Mirror Landau-Ginzburg Research Articles

Tropical Mirror

We describe the tropical curves in toric varieties and define the tropical Gromov-Witten invariants. We introduce amplitudes for the higher topological quantum mechanics (HTQM) on special trees and show that the amplitudes are equal to the tropical Gromov-Witten invariants. We show that the sum over the amplitudes in $A$-model HTQM equals the total amplitude in B-model HTQM, defined as a deformation of the $A$-model HTQM by the mirror superpotential. We derived the mirror superpotentials for the toric varieties and showed that they coincide with the superpotentials in the mirror Landau-Ginzburg theory. We construct the mirror dual states to the evaluation observables in the tropical Gromov-Witten theory.

T-equivariant disc potential and SYZ mirror construction

We develop a G-equivariant Lagrangian Floer theory and obtain a curved A∞ algebra, and in particular a G-equivariant disc potential. We construct a Morse model, which counts pearly trees in the Borel construction LG. When applied to a smooth moment map fiber of a semi-Fano toric manifold, our construction recovers the T-equivariant toric Landau-Ginzburg mirror of Givental. We also study the S1-equivariant Floer theory of a typical singular fiber of a Lagrangian torus fibration (i.e. a pinched torus) and compute its S1-equivariant disc potential via the gluing technique developed in [14,37].

Exotic Lagrangian tori in Grassmannians

We describe an iterative construction of Lagrangian tori in the complex Grassmannian \operatorname{Gr}(k,n) , based on the cluster algebra structure of the coordinate ring of a mirror Landau–Ginzburg model proposed by Marsh and Rietsch (2020). Each torus comes with a Laurent polynomial, and local systems controlled by the k -variables Schur polynomials at the n -th roots of unity. We use this data to give examples of monotone Lagrangian tori that are neither displaceable nor Hamiltonian isotopic to each other, and that support nonzero objects in different summands of the spectral decomposition of the Fukaya category over \mathbb{C} .

The Doran-Harder-Thompson conjecture for toric complete intersections

Given a Tyurin degeneration of a Calabi-Yau complete intersection in a toric variety, we prove gluing formulas relating the generalized functional invariants, periods, and I-functions of the mirror Calabi-Yau family and those of the two mirror Landau-Ginzburg models. Our proof makes explicit the “gluing/splitting” of fibrations in the Doran-Harder-Thompson mirror conjecture. Our gluing formula implies an identity, obtained by composition with their respective mirror maps, that relates the absolute Gromov-Witten invariants for the Calabi-Yaus and relative Gromov-Witten invariants for the quasi-Fanos.

Decompositions of moduli spaces of vector bundles and graph potentials

Abstract We propose a conjectural semiorthogonal decomposition for the derived category of the moduli space of stable rank 2 bundles with fixed determinant of odd degree, independently formulated by Narasimhan. We discuss some evidence for and furthermore propose semiorthogonal decompositions with additional structure. We also discuss two other decompositions. One is a decomposition of this moduli space in the Grothendieck ring of varieties, which relates to various known motivic decompositions. The other is the critical value decomposition of a candidate mirror Landau–Ginzburg model given by graph potentials, which in turn is related under mirror symmetry to Muñoz’s decomposition of quantum cohomology. This corresponds to an orthogonal decomposition of the Fukaya category. We discuss how decompositions on different levels (derived category of coherent sheaves, Grothendieck ring of varieties, Fukaya category, quantum cohomology, critical sets of graph potentials) are related and support each other.

On the fundamental group of open Richardson varieties

We compute the fundamental group of an open Richardson variety in the manifold of complete flags that corresponds to a partial flag manifold. Rietsch showed that these log Calabi-Yau varieties underlie a Landau-Ginzburg mirror for the Langlands dual partial flag manifold, and our computation verifies a prediction of Hori for this mirror. It is log Calabi-Yau as it isomorphic to the complement of the Knutson-Lam-Speyer anti-canonical divisor for the partial flag manifold. We also determine explicit defining equations for this divisor.

