} \ H^3(X, \mathbb{Z})$
• Invariants under change of Kähler structure
  • derived category of coherent sheaves $\mathsf{D}^{\flat}(\text{Coh}(X))$
• Deformation invariants
  • Gromov-Witten, Donaldson-Thomas invariants

Examples

Here are some examples of spaces with agreeing invariants:

1. Let $V$ be a 7-dimensional complex space, and consider the Grassmannian $\text{Gr}_2(V) \subseteq \mathbb{P}(\bigwedge^2 V).$ If $H$ is a hyperplane class in the Grassmannian, the space $X = \text{Gr}_2(V)\cap H^7$ has $K_X = \mathcal{O}_X$, $h^{1,1} = 1, h^{2,1} = 50, H^{3} = 3, c_{2}\cdot H = 84$ (where $H^3 = 3H$ in the cohomology/Chow ring of $X$ and $c_2$ is the second Chern class). Let $\varphi : \mathbb{P}(V) \to \mathbb{P}(V^\vee)$ with $\varphi = -\varphi^\vee$ . Let $Y = D_4(\varphi) = \{x \ | \ \text{rk} \phi(x) \leq 4\}$. Then $X$ and $Y$ are not birational, but nonetheless $\mathsf{D}^{\flat}(\text{Coh}(X)) = \mathsf{D}^{\flat}(\text{Coh}(Y))$.
2. (Reye congruence) Let $V$ be a 5-dimensional complex space, and let $\mathscr{X}$ be the Chow variety of two points in $\mathbb{P}(\text{Sym}^2(V))$. The image of the canonical morphism $\mathbb{P}(V) \xhookrightarrow{} \mathbb{P}(\text{Sym}^2(V))$ is a 4-plane determined by a linear system of 5 quadrics $|Q_1\dots Q_{5}|$, and if $\mathscr{H}$ is the locus of singular quadrics one has $\mathsf{D}^{\flat}(\text{Coh}(\mathscr{X})) = \mathsf{D}^{\flat}(\text{Coh}(\mathscr{H}))$.
3. (Octic double solids) Let $X$ be a smooth complete intersection in $\mathbb{P}^7$, with invariants

Let $Y \xrightarrow{2:1} \mathbb{P}^3$ be a double cover branched over

where $A(x)$ is a matrix associated to $x$ in $M_3(\mathbb{C}[x])$. Then $Y$ is a singular Calabi-Yau threefold having 84 nodal singularities. $Y$ admits a noncompact crepant resolution to a projective space $(\mathbb{P}^3, \mathcal{B})$ equipped with a sheaf of Azumaya algebras, and $\mathsf{D}^{\flat}(\text{Coh}(X)) = \mathsf{D}^{\flat}(\text{Coh}(\mathbb{P}^3, \mathcal{B}))$. For another equivalence, $Y$ has an analytic crepant small resolution to a space $\hat{Y}$ with $\text{Br}(\hat{Y}) \cong \mathbb{Z} / 2\mathbb{Z}$, and with $\alpha$ the nontrivial class in the Brauer group one has $\mathsf{D}^{\flat}(\text{Coh}(X)) = \mathsf{D}^{\flat}(\text{Coh}(\hat{Y}, \alpha)).$

These examples are analysed in the literature using the GLSM formalism: Example 1 is in Hori Tong '07 Example 2 is in Hori '13 Example 3 is in Calderaru et al '07 Another famous example is the Orlov space, this in Hori Herbst Page '08 A feature all of these examples have in common is that $h^{1,1} = 1$, how can we get examples with $h^{1,1} = 2$? $\to$ Genus one fibrations.

The GLSM formalism

A gauged linear sigma model (GLSM) is a 5-tuple of data $(G, \rho, R, W, t)$, where:

• $G$ is a compact Lie group
• $\rho : G \to V$ is a faithful unitary representation of $G$
• $R$ is a faithful unitary representation of $\mathsf{U}(1)$ (same target space as $\rho$)
• $W$ is an element of the $G$-invariants ring $S^G$ of $S = \mathsf{Sym}^\bullet \ V^\vee$ with $R$-weight 2
• $t$ is an element of $\mathcal{T} = \left( \frac{\mathfrak{t}_{\mathbb{C}}^\vee}{2\pi i \rho(\mathfrak{g})} \right)^{\mathcal{W}(T, G)}$ (this is called the FI-theta parameter in the physics literature)

For $\mu : V \to \mathfrak{g}^\vee$ the moment map and $\zeta \in \text{Re}(\mathcal{T})$ a regular value, the Higgs branch is a certain intersection $X_\zeta$ of the preimage $\mu^{-1}(\zeta)$ inside $V$ (what is it?).

