Showing papers on "Mirror symmetry published in 2013"

The Witten equation, mirror symmetry, and quantum singularity theory

Abstract: For any nondegenerate, quasi-homogeneous hypersurface singularity, we describe a family of moduli spaces, a virtual cycle, and a corresponding cohomological eld theory associated to the singularity. This theory is analogous to Gromov-Witten theory and generalizes the theory of r-spin curves, which corresponds to the simple singularity Ar 1. We also resolve two outstanding conjectures of Witten. The rst conjecture is that ADE-singularities are self-dual, and the second conjecture is that the total potential functions of ADE-singularities satisfy corresponding ADE-integrable hierarchies. Other cases of integrable hierarchies are also discussed.

Exact Kähler potential from gauge theory and mirror symmetry

Abstract: We prove a recent conjecture that the partition function of $ \mathcal{N} $ = (2, 2) gauge theories on the two-sphere which flow to Calabi-Yau sigma models in the infrared computes the exact Kahler potential on the quantum Kahler moduli space of the corresponding Calabi-Yau. This establishes the two-sphere partition function as a new method of computation of worldsheet instantons and Gromov-Witten invariants. We also calculate the exact two-sphere partition function for $ \mathcal{N} $ = (2, 2) Landau-Ginzburg models with an arbitrary twisted superpotential W . These results are used to demonstrate that arbitrary abelian gauge theories and their associated mirror Landau-Ginzburg models have identical two-sphere partition functions. We further show that the partition function of non-abelian gauge theories can be rewritten as the partition function of mirror Landau-Ginzburg models.

Time-reversal-invariant topological superconductivity and Majorana Kramers pairs.

Abstract: We propose a feasible route to engineer one- and two-dimensional time-reversal-invariant topological superconductors (SCs) via proximity effects between nodeless ${s}_{\ifmmode\pm\else\textpm\fi{}}$ wave iron-based SCs and semiconductors with large Rashba spin-orbit interactions. At the boundary of a time-reversal-invariant topological SC, there emerges a Kramers pair of Majorana edge (bound) states. For a Josephson $\ensuremath{\pi}$ junction, we predict a Majorana quartet that is protected by mirror symmetry and leads to a mirror fractional Josephson effect. We analyze the evolution of the Majorana pair in Zeeman fields, as the SC undergoes a symmetry class change as well as topological phase transitions, providing an experimental signature in tunneling spectroscopy. We briefly discuss the realization of this mechanism in candidate materials and the possibility of using $s$ and $d$ wave SCs and weak topological insulators.

Monopole operators and Hilbert series of Coulomb branches of 3d N = 4 gauge theories

Abstract: This paper addresses a long standing problem - to identify the chiral ring and moduli space (i.e. as an algebraic variety) on the Coulomb branch of an N = 4 superconformal field theory in 2+1 dimensions. Previous techniques involved a computation of the metric on the moduli space and/or mirror symmetry. These methods are limited to sufficiently small moduli spaces, with enough symmetry, or to Higgs branches of sufficiently small gauge theories. We introduce a simple formula for the Hilbert series of the Coulomb branch, which applies to any good or ugly three-dimensional N = 4 gauge theory. The formula counts monopole operators which are dressed by classical operators, the Casimir invariants of the residual gauge group that is left unbroken by the magnetic flux. We apply our formula to several classes of gauge theories. Along the way we make various tests of mirror symmetry, successfully comparing the Hilbert series of the Coulomb branch with the Hilbert series of the Higgs branch of the mirror theory.

