Showing papers on "Mirror symmetry published in 2015"

Mirror symmetry for log Calabi-Yau surfaces I

Abstract: We give a canonical synthetic construction of the mirror family to pairs (Y,D) where Y is a smooth projective surface and D is an anti-canonical cycle of rational curves. This mirror family is constructed as the spectrum of an explicit algebra structure on a vector space with canonical basis and multiplication rule defined in terms of counts of rational curves on Y meeting D in a single point. The elements of the canonical basis are called theta functions. Their construction depends crucially on the Gromov-Witten theory of the pair (Y,D).

Geometric Engineering, Mirror Symmetry and 6d (1,0) -> 4d, N=2

Abstract: We study compactification of 6 dimensional (1,0) theories on T^2. We use geometric engineering of these theories via F-theory and employ mirror symmetry technology to solve for the effective 4d N=2 geometry for a large number of the (1,0) theories including those associated with conformal matter. Using this we show that for a given 6d theory we can obtain many inequivalent 4d N=2 SCFTs. Some of these respect the global symmetries of the 6d theory while others exhibit SL(2,Z) duality symmetry inherited from global diffeomorphisms of the T^2. This construction also explains the 6d origin of moduli space of 4d affine ADE quiver theories as flat ADE connections on T^2. Among the resulting 4d N=2 CFTs we find theories whose vacuum geometry is captured by an LG theory (as opposed to a curve or a local CY geometry). We obtain arbitrary genus curves of class S with punctures from toroidal compactification of (1,0) SCFTs where the curve of the class S theory emerges through mirror symmetry. We also show that toroidal compactification of the little string version of these theories can lead to class S theories with no punctures on arbitrary genus Riemann surface.

Geometric engineering, mirror symmetry and $ 6{\mathrm{d}}_{\left(1,0\right)}\to 4{\mathrm{d}}_{\left(\mathcal{N}=2\right)} $

Abstract: We study compactification of 6 dimensional (1,0) theories on T 2. We use geometric engineering of these theories via F-theory and employ mirror symmetry technology to solve for the effective 4d $$ \mathcal{N}=2 $$ geometry for a large number of the (1,0) theories including those associated with conformal matter. Using this we show that for a given 6d theory we can obtain many inequivalent 4d $$ \mathcal{N}=2 $$ SCFTs. Some of these respect the global symmetries of the 6d theory while others exhibit SL(2, ℤ) duality symmetry inherited from global diffeomorphisms of the T 2. This construction also explains the 6d origin of moduli space of 4d affine ADE quiver theories as flat ADE connections on T 2. Among the resulting 4d $$ \mathcal{N}=2 $$ CFTs we find theories whose vacuum geometry is captured by an LG theory (as opposed to a curve or a local CY geometry). We obtain arbitrary genus curves of class $$ \mathcal{S} $$ with punctures from toroidal compactification of (1, 0) SCFTs where the curve of the class $$ \mathcal{S} $$ theory emerges through mirror symmetry. We also show that toroidal compactification of the little string version of these theories can lead to class $$ \mathcal{S} $$ theories with no punctures on arbitrary genus Riemann surface.

Spontaneous mirror-symmetry breaking in coupled photonic-crystal nanolasers

Abstract: The observation of symmetry breaking in a coupled nanolaser system could yield new types of switchable devices.

Dirac and Weyl Superconductors in Three Dimensions

Abstract: We introduce the concept of three-dimensional Dirac (Weyl) superconductors (SC), which have protected bulk fourfold (twofold) nodal points and surface Majorana arcs at zero energy. We provide a sufficient criterion for realizing them in centrosymmetric SCs with odd-parity pairing and mirror symmetry. Pairs of Dirac nodes appear in a mirror-invariant plane when the mirror winding number is nontrivial. Breaking mirror symmetry may gap Dirac nodes producing a topological SC. Each Dirac node evolves to a nodal ring when inversion-gauge symmetry is broken, whereas it splits into a pair of Weyl nodes when, and only when, time-reversal symmetry is broken.

