arXiv subjects
Publications and source records attributed to Chiu-Chu Melissa Liu.
We prove the all-genus open mirror symmetry of a toric Calabi-Yau 3-orbifold where the boundary condition is given by multiple Aganagic-Vafa branes. Each brane could be either inner or outer, and admit a fractional framing. The B-model is given by the Chekhov-Eynard-Orantin topological recursion expanded in local coordinates specified by the brane data.
In mirror symmetry, symplectic Landau-Ginzburg models are mirror to a large class of examples, in particular to Fano varieties and hypersurfaces of many Calabi-Yau and Fano varieties. When studying their Fukaya categories on the A-model in homological mirror symmetry, one needs to calculate the weights of pseudo-holomorphic discs bounded by Lagrangian branes. While these calculations simplify for exact and Lefschetz fibrations, we generalize the machinery for computing these weights by dropping the exact and Lefschetz assumptions. For a general symplectic Landau-Ginzburg model, a singular symplectic fibration, we prove that the weights and Lagrangian gradings split into base and fiber components. This is used in many calculations of Fukaya-Seidel categories to provide evidence of Kontsevich's homological mirror symmetry conjecture.
We prove an inversion theorem for recursive formulas satisfied by certain families of converging power series in two variables. These power series are indexed by the Harder-Narasimhan types of principal $G$-bundles of degree $d \in π_1 G$ on a smooth projective curve $X$, where $G$ is a connected complex reductive group. As an application, we obtain a closed formula for the Hodge-Poincaré series of moduli stacks of semistable principal $G$-bundles of degree $d$. We also compute the variation of Hodge structure of the moduli stack of all principal $G$-bundles over $X$, as a function of the period matrix of that curve.
We formulate and prove the Remodeling Conjecture with descendants, which is a version of all-genus equivariant descendant mirror symmetry for semi-projective toric Calabi-Yau 3-orbifolds with integral structures. We construct an isomorphism between the $K$-group of equivariant coherent sheaves on the toric Calabi-Yau 3-orbifold with support bounded in a direction and a certain integral relative first homology group of the equivariant mirror curve. Under this isomorphism, we prove the equivariant mirror symmetric Gamma conjecture which equates quantum cohomology central charges of coherent sheaves and oscillatory integrals along corresponding relative 1-cycles. As a consequence in the non-equivariant setting, we prove a conjecture of Hosono which equates central charges of compactly supported coherent sheaves and period integrals of integral 3-cycles on the Hori-Vafa mirror 3-fold. Furthermore, we establish a correspondence between all-genus equivariant descendant Gromov-Witten invariants with $K$-theoretic framings and oscillatory integrals (Laplace transforms) of the Chekhov-Eynard-Orantin topological recursion invariants along relative 1-cycles on the equivariant mirror curve.
We describe the formulation of Fukaya categories of symplectic manifolds with $B$-fields. In addition, we give a formula for how the $A_\infty$ structure maps change as we deform an object by a Lagrangian isotopy.
We continue the mathematical development of the open/closed correspondence proposed by Mayr and Lerche-Mayr. Given an open geometry on a toric Calabi-Yau 3-orbifold $\mathcal{X}$ relative to a framed Aganagic-Vafa outer brane $(\mathcal{L},f)$, we construct a toric Calabi-Yau 4-orbifold $\widetilde{\mathcal{X}}$ and identify its genus-zero Gromov-Witten invariants with the disk invariants of $(\mathcal{X},\mathcal{L},f)$, generalizing prior work of the authors in the smooth case. We then upgrade the correspondence to the level of generating functions, and prove that the disk function of $(\mathcal{X},\mathcal{L},f)$ can be recovered from the equivariant $J$-function of $\widetilde{\mathcal{X}}$. We further establish a B-model correspondence that retrieves the B-model disk function of $(\mathcal{X},\mathcal{L},f)$ from the equivariant $I$-function of $\widetilde{\mathcal{X}}$, and show that the correspondences are compatible with mirror symmetry in both the open and closed sectors.
We show that Walcher's disk potential for the quintic threefold can be represented as a central charge of a specific Gauged Linear Sigma Model which we call the extended quintic GLSM. This representation provides an open/closed correspondence for the quintic threefold since the central charge is a generating function of closed genus-zero GLSM invariants. We also explain how open Landau-Ginzburg/Calabi-Yau correspondence and open mirror symmetry for the quintic are compatible with wall-crossing and mirror symmetry of the extended GLSM, respectively.