Aspects of 3d N=2 Chern-Simons-Matter Theories

Abstract: We comment on various aspects of the the dynamics of 3d N=2 Chern-Simons gauge theories and their possible phases. Depending on the matter content, real masses and FI parameters, there can be non-compact Higgs or Coulomb branches, compact Higgs or Coulomb branches, and isolated vacua. We compute the Witten index of the theories, and show that it does not change when the system undergoes a phase transition. We study aspects of monopole operators and solitons in these theories, and clarify subtleties in the soliton collective coordinate quantization. We show that solitons are compatible with a mirror symmetry exchange of Higgs and Coulomb branches, with BPS solitons on one branch related to the modulus of the other. Among other results, we show how to derive Aharony duality from Giveon-Kutasov duality.

Refined stable pair invariants for E-, M- and [ p , q ]-strings

Abstract: We use mirror symmetry, the refined holomorphic anomaly equation and modularity properties of elliptic singularities to calculate the refined BPS invariants of stable pairs on non-compact Calabi-Yau manifolds, based on del Pezzo surfaces and elliptic surfaces, in particular the half K3. The BPS numbers contribute naturally to the fivedimensional N =1 supersymmetric index of M-theory, but they can be also interpreted in terms of the superconformal index in six dimensions and upon dimensional reduction the generating functions count N = 2 Seiberg-Witten gauge theory instantons in four dimensions. Using the M/F-theory uplift the additional information encoded in the spin content can be used in an essential way to obtain information about BPS states in physical systems associated to small instantons, tensionless strings, gauge symmetry enhancement in F-theory by [p, q]-strings as well as M-strings.

Exact Results In Two-Dimensional (2,2) Supersymmetric Gauge Theories With Boundary

Abstract: We compute the partition function on the hemisphere of a class of twodimensional (2,2) supersymmetric field theories including gauged linear sigma models. The result provides a general exact formula for the central charge of the D-brane placed at the boundary. It takes the form of Mellin-Barnes integral and the question of its convergence leads to the grade restriction rule concerning branes near the phase boundaries. We find expressions in various phases including the large volume formula in which a characteristic class called the Gamma class shows up. The two sphere partition function factorizes into two hemispheres glued by inverse to the annulus. The result can also be written in a form familiar in mirror symmetry, and suggests a way to find explicit mirror correspondence between branes.

Wall-crossing structures in Donaldson-Thomas invariants, integrable systems and Mirror Symmetry

Abstract: We introduce the notion of Wall-Crossing Structure and discuss it in several examples including complex integrable systems, Donaldson-Thomas invariants and Mirror Symmetry For a big class of non-compact Calabi-Yau 3-folds we construct complex integrable systems of Hitchin type with the base given by the moduli space of deformations of those 3-folds Then Donaldson-Thomas invariants of the Fukaya category of such a Calabi-Yau 3-fold can be (conjecturally) described in two more ways: in terms of the attractor flow on the base of the corresponding complex integrable system and in terms of the skeleton of the mirror dual to the total space of the integrable system The paper also contains a discussion of some material related to the main subject, eg Betti model of Hitchin systems with irregular singularities, WKB asymptotics of connections depending on a small parameter, attractor points in the moduli space of complex structures of a compact Calabi-Yau 3-fold, relation to cluster varieties, etc

Homological mirror symmetry for punctured spheres

Abstract: We prove that the wrapped Fukaya category of a punctured sphere (S with an arbitrary number of points removed) is equivalent to the triangulated category of singularities of a mirror Landau-Ginzburg model, proving one side of the homological mirror symmetry conjecture in this case. By investigating fractional gradings on these categories, we conclude that cyclic covers on the symplectic side are mirror to orbifold quotients of the Landau-Ginzburg model.

Collapsing of abelian fibered Calabi–Yau manifolds

Abstract: We study the collapsing behavior of Ricci-flat Kahler metrics on a projective Calabi–Yau manifold which admits an abelian fibration, when the volume of the fibers approaches zero. We show that away from the critical locus of the fibration the metrics collapse with locally bounded curvature, and along the fibers the rescaled metrics become flat in the limit. The limit metric on the base minus the critical locus is locally isometric to an open dense subset of any Gromov–Hausdorff limit space of the Ricci-flat metrics. We then apply these results to study metric degenerations of families of polarized hyperkahler manifolds in the large complex structure limit. In this setting, we prove an analogue of a result of Gross and Wilson for K3 surfaces, which is motivated by the Strominger–Yau–Zaslow picture of mirror symmetry.