Landau–Ginzburg mirror, quantum differential equations and qKZ difference equations for a partial flag variety

We consider the system of quantum differential equations for a partial flag variety and construct a basis of solutions in the form of multidimensional hypergeometric functions, that is, we construct a Landau–Ginzburg mirror for that partial flag variety. In our construction, the solutions are labeled by elements of the K-theory algebra of the partial flag variety.To establish these facts we consider the equivariant quantum differential equations for a partial flag variety and introduce a compatible system of difference equations, which we call the qKZ equations. We construct a basis of solutions of the joint system of the equivariant quantum differential equations and qKZ difference equations in the form of multidimensional hypergeometric functions. Then the facts about the non-equivariant quantum differential equations are obtained from the facts about the equivariant quantum differential equations by a suitable limit.Analyzing these constructions we obtain a formula for the fundamental Levelt solution of the quantum differential equations for a partial flag variety.

A mirror theorem for multi-root stacks and applications

Let X be a smooth projective variety with a simple normal crossing divisor D:=D_1+D_2+cdots +D_n, where D_isubset X are smooth, irreducible and nef. We prove a mirror theorem for multi-root stacks X_{D,{overrightarrow{r}}} by constructing an I-function lying in a slice of Givental’s Lagrangian cone for Gromov–Witten theory of multi-root stacks. We provide three applications: (1) We show that some genus zero invariants of X_{D,overrightarrow{r}} stabilize for sufficiently large overrightarrow{r}. (2) We state a generalized local-log-orbifold principle conjecture and prove a version of it. (3) We show that regularized quantum periods of Fano varieties coincide with classical periods of the mirror Landau–Ginzburg potentials using orbifold invariants of X_{D,overrightarrow{r}}.

A Landau–Ginzburg mirror theorem via matrix factorizations

Abstract For an invertible quasihomogeneous polynomial 𝒘 {{\boldsymbol{w}}} we prove an all-genus mirror theorem relating two cohomological field theories of Landau–Ginzburg type. On the B-side it is the Saito–Givental theory for a specific choice of a primitive form. On the A-side, it is the matrix factorization CohFT for the dual singularity 𝒘 T {{\boldsymbol{w}}^{T}} with the maximal diagonal symmetry group.

An extension of polar duality of toric varieties and its consequences in Mirror Symmetry

The present paper is dedicated to illustrating an extension of polar duality between Fano toric varieties to a more general duality, called \emph{framed} duality, so giving rise to a powerful and unified method of producing mirror partners of hypersurfaces and complete intersections in toric varieties, of any Kodaira dimension. In particular, the class of projective hypersurfaces and their mirror partners are studied in detail. Moreover, many connections with known Landau-Ginzburg mirror models, Homological Mirror Symmetry and Intrinsic Mirror Symmetry, are discussed.

Landau–Ginzburg mirror symmetry conjecture

We prove the Landau–Ginzburg mirror symmetry conjecture between invertible quasihomogeneous polynomial singularities at all genera. That is, we show that the FJRW theory (LG A-model) of such a polynomial is equivalent to the Saito–Givental theory (LG B-model) of the mirror polynomial.

