The study of 3d mirror symmetry has greatly enhanced our understanding of various aspects of 3d $\mathcal{N}=4$ theories. In this paper, starting with known mirror pairs of 3d $\mathcal{N}=4$ quiver gauge theories and gauging discrete subgroups of the flavour or topological symmetry, we construct new mirror pairs with non-trivial 1-form symmetry. By providing explicit quiver descriptions of these theories, we thoroughly specify their symmetries (0-form, 1-form, and 2-group) and the mirror maps between them.
We study the mirror symmetry of abelian 3d $\mathcal{N}=2$ theories with mixed Chern-Simons levels by turning them into $\mathcal{T}_{A,N}$ theories that are defined as $N$ copies of $U(1)-[1]$ theory coupled together by mixed Chern-Simons levels $k_{ij}$. We find that $\mathcal{T}_{A,N}$ theories have many mirror dual theories with different mixed CS levels and FI parameters. As an example, we analyze $U(1)_k+ N_C \,\mathbf{C} + N_{AC} \, \mathbf{AC}$ theories by transforming these theories into certain $\mathcal{T}_{A,N}$ theories and find many equivalent effective Chern-Simons levels. Final…
We compute supersymmetric indices to test mirror symmetry of three-dimensional $\mathcal{N}=4$ gauge theories and dualities of half-BPS enriched boundary conditions and interfaces in four-dimensional $\mathcal{N}=4$ Super Yang-Mills theory. We find the matching of indices as strong evidences for various dualities of the 3d interfaces conjectured by Gaiotto and Witten under the action of S-duality in Type IIB string theory.
Using a recently proposed duality for $U(N)$ supersymmetric QCD (SQCD) in three dimensions with monopole superpotential, in this paper we derive the mirror dual description of $\mathcal{N}=2$ SQCD with unitary gauge group, generalizing the known mirror dual description of abelian gauge theories. We match the chiral ring of the dual theories and their partition functions on the squashed sphere. We also conjecture a generalization for SQCD with orthogonal and symplectic gauge groups.
Let $(X,\check{X})$ be a mirror pair of a complex torus $X$ and its mirror partner $\check{X}$. This mirror pair is described as the trivial special Lagrangian torus fibrations $X\rightarrow B$ and $\check{X}\rightarrow B$ on the same base space $B$ by SYZ construction. Then, we can associate a holomorphic line bundle $E(s,\mathcal{L})\rightarrow X$ to a pair $(s,\mathcal{L})$ of a Lagrangian section $s$ of $\check{X}\rightarrow B$ and a unitary local system $\mathcal{L}$ along it. In this paper, we first construct the deformation $X_{\mathcal{G}}$ of $X$ by a certain flat gerbe $\mathcal{G}$ …
It is shown that in string theory mirror duality is a gauge symmetry (a Weyl transformation) in the moduli space of $N=2$ backgrounds on group manifolds, and we conjecture on the possible generalization to other backgrounds, such as Calabi-Yau manifolds.
We first give a complete, albeit brief, review of the discovery of mirror symmetry in $N=2$ string/conformal field theory. In particular, we describe the naturality arguments which led to the initial mirror symmetry conjectures and the subsequent work which established the existence of mirror symmetry through direct construction. We then review a number of striking consequences of mirror symmetry -- both conceptual and calculational. Finally, we describe recent work which introduces a variant on our original proof of the existence of mirror symmetry. This work affirms classical--quantum symmet…
We report on recent progress in understanding mirror symmetry. Some of more recent generalizations and applications are also presented. --- A contribution to the Proceedings of ``Strings 2001'' at Mumbai, India.
We survey some algebraic geometric aspects of mirror symmetry and duality in string theory. Some applications of computer algebra to algebraic geometry and string theory are shortly reviewed.