Computation of Open Gromov–Witten Invariants for Toric Calabi–Yau 3-Folds by Topological Recursion, a Proof of the BKMP Conjecture

Abstract: The BKMP conjecture (2006–2008) proposed a new method to compute closed and open Gromov–Witten invariants for every toric Calabi–Yau 3-folds, through a topological recursion based on mirror symmetry. So far, this conjecture has been verified to low genus for several toric CY3folds, and proved to all genus only for $${\mathbb{C}^3}$$ . In this article we prove the general case. Our proof is based on the fact that both sides of the conjecture can be naturally written in terms of combinatorial sums of weighted graphs: on the A-model side this is the localization formula, and on the B-model side the graphs encode the recursive algorithm of the topological recursion. One can slightly reorganize the set of graphs obtained in the B-side, so that it coincides with the one obtained by localization in the A-model. Then it suffices to compare the weights of vertices and edges of graphs on each side, which is done in two steps: the weights coincide in the large radius limit, due to the fact that the toric graph is the tropical limit of the mirror curve. Then the derivatives with respect to Kahler radius coincide due to the special geometry property implied by the topological recursion.

Theory of interacting topological crystalline insulators

Abstract: We study the effect of electron interactions in topological crystalline insulators (TCIs) protected by mirror symmetry, which are realized in the SnTe material class and host multivalley Dirac fermion surface states. We find that interactions reduce the integer classification of noninteracting TCIs in three dimensions, indexed by the mirror Chern number, to a finite group ${Z}_{8}$. In particular, we explicitly construct a microscopic interaction Hamiltonian to gap eight flavors of Dirac fermions on the TCI surface, while preserving the mirror symmetry. Our construction builds on interacting edge states of $U(1)\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ symmetry-protected topological phases of fermions in two dimensions, which we classify. Our work reveals a deep connection between three-dimensional topological phases protected by spatial symmetries and two-dimensional topological phases protected by internal symmetries.

Geometric Engineering, Mirror Symmetry

Abstract: We study compactication of 6 dimensional (1,0) theories on T 2 . We use geometric engineering of these theories via F-theory and employ mirror symmetry technology to solve for the eective 4d N = 2 geometry for a large number of the (1; 0) theories including those associated with conformal matter. Using this we show that for a given 6d theory we can obtain many inequivalent 4d N = 2 SCFTs. Some of these respect the global symmetries of the 6d theory while others exhibit SL(2;Z) duality symmetry inherited from global dieomorphisms of the

The Vertical, the Horizontal and the Rest: | anatomy of the middle cohomology of Calabi-Yau fourfolds | | and F-theory applications |

Abstract: The four-form field strength in F-theory compactifications on Calabi-Yau four-folds takes its value in the middle cohomology group H 4. The middle cohomology is decomposed into a vertical, a horizontal and a remaining component, all three of which are present in general. We argue that a flux along the remaining or vertical component may break some symmetry, while a purely horizontal flux does not influence the unbroken part of the gauge group or the net chirality of charged matter fields. This makes the decomposition crucial to the counting of flux vacua in the context of F-theory GUTs. We use mirror symmetry to derive a combinatorial formula for the dimensions of these components applicable to any toric Calabi-Yau hypersurface, and also make a partial attempt at providing a geometric characterization of the four-cycles Poincare dual to the remaining component of H 4. It is also found in general elliptic Calabi-Yau fourfolds supporting SU(5) gauge symmetry that a remaining component can be present, for example, in a form crucial to the symmetry breaking SU(5) − → SU(3) C × SU(2) L × U(1) Y . The dimension of the horizontal component is used to derive an estimate of the statistical distribution of the number of generations and the rank of 7-brane gauge groups in the landscape of F-theory flux vacua.

A mirror theorem for toric stacks

Abstract: We prove a Givental-style mirror theorem for toric Deligne–Mumford stacks X . This determines the genus-zero Gromov–Witten invariants of X in terms of an explicit hypergeometric function, called the I-function, that takes values in the Chen–Ruan orbifold cohomology of X .

Partition functions of 3d D ^ $$ \widehat{D} $$ -quivers and their mirror duals from 1d free fermions

Abstract: We study the matrix models calculating the sphere partition functions of 3d gauge theories with $$ \mathcal{N}=4 $$ supersymmetry and a quiver structure of a $$ \widehat{D} $$ Dynkin diagram (where each node is a unitary gauge group). As in the case of necklace (Â) quivers, we can map the problem to that of free fermion quantum mechanics whose complicated Hamiltonian we find explicitly. Many of these theories are conjectured to be dual under mirror symmetry to certain unitary linear quivers with extra Sp nodes or antisymmetric hypermultiplets. We show that the free fermion formulation of such mirror pairs are related by a linear symplectic transformation. We then study the large N expansion of the partition function, which as in the case of the  quivers is given to all orders in 1/N by an Airy function. We simplify the algorithm to calculate the numerical coefficients appearing in the Airy function and evaluate them for a wide class of $$ \widehat{D} $$ -quiver theories.