The Coulomb branch is the union of loci in $\mathcal{T}$ where the GLSM is singular (I think).

Set $\widetilde{W}_{\text{eff}}(\sigma, t) = -t(\sigma) + 2\pi i \rho_{W}(\sigma)\cdot\sum_{Q_{j} \in \mathfrak{h}^\vee} Q_{j}(\sigma)(\text{log}(Q_{j}(\sigma) - 1))$, where $t$ is as above and $\sigma \in \mathsf{Z}(\mathfrak{g}_{\mathbb{C}})$. Then the stringy Kähler moduli space is $\mathcal{M}_{K} = \mathcal{T} \setminus \Delta$, where $\Delta$ is the union of critical loci for $\widetilde{W}_{\text{eff}}^H$ over all subgroups $H$ of $G$.

Remark: The real locus of the stringy moduli space has finitely many connected components $P_i$ called phases. By the relation between symplectic and GIT quotients, each phase is a possible linearisation for the GIT quotient. The imaginary locus has smoothness properties allowing one to interpolate through the real phases by passing through the imaginary locus.

GLSM $\to$ D-branes (not used later)

A brane is a 4-tuple $B = (M, Q, \rho, r_*)$, where

• $M$ is a $\mathbb{Z} / 2\mathbb{Z}$-graded free $S$-module ($S$ as above)

• $Q \in \text{End}^1_S(M)$ (this subscript usually indicates a distinguished subgroup of the endomorphism group)

• $\rho_M : G \to GL(M)$ is a complex representation

• $r_{*} : U(1) \to GL(M)$ is a complex representation such that $Q^2 = W$, $Q$ is $G$-equivariant, etc.

• There are limiting points $p_i$ attached to each phase above, which have associated triangulate categories $D_i$ (these are really associated to the Higgs branch of regular values in the phase). There are various conjectural equivalences of these $D_i$ to triangulated categories of algebraic invariants, depending on the branch.

• Conjecture: There are essential surjections $\pi_i : D \to D_i$ (from total triangulated category to triangulated category of the phases) and natural equivalences $D_{i} \xrightarrow{\sim} D_{j}$.

• Define the hemisphere partition function

( not sure what all of the parts here are).

Conjecture: To each limiting point $p_i$, there should be an associated cohomological field theory:

• a state space $H_i = HH_*(D_i)$ (Hochschild homology) with a pairing $\langle \ , \ \rangle: H_{i} \times H_{i} \to \mathbb{C}$
• for all $i$, $Z_{D^2}([B], t) = \langle \text{ch}(\pi_{i}(B)), \hat{\Gamma}_{i} \circ J_{i}(t) \rangle$, where $\text{ch}$ is the Chern character, the $J_i \in H_i[t][[e^t]]$ are certain J-functions and $t \in P_i + (i \mathbb{R}/ \mathbb{Z})^{\text{dim}(\mathcal{M}_{K})}$.

Elliptic Fibrations

• In the following examples, we will use an elliptic normal curve. For our purposes, this is a smooth projective curve $E$ of genus 1 and degree $N \geq 3$, so that $E$ can be immersed in $\mathbb{P}^{N-1}$ with image not contained in a hyperplane.
  • Example: If $N = 3$, $E$ is a cubic in $\mathbb{P}^2$.

Pfaffian varieties

Let $A$ be a $(2r+1)$-dimensional square matrix, skew-symmetric with entries in $(\mathsf{Sym}^\bullet(x_0,\dots, x_{r}))_{1}$ (degree 1 polynomials in $r+1$ variables). Let $M_i$ be the $2r\times 2r$ minor obtained by deleting the $i$th row and column from $A$, $\text{pf}(M_{i})$ its Pfaffian (choice of square-root for $\text{det}(M_{i})$). We define the Pfaffian variety $D_{r}(A)$ of $A$ to be $V(\prod_{i=1}^{2r+1}\text{pf}(M_{i}))$, the vanishing locus of all of the Pfaffians.