Instanton effects and quantum spectral curves

Abstract: We study a spectral problem associated to the quantization of a spectral curve arising in local mirror symmetry. The perturbative WKB quantization condition is determined by the quantum periods, or equivalently by the refined topological string in the Nekrasov-Shatashvili (NS) limit. We show that the information encoded in the quantum periods is radically insufficient to determine the spectrum: there is an infinite series of instanton corrections, which are non-perturbative in \hbar, and lead to an exact WKB quantization condition. Moreover, we conjecture the precise form of the instanton corrections: they are determined by the standard or un-refined topological string free energy, and we test our conjecture successfully against numerical calculations of the spectrum. This suggests that the non-perturbative sector of the NS refined topological string contains information about the standard topological string. As an application of the WKB quantization condition, we explain some recent observations relating membrane instanton corrections in ABJM theory to the refined topological string.

UPt3as a topological crystalline superconductor

Abstract: We investigate the topological aspect of the spin-triplet f-wave superconductor UPt3 through microscopic calculations of edge- and vortex-bound states based on the quasiclassical Eilenberger and Bogoliubov–de Gennes theories. It is shown that a gapless and linear dispersion exists at the edge of the ab-plane. This forms a Majorana valley, protected by the mirror chiral symmetry. We also demonstrate that, with increasing magnetic field, vortex-bound quasiparticles undergo a topological phase transition from topologically trivial states in the double-core vortex to zero-energy states in the normal-core vortex. As long as the d-vector is locked into the ab-plane, the mirror symmetry holds the Majorana property of the zero-energy states, and thus UPt3 preserves topological crystalline superconductivity that is robust against the crystal field and spin–orbit interaction.

Exact Kähler potential for Calabi-Yau fourfolds

Abstract: We study quantum Kahler moduli space of Calabi-Yau fourfolds. Our analysis is based on the recent work by Jockers et al. which gives a novel method to compute the Kahler potential on the quantum Kahler moduli space of Calabi-Yau manifold. In contrast to Calabi-Yau threefold, the quantum nature of higher dimensional Calabi-Yau manifold is yet to be fully elucidated. In this paper we focus on the Calabi-Yau fourfold. In particular, we conjecture the explicit form of the quantum-corrected Kahler potential. We also compute the genus zero Gromov-Witten invariants and test our conjecture by comparing the results with predictions from mirror symmetry. Local toric Calabi-Yau varieties are also discussed.

Mirror symmetry and projective geometry of Reye congruences I

Abstract: Studying the mirror symmetry of a Calabi-Yau threefold $X$ of the Reye congruence in $\mP^4$, we conjecture that $X$ has a non-trivial Fourier-Mukai partner $Y$. We construct $Y$ as the double cover of a determinantal quintic in $\mP^4$ branched over a curve. We also calculate BPS numbers of both $X$ and $Y$ (and also a related Calabi-Yau complete intersection $\tilde X_0$) using mirror symmetry.

Mirror Symmetry between Orbifold Curves and Cusp Singularities with Group Action

Abstract: We consider an orbifold Landau-Ginzburg model $(f,G)$, where $f$ is an invertible polynomial in three variables and $G$ a finite group of symmetries of $f$ containing the exponential grading operator, and its Berglund-Hubsch transpose $(f^T, G^T)$. We show that this defines a mirror symmetry between orbifold curves and cusp singularities with group action. We define Dolgachev numbers for the orbifold curves and Gabrielov numbers for the cusp singularities with group action. We show that these numbers are the same and that the stringy Euler number of the orbifold curve coincides with the $G^T$-equivariant Milnor number of the mirror cusp singularity.