Parity-time symmetry from stacking purely dielectric and magnetic slabs

Abstract: We show that parity-time symmetry in matching electric permittivity to magnetic permeability can be established by considering an effective parity operator involving both mirror symmetry and coupling between electric and magnetic fields. This approach extends the discussion of parity-time symmetry to the situation with more than one material potential. We show that the band structure of a one-dimensional photonic crystal with alternating purely dielectric and purely magnetic slabs can undergo a phase transition between propagation modes and evanescent modes when the balanced gain or loss parameter is varied. The cross matching between different material potentials also allows exceptional points of the constitutive matrix to appear in the long-wavelength limit where they can be used to construct ultrathin metamaterials with unidirectional reflection.

Mirror symmetry and the classification of orbifold del Pezzo surfaces

Abstract: We state a number of conjectures that together allow one to classify a broad class of del Pezzo surfaces with cyclic quotient singularities using mirror symmetry. We prove our conjectures in the simplest cases. The conjectures relate mutation-equivalence classes of Fano polygons with Q-Gorenstein deformation classes of del Pezzo surfaces.

Mirror symmetry: from categories to curve counts

Abstract: We work in the setting of Calabi-Yau mirror symmetry. We establish conditions under which Kontsevich's homological mirror symmetry (which relates the derived Fukaya category to the derived category of coherent sheaves on the mirror) implies Hodge-theoretic mirror symmetry (which relates genus-zero Gromov-Witten invariants to period integrals on the mirror), following the work of Barannikov, Kontsevich and others. As an application, we explain in detail how to prove the classical mirror symmetry prediction for the number of rational curves in each degree on the quintic threefold, via the third-named author's proof of homological mirror symmetry in that case; we also explain how to determine the mirror map in that result, and also how to determine the holomorphic volume form on the mirror that corresponds to the canonical Calabi-Yau structure on the Fukaya category. The crucial tool is the `cyclic open-closed map' from the cyclic homology of the Fukaya category to quantum cohomology, defined by the first-named author in [Gan]. We give precise statements of the important properties of the cyclic open-closed map: it is a homomorphism of variations of semi-infinite Hodge structures; it respects polarizations; and it is an isomorphism when the Fukaya category is non-degenerate (i.e., when the open-closed map hits the unit in quantum cohomology). The main results are contingent on works-in-preparation [PS,GPS] on the symplectic side, which establish the important properties of the cyclic open-closed map in the setting of the `relative Fukaya category'; and they are also contingent on a conjecture on the algebraic geometry side, which says that the cyclic formality map respects certain algebraic structures.

Mirror Symmetry and the Classification of Orbifold del Pezzo Surfaces

Abstract: We state a number of conjectures that together allow one to classify a broad class of del Pezzo surfaces with cyclic quotient singularities using mirror symmetry. We prove our conjectures in the simplest cases. The conjectures relate mutation-equivalence classes of Fano polygons with Q-Gorenstein deformation classes of del Pezzo surfaces.

Noncommutative mirror symmetry for punctured surfaces

Abstract: In 2013, Abouzaid, Auroux, Efimov, Katzarkov and Orlov showed that the wrapped Fukaya categories of punctured spheres and finite unbranched covers of punctured spheres are derived equivalent to the categories of singularities of a superpotential on certain crepant resolutions of toric 3 dimensional singularities. We generalize this result to other punctured Riemann surfaces and reformulate it in terms of certain noncommutative algebras coming from dimer models. In particular, given any consistent dimer model we can look at a subcategory of noncommutative matrix factorizations and show that this category is $ \mathtt {A}_\infty $-isomorphic to a subcategory of the wrapped Fukaya category of a punctured Riemann surface. The connection between the dimer model and the punctured Riemann surface then has a nice interpretation in terms of a duality on dimer models.