Hyperelliptic integrals and mirrors of the Johnson–Kollár del Pezzo surfaces

For all integers k > 0 k>0 , we prove that the hypergeometric function \[ I ^ k ( α ) = ∑ j = 0 ∞ ( ( 8 k + 4 ) j ) ! j ! ( 2 j ) ! ( ( 2 k + 1 ) j ) ! 2 ( ( 4 k + 1 ) j ) ! α j \widehat {I}_k(\alpha )=\sum _{j=0}^\infty \frac {\bigl ((8k+4)j\bigr )!j!}{(2j)!\bigl ((2k+1)j\bigr )!^2 \bigl ((4k+1)j\bigr )!} \ \alpha ^j \] is a period of a pencil of curves of genus 3 k + 1 3k+1 . We prove that the function I ^ k \widehat {I}_k is a generating function of Gromov–Witten invariants of the family of anticanonical del Pezzo hypersurfaces X = X 8 k + 4 ⊂ P ( 2 , 2 k + 1 , 2 k + 1 , 4 k + 1 ) X=X_{8k+4} \subset \mathbb {P}(2,2k+1,2k+1,4k+1) . Thus, the pencil is a Landau–Ginzburg mirror of the family. The surfaces X X were first constructed by Johnson and Kollár. The feature of these surfaces that makes our mirror construction especially interesting is that | − K X | = | O X ( 1 ) | = ∅ |-K_X|=|\mathcal {O}_X (1)|=\varnothing . This means that there is no way to form a Calabi–Yau pair ( X , D ) (X,D) out of X X and hence there is no known mirror construction for X X other than the one given here. We also discuss the connection between our construction and work of Beukers, Cohen and Mellit on hypergeometric functions.

Hodge numbers of Landau–Ginzburg models

We study the Hodge numbers fp,q of Landau–Ginzburg models as defined by Katzarkov, Kontsevich, and Pantev. First we show that these numbers can be computed using ordinary mixed Hodge theory, then we give a concrete recipe for computing these numbers for the Landau–Ginzburg mirrors of Fano threefolds. We finish by proving that for a crepant resolution of a Gorenstein toric Fano threefold X there is a natural LG mirror (Y,w) so that hp,q(X)=f3−q,p(Y,w).

Cyclic Sieving and Cluster Duality of Grassmannian

We introduce a decorated configuration space $\mathscr{C}\!{\rm onf}_n^\times(a)$ with a potential function $\mathcal{W}$. We prove the cluster duality conjecture of Fock-Goncharov for Grassmannians, that is, the tropicalization of $\big(\mathscr{C}\!{\rm onf}_n^\times(a), \mathcal{W}\big)$ canonically parametrizes a linear basis of the homogeneous coordinate ring of the Grassmannian $\operatorname{Gr}_a(n)$ with respect to the Pl\"ucker embedding. We prove that $\big(\mathscr{C}\!{\rm onf}_n^\times(a), \mathcal{W}\big)$ is equivalent to the mirror Landau-Ginzburg model of the Grassmannian considered by Eguchi-Hori-Xiong, Marsh-Rietsch and Rietsch-Williams. As an application, we show a cyclic sieving phenomenon involving plane partitions under a sequence of piecewise-linear toggles.

Fukaya category of Grassmannians: Rectangles

We show that the monotone Lagrangian torus fiber of the Gelfand-Cetlin integrable system on the complex Grassmannian Gr(k,n) supports generators for all maximum modulus summands in the spectral decomposition of the Fukaya category over C, generalizing the example of the Clifford torus in projective space. We introduce an action of the dihedral group Dn on the Landau-Ginzburg mirror proposed by Marsh-Rietsch [27] that makes it equivariant and use it to show that, given a lower modulus, the torus supports nonzero objects in none or many summands of the Fukaya category with that modulus. The alternative is controlled by the vanishing of rectangular Schur polynomials at the n-th roots of unity, and for n=p prime this suffices to give a complete set of generators and prove homological mirror symmetry for Gr(k,p).

Tropical counting from asymptotic analysis on Maurer-Cartan equations

Let X = X Σ X = X_\Sigma be a toric surface and let ( X ˇ , W ) (\check {X}, W) be its Landau-Ginzburg (LG) mirror where W W is the Hori-Vafa potential as shown in their preprint. We apply asymptotic analysis to study the extended deformation theory of the LG model ( X ˇ , W ) (\check {X}, W) , and prove that semi-classical limits of Fourier modes of a specific class of Maurer-Cartan solutions naturally give rise to tropical disks in X X with Maslov index 0 or 2, the latter of which produces a universal unfolding of W W . For X = P 2 X = \mathbb {P}^2 , our construction reproduces Gross’ perturbed potential W n W_n [Adv. Math. 224 (2010), pp. 169–245] which was proven to be the universal unfolding of W W written in canonical coordinates. We also explain how the extended deformation theory can be used to reinterpret the jumping phenomenon of W n W_n across walls of the scattering diagram formed by Maslov index 0 tropical disks originally observed by Gross in the same work (in the case of X = P 2 X = \mathbb {P}^2 ).