Mirror symmetry relates type IIB string theory on a Calabi-Yau 3-fold to type IIA on the mirror CY manifold, whose complex structure and Kaehler moduli spaces are exchanged. We show that the mirror map is a particular case of a more general quantum equivalence, fractional mirror symmetry, between Calabi-Yau compactifications and non-CY ones. As was done by Greene and Plesser for mirror symmetry, we obtain these new dualities by considering orbifolds of Gepner models, of asymmetric nature, leading to superconformal field theories isomorphic to the original ones, but with a different target-spac…
In this paper we conjecture a reformulation of the monomial-divisor mirror map for (2,2) mirror symmetry, valid at a boundary of the moduli space, that is easily extended to also include tangent bundle deformations -- an important step towards understanding (0,2) mirror symmetry. We check our conjecture in a few simple cases, and thereby illustrate how to perform calculations using a description of sheaves recently published by Knutson, Sharpe.
This is a review article on mirror symmetry and aspects of it related to the theory of modular forms. We describe this topic along its historical development and connect to some more recent results toward the end. The article is for publication in a special issue of ICCM Notices.
In this note a prediction of an algebraic mirror construction is checked for elliptic curves of Brieskorn-Pham type via number theoretic methods. It is shown that the modular forms associated to the Hasse-Weil L-series of mirror pairs of such curves are identical.
We study the effect of mirror symmetry for K3 surfaces on D-brane probe physics. The case of elliptically fibered K3 surfaces is considered in detail. In many cases, mirror can transform a singular fiber of Kodaira's type ADE into sets of singular fibers of type I_1 (II) with equal total Euler number, but vanishing contribution to the Picard number of the mirror surface. This provides a geometric model of quantum splitting phenomena. Mirror for three dimensional gauge theories, interchanging Fayet-Iliopoulos and mass terms, is also briefly discussed.
We discuss a K3 and torus from view point of "mirror symmetry". We calculate the periods of the K3 surface and obtain the mirror map, the two-point correlation function, and the prepotential. Then we find there is no instanton correction on K3 (also torus), which is expected from view point of Algebraic geometry.
We use a recent classification of non-degenerate quasihomogeneous polynomials to construct all Landau-Ginzburg (LG) potentials for N=2 superconformal field theories with c=9 and calculate the corresponding Hodge numbers. Surprisingly, the resulting spectra are less symmetric than the existing incomplete results. It turns out that models belonging to the large class for which an explicit construction of a mirror model as an orbifold is known show remarkable mirror symmetry. On the other hand, half of the remaining 15\% of all models have no mirror partners. This lack of mirror symmetry may poin…
(Talk given at the Oskar Klein centenary symposium 19-21 September 1994 in Stockholm, Sweden, to appear in the proceedings.) The first half of this talk is a non-technical discussion of some general aspects of string theory, in particular the problem of compactification. We also give an introduction to mirror symmetry. The second half is a brief account of two recent papers on this subject; one by the author on mirror symmetry for Kazama-Suzuki models and one by P. Berglund and the author on a search for possible mirror pairs of Landau-Ginzburg orbifolds.
We consider orientifolds of Calabi-Yau 3-folds in the context of Type IIA and Type IIB superstrings. We show how mirror symmetry can be used to sum up worldsheet instanton contributions to the superpotential for Type IIA superstrings. The relevant worldsheets have the topology of the disc and ${\bf RP^2}$.
We consider matrix factorizations and homological mirror symmetry on the torus T^2 using a Landau-Ginzburg description. We identify the basic matrix factorizations of the Landau-Ginzburg superpotential and compute the full spectrum, taking into account the explicit dependence on bulk and boundary moduli. We verify homological mirror symmetry by comparing three-point functions in the A-model and the B-model.
We construct quantum mechanical models which mimic many features of string theory. We use these models to gain improved descriptions of B fields and gerbes. We examine analogs of T duality, D branes, and mirror symmetry and derive quantum mechanical analogs of standard phenomena, such as the noncommutative geometry induced by a B field.