Theory of interacting topological crystalline insulators

Abstract: We study the effect of electron interactions in topological crystalline insulators (TCIs) protected by mirror symmetry, which are realized in the SnTe material class and host multivalley Dirac fermion surface states. We find that interactions reduce the integer classification of noninteracting TCIs in three dimensions, indexed by the mirror Chern number, to a finite group Z8. In particular, we explicitly construct a microscopic interaction Hamiltonian to gap eight flavors of Dirac fermions on the TCI surface, while preserving the mirror symmetry. Our construction builds on interacting edge states of U (1) × Z2 symmetry-protected topological phases of fermions in two dimensions, which we classify. Our work reveals a deep connection between three-dimensional topological phases protected by spatial symmetries and two-dimensional topological phases protected by internal symmetries.

Mirror symmetry and the half-filled Landau level

Abstract: We study the dynamics of the half-filled zeroth Landau level of Dirac fermions using mirror symmetry, a supersymmetric duality between certain pairs of $2+1$-dimensional theories. We show that the half-filled zeroth Landau level of a pair of Dirac fermions is dual to a pair of Fermi surfaces of electrically neutral composite fermions, coupled to an emergent gauge field. Thus, we use supersymmetry to provide a derivation of flux attachment and the emergent Fermi-liquid-like state for the lowest Landau level of Dirac fermions. We find that in the dual theory the Coulomb interaction induces a dynamical exponent $z=2$ for the emergent gauge field, making the interactions classically marginal. This enables us to map the problem of $2+1$-dimensional Dirac fermions in a finite transverse magnetic field, interacting via a strong Coulomb interaction, into a perturbatively controlled model. We analyze the resulting low-energy theory using the renormalization group and determine the nature of the BCS interaction in the emergent composite Fermi liquid.

Partition functions of 3d $\hat D$-quivers and their mirror duals from 1d free fermions

Abstract: We study the matrix models calculating the sphere partition functions of 3d gauge theories with $\mathcal{N}=4$ supersymmetry and a quiver structure of a $\hat D$ Dynkin diagram (where each node is a unitary gauge group). As in the case of necklace ($\hat A $) quivers, we can map the problem to that of free fermion quantum mechanics whose complicated Hamiltonian we find explicitly. Many of these theories are conjectured to be dual under mirror symmetry to certain unitary linear quivers with extra Sp nodes or antisymmetric hypermultiplets. We show that the free fermion formulations of such mirror pairs are related by a linear symplectic transformation. We then study the large N expansion of the partition function, which as in the case of the $\hat A$-quivers is given to all orders in 1/N by an Airy function. We simplify the algorithm to calculate the numerical coefficients appearing in the Airy function and evaluate them for a wide class of $\hat D$-quiver theories.

Emergent Geometry and Mirror Symmetry of A Point

Abstract: By considering the partition function of the topological 2D gravity, a conformal field theory on the Airy curve emerges as the mirror theory of Gromov-Witten theory of a point. In particular, a formula for bosonic n-point functions in terms of fermionic 2-point function for this theory is derived.

Formulae in noncommutative Hodge theory

Abstract: We prove that the cyclic homology of a saturated $A_\infty$ category admits the structure of a `polarized variation of Hodge structures', building heavily on the work of many authors: the main point of the paper is to present complete proofs, and also explicit formulae for all of the relevant structures. This forms part of a project of Ganatra, Perutz and the author, to prove that homological mirror symmetry implies enumerative mirror symmetry.

Topological Crystalline Insulator and Quantum Anomalous Hall States in IV-VI based Monolayers and their Quantum Wells

Abstract: Different from the two-dimensional (2D) topological insulator, the 2D topological crystalline insulator (TCI) phase disappears when the mirror symmetry is broken, e.g., upon placing on a substrate. Here, based on a new family of 2D TCIs---SnTe and PbTe monolayers---we theoretically predict the realization of the quantum anomalous Hall effect with a Chern number $\mathcal{C}=2$ even when the mirror symmetry is broken. Remarkably, we also demonstrate that the considered materials retain their large-gap topological properties in quantum well structures obtained by sandwiching the monolayers between NaCl layers. Our results demonstrate that the TCIs can serve as a seed for observing robust topologically nontrivial phases.