E8 spectral curves

I provide an explicit construction of spectral curves for the affine $\mathrm{E}_8$ relativistic Toda chain. Their closed form expression is obtained by determining the full set of character relations in the representation ring of $\mathrm{E}_8$ for the exterior algebra of the adjoint representation; this is in turn employed to provide an explicit construction of both integrals of motion and the action-angle map for the resulting integrable system. I consider two main areas of applications of these constructions. On the one hand, I consider the resulting family of spectral curves in the context of the correspondences between Toda systems, 5d Seiberg-Witten theory, Gromov-Witten theory of orbifolds of the resolved conifold, and Chern-Simons theory to establish a version of the B-model Gopakumar-Vafa correspondence for the $\mathrm{sl}_N$ L\^e-Murakami-Ohtsuki invariant of the Poincar\'e integral homology sphere to all orders in $1/N$. On the other, I consider a degenerate version of the spectral curves and prove a 1-dimensional Landau-Ginzburg mirror theorem for the Frobenius manifold structure on the space of orbits of the extended affine Weyl group of type $\mathrm{E}_8$ introduced by Dubrovin-Zhang (equivalently, the orbifold quantum cohomology of the type-$\mathrm{E}_8$ polynomial $\mathbb{C} P^1$ orbifold). This leads to closed-form expressions for the flat co-ordinates of the Saito metric, the prepotential, and a higher genus mirror theorem based on the Chekhov-Eynard-Orantin recursion. I will also show how the constructions of the paper lead to a generalisation of a conjecture of Norbury-Scott to ADE $\mathbb{P}^1$-orbifolds, and a mirror of the Dubrovin-Zhang construction for all Weyl groups and choices of marked roots.

Extended r -Spin Theory and the Mirror Symmetry for the A r − 1 -Singularity

By a famous result of K. Saito, the parameter space of the miniversal deformation of the $A_{r-1}$-singularity carries a Frobenius manifold structure. The Landau-Ginzburg mirror symmetry says that, in the flat coordinates, the potential of this Frobenius manifold is equal to the generating series of certain integrals over the moduli space of $r$-spin curves. In this paper we show that the parameters of the miniversal deformation, considered as functions of the flat coordinates, also have a simple geometric interpretation using the extended $r$-spin theory, first considered by T. J. Jarvis, T. Kimura and A. Vaintrob, and studied in a recent paper of E. Clader, R. J. Tessler and the author. We prove a similar result for the singularity $D_4$ and present conjectures for the singularities $E_6$ and $E_8$.

Central charges of T-dual branes for toric varieties

Given any equivariant coherent sheaf L \mathcal {L} on a compact semi-positive toric orbifold X \mathcal {X} , its SYZ T-dual mirror dual is a Lagrangian brane in the Landau-Ginzburg mirror. We prove that the oscillatory integral of the equivariant superpotential in the Landau Ginzburg mirror over this Lagrangian brane is the genus-zero 1 1 -descendant Gromov-Witten potential with a Gamma-type class of L \mathcal {L} inserted.

Hodge-theoretic mirror symmetry for toric stacks

Using the mirror theorem [CCIT15], we give a Landau-Ginzburg mirror description for the big equivariant quantum cohomology of toric Deligne-Mumford stacks. More precisely, we prove that the big equivariant quantum D-module of a toric Deligne-Mumford stack is isomorphic to the Saito structure associated to the mirror Landau-Ginzburg potential. We give a GKZ-style presentation of the quantum D-module, and a combinatorial description of quantum cohomology as a quantum Stanley-Reisner ring. We establish the convergence of the mirror isomorphism and of quantum cohomology in the big and equivariant setting.