Gross fibrations, SYZ mirror symmetry, and open Gromov-Witten invariants for toric Calabi-Yau orbifolds

Abstract: For a toric Calabi-Yau (CY) orbifold $\mathcal{X}$ whose underlying toric variety is semi-projective, we construct and study a non-toric Lagrangian torus fibration on $\mathcal{X}$, which we call the Gross fibration. We apply the Strominger-Yau-Zaslow (SYZ) recipe to the Gross fibration of $\mathcal{X}$ to construct its mirror with the instanton corrections coming from genus 0 open orbifold Gromov-Witten (GW) invariants, which are virtual counts of holomorphic orbi-disks in $\mathcal{X}$ bounded by fibers of the Gross fibration. We explicitly evaluate all these invariants by first proving an open/closed equality and then employing the toric mirror theorem for suitable toric (partial) compactifications of $\mathcal{X}$. Our calculations are then applied to (1) prove a conjecture of Gross-Siebert on a relation between genus 0 open orbifold GW invariants and mirror maps of $\mathcal{X}$ -- this is called the open mirror theorem, which leads to an enumerative meaning of mirror maps, and (2) demonstrate how open (orbifold) GW invariants for toric CY orbifolds change under toric crepant resolutions -- an open analogue of Ruan's crepant resolution conjecture.

Topological phases of quasi-one-dimensional fermionic atoms with a synthetic gauge field

Abstract: We theoretically investigate the effect of intertube tunneling in topological superfluid phases of a quasi-one-dimensional Fermi gas with a Rashba-type spin–orbit interaction. It is shown that the effective Hamiltonian is analogous to that of a nanowire topological superconductor with multibands. Using a hidden mirror symmetry in the system, we introduce a new topological number that ensures the existence of non-Abelian Majorana zero modes even in the presence of intertube tunneling. It is demonstrated from the full numerical calculation of self-consistent equations that some of the Majorana modes survive against the intertube tunneling, when the number of one-dimensional tubes is odd in the y-direction. We also discuss a generalization of our consideration to nanowire topological superconductors.

UPt$_3$ as a Topological Crystalline Superconductor

Abstract: We investigate the topological aspect of the spin-triplet $f$-wave superconductor UPt$_3$ through microscopic calculations of edge- and vortex-bound states based on the quasiclassical Eilenberger and Bogoliubov-de Gennes theories. It is shown that a gapless and linear dispersion exists at the edge of the $ab$-plane. This forms a Majorana valley, protected by the mirror chiral symmetry. We also demonstrate that, with increasing magnetic field, vortex-bound quasiparticles undergo a topological phase transition from topologically trivial states in the double-core vortex to zero-energy states in the normal-core vortex. As long as the $d$-vector is locked into the $ab$-plane, the mirror symmetry holds the Majorana property of the zero-energy states, and thus UPt$_3$ preserves topological crystalline superconductivity that is robust against the crystal field and spin-orbit interaction.

All Genus Open-Closed Mirror Symmetry for Affine Toric Calabi-Yau 3-Orbifolds

Abstract: 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.

Mirror-symmetric exact coherent states in plane Poiseuille flow

Abstract: Two new families of exact coherent states are found in plane Poiseuille flow. They are obtained from the stationary and the travelling-wave mirror-symmetric solutions in plane Couette flow by a homotopy continuation. They are characterized by the mirror symmetry inherited from those continued solutions in plane Couette flow. The first family arises from a saddle-node bifurcation and the second family bifurcates by breaking the top–bottom symmetry of the first family. We find that both families exist below the minimum saddle-node-point Reynolds number known to date (Waleffe, Phys. Fluids, vol. 15, 2003, pp. 1517–1534).

Berglund-Hübsch Mirror Symmetry via Vertex Algebras

Abstract: We give a vertex algebra proof of the Berglund-Hubsch duality of nondegenerate invertible potentials. We suggest a way to unify it with the Batyrev-Borisov duality of reflexive Gorenstein cones.