Spectral Theory and Mirror Symmetry

Abstract: Recent developments in string theory have revealed a surprising connection between spectral theory and local mirror symmetry: it has been found that the quantization of mirror curves to toric Calabi-Yau threefolds leads to trace class operators, whose spectral properties are conjecturally encoded in the enumerative geometry of the Calabi-Yau. This leads to a new, infinite family of solvable spectral problems: the Fredholm determinants of these operators can be found explicitly in terms of Gromov-Witten invariants and their refinements; their spectrum is encoded in exact quantization conditions, and turns out to be determined by the vanishing of a quantum theta function. Conversely, the spectral theory of these operators provides a non-perturbative definition of topological string theory on toric Calabi-Yau threefolds. In particular, their integral kernels lead to matrix integral representations of the topological string partition function, which explain some number-theoretic properties of the periods. In this paper we give a pedagogical overview of these developments with a focus on their mathematical implications

Landau-Ginzburg Mirror Symmetry Conjecture

Abstract: 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.

Gamma conjecture via mirror symmetry

Abstract: The asymptotic behaviour of solutions to the quantum differential equation of a Fano manifold F defines a characteristic class A_F of F, called the principal asymptotic class. Gamma conjecture of Vasily Golyshev and the present authors claims that the principal asymptotic class A_F equals the Gamma class G_F associated to Euler's $\Gamma$-function. We illustrate in the case of toric varieties, toric complete intersections and Grassmannians how this conjecture follows from mirror symmetry. We also prove that Gamma conjecture is compatible with taking hyperplane sections, and give a heuristic argument how the mirror oscillatory integral and the Gamma class for the projective space arise from the polynomial loop space.

Non-Kähler SYZ Mirror Symmetry

Abstract: We study SYZ mirror symmetry in the context of non-Kahler Calabi–Yau manifolds. In particular, we study the six-dimensional Type II supersymmetric SU(3) systems with Ramond–Ramond fluxes, and generalize them to higher dimensions. We show that Fourier–Mukai transform provides the mirror map between these Type IIA and Type IIB supersymmetric systems in the semi-flat setting. This is concretely exhibited by nilmanifolds.

Noncommutative homological mirror functor

Abstract: We formulate a constructive theory of noncommutative Landau-Ginzburg models mirror to symplectic manifolds based on Lagrangian Floer theory. The construction comes with a natural functor from the Fukaya category to the category of matrix factorizations of the constructed Landau-Ginzburg model. As applications, it is applied to elliptic orbifolds, punctured Riemann surfaces and certain non-compact Calabi-Yau threefolds to construct their mirrors and functors. In particular it recovers and strengthens several interesting results of Etingof-Ginzburg, Bocklandt and Smith, and gives a unified understanding of their results in terms of mirror symmetry and symplectic geometry. As an interesting application, we construct an explicit global deformation quantization of an affine del Pezzo surface as a noncommutative mirror to an elliptic orbifold.

Ω-deformation of B-twisted gauge theories and the 3d-3d correspondence

Abstract: We study Ω-deformation of B-twisted gauge theories in two dimensions. As an application, we construct an Ω-deformed, topologically twisted five-dimensional maximally supersymmetric Yang-Mills theory on the product of a Riemann surface Σ and a three-manifold M, and show that when Σ is a disk, this theory is equivalent to analytically continued Chern-Simons theory on M. Based on these results, we establish a correspondence between three-dimensional $$ \mathcal{N} $$ = 2 superconformal theories and analytically continued Chern-Simons theory. Furthermore, we argue that there is a mirror symmetry between Ω-deformed two-dimensional theories.

General mirror pairs for gauged linear sigma models

Abstract: We carefully analyze the conditions for an abelian gauged linear -model to exhibit nontrivial IR behavior described by a nonsingular superconformal eld theory determining a superstring vacuum. This is done without reference to a geometric phase, by associating singular behavior to a noncompact space of (semi-)classical vacua. We nd that models determined by reexive combinatorial data are nonsingular for generic values of their parameters. This condition has the pleasant feature that the mirror of a nonsingular gauged linear -model is another such model, but it is clearly too strong and we provide an example of a non-reexive mirror pair. We discuss a weaker condition inspired by considering extremal transitions, which is also mirror symmetric and which we conjecture to be sucient. We apply these ideas to extremal transitions and to understanding the way in which both Berglund-Hubsch mirror symmetry and the Vafa-Witten mirror orbifold with discrete torsion can be seen as special cases of the general combinatorial duality of gauged linear -models. In the former case we encounter an example showing that our weaker condition is still not necessary.