We study the differential equations governing mirror symmetry of elliptic curves, and obtain a characterization of the ODEs which give rise to the integral ${\bf q}$-expansion of mirror maps. Through theta function representation of the defining equation, we express the mirror correspondence in terms of theta constants. By investigating the elliptic curves in $X_9$-family, the identification of the Landau-Ginzburg potential with the spectral curve of Ising model is obtained. Through the Jacobi elliptic function parametrization of Boltzmann weights in the statistical model, an exact Jacobi form…
Recently, mirror symmetry is derived as T-duality applied to gauge systems that flow to non-linear sigma models. We present some of its applications to study quantum geometry involving D-branes. In particular, we show that one can employ D-branes wrapped on torus fibers to reproduce the mirror duality itself, realizing the program of Strominger-Yau-Zaslow in a slightly different context. Floer theory of intersecting Lagrangians plays an essential role.
We prove mirror symmetry for supersymmetric sigma models on Kahler manifolds in 1+1 dimensions. The proof involves establishing the equivalence of the gauged linear sigma model, embedded in a theory with an enlarged gauge symmetry, with a Landau-Ginzburg theory of Toda type. Standard R -> 1/R duality and dynamical generation of superpotential by vortices are crucial in the derivation. This provides not only a proof of mirror symmetry in the case of (local and global) Calabi-Yau manifolds, but also for sigma models on manifolds with positive first Chern class, including deformations of the acti…
We generalize the previously established (0,2) triality of exactly solvable models, Landau-Ginzburg theories and Calabi-Yau manifolds to a number of different classes of (0,2) compactifications derived from (2,2) vacua. For the resulting models we show that the known (2,2) mirror constructions induce mirror symmetry in the (0,2) context.
We investigate the open mirror symmetry of certain non-complete intersection Calabi- Yau 3-folds, so called pfaffian Calabi-Yau. We perform the prediction of the number of disk invariants of several examples by using the direct integration method proposed recently and the open mirror symmetry. We treat several pfaffian Calabi-Yau 3-folds in $\mathbb{P}^6$ and branes with two discrete vacua. Some models have the two special points in its moduli space, around both of which we can consider different A-model mirror partners. We compute disc invariants for both cases. This study is the first applic…
We study the realization of the (super) conformal topological symmetry in two-dimensional field theories. The mirror automorphism of the topological algebra is represented as a reflection in the space of fields. As a consequence, a double BRST structure for topological matter theories is found. It is shown that the implementation of the topological symmetry in non-critical (super)string theories depends on the matter content of the two realizations connected by the mirror transformation.
We study open string mirror symmetry for one-parameter Calabi-Yau hypersurfaces in weighted projective space. We identify mirror pairs of D-brane configurations, derive the corresponding inhomogeneous Picard-Fuchs equations, and solve for the domainwall tensions as analytic functions over moduli space. Our calculations exemplify several features that had not been seen in previous work on the quintic or local Calabi-Yau manifolds. We comment on the calculation of loop amplitudes.
This paper discusses the overlap of the Hori-Vafa formulation of mirror symmetry with some other constructions. We focus on compact Calabi-Yau hypersurfaces \mathcal{M}_G = {G = 0} in weighted complex projective spaces. The Hori-Vafa formalism relates a family {\mathcal{M}_G \in WCP^{m-1}_{Q_1,...,Q_m}[s] | \sum_{i=1}^m Q_i = s} of such hypersurfaces to a single Landau-Ginzburg mirror theory. A technique suggested by Hori and Vafa allows the Picard-Fuchs equations satisfied by the corresponding mirror periods to be determined. Some examples in which the variety \mathcal{M}_G is crepantly resol…
We evaluate half-indices of $\mathcal{N}=(2,2)$ half-BPS boundary conditions in 3d $\mathcal{N}=4$ supersymmetric Abelian gauge theories. We confirm that the Neumann boundary condition is dual to the generic Dirichlet boundary condition for its mirror theory as the half-indices perfectly match with each other. We find that a naive mirror symmetry between the exceptional Dirichlet boundary conditions defining the Verma modules of the quantum Coulomb and Higgs branch algebras does not always hold. The triangular matrix obtained from the elliptic stable envelope describes the precise mirror trans…
We describe mirror symmetry on higher dimensional tori, paying special attention to the behaviour of D-branes under mirror symmetry. To find the mirror D-branes the description of mirror symmetry on D-branes due to Ooguri, Oz en Yin is used. This method allows us to deal with the coisotropic D-branes recently introduced by Kapustin and Orlov. We compare this to the description of mirror symmetry on D-branes using the Fourier-Mukai transform of charges.