A Landau--Ginzburg mirror theorem without concavity

Abstract: We provide a mirror symmetry theorem in a range of cases where the state-of-the-art techniques relying on concavity or convexity do not apply. More specifically, we work on a family of FJRW potentials named after Fan, Jarvis, Ruan, and Witten's quantum singularity theory and viewed as the counterpart of a non-convex Gromov--Witten potential via the physical LG/CY correspondence. The main result provides an explicit formula for Polishchuk and Vaintrob's virtual cycle in genus zero. In the non-concave case of the so-called chain invertible polynomials, it yields a compatibility theorem with the FJRW virtual cycle and a proof of mirror symmetry for FJRW theory.

Supersymmetric defect models and mirror symmetry

Abstract: We study supersymmetric field theories in three space-time dimensions doped by various configurations of electric charges or magnetic fluxes. These are supersymmetric avatars of impurity models. In the presence of additional sources such configurations are shown to preserve half of the supersymmetries. Mirror symmetry relates the two sets of configurations. We discuss the implications for impurity models in 3d $ \mathcal{N} $ = 4 QED with a single charged hypermultiplet (and its mirror, the theory of a free hypermultiplet) as well as 3d $ \mathcal{N} $ = 2 QED with one flavor and its dual, a supersymmetric Wilson-Fisher fixed point. Mirror symmetry allows us to find backreacted solutions for arbitrary arrays of defects in the IR limit of $ \mathcal{N} $ = 4 QED. Our analysis, complemented with appropriate string theory brane constructions, sheds light on various aspects of mirror symmetry, the map between particles and vortices and the emergence of ground state entropy in QED at finite density.

Three dimensional mirror symmetry and partition function on S 3

Abstract: We provide non-trivial checks of $ \mathcal{N} $ = 4, D = 3 mirror symmetry in a large class of quiver gauge theories whose Type IIB (Hanany-Witten) descriptions involve D3 branes ending on orbifold/orientifold 5-planes at the boundary. From the M-theory perspective, such theories can be understood in terms of coincident M2 branes sitting at the origin of a product of an A-type and a D-type ALE (Asymtotically Locally Euclidean) space with G-fluxes. Families of mirror dual pairs, which arise in this fashion, can be labeled as (A m−1 , D n ), where m and n are integers. For a large subset of such infinite families of dual theories, corresponding to generic values of n ≥ 4, arbitrary ranks of the gauge groups and varying m, we test the conjectured duality by proving the precise equality of the S 3 partition functions for dual gauge theories in the IR as functions of masses and FI parameters. The mirror map for a given pair of mirror dual theories can be read off at the end of this computation and we explicitly present these for the aforementioned examples. The computation uses non-trivial identities of hyperbolic functions including certain generalizations of Cauchy determinant identity and Schur’s Pfaffian identity, which are discussed in the paper.

Quantum Periods for 3-Dimensional Fano Manifolds

Abstract: The quantum period of a variety X is a generating function for certain Gromov-Witten invariants of X which plays an important role in mirror symmetry. In this paper we compute the quantum periods of all 3-dimensional Fano manifolds. In particular we show that 3-dimensional Fano manifolds with very ample anticanonical bundle have mirrors given by a collection of Laurent polynomials called Minkowski polynomials. This was conjectured in joint work with Golyshev. It suggests a new approach to the classification of Fano manifolds: by proving an appropriate mirror theorem and then classifying Fano mirrors. Our methods are likely to be of independent interest. We rework the Mori-Mukai classification of 3-dimensional Fano manifolds, showing that each of them can be expressed as the zero locus of a section of a homogeneous vector bundle over a GIT quotient V/G, where G is a product of groups of the form GL_n(C) and V is a representation of G. When G=GL_1(C)^r, this expresses the Fano 3-fold as a toric complete intersection; in the remaining cases, it expresses the Fano 3-fold as a tautological subvariety of a Grassmannian, partial flag manifold, or projective bundle thereon. We then compute the quantum periods using the Quantum Lefschetz Hyperplane Theorem of Coates-Givental and the Abelian/non-Abelian correspondence of Bertram-Ciocan-Fontanine-Kim-Sabbah.

