Mirror Symmetry for Calabi-Yau hypersurfaces in toric varieties is by now well established. However, previous approaches to it did not uncover the underlying reason for mirror varieties to be mirror. We are able to calculate explicitly vertex algebras that correspond to holomorphic parts of A and B models of Calabi-Yau hypersurfaces and complete intersections in toric varieties. We establish the relation between these vertex algebras for mirror Calabi-Yau manifolds. This should eventually allow us to rewrite the whole story of toric Mirror Symmetry in the language of sheaves of vertex algebras. Our approach is purely algebraic and involves simple techniques from toric geometry and homological algebra, as well as some basic results of the theory of vertex algebras. Ideas of this paper may also be useful in other problems related to maps from curves to algebraic varieties. This paper could also be of interest to physicists, because it contains explicit descriptions of A and B models of Calabi-Yau hypersurfaces and complete intersection in terms of free bosons and fermions.
The goal of this paper is to make the vertex operator algebra approach to mirror symmetry accessible to algebraic geometers. Compared to better-known approaches using moduli spaces of stable maps and special Lagrangian fibrations, this approach follows more closely the original line of thinking that lead to the discovery of mirror symmetry by physicists. The ultimate goal of the vertex algebra approach is to give precise mathematical definitions of N=(2,2) superconformal field theories called A and B models associated to any Calabi-Yau variety and then show that thus constructed theories are r…
We give a vertex algebra proof of the Berglund-Hübsch duality of nondegenerate invertible potentials. We suggest a way to unify it with the Batyrev-Borisov duality of reflexive Gorenstein cones.
For three-partition triple Hodge integrals related to the topological vertex, we derive Eynard-Orantin type recursion relations from the cut-and-join equation. This establishes a version of local mirror symmetry for the local $C^3$ geometry with three D-branes, as proposed by Marino and Bouchard-Klemm-Marino-Pasquetti.
We prove the Bouchard-Mariño Conjecture for the framed one-legged topological vertex by deriving the Eynard-Orantin type recursion relations from the cut-and-join equation satisfied by the relevant triple Hodge integrals. This establishes a version of local mirror symmetry for the local $C^3$ geometry with one $D$-brane.
We calculate chiral rings of the N=2 vertex algebras constructed from the combinatorial data of toric mirror symmetry and show that they coincide with the description of stringy cohomology conjectured previously in a joint work with A. Mavlyutov.
A vertex algebra is an algebraic counterpart of a two-dimensional conformal field theory. We give a new definition of a vertex algebra which includes chiral algebras as a special case, but allows for fields which are neither meromorphic nor anti-meromorphic. To any complex torus equipped with a flat Kahler metric and a closed 2-form we associate an N=2 superconformal vertex algebra (N=2 SCVA) in the sense of our definition. We find a criterion for two different tori to produce isomorphic N=2 SCVA's. We show that for algebraic tori isomorphism of N=2 SCVA's implies the equivalence of the derive…
Aganagic and Okounkov proved that the elliptic stable envelope provides the pole cancellation matrix for the enumerative invariants of quiver varieties known as vertex functions. This transforms a basis of a system of $q$-difference equations holomorphic in $\boldsymbol{z}$ with poles in $\boldsymbol{a}$ to a basis of solutions holomorphic in $\boldsymbol{a}$ with poles in $\boldsymbol{z}$. The resulting functions are expected to be the vertex functions of the 3d mirror dual variety. In this paper, we prove that for the cotangent bundle of the full flag variety, the functions obtained in this …
Following the idea of Aganagic--Okounkov \cite{AOelliptic}, we study vertex functions for hypertoric varieties, defined by $K$-theoretic counting of quasimaps from $\mathbb{P}^1$. We prove the 3d mirror symmetry statement that the two sets of $q$-difference equations of a 3d hypertoric mirror pairs are equivalent to each other, with Kähler and equivariant parameters exchanged, and the opposite choice of polarization. Vertex functions of a 3d mirror pair, as solutions to the $q$-difference equations, satisfying particular asymptotic conditions, are related by the elliptic stable envelopes. Vari…
Previously, we introduced a duality transformation for Euler $G$--Frobenius algebras. Using this transformation, we prove that the simple $A,D,E$ singularities and Pham singularities of coprime powers are mirror self--dual where the mirror duality is implemented by orbifolding with respect to the symmetry group generated by the grading operator and dualizing. We furthermore calculate orbifolds and duals to other $G$--Frobenius algebras which relate different $G$--Frobenius algebras for singularities. In particular, using orbifolding and the duality transformation we provide a mirror pairs for …
This is my talk at ICM, Zurich 1994. It contains a short introduction, two basic examples and a refined version of the Mirror Conjecture formulated in terms of homological algebra.