Mirror symmetry of the type II string has a beautiful generalization to the heterotic string. This generalization, known as (0,2) mirror symmetry, is a field still largely in its infancy. We describe recent developments including the ideas behind quantum sheaf cohomology, the mirror map for deformations of (2,2) mirrors, the construction of mirror pairs from worldsheet duality, as well as an overview of some of the many open questions. The (0,2) mirrors of Hirzebruch surfaces are presented as a new example.
We find that the energy spectra of four and five anyons in a harmonic potential exhibit some mirror symmetric (reflection symmetric about the semionic statistics point $θ=π/2$) features analogous to the mirror symmetry in the two and three anyon spectra. However, since the $\ell=0$ sector remains non-mirror symmetric, the fourth and fifth virial coeffients do not reflect this symmetry.
We consider a class of special Lagrangian subspaces of Calabi-Yau manifolds and identify their mirrors, using the recent derivation of mirror symmetry, as certain holomorphic varieties of the mirror geometry. This transforms the counting of holomorphic disc instantons ending on the Lagrangian submanifold to the classical Abel-Jacobi map on the mirror. We recover some results already anticipated as well as obtain some highly non-trivial new predictions.
We develop techniques for computing the equivariant local mirror symmetry of curves, i.e. mirror symmetry for O(k)+O(-2-k) over P^1 for k greater than 0. We also describe related methods for dealing with mirror symmetry of non-nef toric varieties. The basic tools are equivariant I functions and their Birkhoff factorization.
Motivated by recent works on localizations in A-twisted gauged linear sigma models, we discuss a generalization of toric residue mirror symmetry to complete intersections in Grassmannians.
Homological mirror symmetry for crepant resolutions of Gorenstein toric singularities leads to a pair of conjectures on certain hypergeometric systems of PDEs. We explain these conjectures and verify them in some cases.
Mirror Symmetry, Picard-Fuchs equations and instanton corrected Yukawa couplings are discussed within the framework of toric geometry. It allows to establish mirror symmetry of Calabi-Yau spaces for which the mirror manifold had been unavailable in previous constructions. Mirror maps and Yukawa couplings are explicitly given for several examples with two and three moduli.
We prove the Landau-Ginzburg mirror symmetry conjecture between invertible quasi-homogeneous 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.
Associativity of the quantum product ensures flatness of the Dubrovin connection and is the basis for Hodge-theoretic mirror symmetry of Calabi-Yau threefolds. We use ring and module structure on cohomology pertaining to a Lagrangian submanifold to define an extension of the A-model Variation of mixed Hodge structure that matches the predictions from extended mirror symmetry. Our construction makes contact with axioms for open Gromov-Witten theory recently proposed by Solomon-Tukachinsky.
The geometric aspects of mirror symmetry are reviewed, with an eye towards future developments. Given a mirror pair (X,Y) of Calabi-Yau threefolds, the best-understood mirror statements relate certain small corners of the moduli spaces of X and of Y. We will indicate how one might go beyond such statements, and relate the moduli spaces more globally. In fact, in the boldest version of mirror symmetry (the Strominger-Yau-Zaslow conjecture), the Calabi-Yau threefolds X and Y should be directly related to each other through a very geometric construction.