For any positive integer $g$, we introduce the moduli space $\mathcal{A}^F_g =[\mathcal{H}_g/P_g(\mathbb{Z})]$ parametrizing $g$-dimensional principally polarized abelian varieties $V_τ$ together with a Strominger-Yau-Zalsow (SYZ) fibration, where $τ\in \mathcal{H}_g$ is the genus-$g$ Seigel upper half space and $P_g(\mathbb{Z}) \subset \mathrm{Sp}(2g,\mathbb{Z})$ is the integral Siegel parabolic subgroup. We study global SYZ mirror symmetry over the global moduli $\mathcal{H}_g$ and $\mathcal{A}^F_g$, relating the B-model on $V_τ$ and the A-model on its mirror, a compact $2g$-dimensional torus $\mathbb{T}^{2g}$ equipped with a complexified symplectic form. For each $V_τ$, we establish a homological mirror symmetry (HMS) result at the cohomological level over $\mathbb{C}$. This implies core HMS at the cohomological level over $\mathbb{C}$ and a graded $\mathbb{C}$-algebra isomorphism known as Seidel's mirror map. We study global HMS where Floer cohomology groups $HF^*(\hat{\ell}, \hat{\ell}')$ form coherent sheaves over a complex manifold parametrizing triples $(τ, \hat{\ell}, \hat{\ell}')$ where $τ\in \mathcal{H}_g$ defines a complexified symplectic form $ω_τ$ on $\mathbb{T}^{2g}$ and $\hat{\ell}$, $\hat{\ell} '$ are affine Lagrangian branes in $(\mathbb{T}^{2g}, ω_τ)$.
We compute I-functions and central charges for abelian GLSMs using virtual matrix factorizations of Favero and Kim. In the Calabi-Yau case we provide analytic continuation for the central charges by explicit integral formulas. The integrals in question are called hemisphere partition functions and we call the integral representation Higgs-Coulomb correspondence. We then use it to prove GIT stability wall-crossing for central charges.
We study K-theoretic GLSM invariants with one-dimensional gauge group and introduce elliptic central charges that depend on an elliptic cohomology class called an elliptic brane and a choice of level structure. These central charges have an integral representation related to an interpolation problem for the elliptic brane and satisfy natural q-difference equations. Integral representaions lead to the wall crossing statement for the central charges in two different phases of the GLSM. We explain how the orbifold structure leads to differences between wall crossing of the usual (K-theoretic) and elliptic central charges.
We study open-closed orbifold Gromov-Witten invariants of 3-dimensional Calabi-Yau smooth toric Deligne-Mumford (DM) stacks (with possibly non-trivial generic stabilizers and semi-projective coarse moduli spaces) relative to Lagrangian branes of Aganagic-Vafa type. We present foundational materials of enumerative geometry of stable holomorphic maps from bordered orbifold Riemann surfaces to a 3-dimensional Calabi-Yau smooth toric DM stack with boundaries mapped into a Aganagic-Vafa brane. All genus open-closed Gromov-Witten invariants are defined by torus localization and depend on the choice of a framing which is an integer. We also provide another definition of all genus open-closed Gromov-Witten invariants based on algebraic relative orbifold Gromov-Witten theory; this generalizes the definition in Li-Liu-Liu-Zhou [arXiv:math/0408426] for smooth toric Calabi-Yau 3-folds. When the toric DM stack a toric Calabi-Yau 3-orbifold (i.e. when the generic stabilizer is trivial), we define generating functions of open-closed Gromov-Witten invariants or arbitrary genus $g$ and number $h$ of boundary circles; it takes values in the Chen-Ruan orbifold cohomology of the classifying space of a finite cyclic group of order $m$. We prove an open mirror theorem which relates the generating function of orbifold disk invariants to Abel-Jacobi maps of the mirror curve of the toric Calabi-Yau 3-orbifold. This generalizes a conjecture by Aganagic-Vafa [arXiv:hep-th/0012041] and Aganagic-Klemm-Vafa [arXiv:hep-th/0105045] (proved in full generality by the first and the second authors in [arXiv:1103.0693]) on the disk potential of a smooth semi-projective toric Calabi-Yau 3-fold.