We study mirror symmetry of families of elliptic K3 surfaces with elliptic fibers of type $E_6,~E_7$ and $E_8$. We consider a moduli space $\mathsf{T}$ of the mirror K3 surfaces enhanced with the choice of differential forms. We show that coordinates on $\mathsf{T}$ are given by the ring of quasi modular forms in two variables, with modular groups adapted to the fiber type. We furthermore introduce an algebraic group $\mathsf{G}$ which acts on $\mathsf{T}$ from the right and construct its Lie algebra $\mathrm{Lie}(\mathsf{G})$. We prove that the extended Lie algebra generated by $\mathrm{Lie}(…
We construct a cubic field theory which provides all genus amplitudes of the topological A-model for all non-compact Calabi-Yau toric threefolds. The topology of a given Feynman diagram encodes the topology of a fixed Calabi-Yau, with Schwinger parameters playing the role of Kahler classes of Calabi-Yau. We interpret this result as an operatorial computation of the amplitudes in the B-model mirror which is the Kodaira-Spencer quantum theory. The only degree of freedom of this theory is an unconventional chiral scalar on a Riemann surface. In this setup we identify the B-branes on the mirror Ri…
We construct a family of vertex algebras associated with a family of symplectic singularity/resolution, called hypertoric varieties. While the hypertoric varieties are constructed by a certain Hamiltonian reduction associated with a torus action, our vertex algebras are constructed by (semi-infinite) BRST reduction. The construction works algebro-geometrically and we construct sheaves of $\hbar$-adic vertex algebras over hypertoric varieties which localize the vertex algebras. We show when the vertex algebras are vertex operator algebras by giving explicit conformal vectors. We also show that …
We consider the variant of Mirror Symmetry Conjecture for K3 surfaces which relates "geometry" of curves of a general member of a family of K3 with "algebraic functions" on the moduli of the mirror family. Lorentzian Kac--Moody algebras are involved in this construction. We give several examples when this conjecture is valid.
We prove the Landau-Ginzburg Mirror Symmetry Conjecture at the level of (orbifolded) Frobenius algebras for a large class of invertible singularities, including arbitrary sums of loops and Fermats with arbitrary symmetry groups. Specifically, we show that for a quasi-homogeneous polynomial W and an admissible group G within the class, the Frobenius algebra arising in the FJRW theory of [W/G] is isomorphic (as a Frobenius algebra) to the orbifolded Milnor ring of [W^T/G^T], associated to the dual polynomial W^T and dual group G^T.
We explain how deformation theories of geometric objects such as complex structures, Poisson structures and holomorphic bundle structures lead to differential Gerstenhaber or Poisson algebras. We use homological perturbation theory to obtain $A_{\infty}$ algebra structures and some canonically defined deformations of such structures on the cohomology. We formulate the $A_{\infty}$ algebraic mirror symmetry as the identification of the $A_{\infty}$ algebras together with their canonical deformations constructed this way on different manifolds.
