These notes are from a research talk held at the University of Melbourne in February 2026. The talk covered the expected and understood relationships between mirror symmetry invariants, drawing especially from calculations for elliptic fibrations. Any errors and confusions in the notes are my own!

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

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

(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 $\mathsf{D}^{\flat}(\text{Coh}(X)) = \mathsf{D}^{\flat}(\text{Coh}(Y))$.

(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 $\mathsf{D}^{\flat}(\text{Coh}(\mathscr{X})) = \mathsf{D}^{\flat}(\text{Coh}(\mathscr{H}))$.

(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,
• The famous Orlov spaces in Hori Herbst Page '08 is another example.

A feature all of these examples have in common is that $h^{1,1} = 1$. Genus one fibrations give us a way to get examples with $h^{1,1} = 2$!

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

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

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 triangulated 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

( I'm 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.

For $N = 3$, $E$ is the twisted cubic^{H77, p7} 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 5

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 $Y_0$ is an elliptic normal curve.

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 $\pi$, with intersection multiplicity $D\cdot C = N$.

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 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 conjecture about Fourier-Mukai transforms I missed

Open Questions

In general, we usually have a picture as below: We have a mirror space $Y$ and an analytic non-Kähler resolution $\overline{Y}$, but we also have a related space in noncommutative geometry associated to a nonzero element $\alpha \in \text{Br}(Y)$. This space $Y_{\text{nc}, \alpha}$ will have $\mathsf{D}^{\flat}(\text{Coh}(\overline{Y})) \cong \mathsf{D}^{\flat}(\text{Coh}(Y_{\text{nc}, \alpha}))$. We can pass to a smooth deformation $Y'$ of $\overline{Y}$. The correct and precise statement 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?