We establish a correspondence between the disk invariants of a smooth toric Calabi-Yau 3-fold $X$ with boundary condition specified by a framed Aganagic-Vafa outer brane $(L, f)$ and the genus-zero closed Gromov-Witten invariants of a smooth toric Calabi-Yau 4-fold $\widetilde{X}$, proving the open/closed correspondence proposed by Mayr and developed by Lerche-Mayr. Our correspondence is the composition of two intermediate steps: $\circ$ First, a correspondence between the disk invariants of $(X,L,f)$ and the genus-zero maximally-tangent relative Gromov-Witten invariants of a relative Calabi-Yau 3-fold $(Y,D)$, where $Y$ is a toric partial compactification of $X$ by adding a smooth toric divisor $D$. This correspondence can be obtained as a consequence of the topological vertex (Li-Liu-Liu-Zhou) and Fang-Liu where the all-genus open Gromov-Witten invariants of $(X,L,f)$ are identified with the formal relative Gromov-Witten invariants of the formal completion of $(Y,D)$ along the toric 1-skeleton. Here, we present a proof without resorting to formal geometry. $\circ$ Second, a correspondence in genus zero between the maximally-tangent relative Gromov-Witten invariants of $(Y,D)$ and the closed Gromov-Witten invariants of the toric Calabi-Yau 4-fold $\widetilde{X} = \mathcal{O}_Y(-D)$. This can be viewed as an instantiation of the log-local principle of van Garrel-Graber-Ruddat in the non-compact setting.
We study how to use a suitably ample locally free sheaf over a proper Deligne-Mumford stack to furnish an embedding of the stack into a geometric invariant theory (GIT) quotient stack constructed from a finite-dimensional linear representation of the general linear group.
The Remodeling Conjecture proposed by Bouchard-Klemm-Marino-Pasquetti [arXiv:0709.1453, arXiv:0807.0597] relates all genus open and closed Gromov-Witten invariants of a semi-projective toric Calabi-Yau 3-manifolds/3-orbifolds to the Eynard-Orantin invariants of the mirror curve of the toric Calabi-Yau 3-fold. In this paper, we present a proof of the Remodeling Conjecture for open-closed orbifold Gromov-Witten invariants of an arbitrary affine toric Calabi-Yau 3-orbifold relative to a framed Aganagic-Vafa Lagrangian brane. This can be viewed as an all genus open-closed mirror symmetry for affine toric Calabi-Yau 3-orbifolds.
The Remodeling Conjecture proposed by Bouchard-Klemm-Mariño-Pasquetti (BKMP) [arXiv:0709.1453, arXiv:0807.0597] relates the A-model open and closed topological string amplitudes (the all genus open and closed Gromov-Witten invariants) of a semi-projective toric Calabi-Yau 3-manifold/3-orbifold to the Eynard-Orantin invariants of its mirror curve. It is an all genus open-closed mirror symmetry for toric Calabi-Yau 3-manifolds/3-orbifolds. In this paper, we present a proof of the BKMP Remodeling Conjecture for all genus open-closed orbifold Gromov-Witten invariants of an arbitrary semi-projective toric Calabi-Yau 3-orbifold relative to an outer framed Aganagic-Vafa Lagrangian brane. We also prove the conjecture in the closed string sector at all genera.
A smooth GKM stack is a smooth Deligne-Mumford stack equipped with an action of an algebraic torus $T$, with only finitely many zero-dimensional and one-dimensional orbits. (i) We define the stacky GKM graph of a smooth GKM stack, under the mild assumption that any one-dimensional $T$-orbit closure contains at least one $T$ fixed point. The stacky GKM graph is a decorated graph which contains enough information to reconstruct the $T$-equivariant formal neighborhood of the 1-skeleton (union of zero-dimensional and one-dimensional $T$-orbits) as a formal smooth DM stack equipped with a $T$-action. (ii) We axiomize the definition of a stacky GKM graph and introduce abstract stacky GKM graphs which are more general than stacky GKM graphs of honest smooth GKM stacks. From an abstract GKM graph we construct a formal smooth GKM stack. (iii) We define equivariant orbifold Gromov-Witten invariants of smooth GKM stacks, as well as formal equivariant orbifold Gromov-Witten invariants of formal smooth GKM stacks. These invariants can be computed by virtual localization and depend only the stacky GKM graph or the abstract stacky GKM graph. Formal equivariant orbifold Gromov-Witten invariants of the stacky GKM graph of a smooth GKM stack $\mathcal{X}$ are refinements of equivariant orbifold Gromov-Witten invariants of $\mathcal{X}$.
This is a brief introduction to the Bershadsky-Cecotti-Ooguri-Vafa (BCOV) holomorphic anomaly equations and Walcher's extended holomorphic anomaly equations.
We outline various developments of affine and general Landau Ginzburg models in physics. We then describe the A-twisting and coupling to gravity in terms of Algebraic Geometry. We describe constructions of various path integral measures (virtual fundamental class) using the algebro-geometric technique of cosection localization, culminating in the theory of ``Mixed Spin P (MSP) fields" developed by the authors.