Using mirror symmetry as described by Hori and Vafa, we compute the quantum equivariant cohomology ring of toric manifolds. This ring arises naturally in topological gauged sigma-models and is related to the Hamiltonian Gromov-Witten invariants of the target manifold.
We derive the combinatorial representations of Picard group and deformation space of anti-canonical hypersurfaces of a toric variety using techniques in toric geometry. The mirror cohomology correspondence in the context of mirror symmetry is established for a pair of Calabi-Yau (CY) ${\sf n}$-spaces in toric varieties defined by reflexive polytopes for an arbitrary dimension ${\sf n}$. We further identify the Kahler cone of the toric variety and degeneration cone of CY hypersurfaces, by which the Kahler cone and degeneration cone for a mirror CY pair are interchangeable under mirror symmetry.…
Inspired by the homological mirror symmetry conjecture of Kontsevich, we construct new classes of automorphisms of the bounded derived category of coherent sheaves on a smooth Calabi-Yau variety.
We propose a monodromy invariant pairing $K_{hol}(X) \otimes H_3(X^\vee,\ZZ) \to \IQ$ for a mirror pair of Calabi-Yau manifolds, $(X,X^\vee)$. This pairing is utilized implicitly in the previous calculations of the prepotentials for Gromov-Witten invariants. After identifying the pairing explicitly we interpret some hypergeometric series from the viewpoint of homological mirror symmetry due to Kontsevich. Also we consider the local mirror symmetry limit to del Pezzo surfaces in Calabi-Yau 3-folds.
Cluster varieties come in pairs: for any $\mathcal{X}$ cluster variety there is an associated Fock-Goncharov dual $\mathcal{A}$ cluster variety. On the other hand, in the context of mirror symmetry, associated with any log Calabi-Yau variety is its mirror dual, which can be constructed using the enumerative geometry of rational curves in the framework of the Gross-Siebert program. In this paper we bridge the theory of cluster varieties with the algebro-geometric framework of Gross-Siebert mirror symmetry. Particularly, we show that the mirror to the $\mathcal{X}$ cluster variety is a degenerat…
We study a cohomology-valued hypergeometric series which naturally arises in the description of (local) mirror symmetry. We identify it as a central charge formula for BPS states and study its monodromy property from the viewpoint of Kontsevich's homological mirror symmetry. In the case of local mirror symmetry, we will identify a symplectic form, and will conjecture an integral and symplectic monodromy property of a relevant hypergeometric series of Gel'fand-Kapranov-Zelevinski type.
We review various constructions of mirror symmetry in terms of Landau-Ginzburg orbifolds for arbitrary central charge $c$ and \CY\ hypersurfaces and complete intersections in toric varieties. In particular it is shown how the different techniques are related
The first part of this paper is a survey of mathematical results on mirror symmetry phenomena between Hitchin systems for Langlands dual groups. The second part introduces and discusses multiplicity algebras of the Hitchin system on Lagrangians, and considers corresponding conjectural structures on their mirror.
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 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.
We introduce several families of $\mathcal{N}=(2,2)$ UV boundary conditions in 3d $\mathcal N=4$ gauge theories and study their IR images in sigma-models to the Higgs and Coulomb branches. In the presence of Omega deformations, a UV boundary condition defines a pair of modules for quantized algebras of chiral Higgs- and Coulomb-branch operators, respectively, whose structure we derive. In the case of abelian theories, we use the formalism of hyperplane arrangements to make our constructions very explicit, and construct a half-BPS interface that implements the action of 3d mirror symmetry on ga…
We develop the quantum Kodaira-Spencer theory on the elliptic curve and establish the corresponding higher genus B-model. We show that the partition functions of the higher genus B-model on the elliptic curve are almost holomorphic modular forms, which can be identified with partition functions of descendant Gromov-Witten invariants on the mirror elliptic curve. This gives the first compact Calabi-Yau example where mirror symmetry is established at all genera.