Homological projective duality for elliptic normal curves in degree

Let $V$ be a 5-dimensional complex vector space, $L \subseteq \bigwedge^2 V^\vee$ a 5-dimensional subspace, $L^\perp$ its orthogonal complement. Then we have a diamond

of inclusions with $X_0$ our elliptically fibred surface (I think ), the Plücker embedding from the Grassmannian and the induced map of projective spaces from the inclusion $L^\perp \subseteq \bigwedge^2 V$. The dual picture is the diamond

,

and we find that $\text{Pf}(2, V^\vee)$ is dual to the Grassmannian, $\mathsf{D}^{\flat}(\text{Coh}(X_{0})) \cong \mathsf{D}^{\flat}(\text{Coh}(Y_{0})).$

In the GLSM version of this homological duality setup (see Hori Knapp '13), $G = U(2)$ with stringy Kähler space $\mathcal{M}_{K} = \mathbb{C}^* / e^{-t}$, $e^{-t} = \frac{1}{2}(11 \pm 5\sqrt{5})$. The Higgs branch is $Y_0$.

General Elliptic Fibration Picture

The correct setting for an elliptic fibration is a flat proper holomorphic surjection $\pi: X \twoheadrightarrow B$. There is a minimal integer $N > 0$ such that a divisor $D \subseteq X$ exists with $\pi|_{D} : D \xrightarrow{N : 1} B$ an $N$-fold cover. If $N = 1$, we call $\pi$ an elliptic fibration. Equivalently, there is a line bundle $L \to X$ with $\text{deg}(L|_{C}) = N$ for every fibre $C$ of $\pi$, $D\cdot C = N$ (has intersection multiplicity $N$).

Intersection theory characterisation of Calabi-Yau elliptic fibrations (Oguiso '93): Let $X$ be a smooth projective Calabi-Yau threefold, $D \geq 0$ a divisor on $X$ satisfying $D^3 = 0$, $D^2 \not \equiv 0$, $D\cdot c_{2} > 0$. Then there is a base space realising $X$ as an elliptic fibration.

• People are interested in a physics interpretation of these examples for the modular bootstrap and for studying the topological string partition function
• Knapp, Schimanneck and Scheidegger focus especially on the $N = 5$ elliptic normal curve because the Calabi-Yau cannot be realised as a singular toric variety (most known mirrors can)
• They constructed 13 pairs of such spaces and computed their Hodge numbers and other diffeomorphism invariants

Theorem(Knapp, Schimannek, Scheidegger '21): Let $\pi : X \to B$ a smooth projective Calabi-Yau threefold which is an elliptic fibration admitting a 5-section ( this should be equivalent to the fibres being degree 5 elliptic normal curves). Then:

• $X$ is a determinantal subvariety of a Grassmannian bundle
• $X$, $\mathcal{M}_{K}$ can be studied using a $U(M)\times U(1)^n$ GLSM ($M$ a certain $D$-brane)
• these fibrations come in pairs $(X, Y)$, which are conjecturally mirror duals
• there is also a statement about Fourier-Mukai transforms I missed

Open Questions

( this summary should be close to correct but I'm kind of guessing what the notation means) In general, we so far usually have a picture as below: We have a mirror space $Y$ and an analytic non-Kähler resolution $\overline{Y}$, but we can get an associated algebro-geometric space by taking the noncommutative space associated to a nonzero element $\alpha \in \text{Br}(Y)$ which will have $\mathsf{D}^{\flat}(\text{Coh}(\overline{Y})) \cong \mathsf{D}^{\flat}(\text{Coh}(Y_{\text{nc}, \alpha}))$. Then we can pass to a smooth deformation $Y'$ of $\overline{Y}$. The correct and precise version of this is in Thomas Calabrese '16.

The things we want to know are:

• What are the bounded derived categories of singular elliptic fibrations?
• Is it possible to calculate topological invariants using only the data of the $D$-brane?
• Can we work out $\text{Br}(Y)$ and the noncommutative scheme structure from the GLSM?
• Can we work out the Fourier-Mukai transform using matrix factorisation in the GLSM?