The exterior algebra $E$ on a finite-rank free module $V$ carries a $\mathbb{Z}/2$-grading and an increasing filtration, and the $\mathbb{Z}/2$-graded filtered deformations of $E$ as an associative algebra are the familiar Clifford algebras, classified by quadratic forms on $V$. We extend this result to $A_\infty$-algebra deformations $\mathcal{A}$, showing that they are classified by formal functions on $V$. The proof translates the problem into the language of matrix factorisations, using the localised mirror functor construction of Cho-Hong-Lau, and works over an arbitrary ground ring. We a…
The existence of a flat torsion-free connection, or left symmetric algebra structure on a Lie algebra g gives rise to a canonically defined complex structure on g+g and a symplectic structure on g+g^*. We verify that the associated differential Gerstenhaber algebras controlling the deformation theories of the complex and symplectic form are isomorphic. This provides a class of examples of "weak mirror symmetry" as suggested by Merkulov. For nilpotent algebras in dimension 4 and 6 the isomorphism classes of the semi-direct products g+g and g+g^* are listed. A one-parameter family of inequivalen…
We show that complex multiplication on abelian varieties is equivalent to the existence of a constant rational Kähler metric. We give a sufficient condition for a mirror of an abelian variety of CM-type to be of CM-type as well. We also study the relationship between complex multiplication and rationality of a toroidal lattice vertex algebra.
Let $X$ be a Nakajima quiver variety and $X'$ its $3d$-mirror. We consider the action of the Picard torus $\mathsf{K}=\mathrm{Pic}(X)\otimes \mathbb{C}^{\times}$ on $X'$. Assuming that $(X')^{\mathsf{K}}$ is finite, we propose a formula for the $\mathsf{K}$-character of the tangent spaces at the fixed points in terms of certain enumerative invariants of $X$ known as vertex functions.
Any vertex algebra has a canonical decreasing filtration, called Li filtration, whose associated graded space has a natural structure of a vertex Poisson algebra. In this note, we introduce an analogous filtration for any SUSY vertex algebra, which was introduced by Heluani and Kac as a superfield formalism of a supersymmetric vertex algebra. We prove that the associated graded superspace of our filtration has a structure of SUSY vertex Poisson algebras. We also introduce and discuss related notions, such as Zhu's $C_2$-Poisson superalgebras, associated superschemes and singular supports, for …
We recall Borcherds's approach to vertex algebras via "singular commutative rings", and introduce new examples of his constructions which we compare to vertex algebras, chiral algebras, and factorization algebras. We show that all vertex algebras (resp. chiral algebras or equivalently factorization algebras) can be realized in these new categories $\text{VA}(A,H,S)$, but we also show that the functors from $\text{VA}(A,H,S)$ to vertex algebras or chiral algebras are not equivalences: a single vertex or chiral algebra may have non-equivalent realizations as an $(A, H,S)$-vertex algebra.
We study moduli spaces of mirror non-compact Calabi-Yau threefolds enhanced with choices of differential forms. The differential forms are elements of the middle dimensional cohomology whose variation is described by a variation of mixed Hodge structures which is equipped with a flat Gauss-Manin connection. We construct graded differential rings of special functions on these moduli spaces and show that they contain rings of quasi-modular forms. We show that the algebra of derivations of quasi-modular forms can be obtained from the Gauss--Manin connection contracted with vector fields on the en…
In this paper we prove a formula relating the equivariant Euler characteristic of $K$-theoretic stable envelopes to an object known as the index vertex for the cotangent bundle of the full flag variety. Our formula demonstrates that the index vertex is the power series expansion of a rational function. This result is a consequence of the 3d mirror self-symmetry of the variety considered here. In general, one expects an analogous result to hold for any two varieties related by 3d mirror symmetry.
We introduce the notion of irregular vertex (operator) algebras. The irregular versions of fundamental properties, such as Goddard uniqueness theorem, associativity and operator product expansions are formulated and proved. We also give some elementary examples of irregular vertex operator algebras.