Tropical mirror symmetry for elliptic curves

Abstract: Mirror symmetry relates Gromov-Witten invariants of an elliptic curve with certain integrals over Feynman graphs. We prove a tropical generalization of mirror symmetry for elliptic curves, i.e., a statement relating certain labeled Gromov-Witten invariants of a tropical elliptic curve to more refined Feynman integrals. This result easily implies the tropical analogue of the mirror symmetry statement mentioned above and, using the necessary Correspondence Theorem, also the mirror symmetry statement itself. In this way, our tropical generalization leads to an alternative proof of mirror symmetry for elliptic curves. We believe that our approach via tropical mirror symmetry naturally carries the potential of being generalized to more adventurous situations of mirror symmetry. Moreover, our tropical approach has the advantage that all involved invariants are easy to compute. Furthermore, we can use the techniques for computing Feynman integrals to prove that they are quasimodular forms. Also, as a side product, we can give a combinatorial characterization of Feynman graphs for which the corresponding integrals are zero. More generally, the tropical mirror symmetry theorem gives a natural interpretation of the A-model side (i.e., the generating function of Gromov-Witten invariants) in terms of a sum over Feynman graphs. Hence our quasimodularity result becomes meaningful on the A-model side as well. Our theoretical results are complemented by a Singular package including several procedures that can be used to compute Hurwitz numbers of the elliptic curve as integrals over Feynman graphs.

Symmetry in context: Salience of mirror symmetry in natural patterns

Abstract: Symmetry is a biologically relevant, mathematically involving, and aesthetically compelling visual phenomenon. Mirror symmetry detection is considered particularly rapid and efficient, based on experiments with random noise. Symmetry detection in natural settings, however, is often accomplished against structured backgrounds. To measure salience of symmetry in diverse contexts, we assembled mirror symmetric patterns from 101 natural textures. Temporal thresholds for detecting the symmetry axis ranged from 28 to 568 ms indicating a wide range of salience (1/Threshold). We built a model for estimating symmetry-energy by connecting pairs of mirror-symmetric filters that simulated cortical receptive fields. The model easily identified the axis of symmetry for all patterns. However, symmetry-energy quantified at this axis correlated weakly with salience. To examine context effects on symmetry detection, we used the same model to estimate approximate symmetry resulting from the underlying texture throughout the image. Magnitudes of approximate symmetry at flanking and orthogonal axes showed strong negative correlations with salience, revealing context interference with symmetry detection. A regression model that included the context-based measures explained the salience results, and revealed why perceptual symmetry can differ from mathematical characterizations. Using natural patterns thus produces new insights into symmetry perception and its possible neural circuits.

Refined stable pair invariants for E-, M- and [p,q]-strings

Abstract: We use mirror symmetry, the refined holomorphic anomaly equation and modularity properties of elliptic singularities to calculate the refined BPS invariants of stable pairs on non-compact Calabi-Yau manifolds, based on del Pezzo surfaces and elliptic surfaces, in particular the half K3. The BPS numbers contribute naturally to the five-dimensional N=1 supersymmetric index of M-theory, but they can be also interpreted in terms of the superconformal index in six dimensions and upon dimensional reduction the generating functions count N=2 Seiberg-Witten gauge theory instantons in four dimensions. Using the M/F-theory uplift the additional information encoded in the spin content can be used in an essential way to obtain information about BPS states in physical systems associated to small instantons, tensionless strings, gauge symmetry enhancement in F-theory by [p,q]-strings as well as M-strings.